Introduction to Scanning Tunneling Microscopy [3 ed.] 9780198856559, 0198856555

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MONOGRAPHS ON THE PHYSICS AND CHEMISTRY OF MATERIALS General Editors

Richard J. Brook Anthony Cheetham Arthur Heuer Sir Peter Hirsch Tobin J. Marks David G. Pettifor Manfred Ruhle John Silcox Adrian P. Sutton Matthew V. Tirrell Vaclav Vitek

MONOGRAPHS ON THE PHYSICS AND CHEMISTRY OF MATERIALS Theory of dielectrics H. Frohlich Strong solids (Third edition) A. Kelly and N. H. Macmillan Optical spectroscopy of inorganic solids B. Henderson and G. F. Imbusch Quantum theory of collective phenomena G. L. Sewell Principles of dielectrics B. K. P. Scaife Surface analytical techniques J. C. Rivi`ere Basic theory of surface states Sydney G. Davison and Maria Steslicka Acoustic microscopy G. A. D. Briggs Light scattering: principles and development W. Brown Quasicrystals: a primer (Second edition) C. Janot Interfaces in crystalline materials A. P. Sutton and R. W. Balluffi Atom probe field ion microscopy M. K. Miller, A. Cerezo, M. G. Hetherington, and G. D. W. Smith Rare-earth iron permanent magnets J. M. D. Coey Statistical physics of fracture and breakdown in disordered systems B. K. Chakrabarti and L. G. Benguigui Electronic processes in organic crystals and polymers (Second edition) M. Pope and C. E. Swenberg NMR imaging of materials B. Bl¨ umich Statistical mechanics of solids L. A. Girifalco Experimental techniques in low-temperature physics (Fourth edition) G. K. White and P. J. Meeson High-resolution electron microscopy (Third edition) J. C. H. Spence High-energy electron diffraction and microscopy L.-M. Peng, S. L. Dudarev, and M. J. Whelan The physics of lyotropic liquid crystals: phase transitions and structural properties A. M. Figueiredo Neto and S. Salinas Instabilities and self-organization in materials, Volume 1: Fundamentals of nanoscience, Volume 2: Applications in materials design and nanotechnology N. Ghoniem and D. Walgraef Introduction to scanning tunneling microscopy (Second edition) C. J. Chen

Introduction to Scanning Tunneling Microscopy Third Edition C. Julian Chen Department of Applied Physics and Applied Mathematics, Columbia University, New York

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c C. Julian Chen 2021  The moral rights of the author have been asserted Third Edition published in 2021 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020952115 ISBN 978–0–19–885655–9 DOI: 10.1093/oso/9780198856559.001.0001 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

TO LICHING, WINSTON, KRISTIN, MARCUS, AND NORA

Contents Preface to the Third Edition Preface to the Second Edition Preface to the First Edition

xxv xxix xxxiii

Chapter 1: Overview

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1.1

The scanning tunneling microscope . . . . . . . . . . . . . . .

1.2

The concept of tunneling . . . . . . 1.2.1 Transmission coefficient . . 1.2.2 Semiclassical approximation 1.2.3 The Landauer theory . . . . 1.2.4 Tunneling conductance . . .

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1.3

Probing electronic structure at atomic scale . 1.3.1 Experimental observations . . . . . . . 1.3.2 Origin of atomic resolution in STM . . 1.3.3 Observing and mapping wavefunctions

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The atomic force microscope . . . . . . . . . . . . . . . . . . 22 1.4.1 Atomic-scale imaging by AFM . . . . . . . . . . . . . 22 1.4.2 Role of covalent bonding in AFM imaging . . . . . . . 25

1.5

Illustrative applications . . . . 1.5.1 Self-assembled molecules 1.5.2 Electrochemistry STM . 1.5.3 Catalysis research . . . 1.5.4 Atom manipulation . .

Part I

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Principles

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26 26 30 34 38

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Chapter 2: Tunneling Phenomenon

45

2.1

The metal–insulator–metal tunneling junction . . . . . . . . . 46

2.2

The Bardeen theory of tunneling . . . . . . . . . . . . . 2.2.1 One-dimensional case . . . . . . . . . . . . . . . 2.2.2 Tunneling spectroscopy . . . . . . . . . . . . . . 2.2.3 Energy dependence of tunneling matrix elements 2.2.4 Asymmetry in tunneling spectrum . . . . . . . . 2.2.5 Three-dimensional case . . . . . . . . . . . . . . 2.2.6 Error estimation . . . . . . . . . . . . . . . . . . 2.2.7 Wavefunction correction . . . . . . . . . . . . . .

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48 48 53 54 55 57 59 60

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2.2.8 The transfer-Hamiltonian formalism . . . . . . . . . . 61 2.2.9 The tunneling matrix . . . . . . . . . . . . . . . . . . 63 2.2.10 Relation to the Landauer theory . . . . . . . . . . . . 64 2.3

Inelastic tunneling . . . . . . . . . 2.3.1 Experimental facts . . . . . 2.3.2 Frequency condition . . . . 2.3.3 Effect of finite temperature

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64 65 66 67

2.4

Spin-polarized tunneling . . . . . 2.4.1 General formalism . . . . 2.4.2 The spin-valve effect . . . 2.4.3 Experimental observations

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69 70 72 76

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Chapter 3: Tunneling Matrix Elements

77

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2

Tip wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.1 General form . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.2 Tip wavefunctions as Green’s functions . . . . . . . . 81

3.3

The derivative rule: individual cases 3.3.1 s-wave tip state . . . . . . . . 3.3.2 p-wave tip states . . . . . . . 3.3.3 d -wave tip states . . . . . . .

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The derivative rule: general case . . . . . . . . . . . . . . . . 85

3.5

Tips with axial symmetry . . . . . . . . . . . . . . . . . . . . 90 3.5.1 Lateral effects of tip states . . . . . . . . . . . . . . . 91

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Chapter 4: Atomic Forces

83 83 84 84

93

4.1

Van der Waals force . . . . . . . . . . . . . . 4.1.1 The van der Waals equation of state . 4.1.2 The origin of van der Waals force . . . 4.1.3 Van der Waals force between a tip and

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4.2

Pauli repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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The ionic bond . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4

The chemical bond . . . . . . . . . . . . . . . . . . . . . . . . 100 4.4.1 The concept of the chemical bond . . . . . . . . . . . 100 4.4.2 Bonding energy as a Bardeen surface integral . . . . . 102

4.5

The hydrogen molecular ion . . . . . . . . . . . . 4.5.1 Van der Waals force . . . . . . . . . . . . 4.5.2 Evaluation of the Bardeen surface integral 4.5.3 Compare with the exact solution . . . . .

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4.6

Chemical bonds of many-electron atoms . . . . 4.6.1 The muffin-tin potential approximation 4.6.2 The black-ball model of atoms . . . . . 4.6.3 Wavefunctions outside the atomic core . 4.6.4 Types of chemical bonds . . . . . . . . . 4.6.5 Comparing with experimental data . . . 4.6.6 A brief summary . . . . . . . . . . . . .

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Chemical bond as resonance and tunneling . . . . . . . . . . 123 4.7.1 Heisenberg’s model of resonance . . . . . . . . . . . . 123 4.7.2 Resonance energy as tunneling matrix element . . . . 126

Chapter 5: Atomic Forces and Tunneling

112 112 114 116 117 120 123

131

5.1

The principle of equivalence . . . . . . . . . . . . . . . . . . . 131

5.2

An experimentally verifiable theory 5.2.1 Case of elastic tunneling . . 5.2.2 A measurable consequence . 5.2.3 Van der Waals force . . . . 5.2.4 Repulsive force . . . . . . .

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134 134 137 138 138

Experimental verifications . . . . . . . . . . . . 5.3.1 Early experiments on metal surfaces . . 5.3.2 Experiments with frequency-modulation 5.3.3 Experiments with static AFM . . . . . . 5.3.4 Silicon tip and silicon sample . . . . . . 5.3.5 Noncontact atomic force spectroscopy .

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138 138 140 142 143 145

5.4

Mapping wavefunctions with AFM . . . . . . . 5.4.1 Case of an s-wave tip . . . . . . . . . . 5.4.2 Case of a CO-functionalized tip . . . . . 5.4.3 Viewpoint of reciprocity . . . . . . . . . 5.4.4 An intuitive explanation . . . . . . . . . 5.4.5 Pauli repulsion and van der Waals force

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147 147 149 151 153 153

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Threshold resistance in atom manipulation . . . . . . . . . . 154

5.6

General theoretical arguments . . . . . . . . . . . . . . . . 5.6.1 The double-well problem . . . . . . . . . . . . . . . 5.6.2 Canonical transformation of transfer Hamiltonian . 5.6.3 Diagonizing the tunneling matrix . . . . . . . . . .

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The Hofer–Fisher theory . . . . . . . . . . . . . . . . . . . . . 163

Chapter 6: Nanometer-Scale Imaging

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157 157 160 161

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Types of STM and AFM images . . . . . . . . . . . . . . . . 167

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The Tersoff–Hamann model . . . . . . . . . . . . . . . . . . . 169

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6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6 6.2.7 6.3

The concept . . . . . . . . . . . . The original derivation . . . . . . Profiles of surface reconstructions Extension to finite bias voltages Surface states: the concept . . . Surface states: STM observations Heterogeneous surfaces . . . . . .

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169 170 173 176 178 180 184

Limitations of the Tersoff–Hamann model . . . . . . . . . . . 184

Chapter 7: Atomic-Scale Imaging

187

7.1

Experimental facts . . . . . . . . . . . . . . 7.1.1 Universality of atomic resolution . . 7.1.2 Corrugation inversion . . . . . . . . 7.1.3 Tip-state dependence . . . . . . . . 7.1.4 Distance dependence of corrugation

7.2

Intuitive explanations . . . . . 7.2.1 Sharpness of tip states . 7.2.2 Phase effect . . . . . . . 7.2.3 Arguments based on the

7.3

Analytic treatments . . . . . . . . . . . . 7.3.1 A one-dimensional case . . . . . . 7.3.2 Surfaces with hexagonal symmetry 7.3.3 Corrugation inversion . . . . . . . 7.3.4 Profiles of atomic states as seen by 7.3.5 Independent-orbital approximation

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188 188 188 189 191

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192 192 193 195

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First-principles studies: tip electronic states . . . . . 7.4.1 W clusters as STM tip models . . . . . . . . 7.4.2 DFT study of a W–Cu STM junction . . . . 7.4.3 Transition-metal pyramidal tips . . . . . . . . 7.4.4 Transition-metal atoms adsorbed on W slabs

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First-principles studies: the images . . 7.5.1 Transition-metal surfaces . . . 7.5.2 Atomic corrugation and surface 7.5.3 Atom-resolved AFM images . .

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220 220 222 223

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Spin-polarized STM . . . . . . . . . . . . . . . . . . . . . . . 227

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Chemical identification of surface atoms . . . . . . . . . . . . 230

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The principle of reciprocity . . . . . . . . . . . . . . . . . . . 231

Chapter 8: Imaging Wavefunctions 8.1

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235

Use of ultrathin insulating barriers . . . . . . . . . . . . . . . 237

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8.2

Imaging wavefunctions with STM . . . . 8.2.1 Imaging atomic wavefunctions . 8.2.2 Imaging molecular wavefunctions 8.2.3 Imaging nodal structures . . . .

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Imaging wavefunctions with AFM

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Meaning of wavefunction observation . . 8.4.1 Interpretations of wavefunctions 8.4.2 Wavefunction as a physical field 8.4.3 Born’s statistical interpretation .

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238 238 240 241

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Chapter 9: Nanomechanical Effects

247 248 249 251 253

9.1

Mechanical stability of the tip-sample junction . . 9.1.1 Experimental observations . . . . . . . . . . 9.1.2 Condition of mechanical stability . . . . . . 9.1.3 Relaxation and the apparent G ∼ z relation

9.2

Mechanical effects on observed corrugations . . . . . . . . . . 265 9.2.1 Soft surfaces . . . . . . . . . . . . . . . . . . . . . . . 265 9.2.2 Hard surfaces . . . . . . . . . . . . . . . . . . . . . . . 267

9.3

Force in tunneling-barrier measurements . . . . . . . . . . . . 270

Part II

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Instrumentation

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254 254 257 263

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Chapter 10: Piezoelectric Scanner

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10.1 Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . 277 10.1.1 Piezoelectric effect . . . . . . . . . . . . . . . . . . . . 277 10.1.2 Inverse piezoelectric effect . . . . . . . . . . . . . . . . 278 10.2 Piezoelectric materials in STM and AFM . . . . . . . . . . . 281 10.2.1 Quartz . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.2.2 Lead zirconate titanate ceramics . . . . . . . . . . . . 282 10.3 Piezoelectric devices in STM and AFM . . . . . . . . . . . . 286 10.3.1 Tripod scanner . . . . . . . . . . . . . . . . . . . . . . 286 10.3.2 Bimorph . . . . . . . . . . . . . . . . . . . . . . . . . . 287 10.4 The tube scanner . . . . . . . . . . . . . . . . 10.4.1 Deflection . . . . . . . . . . . . . . . . 10.4.2 In situ testing and calibration . . . . . 10.4.3 Resonant frequencies . . . . . . . . . . 10.4.4 Tilt compensation: the s-scanner . . . 10.4.5 Repolarizing a depolarized tube piezo

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289 290 292 295 296 297

10.5 The shear piezo . . . . . . . . . . . . . . . . . . . . . . . . . . 297

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Chapter 11: Vibration Isolation

299

11.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 299 11.2 Environmental vibration . . . . . . . . . . . . . . . . . . . . . 303 11.2.1 Measurement method . . . . . . . . . . . . . . . . . . 304 11.2.2 Vibration isolation of the foundation . . . . . . . . . . 305 11.3 Vibrational immunity of STM . . . . . . . . . . . . . . . . . . 307 11.4 Suspension-spring systems . . . . . . 11.4.1 Analysis of two-stage systems 11.4.2 Choice of springs . . . . . . . 11.4.3 Eddy-current damper . . . .

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308 308 310 311

11.5 Pneumatic systems . . . . . . . . . . . . . . . . . . . . . . . . 312 Chapter 12: Electronics and Control 12.1 Current amplifier . . . . . . . . . . . 12.1.1 Johnson noise and shot noise 12.1.2 Frequency response . . . . . . 12.1.3 Microphone effect . . . . . . 12.1.4 Logarithmic amplifier . . . .

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313 314 316 317 318

12.2 Feedback circuit . . . . . . . . . . . . . . . . . . . . . . . . . 319 12.2.1 Steady-state response . . . . . . . . . . . . . . . . . . 320 12.2.2 Transient response . . . . . . . . . . . . . . . . . . . . 322 12.3 Computer interface . . . . . . . . . . . . . . . . . . . . . . . . 326 12.3.1 Automatic approaching . . . . . . . . . . . . . . . . . 328 Chapter 13: Mechanical Design

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13.1 The louse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 13.2 The pocket-size STM . . . . . . . . . . . . . . . . . . . . . . . 330 13.3 The single-tube STM . . . . . . . . . . . . . . . . . . . . . . . 331 13.4 The Besocke-type STM: the beetle . . . . . . . . . . . . . . . 332 13.5 The walker . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.6 The kangaroo . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 13.7 The Inchworm

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13.8 The match . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Chapter 14: Tip Treatment

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14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 14.2 Electrochemical tip etching . . . . . . . . . . . . . . . . . . . 344

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14.3 Ex situ tip treatments . . . . . . . . . . . . . . . . . . . . . . 347 14.3.1 Annealing . . . . . . . . . . . . . . . . . . . . . . . . . 347 14.3.2 Field evaporation and controlled deposition . . . . . . 348 14.4 In situ tip treatments . . . . . . . . . . . . . . . . . . . . . . 349 14.4.1 High-field treatment . . . . . . . . . . . . . . . . . . . 350 14.4.2 Controlled collision . . . . . . . . . . . . . . . . . . . . 351 14.5 Tip treatment for spin-polarized STM . . . . . . . . . . 14.5.1 Coating the tip with ferromagnetic materials . . 14.5.2 Coating the tip with antiferromagnetic materials 14.5.3 Controlled collision with magnetic surfaces . . .

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351 351 353 353

14.6 Tip preparation for electrochemistry STM . . . . . . . . . . . 354 14.7 Tip functionalization . . . . . . . . . . . . . . . . . . . . . . . 355 14.7.1 Tip functionalization with Xe atom . . . . . . . . . . 355 14.7.2 Tip functionalization with CO molecule . . . . . . . . 356

Part III

Related Methods

359

Chapter 15: Scanning Tunneling Spectroscopy

363

15.1 Electronics for scanning tunneling spectroscopy . . . . . . . . 363 15.2 Nature of the observed tunneling spectra . . . . . . . . . . . . 364 15.3 Tip treatment for spectroscopy studies . . . . . . . . . . . . . 366 15.3.1 Annealing . . . . . . . . . . . . . . . . . . . . . . . . . 366 15.4 Inelastic scanning tunneling spectroscopy 15.4.1 Instrumentation . . . . . . . . . . 15.4.2 Tip treatment for STM-IETS . . . 15.4.3 Effect of finite modulation voltage 15.4.4 Experimental observations . . . . .

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368 369 369 371 372

15.5 High-Tc superconductors . . . . . . . . . . . . . . . . . . . . . 373 15.5.1 Measuring the energy gap . . . . . . . . . . . . . . . . 374 15.5.2 The Abrikosov flux lattice . . . . . . . . . . . . . . . . 375 Chapter 16: Atomic Force Microscopy

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16.1 Static mode and dynamic mode . . . . . . . . . . . . . . . . . 380 16.2 Cantilevers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 16.2.1 Basic requirements . . . . . . . . . . . . . . . . . . . . 381 16.2.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 382 16.3 Static force detection . . . . . . . . . . . . . . . . . . . . . . . 384 16.3.1 Optical beam deflection . . . . . . . . . . . . . . . . . 384

Contents

xiv

16.3.2 Optical interferometry . . . . . . . . . . . . . . . . . . 386 16.4 Tapping-mode AFM . . . . . . . . . . . . . . . . . . . . . . . 387 16.4.1 Acoustic actuation in liquids . . . . . . . . . . . . . . 388 16.4.2 Magnetic actuation in liquids . . . . . . . . . . . . . . 389 16.5 Noncontact AFM . . . . . . . . . . . . . . . . 16.5.1 Case of small amplitude . . . . . . . . 16.5.2 Case of finite amplitude . . . . . . . . 16.5.3 Response function for frequency shift . 16.5.4 Second harmonics . . . . . . . . . . . 16.5.5 Average tunneling current . . . . . . . 16.5.6 Implementation . . . . . . . . . . . . .

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391 391 393 395 396 398 399

Appendix A: Green’s Functions

401

Appendix B: Real Spherical Harmonics

403

Appendix C: Spherical Modified Bessel Functions

407

Appendix D: Plane Groups and Invariant Functions

411

D.1 A brief summary of plane groups . . . . . . . . . . . . . . . . 411 D.2 Invariant functions . . . . . . . . . . . . . . . . . . . . . . . . 413 Appendix E: Elementary Elasticity Theory

417

E.1 Stress and strain . . . . . . . . . . . . . . . . . . . . . . . . . 417 E.2 Small deflection of beams . . . . . . . . . . . . . . . . . . . . 419 E.3 Vibration of beams . . . . . . . . . . . . . . . . . . . . . . . . 422 E.4 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 E.5 Helical springs . . . . . . . . . . . . . . . . . . . . . . . . . . 425 E.6 Contact stress: The Hertz formulas . . . . . . . . . . . . . . . 426 Bibliography

429

Index

448

List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35

The scanning tunneling microscope in a nutshell . . . . . . . . Gray-scale image and contour plot . . . . . . . . . . . . . . . . The difference between classical theory and quantum theory . . A one-dimensional metal–vacuum–metal tunneling junction . . The semiclassical approximation . . . . . . . . . . . . . . . . . The Landauer theory of tunneling . . . . . . . . . . . . . . . . Experimental observation of conductance quantum . . . . . . . Statistical results of experimental observation of conductance quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tunneling through a controllable vacuum gap . . . . . . . . . . Distance dependence of tunneling conductance for three metals Dalton’s atoms and Schr¨odinger’s atomic wavefunctions . . . . Estimation of the lateral resolution in STM . . . . . . . . . . . The structure of Si(111)-7×7 resolved in real space . . . . . . . Electronic states and DAS model of Si(111)-7×7 . . . . . . . . Four STM images of 4Hb-TaS2 . . . . . . . . . . . . . . . . . . Corrugation reversal during a scan . . . . . . . . . . . . . . . . Dependence of corrugation on tip–sample distance . . . . . . . Microscopic view of STM imaging mechanism . . . . . . . . . . Electronic states on W clusters . . . . . . . . . . . . . . . . . . Observing nodal structures in molecular wavefunctions . . . . . The atomic force microscope (AFM) . . . . . . . . . . . . . . . A schematic of the dynamic-mode AFM . . . . . . . . . . . . . Atomic resolution on Si(111)7×7 surface by AFM . . . . . . . . Theoretical explanation of atomic resolution by AFM . . . . . . Large-scale STM image of linear molecules on graphite . . . . . Coadsorption of solvant molecules . . . . . . . . . . . . . . . . . Bias dependence of STM images . . . . . . . . . . . . . . . . . The four-electrode electrochemical cell with STM . . . . . . . . Au(111) surface imaged by STM in liquids . . . . . . . . . . . . Voltammogram and STM image of Au in 0.1M H2 SO4 . . . . . Self-assembled monolayer of cysteine on Au(111) . . . . . . . . STM topographical images of the Ni-Au system . . . . . . . . . Conversion rates of the Ni catalyst and the Ni-Au catalyst . . . Reaction of H and thiophene with a MoS2 nanocluster . . . . . The basic steps of atom manipulation . . . . . . . . . . . . . .

1 2 3 5 6 7 8

2.1 2.2 2.3

Metal–insulator–metal tunneling junction . . . . . . . . . . . . 46 Tunneling spectroscopy in classic tunneling junctions . . . . . . 47 The Bardeen tunneling theory: one-dimensional case . . . . . . 49

9 11 12 13 14 15 16 17 18 19 20 20 21 22 23 24 25 27 28 29 30 31 32 33 34 35 37 38

xvi

List of Figures

2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15

The origin of the elastic-tunneling condition . . . . . . . . . . Energy dependence of tunneling matrix element . . . . . . . . Bardeen tunneling theory: three-dimensional case . . . . . . . A schematic of the inelastic electron tunneling spectroscopy experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observed IETS of acetic acid molecule . . . . . . . . . . . . . Frequency condition and energy diagram of IETS . . . . . . . Line profile of IETS due to finite temperature . . . . . . . . . The Bardeen theory of spin-polarized tunneling . . . . . . . . The spin-valve effect . . . . . . . . . . . . . . . . . . . . . . . The Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . The ferromagnet–insulator–ferromagnet tunneling junction . . Angular dependence of tunneling resistance . . . . . . . . . .

3.1 3.2 3.3

Derivation of tunneling matrix elements . . . . . . . . . . . . . 78 Derivation of the derivative rule: general case . . . . . . . . . . 86 Lateral effects of tip states . . . . . . . . . . . . . . . . . . . . . 91

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21

Quantum mechanics of van der Waals force . . . . . . . . . . . 95 Van der Waals force between a paraboloidal tip and a flat sample 97 The ionic bond energy of NaCl . . . . . . . . . . . . . . . . . . 99 Concept of the chemical bond . . . . . . . . . . . . . . . . . . . 101 Potential curve for the hydrogen molecular ion . . . . . . . . . 105 Perturbation treatment of the hydrogen molecular ion . . . . . 106 Wavefunctions of the hydrogen molecular ion . . . . . . . . . . 107 Evaluation of the correction factor . . . . . . . . . . . . . . . . 108 Accuracy of the perturbation treatment of hydrogen molecular ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Muffin-tin potential for period potential problem . . . . . . . . 113 The black-ball approximation for the chemical bond . . . . . . 114 Wavefunctions outside the atomic core . . . . . . . . . . . . . . 117 Molecular orbitals built from two s-type atomic orbitals . . . . 118 The pσ and pσ ∗ molecular orbitals . . . . . . . . . . . . . . . . 119 The pπ and pπ ∗ molecular orbitals . . . . . . . . . . . . . . . . 119 Covalent bond energy and Morse function . . . . . . . . . . . . 121 Comparing with experimental data . . . . . . . . . . . . . . . . 122 Heisenberg’s resonance: case of tunneling . . . . . . . . . . . . 124 Heisenberg’s resonance: case of energy-level split . . . . . . . . 125 Heisenberg’s resonance: transfer of oscillation amplitude . . . . 126 Three regimes of interaction in the hydrogen molecular ion . . 127

5.1 5.2 5.3

Simultaneous measurement of force and tunneling conductance 132 Perturbation treatment of force and tunneling . . . . . . . . . . 135 Correlation between tunneling conductance and force . . . . . . 139

. 52 . 55 . 58 . . . . . . . . .

65 66 67 68 70 72 74 75 76

List of Figures 5.4

xvii

5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19

Verifying the relation between tunneling current and chemical force 1: extracting the decay length of tunneling current . . . . Verifying the relation between tunneling current and chemical force 2: subtraction of the van der Waals force from the total force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of simultaneous measurement of force and current . . . Verification of the quadratic relation between tunneling current and force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Force and tunneling conductance with a Si tip and a Si sample Interaction force between a Si tip and a Si sample . . . . . . . Resonance force observed by noncontact AFM . . . . . . . . . . The 4s wavefunction of Cu atom . . . . . . . . . . . . . . . . . Predicted AFM image of a Cu atom with a CO tip . . . . . . . Viewpoint of reciprocity . . . . . . . . . . . . . . . . . . . . . . An intuitive explanation . . . . . . . . . . . . . . . . . . . . . . The basic steps of atom manipulation . . . . . . . . . . . . . . Threshold resistance in atom manipulation . . . . . . . . . . . The double-well problem in quantum mechanics . . . . . . . . . Force and tunneling current in the low-current regime 1 . . . . Force and tunneling current in the low-current regime 2 . . . .

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

The Tersoff–Hamann model of STM . . . . . . . . . . . Calculated ρ(r, EF ) of Au(110) surface . . . . . . . . . . A metal surface with one-dimensional periodicity . . . . Features of the Fermi-level LDOS corrugation amplitude The concept of surface states . . . . . . . . . . . . . . . Surface state observed on Au(111) . . . . . . . . . . . . Surface state on Cu(111), scattered by an adatom . . . . Surface state on Cu(111), scattered by an edge . . . . .

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171 174 175 176 179 180 182 183

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14

Atomic resolution on Au(111) by STM . . . . . . . . . . . Atomic-scale images of Dy on W(110) . . . . . . . . . . . Inverted corrugation and atomic defects of Dy on W(110) Effect of tip state on STM images . . . . . . . . . . . . . . Changes of image pattern due to tip-state switching . . . Distance dependence of corrugation . . . . . . . . . . . . . STM corrugation observed on Si(111)7×7 surface . . . . . Role of dz2 tip states in STM imaging . . . . . . . . . . . The phase effect in STM imaging . . . . . . . . . . . . . . Origin of atomic resolution on metal surfaces . . . . . . . A metal surface with one-dimensional periodicity . . . . . Tip-induced corrugation enhancements . . . . . . . . . . . Geometric structure of a close-packed metal surface . . . . The hexagonal cosine functions . . . . . . . . . . . . . . .

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187 188 189 189 190 191 192 193 194 195 197 198 201 202

5.5

5.6 5.7

140

141 142 143 144 145 146 149 150 152 153 154 155 158 164 165

xviii

List of Figures

7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28 7.29 7.30 7.31 7.32 7.33 7.34 7.35 7.36 7.37

Charge-density contour plot of Al(111) film . . . . . . . . . . Interpretation of atom-resolved STM images on Al(111) . . . LDOS of several tip electronic states . . . . . . . . . . . . . . Enhancement factor for different tip states . . . . . . . . . . . Apparent radius for a spherical conductance distribution . . . Apparent radius as a function of tip–sample distance . . . . . Close-packed surface with tetragonal symmetry . . . . . . . . Electronic states of W clusters near the Fermi level . . . . . . A model STM system: a W tip and a Cu sample . . . . . . . Electronic states of a W pyramid . . . . . . . . . . . . . . . . Electronic states of a transition-metal atom adsorbed on a W slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental and theoretical corrugations on W(110) . . . . Corrugation inversion on W(110) surface . . . . . . . . . . . . Atomic corrugation and surface wave on Be(10¯10) . . . . . . Silicon clusters as models for non-contact AFM tips . . . . . Van der Waals force between a Si tip and a Si surface . . . . Fitting of the chemical-bond forces by Morse functions . . . . Chemical-bond force between a Si tip and a Si surface . . . . Spin-polarized STM in independent orbital approximation . . Chemical identification of surface atoms . . . . . . . . . . . . Observation of a tip state by AFM . . . . . . . . . . . . . . . Observation of a tip state by STM . . . . . . . . . . . . . . . Quantitative profile of a tip state observed by STM . . . . . .

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203 204 205 207 209 211 213 217 218 219

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220 221 221 223 224 224 225 226 228 231 232 233 234

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15

Ultrathin insulating barrier for imaging wavefunctions . . . . . Wavefunction of ground-state hydrogen atom . . . . . . . . . . STM images of Au atoms on NaCl film . . . . . . . . . . . . . . Explanation of the charge-state switching of an Au atom . . . . Charge density and STM image of naphthalocynine . . . . . . . STM images of HOMO of pentacene, s-wave tip and p-wave tip STM images of LUMO of pentacene, s-wave tip and p-wave tip Mechanism of STM imaging with a p-wave tip . . . . . . . . . . Imaging naphthalocynine with a Cu tip and a CO tip . . . . . Imaging naphthalocynine with a Cu tip and a CO tip . . . . . AFM images of a pentacene molecule with different tips . . . . Three-dimensional electrical charge map inside a molecule . . . Observing and mapping electrical fields . . . . . . . . . . . . . Observing and mapping magnetic fields . . . . . . . . . . . . . Double-slit experiment with single-photon detectors . . . . . .

237 238 239 240 241 242 242 243 244 245 246 247 249 250 251

9.1 9.2 9.3

A combined STM and FIM . . . . . . . . . . . . . . . . . . . . 255 Records of three sets of approaching and retracting curves . . . 256 Stiffness of a W(100) tip . . . . . . . . . . . . . . . . . . . . . . 258

List of Figures

xix

9.4 9.5 9.6 9.7 9.8 9.9 9.10

Relaxation of a W–Au STM junction . . . . . . . . . . . . . Stability of STM and the rigidity of surfaces . . . . . . . . . The effect of relaxation on the apparent G ∼ z relation . . . Amplification of corrugation amplitude by deformation . . . Dependence of corrugation on tunneling conductance . . . . Atomic force between an Al tip and an Al sample . . . . . . Variation of measured apparent barrier height with distance

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262 263 264 266 267 268 271

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18

Piezoelectric effect . . . . . . . . . . . . . . . . . . . . . . The inverse piezoelectric effect . . . . . . . . . . . . . . . Definition of piezoelectric constants . . . . . . . . . . . . . A photograph of an artificial quartz crystal . . . . . . . . Axes and various cuts of quartz crystal . . . . . . . . . . . PZT: piezoelectric properties and composition . . . . . . . Variation of piezoelectric constant with temperature . . . Tripod scanner . . . . . . . . . . . . . . . . . . . . . . . . Bimorph . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflection of a bimorph . . . . . . . . . . . . . . . . . . . The tube scanner . . . . . . . . . . . . . . . . . . . . . . . Deflection of a tube scanner . . . . . . . . . . . . . . . . . Accuracy of the analytic expression . . . . . . . . . . . . . Double piezoelectric response . . . . . . . . . . . . . . . . Measuring circuit for the double piezoelectric response . . Measured double piezoelectric response of a tube scanner The s-scanner . . . . . . . . . . . . . . . . . . . . . . . . . The shear piezo . . . . . . . . . . . . . . . . . . . . . . . .

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277 278 279 280 281 283 284 286 287 288 289 290 291 293 294 295 297 298

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11.1

A vibrating system with one degree of freedom and its transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Schematic and working mechanism of the Hall–Sears seismometer, HS-10-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Vibration spectra of laboratory floors . . . . . . . . . . . . . . 11.4 Vibration isolation of the foundation . . . . . . . . . . . . . . 11.5 A photograph of the foundation with vibration isolation . . . 11.6 Vibrational immunity of STM . . . . . . . . . . . . . . . . . . 11.7 A two-stage suspension-spring vibration isolation system . . . 11.8 Transfer functions for two-stage vibration isolation system . . 11.9 Eddy-current damper . . . . . . . . . . . . . . . . . . . . . . . 11.10 Dimensionless constant in the calculation of the damping constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 12.2 12.3 12.4

Two basic types of current amplifiers . . . . . . . . . . Broad-band current amplifiers . . . . . . . . . . . . . . The influence of the input capacitance on output noise Logarithmic amplifier . . . . . . . . . . . . . . . . . .

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. 300 . . . . . . . .

304 305 306 306 307 308 309 311

. 312 . . . .

314 316 317 319

xx

List of Figures

12.5 12.6 12.7 12.8

A schematic of the feedback loop in an STM . . . . . . . . . A simple feedback electronics with integration compensation . Transient response of the STM feedback system . . . . . . . . The essential elements of a computer-controlled STM . . . . .

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320 323 325 327

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14

The piezoelectric steppe: The louse . . . . . . . . . . The pocket-size STM . . . . . . . . . . . . . . . . . . Single-tube STM . . . . . . . . . . . . . . . . . . . . The Besocke-type STM . . . . . . . . . . . . . . . . Working principle of the Besocke design . . . . . . . Tip approaching and retrieving of the Besocke design A Besocke-type variable-temperature STM . . . . . . STM with a walker . . . . . . . . . . . . . . . . . . . Working principle of the walker . . . . . . . . . . . . The stick-slip stepper . . . . . . . . . . . . . . . . . . The Inchworm . . . . . . . . . . . . . . . . . . . . . An STM with an Inchworm as coarse positioner . . . The Aarhus STM design . . . . . . . . . . . . . . . . Photographs of the Aarhus STM . . . . . . . . . . .

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329 330 331 333 333 334 335 335 336 337 338 339 340 341

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15

Electrochemical etching of tungsten tips . . . . . . . . Dependence of tip radius of curvature with cutoff time FIM image of a W tip immediately after etching . . . The phase diagram of the W–O system . . . . . . . . . Tip annealing methods . . . . . . . . . . . . . . . . . . Tip formation by field evaporation . . . . . . . . . . . Mechanism of tip sharpening by an electric field . . . . Mechanism of tip sharpening by controlled collision . . The spin-density orientation of Gd- and Fe-coated tips Bias-dependence of spin-density orientation of tips . . Tip preparation for electrochemistry studies . . . . . . Transferring a Xe atom to and from the tip . . . . . . Picking up and putting down a CO molecule . . . . . Images before and after picking up a CO molecule . . Images of CO molecules with a higher dosage . . . . .

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344 346 347 348 348 349 350 351 352 353 354 356 357 357 358

15.1 15.2 15.3 15.4 15.5 15.6 15.7

Electronics for scanning tunneling spectroscopy . . . . . . . . Nature of the observed tunneling spectrum . . . . . . . . . . Tip treatment for tunneling spectroscopy . . . . . . . . . . . Structure of some low-Miller-index W surfaces . . . . . . . . . A schematic of STM-IETS . . . . . . . . . . . . . . . . . . . . Instrumentation of inelastic electron tunneling spectroscopy . Broadening of the spectral peak owing to a finite modulation amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observed STM-IETS, showing isotope effect . . . . . . . . . .

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364 366 367 368 369 370

15.8

. 371 . 372

List of Figures

xxi

15.9 15.10 15.11 15.12

Tunneling spectroscopy of superconductors . . . . . . . . . Abrikosov flux lattice of NbSe2 imaged by STM . . . . . . . Tunneling spectra at different points of the vortex . . . . . Vortex lattice and scanning spectroscopy of Bi2 Sr2 CaCu2 Ox

16.1 16.2

16.11 16.12 16.13 16.14 16.15

Schematic of the AFM . . . . . . . . . . . . . . . . . . . . . . Fabrication of silicon nitride microcantilevers with integrated tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photographs of microcantilevers . . . . . . . . . . . . . . . . . Detection of cantilever deflection by optical beam deflection . Detection of cantilever deflection using optical interferometer Principle of tapping-mode AFM . . . . . . . . . . . . . . . . . Instrumentation of tapping-mode AFM . . . . . . . . . . . . . Excitation spectrum of the cantilever in liquid . . . . . . . . . Tapping-mode AFM in liquid-phase using magnetic actuation Cantilever response spectra from acoustic and magnetic actuation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of force on the resonance frequency of a cantilever . . . Dependence of frequency shift on vibrational amplitude . . . Second harmonics in the dynamic mode AFM . . . . . . . . . Average current in dynamic mode AFM-STM . . . . . . . . . A schematic of the frequency-modulation AFM . . . . . . . .

B.1

Real spherical harmonics . . . . . . . . . . . . . . . . . . . . . . 405

C.1

Spherical modified Bessel functions . . . . . . . . . . . . . . . . 408

D.1 D.2 D.3 D.4

The plane groups I . . The plane groups II . Relations among plane The Si(110) surface . .

E.1 E.2 E.3 E.4 E.5 E.6 E.7 E.8 E.9

Normal stress and normal strain . . . . . . . . Shear stress and shear strain . . . . . . . . . . Bending of a beam . . . . . . . . . . . . . . . . Deformation of a segment of a beam . . . . . . Vibration of a beam . . . . . . . . . . . . . . . Torsion of a circular bar . . . . . . . . . . . . . Torsion and torsional vibration of a rectangular Stiffness of a helical spring . . . . . . . . . . . . Contact stress and contact deformation . . . .

16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10

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374 375 376 377

. 379 . . . . . . . .

383 384 385 386 387 388 389 390

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390 392 396 397 398 400

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412 413 414 415

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cantilever . . . . . . . . . . . .

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417 418 420 421 422 424 425 426 427

List of Tables 1

International STM Conferences . . . . . . . . . . . . . . . . xxxiii .

1.1 1.2 1.3

Work functions and decay constants . . . . . . . . . . . . . . 5 Tunneling conductance and tip–sample distance . . . . . . . . 10 Corrugation amplitude and tip–sample distance . . . . . . . . 19

3.1 3.2 3.3 3.4

Tip wavefunctions . . . . . . . . . . . . . Tip wavefunctions and Green’s functions . Tunneling matrix elements . . . . . . . . . Tip states and tunneling matrix elements

4.1 4.2 4.3

Potential curve of hydrogen molecular ion . . . . . . . . . . . 111 Examples of atomic wavefunction data . . . . . . . . . . . . . 115 Parameters of homonuclear diatomic molecules . . . . . . . . 120

5.1 5.2 5.3 5.4

Tunneling conductance and chemical bond force . . . Slater parameters of the 4s wavefunction of Cu atom Threshold resistances Rt in atom manipulation . . . Diffusion activation energy measured by FIM . . . .

6.1 6.2

Experimental parameters of surface states . . . . . . . . . . . 181 Parameters of the scattered surface waves . . . . . . . . . . . 182

7.1 7.2 7.3 7.4

Tunneling matrix elements for d-type type tip states Apparent curvature of individual atomic states . . . The independent-orbital model . . . . . . . . . . . . Parameters of the Morse functions . . . . . . . . . .

8.1 8.2

Wavelength and electron energy . . . . . . . . . . . . . . . . . 236 Wavefunction radii and energy levels . . . . . . . . . . . . . . 239

9.1 9.2

Stiffness of various pyramidal tips . . . . . . . . . . . . . . . . 259 Stiffness of close-packed metal surfaces . . . . . . . . . . . . . 261

. . . .

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. . . .

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. . . .

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. . . .

. . . .

80 82 85 90

133 148 155 156

206 210 215 225

10.1 Physical constants of quartz . . . . . . . . . . . . . . . . . . . 282 10.2 Properties of PZT ceramics . . . . . . . . . . . . . . . . . . . 285 11.1 Properties of common spring materials . . . . . . . . . . . . . 310

Preface to the Third Edition More than ten years have passed since the publication of the second edition of Introduction to Scanning Tunneling Microscopy (STM). Significant advances in this research field have been made during that decade. One of the most important advances is the direct experimental observation of the wavefunctions of atoms and molecules (through field quantities representing local values of wavefunctions) down to picometer resolution with negligible disturbance. This advance was a result of two breakthroughs. The first breakthrough came about 2005 when a group at IBM Zurich Laboratory discovered a method to image the wavefunctions of single atoms and organic molecules in pristine state using STM by separating the molecule and the metal substrate with an ultrathin film of insulator, typically NaCl [1, 2]. By using different biases, images of highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are clearly observed, agreeing with the charge density contours of those wavefunctions calculated from first-principle quantum mechanical computations. The second breakthrough took place around 2011, where the same group at IBM Zurich Laboratory imaged the organic molecules sitting on an insulating film using an STM tip functionalized with a CO molecule [3]. The STM images did not resemble the charge density contour at all, but rather the squares of the lateral derivatives of the molecular wavefunctions, which peak at the nodal structures of the molecular wavefunctions. For the first time in science history, the nodal structures inside molecular wavefunctions are directly observed and mapped in real space. That Physics Review Letter was reviewed by a Viewpoint article in Physics [4], entitled Visualizing Quantum Mechanics, which commented that the direct observation of the nodal structures inside molecular orbitals “will help future generations of chemists in obtaining an intuitive understanding of molecular properties that will guide them to novel solutions in all areas of chemistry.” The direct observation of the wavefunctions and their derivatives touches a fundamental scientific question unresolved over almost one century: the interpretation of wavefunctions. In A Brief History of Time [5], Stephen Hawking said: “Quantum mechanics underlies all of modern science and technology. It governs the behavior of transistors and integrated circuits, and is the basis of modern chemistry and biology.” On the other hand, as Richard Feynman famously said, “If you think you understand quantum mechanics, you don’t understand quantum mechanics”—because the interpretation of its central subject, wavefunction, was highly controversial [6, 7]. The direct experimental observation of wavefunctions shows that they are observable physical reality, similar to Maxwell’s electromagnetic fields. This new edition features an added chapter on Imaging Wavefunctions, including a section Meaning of Wavefunction Observation.

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The discovery also deepens the understanding of imaging mechanics of STM. The observation of nodal structures in molecular orbitals was attributed to the px and py states of the CO molecule through the application of the derivative rule [3, 4]. To verify the imaging mechanism, a number of theoretical studies on the derivative rule were published [8, 9, 10, 11, 12]. Those papers represent further developments to the derivative rule over the original publications [13, 14, 15]. In this new edition, the presentation of the derivative rule in Chapter 3 is improved. During 1994 to 2003, I was assigned by IBM to a completely different research field, human language. I received much more corporate recognition for my human language research than on STM and AFM. But I was anxious to resume research in STM and AFM. In December 2003, I received a kind invitation from Professor Roland Wiesendanger to become a guest scientist at the Institute for Applied Physics of Hamburg University, one of the largest center of STM and AFM research in the world. By looking back, I believe that during the three years at Hamburg, my most important publication was in Physicsl Review B coauthored with Wiesendanger[16]. Here is a synopsis of the tutorial as presented in Section 4.6. In 1961, Conyers Herring [17] and Lev Landau [18] independently developed a perturbation theory of the chemical bond in terms of a surface integral of the wavefunctions of the two parties, similar to the surface integral of the tunneling matrix element in Bardeen’s tunneling theory. For the simplest molecule, the hydrogen molecular ion, the surface integral can be evaluated analytically, which generates an accurate asymototic expression of the chemical-bond energy. However, the Herring–Landau theory has not been tested beyond the hydrogen atom. Because the atomic wavefunctions have been computed by the Hartree–Fock method with high accuracy, and the surface integral of the Bardeen tunneling matrix element can be evaluated by the derivative rule [13, 14, 15], can we extend the Herring–Landau theory to many-electron atoms? The Brief Report submission tested the theory on all homonuclear diatomic molecules of atoms of the first two rows of the periodic table, and concluded with a definitive answer [16]. Conyers Herring was the referee. He recommended publishing it immediately. However, that Brief Report [16] was too brief. Formulated in timedependent perturbation theory, the mathematics was difficult. A tutorial presentation is provided in the revised Chapter 4 using mathematics of undergraduate level on elements in the first row of the periodic table, especially carbon, nitrogen, and oxygen. An immediate application is the analysis of AFM images by a CO-functionalized tip, presented in Section 5.4, Mapping wavefunctions with AFM. Very simple analytic expressions for the force distribution, and consequently the AFM images, is obtained, which can be applied immediately to compare with experimnetal observations. Numerical results of the chemical bond energies of diatomic molecules and the AFM imaging with a CO-functionalized tip are presented in detail.

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Furthermore, according to the Bardeen tunneling theory, the tunneling conductance G is proportional to the square of the tunneling matrix element |M |. On the other hand, according to the Herring-Landau theory, the chemical bond energy ΔE equals the absolute value of the tunneling matrix element |M |. Therefore, there should be a universal relation between tunneling conductance and chemical bond energy, G ∝ |ΔE|2 ,

(1)

which was first published in 1991 [19]. This relation explained already existing observations [20, 21] and was reconfirmed by several dedicated experiments with metal tips and metal surfaces [22, 23]. In 2013, a research group led by Yoshiaki Sugimoto published a report of carefully executed study of that relation on the classical system for STM and AFM, a silicon tip with a silicon sample. A large number of independent measurements are performed. Statistically overwhelming data supported the quadratic relation, Eq. 1 [24]. Motivated by the new experiments, Chapter 5, Atomic Forces and Tunneling, is thoroughly rewritten to make the mathematics accessable at the undergraduate level and thus pedagogically sound. Nevertheless, sophisticated arguments based on advaced quantum mechanics are retained to satisfy readers with a lofty taste of profound mathematics. Other additions are responding to the editors and reviewers. The first chapter now features the imaging of self-assembled molecules at a liquid-solid interface. Using very simple STM instrumentation working in liquid at room temperature, shockingly beautiful and scientifically informative images are obtained. It can be an entry-level STM experiment even as a high-school student project. The scanning tunneling spectra of superconductors, especially high-Tc superconductors is now the last section of Chapter 15, Scanning Tunneling Spectroscopy. With no attempt to achieve true atomic resolution, the unique instrumentation of STM enables accurate measurement of the density of states of the superconductors, as well the direct imaging of Abrikosov flux lattice in real space. It is not only a beautiful work, but also essential to solve the riddle of the mechanism of high-Tc superconductors. During the writing of the new edition, an unavoidable difficulty was how to present the experimental observation of wavefunctions. According to the orthodox view of quantum mechanics, represented by the Copenhagen doctrine and the von Neumann axioms [7], wavefunction is not observable. Regarding this question, I communicated extensively with Franck Lalo¨e, the author of Do We Really understand Quantum Mechanics? [25], and the coauthor of the well-known textbook Quantum Mechanics with Nobelist Claude Cohen-Tannoudji [26]. In the foreword of the 2019 edition of Lalo¨e’s book [25], Cohen-Tannoudji said that it “provides clear and objective presentation of the alternative formulations that have proposed to replace the traditional orthodoxy”. Accordingly, until about 1970 or 1980, most physi-

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cists took the Copenhagen interpretation as the orthodoxy. Nowadays, the attitude of physicists is more open concerning the matters [25]. Lalo¨e recom´ mended the three-volume masterpiece M´ecanique Quantique [27] by Emile Durand, a graduate student of Louis de Briglie and a long-term dean of Toulouse University. Published in the 1970s, it started with a formulation that all wavefunctions are real, including Pauli and Dirac spinors [27]. By identifying real wavefunctions as the physical reality, the meaning of wavefunction observation becomes natural, as shown in Section 8.4. The Author sincerely thanks Yoshiaki Sugimoto for pointing out a number of typos in the second edition when he used it as a graduate-level textbook, and for reviewing an manuscipt of the third edition. The Author highly appreciates inspiring discussions with Professor Magnus Paulsson and Dr. Alexander Gustaffson of Linnaeus University regarding theoretical issues regarding imaging wavefunctions. The Author is greatly benefitted through numerous discussions with Leo Gross on the experimental conditions of the observation of wavefunctions, especially the observation of nodel structures. The Author is deeply indebted to Professor Roland Wiesendanger for inviting me to join the Physics Department of Hamburg University in December 2003. In the three years as a visiting scientist, I resumed research on STM and AFM full time, and prepared the materials for the second edition of this book. The Author is especially thankful for Professor Richard Osgood for inviting me to join the Department of Applied Physics and Applied Mathematics for an adjunct teaching and research position in late 2006. For more than a decade, the supurb academic atmosphere and the freedom for intellectural pursuit has enabled me to finish the second and thrid editions of this book, as well as other monographs.

C. Julian Chen Columbia University New York October 2019

Preface to the Second Edition In a 1959 speech entitled There’s Plenty of Room at the Bottom [28], Richard Feynman invited scientists to a new field of research: to see individual atoms distinctly, and to arrange the atoms the way we want. Feynman envisioned that, by achieving those goals, one could synthesize any chemical substance that the chemist writes down, resolve many central and fundamental problems in biology at the molecular level, and dramatically increase the density of information storage. Some 20 years later, those goals began to be achieved through the invention and application of the scanning tunneling microscope (STM) [29, 30] and the atomic force microscope (AFM) [31]. The inventors of STM, two physicists at IBM Research Division, Gerd Binnig and Heinrich Rohrer, shared the 1986 Nobel Prize in physics [32, 33]. At that time, I was fortunate to be in the Department of Physical Sciences of IBM Research Division, and had the opportunity to design, build, and run those fascinating instruments. Partially based on my personal experience and understanding, in 1993, the first edition of this book was published [34]. In the decade following, Feynman’s foresight has grown into a vast field of research, nanoscience and nanotechnology. As a result, tremendous advances have been achieved in the understanding of the basic physics as well as instrument design and operation of STM and AFM. It is time to publish a second edition to include those recent advances, and to satisfy the urgent need for an updated, unified, accurate, and pedagogically assessable textbook and reference book on STM and AFM. During the years of 1994 to 2003, I was concentrating on the research of human voice and languages, which were my favorite subjects ever since my college years. And I received more corporate recognition than for my basic research in physics [35]. However, the news about the advancements in STM and AFM constantly called me to come back to nanoscience and nanotechnology. In December 2003, I received a kind invitation from Professor Roland Wiesendanger, the Director of the Institute of Applied Physics at Hamburg University, to become a guest scientist. This is one of the largest and most productive centers of STM and AFM research, especially in spinpolarized STM and non-contact AFM. And for the first time in my life, I could concentrate 100% of my time on nanoscience and nanotechnology research. In the summer semester of 2005, in a graduate-level course in nanostructure physics jointly given to the Department of Physics and the Department of Chemistry at Hamburg University, Professor Roland Wiesendanger lectured the analyses of various nanostructures, and I lectured the principles and instrumentation of STM and AFM. The lecture notes then became the blueprint of the second edition of the STM book. Following are some examples of the additions to the second edition:

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Atomic force microscopy, with a refined frequency-modulation mode, has achieved true atomic resolution in the attractive atomic force regime, often referred to as the non-contact AFM. In some cases, its resolution has even surpassed that of STM. The observed bias-dependence of atomic forces provides information about the details of electronic structure. This new technique enables atomic-scale imaging and characterization not only for conductors, but also for insulators. Significant breakthrough in spin-polarized STM has enabled the observation of local magnetic phenomena down to atomic scale. Such advancement was to drive the development of nanomagnetism, which would have deep impact on the technological applications of magnetism. Inelastic electron tunneling spectroscopy (IETS), initially discovered in metal-insulator-metal tunneling junctions to observe vibrational frequencies of embedded molecules, was advanced to STM junctions, enabling the observation of vibrational states of individual molecules. The successful demonstration of STM-IETS elevated the field of single-molecule chemistry to an unprecedented level. At the time that the first edition was written, atom manipulation was still a highly specialized personal art. In the later years, the underlying physics has gradually been discovered, and the atom-manipulation process is becoming a precise science. Besides single atoms, molecules are also subject to manipulation. It was often said that STM is to nanotechnology what the telescope was to astronomy. Yet STM is capable of manipulating the objects it observes, to build nanoscale structures never existed in Nature. No telescope is capable of bringing Mars and Venus together. In the process of further improving the resolution of STM and AFM, the understanding of its basic physics has been advanced. Numerous convincing theoretical and experimental studies have shown that the imaging mechanism of both STM and AFM at atomic resolution can be understood as a sequence of making and breaking of partial covalent bonds between the anisotropic quasi-atomic orbitals on the tip and those on the sample. The nature of STM and AFM, including those with spin-polarized tips, can be understood with a unified perspective based on Heisenberg’s concept of resonance in quantum mechanics. Commercialization of STM and AFM has been greatly advanced. Owing to the rapidly expanding research in nanotechnology, especially in molecular biology and in materials science, AFM with tapping mode operating in air or in liquid now constitutes the largest market share. Therefore, a brief presentation of its basic principles is included. In spite of the availability of commercial STMs and AFMs, research groups worldwide continue to design and build customized instruments to achieve advanced features and to serve special experimental needs. Often, those new designs are then adapted by instrument manufacturers to become products. Although the basic principles of the design and construction

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of STM and AFM was laid down in the second part of the first edition, Instrumentation, new trends and ideas have been added. The basic organization of the second edition is essentially identical to the first edition. All the materials in the first edition proven to be useful are preserved. Some of the less important materials are eliminated or converted to Problems at the ends of various chapters. The first chapter, Overview, is preserved but updated. Several recent applications of STM and AFM, probably of interest to general readers, are added. The title of the first Part is changed to Principles from Imaging Mechanism because of the inclusion of the physics of atom and molecule manipulation. The Gallery of STM Images is updated, with more historical photos and AFM images added, which is now entitled simply Gallery. To preserve the classical style of the first edition and to reduce the cost of printing, all photographs and images are in black-and-white. Similar to the first edition, only a few illustrative applications of STM and AFM are presented, because there are already many excellent books on various applications. For example, the monograph Scanning Probe Microscopy and Spectroscopy: Methods and Applications by R. Wiesendanger [36]; the book series Scanning Tunneling Microscopy I, II, and III edited by R. Wiesendanger and H.-J. G¨ untherodt [37]; Scanning Tunneling Microscopy edited by J. A. Stroscio and W. J. Kaiser [38]; the second edition of the monograph Scanning Tunneling Microscopy and its Applications by C. Bai [39]; and the second edition of Scanning Probe Microscopy and Spectroscopy: Theory, Techniques, and Applications edited by D. Bonnell [40]. Similar to the first edition, to ensure pedagogical soundness, the focus is on simple but useful theories, with every derivation presented in full detail. Many new figures are added to illustrate the concepts in physics. In the second edition, care has been taken to use SI units as much as possible. That would provide a unified order-of-magnitude mental picture of the physical quantities involved. Because of the enormous growth of the size of literature, the reference list at the back of the second edition only includes those cited by the text, selected and arranged automatically by LaTeX. In the age of the Internet, exhaustive reference lists can be obtained by searching on the web. The author is deeply grateful to H. Rohrer for commenting extensively on the manuscript of the first edition, and publicly recommending the book to newcomers as well as experts. The author is equally grateful to Ch. Gerber, an architect of both the first STMs and the first AFMs, for comments on the second edition, and especially for providing a number of precious original photographs and images of historical interest. An early manuscript of the second edition was thoroughly reviewed by a number of experts in that field. Corrections and improvements were made upon their comments. Following is an incomplete list. The entire book by K.-H. Rieder of Swiss Federal Laboratories for Materials Research. Part I and Chapter 16 by R. P´erez of Universidad Autonoma de Madrid. Part II, especially Chapter 16, by

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F. Giessibl of University of Regensburg. All chapters related to scanning tunneling spectroscopy and spin-polarised STM by O. Pietzsch of Hamburg University. The main part of Chapter 1 and Chapter 13 by F. Besenbacher, J. V. Lauritsen, and E. Laegsgaard of University of Aarhus. Chapter 10 by W. Coburn and P. Stokes of EBL Products, Inc., and M. Ordillas of Morgan Electro Ceramics, Inc. Chapter 15 by R. Feenstra of Carnegie Melon University. Chapter 16, especially Section 16.4, by C. Prater of Veeco Instruments Inc. Section 1.5.2 and Section 15.6 by the group of J. Ulstrup of Technical University of Denmark. Section 14.3.5 by R. A. Wolkow of University of Alberta. Last but not least, Section 5.6 was thoroughly reviewed by W. Hofer, now at Newcastle University. The Gallery is an integrated part of the book. The author is indebted to the following authors who contributed high-resolution black-and-white original photographs and images for the publication of this book: Plates 2, 3, and 4, Ch. Gerber. Plate 5, R. Wiesendanger. Plates 7, 11, 12, and 16, J. Boland. Plates 8, 10, and 15, R. Feenstra. Plate 9, J. V. Barth. Plate 13, D. J. Thomson. Plate 14, J. Repp and G. Meyer. Plate 17, M. Bode. Plate 18, L. Berbil-Bautista. Plate 19, O. Pietzsch. Plate 20, A. R. Smith. Plates 21 and 22, S. F¨ oulsch. Plate 23, K. F. Braun. Plates 24, P. Weiss. Plate 25, C. F. Hirjibehedin. Plate 27, F. J. Giessibl. Plate 28, J. V. Lauritsen and F. ` Custance. Plate 30, O. ` Custance. Besenbacher. Plate 29, R. P´erez and O.

C. Julian Chen Columbia University New York July 2007

Preface to the First Edition It has been more than 10 years since the scanning tunneling microscope (STM) made its debut by resolving the structure of Si(111)-7×7 in real space [41]. This new instrument has proved to be an extremely powerful tool for many disciplines in condensed-matter physics, chemistry, and biology. The STM can resolve local electronic structure at an atomic scale on literally every kind of conducting solid surface, thus also allowing its local atomic structure to be revealed. An extension of scanning tunneling microscopy, atomic force microscopy (AFM) [31], can image the local atomic structure even on insulating surfaces. The ability of STM and AFM to image in various ambiances with virtually no damage or interference to the sample made it possible to observe processes continuously. For example, the entire process of a living cell infected by viruses was investigated in situ using AFM [42]. The field of scanning tunneling microscopy has enjoyed a rapid and sustained growth, phenomenal for a new branch of science. The growing number of papers presented at the six International STM Conferences documents the rising interest in this field, as shown in Table 1. Similarly, STMs are being developed commercially at an astonishing speed. As of summer 1991, over 30 companies have manufactured and marketed STMs and parts. Many companies dedicated to STMs and AFMs are seeing their business expand rapidly. The recent performance of Digital Instruments, Park Scientific Instruments, WA Technology, Angstrom Technology, TopoMetrix, and RHK Technology are typical of such companies. Likewise, a number of major manufacturers of scientific instruments have expanded their wares to include STMs. Vacuum Generators, Newport Instruments, Omicron Vacuumphysik GmbH, Leica, JEOL, Nikon, and Seiko number among these manufacturers. Several other companies are supplying STM parts and accessories, such as piezoelectric elements, steppers, and probe tips. Because of their broad endorsement by the scientific and com-

Table 1: International STM Conferences Year

Date

Location

Papers

1986 1987 1988 1989 1990 1991

July 14–18 July 20–24 July 4–8 July 9–14 July 23–27 August 12–16

Santiago de Compostela, Spain Oxnard, California Oxford, UK Oarai, Japan Baltimore, Maryland Interlaken, Switzerland

59 110 157 213 357 580

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mercial spheres, STMs and AFMs will stand alongside optical and electron microscopes for the foreseeable future as important instruments in laboratories around the world. Ten years in contemporary science is a long, long period. Myriad startling developments can take place in that amount of time. As Ziman says in Principles of the Theory of Solids: “Today’s discovery will tomorrow be part of the mental furniture of every research worker. By the end of next week it will be in every course of graduate lectures. Within the month there will be a clamor to have it in the undergraduate curriculum. Next year, I do believe, it will seem so commonplace that it may be assumed to be known by every schoolboy.” With the prolific activity in the field of scanning tunneling microscopy over the past 10 years, there remains a visible gap in the published material on that topic. Several edited collections of review articles on STM have already been published (for example, Behm, Garcia, and Rohrer, editors, Scanning Tunneling Microscopy and Related Methods, Kluwer, 1990, G¨ untherodt and Wiesendanger, editors, Scanning Tunneling Microscopy, Vols. I through III, Springer Verlag, 1991/1992, and a number of others.) A textbook on scanning force microscopy has already appeared (Sarid, Scanning Force Microscopy, Oxford University Press, 1991). Nevertheless, a coherent treatise or textbook on STM is still lacking. We have seen the use of STM and the information derived from it continue to expand rapidly. We have seen the potential of the STM in the microelectronics and chemical industries for process control and diagnostics gradually becomes a reality. The need for a basic reference book and a textbook on STM is clearly evident. To satisfy this need is my goal in writing this work. In the iterative process of planning and writing the chapters of this book, I have encountered a number of unexpected difficulties. First, a treatise or a textbook must be presented in a logical sequence starting from a common knowledge background, for example, the standard undergraduate physics courses. However, there are many inconsistencies and discontinuities in the existing STM literature. To make a logically coherent presentation, the materials have to be digested carefully, and the numerous gaps have to be filled. A second unexpected difficulty springs from the fact that the field of STM is inherently cross-disciplinary. The roots of STM run deeply into quantum mechanics, solid-state physics, chemical physics, electronic engineering, mechanical engineering, and control theory. To organize the necessary information in a comprehensible and coherent manner for a broad spectrum of readers is no trivial task. Finally, the field of STM is evolving so rapidly that a painstakingly written treatise or textbook might be obsolete just as it has been printed. Based on those considerations, in this book I have chosen, to the best of my knowledge and judgement, only the topics that are fundamental, indispensable, and having a lasting value. The organization of the book is as follows.

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The first chapter, Overview, describes the basic experimental facts and theoretical concepts. This chapter is written at a general physics level and can be read as an independent unit. Part I of the book is a systematic presentation of much of the fundamental physics in STM and AFM from a single starting point, a time-dependent perturbation theory that is a modified version of the Bardeen approach. The Bardeen approach to tunneling phenomena is a perturbation theory for the understanding of the classical tunneling junction experiments. For atomicscale phenomena as in STM and AFM, modifications to the original Bardeen approach are necessary. This approach can provide conceptual understanding and analytic predictions for both tunneling current and attractive atomic forces, as well as a number of tip–sample interaction effects. All essential derivations are given in full detail, to make it suitable as a textbook and a reference book. To understand Part I, some familiarity with elementary quantum mechanics and elementary solid-state physics is expected. For example, the reader is assumed to know elementary quantum mechanics equivalent to the first seven chapters of the popular Landau–Lifshitz Quantum Mechanics (Nonrelativistic Theory) and elementary solid-state physics equivalent to the first ten chapters of Ashcroft and Mermin’s Solid State Physics. Part II of the book deals with basic physical principles of STM instrumentation and applications, with many concrete working examples. The reader is expected to know relevant materials routinely taught in the standard undergraduate curricula in the fields of physics, chemistry, materials science, or related engineering disciplines. Effort is taken to make every chapter independent, such that each chapter can be understood with little reference to other chapters. Piezoelectricity and piezoelectric ceramics, which are not taught routinely in colleges, are presented from the basic concepts on. Elements of control theory, necessary for the understanding of STM operation, are presented for physicists and chemists in the main text and in an Appendix. As in Part I, all essential derivations are given in full detail. For specific applications, preliminary knowledge in the specific fields of the reader’s interest (such as physics, chemistry, electrochemistry, biochemistry, or various engineering sciences) is assumed. The Appendices cover a number of topics that are not standard parts of an average science or engineering undergraduate curriculum, and are relatively difficult to glean from popular textbooks. To understand those Appendices, the reader is assumed to have the average undergraduate background of a science or engineering major. This introductory book, moderate in size and sophistication, is not intended to be the ultimate STM treatise. The first part of this book, especially, is not intended to be a comprehensive review of all published STM theories. More sophisticated theoretical approaches, such as those directly based on first-principles numerical calculations, are beyond the scope of

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this introductory book. With its moderate scope, this book is also not intended to cover all applications of STM. Rather, the applications presented are illustrative in nature. Several excellent collections of review articles on STM applications have already been published or are in preparation. An exhaustive presentation of STM applications to various fields of science and technology needs a book series, with at least one additional volume per year. Moreover, this book does not cover the numerous ramifications of STM, except a brief chapter on AFM. The references listed at the back of the book do not represent a catalog of existing STM literature. Rather, it is a list of references that would have lasting value for the understanding of the fundamental physics in STM and AFM. Many references from related fields, essential to the understanding of the fundamental processes in STM and AFM, are also included. A preliminary camera-ready manuscript of this book was prepared in 1991. To ensure that its content was well-rounded, reasonably truthful and accurate, the book manuscript was sent to many fellow scientists for reviewing, mostly arranged by Oxford University Press. I am greatly indebted to those reviewers who have spent a substantial amount of time in scrutinizing it in detail and providing a large number of valuable comments and suggestions, both as reviewing reports and as marked on the manuscript. Those comments greatly helped me to correct omissions, inaccuracies, and inconsistencies, as well as improving the style of presentation. Among the reviewers are, in alphabetical order, Dr A. Baratoff (IBM Zurich Laboratory), Dr I. P. Batra (IBM Almaden Laboratory), Professor A. Briggs (University of Oxford), Dr S. Chiang (IBM Almaden Laboratory), Dr R. Feenstra (IBM Yorktown Laboratory), Professor N. Garcia (Universidad Autonoma de Madrid), Professor R. J. Hamers (University of Wisconsin, Madison), Professor J. B. Pethica (University of Oxford), Professor C. F. Quate (Stanford University), Dr H. Rohrer (IBM Zurich Laboratory), Professor W. Sacks (Universit´e Pierre et Marie Curie), Professor T. T. Tsong (Pennsylvania State University), and Dr R. D. Young (National Institute of Standard and Technology). In addition, a number of fellow scientists reviewed specific parts of it. Also, in alphabetical order, Dr N. Amer (IBM Yorktown Laboratory, Chapter 15), R. Borroff (Burleigh Instrument Inc., Chapter 12), Dr F. J. Himpsel (IBM Yorktown Laboratory, Chapter 3), R. I. Kaufman (IBM Yorktown Laboratory, Chapter 11), Dr V. Moruzzi (IBM Yorktown Laboratory, Chapter 3), C. Near (Morgan Matroc Inc., Vernitron Division, Chapter 9), Dr E. O’Sullivan (IBM Yorktown Laboratory, sections on electrochemistry), R. Petrucci (Staveley Sensors Inc., EBL Division, Chapter 9), Dr D. Rath (IBM Yorktown Laboratory, sections on electrochemistry), and Dr C. Teague (National Institute of Standard and Technology, Chapter 1). I would also like to thank the senior editor in Physical Sciences, Mr. J. Robbins, and Miss A. Lekhwani of Oxford University Press in assist-

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ing in the publication, especially the painstaking reviewing process of the manuscript. Also, special thanks to Elizabeth McAuliffe of IBM Yorktown Laboratory for her development and assistance in the BookMaster software enabling me to prepare the camera-ready manuscript, which was essential for the preprinting reviewing process. Numerous colleagues provided original STM and AFM images, which enabled me to compile a gallery of 30 spectacular images in the high-quality printing section, which are acknowledged on each page. Finally, I admit that, even with such extensive reviews and revisions, the book is still neither perfect nor does it represent the last word. Especially in such an active field as STM and AFM, new concepts and new measurements come out every day. I expect that substantial progress will be made in the years to come. Naturally, I am looking forward to future editions. I am anxious to hear any comments and suggestions from readers, with whose help, the future editions of this book would be more useful, more truthful, and more accurate. Thus spake Johann Wolfgang von Goethe: Oft, wenn es erst durch Jahre durchgedrungen, Erscheint es in vollendeter Gestalt. Was gl¨ anzt, ist f¨ ur den Augenblick geboren, Das Echte bleibt der Nachwelt unverloren.1

October 1992, at Yorktown Heights, New York C. Julian Chen

1 Often, after years of perseverance, it emerges in a completed form. What glitters, is born for the moment. The Genuine lives on to the afterworld. Faust, Vorspiel auf dem Theater.

Gallery Historical photographs Plate 1. The IBM Zurich Laboratory soccer team Plate 2. A humble gadget that shocked the science community Plate 3. The creators of the atomic force microscope Plate 4. The first atomic force microscope STM studies of surface structures Plate 5. ‘Stairway to Heaven’ to touch atoms Plate 6. Zooming into atoms Plate 7. Underneath the Si(111)-7×7 surface Plate 8. STM image of a GaN(000¯ 1) surface √ Plate 9. Large-scale image of the Au(111)-22× 3 structure Plate 10. Large-scale image of the Ge(111) surface Plate 11. Details of the Ge(111)-c(2×8) surface Molecular Plate 12. Plate 13. Plate 14. Plate 15. Plate 16.

orbitals and chemistry Individual π and π ∗ molecular orbitals observed by STM Organic molecules observed by STM Observation of the HUMO and LOMO of an organic molecule Voltage-dependent images of the Si(111)-2×1 surface Chemical vapor deposition of the Si(100) surface

Spin-polarized STM Plate 17. SP-STM images of an Fe island Plate 18. SP-STM images of Dy films Plate 19. SP-STM studies of nanoscale Co islands on Cu(111) Plate 20. SP-STM studies of antiferromagnetic crystal Mn3 N2 Atom manipulation Plate 21. Construction of Cun chains by atom manipulation using STM Plate 22. Quantum states observed on Cun chains Plate 23. The triangular quantum corral Plate 24. Manipulating hydrogen atoms underneath palladium surface Plate 25. An artificial atomic-scale rock garden Atomic force microscopy Plate 26. Si(111)-7×7 structure resolved by AFM Plate 27. Current images and higher-harmonics force images on graphite Plate 28. Tip dependence of non-contact AFM images of TiO2 Plate 29. Chemical identification of individual surface atoms using AFM Plate 30. Atom manipulation using AFM

Gallery

Plate 1. The IBM Zurich Laboratory soccer team. On October 15, 1986, the soccer team of IBM Zurich Laboratory and Dow Chemical played a game which had been arranged earlier. To everyone’s surprise, a few hours before the game, the Swedish academy announced the Nobel Prize for Gerd Binnig (right, holding flowers) and Heinrich Rohrer (left, holding flowers). Newspaper reporters rushed in for a press conference. Towards the end of the press conference, Binnig and Rohrer said that they must leave immediately because both were members of the laboratory soccer team. The reporters followed them to the soccer field. A photographer for the Swiss newspaper Blick took this photograph before the game started. At the center of the photograph, holding a soccer ball is Christoph Gerber, responsible for building the first scanning tunneling microscope as well as the first atomic force microscope. (Original photograph by courtesy of IBM Zurich Laboratory.)

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Plate 2. A humble gadget that shocked the scientific community. Upper, a photograph of an earlier version of the scanning tunneling microscope that enabled real-space imaging of the Si(111)7×7 structure. Lower, a schematic of it. The instrument, built with modest means, has been able to fulfill the ambitious goals proposed by Richard Feynman in 1959 in a famous speech There’s Plenty of Room at the Bottom [28]: to see individual atoms distinctly, and to arrange the atoms the way we want. It became a cornerstone of nanotechnology as we know it. (Original photograph by courtesy of Ch. Gerber.)

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Plate 3. The creators of the atomic force microscope (AFM). From the left, Calvin Quate, Gerd Binnig, and Christoph Gerber. In 1985, Gerd Binnig and Christoph Gerber took a sabbatical leave from IBM Zurich Laboratory to become adjunct faculty members of Stanford University. Together with Calvin Quate, they designed and built the first AFM. Just before the Christmas of 1985, they submitted a manuscript entitled Atomic Force Microscope to Physical Review Letters. It was then published on March 3, 1986 [31]. In the following two decades, as of June 2007, the paper was cited more than 4900 times, which makes it the second most cited paper published in Physical Review Letters. They are now among the 100 most cited scientists worldwide. (Original photograph by courtesy of Ch. Gerber.)

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Plate 4. The first atomic force microscope (AFM). (a) A photograph of the first AFM, now on permanent exhibition in the Science Museum of London. (b) A schematic, adapted from a figure in the first Physical Review Letters on AFM [31]. As shown, the first AFM was built with modest means. Originally designed for imaging the surfaces of non-conducting materials with nanometer or atomic resolution, AFM has been adapted to various environments (for example, in liquids, at low temperatures, in high magnetic fields), and found applications to materials sciences, chemistry, and molecular biology. (Original photograph by courtesy of Ch. Gerber. See [43] for details.)

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Plate 5. ‘Stairway to Heaven’ to touch atoms. Since the time of John Dalton, for more than 150 years, atoms and molecules were entities in the theoretical scientist’s heaven. As recently as 1984, in an article in Physics Today, entitled A Theorist’s Philosophy of Science, atoms and molecules were still characterized as entities that cannot be perceived, by anyone, ever [44]. Since the invention of STM, individual atoms and molecules as well as the electronic states therein became a tangible reality, indubitability. Image showing Si(111)-7×7 surface with steps [45]. Size of image: 32×36 nm. The height of each step is 1.2 nm. (Original image by courtesy of R. Wiesendanger.)

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Plate 6. Zooming into atoms. Three constant-current topographic STM images of 1T -TaS2 . (a) On a large scale, the charge density wave, with a periodicity of 1.18 nm, is the prominent feature. Such an observation depends weakly on tip electronic structure. (b) With good tip conditions, the atomic structure, with periodicity of 0.335 nm, can be clearly identified. (c) The amplitude of the atomic corrugation can be greater than 100 pm. Single atom defects are clearly resolved. Images obtained as calibration of an early STM designed and built by the author, using a tip mechanically cut from a Pt-Ir wire, with bias 20 mV and set current 2 nA.

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Plate 7. Underneath the Si(111)-7×7 surface. By stripping the first layer of Si atoms off the Si(111)-7×7 surface, the underlying layer is exposed. The Si atoms on the second layer are more closely packed. (a) The usual STM image of the Si(111)-7×7 surface, with nearest-neighbor distance of 0.768 nm. (b) After reacting with chlorine, the top-layer Si atoms are stripped off. The nearest-neighbor distance of the Si atoms on this layer is 0.384 nm. Such a resolution is now routinely achieved after a proper treatment of the tip. (Original images by courtesy of J. Boland. For details, see [46].)

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¯ surface. STM image of a screw Plate 8. STM image of a GaN(0001) dislocation intersecting an MBE-grown GaN(000¯1) surface, with various domains of a c(6×12) surface reconstruction surrounding the dislocation. GaN growth occurs by step flow at the steps emanating out from the dislocation core, resulting in a spiral growth mound forming around the dislocation. Image was taken with sample bias 1.0 V and tunneling current 0.05 nA. Size of the image: 75×100 nm2 . (Original image by courtesy of A. R. Smith and R. Feenstra. See Ref. [47] for details.)

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√ Plate 9. Large-scale image of the Au(111)-22× 3 structure. The √ Au(111) surface reconstructs at room temperature to form a 22 × 3 structure, which has a twofold symmetry. On a large scale, three equivalent orientations of such reconstruction coexist on the surface. Furthermore, on an intermediate scale, a herringbone pattern is formed. (Original image by courtesy of J. V. Barth. See [48] for details.)

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Plate 10. Large-scale image of the Ge(111) surface. The Ge sample was cleaved in ultra-high vacuum, and annealed at about 400◦ C. On most of the areas of the surface, a c(2 × 8) reconstruction is observed. Because the nascent Ge(111) surface has a threefold symmetry and the c(2 × 8) reconstruction has two-fold symmetry, there are three equivalent orientations. In the transition regions, a 5 × 5 reconstruction appears. Image taken with bias +1.8V and set current 0.1 nA. Size: 130×170 nm2 . (Original image by courtesy of R. M. Feenstra. See [49] for details.)

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Plate 11. Details of the Ge(111)-c(2×8) surface. (a) The dangling bonds on the top-layer Ge atoms are revealed by STM. (b) After reacting with hydrogen, the top-layer Ge atoms are removed. The underlying Ge atoms are arranged in a bulk-like structure. The distance between the Ge atoms is 0.399 nm. (Original images by courtesy of J. Boland. See [50] for details.)

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Plate 12. Individual π and π ∗ molecular orbitals observed by STM. On the nascent Si(100) surface, every Si atom has a pz -type dangling bond. By saturating the Si(100) surface with hydrogen, all the pz -type dangling bonds are capped with a hydrogen atom. By heating the H-saturated Si(100) surface carefully, a small fraction of the hydrogen desorbed. The pz type dangling bonds are paired to form bonding and antibonding orbitals. (a) The filled π-orbital. (b) The empty π-orbital. As expected, the empty π-orbital exhibits a nodal plane in between. (Original image by courtesy of J. Boland. See [51, 52] for details.)

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Plate 13. Organic molecules observed by STM. Upper, an STM image of n-C32 H66 adsorbed on a graphite surface. The length of these lineal molecules is approximately 4 nm. Note the high degree of two-dimensional ordering: the long molecule axes are parallel to each other, and troughs are formed where the ends of the molecules abut. The image is about 10×10 nm2 . The bias is 0.4 V. The set current is 1 nA. Lower, a proposed model. (a) Graphite surface with n-C32 H66 molecules superimposed. (b) Careful observation of the angle and spacing in the STM images shows that its features correspond to those expected for graphite rather than n-C32 H66 molecules. (Original image by courtesy of D. J. Thomson. See [53] for details.)

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Plate 14. Observation of the HOMO and LUMO of an organic molecule. By depositing an extra-thin insulating layer (for example, two atomic layers of NaCl) on a conducting surface (for example, Cu(111)), the organic molecules adsorbed on top exhibit features similar to free molecules. Using STM, especially the dynamic-conductance imaging, the electronic states can be probed. (a) The STM images of pentacene molecules at biases corresponding to the energy level of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). With a pentecene molecule adsorbed on the tip, the images becomes sharper. Also shown is the theoretical charge-density contours at the HOMO and LUMO energy levels, and the atomic structure of pentecene. (b) The tunneling spectrum (solid), and the dynamic-conductance curve (dotted) of pentecene, showing discrete energy levels. (Original images by courtesy of J. Repp and G. Meyer. See [2] for details.)

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Plate 15. Voltage-dependent images of the Si(111)-2×1 surface. (a) The topographic images of Si(111)-2×1 at a positive bias looks different from the topographic image of the same surface at a negative bias. In particular, in the (0¯ 11) direction, the positions of the peaks are reversed. This becomes even clearer in (b), where the topographic contours of the images at both biases are displayed together. The STM images reflect the electronic structure rather than the positions of the nuclei. (Original images by courtesy of R. Feenstra. See [54] for details.)

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Plate 16. Epitaxial growth of the Si(100) surface by chemical vapor deposition. By exposing a Si(100) surface at 400◦ C to disilane (Si2 H6 ), new Si layers are grown. During this process, disilane decomposes into SiH2 groups with pair up, eliminates hydrogen, and forms the dimer structure characteristic of the Si(100) surface. Upper, as STM image of a Si(100) surface after exposure to hydrogen and annealed at 650K for 10 sec. The area shown is 25×25 nm2 . Lower, schematic of the decomposition of the SiH2 fragments to yield the epitaxial monohydride surface. (Original image by courtesy of J. Boland. See [55] for details.)

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Plate 17. Spin-polarized STM images of an Fe island. Using a spin-polarized tip, the magnetic properties of nanometer-scaled structures can be imaged. (a) The topographic image of an Fe island on a W(110) surface. It is essentially featureless. (b) The dynamic conductance image of the same Fe island. A magnetic vortex is clearly shown. (c) A schematic of the magnetic vortex core. (Original images by courtesy of M. Bode. See [56, 57] for details.)

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Plate 18. Spin-polarized STM images of Dy films. Using a spinpolarized tip, the STM images show clear magnetic domains on Dy films grown on W(110) surfaces. The samples were prepared by vacuum depositing Dy on W(110) surface at room temperature, followed by annealing at 490K to 680K for 4 min. (a) A topographic image of a 90 ML Dy film on W(110) substrate, showing essentially featureless surface except some dislocation lines. (b) Using a tip with a Dy cluster picked up by controlled collision, the magnetic domain structure becomes apparent. (c) A topographic image of a thick (450 ML) Dy film on a W(110) surface, showing thickness variation and dislocation lines. (d) Using a spin-polarized tip, the magnetic domains become apparent. Note that there is only a weak correlation between the dislocation lines and the magnetic domain structure. (Original image by courtesy of L. Berbil-Bautista. See [58] for details.)

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Plate 19. Spin-polarized STM studies of nanoscale Co islands on Cu(111). Co/Cu multilayers are widely used in devices based on the giant magnetoresistance effects. The interface can be studied by depositing Co on Cu surfaces. Using STM, very rich phenomena were observed. (a) The epitaxial Co adlayers form two types of triangular islands. (b) Laterally resolved spin-polarized STM images show that each of the triangular islands is a ferromagnet, with reversible magnetization vectors. (c)−(f) At different bias voltages, the magnetic contrast and the standing waves exhibit a variety of patterns. (Original images by courtesy of O. Pietzsch. See [59, 60] for details.)

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Plate 20. Spin-polarized STM studies of antiferromagnetic crystal Mn3 N2 . On the Mn3 N2 (010) surface, the adjacent rows of Mn atoms have alternating spin polarization orientations. Using a spin-polarized tip, the topographic STM image (a) contains an electronic component and a magnetic component, see (b) and (c). In (b), the tip is coated with Mn, which is antiferromagnetic. In (c), the tip is coated with Fe, which is ferromagnetic. The results are similar. (Original images by courtesy of A. R. Smith. See [61, 62] for details.)

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Plate 21. Construction of Cun chains by atom manipulation using STM. Using the STM, atoms can be arranged the way we want. Here, a sequence of constant-current STM images (13.5×13.5 nm2 , 1 nA, 1 V) demonstrate the assembly of a monatomic Cu20 chain on a Cu(111) surface by lateral manipulation at 7K. Starting from a group of discrete Cu adatoms (top left image), a Cu dimer is formed by joining two adatoms (top right image), and straight monatomic chains are built up by repeated incorporation of Cu adatoms, one atom at a time. Each manipulation step increases the apparent chain length by an increment of a0 = 0.255 nm (Cu bulk lattice constant a0 = 0.361 nm), indicating that the Cu chain atoms reside on nearest-neighbor surface lattice sites. The unstable and roundshaped appearance of the dimer shown in the inset (1.1×1.1 nm2 , 1 nA, 1 V) is due to local intra-cell diffusion of the Cu/Cu(111) dimer present at temperatures > 5K. (Original images by courtesy of S. F¨olsch.)

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Plate 22. Quantum states observed on Cun chains. Top right panel: Differential tunneling conductance spectra measured for monatomic Cu/Cu(111) adatom chains in the regime of unoccupied sample states at a constant STM tip height of about 0.6 nm; for clarity the spectra are offset relative to the bottom curve showing the dI/dV spectrum of the bare Cu(111) surface. While a single peak is observed at 3.3 V for the discrete Cu adatom, a series of chain-localized quantum states develops with increasing chain length. The gray-scale images show the spatial variation of the dI/dV signal (as a measure of the local density of states) recorded at the respective biases at which peaks are observed in the dI/dV spectra. Rows from top to bottom correspond to chains of three to nine atoms while columns include states of order n (n: number of lobes of the squared wavefunction). The data reveal the on-chain localization of the quantum states which arise from interatomic coupling and provide a clear-cut example of quasi-one-dimensional confinement in an atomic-scale nanostructure. (Original images by courtesy of S. F¨olsch. See [63] for details.)

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Plate 23. The triangular quantum corral. (a) The making of a triangular quantum corral. Using 51 Ag atoms, a triangular quantum corral is assembled using STM. A series of six STM images (50×50 nm2 , at U = 39 mV, I = 1.1 nA) show the progress of the construction. (b) Standing waves of the surface states confined inside the triangular quantum corral, and the scattered waves outside the triangular quantum corral. (Original images by courtesy of K.-F. Braun. See [64] for details.)

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Plate 24. Manipulating hydrogen atoms beneath the surface of palladium. Besides manipulating atoms adsorbed on top of solid surfaces, the STM is demonstrated to be able to manipulate atoms beneath the surface. An example is hydrogen in palladium. The interaction of H in Pd is of fundamental importance to catalysis, fuel cells, and hydrogen storage, etc. a and b: Using STM, the local concentration of hydrogen atoms beneath the metal surface can be altered, and than can be mapped as STM images. c: Three letters, the initials of Pennsylvania State University, with a nittany-lion paw, the symbol of PSU, written using the subsurface hydrogen concentration in palladium as the ink. (Original images by courtesy of P. Weiss. See [65] for details.)

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Plate 25. An artificial atomic-scale rock garden. Top, a perspective view of the STM image. Below, a cross-sectional schematic of the atomic structure. The ‘rocks’ are made of copper. The monoatomic steps of the Cu(100) surface look smooth where the individual atoms are not resolved. The ‘soil’ is made of CuN, a monolayer of nitrogen atoms sitting on top of the copper surface. Although geometrically, the N layer is higher than the Cu surface, in the STM image, CuN looks lower because of the low local density of states due to its insulating character. Various types of ‘plants’ – Mn chains – are assembled by atom manipulation using STM, mostly on the CuN islands, as illustrated in the schematic. Species include chains of 7 Mn atoms (Mn7 ) and 10 Mn atoms (Mn10 ). There are several types of dimers, Mn2 : A dimer along a vertical N row (V); dimers along horizontal N rows (H); a dimer of nearest-neighbor diagonal Cu sites (D); and a dimer with the Mn atoms on top of adjacent N sites (N). (Original image by courtesy of C. F. Hirjibehedin. The schematic is designed by the author. See [66] for details.)

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Plate 26. Si(111)-7×7 resolved by atomic force microscopy. Similar to STM, the dynamic-mode AFM resolves the atomic structure of the Si(111)-7×7 surface. However, AFM can resolve not only the adatoms, but also the rest atoms. This is because the attractive atomic force can reach farther than tunneling conductance. (a) A 10×7 nm2 image obtained in the topographic mode. (b) A 5×5 nm2 image, showing more details. (c) The DAS model of the Si(111)-7×7 surface. (d) The topographic profile taken along the dashed line in (b). (e) The topographic profile taken along the solid line in (b). Here, ch, ca, ra, and ct-a denote the corner hole, corner adatom, rest atom, and center atom, respectively. (Adapted with permission from [67]. Copyrighted (2000) by the American Physical Society.)

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Plate 27. Constant-height current images and higher-harmonics force images on graphite (HOPG). Recorded with an oscillating tungsten tip mounted on a qPlus sensor. The left column shows the averaged tunneling current, revealing a typical STM image of HOPG. The center column shows the magnitude of the higher harmonics signal. The tungtsten tip oscillates in the highly nonlinear force field of the graphite sample, and a large magnitude of the higher harmonics (bright in center column) points to a large magnitude of the higher order force gradients. The three higher harmonic images in the center column along with there tunneling current companions in the left column were selected from a large data set where the W tip was altered by applying voltage pulses or slight collisions. The symmetry of the higher harmonic images points to the orbital structure of the W front atom. Tungsten is a bcc material, therefore there are three highsymmetry crystal directions (right column): [110] with a two-fold symmetry (top), [111] with a three-fold symmetry (middle) and [100] with a four-fold symmetry (bottom). (Original image by curtesy of F. J. Giessibl. See [68] for detials.)

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Plate 28. Tip dependence of non-contact AFM images of TiO2 . The non-contact AFM images of ionic crystals originate from the Coulomb interactions between the partially ionized atoms at the sample surface and the outermost atom of the tip apex. Depending on the polarity of tip apex, two basic types of images are observed. (a) Sometimes, a sponteneous tip restructuring would cause a switch between the two types of images. (b) With a negatively terminated tip, the oxygen rows look dark, the titanium rows look bright, and the OH group looks very bright. (c) With a positively terminated tip, the oxygen rows look bright, the titanium rows look dark, and the OH group looks very dark. (d) A ball model of the TiO2 surface. (e)-(f) Computer simulations of the NC-AFM images of the TiO2 surface, especially the OH site with different tip polarities. (Original figures by courtesy of J. V. Lauritsen and F. Besenbacher. See [69] for details.)

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Plate 29 (opposite page). Chemical identification of individual surface atoms using atomic force microscopy. By quantifying the short-range chemical interaction force between the different atoms of a heterogeneous surface and the outermost atom of the tip, it is possible to reveal the chemical composition of a surface area at atomic scale. Here, there is an example where the local composition of a surface alloy containing Si, Sn, and Pb atoms blended in equal proportions on a Si(111) substrate [(a) and (e)] has been disclosed [(c) and (g)] by quantifying the maximum attractive forces over each surface atom and comparing them with the relative interaction ratio for Sn and Pb calibrated against Si [(d) and (h)]. This relative interaction ratio is a magnitude independent of the tip-apex structure and chemical termination that reflects the relative strength pair of surface atoms have for making a chemical bond with the outermost atom of the tip. Notice that some of these surface atoms show identical topographic signals [(b) and (f)] upon the local environment, making it impossible to identify them by counting solely on the topography. Image dimensions are (4.3×4.3) nm2 . (Original figures in black and white by courtesy of R. P´erez ´ Custance. Reprinted with permission [70]. Copyrighted (2007) by and O. Macmillan Publishers Ltd.) Plate 30 (above). Atom manipulation using the atomic force microscope. The atomic force microscope can be used for atom-by-atom assembly of designed nanostructures. This image represents a symbol of the Tin element, created at room temperature by the manipulation of more than 120 individual Sn atoms at a Ge(111)-c(2×8) surface. Image dimensions are (7.7×4.8) nm2 . (Original image in black and white by courtesy ´ Custance. Reprinted with permission [71]. Copyrighted (2005) by of O. Macmillan Publishers Ltd.)

Chapter 1 Overview 1.1

The scanning tunneling microscope

The scanning tunneling microscope (STM) was invented by Binnig and Rohrer and implemented by Binnig, Rohrer, Gerber, and Weibel [29, 30]. Figure 1.1 shows its essential elements. A probe tip, usually made of W or Pt–Ir alloy, is attached to a piezodrive, which consists of three mutually perpendicular piezoelectric transducers: x piezo, y piezo, and z piezo. Upon applying a voltage, a piezoelectric transducer expands or contracts. By applying a sawtooth voltage on the x piezo and a voltage ramp on the y piezo, the tip scans on the xy plane. Using the coarse positioner and the z piezo, the tip and the sample are brought to within a fraction of a nanometer each other. The electron wavefunctions in the tip overlap electron wavefunctions in the sample surface. A finite tunneling conductance is generated. By applying a bias voltage between the tip and the sample, a tunneling current is generated. The concept of tunneling will be presented in Section 1.2.

Fig. 1.1. The scanning tunneling microscope in a nutshell. The scanning waveforms, applying on the x and y piezos, make the tip raster scan on the sample surface. A bias voltage is applied between the sample and the tip to induce a tunneling current. The z piezo is controlled by a feedback system to maintain the tunneling current constant. The voltage on the z piezo represents the local height of the topography. To ensure stable operation, vibration isolation is essential.

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2

Overview

A widely used convention of the polarity of bias voltage is that the tip is virtually grounded. The bias voltage V is the sample voltage. If V > 0, the electrons are tunneling from the occupied states of the tip into the empty states of the sample. If V < 0, the electrons are tunneling from the occupied states of the sample into the empty states of the tip. The tunneling current is converted to a voltage by the current amplifier, which is then compared with a reference value. The difference is amplified to drive the z piezo. The phase of the amplifier is chosen to provide a negative feedback: if the absolute value of the tunneling current is larger than the reference value, then the voltage applied to the z piezo tends to withdraw the tip from the sample surface, and vice versa. Therefore, an equilibrium z position is established. As the tip scans over the xy plane, a two-dimensional array of equilibrium z positions, representing a contour plot of the equal tunneling-current surface, is obtained, displayed, and stored in the computer memory. The topography of the surface is displayed on a computer screen, typically as a gray-scale image, see Fig. 1.2(a). The gray-scale image is similar to a black-and-white television picture. Usually, the bright spots represent high z values (protrusions), and the dark spots represent low z values (depressions). The z values corresponding to the gray levels are indicated by a scale bar. For a more quantitative representation of the topography, a contour plot along a given line is often provided, as shown in Fig. 1.2(b). The most convenient unit for x and y is nanometer (nm, 10−9 m), and the most convenient unit for z is picometer (pm, 10−12 m). To achieve atomic resolution, vibration isolation is essential. This is achieved by making the STM unit as rigid as possible, and by reducing the influence of environmental vibration to the STM unit. See Chapter 10.

Fig. 1.2. Gray-scale image and contour plot. (a) A 5nm×5nm gray-level topographic image of Si(111)7×7. The bright spots represent protrusions, and the dark spots represent depressions. The z values corresponding to the gray levels are indicated by a scale bar. (b) The topographic contour along a line in (a), for a more quantitative representation. By courtesy of F. Giessibl.

1.2 The concept of tunneling

1.2

3

The concept of tunneling

In this section, we present elementary theories of tunneling through a onedimensional potential barrier, which will provide us with an understanding of the basic physics of STM. 1.2.1

Transmission coefficient

In classical mechanics, an electron in a potential U (z) is described by p2 + U (z) = E, (1.1) 2m where m is the electron mass, and E is the energy. In regions where E > U (z), the electron has a non-zero momentum p = [2m(E − U )]1/2 . On the other hand, the electron cannot penetrate into any region with E < U (z), or a potential barrier. In quantum mechanics, the electron is described by a wavefunction ψ(z), which satisfies the Schr¨ odinger equation, 2 d2 ψ(z) + U (z)ψ(z) = Eψ(z). (1.2) 2m dz 2 Consider the case of a piecewise-constant potential, as shown in Fig. 1.3. In the classically allowed region, E > U, Eq. 1.2 has solutions −

ψ(z) = ψ(0)e±ikz ,

(1.3)

 2m(E − U ) k= 

(1.4)

where

Fig. 1.3. The difference between classical theory and quantum theory. In classical mechanics, a particle cannot pass through a potential barrier. In quantum mechanics, a particle has a non-zero probability of tunneling through a potential barrier. After an idea of Van Vleck. See [72].

4

Overview

is the wave vector. The electron is moving (in either a positive or negative direction) with a constant momentum p = k, or a constant velocity v = p/m, the same as the classical case. In the classically forbidden region, Eq. 1.2 has a solution ψ(z) = ψ(0)e−κz ,

(1.5)

 2m(U − E) κ= 

(1.6)

where

is the decay constant. It describes an electron penetrating through the barrier into the +z direction. The probability density of observing an electron near a point z is proportional to |ψ(0)|2 e−2κz , which has a non-zero value in the barrier region, thus has a non-zero probability penetrating a barrier. Another solution, ψ(z) = ψ(0) eκz , describes an electron state decaying in the −z direction, indicating that tunneling is bidirectional. Starting from this elementary model, we can explain some basic features of metal–vacuum–metal tunneling, as shown in Fig. 1.4. The work function φ of a metal surface is defined as the minimum energy required to remove an electron from the bulk to the vacuum level. In general, the work function depends not only on the material, but also on the crystallographic orientation of the surface. Neglecting the thermal excitation, the Fermi level is the upper limit of the occupied states in a metal. Taking the vacuum level as the reference point of energy, we have EF = −φ. To simplify discussion, we assume that the work functions of the tip and the sample are equal. The electron in the sample can tunnel into the tip and vice versa. By applying a bias voltage V, a net tunneling current occurs. A sample state ψn with energy level En lying between EF − eV and EF has a chance tunneling into the tip. We assume that the bias is much smaller than the value of the work function, that is, eV  φ. Then the energy levels of all the sample states of interest are very close to the Fermi level, that is, En ≈ −φ. Here we define a transmission coefficient T, which is the ratio of the tunneling current at the tip surface z to the impinging current at z = 0,

T ≡

I(z) = e−2κz , I(0) √

where

(1.7)

2mφ (1.8)  is the decay constant of a sample state near the Fermi level in the barrier region. Using electron-volt as the unit of the work function, and nm−1 as the unit of the decay constant, the numerical value of Eq. 1.8 is κ=

1.2 The concept of tunneling

5

Fig. 1.4. A one-dimensional metal–vacuum–metal tunneling junction. (a) The sample, left, and the tip, right, are modeled as semi-infinite pieces of free-electron metal. When the distance z is small, the vacuum tail of a sample state can penetrate into the region of the tip. By applying a bias voltage eV , the sample states between energy level EF − eV and EF can tunnel to the tip, generating a tunneling current proportional to the bias voltage V . Once the atomic configuration of an STM junction is fixed, the tunneling conductance G = I/V is fixed. (b) The tip–sample distance z is defined as the z-coordinate of the nucleus of the topmost tip atom with respect to the nucleus of the nearest atom of the sample. (c) The equilibrium tip–sample distance ze is defined as the distance with zero net force. At z = ze , the tip is in a single-atom contact with the sample. At the absolute scale of z, the typical value of ze is about 0.25 nm. For free-electron metals, the conductance of such a single-atom contact is roughly G0 = e2 /π = 2e2 /h ≈ 77.48μS. See Sections 1.2.3 and 1.2.4.

κ = 5.123

 φ(eV) nm−1 .

(1.9)

The typical values of work functions of materials used in STM experiments, together with the decay constants, are listed in Table 1.1. The average value is about φ ≈ 5 eV, which gives a typical value of the decay constant κ ≈ 11.4 nm−1 . According to Eq. 1.7, the current decays 9.78 times, or one order of magnitude, per 0.1 nm. From Eq. 1.7 and 1.8, one can derive an expression for the tunneling barrier height through experimental measurements: φ=

2 8m



d ln I dz

2

 ≈ 95

d ln I dz

2 .

(1.10)

Eq. 1.10 is the definition of the apparent barrier height. See Chapter 8 for details.

Table 1.1: Work functions and decay constants Element

Al

Au

Cu

Ir

Ni

Pt

Si

W

φ (eV)

4.1

5.4

4.6

5.6

5.2

5.7

4.8

4.8

10.3

11.9

10.9

12.1

11.6

12.2

11.2

11.2

κ (nm

−1

)

6

1.2.2

Overview

Semiclassical approximation

Only for a handful of potential barriers does the Schr¨ odinger equation have analytic solutions. For general potential barriers, the semiclassical approximation, or the Wentzel–Kramers–Brilliouin (WKB) approximation, is often useful, see Fig. 1.5. In many cases, analytic expressions for the transmission coefficients can be obtained [18]. In the classically forbidden region [E − U (z)] < 0, define  2m [U (z) − E] κ(z) = . (1.11)  The transmission coefficient is    z2 κ(z)dz . (1.12) T = exp −2 z1

Applying to a square potential barrier, Eq. 1.7 is recovered. An example of the application of the WKB formula is the derivation of the Fowler–Nordheim equation for field emission. It is left as an exercise. Fig. 1.5. The semiclassical approximation. A one-dimensional potential barrier. From z = z1 to z = z2 , the barrier is classically forbidden. The transmission coefficient T is the square of the ratio between the amplitudes of the impinging wave and the transmitted wave. The reflection coefficient R equals 1 − T .

1.2.3

The Landauer theory

The elementary solution of the square-barrier problem and the WKB approximation provide us the transmission coefficient, as well as the variation of tunneling conductance with distance. However, those theories do not provide information about the absolute value of the tunneling conductance with respect to absolute tip–sample distance. And this is important from both theoretical and experimental points of view. First, we need an experimentally verifiable definition of the tip–sample distance. From the theoretical point of view, one could define the absolute tip–sample distance z as the z-coordinate of the nucleus of the topmost tip atom with respect to the nucleus of the nearest atom of the sample, see Fig. 1.4. This distance is not experimentally verifiable. Experimentally, an equilibrium distance ze can be defined, as the distance with zero net force. The absolute tip–sample distance at equilibrium, ze , is roughly the equilibrium internuclear distance of a diatomic molecule, consisting of the apex atom of the tip and the nearest atom on the sample surface. For

1.2 The concept of tunneling

7

Fig. 1.6. The Landauer theory of tunneling. Two electrodes: each is described as a perfect one-dimensional potential well, is separated by a potential barrier. The electron wavefunctions satisfy the Schr¨ odinger equation of the square potential well. The bias voltage between the two electrodes equals the difference of the Fermi levels of the two electrodes. For electrons with energy levels between the two Fermi levels, tunneling could take place.

typical materials in STM and AFM, it is approximately 0.22 to 0.28 nm. On average, it is about 0.25 nm. Experimentally, ze can be determined using the z-piezo displacement, especially with a combined STM and AFM. When z is slightly greater than ze , there is an attractive atomic force between the two atoms. When z is approaching ze , the hard-core repulsive force starts to show up. At z = ze , the attractive force equals the repulsive force. The net force becomes zero. When z < ze , the net force is repulsive. In well-controlled experiments, the equilibrium tip–sample distance can be measured reliably. It has been shown repeatedly that a well-structured tip–sample junction could tolerate a mild repulsive force without causing irreversible inelastic deformation, see Chapter 8. At the equilibrium point, the tip is in contact with the sample with a single atom or a few atoms. In 1957, Landauer [73] developed a theory about the absolute value of tunneling conductance based on a one-dimensional semiclassical model, and predicted the existence of a conductance quantum. In the decades following, experimentalists found that Landauer’s conductance quantum agrees well with the measured conductance of a single-atom contact. Landauer’s theory is widely applied as the physical picture of electron transfer phenomena in mesoscopic systems [74, 75]. Landauer’s derivation was based on two assumptions: First, the electrodes can be described by a one-dimensional free-electron gas in an ideal square potential well. Assuming the width of electrode A is b, see Fig. 1.6, the n-th wavefunction is  2 nπz ψn (z) = sin , (1.13) b b which are normalized as usual, 

b 2

|ψn (z)| dz = 1. 0

(1.14)

8

Overview

The energy eigenvalue of the n-th state is 2  nπ 2 . (1.15) E= 2m b From that expression, one can estimate the number of electrons per unit energy per unit volume ρ(E) (per unit length in the one-dimensional case). Bear in mind that the electron has two spin states,  m 2 2 4 2 ∂n = = = . (1.16) ρ(E) = b ∂E π 2E πv hv Here v is the classical velocity of the electron,  2E . v= m

(1.17)

Second, the current is ballistic, which is the product of the density of the electron and the classical velocity v. Electrons in both electrodes could tunnel into the other side. Therefore, without a bias voltage, there is no net tunneling current. The electrons are fermions. If the energy scale of interest is greater than kT , then all the states with E < EF are occupied, and all the stated with E > EF are empty. The bias voltage V generates a difference of the Fermi levels in the two electrodes, as shown in Fig. 1.6, (A)

EF

(B)

− EF

= eV.

(1.18)

Assuming that the bias voltage V is small, such that the electron density per unit energy ρ(E) does not vary appreciably within eV , the net impinging current from electrode A to the tunneling barrier is  1  1 (A) (B) Ii = ev ρ(EF ) EF − EF = e2 v ρ(EF ) V. (1.19) 2 2

Fig. 1.7. Experimental observation of conductance quantum. (a) By gradually stretching a thin gold wire to form an atom-sized bridge, the conductance is reduced by steps of conductance quantum, e2 /π. (b) Models of the atom-sized bridges. Case (A): two rows of gold atoms, generating two channels of elementary conductance. Case (B): one row of gold atoms, generating a single channel of elementary conductance. (Reproduced with permission from [76]. Copyright 1988 Macmillan Publishers Ltd.)

1.2 The concept of tunneling

9

Fig. 1.8. Statistical results of experimental observation of conductance quantum. The values of observed conductance are displayed as a histogram. As shown, the peak at G = G0 is the sharpest and strongest. The peaks at G = 2G0 and G = 3G0 are also clear. (Adapted with permission from [77]. Copyright 1995 American Physical Society.)

Because one half of the electrons go to the right, and other half go to the left, see Fig. 1.6(a), there is a factor of 1/2. Substituting Eq. 1.16 into Eq. 1.19, the net current impinging the tunneling barrier is Ii =

2e2 e2 V = V. π h

(1.20)

With a transmission coefficient T , the tunneling current is It = T Ii =

2e2 e2 VT = V T. π h

(1.21)

Therefore, the tunneling conductance G is G=

e2 2e2 It = T = T ≡ G0 T. V π h

(1.22)

The constant G0 , defined as G0 ≡

2e2 e2 = ≈ 77.48 μS, π h

(1.23)

is the conductance quantum. Experiments using nearly free-electron metal electrodes, especially gold, showed that the conductance quantum defined in Eq. 1.23 is indeed the conductance of single-atom contact [76, 77]. For a review, see Agra¨ıt et al. [78]. Gold is inert and malleable. Very thin gold bridges down to a few atoms can be easily made. As shown in Fig. 1.7(a), by stretching a thin gold wire with a piezo and watching the variation of conductance, at the very last stage of stretching before it breaks, the conductance shows a stepwise behavior with step sizes of the conductance quantum,

10

Overview

77.48 μS. Figure 1.7(b) shows models of the atom-sized gold bridges. Case A shows two single-atom gold bridges, corresponding to a conductance of 2G0 . Case B shows one single-atom gold bridge, corresponding to a conductance of G0 . The experiments have been repeated numerous times. Often, the results are represented as a histogram of conductance values. Figure 1.8 is an example with gold [77]. As shown, the peak at G = G0 is the sharpest and strongest. The peaks at G = 2G0 and G = 3G0 are also clear. Experiments on other materials showed similar behavior, but the step sizes are often different. For example, measurements with a Co-Ge junction show conductance plateaus at 2.01 G0 , 2.66 G0 , and 3.45 G0 [79]. However, the order of magnitude is always correct. 1.2.4

Tunneling conductance

Combining the Landauer theory of the conductance quantum and the explicit expression of transmission coefficient, Eq. 1.7, for free-electron metals, a general relation between the tunneling conductance and the tip–sample distance can be derived. Because at z = ze , the tip is in a single-atom contact with the sample, the conductance should be of the order of G0 . Therefore, the tunneling conductance on the entire range is G = G0 e−2κ(z−ze ) .

(1.24)

As discussed, the equilibrium tip–sample distance ze is defined as the point with a zero net force between the tip and the sample, see Fig. 1.4. That point, on the scale of z-piezo displacement, can be experimentally determined, especially with a combined STM and AFM. Therefore, Eq. 1.24 provides a one-to-one relation between the tunneling conductance G and the relative tip–sample distance, z − ze . Table 1.2 shows the relation between tunneling conductance and tip– sample distance, following Eq. 1.24, assuming that the work function is about 5 eV. The first row is the tunneling resistance. The second row is the tunneling conductance. The third row is the absolute distance, defined as the vertical distance between the nucleus of the apex atom of the tip and

Table 1.2: Tunneling conductance and tip–sample distance RT (MΩ)

0.01

0.1

1

10

100

1000

10000

G (μS)

100

10

1

0.1

0.01

0.001

0.0001

zabs (nm)

0.24

0.34

0.44

0.54

0.64

0.74

0.84

−0.01

0.09

0.19

0.29

0.39

0.49

0.59

z − ze (nm)

1.2 The concept of tunneling

11

Fig. 1.9. Tunneling through a controllable vacuum gap. A historical experiment to establish the relation between tunneling conductance and tip–sample distance as the foundation of STM. The exponential dependence I ∼ V is observed over four orders of magnitude. On clean surfaces, an apparent barrier height of 3.5 eV was observed. Notice that, even in the first experiment, the observed high values of tunneling conductance are close to or even greater than 0.1×G0 ∼ = 7.75μS, or a tunneling resistance less than 129 kΩ. It corresponds to an absolute tip–sample distance of 0.35 nm or less, or a relative tip–sample distance z − ze of 0.10 nm or less. (Reprinted with permission from [29]. Copyright 1982 American Institute of Physics.)

the nearest nucleus of the sample surface. The fourth row is the relative tip–sample distance, measured from the equilibrium point ze . Despite its extreme simplicity, Eq. 1.24 is verified by numerous experiments with nearly free-electron metals with reasonable accuracy. Historically, as a first step towards a working STM, Binnig et al. demonstrated the exponential distance dependence of tunneling current over four orders of magnitude [30, 29]. Their pioneering experiments are summarized in Fig. 1.9. The value of the work function was found to depend sharply on the condition of the surfaces. Initially, the measured values were around 0.6–0.7 eV. After repeated cleaning, the slope became much steeper. A value of 3.2 eV was obtained, which can last for several minutes. Since then, the experimental observations of the G ∼ z relation have been repeated numerous times. An example is a study of three metals by the Besenbacher group [80]. Figure 1.10 shows the relations of tunneling conductance versus tip–sample distance for three metals. For all three metals, almost perfect exponential relation was observed up to within 0.05 nm from a mechanical contact, characterized by the conductance of single atom contact, 77.48 μS. Beyond that point, a firm mechanical contact was ob-

12

Overview

Fig. 1.10. Distance dependence of tunneling conductance for three metals. An exponential dependence of G ∼ z is observed over three orders of magnitude. For all three metals, the observed high values of tunneling conductance is greater than 0.1×G0 ∼ = 7.75μS. In the case of Ni(100), the observed highest tunneling conductance is close to G0 . It corresponds to an absolute tip–sample distance of 0.35 nm or less, or a relative tip–sample distance z − zc of 0.10 nm or less. (Adapted with permission from Fig. 1(b) of [80]. Copyright 1996 American Physical Society.)

served. The work functions obtained from the decay constants are consistent with theoretical values. Near the contact point z ≈ ze , depending on the condition of the tip and the sample as well as the process of approaching, hysteresis and nonreversable deformations have been observed experimentally. However, if the tip and the sample are well-prepared, then the junction is completely reversible with no or only a minor hysteresis. See Chapter 8 for details.

1.3

Probing electronic structure at atomic scale

The invention of the STM realized a dream of physicists and chemists almost two centuries old: to visualize individual atoms and their internal structures in real space. Since John Dalton published his first volume of A New System of Chemical Philosophy in 1808, the concept of atoms and molecules has been the cornerstone of modern physics and chemistry. The upper half of Fig. 1.11 is reproduced from Dalton’s book. Even in contemporary chemistry and physics, Dalton’s initial dogma remains quite accurate, except that the atoms (derived from Greek, ατ oμoς, indivisible), are divisible. According to the modern theory of atomic systems, quantum mechanics, an atom consists of a number of electrons in a series of stationary states surrounding an extremely small, positively charged nucleus. Each of these electron states has a well-defined energy level. Apart from electron spin, the electron states are classified according to the quantum numbers l and m. Each individual

1.3 Probing electronic structure at atomic scale

13

Fig. 1.11. Dalton’s atoms and Schr¨ odinger’s atomic wavefunctions. (a) a chart in Dalton’s A New System of Chemical Philosophy, published in 1808. In modern symbols, these atoms are: 1, H; 2, N; 3, C; 4, O; 5, P; 6, S; 7, Mg; 8, Ca; 9, Na; 10, K; 11, Sr; 12, Ba; 13, Fe; 14, Zn; 15, Cu; 16, Pb; 17, Ag; 18, Pt; 19, Au; 20, Hg. The major modern modification to Dalton’s theory is that the atoms are divisible. According to quantum mechanics, each atom has a series of atomic states, or wavefunctions. The contour maps in (b) represent typical electronic states in atoms. The outermost contour on each map represents a density of 10−6 nm−3 . The successive contours represent an increase of a factor of 2. The regions with dashed-curve contours have opposite phases in the wavefunction from those with solid-curve contours.

14

Overview

electronic state has its characteristic probability distribution, or electron density distribution, as shown in the lower half of Fig. 1.11. In molecules or solids, the electronic structure can often be represented as linear combinations of atomic states. However, until very recently, the information about individual atoms and their electronic structures – the individual electronic states – was inferred from indirect measurements. Now, STM has made it technically possible for scientists to probe directly the electronic structures of various materials at an atomic scale (≈ 0.2 nm). At the beginning of their experimentation, Binnig and Rohrer anticipated that the realization of controllable vacuum tunneling would result in a lateral resolution much smaller than the radius of the tip end. Their original argument was recorded on Binnig’s 1978 laboratory notebook, which was reproduced in a Physics Today article by Quate [81]. Schematically, it is shown in Fig. 1.12. If the distance between the tip end and the sample surface is much smaller than the tip radius, near the tip end, the current lines are almost perpendicular to the sample surface. At a point Δx on the tip, the distance to the sample surface, Δz, is increased by Δz ≈

Δx2 . 2R

(1.25)

Assuming that at each point the tunneling current density follows the formula for the one-dimensional case, Eq. 1.7, the lateral current distribution is   Δx2 . (1.26) I(Δx) = I0 exp −2κ 2R Typically, κ ≈ 10nm−1 . For R ≈ 1 nm, at Δx ≈ 0.3 nm, the current

Fig. 1.12. Estimation of the lateral resolution in STM. In 1978, Binnig made an estimation of the possible lateral resolution of STM with a simple spherical-tip model. The tip end, with radius R, is very close to the sample surface. The tunneling current is concentrated in a small region around the origin, x = 0. In order to justify the validity of such a macroscopic picture of the tip, the radius of curvature of the tip should be at least as large as several atoms, for example, 0.9 nm. According to Eq. 1.26, the width of the tunneling current column is approximately 2Δx ≈ 0.6 nm [81].

1.3 Probing electronic structure at atomic scale

15

drops by a factor of e−2 , that is, about one order of magnitude. The diameter of such a current column is the resolution limit, which is about 0.6 nm. Therefore, with moderate means, a very high lateral resolution is expected. The actual achievement of STM greatly exceeds this expectation. Details of surface electronic structures with a spatial resolution of 0.2 nm are now routinely observed. Atomic structures of surfaces and adsorbates of a large number of systems are determined. Furthermore, the active role of the STM tip through the tip–sample interactions enables real-space manipulation and control of individual atoms. An era of experimenting and working on an atomic scale arises. 1.3.1

Experimental observations

The first success of STM was the real-space imaging of the Si(111)-7×7 structure [41], as shown in Fig. 1.13. The 7×7 reconstruction of the Si(111) surface, one of the most intriguing phenomena in surface science, was discovered by low-energy electron diffraction (LEED) in 1959. The LEED diffraction pattern clearly shows that the unit cell of this reconstructed surface is constituted of 49 silicon atoms on the original Si(111) surface, and exhibits a p6mm symmetry. However, the details of the arrangement of the 49 Si atoms in each layer of one unit cell cannot be determined unambiguously from the LEED pattern. In the 1960s and 1970s, a large number of models were proposed. There was not enough experimental evidence to determine which one was correct. The STM image of that surface taken by Binnig et al. [41] clearly showed that there are 12 adatoms and one large hole in each unit cell. This experimental fact contradicted all the previous models.

Fig. 1.13. The structure of Si(111)-7×7 resolved in real space. The historical event of resolving individual atoms on a solid surface. (Original high-resolution photograph courtesy of Ch. Gerber. Reproduced with permission [41]. Copyright 1983 American Physical Society.)

16

Overview

Based on the real-space observation, Binnig et al. proposed the 12adatom model of the Si(111)-7×7 reconstruction. In addition, they proposed that the surface does not have a six-fold symmetry. The two triangular halves are different. The true symmetry is p3m1 instead of p6mm. The complete details of the reconstruction were worked out by Takayanagi et

Fig. 1.14. Electronic states and DAS model of Si(111)-7×7. (I) By varying the bias voltage and then acquire the dynamic conductance at each point, details of the electronic structure of Si(111)-7×7 for a range of energy levels can be precisely measured. (a) The topographic image. (b) A schematic. (c) – (f) Normalized dynamic conductance images at different bias voltages, acquired at liquid helium temperature (4.2 K). (Original high-resolution images courtesy of J. Mysliveˇ cek. Reproduced with permission [82]. Copyright 2006 American Physical Society.) (II) The DAS model. In each unit cell, there are 9 dimers, 12 adatoms, and a stacking-fault layer. There are 19 dangling bonds in each unit cell. (After [83].)

1.3 Probing electronic structure at atomic scale

17

Fig. 1.15. Four STM images of 4Hb-TaS2 at 4.2 K. These images were taken during a period of about 2 hours on the same area of the surface under identical tunneling conditions (I=2.2 nA, V =25 mV). These images demonstrate the role of tip electronic states on the STM images. (Reproduced with permission from [84]. Copyright 1988 Taylor and Francis.)

al. [83], as shown in Fig. 1.14. The DAS model has the fewest number of dangling bonds (19) among all the models ever proposed. There are 12 dangling bonds at the adatoms, 6 at the rest atoms, and 1 at the center atom deep in the corner hole. The 19 dangling bonds are at different energy levels. The huge corrugations observed on silicon surfaces have surprised experimentalists and theoristsl. Soon, the ubiquitous atomic resolution observed on low-Miller-index metal surfaces has exceeded even the wildest expectations. The observed corrugations amplitudes were one to two orders of magnitudes greater than the predictions of the Tersoff–Hamann model, see Section 6.2. According to the Tersoff–Hamann model, an STM image is a contour of Fermi-level local density of states (LDOS) at the center of curvature of the tip. For metals, it is essentially a charge-density contour. And atomic corrugations on low-Miller-index metal surfaces are too small to be observed. Atomic resolution has been observed on a large number of clean metal surfaces, including Au(111), Au(001), Al(111), Cu(111), Cu(110), Cu(001), Pt(001), Pt(111), Ru(0001), Ni(110), Ni(001), etc. (See the review article by Behm [85].) The observation of atomic resolution on metal surfaces was recorded as early as 1983, see Chapter 7. The first reports were published in

18

Overview

Fig. 1.16. Corrugation reversal during a scan. The sample is Au(111). The upper part of the image exhibits a positive corrugation. The image changes abruptly into negative corrugation. Individual Au atoms on both parts are clearly resolved. The corrugation reversal is caused by a spontaneous tip restructuring. (Original high-resolution figure by courtesy of J. V. Barth. Reproduced with permission [48]. Copyright 1990 American Physical Society.)

1987 at the Second International Conference on STM [86, 87]. The reported atomic resolution on Au(111) surface, with a corrugation amplitude 30 pm, was a pleasant surprise at that time. Later, similar atom-resolved images were observed on virtually every clean metal surface [85]. 1.3.2

Origin of atomic resolution in STM

The ability of resolving every single atom on solid surfaces is the most significant achievement of STM. It is not a surprise that the origin of atomic resolution in STM is a consequence of the convolution and interaction of the atomic states of the tip with the atomic states of the sample atoms. The occurrence of atomic resolution relies on the occurrence of optimum tip electronic states. Here are some experimental facts: First, the atomic resolution is not always observable. In order to achieve atomic resolution, certain tip-sharpening procedures must be carried out. Examples of such procedures include a controlled collision with the sample, and an electric pulse applied between the tip and the sample. Second, during the scanning, the image often shows spontaneous changes, and the atomic resolution could appear or disappear unexpectedly. As the sample and the tunneling conditions remain identical, the only explanation is that the tip structure has changed. Figure 1.15 is an example. Third, in many cases, the atomic corrugation is inverted. In other words, the atomic sites appear as depressions (minima) on the topographic image, rather than protrusions (maxima). Upon a spontaneous tip restructuring, a positive image could suddenly change into a negative image, and vice versa. Figure 1.16 is a recorded case of corrugation inversion during a scan. Fourth, the atomic corrugation has an almost exponential dependence on the tip–sample distance. The highest atomic corrugations are always

1.3 Probing electronic structure at atomic scale

19

Fig. 1.17. Dependence of corrugation on tip–sample distance. The observed corrugation amplitude on a close-packed metal surface can be more than 20 times greater than the maximum corrugation amplitude of the charge density contour of the metal surface, as expected from the Tersoff-Hamann model of STM. (Reproduced with permission from [88]. Copyright 1989 American Physical Society.)

observed at very short tip–sample distances. Figure 1.17 is an example. Table 1.3 shows some reported data of the relation between corrugation amplitude and the tunneling conductance, which is related to tip–sample distance through Eq. 1.24. In 1984, Baratoff [91] proposed that the atomic resolution in STM is due to a dangling bond state protruding from the tungsten tip, see Fig. 1.18. Near the Fermi level, the density of states of tungsten is dominated by various d-states. Especially, the dz2 -like states often dangling out from the surface. To investigate the STM imaging mechanism further, Ohnishi and Tsukada [92] made a first-principles calculation of the electronic states for a number of W clusters. From the calculations, they found that, on the apex atom of many W clusters, there is a dz2 -like state protruding from the apex atom, energetically very close to the Fermi level. Using Green’s function methods, they also found that the tunneling current is predominately contributed by this d-state. Ohnishi and Tsukada [92] proposed that such an orbital would be advantageous for a sharp STM image. Figure 1.19 shows the electronic states near the Fermi level on W4 and W5 clusters.

Table 1.3: Corrugation amplitude and tip–sample distance. Ref.

Surface

G (μS)

zabs (nm)

z − zc (nm)

Δz (pm)

[88] [48] [89] [90]

Al(111) Au(111) Cu(100) W(110)

0.02-2 9.1 0.5-3 11

0.37-0.57 0.35 0.39-0.50 0.34

0.12-0.32 0.10 0.14-0.25 0.09

10-40 20 5-20 15

20

Overview

Fig. 1.18. Microscopic view of STM imaging mechanism. An atomic state at the tip end, exemplified by a dz2 state protruding from the apex of a W tip, interacts with a two-dimensional array of atomic states, exemplified by sp3 states on the Si surface. This results in a highly corrugated tunneling current distribution. (Reproduced with permission from [19]. Copyright 1991, AVS The Science and Technology Institute.)

In 1989, based on experimental observations, Demuth et al. [94] proposed that the sp3 dangling bond of a silicon cluster adsorbed on the tungsten tip is the origin of atomic resolution observed on Si(111)-7×7. Experimentally, even the best-prepared clean tungsten tips usually do not immediately produce the highest resolution on the Si surface. Most often the highest resolution is achieved after long periods of scanning or controlled tip crashing. When there is no atomic resolution, an effective procedure to achieve atomic resolution is to collide the tip mildly with the Si surface. After such a controlled crashing, a crater is found on the Si surface, which shows that a Si cluster has been picked up by the tip. Atomic resolution is then often achieved. They concluded that tip treatment is one half of the STM experiment, because tunneling is determined by the convolution of the tip electronic states and the sample electronic states. The understanding of the origin of atomic resolution in STM and AFM is of fundamental importance. A full discussion is given in Chapter 7.

Fig. 1.19. Electronic states on W clusters. The electronic states near the Fermi level on tungsten clusters, W4 and W5 , were calculated by Ohnishi and Tsukuda (1989). At low bias, these d-like tip states contribute more than 90% of the tunneling current. (Reproduced with permission from [93]. Copyright 1990, AVS The Science and Technology Institute.)

1.3 Probing electronic structure at atomic scale

1.3.3

21

Observing and mapping wavefunctions

In the first quarter century of STM, the focus was to probe and image atomic structures through the interactions between the tip wavefunction and the sample wavefunction. Since early 2000s, it was found that STM is capable of directly observe and map atomic and molecular wavefunctions themselves, see Fig. 1.20. For details, see Chapter 8. Although atoms and molecules adsorbed on metal surfaces have been observed by STM since early 1980s, those objects were severly perturbed by the substrate. Around 2005, a new method for imaging atomic and molecular wavefunctions was invented: by placing an ultrathin insulating buffer layer between the metal substrate and the atom or molecule under observation, typicaly two atomic layers of NaCl, the wavefunction under observation keeps its prestine state. Using this new technology, wavefunctions of atoms and molecules, including the HOMO wavefunciton and the LUMO wavefunction, are observed and mapped. The images match the theoretical charge density distributions, that is, the square of wavefunctions. In 2011, by using a CO-functionalized tip to image organic molecules lying on top of an ultrathin buffer layer, completely different images were observed. It has been known for decades that the CO molecule possesses a degenerate pair of 2π ∗ orbitals, which is formed by a pair of degenerate px and py wavefunctions of the O atom. Those p-wavefunctions generate an image of the square of the lateral derivatives of the sample wavefunction. The tunneling current peaks at places with the greatest lateral derivative, typically where the wavefunction changes sign. Those places are the locations of the nodal surfaces of the molecular wavefunctions. Therefore, the nodal structures of wavefunctions are observed and mapped.

Fig. 1.20. Observing nodal structures in molecular wavefunctions. By seperating the molecule from the conducting substrate with an ultrathin insulating buffer layer, typically two atomic layers of NaCi, the molecular wavefunction keeps in its prestine state. By using a CO-functionalized tip, the nodal structure inside the molecular wavefunction is observed and mapped. After Bartel [4].

22

Overview

Those experiments showed that wavefunctions including their nodal structures are observable physical reality. The implications to the interpretation of quantum mechanics is discussed in Section 8.4.

1.4 1.4.1

The atomic force microscope Atomic-scale imaging by AFM

The STM resolves individual atoms on conducting surfaces. For resolving individual atoms on insulating surfaces, Binnig, Quate, and Gerber [31] introduced an instrument similar to the STM, the atomic force microscope (AFM). Instead of using a conducting tip to raster scan over the sample surface and utilizing the tunneling current as the feedback signal, AFM uses a mechanically sharp tip to sense the force between the tip and the sample, see Fig. 1.21. The tip is attached to a flexible cantilever. The force between the tip and the sample surface causes minute deflections of the cantilever. The deflection of the cantilever is detected and utilized as the feedback signal. By keeping the force constant, a topographic image of constant force is obtained. The AFM can also be operated in forceimage mode, similar to the current-image mode in STM. And there are two definitions of force spectroscopy in the literature: the distance-dependence of force, and bias-voltage dependence of force, respectively. Fig. 1.21. The atomic force microscope (AFM). The sample is mounted on top of a tube scanner. Using the coarse advance screws, the tip, mounted on a cantilever, is brought into gentle contact with the sample surface. By actuating the tube scanner, the tip is raster scanning the sample surface. The force between the tip and the sample causes a deflection of the cantilever. The deflection of the cantilever is then detected by a deflection sensor. The detected force signal is amplified, then compared with a reference value. The difference signal is again amplified to drive a feedback circuit, to adjust the z piezo to keep the force constant. A constant-force topographic image of the surface structure is obtained. (Reproduced with permission from [95]. Copyright 1988 the American Association for the Advancement of Science.)

1.4 The atomic force microscope

23

The ultimate goal of the AFM—to detect and measure the force between two single atoms and then to use the interatomic force as a signal to detect individual atoms—, was first achieved by the frequency modulation method of dynamic-mode AFM [96]. A schematic is shown in Fig. 1.22. The details are presented in Chapter 15. Instead of letting the tip in firm contact with the sample, the cantilever is oscillating, actuated by an external piezo. And its vibration is detected by a deflection sensor, such as a laser and a split-diode, see Fig. 1.22. The tip is in the proximity of the sample surface. The weak attractive interaction between the tip and the sample causes a minute shift of the resonance frequency. The controller unit, a high-precision oscillator, set the cantilever to oscillate at its resonance frequency with a predetermined amplitude by a positive-feedback circuit. The frequency shift is then converted into an analog signal, which is used in analogy to the tunneling current in STM, see Fig. 1.22. The constant-force topographic image, or more precisely, a constant-frequency-shift topographic image, is obtained. Here we briefly describe the concept of frequency-modulation AFM. The cantilever can be represented as a simple harmonic oscillator with spring constant k and mass m. Without external force, the resonance frequency of the cantilever f0 is

Fig. 1.22. A schematic of the dynamic-mode AFM. The cantilever is actuated by an external piezo, and the vibration is detected by a deflection sensor, such as a laser and a split-diode. With a controller unit, the cantilever is set to oscillate at its resonance frequency with a predetermined amplitude by a positive-feedback circuit. The interaction between the tip and the sample causes a shift of the resonance frequency. The frequency shift is used as the signal, similar to tunneling current in STM. The acronym SPM means scanning probe microscope, applicable to various types of proximity probes, including tunneling and atomic force.

24

Overview

 k 1 . (1.27) f0 = 2π m When the tip is approaching the sample, a force gradient F  is added to the spring constant. The resonance frequency is changed to  k + F 1 f0  f= ≈ f0 + F ≡ f0 + Δf. (1.28) 2π m 2k Because frequency shift can be measured with extremely high accuracy, even a very small change of force gradient can be detected by the change of frequency shift, Δf = (f0 /2k) F  . Here is a numerical example. Typically, the atomic force changes 1 nanonewton per 0.1 nanometer. Suppose the resonance frequency of the cantilever is 100 kHz, and its spring constant is 100 N/m, the frequency shift is 100, 000 0.1 ≈ 50 Hz. (1.29) 2 × 100 Since modern electronic instruments can detect frequency changes of a fraction of one hertz, the atomic force gradient can be comfortably measured. The first true atomic-resolution image by FM-AFM was obtained by Giessibl in 1995 [97], as shown in Fig. 1.23. The image clearly shows the effect of tip electronic states. At time before T1 , there is no atomic Δf =

Fig. 1.23. Atomic resolution observed on Si(111)7×7 surface by AFM. The image has several sections. At time before T1 , there is no atomic resolution. Suddenly, at T1 , the atomic resolution with high corrugation amplitude is turned on spontaneously, After a while, at T2 , the atomic resolution is tuned off spontaneously. Later, at T3 , the atomic resolution is resumed, but with a smaller corrugation amplitude. It clearly shows the effect of tip electronic states. (Reproduced with permission from [97]. Copyright 1995 the American Association for the Advancement of Science.)

1.4 The atomic force microscope

25

resolution. Suddenly, at T1 , the atomic resolution with high corrugation amplitude is turned on spontaneously. After a while, at T2 , the atomic resolution is turned off spontaneously. Later, at T3 , the atomic resolution is resumed, but with a smaller corrugation amplitude. 1.4.2

Role of covalent bonding in AFM imaging

A quantitative theory of the atomic resolution of AFM was developed by P´erez et al. [98, 99]. Three types of AFM tips were tried, as shown in Fig. 1.24. All the tip models are part of a regular silicon crystal. The first one, (a), has four Si atoms. But it does not provide atomic resolution as observed. The second and the third show a clear sp3 -type dangling bond state at the apex. Detailed numerical simulation shows corrugation amplitudes which match well with experimental observations. An intuitive explanation is shown in the right side of Fig. 1.24. Obviously, the tip structures that are needed to provide atomic resolution are not always present. To achieve atomic resolution, one must either carry out special tip treatment procedures to generate various structures and then try it out, or rely on the graciousness of Nature to endow such structures. Anyway, the process is not always under control, and is not always reproducible. The field of frequency-modulation AFM (FM-AFM), often also called non-contact AFM (NC-AFM), has grown into a sub-field of scientific inquiry, and growing fast. Currently, atomic resolution using FM-AFM has been achieved on a lot of conducting and insulating surfaces. The achievement of atomic resolution on insulating surfaces is valuable because STM is not capable of image on insulating surfaces.

Fig. 1.24. Theoretical explanation of atomic resolution by AFM. Three models of the tip are tried. (a) Four Si atoms: does not explain the observed atomic resolution; (b) and (c), Four and ten Si atoms with hydrogen terminated bonds. Both show corrugation amplitudes matching well with experimental observations. The right-side shows an intuitive explanation of the atomic resolution. (Adapted woth permission from [99]. Copyright 1998 American Physical Society.)

26

Overview

At the mean time, the design of AFM has evolved into an incredibly versatile tool to image surfaces down to a resolution of a couple of nanometers, which is several orders of magnitude better than the stylus profiler. Among all the instruments in the scanning probe microscope family, AFM accounts for more than three quarters in number and in revenue. Especially, it is used by the biochemistry and materials-science communities because of its universality and non-destructive features. The most used force detection method is the tapping mode, or amplitude-modulation mode. The setup is similar to that shown in Fig. 1.22. Instead of using the frequency shift, the change of vibrational amplitude is utilized as the signal.

1.5

Illustrative applications

Over the last decades, the applications of STM and AFM have grown so fast and so vast that a thorough review would take many volumes. There are already many good books as well as review articles published on applications of STM and AFM in various fields, and so in this section, a few examples are presented as illustrations. 1.5.1

Self-assembled molecules at a liquid-solid interface

Many applications of STM require ultrahigh vacuum and low temperature. The instrument is expensive and difficult to maintain, and the operation is highly demanding. However, to image self-assembled molecules at liquidsolid interfaces, a modest investment in instrumentation suffices. Furthermore, it may operate in a dirty environment at room temperature. Nevertheless, astonishingly beautiful images and overwhelmingly rich information can be acquired even for newcomers and amateurs [100, 101, 102]. Most STM experiments on self-assembled organic molecules use highly oriented pyrolytic graphite (HOPG) as the substrate, which can be easily cleaved to generate atomically flat surfaces. The cleaved HOPG surfaces are extremely inert in air and organic solvants. It is also one of the easiest surface to image with a STM, see Chapter 9, especially Section 9.2. An early example of the organic molecule studied is n-C32 H66 , see Plate 13 [53]. To deposit the organic molecules on the substrate, a drop of solution of 16 mg of n-C32 H66 in 10 ml of n-dectane is applied to the HOPG surface. A stable liquid-solid system is thus formed. The tip is made of either mechanically cut Pt-Ir wire, or an electrochemically etched W wire, see Chapter 14. Sometimes, once the tip approaches through the liquid drop and starts scanning, the image appears. The typical operational bias voltage on the sample is +0.4 to +1 V, with a tunneling current of 0.5 to 1 nA. If no atomic-resolution image is observed, applying an electrical pulse of 2 V usually results in a good image, see Chapter 14.

1.5 Illustrative applications

27

Fig. 1.25. Large-scale STM image of linear molecules on graphite. STM image of CH3 (CH2 )29 OH in phenyloctane on graphite. Because graphite has a hexagonal symmetry, each domain could have one of the three orientations. The black bar indicates the length of a molecule. The arrows point to the lattice lines of graphite [101].

Plate 13 shows an STM image of the self-assembled n-C32 H66 molecules. Note the striking degree of ordering and the apparent atomic resolution obtained under easily operable conditions. The highly regular pattern is partly because of a fortuitous match of the distance between the centers of the hexagons of the graphite lattice, 246 pm, and the distance between alternate methylene groups of the hydrocarbone backbone, 251 pm. See Plate 13 (a) and (b). Close inspection of the STM images reveals a varying contrast along the length of an individual molecule, apparently created by the slight mismatch between the graphite lattice and the methylene groups. Therefore, the STM images reveal not only the molecules, but also the interaction between the molecule and the substrate. Because the cleaved HOPG surface has a sixfold symmetry, on a large scale, the pattern of self-assembled linear molecules shows three types of domains with three orientations [101], see Fig. 1.25. Role of solvents The solvent, or the liquid for STM imaging of self-assembled molecules, needs to be carefully selected [101, 102]. First, the solvent must be nonpolar to allow the tunneling current to be measured while the tip is in the solution. In polar solvents the ionic current could be large enough to obscure the tunneling current. Second, the solvent must have a low vapor pressure to prevent excess evaporation during STM measurement. Third, the solvent must be less strongly bound to the substrate. Examples of suitable solvents include 1-phenyloctane, n-decane, and 1-octanol.

28

Overview

The solvent could have significant effects on the two-dimensional pattern. Figure 1.26(A) shows the molecules involved: the solvants 1-phenyloctane and 1-octanol, and the organic molecule under investigation. Figure 1.26(B) shows the STM image of the molecules in 1-octanol. The pattern is formed by groups of tetramers. The long alkyl chains are seperated by approximately 0.46 nm. Figure 1.26(C) is a tentative molecular model. Figure 1.26(D) shows the image obtained in solvant 1-phenyloctane. The space between the long alkyl chains are packed with coadsorbed 1-phenyloctane molecules. Figure 1.26(E) is a tentative molecular model. The molecular sizes of 1-octanol and 1-phenyloctane are similar. However, 1-phenyloctane tends to adsorb on the graphite surface. In certain cases, to avoid coadsorption, 1-octanol is preferred.

Fig. 1.26. Coadsorption of solvant molecules. (A) The molecules involved: the solvants 1-phenyloctane and 1-octanol, and the molecule under investigation. (B) with 1-octanol as solvant, only the giant molecules are adsorbed on graphite surface. Image size, 8.4 nm × 8.4 nm. A tentative model is shown in (C). (D) An image of size 12.8 nm × 12.8 nm, using 1-phenyloctane as solvant. The solvant moleules coadsorbed on graphite surface. A tentative model is shown in (E). After Feyter et al. [102].

1.5 Illustrative applications

29

Bias voltage and electronic effects By changing the bias voltage, the image could change appreciably, as shown in Fig. 1.27 [102, 103]. The molecule under investigation, shown in Fig. 1.27(1), is unique. It contains two electron donor components oligo-p-phenylene vinylene D, and an electron acceptor component perylene diimide A. The three parts are covalently linked to each other but well-separated in space. The STM images depend on the bias voltage V , as shown in Fig. 1.27(2). With a negative bias, the donor component D becomes prominant. With a positive bias, the acceptor component A becomes prominant. With a small bias, either positive or negative, both components D and A are visible. A tentative explanation is shown in Fig. 1.27(3). The HOMO of component D and the HOMO of component A are shifed by a fraction of an eV. A change of bias voltage shifts the focus of imaging [102, 103].

Fig. 1.27. Bias dependence of STM images. (1) The molecule under investigation has two donor components D and an acceptor component A. (2) The STM images at different bias voltages. (3) A tentative explanation. See Feyter et al. [102, 103].

30

1.5.2

Overview

Electrochemistry STM

The STM imaging of individual molecules at the liquid-solid interface can also been practiced in conducting liquids, especially in an electrochemistry environment. Electrochemistry and the related chemical industry is based on atomic-scale reactions at the interface of a solid and an electrolyte, a conducting liquid. Most of the biological objects are only active in conducting aqueous solutions. However, the traditional method for characterizing the surface relies on ultra high vacuum. The previous section showed that STM can image atomic-scale phenomena at the interface of solid and a nonconducting liquid. By using a special partially insulated tip, atomic resolution can be readily achieved in an electrolyte. Furthermore, the well-established methods of electrochemistry can be applied to change the surface and the adsorbed atoms and molecules. The pioneering works in this field [105, 106] showed a great potential of operating STM in liquids by using a two-electrode system: the substrate and the STM tip. Later on, by combining STM with the standard methods of electrochemistry [107, 108], a four-electrode system was introduced [109, 110], which provides a powerful method to study the liquid-solid interface and a large number of electrochemical processes down to atomic level. Figure 1.28 is a schematic of an electrochemistry STM. The standard electrochemistry cell contains three electrodes [111]. The working electrode (WE) is the substrate under investigation. The reference electrode (RE) provides the reference potential, with insignificant electrical current flow. Typically, a saturated calomel electrode (SCE) is used as the reference electrode [111]. The counter electrode (CE) supports the Faradaic current to or

Fig. 1.28. The four-electrode electrochemical cell with STM. The standard electrochemical cell has three electrodes: the working electrode (WE), the reference electrode (RE), and the counter electrode (CE). By ramping the potential of the working electrode back and forth, a cyclic current-potential curve can be generated. The STM tip is the fourth electrode, to provide tunneling with the working electrode. To prevent the harmful effect of oxygen, the entire cell is placed in argon atmosphere. (By courtesy of J. Ulstrup’s group of Technical University of Denmark. See [104] for details.)

1.5 Illustrative applications

31

Fig. 1.29. √ Au(111) surface imaged by STM in liquids. (a) At relatively large scale, the 22× 3 reconstruction should be observed readily. (b) At smaller scale, individual atoms should be observed. It provides a calibration of the x, y scales. (Original images by courtesy of J. Ulstrup’s group of Technical University of Denmark.)

from the working electrode. A routine experiment in electrochemistry consists of a cycle to ramp the potential of the working electrode up and down with regard to the reference electrode. The Faradaic current is recorded as a function of time, thus also a function of potential. The cyclic currentpotential curve, the so-called cyclic voltammogram, contains rich information regarding the electrochemistry at the liquid-solid interface. For STM experiments, a fourth electrode, the STM tip, is introduced. The tip should be covered with an insulating film except on the very end of the tip. The technique of making such a tip is described in Section 13.6. Oxygen often reacts with the adsorbates of interest, especially biological molecules. To improve the cleanness of the liquid-solid interface, oxygen should be removed from the electrolyte. An effective method is to enclose the entire system in an inert-gas atmosphere, such as nitrogen or argon. To reduce the vibration caused by the flow of inert gas, a buffer bottle and two valves are installed to limit the flow rate, see Fig 1.28. As in all STM-related experiments, it is important to start with an atomically flat, well-defined and well-understood substrate surface. The Au(111) surface is an excellent substrate for the operation of STM in liquid. Small pieces of single crystal gold can be generated by flame annealing using a H2 flame. The single crystal, typically a few mm in diameter, shows different crystallographic planes on its surface. After identified, a small piece of gold chip is cut off from the sphere then mounted on a sample holder. Au(111) samples can also be made by vacuum depositing gold on mica, followed by annealing. The advantage of gold is that it does not oxidize in air, thus the samples can √ be transferred through air. If the STM system works properly, the 22× 3 reconstruction should be observed readily, see Fig. 1.29(a). This provides a natural calibration for the x, y scales, and the identification of the crystallographic orientations of the gold surface

32

Overview

Fig. 1.30. A voltammogram and an STM image of Au in 0.1M H2 SO4 . (a) A voltammogram of Au in 0.1M H2 SO4 . It starts at -0.35 V with a freshly prepared √ The broad peak at 0.35 V signals the transition from Au(111)-22× 3 reconstruction. √ the Au(111)-22× 3 reconstruction to a 1×1 structure. The very sharp peak at around 0.73 V corresponds to the formation of an ordered adlayer of sulfate. (b) An STM image of the Au(111) surface, taken at 0.65 V first, then changed the potential to 0.8 V. An order adlayer of sulfate is formed. (Adapted with permission from Kolb [110]. Copyright Wiley-VCH Verlag GmbH & Co. KGaA.)

with regard to the scan directions. If the tip is in good condition, atomic resolution is achievable, see Fig. 1.29(b). In the UHV environment, changing the surface reconstruction, depositing or removing a layer of atoms or molecules at the surface, time-consuming and often irreversible annealing and vacuum evaporation processes. In an electrochemical cell, those processes could be accomplished within seconds by simply changing the potential, and are almost always reversible. For example, the potential-induced reconstruction has been observed on all three low-Miller-index gold surfaces. In 0.1M H2 SO4 , for potential Us 0.73 V, a full ordered there is no stable SO2− 4 adlayer of SO2− is formed. Figure 1.30(a) shows a voltammogram of the 4 Au surface in 0.1M H2 SO4 . The √ first peak at about 0.35 V corresponds to the transition from the 22× 3 reconstruction to the Au(111)-1×1 structure. The second peak around 0.73 V corresponds to the formation of an ordered adlayer of sulfate. The very narrow peak means that the ordered adlayer is formed within a very small potential range. Figure 1.30(b) is an STM image of the Au(111) surface, recorded at different potentials. The upper half, recorded at a potential of 0.65 V, shows atomic resolution of the Au(111)-1×1 surface. The lower half, recorded at a potential of 0.8 V, shows an ordered adlayer of sulfate. In order to observe large biomolecules, e.g. nucleotides and proteins with STM or AFM in aqueous solutions, a key step is immobilization: to adsorb the molecule on a flat surface without changing its functions. A commonly used method is to start with self-assembled monolayers of thiols on atomically flat gold surfaces, for example, Au(111). The thiols, the or-

1.5 Illustrative applications

33

ganic molecules with a thiol functional group -SH, can form a strong S-Au bond on gold surfaces (about 1 eV), leading to self-assembled monolayers. The other functional groups in that molecule, such as -OH, -CH3 , -CHO, could provide immobilizing anchors to the molecules of interest. Alkanethiols Cn H2n+1 SH are the most commonly used thiols to form self-assembled monolayers. Organic molecules containing carboxyl (-COOH) and amine (-NH2 ) functional groups are important for immobilization of biological macromolecules. Among the thiols, cysteine is of particular interest [112]. Cysteine (HS-CH2 -CHNH2 -COOH) is one of the 20 amino acids in proteins, an indispensable building block of biological systems. Cysteine is a particularly important building block in the formation of human hair, fingernails, skin, and wool, and a ligand in many metalloproteins. Cysteine is one of the two amino acids which contain sulfur, and the only amino acid which contains a thiol group. A self-assembled monolayer of cysteine on gold surface provides both the carboxyl group and the amine group for the immobilization of biological macromolecules, especially proteins. Self-assembled monolayers of cysteine are formed by exposing Au(111) to a dilute solution of cysteine in 50 mM aqueous buffer electrolyte of ammonium acetate (CH3 COONH4 ) [112]. The advantage of using ammonium acetate is that its pH is around 4.6, mild to biomolecules, and it does not adsorb on Au(111) surface over a quite wide potential range. A cysteine concentration of 1 × 10−6 M is sufficient to build a highly ordered cysteine monolayer under potential control. A gradual disappearance of the herringbone pattern of the STM image signifies its formation. A typical STM image of a monolayer of cysteine thus obtained is shown in Fig. 1.31(a). A structural model derived from the experiments is shown in Fig. 1.31(b). A ball-and-stick model of cysteine is shown in Fig. 1.31(c). As shown, the sulfur atom is bonded to a hollow site of the Au(111) surface. The lateral interactions of cysteine generate an ordered pattern.

Fig. 1.31. Self-assembled monolayer of cysteine on Au(111). (a) A STM image of the self-assembled monolayer of cysteine on Au(111). (b) Proposed model of its structure. The sulfur atom is bonded to the Au(111) surface at the hollow site. (c) A ball-and-stick model of cysteine. (By courtesy of J. Ulstrup’s group of Technical University of Denmark. Reproduced with permission [112].)

34

1.5.3

Overview

Catalysis research

Catalysis is the backbone of synthetic chemistry. It is a basic process in the production of fabrics, fuels, fertilizers and pharmaceuticals. Catalysis is also essential in environmental protection. For example, required by law, each automobile must have a catalytic converter to remove toxic exhaust gases, notably CO and NOx . Although it is well known that catalytic reactions often take place at specific active atomic sites on the catalyst surface, the research in catalysis have been mainly relying on test-anderror experimentation involving macroscopic parameters. The invention of STM enables detailed studies of the electronic structures of the active sites at catalyst surfaces. STM can even operate in an atmosphere with actual reactants, thus to observe the transformation of molecules at active atomic sites on real time. Therefore, STM research can help facilitate a full understanding and control of the constituents at the molecular and atomic levels, and to open up the possibility of designing the catalysis process at the single molecule level [113]. Here are two examples. Ni-Au catalyst for steam reforming Steam reforming is a method in the chemical industry to produce hydrogen from hydrocarbons. For example, in the presence of a catalyst, alkanes react with steam (overheated water vapor) to generate H2 and CO: Cn H2n+2 + nH2 O → n CO + (2n + 1) H2 .

(1.30)

Fig. 1.32. STM topographical images of the Ni-Au system. (a) An image of Ni(111) with 2% Au. The Au atom appears as depressions because the LDOS at the Fermi level is lower. The Ni atoms surrounding an Au atom appear as protrusions because of the enhancement of LDOS at the Fermi level, indicating a higher chemical reactivity. For the Ni atom between two Au atoms, the LDOS enhancement is even higher. (b) An image of Ni(111) surface with 7% of Au. The number of Ni atoms with doubly enhanced Fermi-level LDOS is increased. (Original figure in black-and-white with high resolution by courtesy of F. Besenbacher and J. V. Lauritsen. Reproduced with permission [114]. Copyright 1998 American Association for the Advancement of Science.)

1.5 Illustrative applications

35

The most important case is the reaction of methane with steam, CH4 + H2 O → CO + 3 H2 .

(1.31)

It is the most common and least expensive method to produce hydrogen. The typical catalyst is nickel nanoclusters supported on a MgAl2 O4 substrate. Nevertheless, in parallel with the steam reforming process, nickel also catalyzes the production of graphite, which impedes the activity due to the formation of a blocking carbon layer on the nickel surface (coking) and may eventually lead to the breakdown of the catalyst. An existing solution to that problem is by mixing a minute amount of H2 S to the reactants, which poisons the nickel catalyst. Because sulfur poisons the formation of graphite more than it poisons the steam formation reaction, the lifetime of the catalyst is prolonged. However, the overall rate is lower, and the inclusion of H2 S is harmful to the subsequent processes and the environment. A new idea comes from the STM studies of the Ni-Au system [114]. Ni and Au are immiscible in the bulk. No binary alloy can be formed. However, STM studies showed that by mixing a few percents of Au to Ni, after annealing at 800 K, a surface binary alloy is formed: many Ni atoms on the surface are substituted by Au atoms, see Fig. 1.32. The topographic STM image shows that as if the Au atoms are darker (deeper), and the Ni atoms immediately adjacent to an Au atom are brighter (higher). It is not because the Au atoms are depressed and the adjacent Ni atoms are protruded, but is an electronic effect inherent to STM. According to the Tersoff–Hamann model of STM imaging [115, 116] (see Chapter 6), a topographic STM image is the contour of equal Fermi-level local density

Fig. 1.33. Conversion rates of the Ni catalyst and the Ni-Au catalyst. The conversion rates of steam reformation of n-butane and water vapor using different catalysts. Gray curve: pure Ni catalyst. the rate deteriorates with time as a result of the parasitic graphite formation process. Black curve: Ni-Au catalyst, which shows no deterioration of reaction rate. (Original figure in black-and-white with high resolution by courtesy of F. Besenbacher and J. V. Lauritsen. Reproduced with permission [114]. Copyright 1998 American Association for the Advancement of Science.)

36

Overview

of states of the sample, measured at the center-of-curvature of the tip r0 , ρ(EF , r0 ). On the Au atoms, the value of ρ(EF , r) is lower than that of the normal Ni atoms. On the other hand, the value of ρ(EF , r) of a Ni atom adjacent to a Au atom is higher owing to the perturbation by the Au atom. For a Ni atom with two Au neighbors, the perturbation is even stronger, see Fig. 1.32. Because the Fermi-level local density of states is closely related to catalytic reactivity, those Ni atoms provide a higher reactivity than the normal Ni atoms. Furthermore, the carbon atoms are much less likely to adsorb on the Ni atoms adjacent to a Au atom. The results are: by mixing a few percents of Au to Ni, the reactivity of the steam formation process is enhanced, and the graphite formation process is reduced. While the conversion rate of the pure Ni catalyst decreases with time, the conversion rate of the Ni-Au catalyst remains constant. Therefore, the Ni-Au catalyst could be active for a much longer time, see Fig. 1.33. Based on the STM study, a new catalyst, with about 16.5 weight percents of Ni and 0.3 weight percents of Au on a MgAl2 O4 matrix, is designed, which substantially improves the steam formation process. Understand and improve the MoS2 catalyst Petroleum contains a wide range of sulfur compounds, for example, alkanethiol (Cn H2n+1 SH) and thiophene (C4 H4 S). Sulfur is harmful for the chemical process and for the environment. The process to remove sulfur, hydrodesulphurization (HDS), is an essential step in petrochemical industry. The typical catalyst is molybdenum disulfate (MoS2 ). In the presence of hydrogen, the HDS reaction converts alkanethiol into alkane, Cn H2n+1 SH + H2 → Cn H2n+2 + H2 S,

(1.32)

and the gas-phase H2 S is easily removed. MoS2 is a layered material, similar to graphite. The bulk of MoS2 consists of thin sheets of S-Mo-S sandwiches, and those sheets are held together by van der Waals forces. The basal planes of MoS2 are chemically inert. Therefore, catalytic reactivity must reside in the rims of the sheets. However, before the application of STM, the preferred structure and the active sites of MoS2 crystallites, as well as the reaction paths of the HDS process, were only vaguely guessed based on macroscopic evidences. A systematic study of the structure, active sites, and reaction paths is carried out with STM experiments and density-functional theory (DFT) computations [117]. In the industry, MgAl2 O4 is usually used as substrates for the MoS2 clusters. However, it is not appropriate for STM studies because it is an insulator. For STM studies, single crystal gold is chosen as the substrate, especilly the Au(111) surface. Gold is relatively inert and allows for investigation of the intrinsic properties of the catalysts.

1.5 Illustrative applications

37

The nanoclusters of MoS2 were prepared by depositing molybdenum on Au(111) surface in an atmosphere of H2 S at a pressure of 1×10−6 mbar, then the substrate was annealed to 673–723K for 15 minutes. A fairly uniform distribution of crystalline MoS2 nanoclusters was formed. Most of the MoS2 nanoclusters appear to have a triangular shape, with an average area of 5 nm2 . The side length of each triangular nanocluster is about 3 nm. From a combination of STM experiments and first-principles computations, it was concluded that the edges are saturated with sulfur atoms [117]. By exposing the MoS2 nanoclusters with various substances, the reactivity of MoS2 is studied with STM. It was found that MoS2 neither reacts with organic molecules containing sulfur, such as alkanethiol (Cn H2n+1 SH) and thiophene (C4 H4 S), nor reacts with H2 . Only after exposing MoS2 with predissociated (atomic) hydrogen, are visible changes observed. Sulfur vacancies are created at the rims. Further exposing the MoS2 nanoclusters already treated with atomic hydrogen to thiophene (C4 H4 S) initiates a strong reaction, see Fig. 1.34. As

Fig. 1.34. Reaction of H and thiophene with a MoS2 nanocluster. (a) The triangular MoS2 nanocluster, after exposure to atomic hydrogen and then thiophene at elevated temperature, is cooled down and characterized by STM. Image size: 5×5 nm2 . Bean-like structures, identified as the intermediate product of thiophene, cis-but-2-enethiolate (CH3 -CH=CH-CH2 -S− ), are found on the bright rim. Also, there is a prominent change in the outermost rim protrusions, identified as sulfur vacancies. Part of the rim is highlighted in (b). Details of a line scan of the rim is shown in (c). Solid curve (I): STM line scan along the dashed line in (b). Dashed curve (II): the line scan of the correspondent unreacted area, shown for comparison. (Original high-resolution image in black-and-white by courtesy of J. V. Lauritsen. See [113, 117] for details.)

38

Overview

shown, parts of the rim are significantly altered. The outermost protrusions of the rims are markedly shifted, and bean-like structures appear adjacent to the bright rim. The STM study revealed the mechanism of catalytic reaction. First, a hydrogen atom reacts with the rim of a MoS2 nanocluster and strips off a sulfur atom to expose a molybdenum atom. The molybdenum atom becomes an active site. An alkanethiol molecule can chemically adsorb on the active site by forming a S-Mo bond. After picking up a H atom, the hydrocarbon molecule becomes free, and the sulfur atom occupies the S vacancy at the rim of the MoS2 nanocluster. The desulphurization of thiophene is more complex. At an active site of MoS2 , it is first converted to an intermediate product, cis-but-2-ene-thiolate (CH3 -CH=CH-CH2 -S− ), and adsorbs on the bright rim. Then it reacts further with hydrogen to become H2 S and alkenes. The detailed understanding of the catalytic reaction at the atomic level has resulted in improvements of the hydrodesulphurization catalyst. 1.5.4

Atom manipulation

In the speech entitled There’s Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics [28], Richard Feynman posed a question “Why cannot we write the entire 24 volumes of the Encyclopedia Britannica on the head of a pin?” Such a dense writing requires that each dot of the letters is 8 nm in diameter, a length of about 32 atoms in an ordinary metal.

Fig. 1.35. The basic steps of atom manipulation. First, position the tip to above the adatom to be moved. Then, following the three steps: Step A, gradually increase the set tunneling current to move the tip towards the adatom, until the interaction energy between the tip and the adatom reaches the diffusion activation energy, the energy required to move the adatom across the ridge between two adjacent stable positions. Step B, pull the adatom to a desired location. Step C, gradually decrease the set tunneling current to move the tip away from the adatom.

1.5 Illustrative applications

39

In other words, each dot would contain in its area about 1,000 atoms. At that time, such a dense writing was a fantasy. The invention of STM has completely changed the scenario. Using STM as a writing tool, features made of single atoms can be generated, far exceeding Feynman’s goal. With such a density, the publications of the entire US Congress Library could be written on the head of a pin. The basic operation of atom manipulation is to use the STM tip to move an adatom from an initial position to a new position on a substrate, as shown in Fig. 1.35. At the beginning, the atom for the designed structure is deposited randomly on a substrate. Each atom is then an adatom on the substrate at a random position. In order to move an adatom to another stable location, an activation energy E must be applied to lift it across a ridge to reach another stable position. There are three steps for a move, see Fig. 1.35. Step A is to place the tip at the top of an adatom to be moved, then gradually increase the set tunneling current. As a result, the tip moves towards the adatom. A partial chemical bond is formed. When the chemical bond energy equals the barrier energy, the tip should be able to pull the adatom over the ridge. Step B is to move the tip sideways to pull the adatom to a desired location. Step C is to gradually decrease the set tunneling current. As a result, the tip moves away from the adatom and leaves it at the new position. During the process of atom manipulation, neither the interaction energy between the tip and the adatom nor the distance between the apex atom of the tip and the adatom is known. The pioneers of atom manipulation found from experience that this can be controlled by tunneling conductance, or its inverse, the tunneling resistance. The threshold interaction energy to allow the tip to pull the adatom over a diffusion barrier corresponds to a threshold conductance, or equivalently, a threshold resistance [118, 119]. There is a general and fundamental relation between interaction energy and tunneling conductance, which is discussed in Chapter 5.

Part I Principles

Part I: Principles The first part of the growth of a physical science consists in the discovery of a system of quantities on which its phenomena may be conceived to depend. The next stage is the discovery of the mathematical form of the relations between these quantities. After this, the science may be treated as a mathematical science, and the verification of the laws is effected by a theoretical investigation of the conditions under which certain quantities can be most accurately measured, followed by an experimental realization of these conditions, and actual measurement of the quantities. On the Mathematical Classification of Physical Quantities. James Clerk Maxwell Distinguished Professor of Experimental Physics Cambridge University

Bardeen’s perturbation theory of tunneling [120] has been the cornerstone of the theoretical understanding of Giaever’s classical tunneling junction experiments [121], and has played an important role in the interpretation of STM. Bardeen’s approach is different from the traditional Reileigh– Schr¨odinger perturbation theory, it starts with two complete orthonormal systems of wavefunctions, and the perturbation originates from the interaction of the two subsystems, rather than an external potential. Yet Bardeen’s tunneling theory is enormously powerful: it could provide analytic expressions for the tunneling current, which facilitates conceptual understanding as well as quantitative comparison with experimental measurements [14, 15, 122]. A good example of its application is the derivation of the well-known formula of the Tersoff–Hamann model [115, 116]. The deficiency of the original Bardeen perturbation theory of tunneling can be corrected by including the modifications of the wavefunctions of one party (for example, the sample) due to the existence of another party (for example, the tip). Actually, the concept and method for such modifications were proposed by Herring [17] and Landau [18] in the treatment of the covalent bond, the archetypal atomic force.1 By including such corrections, a modified Bardeen approximation is established. By applying it to the only 1 The perturbation treatment of the hydrogen molecular ion problem is presented in Quantum Mechanics (Non-relativistic theory) of Landau and Lifshitz, Section 81, Valency. It refers to a paper authored by L. Landau, and another paper authored by C. Herring, both published in 1961.

44

Part I

tunneling problem in Nature that has an exact analytical solution, the hydrogen molecular ion, a quantitative description of the exchange interaction as well as high-accuracy expressions for the asymptotic potential curves is found. Furthermore, by applying the Herring–Landau method to molecules composed of many-electron atoms, satisfying quantitative agreement with experimental data is also demonstrated [16]. Therefore, the accuracy of the modified Bardeen approximation is justified. For many practical cases in STM, Bardeen’s surface integral results in simple analytical forms. The correction to the wavefunctions, in many practical cases in STM and AFM, can be managed analytically. The similarity of the Bardeen theory of tunneling and the Herring–Landau theory of covalent bond implies that, tunneling conductance and chemical bond force could be equivalent [123]. The measurable consequences of such an equivalence have been verified experimentally. In spite of its simplicity, the perturbation theory of tunneling produces results that account for much of the physics in STM and AFM, and give reasonably accurate quantitative agreement with measurements, including topographic and current images, tunneling spectra, tunneling barrier measurements, and the effect of force between the tip and the sample. Of course, it does not mean that the first-principles calculations are not important. For any practical system of interest, e.g. both the tip and the sample, an allelectron first-principles calculation, instead of simplified models, is essential for achieving a thorough understanding.

Chapter 2 Tunneling Phenomenon The tunneling phenomenon, one of the fundamental features of quantum mechanics that distinguishes it from classical mechanics, has played a vital role in many branches of modern physics. One of the earliest applications of the concept of tunneling was in nuclear physics, where Gamow elucidated the α-decay in 1928 [124]. In the same year, in the field of atomic physics, Oppenheimer explained field-ionization of the hydrogen atom with tunneling [125]. An application in condensed-matter physics also appeared in the same year: the theory of field emission by Fowler and Nordheim [126]. Four Nobel Prizes in physics have been awarded to discoveries and inventions related to tunneling: Giaever’s tunneling junction experiment [121, 127], the discovery of the Josephson effect [128, 129], Esaki’s invention of tunnel diode [130], and of course, STM [29, 30]. The most widely applied theory for the tunneling phenomenon in solids as well as in STM is Bardeen’s first-order perturbation theory. Historically, Giaever’s tunneling experiments [121, 127] were aimed at verifying the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity [131]. At the beginning, the relation between their experimental discovery and the energy spectrum of superconductor was not clear. Bardeen [120] developed a tunneling theory to make a connection. His theory is perturbative in nature, but different from the standard perturbation theory in quantum mechanics. In a way, Bardeen’s tunneling theory is an ingenious creation by itself. This Chapter presents a detailed, step-by-step derivation of the Bardeen theory. First, we explore a one-dimensional single-electron version in the spirit of Oppenheimer’s treatment of field ionization [125], which is equivalent to Bardeen’s original theory [120]. Then we discuss a three-dimensional version pertinent to the physics of STM. In the derivation of the threedimensional case, we go beyond the original Bardeen approximation, to include a correction factor following the idea of Herring [17] and Landau [18]. We show, by introducing such a correction factor, the accuracy of the Bardeen approximation is well justified. In addition, we present the Bardeen theory in second-quantization format, the so-called transfer-Hamiltonian formalism; and the matrix format, which is the origin of the term “tunneling matrix element”. To evaluate an approximation method, the best approach is to compare its result with that of an analytically resolvable case. Nature, ever the provider, supplies us with an analytically resolvable and experimentally *OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

46

Tunneling Phenomenon

verifiable problem in quantum mechanics, which can also be treated using a perturbation method similar to that of Bardeen: the hydrogen molecular ion [18, 17]. Chapter 4 discuss this in more detail. We show that the perturbation method of Bardeen, after certain corrections, not only generates an exact asymptote, but is fairly accurate in the intermediate regime where most of the STM and AFM experiments are conducted. In addition to the basics, we present extensions of Bardeen’s perturbation theory to inelastic electron tunneling spectroscopy (IETS) and spinpolarized tunneling.

2.1

The metal–insulator–metal tunneling junction

In order to verify the BCS theory of superconductivity [131], Giaever [121, 127, 132] designed and demonstrated an experiment of tunneling spectroscopy with metal–insulator–metal (MIM) tunneling junctions. His MIM tunneling experiment provided a direct measurement of the energy gap in superconductors, which was a critical evidence for the BCS theory. As a result of the study of MIM tunneling junctions, many new concepts on tunneling phenomena have been developed (see, for example, Hansma [133], Duke [134], Burstein and Lundquist [135], and Wolf [136]). Those concepts developed through the study of MIM tunneling junctions are instrumental in the understanding of the tunneling phenomena in STM, as well as the scanning tunneling spectroscopy (STS). The realization of MIM tunneling is, to a great extent, owing to a gift from Nature: a superb and easy-to-form insulating film of Al2 O3 (sapphire is Al2 O3 in crystalline form). By heating a piece of aluminum in air, a

Fig. 2.1. Metal–insulator–metal tunneling junction. The junction is made through the following steps. (a) A glass slide with indium contacts. (b) An aluminum strip is vacuum deposited. (c) The aluminum strip is heated in air to form a thin Al2 O3 film. (d) A lead film is deposited across the aluminum strip, forming an Al-Al2 O3 -Pb sandwich. (Reproduced with permission from Giaever and Megerle [132]. Copyright 1961 American Physical Society.)

2.1 The metal–insulator–metal tunneling junction

47

Fig. 2.2. Tunneling spectroscopy in classic tunneling junctions. (a) If both electrodes are metallic, the I/V curve is linear. (b) If one electrode has an energy gap, an edge occurs in the I/V curve. (c) If both electrodes have energy gaps, two edges occur. A ‘negative differential conductance’ appears. (Reproduced with permission from Giaever and Megerle [132]. Copyright 1961 American Physical Society.)

thin film of Al2 O3 is produced on its surface. Even as thin as 3 nm, it has a very high electrical resistance, and is often free from pinholes. By evaporating another metal on the oxide-covered Al piece, typically Pb, an MIM tunneling junction is formed, see Fig. 2.1. The standard theoretical method for the understanding of the MIM tunneling junction is the time-dependent perturbation approach developed by Bardeen [120]. It is sufficiently simple for treating many realistic cases, and has been successfully used for describing a wide variety of effects. Instead of trying to solve the Schr¨odinger equation of the combined system, Bardeen considers two separate subsystems first. The electronic

48

Tunneling Phenomenon

states of the individual subsystems are obtained by solving the stationary Schr¨ odinger equations of the individual subsystems. For many practical systems, those solutions are much easier to obtain than the combined systems. The transmission rate of electrons from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by a surface integral of the unperturbed wavefunctions of the two subsystems at a separation surface. The choice of the separation surface does not affect the results appreciably. In the case of planar tunneling junctions, the separation surface is simply a plane between the two electrodes at an arbitrary position z = z0 . Furthermore, if M does not change appreciably in the energy interval of interest, then the tunneling spectrum is determined by the convolution of the density of states (DOS) of the two electrodes: (see Fig. 2.2): 

eV

I ∝

ρA (EF − eV + ) ρB (EF + ) d .

(2.1)

0

Therefore, the experimental results of the tunneling experiment provide a direct verification of the energy gaps predicted by the BCS theory.

2.2

The Bardeen theory of tunneling

The most widely used theory of tunneling regarding to STM was developed by Bardeen [120] initially for explaining Giaever’s tunneling experiment [132], a key bit of experimental evidence for the theory of superconductors. 2.2.1

One-dimensional case

The planar tunneling junction problem treated by Bardeen is schematically shown in Fig. 2.3. When the two electrodes are far apart, as shown in Fig. 2.3(a), the wavefunctions of electrode A satisfies Schr¨ odinger equation of the free electrode A,

∂Ψ 2 ∂ 2 i (2.2) = − + U A Ψ, ∂t 2m ∂z 2 where UA is the potential function of electrode A, and Ψ depends on both time and spatial coordinates. The stationary states are Ψ = ψμ e−iEμ t/ , with the spatial wavefunctions and energy eigenvalues satisfying

2 ∂ 2 − ψμ = Eμ ψμ . + U A 2m ∂z 2

(2.3)

(2.4)

2.2 The Bardeen theory of tunneling

49

Fig. 2.3. The Bardeen tunneling theory: one-dimensional case. (a). When the two electrodes are far apart, the wavefunctions of both electrodes A and B decay into the vacuum. (b). By bringing the two electrodes together, tunneling takes place. The tunneling current can be evaluated using a time-dependent perturbation theory.

Similarly, for the free electrode B, the Schr¨odinger’s equation is

2 ∂ 2 ∂Ψ = − + UB Ψ, i ∂t 2m ∂z 2

(2.5)

and the stationary states are Ψ = χν e−iEν t/ , with the spatial wavefunctions χν satisfying

2 ∂ 2 − + UB χν = Eν χν . 2m ∂z 2

(2.6)

(2.7)

In the space between electrode A and electrode B, the wavefunctions of both electrodes decay into the vacuum. By bringing the two electrodes together, as shown in Fig. 2.3(b), the Schr¨ odinger equation of the combined system is

2 ∂ 2 ∂Ψ = − (2.8) + U + U i A B Ψ. ∂t 2m ∂z 2 With the presence of the combined potential, a state ψμ described by Eq. 2.4 at t = 0 will not evolve according to Eq. 2.2. Instead, it has a probability of transferring to the states of electrode B. In other words, we assume Ψ = ψμ e−iEμ t/ +

∞ 

cν (t)χν e−iEν t/ ,

(2.9)

ν=1

where cν (t) are coefficients to be determined by Eq. 2.8, with cν (0) = 0.

(2.10)

It is important to note that each set of wavefunctions ψμ and χν originate from different Hamiltonians. Neither ψμ nor χν is an eigenfunction of the

50

Tunneling Phenomenon

Hamiltonian of the combined system. A basic assumption (or basic approximation) of Bardeen’s tunneling theory is that the two sets of wavefunctions are approximately orthogonal,  (2.11) ψμ∗ χν d3 r ∼ = 0. Therefore, the trial wavefunction, Eq. 2.9, is still normalized up to a secondorder infinitesimal quantity proportional to |cν |2 . Inserting Eq. 2.9 into Eq. 2.8, we obtain i

∞  dcν (t) ν=1

dt

χν e−iEν t/ = UB ψμ e−iEμ t/ + UA

∞ 

cλ (t)χλ e−iEλ t/ .

λ=1

(2.12) The second term on the right-hand side of Eq. 2.12 is a second-order infinitesimal quantity. Therefore, to first-order,  dcν (t) = ψμ UB χ∗ν d3 r e−i(Eμ −Eν )t/ . (2.13) i dt z>z0

Since UB is non-zero only in the volume of electrode B, z > z0 , the integral is evaluated only in the right-hand side of the separation surface. By defining a tunneling matrix element as  ψμ UB χ∗ν d3 r , (2.14) Mμν = z>z0

an explicit expression of the tunneling current can be derived. Integrating Eq. 2.13 over time, the amplitude of the ν-th state of electrode B at time t is e−i(Eμ −Eν )t/ − 1 cν (t) = Mμν . (2.15) Eμ − Eν Starting with the μ-th state of electrode A, the probability of having the ν-th state of electrode B at time t is pμν (t) ≡ |cν (t) |2 = |Mμν |2

4 sin2 [(Eμ − Eν ) t/2] (Eμ − Eν )

2

.

(2.16)

The function f (t), defined as f (t) ≡

4 sin2 [(Eμ − Eν ) t/2] (Eμ − Eν )

2

,

(2.17)

reaches its maximum at Eμ = Eν , and rapidly approaches zero for Eμ = Eν . Thus, the tunneling current depends on how many states near the energy

2.2 The Bardeen theory of tunneling

51

value of a state in electrode A into which can electrode B effectively tunnel. Introducing an important quantity, the density of states of electrode B at energy E, ρB (E), the total probability of tip states that the original sample state can tunnel into in time t is 2π pμν (t) = |Mμν |2 ρB (Eμ ) t, (2.18)  where the mathematical identity ∞ −∞

sin2 au du = 1 πau2

(2.19)

is used. The integrand approaches a delta function δ(x) for large a, see Fig. 2.4. Since the time of tunneling is much greater than /ΔE, where ΔE is the energy resolution, Eq. 2.18 implies a condition of elastic tunneling: Eμ = Eν .

(2.20)

It means that a state in one electrode can only tunnel into states in another electrode of the same energy value. The opposite case is inelastic tunneling, which we will discuss later. From electrode A, the number of available states is defined by the density of states of electrode A at energy E, ρA (E), and the energy interval defined by the bias voltage V . If the density of states of both electrodes does not vary appreciably near the Fermi level on the range of the applied bias voltage, the tunneling current is 2πe2 |Mμν |2 ρB (EF ) ρA (EF ) V. (2.21)  The tunneling current I is proportional to bias voltage V . It can be conveniently written in terms of tunneling conductance G = I/V as I=

G = 2π 2 G0 |Mμν |2 ρB (EF ) ρA (EF ) ,

(2.22)

2

where the conductance quantum, Eq. 1.23, is e /π = 77.48μS. Using Eq. 2.4 and 2.7, the integral in Eq. 2.14 can be converted into a surface integral only depending on the unperturbed wavefunctions of the two electrodes at the separation surface. By applying Eq. 2.7, we have    2 ∂ 2 Mμν = ψμ Eν + (2.23) χ∗ν d3 r . 2m ∂z 2 z>z0

Because of the elastic tunneling condition, Eμ = Eν , Eq. 2.14 can be converted into    2 ∂ 2 ∗ χ (2.24) χ∗ν Eμ ψμ + ψμ d3 r . Mμν = 2m ∂z 2 ν z>z0

52

Tunneling Phenomenon

Fig. 2.4. The origin of the elastic-tunneling condition. The integrand in Eq. 2.19, with a=31.4, which approaches a delta function when a approaches infinity. If the time of the experiment is not too short, the condition of elastic tunneling is valid.

Using Eq. 2.4, and noticing that, on the tip side, the sample potential UA is zero, we obtain 

2 =− 2m



ψμ ∂ 2 χ∗ν − ψ μ ∂z 2 ∂z 2

∂ χ∗ν

2

 d3 r .

(2.25)



∂ 2 ψμ ∂ 2 χ∗ν ∂χ∗ν ∂ ∗ ∂ψμ χ − ψ , − ψ ≡ μ μ ∂z 2 ∂z 2 ∂z ν ∂z ∂z

(2.26)

Mμν

z>z0

With the identity χ∗ν

the integration over z can be carried out to obtain Mμν

2 = 2m





∂χ∗ν ∗ ∂ψμ − χν dxdy. ψμ ∂z ∂z

(2.27)

z=z0

Equation 2.27 is Bardeen’s tunneling matrix element in one-dimensional form. It is a surface integral of the wavefunctions (and its normal derivatives) of the two free electrodes, evaluated at the separation surface. The potential barrier information does not appear explicitly, and only the information of the wavefunctions at the separation surface is required. Furthermore, the formula is symmetric with regards to both electrodes. It is the basis of the reciprocity principle in STM and AFM, that has important consequences in designing and interpreting experimental results.

2.2 The Bardeen theory of tunneling

2.2.2

53

Tunneling spectroscopy

Bardeen’s theory provids a sound basis for the interpretation of the observed tunneling spectroscopy. Using Eq. 2.21, the tunneling current at a bias voltage V can be evaluated by summing over all relevant states. At a finite temperature, the electrons in both electrodes follow the Fermi distribution [137]. With a bias voltage V , the total tunneling current is I =

4πe 



∞ −∞

[ f (EF − eV + ) − f (EF + )] (2.28)

×ρA (EF − eV + ) ρB (EF + ) | M |2 d , where f (E) = (1+exp[(E −EF )/kB T ])−1 is the Fermi distribution function. The quantities ρA (E) and ρB (E) are the DOS of electrode A and electrode B, respectively. If kB T is smaller than the energy resolution required in the measurement, then the Fermi distribution function can be approximated by a step function. In this case, the tunneling current is (see Fig. 2.3): I=

4πe 



eV

ρA (EF − eV + ) ρB (EF + )|M |2 d .

(2.29)

0

In the interpretation of the experiment of Giaever [121, 127], Bardeen [120] further assumed that the magnitude of the tunneling matrix element |M | does not change appreciably in the energy interval of interest. Then, the tunneling current is determined by the convolution of the DOS of two electrodes: 

eV

ρA (EF − eV + ) ρB (EF + ) d .

I ∝

(2.30)

0

Clearly, according to the Bardeen formula, Eq. 2.30, the electronic structure of the two participating electrodes enters into the formula in a symmetric way. In other words, they are interchangeable. In determining the tunneling current, the DOS of one electrode ρA and the DOS of the other electrode ρB contribute equally in determining the tunneling current. This point was well verified in the classic tunneling junction experiment [121, 127, 132]. As shown in Fig. 2.2, if both electrodes are metals in their normal state, then the I − V curve is a straight line. If one of the metals is superconducting (that is, there is an energy gap and the DOS has a sharp peak), then the I − V curve shows a threshold. If both metals are superconducting (that is, energy gaps exist in the DOS of both), then there are two thresholds with an interval of negative differential conductance.

54

Tunneling Phenomenon

2.2.3

Energy dependence of tunneling matrix elements

Bardeen’s assumption that, the tunneling matrix element |M | is a constant, is reasonable for the interpretation of Giaever’s experiment [121, 127], because the magnitude of the energy gap in superconductors is only a few meV. However, in STS experiments, the energy scale can be as large as ±2 eV. The energy dependence of the tunneling matrix element cannot be overlooked. The variation of |M | with energy can be evaluated from the Bardeen formula, Eq. 2.27. In the gap region, the wavefunction of electrode A is (Fig. 2.3), ψμ (z) = ψμ (0) e−κμ z ,

(2.31)

 where κμ = 2m|Eμ |/ is the decay constant corresponding to the energy eigenvalue of ψμ . Similarly, in the gap region, the wavefunction of electrode B is χν (z) = χν (s) eκν (z−s) .

(2.32)

Because of the condition of elastic tunneling, Eq. 2.20, the two decay constants are equal,  κν = κμ =

2mEμ . 

(2.33)

Inserting Eqs 2.31 and 2.32 into Eq. 2.27, we obtain Mμν



2 = 2m ⎡

2κμ ψμ (0)χν (s) e−κμ z0 eκμ (z0 −s) dxdy

z=z0

 =⎣ 2m 2



⎤ 2κμ ψμ (0)χν (s) dxdy ⎦ e

(2.34) −κμ s

.

z=z0

As expected, the tunneling matrix element is independent of the position of the separation surface, z = z0 . The expression in the square bracket is a constant, because χν (s) is the value of the wavefunction of electrode B at its surface. The energy dependence of M is through the decay constant κμ . Qualitatively, the effect of the energy dependence of the tunneling matrix element is as follows, see Fig. 2.5. The value of e−κμ s near the top of the interval eV is bigger than its value near the bottom. Therefore, the energy spectrum of electrode A near the Fermi level and the empty state energy spectrum of electrode B about eV above the Fermi level are the dominant contributor to the integral in Eq. 2.29. A quantitative treatment is in the following section.

2.2 The Bardeen theory of tunneling

55

Fig. 2.5. Energy dependence of tunneling matrix element and the asymmetry of tunneling spectrum. A positive bias voltage V is applied. The electrons tunnel from the occupied states of the tip into the empty states of the sample. For states near the Fermi level of the tip, the decay constant is smaller, and the tunneling matrix element is larger. For states near the Fermi level of the sample, the decay constant is larger, and the tunneling matrix element is smaller. The tunneling spectrum is primarily determined by the tip DOS near the Fermi level and the sample DOS about eV above the Fermi level.

2.2.4

Asymmetry in tunneling spectrum

In this section, we apply the energy dependence of tunneling matrix elements to scanning tunneling spectroscopy, to predict the existence of an assymetry, or a polarity dependence, on the result of measurements. Figure 2.6 shows an energy diagram of a tip-sample junction. Upon application of a positive bias voltage V , the Fermi level of the sample is lower than that of the tip by eV . The electrons tunnel from the occupied states of the tip into the empty states of the sample. To facilitate analysis, an average work function φ¯ is introduced, which is the energy difference of the vacuum level to the middle of the two Fermi levels, or the middle point of the bias interval. To simplify notation, we omit the subscripts, and rewrite the Bardeen formula of tunneling current, Eq. 2.29, into a form symmetric with respect to the tip and the sample, 4πe I= 



1 2 eV

− 12 eV

1 1 ρS (EF + eV + ) ρT (EF − eV + ) |M ( )|2 d . (2.35) 2 2

For a tip state with energy eigenvalue E, the decay constant is  √ 2mE 2m(φ¯ − ) = , (2.36) κ=   where is the variable for integration in Eq. 2.35. Because under actual experimental conditions, is less than one half of the energy interval eV ,

56

Tunneling Phenomenon

¯ Equation 2.36 can be expanded to which is always much smaller than φ. become      2mφ¯ (2.37) 1 − ¯ ≡ κ0 1 − ¯ , κ≈  2φ 2φ where κ0 is the decay constant corresponding to the average work function,  2mφ¯ . (2.38) κ0 =  Following Eq. 2.34, we find that, the tunneling matrix element shows a dependence on the energy parameter :   κ0 s , (2.39) M ( ) = M (0) exp 2φ¯ Inserting Eq. 2.39 into Eq. 2.35, we obtain  1 4πe 2 eV 1 1 I = ρS (EF + eV + ) ρT (EF − eV + ) |M (0)|2  − 12 eV 2 2   κ0 s d . × exp φ¯

(2.40)

Apparently, the exponential factor makes the positive half of the bias interval more important that the negative half. Here is a numerical estimation of the magnitude of the effect. Typically, κ0 is 11 nm−1 , φ¯ is 5 eV, s is 1 nm, and the maximum value of is 0.5 eV. The argument of the exponential is about 1.1. Therefore, the ratio of the maximal contribution (at = 0.5 eV) and the minimal contribution (at = −0.5 eV) is e2.2 ≈ 9, or about one order of magnitude. In the limiting case, while s is large, the main contribution of the integral, Eq. 2.40, comes from a small energy interval near ≈ eV/2. In this case, the tunneling conductance is approximately   dI ≈ ρS (EF + eV ) ρT (EF ). (2.41) dU U =V In other words, by using a positive bias to probe the unoccupied states of the sample, the tunneling current at bias voltage V is proportional to the sample DOS at energy EF + eV , as if the tip DOS is concentrated at the Fermi level. Similarly, by applying a negative bias, such that the electrons from the occupied states of the sample tunnel to the unoccupied states of the tip, the energy spectrum of the tip plays a leading role. In this case, in order to obtain the energy spectrum of the sample, great care must be taken to ensure that the tip DOS is flat.

2.2 The Bardeen theory of tunneling

57

These results were obtained using a one-dimensional WKB model [138, 139]. However, the WKB approach is heuristic rather than logical, because the connection between tunneling conductance and DOS cannot be derived from the WKB method. The motivation of Bardeen to develop his tunneling theory was to provide a logical and quantitative connection between tunneling conductance and DOS to replace the WKB method [120]. 2.2.5

Three-dimensional case

This section presents a step-by-step derivation of the Bardeen tunneling theory in three dimensions. The sample and the tip are treated somewhat differently, especially regarding the evaluation of tunneling matrix elements, which we present in Chapter 3. We show that, by evaluating Bardeen’s tunneling matrix element from properly modified wavefunctions, the perturbation theory is accurate even if the barrier collapses. Figure 2.6 is the energy schematic of the combined system. As the tip and the sample approach each other with a finite bias V, the potential U in the barrier region becomes different from the potentials of the free tip and the free sample. To make perturbation calculations, we draw a separation surface Σ between the tip and the sample, then define a pair of subsystems with potential surfaces US and UT , respectively. As we show later on, the exact position of the separation surface is not important. As shown in Fig. 2.6, we define the potentials of the individual systems to satisfy two conditions. First, the sum of the two potentials of the individual systems equals the potential of the combined system, that is, US + UT = U.

(2.42)

Second, the product of the two potentials is zero throughout the entire space, US UT = 0.

(2.43)

The reference point of energy, the vacuum level, is well defined in the STM problem. The entire system is neutral. Therefore, at infinity, there is a welldefined vacuum potential. In the vicinity of the apex of the tip, the potential barrier in the gap is substantially lowered. However, the barrier lowering is confined to a small region near the tip end. Outside the interaction region, the potential in the space equals the vacuum level. This condition, Eq. 2.43, minimizes Oppenheimer’s error estimation term [125]. As seen in Fig. 2.6, the potentials US and UT are different from the potentials of the free tip, UT 0 , and the free sample, US0 , respectively. The effect of the differences (VS = US0 − US and VT = US0 − US ), can be taken into account by time-independent perturbation methods. We return to this point later.

58

Tunneling Phenomenon

The Schr¨odinger equation of the free sample is   2 2 ∇ + US ψμ = Eμ ψμ , − 2m

(2.44)

and for the free tip,   2 2 ∇ + UT χν = Eν χν . − 2m

(2.45)

The time-dependent Schr¨odinger equation of the combined system is   2 2 ∂Ψ (2.46) = − ∇ + US + UT Ψ. i ∂t 2m

Fig. 2.6. The Bardeen tunneling theory: three-dimensional case. (a) A separation surface Σ is drawn between the two subspaces ΩS and ΩT . The precise location is not critical. (b) The potential surface of the combined system. (c) and (d) The potential surfaces of the subsystems. The direction of the surface element dS is pointing into the tip subspace ΩT , as shown.

2.2 The Bardeen theory of tunneling

59

Part of the derivation is identical to the one-dimensional case, up to Eq. 2.14. The tunneling matrix element is now defined as  Mμν = χ∗ν UT ψμ d3 r . (2.47) ΩT

Since UT is non-vanishing only in the subspace of the tip, the matrix element Mμν is evaluated only in that subspace, ΩT . Similarly, we have  φ∗μ US χν d3 r . (2.48) Mνμ = ΩS

Furthermore, since US is nonvanishing only in the subspace of the tip, the matrix element Mνμ is evaluated only in that subspace, ΩS . Following the steps of Eqs 2.23 through 2.25, we find an expression of the tunneling matrix element in which the potential functions are eliminated,   ∗ 2  2 Mμν = χ ∇ ψμ − ψμ ∇2 χ∗ν d3 r . (2.49) 2m ΩT ν Using Green’s theorem [140],    ∗ 2  χν ∇ ψμ − ψμ ∇2 χ∗ν d3 r = − (χ∗ν ∇ ψμ − ψμ ∇χ∗ν ) · dS, ΩT

(2.50)

Σ

the tunneling matrix element can be converted into a surface integral on the separation surface Σ between the sample and the tip,  2 (ψμ ∇χ∗ν − χ∗ν ∇ ψμ ) · dS. (2.51) Mμν = 2m Σ using the same procedure, we find  2 ∗ (ψμ ∇χ∗ν − χ∗ν ∇ ψμ ) · dS = Mμν . Mνμ = 2m Σ 2.2.6

(2.52)

Error estimation

It is important to know the accuracy of the transition probability from firstorder time-dependent perturbation theory. Following Oppenheimer [125], we show that our choice of unperturbed potentials minimizes the error estimation term. Therefore, it is the optimum choice. To make an error estimation, we evaluate the second-order transition probability by taking into account the second term in Eq. 2.8, 2  2π  (χν , US χλ ) (χλ , UT ψμ )  (2) (2.53) wμν =  δ(Eμ − Eν ).     Eλ − Eν λ

60

Tunneling Phenomenon

Oppenheimer estimated the error as follows. Since Eμ , Eν are always several eV below vacuum level, the tip wavefunctions with energy levels close to that of the sample decay quickly outside the tip body. Thus, the matrix elements (χν , US χλ ) for Eλ ≤ Eμ are much smaller than those with Eλ ≈ 0. On the other hand, while Eλ > 0, the tip wavefunction oscillates quickly, and the matrix elements become small again. Therefore, the numerator must have ¯ . By replacing Eλ by E ¯ in the a sharp maximum somewhere at Eλ ≈ E denominator of Eq. 2.53 we obtain the Oppenheimer error estimation term: (2) wμν

2π ≈ 

   (χ , U χ ) (χ , U ψ ) 2  ν S λ λ T μ   δ(Eμ − Eν )  ¯ − Eν   E λ

 2 2π  (χν , US UT ψμ )  =  δ(Eμ − Eν ) = 0 ¯ − Eν   E

(2.54)

because of Eq. 2.43. Therefore, if the conditions 2.42 and 2.43 are fulfilled, the accuracy of the first-order perturbation is optimized. 2.2.7

Wavefunction correction

The wavefunctions in Eq. 2.51 are different from the wavefunctions of the free tip and free sample. There is a distortion potential: V = US − US0 on the sample side, and V = UT − UT 0 on the tip side. Its effect can be treated as a time-independent perturbation. The following presents an approximate method based on the Green’s function of the vacuum (see Appendix A). To first order, the distorted wavefunction ψ is related to the undistorted one, ψ0 , by  ψ(r) = ψ0 (r) +

G(r, r )V (r )ψ0 (r ) d3 r  .

The Green’s function is defined by   2 2 ∇ + US0 − E G(r, r ) = −δ(r − r ). − 2m

(2.55)

(2.56)

Using the properties of the Green’s function (see Appendix A), the evaluation of the effect of distortion on transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive, see Eq. 2.55. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green’s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier.

2.2 The Bardeen theory of tunneling

Consider the one-dimensional case. From Eq. 2.56  m ψ(z0 ) = e−κz0 − 2 e−κ|z0 −z| e−κz V (z) dz. κ

61

(2.57)

In the region z > z0 , the integrand vanishes rapidly. The integration in region z < z0 gives    z0 m ψ(z0 ) = e−κz0 1 − 2 V (z) dz . (2.58) κ −∞ For z ≥ z0 , the distorted wavefunction has the same exponential dependence on z as the free wavefunction. Thus, ∂ψ/∂z gains the same factor. The tip wavefunction χ gains a similar factor. Therefore, the transmission probability becomes  2  2  z0  ∞ m m T = T0 1 − 2 1− 2 V (z) dz V (z) dz . (2.59) κ −∞ κ z0 2.2.8

The transfer-Hamiltonian formalism

The Bardeen tunneling theory is often called the Transfer-Hamiltonian formalism. From the discussions hitherto, there is no clue for such a nickname. In fact, the name came from a follow-up paper by Cohen, Falikov, and Phillips [141]. In Bardeen’s original paper [120], the problem was formulated in the occupation-number representation, or the second-quantization format. Instead of using the coordinates r as the variable, the occupation number of each state of the system, n1 , n2 , ... nj , ... is used to describe a state of a many-body system [18, 142]. In general, a state is represented by |Φ = |n1 , n2 , ... nj , ... .

(2.60)

For a system of fermions, nj can only take either 0 or 1. The basic building block of such a system is a creation operator a ˆ†j , which increases the number of occupation of state j by one if it is originally zero; and an annihilation operator a ˆj , which decreases the number of occupation of state j by one if it is originally one. To test if a state is occupied, one could first apply an annihilation operator, followed by applying a creation operator. After the two operations, the state stays intact. If originally the state was occupied, then it returns 1. Otherwise it returns 0, because the application of an annihilation operator to an empty state is meaningless. In other words, ˆj |n1 , n2 , ... nj , ... = nj |n1 , n2 , ... nj , ... . a ˆ†j a

(2.61)

Similarly, if originally, nj = 0, we have ˆ†j |n1 , n2 , ... nj , ... = (1 − nj ) |n1 , n2 , ... nj , ... . a ˆj a

(2.62)

62

Tunneling Phenomenon

Combining Eqs 2.61 and 2.62, we obtain the anticommutation relation, ˆj + a ˆj a ˆ†j = 1. a ˆ†j a

(2.63)

The Hamiltonian, or the total energy operator of the system is, obviously,  ˆ = Ej a ˆ†j a ˆj . (2.64) H j

A many-body state with only one single-electron state m occupied can be constructed using the creation operator a ˆ†m acting on an empty state, |m = a ˆ†m |0 . In fact, this can be tested using the Hamiltonian operator,  ˆ Ej a ˆ†j a ˆj a ˆ†m |0 = Em |m . H|m =

(2.65)

(2.66)

j

Cohen, Falikov, and Phillips [141] showed that, in the occupation-number representation, tunneling can be described by an effective Hamiltonian conˆ A, sisting of three components: the Hamiltonian of the first subsystem H ˆt ˆ the Hamiltonian of a second subsystem HB , and a transfer Hamiltonian H which describes electron transfer, or tunneling: ˆB + H ˆ t, ˆ =H ˆA + H H

(2.67)

where the Hamiltonians of the individual unperturbed subsystems are  ˆA = Eμ a ˆ†μ a ˆμ , H μ

ˆB = H



Eν ˆb†ν ˆbν ,

(2.68)

ν

and the transfer-Hamiltonian is    ∗ ˆ† ˆt = δ(Eμ − Eν ) Mμν a ˆ†μˆbν + Mμν ˆμ . bν a H

(2.69)

μν

Here Eμ and Eν are identical to those in Eqs 2.44 and 2.45, and the tunneling matrix element Mμν is defined by Eq. 2.51. By writing the total Hamiltonian in the form of Eq. 2.67, we assumed ˆ B are complete; only ˆ A or the solutions of H that neither the solutions of H the union of both sets of wavefunctions is complete, as stated in Bardeen’s original paper [120]. His argument is that the wavefunctions of electrode A decays into the vacuum, and are reflected back on the surface of electrode B. Thus, in the volume of electrode B, the wavefunctions of electrode

2.2 The Bardeen theory of tunneling

63

A vanishe, and vice versa. It is impossible to express a wavefunction in electrode B by a linear combination of wavefunctions in electrode A, and vice versa. Therefore, to represent an arbitrary wavefunction in the entire space, one must take the entirety of both sets of wavefunctions as the basis. ˆ t is intuitively obvious. The first term The meaning of the two terms in H annihilates a state in electrode B and creates a state of the same energy in electrode A. The second term annihilates a state in electrode A and creates a state of the same energy in electrode B. It represents a tunneling process. The occurrence of the delta function δ(Eμ − Eν ) is because, during the derivation of the tunneling matrix element, the elastic tunneling condition, Eq. 2.20 is used. Cohen, Falikov, and Phillips’ second-quantization formalism [141] is but another way of presenting Bardeen’s perturbation theory of tunneling. Using the second-quantization formalism, the rich weaponry in quantum field theory, such as Green’s function method, propagators and Feynman diagrams, can be applied [134]. In the low-bias-voltage limit, the usual Bardeen formula, Eq. 2.21, is recovered: I=

2πe2  2 V |Mμν | δ(Eν − EF ) δ(Eμ − EF )  μν

(2.70)

2πe2 V |Mμν |2 ρB (EF ) ρS (EF ). =  2.2.9

The tunneling matrix

Bardeen’s tunneling theory can also be written in matrix form. Similar to the transfer-Hamiltonian formalism of Cohen, Falikov, and Phillips [141], by taking all the states in both electrodes as an entire system, as did Bardeen [120], the Hamiltonian of the entire system, in matrix form, can be sectored into blocks of equal energy eigenvalues. In general, the states could be degenerate. If, at energy level E0 , the states in electrode A are m-fold degenerate, and the states in electrode B are n-fold degenerate, that block of the tunneling matrix is ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ˆ =⎜ H ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

E0 0 .. .

0 E0 .. .

0

0

∗ M11 ∗ M21 .. .

∗ M12 ∗ M22 .. .

∗ Mm1

∗ Mm2

··· ··· · ··· ··· ··· · ···

0 0 .. .

M11 M21 .. .

M12 M22 .. .

E0

Mm1

Mm2

∗ M1n ∗ M2n .. .

E0 0 .. .

0 E0 .. .

0

0

∗ Mmn

··· ··· · ··· ··· ··· · ···

M1n M2n .. . Mmn 0 0 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(2.71)

E0

That format explains the origin of the term tunneling matrix element.

64

2.2.10

Tunneling Phenomenon

Relation to the Landauer theory

As pointed by Datta [74], the general result of Bardeen’s tunneling theory, Eq. 2.22, G = 2π 2 G0 |Mμν |2 ρB (EF ) ρA (EF ), (2.72) can be cast in the formalism of the Landauer theory, Eq. 1.22, G = G0 T ≡

e2 T, π

(2.73)

by defining the transmission coefficient as T = 2π 2 |Mμν |2 ρB (EF ) ρA (EF ).

(2.74)

Notice that the dimension of the density of states is the inverse of energy; the transmission coefficient is dimensionless. Combining Eq. 2.74 with Eq. 1.24 gives an approximate expression T = 2π 2 |Mμν |2 ρB (EF ) ρA (EF ) = e−2κ(z−ze ) ,

(2.75)

where ze is the equilibrium tip–sample distance. This treatment only takes one pair of states into account. By using the concept of channels or modes in the Landauer theory [74], for tunneling involving multiple states on the sample or on the tip, each pair can be treated as a channel. Equation 2.73 is then extended to  G = G0 TN , (2.76) N

where, for the N -th channel, the transmission coefficient TN can be represented by an expression similar to Eq. 2.74.

2.3

Inelastic tunneling

Section 2.1 presented the classical MIM tunneling experiment. If both electrodes are normal metals, the tunneling spectrum should be structureless. In 1966, by including certain molecules in the tunneling junction, Jaklevic and Lambe [144] found the tunneling electrons interacted with the vibrational states of the molecule. At the bias voltage corresponding to a vibrational frequency of the molecule, a kink in the tunneling conductance G was observed. In other words, sharp peaks in dG/dV = d2 I/dV 2 were found at bias voltages V = ω/e. To identify the origin of those peaks unambiguously, an isotope substitution experiment was performed [145], and their assumption was beautifully comfirmed. Within several years, the new technique, Inelastic Electron Tunneling Spectra (IETS), has grown into a significant field of research [143, 133].

2.3 Inelastic tunneling

2.3.1

65

Experimental facts

A schematic of the IETS experiment is shown in Fig. 2.7. Briefly, the experimental procedure is as follows. First, vacuum-evaporate a stripe of Al film on a glass slide. Then oxidize it to form an insulating layer of Al2 O3 , typically 3 nm thick. The desired molecules are doped on top of the Al2 O3 stripe. Finally, a second electrode, also in the shape of a stripe, but crossing the Al stripe, typically Pb, is deposited to form a junction. The width of both electrodes is approximately 0.2 mm. The typical size of the junction is 0.2 mm × 0.2 mm. For reasons described further on, tunneling spectra can only be obtained at liquid helium temperatures (4.2 K and below). One of the first molecules probed was acetic acid (CH3 COOH). A set of strong peaks in d2 I/dV 2 at about 360 mV were observed. It was identified as the excitation of the C-H stretching mode, quantitatively consistent with the vibrational frequency observed by infrared optical spectroscopy. Later, the deuterated acetic acid was also probed. The peaks at 360 mV was shifted at 275 mV, see Fig. 2.8. Because the chemical bonds of both molecules should be identical, so is the spring constant k. Thus, the ratio of the vibrational frequencies should depend only on the ratio of the masses of H and D. Actually, since approximately, 1 f= 2π



k (m + M ) , mM

(2.77)

where m is the mass of H or D, and M is the mass of C, the frequency ratio should be approximately 1.36, which matches well with the observed ratio. The isotope effect showed unambiguously that IETS is probing the vibration modes of molecules. Fig. 2.8 shows IETS of acetic acid (CH3 COOH) and the deuterated version (CD3 COOD), demonstrating the isotope effect. (Adapted from Lambe and Jaklevic [145].) A brief theory of inelastic tunneling is presented as follows:

Fig. 2.7. A schematic of the inelastic electron tunneling spectroscopy experiment. (Adapted with permission from [143]. Copyright 1977 Elsevier.)

66

Tunneling Phenomenon

Fig. 2.8. Observed IETS of acetic acid molecule. The isotope effect is clearly shown. (Reproduced with permission from [145]. Copyright 1968 Americal Physical Society.)

2.3.2

Frequency condition

Consider that the potential energy function of electrode B UB contains that of a vibrating polar molecule, as shown in Fig. 2.7. It is time-dependent U = U0 cos ωt,

(2.78)

where U0 is the amplitude of the dipole field, and ω is the circular vibrational frequency. The tunneling matrix element from that vibrating molecule is then  Mμν = ψμ (U0 cos ωt) χ∗ν d3 r = mμν cos ωt, (2.79) z>z0

where mμν is the amplitude of the tunneling matrix element. Now Eq. 2.13 becomes dcν (t) = mμν cos ωt e−i(Eμ −Eν )t/ . (2.80) dt Using the Euler formula, Eq. 2.80 becomes  mμν  −i(Eμ −Eν +ω)t/ dcν (t) (2.81) = e + e−i(Eμ −Eν −ω)t/ . i dt 2 The two terms can be treated separately. Following the previous steps, Eqs 2.16 through 2.18, we obtain i

pμν (t) ≡ |cν (t) |2 = |mμν |2

sin2 [(Eμ − Eν ± ω) t/2] (Eμ − Eν ± ω)

2

.

(2.82)

2.3 Inelastic tunneling

67

Fig. 2.9. Frequency condition and energy diagram of IETS. (a) The frequency condition. (b) The energy diagram.

For large t, it approaches pμν (t) =

π |mμν |2 δ (Eμ − Eν ± ω). 2

(2.83)

Therefore, in addition to the elastic tunneling term, Eq. 2.20, we have an additional term while the energy level of the electronic states changes by an amount of ω. 2.3.3

Effect of finite temperature

Besides the intrinsic width due to the interaction of the vibration mode with the surface, the width of observed vibrational peaks is influenced by two factors: finite temperature and finite modulation voltage. Because the three factors are independent of each other, the observed width W is related to the intrinsic width WI , the thermal broadening WB , and the modulation broadening WM by 2 W 2 = WI2 + WB2 + WM

(2.84)

We will present the modulation broadening in Subsection 15.4.3. The thermal broadening follows an argument based on Fermi distribution as follows, see Fig. 2.9. Consider an initial state at energy E, and a final state with a vibrational excitation. The occupation probability of the initial-state electrons is given by the Fermi function f1 (E) =

1 ,  E − EF 1 + exp kT

(2.85)

where EF is the Fermi level. With a bias voltage V , the Fermi level of the other electrode is shifted by eV. The electron energy could excite a vibrational state of energy ω. To facilitate such a transition, the final

68

Tunneling Phenomenon

state in the second electrode should be unoccupied. The probability of that condition is 

f2 (E) = 1 − 1 + exp

1 . E − ω − (EF − eV ) kT

(2.86)

Assuming that both electrodes are normal metal, then the tunneling matrix elements are constant over the energy range of interest. Integrating over all the initial electronic states, the tunneling current to the final states with vibrationally excitation is proportional to ∞ I∝

dE f1 (E) f2 (E).

(2.87)

f1 (E) = 1 − η (E − EF ),

(2.88)

f2 (E) = η (E + eV − ω − EF ).

(2.89)

−∞

At T=0, and The integral Eq. 2.87 becomes 

EF f1 (E) f2 (E)dE =

I= EF −eV +ω

0, if eV < ω eV − ω, if eV ≥ ω

 .

(2.90)

By taking the first derivative, the tunneling conductance G = dI/dV is zero for eV < ω and a constant for eV > ω. The second derivative dG/dV = d2 I/dV 2 should have a sharp peak at eV = ω and zero elsewhere. This is the ideal line shape without intrinsic broadening.

Fig. 2.10. Line profile of IETS due to finite temperature. A plot of Eq. 2.92. The I/V curve is also shown. The dashed line represents I/V curve at zero absolute temperature, where the line width vanishes.

2.4 Spin-polarized tunneling

69

For T > 0, the integral Eq. 2.87 can be evaluated analytically. Denoting x=

eV − ω , kT

(2.91)

then taking the second derivative of I with regard to V , we obtain the line profile of a single vibration excitation due to thermal broadening, x d2 I x (x − 2) e + x + 2 ∝ 6e . 3 dV 2 (ex − 1)

(2.92)

The number 6 is added for normalization. Fig. 2.10 shows the line profile due to finite temperature. The line width at half maximum, or at d2 I/dV 2 = 0.5 is δx = 5.4, or Δ (eV − ω) = 5.4 kT. (2.93)

2.4

Spin-polarized tunneling

For tunneling junctions formed by two ferromagnets separated by a insulating barrier (ferromagnet–insulator–ferromagnet, FM–I–FM junction), the tunneling conductance is found to be dependent on the relative orientation of the magnetizations of the ferromagnets. The relative orientation of the magnetization is controlled by an external magnetic field. Usually, the two ferromagnets have different coercive forces. By applying an external magnetic field greater than the coercive force of electrode A, and smaller than that of electrode B, the magnetization of electrode A can be altered relative to that of electrode B. It was demonstrated first by Julliere [146]. A theory based on the square tunneling barrier was developed by Slonczewski [147], who showed that the tunneling conductance should depend on the cosine of the magnetization orientations of the ferromagnets. This phenomenon is based on the spin of the electrons. The spin angular momentum and the magnetic moment of the electron is quantized in space, which can have two eigenstates, often denoted as spin-up and spin-down. For ferromagnets, near the Fermi level, DOS of electrons depends on the state of spin, ρ↑ (EF ) = ρ↓ (EF ).

(2.94)

In STM experiments, if the tip state is spin-polarized, in other words, for a reference direction in the space the spin-up tip DOS and the spin-down tip DOS are different, then the local magnetic properties of the sample can be imaged. Nearly atomic resolution has been achieved. This is the concept of spin-polarized STM. We present the details in Section 7.6. For a review, see Wiesendanger and Bode [148].

70

Tunneling Phenomenon

The next section presents a general theory of spin-polarized tunneling based on the Bardeen theory of tunneling. 2.4.1

General formalism

The Bardeen tunneling theory can be extended to include spin. Instead of using a single-component wavefunction, two components are necessary to describe a state of an electron with spin. For example, in general, a sample state can be represented by a spinor.   ψμ↑ (r) Ψ= (2.95) e−iEμ t/ , ψμ↓ (r) The angular momentum of the electron spin as a dynamic variable is defined through the Pauli matrix (a 2 × 2 matrix is marked with a hat), ˆ = m

 ˆ σ, 2

ˆ are where the components of the Pauli matrix σ       0 1 0 −i 1 0 , σ ˆy = , σ ˆz = . σ ˆx = 1 0 i 0 0 −1

(2.96)

(2.97)

For example, the expectation value of the z-component of the spin angular momentum is   Ψ† σ ˆ z Ψ d3 r , (2.98) mz = 2 ∗ ∗ where Ψ† = (ψμ↑ , ψμ↓ ) eiEμ / . Obviously, if ψμ↓ = 0, mz = +/2, and if ψμ↑ = 0, mz = −/2. Those states are called spin-up and spin-down states, respectively.

Fig. 2.11. The Bardeen theory of spin-polarized tunneling. Two electrodes with spin polarization, A and B, are separated by an insulator or vacuum. The directions of spin polarization of the two electrodes could be different. Taking the spin polarization direction of one of the electrode, say A, as the reference, the expressions for the tunneling matrix elements are simplified.

2.4 Spin-polarized tunneling

The time-dependent Pauli equation of the combined system is

2 2 ˆ ∂Ψ ˆB Ψ. = − ∇ + UA + U i ∂t 2m

71

(2.99)

Note that the wavefunction Ψ is now a two-component spinor, similar to Eq. 2.95; and the potential functions are now 2 × 2 matrices,     UA↑↑ UA↑↓ UB↑↑ UB↑↓ ˆ ˆ , UB = , (2.100) UA = UA↓↑ UA↓↓ UB↓↑ UB↓↓ ˆB is non-zero only in ΩB . ˆA is non-zero only in ΩA , and U where U Bardeen’s treatment for the tunneling problem in Section 2.2 is extended by Reittu in 1997 to include spin [149], see Fig. Therefore, the two components of the wavefunction of electrode A can be treated separately to satisfy the following equations,

2 2 − ∇ + UA↑↑ χν↑ (r) = Eν χν↑ (r), (2.101) 2m

2 2 ∇ + UA↓↓ χν↓ (r) = Eν χν↓ (r). (2.102) − 2m In general, the spinor of electrode B satisfies the Pauli equation,       2 2 ψμ↑ (r) UB↑↑ UB↑↓ ψμ↑ (r) − ∇ + = Eμ . UB↓↑ UB↓↓ ψμ↓ (r) ψμ↓ (r) 2m

(2.103)

However, in the reference frame of electrode A, the state of electrode B is, in general, not diagonized with respect to spin. The evolution of a state, initially as a spin-up tip state, can be written as  Ψ=

χν↑ (r) 0

 e

−iEν t/

 ∞   cν↑μ↑ (t)ψμ↑ (r) + e−iEμ t/ . cν↑μ↓ (t)ψμ↓ (r)

(2.104)

μ=1

Similarly, for a state initially as a spin-down tip state,  Ψ=

0 χν↓ (r)



−iEν t/

e

 ∞   cν↓μ↑ (t)ψμ↑ (r) e−iEμ t/ . + cν↓μ↓ (t)ψμ↓ (r)

(2.105)

μ=1

where the four coefficients cν↑μ↑ (t), etc. are determined by the time-dependent Pauli equation. For example, inserting Eqs 2.104 into Eq. 2.99, to the first order, the differential equation for the coefficient cν↑μ↑ (t) is  d χ∗ν↑ UA↑↑ ψμ↑ (r)d3 r e−i(Eν −Eμ )t/ . (2.106) i cν↑μ↑ (t) = dt ΩA

72

Tunneling Phenomenon

Similar to the spin-averaged case, Eq. 2.14, we define four sets of spindependent tunneling matrix elements  ψμ↑ UA↑↑ χ∗ν↑ d3 r , (2.107) Mμ↑ν↑ = ΩA

and so on. Furthermore, following the steps of Eqs. 2.23 through 2.27, we obtain the surface-integral expressions for those matrix elements, Mμ↑ν↑

2 = 2m





 ψμ↑ ∇χ∗ν↑ − χ∗ν↑ ∇ψμ↑ · dS,

(2.108)

Σ

and so on. 2.4.2

The spin-valve effect

As an example, here we present the problem of tunneling between two ferromagnetic thin films. Experimentally, it is observed that the tunneling conductance depends on the relative orientation of the magnetization density vectors, which is called the spin-valve effect. It originates from an unequal DOS distribution at the Fermi level with respect to spin, or spin polarization. It was first demonstrated by Julli`ere [146]. He also provided a simple two-current model. Later, it was treated by Slonczewski using an exact solution of the one-dimensional rectangular-barrier tunneling problem with spin [147]. Here we present a treatment using Bardeen’s approach, similar to that of Reittu [149], see Fig. 2.12.

Fig. 2.12. The spin-valve effect. Two pieces of ferromagnet, A and B, are separated by an insulator or vacuum of thickness d. The directions of magnetization are different. By taking the direction of the magnetization of A as the reference, or z-axis, the magnetization of B has an angle θ with respect to the reference direction. The tunneling conductance depends on the angle θ.

2.4 Spin-polarized tunneling

73

Inside the magnet, the electron is subject to a molecular magnetic field B, related to the magnetization vector M by B = λM,

(2.109)

where λ is a constant. It results in a spin-dependent term in the potential energy, ˆ = −gμB B · σ ΔU ˆ, (2.110) ˆ repwhere g is the gyromagnetic factor, μB is the Bohr magneton, and σ resents the Pauli matrices. By choosing the direction of the magnetic field in magnet A as the reference direction z, the spin-dependent Hamiltonian in magnet A is diagonized with respect to the spin degree of freedom,   Bz 0 ˆ . (2.111) ΔU = −gμB 0 −Bz For each spatial wavefunction ψ(r), there are two states with spinors:     1 0 ΨA↑ = ψ(r) , ΨA↓ = ψ(r) . (2.112) 0 1 The spin-dependent Hamiltonian in magnet B can also be diagonized, but with a different z-axis. In the local reference frame, the spinors are     1 0 , ΨB↓ = χ(r) . (2.113) ΨB↑ = χ(r) 0 1 In the reference frame of magnet A, the spinors in magnet B can be transformed using the spinor transformation [18]. An arbitrary rotation in threedimensional space can be decomposed into three consecutive elementary rotations using three Euler angles, ψ, θ, and φ, see Fig. 2.13. The three elementary unitary transformation matrices are   iψ/2  0 e iψ ˆ (ψ) = exp σ ˆz = , (2.114) U 2 0 e−iψ/2    cos(θ/2) i sin(θ/2) iθ ˆ U (θ) = exp σ ˆx = , (2.115) 2 i sin(θ/2) cos(θ/2)   iφ/2  0 e iφ ˆ (φ) = exp σ ˆz = . (2.116) U 2 0 e−iφ/2 The complete transformation matrix is ˆ (φ, θ, ψ) = U ˆ (φ)U ˆ (θ)U ˆ (ψ). U

(2.117)

74

Tunneling Phenomenon

Fig. 2.13. The Euler angles. The classical representation of an arbitrary rotation in three-dimensional Euclidean space is through three consecutive elementary rotations, according to Leonhard Euler (1707– 1783). (1) A rotation about the z-axis by an angle φ. The x-axis now becomes ξ  . (2) A rotation about the ξ  axis by θ. The y-axis becomes η  , and the z-axis becomes z  . (3) A rotation about the z  axis by an angle ψ.

The spinors of electrode B, in the reference coordinate system of electrode A, are  ei(φ+ψ)/2 cos(θ/2) ΨB↑ = χ(r) , (2.118) iei(φ−ψ)/2 sin(θ/2)  iei(−φ+ψ)/2 sin(θ/2) ΨB↓ = χ(r) . (2.119) e−i(φ+ψ)/2 cos(θ/2) The tunneling matrix in the coordinate system of magnet A is   iei(φ−ψ)/2 sin(θ/2) ei(φ+ψ)/2 cos(θ/2) M↑↑ M↑↓ = M0 , M↓↑ M↓↓ iei(−φ+ψ)/2 sin(θ/2) e−i(φ+ψ)/2 cos(θ/2) (2.120) where  2 [ψ ∇χ∗ − χ∗ ∇ψ] · dS. (2.121) M0 = 2m Σ Similar to Eq. 2.22, the total tunneling conductance is  G = 2π 2 G0 ρA↑ ρB↑ |M↑↑ |2 + ρA↑ ρB↓ |M↑↓ |2

 +ρA↓ ρB↑ |M↓↑ |2 +ρA↓ ρB↓ |M↓↓ |2 .

(2.122)

Equation 2.122 can be written into a convenient form by introducing the spin-averaged density of states ρA = ρA↑ + ρA↓ ,

ρB = ρB↑ + ρB↓ ,

(2.123)

and the spin-polarized density of states mA = ρA↑ − ρA↓ ,

mB = ρB↑ − ρB↓ .

(2.124)

Substituting Eqs 2.120, 2.123, and 2.124 into Eq. 2.122, using the identities θ θ + sin2 = 1, 2 2

(2.125)

θ θ − sin2 = cos θ, 2 2

(2.126)

cos2 cos2

2.4 Spin-polarized tunneling

75

Fig. 2.14. The ferromagnet–insulator–ferromagnet tunneling junction. The first FM electrode, typically 8–15 nm thick and 0.2 mm wide, is deposited on a glass substrate. An insulating film, typically Al2 O3 or MgO, is then formed by evaporation of Al (or Mg) followed by oxygen glow discharge. The second FM electrode is deposited on top of it. (Reproduced with permission from Moodera and Mathon [151]. Copyright 1999 Elsevier.)

we obtain an expression for the tunneling conductance, G = 2π 2 G0 |M0 |2 ( ρA ρB + mA mB cos θ ).

(2.127)

The first term in the parenthesis is the spin-averaged combined density of states, and the second term is the spin-polarized combined density of states. Alternatively, Eq. 2.127 can be simplified by introducing two dimensionless spin polarization parameters, PA and PB , defined as mA mB PA ≡ , PB ≡ . (2.128) ρA ρB Then the tunneling conductance is G = 2π 2 G0 |M0 |2 ρA ρB (1 + PA PB cos θ).

(2.129)

For processing experimental data, Eq. 2.129 is often written as G = Gsa + Gsp cos θ,

(2.130)

where Gsa is the spin-averaged tunneling conductance, and Gsp is the spinpolarized tunneling conductance. The definitions of those two quantities are obvious by comparing Eq. 2.130 with Eq. 2.129. As we have described in the beginning of this section, the orientation of the magnetization of the ferromagnet with weaker coercive force is controlled by an external magnetic field. Therefore, the ferromagnetic tunneling junction is also a magnetoresistance device. Julliere [146] defined a dimensionless junction magnetoresistance (JMR) as JM R ≡

Gmax − Gmin . Gmax

(2.131)

2PA PB . 1 + P A PB

(2.132)

Using Eq. 2.129, we have JM R =

76

2.4.3

Tunneling Phenomenon

Experimental observations

Equation 2.129 has been verified by numerous experiments using FM–I–FM tunneling junctions. For a review, see for example Moodera and Kinder [152], Moodera and Mathon [151]. A schematic of the geometry of the FM-I-FM junction is shown in Fig. 2.14. A typical fabrication process is as follows. First, a Si film of 1 nm thick is deposited on a glass substrate at liquid nitrogen temperature as the nucleating layer. The first FM electrode, typically 8–15 nm thick and 0.2 mm wide, is deposited, followed by an evaporation of Al (or Mg) of 1.2–2.0 nm. The substrate is then warmed to room temperature and subjected to oxygen glow discharge to create an insulating layer of Ai2 O3 or MgO. Finally, the second FM electrode, typically 8–25 nm thick and 0.3 mm wide, is deposited on top of it. Figure 2.15 shows a typical result of the measurement of the angular dependence of tunneling resistance, using a CoFe/Ai2 O3 /Co FM–I–FM junction. A high field is first applied to align the magnetization of the two electrodes. Then, the magnitude of the magnetic field is reduced to below the coercive force of CoFe but greater than that of Co. The tunneling resistance is recorded every 5◦ to 10◦ as the sample is rotated with respect to the magnetic field. A clear angular dependence of the tunneling conductance with respect to the applied magnetic field is observed. The cosine relation, first predicted by Slonczewski [147], is well verified.

Fig. 2.15. Angular dependence of tunneling resistance on the orientation of the applied magnetic field. (Reproduced with permission from Moodera and Kinder [152]. Copyright 1996 American Institute of Physics.)

Chapter 3 Tunneling Matrix Elements 3.1

Introduction

Chapter 2 presented Bardeen’s tunneling theory as the foundation of understanding of the essential physics of STM. A central issue of the application of Bardeen’s tunneling theory is the evaluation of the tunneling matrix elements, which is a surface integral of the wavefunctions of the tip and the sample on a separation surface, as shown in Fig. 3.1: M = −

2 2m



(χ∗ ∇ ψ − ψ ∇χ∗ ) · dS,

(3.1)

Σ

where ψ is a wavefunction of the sample, and χ is a wavefunction of the tip. As shown in Eq. 3.1, only the values of these wavefunctions at the separation surface, which is located roughly in the middle of the gap, are relevant. Certain corrections to the free wavefunctions will improve the accuracy. As we showed in Chapter 2, for the case of a flat surface and a single atom, the correction due to the image force is essentially a constant multiplier. The matrix element has the dimension of energy. In Chapters 4 and 5, we show that the physical meaning of Bardeen’s matrix element is the interaction energy due to the overlap of the two unperturbed states. In other words, it is the chemical bond energy first introduced by Pauling [153], the dominant component of the force between the tip and the sample under normal STM operating conditions. Therefore, an explicit form for the tunneling matrix elements described by Eq. 3.1 is important in the quantitative understanding of STM and AFM. The problem of evaluating the tunneling matrix elements with respect to STM has been investigated by many authors (Tersoff and Hamann [115, 116]; Baratoff [91]; Chung, Feuchtwang, and Cutler [154]; Chen [13, 14, 15]; Lawunmi and Payne [155]; Sacks and Noguera [156, 157]; and Sacks [158]). All those authors used a spherical-harmonic expansion to represent tip wavefunctions in the gap region, which is a natural choice. The sphericalharmonic expansion is used extensively in solid-state physics as well as in quantum chemistry for describing and classifying electronic states [159, 140]. In problems without a magnetic field, the real spherical harmonics are preferred [159], as described in Appendix B. *OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

78

Tunneling Matrix Elements

Fig. 3.1. Derivation of tunneling matrix elements. A separation surface is placed between the tip and the sample. The exact position and the shape of the separation surface is not important. The coordinates for the Cartesian coordinate system and spherical coordinate system are shown, except y and φ. (Reproduced with permission from [15]. Copyright 1990 American Physical Society.)

In this chapter, we present step-by-step derivations of the explicit expressions for matrix elements based on the spherical-harmonic expansion of the tip wavefunction in the gap region. The result, the derivative rule, is simple and intuitively understandable. Two independent proofs are presented. The mathematical tool for the derivation is the spherical modified Bessel functions, which are actually elementary functions, and probably the simplest of all Bessel functions; see Appendix C. A derivation based on parabolic coordinate system is also briefly summarized.

3.2 3.2.1

Tip wavefunctions General form

From Eq. 3.1, the tunneling matrix elements are determined by the wavefunctions of the tip and the sample at the separation surface, roughly in the middle of the vacuum gap, as shown in Fig. 3.1. For both tip states and sample states near the Fermi level, the wavefunctions on and beyond the separation surface satisfy Schr¨ odinger’s equation in the vacuum, (∇2 − κ2 )χ(r) = 0,

(3.2)

where κ = (2mφ)1/2 −1 is the decay constant, determined by the work function φ. Near the center of the apex atom, Eq. 3.2 is not valid. Nevertheless,

3.2 Tip wavefunctions

79

for calculating tunneling matrix elements, we only need the wavefunctions near the separation surface, which satisfy Eq. 3.2. The tip wavefunctions can be expanded into the spherical-harmonic components, Ylm (θ, φ), with the nucleus of the apex atom as the origin. Each component is characterized by quantum numbers l and m. In other words, we are looking for solutions of Eq. 3.2 in the form  Clm flm (κρ) Ylm (θ, φ), (3.3) χ(r) = l,m

where ρ ≡ |r − r0 | =

 (x − x0 )2 + (y − y0 )2 + (z − z0 )2 ,

(3.4)

and r0 ≡ (x0 , y0 , z0 ) is the center of the apex atom. Substituting Eq. 3.3 into Eq. 3.2, we obtain the differential equation for flm (u),     d 2 df (u) u − u2 + l(l + 1) f (u) = 0. (3.5) du du As shown, the radial functions depend only on l. The standard linear-independent solutions for Eq. 3.5 are the spherical modified Bessel functions of the first kind il (u), and the second kind kl (u) [140]. A brief introduction to them is provided in Appendix C. Those socalled special functions are actually elementary functions:   sinh u d l , (3.6) il (u) = u u du u 

and l

kl (u) = (−1) u

l

d u du

l

e−u . u

(3.7)

The functions il (u) diverge at large u, not appropriate to represent tip wavefunctions. The functions kl (u) are regular at large u, which satisfies the desired boundary condition. The first three are as follows: 1 −u e , u   1 1 + e−u , k1 (u) = u u2   1 3 3 + 2 + 3 e−u . k2 (u) = u u u k0 (u) =

(3.8)

(3.9)

(3.10)

Therefore, a component of tip wavefunction with quantum numbers l and m has the general form χlm (r) = Clm kl (κρ) Ylm (θ, φ).

(3.11)

80

Tunneling Matrix Elements

Table 3.1: Tip wavefunctions State

Wavefunction

Formula

s

χs

Cs k0 (κρ)

pz

χpz

Cpz k1 (κρ)

px

χpx

py

χpy

dz 2

χz 2

dxz

χxz

Cxz k2 (κρ)

(x − x0 )(z − z0 ) ρ2

dyz

χyz

Cyz k2 (κρ)

(y − y0 )(z − z0 ) ρ2

dxy

χxy

Cxy k2 (κρ)

(x − x0 )(y − y0 ) ρ2

dx2 −y2

χx2 −y2

Cx2 −y2 k2 (κρ)

z − z0 ρ x − x0 Cpx k1 (κρ) ρ y − y0 Cpy k1 (κρ) ρ

(z − z0 )2 1 2 Cz k2 (κρ) − ρ2 3

(x − x0 )2 − (y − y0 )2 ρ2

According to the Wigner theorem, if the Hamiltonian is time-reversal invariant, all wavefunctions can be chosen real, see Appendix B. For simplicity, we use real tip wavefunctions and sample wavefunctions. Using the relation between spherical coordinates and Cartesian coordinates, x − x0 = ρ sin θ cos φ, y − y0 = ρ sin θ sin φ, z − z0 = ρ cos θ,

(3.12)

the spherical harmonics can be written in terms of x, y, and z. Because the tip wavefunctions are needed only around the separation surface, the normalization factors in the original definitions of spherical harmonics can be discarded. The coefficients in tip wavefunctions are defined in a natural way, which can be obtained by comparing with the results of first-principle −3/2 ]. computations, see Table 3.1. The dimension of the wavefunction is [L −3/2 ]. Therefore, the dimension of the coefficients is also [L

3.2 Tip wavefunctions

3.2.2

81

Tip wavefunctions as Green’s functions

The Green’s function for the Schr¨ odinger equation in vacuum, Eq. 3.2, is defined by the differential equation [140]  ∇2 − κ2 G(r − r0 ) = −δ(r − r0 ),



(3.13)

with the boundary condition that it is regular at |r − r0 | → ∞, the explicit form of the Green’s function is (see Appendix A): G(r − r0 ) =

exp(−κ|r − r0 |) , 4π|r − r0 |

(3.14)

which can be verified by direct substitution. Denoting ρ = |r − r0 |, the Green’s function can be written in terms of a spherical modified Bessel function of the second kind, G(r − r0 ) =

κ k0 (κρ). 4π

(3.15)

As shown in Table 3.2, the s-wave tip wavefunction equals the Green’s function up to a constant, with the center of the apex atom taken as r0 , χs (r) =

4πCs G(r − r0 ). κ

(3.16)

By taking the derivative of both sides of Eq. 3.15 with respect to z0 , and using the relation (see Appendix C): d k0 (u) = −k1 (u). du

(3.17)

κ z − z0 ∂ k1 (κρ). G(r − r0 ) = κ∂z0 4π ρ

(3.18)

we obtain

By comparing with Table 3.1, the tip wavefunction for the pz state is χpz (r) =

4πCpz ∂ G(r − r0 ). κ κ∂z0

(3.19)

χpx (r) =

4πCpx ∂ G(r − r0 ), κ κ∂x0

(3.20)

χpy (r) =

4πCpy ∂ G(r − r0 ). κ κ∂y0

(3.21)

Similarly, we have

82

Tunneling Matrix Elements

Table 3.2: Tip wavefunctions and Green’s functions State

Wavefunction

Formula

s

χs

pz

χpz

px

χpx

py

χpy

dz 2

χz 2

dxz

χxz

4πCxz ∂2 G(r − r0 ) κ κ2 ∂x0 ∂z0

dyz

χyz

4πCyz ∂2 G(r − r0 ) κ κ2 ∂y0 ∂z0

dxy

χxy

dx2 −y2

χx2 −y2

4πCs G(r − r0 ) κ 4πCpz ∂ G(r − r0 ) κ κ∂z0 4πCpx ∂ G(r − r0 ) κ κ∂x0 4πCpy ∂ G(r − r0 ) κ κ∂y0 2

∂ 4πCz2 1 G(r − r G(r − r ) − ) 0 0 κ κ2 ∂z02 3

4πCxy ∂2 G(r − r0 ) 2 κ κ ∂x0 ∂y0

4πCx2 −y2 ∂2 ∂2 − 2 2 G(r − r0 ) κ κ2 ∂x20 κ ∂y0

By taking derivative with respect to x0 on Eq. 3.18, and noticing that   d k1 (u) k2 (u) =− , (3.22) du u u we obtain ∂2 κ x − x0 z − z 0 k2 (κρ). G(r − r0 ) = 2 κ ∂x0 ∂z0 4π ρ ρ

(3.23)

By comparing with Table 3.1, we obtain χxz (r) =

∂2 4πCxz G(r − r0 ). 2 κ κ ∂x0 ∂z0

(3.24)

Similar results can be obtained for the wavefunctions of dyz and dxy states. By taking the derivative with respect to z0 on both sides of Eq. 3.18, an extra term containing k1 (κρ)/ρ is generated:

∂2 κ (z − z0 )2 1 k G(r − r ) = k (κρ) − (κρ) . (3.25) 0 2 1 κ2 ∂z02 4π ρ2 ρ

3.3 The derivative rule: individual cases

83

Remembering that (see Appendix C) 3k1 (u) = k0 (u) − k2 (u), u and combining with Eq. 3.16, we obtain 2

∂ 4πCz2 1 χz2 (r) = G(r − r G(r − r ) − ) 0 0 . κ κ2 ∂z02 3

(3.26)

(3.27)

For the wavefunction of the dx2 −y2 tip state, the extra term generated by taking the second derivative cancels. Therefore,

4πCx2 −y2 ∂2 ∂2 − (3.28) G(r − r0 ). χx2 −y2 (r) = κ κ2 ∂x20 κ2 ∂y02 A list of tip wavefunctions in terms of Green’s functions is shown in Table 3.2. We will use these formulas to evaluate tunneling matrix elements.

3.3 3.3.1

The derivative rule: individual cases s-wave tip state

As shown in Eq. 3.1, the tunneling matrix elements are determined by the Bardeen integral of the tip wavefunctions and the sample wavefunctions on a separation surface. For an s-wave tip state, using Eq. 3.16,  2 (χs (r)∇ψ(r) − ψ(r)∇χs (r)) · dS Ms = 2m Σ (3.29)  2π2 Cs [G(r − r0 )∇ψ(r) − ψ(r)∇G(r − r0 )] · dS. = κm Σ Using Green’s theorem, it can be converted into a volume integral in ΩT , the tip side of the separation surface. In ΩT ,, the sample wavefunction ψ(r) satisfies Schr¨ odinger’s equation, Eq. 3.2. The Green’s function satisfies Eq. 3.13. Therefore,    2π2 Cs G(r − r0 )∇2 ψ(r) − ψ(r)∇2 G(r − r0 ) d3 r Ms = κm ΩT  2 2π Cs (3.30) = ψ(r)δ(r − r0 ) d3 r κm ΩT 2π2 Cs ψ(r0 ), κm which was derived first by Tersoff and Hamann [115, 116] using the twodimensional Fourier transform method, see Section 6.2. As seen, the simplicity of this result is because the s-wave tip wavefunction is the Green’s function of the Schr¨odinger equation in vacuum. =

84

Tunneling Matrix Elements

p-wave tip states

3.3.2

Taking the derivative with respect to z0 on both sides of Eq. 3.29, and noticing that z0 is a parameter in the integral (not involving the integration process), and the expression of the pz tip wavefunction, Eq. 3.19, we find 2π2 Cpz ∂ψ(r0 ) κm ∂z0  2 2π Cpz ∂ = [G(r − r0 )∇ψ(r) − ψ(r)∇G(r − r0 )] · dS κm ∂z0 Σ

 ∂G ∂G 2π2 Cpz ∇ψ(r) − ψ(r)∇ · dS = κm ∂z0 Σ ∂z0  2 [χpz (r)∇ψ(r) − ψ(r)∇χz (r)] · dS = 2m Σ = Mz .

(3.31)

The tunneling matrix element for a pz tip state is proportional to the z derivative of the sample wavefunction at the center of the apex atom. The matrix elements for px and py tip states are similar. d -wave tip states

3.3.3

The tunneling matrix elements from the d-wave tip states can also be derived using the relation between the tip and Green’s functions established in the previous section. For example, for the dxz tip state, 2π2 Cxz ∂ 2 ψ(r0 ) κm κ2 ∂x0 ∂z0

 ∂2 2π2 Cxz [G(r − r0 )∇ψ(r) − ψ(r)∇G(r − r0 )] · dS κm κ2 ∂x0 ∂z0 Σ

 ∂2G ∂2G 2π2 Cxz · dS ∇ψ(r) − ψ(r)∇ 2 = 2 κm κ ∂x0 ∂z0 Σ κ ∂x0 ∂z0  2 [χxz (r)∇ψ(r) − ψ(r)∇χxz (r)] · dS = 2m Σ =

= Mxz . (3.32) The derivation of the tunneling matrix elements for other d-wave tip states follows that of the dxz tip state. For dz2 and dx2 −y2 states, the expressions of the tip wavefunctions have two terms. The same procedure can be applied. Table 3.3 lists the results.

3.4 The derivative rule: general case

85

Table 3.3: Tunneling matrix elements Tip state

Matrix element 2π2 Cs ψ(r0 ) κm 2π2 Cpz ∂ψ (r0 ) κm κ∂z 2π2 Cpx ∂ψ (r0 ) κm κ∂x 2π2 Cpy ∂ψ (r0 ) κm κ∂y

s pz px py

2π2 Cxz ∂ 2 ψ (r0 ) κm κ2 ∂z∂x 2π2 Cyz ∂ 2 ψ (r0 ) κm κ2 ∂z∂y

dxz dyz dxy dz 2 dx2 −y2

2π2 Cxy ∂ 2 ψ (r0 ) κm κ2 ∂x∂y

∂2ψ 2π2 Cz2 1 − ψ (r0 ) κm κ2 ∂z 2 3

2 2 2π Cx2 −y2 ∂ ψ ∂2ψ (r0 ) − κm κ2 ∂x2 κ2 ∂y 2

The results can be summarized as a derivative rule: Write the angle dependence of the tip wavefunction in terms of x, y, and z. Replace them with the simple rule, x→

∂ ; κ∂x

y→

∂ ; κ∂y

z→

∂ ; κ∂z

(3.33)

and act on the sample wavefunction. The term r2 corresponds to ∇2 . The Schr¨ odinger equation (Eq. 3.2) implies ∇2 ψ = κ2 ψ. Therefore, it generates a constant term. A list of tunneling matrix elements is in Table 3.3.

3.4

The derivative rule: general case

In this section, we present an alternative proof of the derivative rule, which provides an expression for the transmission matrix element from an arbitrary tip state expanded in terms of spherical harmonics. In the previous

86

Tunneling Matrix Elements

sections, we have expanded the tip wavefunction on the separation surface in terms of spherical harmonics. In general, the expansion is χ=



βlm kl (κρ) Ylm (θ, φ).

(3.34)

l,m

The coefficients βlm are determined by fitting the tip wavefunction on and beyond the separation surface. Inside the tip body, the actual tip wavefunction does not satisfy Eq. 3.2, and the expansion (Eq. 3.34) does not represent the actual tip wavefunction. Instead, it represents the vacuum continuation of the vacuum tail of the tip wavefunction into the tip body. Actually, except for the center of the apex atom, that is, the origin of the spherical harmonics, the expansion Eq. 3.34 satisfies the Schr¨odinger equation in the vacuum, Eq. 3.2. On the other hand, the sample wavefunction in the entire volume of the tip body should satisfy the Schr¨odinger equation, Eq. 3.2. Especially, it should be regular at the origin. Therefore, in the tip body, the sample wavefunction must have the form ψ=



αlm il (κρ) Ylm (θ, φ),

(3.35)

l,m

where the coefficients αlm are determined by the sample wavefunction in the vacuum.

Fig. 3.2. Derivation of the derivative rule: general case. The expansion in Eq. 3.34 satisfies the Schr¨ odinger equation in the vacuum except near the nucleus of the apex atom. The surface on which the Bardeen integral is evaluated is a surface that encloses the nucleus of the apex atom. (Reproduced with permission from [15]. Copyright 1990 American Physical Society.)

3.4 The derivative rule: general case

87

To proceed with the proof, we first state a property of the Bardeen integral: if both functions involved, ψ and χ, satisfy the same Schr¨ odinger equation with the same energy eigenvalue in a region Ω, then the Bardeen integral J on the surface enclosing that volume Ω vanishes. Actually, using Green’s theorem, Eq. 3.1 becomes   2 ∗ 2 2 ∗ (χ ∇ ψ − ψ∇ χ ) dΩ = (χ∗ T ψ − ψT χ∗ ) dΩ. (3.36) J =− 2m Ω Ω Because ψ and χ satisfy the same Schr¨odinger equation with the same energy eigenvalue, Eq. 3.36 becomes  (3.37) J = (χ∗ U ψ − ψU χ∗ ) dΩ = 0. Ω

As we have shown, except at the nucleus of the apex atom, or the origin of the spherical harmonics, the expansion form of the tip wavefunction χ, Eq. 3.34, and the sample wavefunction ψ satisfy the same Schr¨odinger equation, Eq. 3.13. Therefore, we can take any surface enclosing the nucleus of the apex atom to evaluate the transmission matrix element, Eq. 3.1, especially, a sphere of arbitrary radius r0 centered at the nucleus of the apex atom. Substituting Eq. 3.34 into Eq. 3.1, 2   αlm βl∗ m 2m l,m l m

dil (κr) dkl (κr) kl (κr) − il (κr) × dr dr  × r2 sin θ dθ dφ Ylm (θ, φ)Yl∗ m (θ, φ).

M =

(3.38)

The integral in Eq. 3.38 is δll δmm due to the orthogonality property of the spherical harmonics. The second line is the Wronskian of the spherical modified Bessel functions, 1 . (3.39) u2 The r−2 factor in the Wronskian cancels the r2 factor of the surface integral. As expected, the value of the surface integral is independent of the actual location of the surface. Therefore, we obtain a general expression for the tunneling matrix elements, the sum rule, il (u) kl (u) − il (u) kl (u) = −

M=

2  ∗ αlm βlm . 2mκ

(3.40)

l,m

The steps from Eq. 3.38 to Eq. 3.40 simply mean that for each component of the tip wavefunction with angular dependence characterized by l

88

Tunneling Matrix Elements

and m, the tunneling matrix element is proportional to the corresponding component of the sample wavefunction with the same angular dependence. In the following, we show that the coefficients αlm in Eq. 3.35 are related to the derivatives of the sample wavefunction ψ with respect to x, y, and z at the nucleus of the apex atom in an extremely simple way. (To simplify the notation, we take the nucleus of the apex atom as the origin of the coordinate system, i.e., x0 = 0, y0 = 0, and z0 = 0.) This is similar to the well-known case that the expansion coefficients for a power series are simply related to the derivatives of the function at the point of expansion, the so-called Taylor series or MacLaurin series. We will then obtain the derivative rule again, from a completely different point of view. The key of the proof is the properties of the spherical modified Bessel function of the first kind, il (u). For small values of u, the function il (u) has the following form: il (u) =

  ul 1 + O(u2 ) . (2l + 1)!!

(3.41)

By multiplying il (κρ) with a spherical harmonic of order l, this term becomes a homogeneous polynomial of x, y, and z of order l. By taking a partial derivative of il (κr) with respect to x, y, and z with (summed) order n, and taking the value at r=0, all the terms with powers of x, y, and z of l > n drop off. In particular, for the cases of l=0 and 1, there is only one term left in the derivative at ρ=0, that is, a term containing only one coefficient in Eq. 3.40. For l=2, the derivative may contain the second term in i0 (κρ), which should be subtracted off to obtain the coefficient for an l=2 component. We start our derivation by writing down the explicit form of the vacuum asymptote of a tip wavefunction (as well as its vacuum continuation in the tip body). As we explain later in Section 5.3, for the simplicity of relevant mathematics, the rather complicated normalization constants of the spherical harmonics are absorbed in the expression of the sample wavefunction. Up to l=2, we define the coefficients of the expansion by the following expression:  χ = β00 k0 (κρ) + β10

z x y + β11 + β12 ρ ρ ρ

  2 z xz yz 1 + β21 2 + β22 2 − + β20 ρ2 3 ρ ρ  2 

2 x xy y +β23 2 + β24 − 2 k2 (κρ). 2 ρ ρ ρ

 k1 (κρ) (3.42)

3.4 The derivative rule: general case

89

Similarly, for the sample wavefunction, up to the lowest significant term in the power expansion of the spherical modified Bessel function of the first kind, il (κρ),

1 2 4πψ = α00 1 + (κρ) + α10 κz + α11 κx + α12 κy 6   3κ2 1 2 2 +α20 z − ρ + α21 (κ2 xz) + α22 (κ2 zy) 4 3   +α23 (κ2 xy) + α24 κ2 (x2 − y 2 ) + O(ρ3 ).

(3.43)

The factor 4π is introduced for convenience. Now, it is straightforward to obtain a relation between the coefficients αlm and the derivatives of sample wavefunctions. For example, because of ρ2 = x2 + y 2 + z 2 , we have 4π ∂ 2 1 ψ(r0 ) = α20 + α00 . κ2 ∂z 2 3

(3.44)

4πψ(r0 ) = α00 ,

(3.45)

Noticing that

we obtain  α20 = 4π

1 ∂2 1 − κ2 ∂z 2 3

 ψ(r0 ).

(3.46)

The derivations for other components are straightforward. Therefore, the tunneling matrix element for an arbitrary tip state, up to l = 2, is ∂ψ ∂ψ ∂ψ 2π2 β00 ψ + +β10 + β11 + β12 M = mκ κ∂z κ∂x κ∂y   2 ∂ ψ ∂2ψ ∂2ψ 1 ψ + β + β − +β20 21 22 κ2 ∂z 2 3 κ2 ∂x∂z κ2 ∂y∂z  2 

∂ ψ ∂2ψ ∂2ψ + β23 2 + β24 − . κ ∂x∂y κ2 ∂x2 κ2 ∂y 2

(3.47)

The values of the derivatives are taken at r0 . Again, we obtain the derivative rule from a completely different point of view. The second derivation gives a general formula to calculate the tunneling matrix element for an arbitrary tip wavefunction, with its vacuum tail expanded in terms of spherical harmonics.

90

Tunneling Matrix Elements

3.5

Tips with axial symmetry

The spherical coordinates work for atomic states, when a wavefunction can be expanded into spherical-harmonics components. A more realistic description of a tip is with an axial symmetry. The parabolic coordinate system (ξ, η, φ) provides a better description [13, 15]. A well-known application of the parabolic coordinate system is the treatment of Stark effect of a hydrogen atom in a 1926 papers of Schr¨odinger [160]. It is defined as √ x = ξη cos φ √ y = ξη sin φ (3.48) ξ−η . z= 2 The radius is defined as

ξ+η . 2 The Schr¨odinger equation in vacuum is    

∂ 4 ∂ψ ∂ ∂ψ 1 ∂2ψ ξ + η + = κ2 ψ. ξ + η ∂ξ ∂ξ ∂η ∂η ξη ∂φ2 r=

(3.49)

(3.50)

With the substitution ψ = e−κ(ξ+η)/2 (ξη)|m|/2 f (η) g(ξ)eimφ ,

(3.51)

one finds that both f (η) and g(ξ) satisfy the confluent hypergeometric equation, or Kummar’s equation. The solutions of the Kummar’s equation is well known. In analogy to the solutions in spherical coordinate system, the tip wavefunctions should be regular at large distances, and the sample wavefunction should be regular near the center [13, 15]. The tip states

Table 3.4: Tip states and tunneling matrix elements Symmetry

Tip state

Tunneling matrix element

σ

e−κr 2r

ψ(r0 )

πx

e−κr x 4r2

∂ψ(r0 ) ∂x

πy

e−κr y 4r2

∂ψ(r0 ) ∂y

3.5 Tips with axial symmetry

91

and sample states are classified with the irreducible representations of the point group C∞v , the symmetry group for a linear molecule. The tunneling matrix elements can also be reduced to analytic forms. For lateral dimensions, x and y, the derivative rule also applies. For several important cases, the asymptotic expressions of tip wavefunctions and the tunneling matrix elements are listed in Table 3.4. 3.5.1

Lateral effects of tip states

In both spherical coordinate system and the parabolic coordinate system, the lateral components of the tip wavefunction will make STM images inverted. Here we show the effect of the px and py tip wavefunctions in the spherical coordinate system, as well as the πx and πy tip wavefunctions in the parabolic coordinate system, see Fig. 3.3. Shown in Fig. 3.3(A) through (C) is a case that the sample is a metal atom which has a σ state. In (A), if the tip is a metal atom adsorbed on a metal surface, the tip has a σ tip state. The image is a single protrusion. If the tip state is πx , see (B), the image splits into a pair of oblong-shaped protrusions in the x-direction. And as shown in (C), if the tip state is πy ,

Fig. 3.3. Lateral effects of tip states. (A) With a σ-type tip, a metal atom on the sample, represented by a σ-type wavefunction, the image is a round bump. (B) With a πx -type tip, the image of a metal atom would look like two oblong peaks on both sides in the x-direction. (C) With a πy -type tip, the image of a metal atom would look like two oblong peaks on both sides in the y-direction. If the tip has both πx -type and πy -type wavefunctions, the two oblong bumps would merge and become a donut-shaped image. (E), with a a πy -type tip and a a πy -type wavefunction on the sample, the image would appear as a single round bump. Note that because the Bardeen tunneling theory os symmetric to the tip and the sample, of the tip state and the sample state swap place, the image would be identical.

92

Tunneling Matrix Elements

the pair of oblong-shaped protrusions aline to y-direction. Consequently, if the tip has both πx state and πy state, the four oblong-shaped protrusions will coalesce into a donut-shaped feature. On the other hand, if both the tip state and the sample state are πx type, as in (E), or both are πy -type, as in (I), the image is a single protrusion. Such images are observed on systems with Cu atoms and CO molecules. As discussed in Chapter 2, a direct consequence of the Bardeen tunneling theory (or the extension of it) is the reciprocity principle: if the electronic state of the tip and the sample state under observation are interchanged, the image should be the same. An alternative wording of the same principle is that an image of microscopic scale may be interpreted either by probing the sample state with a tip state or by probing the tip state with a sample state. The validity of this criterion, as applied to the derivative rule, can be tested by direct calculation. Actually, if one of the two states is an s state, the reciprocity principle is a direct consequence of the Green’s function representation of the other tip states. Also as shown in Fig. 3.3, for the explanation of the lateral effect of tip states, the classification of the tip wavefunction and the sample wavefunction can be based on axial symmetry: σ, πx , or πy ; as well as based on spherical symmetry: s, px , or py .

Chapter 4 Atomic Forces In a widely cited Physical Review Letter [31], Binnig, Quate, and Gerber defined atomic force as the focus of their novel microscope. Their choice was not incidental. Force is a universal phenomenon in Nature, including gravitational force, electromagnetic force, elastic force, frictional force, hydraulic force, pneumatic force, and nuclear forces of weak interactions and strong interactions, etc. Although the technique of AFM has been used for other types of forces, the focus was, and still is, on atomic forces. In this chapter, we provide an overview of the various types of atomic force, including van der Waals force, Pauli repulsion, the ionic bond, and the covalent bond, also known as the chemical bond.

4.1

Van der Waals force

The name van der Waals force originates from the van der Waals equation of state for gases and liquids. The van der Waals force is universal between atoms, molecules, and solids. London [161] has shown that the van der Waals force is a general consequence of the zero-point energy in quantum mechanics, following Heisenberg’s uncertainty principle. 4.1.1

The van der Waals equation of state

In 1873, Dutch physicist Johannes Diderik van der Waals proposed an equation of state for gases,  P+

a  (V − b) = RT. V2

(4.1)

Here, P is the pressure, V is the volume of one mole of the gas, R is the gas constant, and T is the absolute temperature. The constant a represents the attractive force between a pair of molecules, and b represents the finite volume of a molecule. Equation 4.1 played a vital role in understanding the transition between the gas phase and the liquid phase, and guided the experiments of liquefaction of gases. After Heike Kamerlingh Onnes, another Dutch physicist, successfully liquefied helium in 1908, van der Waals was awarded the 1910 Nobel Prize in physics. *OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

94

Atomic Forces

The origin of the constant b is related to the total volume of the gas molecules. Take an example of spherical molecules. Suppose that any two molecules cannot approach each other closer than a distance re , the impenetrable volume of a single molecules is (π/6)re3 . However, the actual volume of N molecules depends on how those molecules are packed. Here are three common cases: √ fcc : b = (1/ 2)N re3 ∼ = 0.707N re3 , √ (4.2) bcc : b = (4/3 3)N re3 ∼ = 0.770N re3 , :

sc

b = N re3 ,

where N is Avegadro’s constant; fcc stands for face-centered cubic, bcc stands for body-centered cubic, and sc stands for simple cubic lattices, respectively [137]. The origin of constant a can be understood by the following example. We are assuming that, when the intermolecular distance r (from center to center) is greater than re , there is a weak attractive interaction energy following a power law, −C/ren . Here, C is the force constant, and n is a number greater than 3. The total interaction energy in a mole of gas can be estimated as follows. The number density of gas molecules is N/V . The number of gas molecules between r and r + dr is 4π(N/V ) r2 dr. The interaction energy of all molecules in the volume with a single molecule is  =−

∞ re

4πN Cr2 1 4πN C dr = − . rn (n − 3)V ren−3

(4.3)

The condition n > 3 allows us to replace the upper limit of the integral by ∞. Otherwise the integral is divergent, indicating that no gas phase could exist. The total interaction energy is E = N = −

4πN 2 C 1 . (n − 3)V ren−3

(4.4)

Because the partial pressure generated by the interaction energy is p = E/V , the constant a is then a = −pV 2 =

4πN 2 C 1 . (n − 3) ren−3

(4.5)

The constants in the van der Waals equation have been accurately measured for many gases. For inert gases, the measured values agrees well with theoretical expectations. 4.1.2

The origin of van der Waals force

The origin of van der Waals force in terms of quantum mechanics can be illustrated by the example of two hydrogen atoms [162]. The method has

4.1 Van der Waals force

95

Fig. 4.1. Quantum mechanics of van der Waals force. Two atoms are separated by a relatively large distance R. The overlap of wavefunctions can be neglected. Using second-order perturbation theory, a weak attractive force proportional to R−6 is predicted.

been also applied to all inert gas atoms, and achieved high accuracy. Figure 4.1 shows a schematic. As shown, the Hamiltonian of the system is H = H1 (r1 ) + H2 (r2 ) + W (R, r1 , r2 ).

(4.6)

Here the solutions of H1 (r1 ) and H2 (r2 ) are known, H1 (r1 )ψn (r1 ) = En ψn (r1 ),

(4.7)

H2 (r2 )ψm (r2 ) = Em ψm (r2 ). The perturbation potential W is a dipole-dipole interaction. Assigning the line connecting the two atoms as the z axis, it is e2 W ∼ (4.8) = 3 [x1 x2 + y1 y2 − 2z1 z2 ] . R For the interaction between two atoms in the ground state, it is convenient to denote the following perturbation matrix elements, which are independent of R:  wnm =

ψ0∗ (r1 )ψ0∗ (r2 ) [x1 x2 + y1 y2 − 2z1 z2 ] ψn (r1 )ψm (r2 )d3 r1 d3 r2 . (4.9)

Because the first-order perturbation term is zero, the interaction energy is determined by the second-order perturbation term, (R) =

e4 R6



|wnm |2 , 2E0 − En − Em n>0,m>0

(4.10)

since for n > 0, En > E0 , the interaction energy is always negative. The functional dependence implies that the force is always attractive. Using the wavefunctions of the hydrogen atom, the above expression predicts an accurate value of the van der Waals interaction energy. Applications to inert gas

96

Atomic Forces

atoms and small molecules also produces satisfactory results. For reactive atoms, van der Waals interaction does not provide a satisfactory description of the interatomic forces: the covalent bond interaction dominates. For hydrogen atoms, the matrix elements can be evaluated explicitely, and the van der Waals interaction potential between two hydrogen atoms is (R) = −

e2 r5 C = −6.47 6B . 6 R R

(4.11)

Here rB = 0.0529 nm is the Bohr radius. There are several general features of the van der Waals force which are noticeable from the above derivation. First, the force is long-range in nature. In comparison, the covalent-bond interaction has an exponential dependence with distance, and the repulsive force has an even sharper distance dependence. The minus sixth power dependence on distance is universal. Second, it is mostly isotropic, or non-directional. The covalent-bond interaction, with its strong directionality, is markedly different. Third, the van der Waals force is additive; between two groups of atoms be evaluated as the sum of the forces from each atom pair in the two groups. 4.1.3

Van der Waals force between a tip and a sample

The minus sixth power dependence of the van der Waals force is valid for the interaction between two individual atoms or molecules. Because of its long-range nature, in STM and AFM, not only the atoms in the immediate vicinity of the tip apex, but also a sizeable volume of atoms in the tip body and the sample body contributes to the observed total van der Waals interaction. Therefore, in STM and AFM, the minus sixth power dependence of the van der Waals force between two isolated atoms is not observable. The measurable interaction is the total van der Waals interaction between the tip and the sample. Because the van der Waals interaction is additive, the total interaction energy can be obtained by integration. The van der Waals interaction energy between an elementary volume of dv of the tip and an elementary volume du of the sample can be written as dU = −

C ρ1 ρ 2 du dv, R6

(4.12)

where ρ1 and ρ2 are the density of atoms of the tip and the sample, respectively. The sample can be described as a uniform semi-infinite continuum, see Fig. 4.2(a). Integrating over the volume of the sample gives

4.1 Van der Waals force

97

Fig. 4.2. Van der Waals force between a paraboloidal tip and a flat sample. (a) The van der Waals force between an elementary volume and a semi-infinite sample (b) The total force, by describing the tip as a paraboloid with local radius Rc .

 dU = −Cρ1 ρ2 dv



∞

∞

dζ ζ=0

ρ=0

2πρdρ [ρ2

+ (z + ζ)2 ]

3

(4.13) πCρ1 ρ2 =− dv. 6z 3 In most cases, the tip can be described adequately as a paraboloid with a local radius of curvature Rc near the apex. Since the van der Waals interaction of an elementary volume only depends on its distance to the sample surface, the total interaction can be calculated slice by slice. The volume of a thin slice of the tip is (see Fig. 4.2(b)), dv = πx2 dz = 2πRc zdz.

(4.14)

Combining with Eq. 4.13, the total interaction energy is  πCρ1 ρ2 ∞ 2πRc zdz U =− 3 6 z=0 (R + z)

(4.15) π 2 Cρ1 ρ2 Rc , =− 6R where R is the distance from the tip apex to the sample surface, see Fig. 4.2. Defining the Hamaker constant as H ≡ π 2 Cρ1 ρ2 ,

(4.16)

the van der Waals interaction energy between a tip of local radius Rc and a sample surface is HRc U =− . (4.17) 6R

98

Atomic Forces

∼ 2–3 eV. For more details For typical metals, the Hamaker constant is H = about van der Waals force, see Israelachvili [163].

4.2

Pauli repulsion

The Pauli repulsion between molecules is an indispensible component of the van der Waals equation of state. Owing to Pauli’s exclusion principle, at very short interatomic distances, the repulsion between the electron clouds of atoms always arises. Such repulsive forces are often isotropic and characterized by a steep gradient. There is no general mathematical expression available from quantum mechanics. Instead, a number of empirical potential functions are used. The simplest is a hard-sphere potential, which has only one parameter, the hard-sphere radius re : r < re

:

U = ∞.

(4.18)

The second is a power-law repulsive potential, such as the minus 12th power function in the Lennard-Jones potential,  r 12 e

. (4.19) r Here a new parameter appears, the dissociation energy U0 . The Lennard– Jones potential describes fairly accurately the interaction energy between inert-gas atoms. In STM and AFM experiments, the most appropriate discription is an exponential function as in the Morse potential, also called Morse function, U = U0

U = U0 e−2β(r−re ) .

(4.20)

It includes a third parameter, the decay constant β. The decay constant of the repulsive potential is twice as that of the attractive potential. Although more complicated, it is the only one which has some theoretical justification from quantum mechanics [164]. Additionally, it often provides an accurate approximation to the real potential curves.

4.3

The ionic bond

The van der Waals interaction is weak. The typical adhesion energy is of the order of 0.01–0.02 eV. The ionic bond, resulting from the electrostatic interaction of oppositely charged ions, is often much stronger. For many ionic crystals, the Coulomb interaction, plus a Pauli repulsion term, provides a fairly accurate quantitative picture. The first step in understanding the ionic bond is that because of the very long range of the Coulomb force, in any ionic crystal, the total interaction

4.3 The ionic bond

99

energy is the sum of a very large number of ions. And the result depends not only on the nearest-neighbor distance, but also on the crystal structure [137]. The electrostatic interaction per ion pair has the form UCoulomb = −α

e2 , r

(4.21)

where r is the nearest-neighbor distance, and α is the Madelung constant. For the rock salt structure, α = 1.7476. The next step is to add a repulsive term to the interaction energy, otherwise the crystal will collapse. An exponential term with two parameters, URepulsive = λe−βr ,

(4.22)

is commonly used. The two parameters, λ and β, are obtained by matching with the crystallographic data and the Young’s module. The total interaction energy for N pairs of atoms is then

e2 UTotal = N λe−βr − α . r

(4.23)

Fig. 4.3. The ionic bond energy of NaCl. The main part of the interaction energy can be modeled by the electrostatic interactions between ions. It accounts for about 85% of the interaction energy. However, without repulsive interaction energy, the system would collapse. A properly chosen repulsive-force term, in the form of an exponential function, provides a complete picture of the ionic bonding.

100

Atomic Forces

The variation of the ionic bond energy with nearest-neighbor distance r is shown in Fig. 4.3. As shown, the Coulomb interaction accounts for about 85% of the interaction energy. The electrostatic model works fine.

4.4

The chemical bond

The electrostatic force between atoms, or the ionic bond, is stronger than the van der Waals interaction. However, the electrostatic force alone cannot explain a basic fact in chemistry: two identical atoms often form a molecule with a bond stronger than the ionic bond, such as the nitrogen molecule and the oxygen molecule. The strongest bonds are often formed between identical atoms, for example in diamond. Such type of bond is called a covalent bond, or a chemical bond [153]. Regarding to STM and AFM, the chemical bond energy and force are the most relevant. 4.4.1

The concept of the chemical bond

The chemical bond is the central concept of modern chemistry. Linus Pauling received his first Nobel prize for exlaining the concept of the chemical bond based on quantum mechanics [153]. In most quantum mechanics and quantum chemistry books, the concept of the chemical bond is introduced by the example of a hydrogen molecule using the Heitler–London method [165]. However, the mathematics is cumbersome and provides no analytic results. Furthermore, as pointed out by Herring [17], the Heitler–London method predicts a wrong sign of the bond energy at large distances, besides a 33% error in the dissociation energy. In the 1960s, Herring [17] and Landau [18] independently developed a perturbation method for the chemical bond, resulting in a surface-integral expression of the chemical bond. For the hydrogen molecular ion, an analytic expression of the first term of the asymptotic expansion of the exact chemical bond energy is derived. The method has been extended to the hydrogen molecule [18, 166]. Here we make a mathematically transparent presentation following the logic of Herring and Landau. Besides the hydrogen molecular ion poblem, we show that the method can be extended to molecules from many-electron atoms, see Section 4.6, and the atomic forces in STM and AFM, see Chapter 5. The origin of the chemical bond can be understood in terms of linear superposition of wavefunctions. Shown in Fig. 4.4 (A) and (B) are two atoms, each has a wavefunction, or atomic orbital (AO), at the same energy E0 . When the two atoms come together to become a diatomic molecule, molecular wavefunctions, or molecular orbitals (MO), are formed. Because the Schr¨ odinger equation is linear, a molecular orbital can be a linear combination of atomic orbitals (LCAO), see Fig. 4.4(C) and (D). The term orbital refers to the wavefunction of a single electron state.

4.4 The chemical bond

101

Fig. 4.4. Concept of the chemical bond. When two atoms (A) and (B) approach each other to become a diatomic molecule, the molecular wavefunctions are linear superpositions of atomic wavefunctions. Two types of superpositions are possible. The symmetric superposition, (C), makes a bonding wavefunction, or a bonding molecular orbital. The antisymmetric superposition, (D), makes an antibonding wavefunction, or an antibonding molecular orbital. The bonding orbital does not have any additional node, and its energy is lower than the energy of individual atomic orbitals. The antibonding orbital creates an additional node, and its energy is higher.

The chemical bond can be better understood using real wavefunctions. According to the Wigner theorem presented in Appendix B, if the Hamiltonian is time-reversal invariant, all wavefunctions can be chosen real. Because the overall sign of a wavefunction is irrelevant, the sign of atomic wavefunctions near the interface can always be assigned positive. There are two types of molecular orbitals. The molecular orbital in Fig. 4.4(C) is the sum of the atomic orbitals, which is called a bonding molecular orbital ψb . In the interface region, the amplitude of the molecular orbital is greater than that of the atomic orbitals. The productive linear superposition of atomic wavefunctions lowers the energy level of the molecule to Eb . It gains a bonding energy, ΔE = E0 − Eb .

(4.24)

In Fig. 4.4(D), the molecular orbital is the difference of the two atomic orbitals, which is called an antibonding molecular orbital ψa . The molecular wavefunction changes sign near the interface. The surface where the amplitude of wavefunction vanishes is called a nodal surface, a nodal plane, or simply a node. Due to the destructive linear superposition, the energy level of the molecule Ea is higher than that of the atoms. The difference, an antibonding energy, approximately equals the bonding energy: ΔE = Ea − E0 .

(4.25)

Assuming that the overlap of the atomic orbitals is small, a simple and general expression of the bonding energy in terms of the atomic wavefunc-

102

Atomic Forces

tions near the seperation surface x = 0 can be derived. We begin with the normalized wavefunctions of the atoms,   2 3 ψ1 d r = ψ22 d3 r = 1. (4.26) The condition of small overlap means  ψ1 ψ2 d3 r ≈ 0. 4.4.2

(4.27)

Bonding energy as a Bardeen surface integral

Using a perturbation method similar to Bardeen’s tunneling theory, an expression of chemical bond energy in terms of the atomic wavefunctions near the separation suface can be derived, see Fig. 4.4. The idea is similar to the treatment of the hydrogen molecular ion on pages 314-316 of Landau and Lifshitz [18]. Here the mathematics is rigorous and transparent. The atomic wavefunction of atom 1 satisfies the Schr¨odinger equation

2 2 − ∇ + U1 ψ1 = E0 ψ1 , (4.28) 2m and the atomic wavefunction of atom 2 satisfies

2 2 − ∇ + U2 ψ2 = E0 ψ2 . 2m

(4.29)

In the molecule, the bonding wavefunction is 1 ψb = √ (ψ1 + ψ2 ). 2

(4.30)

Because the overlap is small, as shown in Eqs. 4.26 and 4.27, the molecular wavefunction is approximately normalized: 

   1 2 3 2 3 3 2 3 ψb d r = ψ1 d r + 2 ψ1 ψ2 d r + ψ2 d r ≈ 1. (4.31) 2 It satisfies the Schr¨odinger equation of the molecule, Fig. 4.4(C):

2 2 ∇ + U1 + U2 ψb = Eb ψb . − 2m

(4.32)

Multiply Eq. 4.28 by ψb , integrate over the left region, one obtains

  2 2 ∇ + U 1 ψ 1 d 3 r = E0 ψb − ψb ψ1 d3 r. (4.33) 2m x 0, m = 0, can be approximated by applying a factor to the tunneling matrix element l  q2 |M |2 → 1 + 2 |M |2 , (6.56) κ

Fig. 6.7. Surface state on Cu(111), scattered by an adatom. (a) Dynamic tunneling conductance as a function of distance from the adatom and bias voltage, showing good agreement with Eq. 6.52. (b) The dynamic tunneling conductance images at four different bias voltages. (Original image courtesy of Paul Weiss. Reproduced from the Ph.D. Thesis of S. U. Nanayakkara, Pennsylvania State University, with the permission of Pennsylvania State University.)

6.2 The Tersoff–Hamann model

183

√ where κ = 2mφ/ is the decay constant. Since the work function φ of Cu(111) is 4.6 eV, κ = 10.9 nm−1 . From Table 6.2, the largest value of wave vector is q ≈ 2.68 nm−1 . From Eq. 6.56, one finds l

|M |2 → (1 + 0.06) |M |2 .

(6.57)

Therefore, the pz state and the dz2 states gives essentially the same result, and the contribution of m = 0 states is negligible. The Tersoff–Hamann formula is almost perfectly accurate. One should expect  G(V ) ≡

dI dU

 ∝ U =V

1 sin2 (qρ + φ). ρ

(6.58)

Figure 6.7 shows the dynamic conductance images of the surface waves scattered by an adatom, observed on Cu(111). Figure 6.8 shows the dynamic conductance images of the surface waves scattered by an edge.

Fig. 6.8. Surface state on Cu(111), scattered by an edge. (a) Dynamic tunneling conductance as a function of distance from the adatom and bias voltage, showing good agreement with Eq. 6.52. (b) The dynamic tunneling conductance images at five different bias voltages. (Original image by courtesy of Paul Weiss. Reproduced from the Ph.D. Thesis of S. U. Nanayakkara, Pennsylvania State University, with the permission of Pennsylvania State University.)

184

6.2.7

Nanometer-Scale Imaging

Heterogeneous surfaces

For heterogeneous surfaces, for example, surfaces with defects, substitutional atoms, or adsorbates, the Tersoff-Hamann model often correctly predicts the interatomic contrast. For example, for a non-metal atom adsorbed on a metal surface, the value of Fermi-level LDOS at the center of that adsorbate is often much lower than those on the clean areas. Experimentally, in the topographical images, it is almost always observed that the non-metal adsorbates look deeper than the clean part of the surface. The substitutional atoms or adsorbates also perturb the electronic properties of the neighboring atoms, which makes a disturbance in an area many times greater than that of a single atom. In those cases, the feature size is always greater than 1 nm, and Eq. 6.26 is always satisfied. The tunneling current distributions from l > 0, m = 0 tip states are similar to those with the l = 0 state, and the tunneling currents from m = 0 states are much smaller than those from the m = 0 tip states. For example, the interatomic contrast of the STM topographic images in catalysis research, see Section 1.5.3, can be well described by the Tersoff-Hamann model. The above statement can be extended to the case of finite bias voltages. If the tip DOS is basically independent of energy level in the interval of interest, the images of the dynamic conductance at a given bias voltage V is proportional to the sample LDOS at E = EF + eV at the center of curvature of the tip, ρ(EF + eV, r0 ). For example, for organic molecules, the dynamic-conductance images often resemble the charge distributions of HOMO or LUMO. Plate 14 is an example.

6.3

Limitations of the Tersoff–Hamann model

The limitations of the Tersoff–Hamann model have been precisely pointed out by the original authors [115, 116], as shown in Eqs 6.25 and 6.26. For feature sizes of the order of 0.3 nm and smaller, tip wavefunctions other than the s-wave could play a dominating role, and the STM images could be very different from the predictions of the s-wave-tip model. Actually, for most materials (except probably the alkali metals), the typical nearest interatomic distances is about 0.20 to 0.25 nm. The typical wave vector related to atomic features, according to Eq. 6.26, q=

π ∼ = 14.3 nm−1 . 0.22 nm

(6.59)

Then, using Eq. 6.56, for a d-state with l = 2, m = 0, 2

|Md |2 → (2.31) |Ms |2 ≈ 5.34 |Ms |2 .

(6.60)

Therefore, the tunneling conductance due to an l = 2 tip state could be much larger than that from an s-wave tip state. To quantitatively inter-

6.3 Limitations of the Tersoff–Hamann model

185

pret and predict STM images with atomic resolution, different types of tip electronic states are often the key, see Chapter 7. In fact, in the studies of surface states, sometimes the underlying atomic structure also appears. In order to understand the experimentally observed combined surface wave and atomic corrugation, tip states other than the s-wave states must be considered. We will come to this point in Section 7.5. On the other hand, the extension of the Tersoff–Hamann model to finite bias voltages is not unconditional. Even for features larger than nanometer scale where the effect of non-s-tip states is not prominent, the third condition, the tip DOS is a constant over the energy interval of interest, is often not true. In actual STS experiments, a crucial art is to treat the tip to make that condition happen. For example, by touching the tip with a nearly-free-electron metal surface, such as Cu(100), the tip often picks up a cluster of copper atoms. The energy spectrum of the tip is often flattened over the range of interest. This condition can be further verified by doing STS experiments on bare copper surfaces. However, unless the tip is treated purposely, the electronic spectrum of the tip is seldom featureless. Especially, as shown in Section 2.2.4, because of the energy dependence of the tunneling matrix element, under a negative bias voltage when the electrons tunnel from the occupied states of the sample near the Fermi level into the empty states of the tip, the tunneling spectrum is predominately determined by the features of the energy spectrum of the tip. This effect has been observed repeatedly in experiments, see Section 15.2. In such cases, to make the experiment reproducible and reliable, experimental determination of the energy spectrum of the tip is important. The important role of the tip LDOS in STS is probably the reason Tersoff and Hamann did not make a definitive conclusion about extending their result to the case of finite bias in their original publication, although certain discussions were made [115, 116]. We discussed this point in Chapter 2 and will come back to it in Chapter 15.

Chapter 7 Atomic-Scale Imaging Besides imaging nanometer-scale features, STM and AFM have demonstrated the capability of imaging individual atoms and subatomic features. Because the typical nearest-neighbor distance of adjacent atoms is about 0.2–0.3 nm, the resolution limit of STM and AFM is much finer than one nanometer. Furthermore, STM and AFM are capable of identifying the chemical nature of atoms and mapping individual chemical bonds. The physics, or the mechanism of atom-scale imaging in STM and AFM, is different from that of nanometer-scale imaging. At nanometer scale, the images can be treated as being independent of the tip electronic states, see Chapter 6. However, in atomic-scale imaging, the tip electronic states, determined by the chemical and structural nature at the very end of the tip, are also a key factor in the interpretation of the images. A key evidence of the role of tip electronic states in STM is the atomic resolution on close-packed metal surfaces such as Au(111). It was observed as early as 1983. Several serious publications with extensive discussions appeared after 1987 [86, 88, 229]. Typically, the atomically resolved images were obtained with relatively small tunneling resistances, usually equal to or smaller than 10 MΩ. Typically the corrugation amplitude varies from 5 pm to 50 pm, but 30 pm is easily achievable. The observed corrugation is at least one order of magnitude greater than the corrugation of the Fermi-level LDOS: 1 pm or less. Single-atom defects are clearly observable. Therefore, interpretations based on mechanical artifacts can be excluded.

Fig. 7.1. Atomic resolution on Au(111) by STM. The early STM images of atomic-resolution on Au(111), observed at Stanford University in 1983 and 1987. In the 1987 STM image, the effect of a spontaneous tip structuring was recorded. (By courtesy of S. I. Park of Park Scientific Instruments Asia.)

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7.1

Atomic-Scale Imaging

Experimental facts

Atomic and subatomic resolution have been achieved on virtually every type of conducting solid surface by STM, and then later extended to nonconducting solid surfaces by non–contact AFM. In this section, we give a brief summary of the experimental facts. 7.1.1

Universality of atomic resolution

Atomic-scale resolution with corrugation amplitude greater than 10 pm, sometimes as high as 50 pm, has been routinely achieved by STM laboratories worldwide. It is independent of the type of material. It ranges from soft metals (Al, Cu, Au) to very hard metals (W, Ir, Rh, and rare-earth metals). It is observed on all crystallographic orientations, for example the (111), (110), (100) surfaces of cubic crystals. Fig. 7.2(a) shows an atomicscale image of Dy film grown on W(110) surface. The corrugation profiles are shown in Fig. 7.2(b). Along three different crystallographic orientations, corrugations of 10 to 15 pm are observed. 7.1.2

Corrugation inversion

The qualitative feature of the atomic-scale corrugation changes with experimental condition, and may change within a single scan under identical tunneling conditions. Often, the observed atomic corrugation is inverted : the atomic sites appear as depressions instead of protrusions. Figure 7.3 was obtained from the same experiment under identical tunneling conditions as Fig. 7.2. After a tip-sharpening procedure, the corrugation is inverted. And this could also happen spontaneously.

Fig. 7.2. Atomic-scale images of Dy on W(110). (a) The sample: 90 ML of Dy, vacuum-deposited on a single-crystal W(110). Tunneling parameters: U = –2.4 mV, I = 330 nA. (b) Corrugation profiles along three close-packed directions, with amplitudes 10 to 15 pm. (Original image by courtesy of L. Berbil-Bautista. Reproduced with permission [58]. Copyright 2006 Hamburg University.)

7.1 Experimental facts

189

Fig. 7.3. Inverted corrugation and atomic defects of Dy on W(110). (a) The sample was exactly the same as in Fig. 7.2. Tunneling parameters are identical to those of Fig. 7.2. (b) Corrugation profiles along two lines. The corrugation is inverted: the atoms appear as depressions instead of protrusions. Corrugation amplitudes of 10 to 15 pm are observed. Adapted from the Ph.D. thesis of Luis Berbil-Bautista, Hamburg University, 2006. (Original image by courtesy of L. Berbil-Bautista. Reproduced with permission [58]. Copyright 2006 Hamburg University.)

7.1.3

Tip-state dependence

A convincing evidence of the tip-state effect is the spontaneous switching of imaging mode during a single scan. It often occurs within a fraction of a millisecond, or within a single line of scan. After the spontaneous tip restructuring, the image could be stable for some time. Then, it could again happen spontaneously. This phenomenon has been frequently observed and repeatedly reported. Figure 7.4 shows that the corrugation amplitude can be quite different before and after the tip change.

Fig. 7.4. Effect of tip state on STM images. The experiment is a combined STM and frequency-modulation AFM. (a) During a scan, a tip change leads to an increased corrugation from 16 pm to 39 pm. (b) The simultaneously acquired image of excitation amplitude also shows a substantial change. (c) The original line scans for the topography and the excitation amplitude. (Reproduced with permission from Loppacher et al. [22]. Copyright 2000 American Physical Society.)

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Atomic-Scale Imaging

A tip change could alter not only the corrugation amplitude and √ polarity, √ but also the qualitative appearance of the images. The Ag/Si(111)- 3 × 3 surface is an example. Two types of STM images of the same surface were reported: the honeycomb pattern, Fig. 7.5(a); and the inequivalent triangle pattern, Fig. 7.5(b). Based on those STM observations, a theory about a phase transition at an unknown temperature was proposed [231, 232]. However, such a phase transition was not supported by photoemission studies over the temperature range of interest [233]. Then, Zhang et al. [230] reported an observation √ of√two spontaneous tip restructuring in a single scan of the Ag/Si(111) 3 × 3 surface, resulting in three types of STM images in one frame, see Fig. 7.5(c). The image changed from a honeycomb pattern (bottom) to a unstable pattern for a fraction of a second (middle), then changed to a third pattern (top). The third type of image, Fig. 7.5(d), shows

Fig. 7.5. Changes of image pattern due to switching. Two types √ √ tip-state of STM images were observed on the Ag/Si(111)- 3 × 3 surface: (a) the honeycomb pattern, and (b) the inequivalent triangle pattern. For years, it was suspected that there wass a phase transition on that surface at an unspecified temperature. However, a single image taken at the identical tunneling condition, (c), shows that the different images are due to a difference in tip electronic structure. Two spontaneous tip changes (marked by arrows) occurred during a single scan. A third type of image, (d), is observed, obviously also due to the tip-state effect. (Adapted with permission from Zhang et al. [230]. Copyright 2000 American Physical Society.)

7.1 Experimental facts

191

bright hexagonal protrusions surrounded by a honeycomb √ pattern. √ Therefore, the different STM images observed on the Ag/Si(111) 3 × 3 surface are due to different tip states, whereas the sample states are unchanged. 7.1.4

Distance dependence of corrugation

It was universally observed that the corrugation amplitude depend exponentially on tip–sample distances [88, 89]. In other words, the corrugation amplitude decays exponentially with tip–sample distance. To achieve a favorable condition to observe atomic corrugation, the tip must be placed as close as possible to the sample surface, usually within 0.2 nm of a mechanical contact. Experimentally, the distance is determined by the tunneling conductance. According to Eq. 1.24, the tunneling conductance is G = G0 e−2κ(z−ze ) ,

(7.1)  where G0 = 77.48 μS is the conductance quantum, and κ = 5.1 φ(eV) nm−1 is the decay constant, typically 11 nm−1 . That means that the tunneling conductance is close to or greater than 1 μS, or the tunneling resistance is close to or smaller than 1 MΩ. At very small tip–sample distances, the corrugation no longer increases exponentially, because the tip–sample interaction effect reduces the observed corrugation, see Section 9.2.2.

Fig. 7.6. Distance dependence of corrugation. At normal tip–sample separations, the corrugation amplitude shows an exponential dependence on tip–sample separation. At very short tip–sample separations, the effect of tip–sample interaction reduces the apparent corrugation, as measured through the z-piezo displacement. The Figure also shows the effect of tip structure. The corrugation amplitudes from two tips show a similar dependence on tip–sample separation, but the corrugation amplitudes differ by one order of magnitude. (Adapted with permission from Clarke et al. [89]. Copyright 1996 American Physical Society.)

192

7.2

Atomic-Scale Imaging

Intuitive explanations

It is clear from the experimental facts that, in order to understand the atomic-scale imaging, tip electronic states and tip–sample interactions are the keys. In this section, we present intuitive explanations of how tip states affect STM and AFM imaging. 7.2.1

Sharpness of tip states

The profile of reconstruction of the Si(111)-7×7 surface has a spacing of about 0.6 nm, which is near the resolution limit of the Tersoff–Hamann model. The observed corrugations on the Si(111)7×7 surface, often exceed 100 pm, sometimes as high as 200 pm, are much greater than the predictions the Tersoff–Hamann model, see Fig. 7.7. In 1988, Demuth et al. proposed an explanation [94], based on their experimental observations. Their experience showed that even the bestprepared clean tungsten tips usually do not immediately produce the highest

Fig. 7.7. STM corrugation observed on Si(111)7×7 surface and conjectured imaging mechanism. (a) Examples of corrugations observed on Si(111)7×7 surface by STM, which often exceed 100 pm, much greater than the predictions the s-wave tip model. (2) Demuth et al. conjectured that the observed corrugations are due to an sp3 dangling-bond state of a silicon cluster adsorbed on the apex of the tip. (Adapted with permission from [94]. Copyright 1988 Wiley-Blackwell.)

7.2 Intuitive explanations

193

Fig. 7.8. Role of dz2 tip states in STM imaging. The tip is usually made of tungsten. It is well known that the dz2 state often exists on W surfaces. The dz2 states are much ‘sharper’ than the s-states, which could provide a stronger corrugation. (After Baratoff [91].)

resolution on the Si surface. Most often the highest resolution is achieved after long periods of scanning or controlled tip crashing. When there is no atomic resolution, an effective procedure to achieve atomic resolution is to collide the tip mildly with the Si surface. After such a controlled crashing, a crater is found on the Si surface, which shows that a Si cluster has been picked up by the tip. Atomic resolution is then often achieved. They conjectured that the Si cluster often has an sp3 dangling bond at its apex, which should provide better corrugation than an s-state (Fig. 7.7). They concluded that tip treatment is one half of the STM experiment [94]. Based on another experimental fact, that the tips are usually made of tungsten, and the existence of a dz2 state at the apex of the tungsten tip, Baratoff [91] made a conjecture as early as 1984 that the observed high corrugation on the Si(111)-7×7 surface it is due to the dz2 states protruding into the vacuum (Fig. 7.8). The dz2 state has a much narrower angle of focus, which could qualitatively explain the observed high resolution. 7.2.2

Phase effect

A more quantitative understanding of the effect of different tip states can be established through the phases of the different lobes of the tip wavefunctions. Consider a wavefunction of the sample surface which has two components: a uniform component which is independent of the lateral coordinates, and a sinusoidal component which is the origin of corrugation:

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Atomic-Scale Imaging

Fig. 7.9. The phase effect in STM imaging. Except for the s-tip state, the tip wavefunctions comprise multiple lobes with opposite phases. The tunneling amplitude depends on the relative phase. Therefore, for tip states other than the s-state, the variations of the sample wavefunctions are highlighted.

ψ(x, z) = f0 (z) + cos kx f1 (z).

(7.2)

It should satisfy the Schr¨odinger equation in vacuum:

∂2 ∂2 + 2 ψ(x, z) = κ2 ψ(x, z). ∂x2 ∂z

(7.3)

Substituting Eq. 7.2 into Eq. 7.3, using the condition that the wavefunction should decay in the positive z direction, we find ψ(x, z) = A e−κz + B cos kx e−

√

κ2 +k2 z

,

(7.4)

where A and B are constants which depend on the nature of the surface. The image is formed by an interaction of the tip state and the sample state. If the sign of a part of the wavefunction is changed, the sign of the interaction is also changed. For an s-wave tip state, there is no phase change. For a pz state, the phase changes at a z-plane. Therefore, the tunneling amplitude is now related to the z-derivative of the sample wavefunction. For a dz2 state, the tunneling matrix element is related to the second derivative of the sample wavefunction with respect to z. For a px or py state, the tunneling amplitude is related to the x or y derivative of the sample wavefunction. Therefore, with a non-s-tip state, the variation of the sample wavefunction is highlighted. And the corrugation is enhanced. The possible corrugation enhancement due to pz and dz2 tip states related to phase effect has already been discussed in the first publications on the Tersoff–Hamann model [115, 116]. In their derivation process, Tersoff and Hamann made an estimate of the effect of non-s-wave tip states, with l > 0, where l is the angular momentum of the tip wavefunction. The effect of tip wavefunctions with higher l can be represented by taking z-derivatives

7.2 Intuitive explanations

195

from Eq. 6.17, which introduces a factor to the tunneling matrix element  M→

q2 1+ 2 κ

l/2 M.

(7.5)

Therefore, the importance of higher angular momentum contributions depends on the magnitude of the factor q2 /κ2 . For a typical metal surface, the work function is 5 eV, and the decay constant is 11.4 nm−1 . The quantity q is the Fourier component of the feature size a of the sample, with a relation | q |= π/a. Therefore, when a=

π π ≤ ∼ = 0.3 nm, |q| κ

(7.6)

a corrugation enhancement should occur. For the case of Si(111)-7×7, the enhancement is noticeable, but not significant. For low-Miller-index metal surfaces, where the interatomic distances are smaller than 0.3 nm, the enhancement could be very high. 7.2.3

Arguments based on the reciprocity principle

The effect of pz or dz2 dangling bonds on STM resolution can be understood in the light of the reciprocity principle [13], which is the fundamental microscopic symmetry between the tip and the sample: by interchanging the ‘acting’ electronic state of the tip and the sample state under observation, the image should be the same. The discrepancy between the sharp STM image and the low corrugation of the charge density on low-Miller-index metal surfaces can be explained in

Fig. 7.10. Origin of atomic resolution on metal surfaces According to the reciprocity principle, the image taken with a dz2 tip state (which exists on a W tip) on a free-electron-metal surface is equivalent to an image taken with a point tip on a fictitious sample surface with a dz2 state on each top-layer atom, which obviously has a stronger corrugation. (Reproduced with permission from [14]. Copyright 1990 American Physical Society.)

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Atomic-Scale Imaging

light of the reciprocity principle as follows. Figure 7.10 shows a qualitative explanation of the effect of a dz2 tip state. For an s-wave tip state, the STM image of a metal surface is the charge-density contour, which can be evaluated using atomic-charge superposition (that is, as a sum of the charge densities of individual atoms, each made of s-states, see Section 6.2.3 especially Eq. 6.35). According to the reciprocity principle (Fig. 7.10), with a dz2 tip state, the tip no longer traces the contour of the Fermi-level LDOS. Instead, it traces the charge-density contour of a fictitious surface with a dz2 state on each atom. Because the contour of each dz2 state is much sharper than that of an s-state, the overall contour exhibits much stronger atomic corrugation than that of the Fermi-level LDOS.

7.3

Analytic treatments

The effect of different tip electronic states on STM images can be treated using analytic methods which result in simple close-form formulas. Those formulas, based on the explicit expressions of tunneling matrix elements (Section 3.3) and the leading Bloch wave approximation, have been verified by first-principles numerical studies and experimental measurements. 7.3.1

A one-dimensional case

We start with the simplest case of a one-dimensional metal surface. The concept of the leading Bloch wave approximation and the application of the derivative rule are demonstrated with this case. The spirit of the leading Bloch wave approximation is to fit the zdependence of both the uncorrugated and the corrugated components of the surface charge density by an exponential function, and then to find the coefficients from the results of first-principle computations. Consider a metal surface of a one-dimensional periodicity a with a reflection symmetry at x = 0 (Fig. 7.11). The general formula for the electron charge density distribution is ρ(x, z) = C0 e−αz + C1 e−γz cos2 (qx)   = C0 e−αz 1 + (C1 /C0 )e−βz cos2 (qx) ,

(7.7)

where β ≡ γ − α, and π 1 g= , (7.8) 2 a and g is the primitive reciprocal lattice vector. The constants C0 , C1 , α, and γ are determined by fitting with results from first-principles calculations. The second term in Eq. 7.7, the corrugated term, is much smaller than the first term, the constant term. The factor cos gx in Eq. 7.7 is rewritten as q=

7.3 Analytic treatments

197

Fig. 7.11. A metal surface with one-dimensional periodicity. The lowest Fourier components of the charge-density distribution are determined by the Bloch functions at ¯ and K ¯ points in reciprocal space. the Γ

cos2 (qx) for convenience. (Actually, for cos2 (qx), the corrugation amplitude is 1 rather than 2 as for cos(gx), and the origin of the corrugation from the relevant Bloch wave becomes more explicit, as will be shown.) In STM, only the electron states near the Fermi level are involved. The constants α and γ in Eq. 7.7 can be obtained from general considerations of the surface wavefunctions. Near the Fermi level, the surface wavefunctions in the vacuum region satisfy the Schr¨ odinger equation in the vacuum: (∇2 − κ2 )ψ = 0,

(7.9) √ where, as usual, κ = 2mφ/, and φ is the work function of the sample. As shown in Fig. 7.11, the first term in Eq. 7.7 originates from the constant ¯ From term in the Bloch functions, that is, the Bloch functions near Γ. Eq. 7.9, the lowest Fourier component is: ψΓ¯ = const. × e−κz , which makes the first term C0 e constant in Eq. 7.7,

−2κz

(7.10)

. Therefore, we identified the first decay

α = 2κ.

(7.11)

¯ points have a longer decay length and The Bloch functions near the K contribute to the second term of Eq. 7.7. In general, following Eq. 7.9, a surface Bloch function at that point has the form: iqx ψK ¯ =e

∞ 

cn e−

√

κ2 +(q+2nq)2 z

e2inqx .

(7.12)

n=−∞

In addition to the term with n = 0, the term with n = −1 has the same decay length, and thus has the same magnitude. Also, the Bloch function that generates the symmetric charge density must also be symmetric. The ¯ is: lowest-order symmetric Fourier sum of the Bloch function near K √ 2 2 − κ +q z ψK cos qx. (7.13) ¯ ∝ e

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Atomic-Scale Imaging

Fig. 7.12. Tip-induced corrugation enhancements. Enhancement of tunneling matrix elements arising from different tip states. The tunneling current is proportional to the square of the tunneling matrix element. Therefore, the enhancement factor for the corrugation amplitude is the square of the enhancement factor for the tunneling matrix element. (Reproduced with permission from [14]. Copyright 1990 American Physical Society.) 2

The charge density is proportional to |ψK ¯ | . We then find the second constant in Eq. 7.7,  (7.14) γ = 2 κ2 + q 2 . Therefore, an approximate expression for the Fermi-level LDOS is ρ(r, EF ) =

EF + 1  2 |ψμ (r)| Eμ =EF

= C0 e−2κz + C1 e−2

√

(7.15) κ2 +q 2 z

cos2 qx.

The corrugation amplitude of the Fermi-level LDOS for a metal surface with one-dimensional corrugation can be obtained using Eqs 6.6 and 7.15, Δz =

C1 e−βz , 2κC0

(7.16)

where β ≡ γ−α=2

 κ2 + q 2 − 2κ.

(7.17)

The expression of γ for metals with one-dimensional periodicity, Eq. 7.7, was obtained first by Tersoff and Hamann [116]. The ratio C1 /C0 can be obtained from first-principles computations. Using the expressions of the tunneling matrix elements derived in Chapter 3, theoretical STM images can be calculated. In this section, we discuss the theoretical STM images of the metal surface with one-dimensional corrugation.

7.3 Analytic treatments

199

s-wave tip state The tunneling matrix element for an s-wave tip state is proportional to the amplitude of the sample wavefunction at the nucleus of the apex atom. The tunneling current is: I = const. × 

EF 

|ψ(r0 )|

EF −eV

∝ V C0 e

−2κz

+ C1 e

−2

2

√

κ2 +q 2 z

2



(7.18)

cos qx .

For free-electron metals, the local density of states near the Fermi level is proportional to the total valence-electron charge density. Therefore, up to an overall constant depending on the bias V , the tunneling current is proportional to the charge density at the nucleus of the apex atom: I = const. × ρ(x, z).

(7.19)

The topographic image can be calculated from Eq. 6.6. It is Δz(x) =

C1 e−βz cos2 gx. 2κC0

(7.20)

The corrugation of the charge density on metal surfaces can be obtained from first-principles computations or helium scattering experiments. A helium atom can reach to about 0.25 nm from the top-layer nuclei. At that distance, the repulsive force between the helium atom and the surface is already strong. The corrugation at that distance is about 3 pm, from both theory and experiments. For STM, with a pure s-wave tip state, if the nucleus of the apex atom reaches as deep as a helium atom can reach into the electron cloud at the surface, a corrugation of about 3 pm would be observed. At that distance, a strong repulsive force is present. However, experimentally, corrugations greater than 30 pm are routinely observed in the weak attractive force regime. Therefore, the observed atomic resolution must be due to other tip states. pz -tip state According to the derivative rule, the tunneling matrix element for the sur¯ from a pz tip state is identical to that from a face wavefunction at point Γ ¯ the tunneling spherical tip state. However, for a surface wavefunction at K, matrix element from a pz tip state is: MK ¯ ∝

 1/2 q2 1+ 2 ψK ¯, κ

and the topographic image arising from a pz tip state is:

(7.21)

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Atomic-Scale Imaging

 Δz(x) =

1+

q2 κ2



C1 −βz e cos2 qx. 2κC0

(7.22)

Therefore, the corrugation amplitude arising from a pz tip state gains a factor of [1 + (q 2 /κ2 )] over that of the charge-density contour (Fig. 7.12). This is the quantitative explanation of the resolution enhancement due to p-like tip states, as proposed by Demuth et al. [94]. dz2 -tip state Using the expression for the tunneling matrix element of a dz2 tip state, ¯ , it picks up a factor of 2/3, whereas for a for a sample wavefunction at Γ ¯ sample wavefunction at K, it picks up a factor of [(2/3) + (q 2 /κ2 )]. Similar to the case of pz tip state, we find the topographic image to be: Δz(x) =

2  C1 −βz 3q 2 e cos2 qx. 1+ 2 2κ 2κC0

(7.23)

The enhancement for the tunneling matrix element is shown in Fig. 7.12. The enhancement factor for the corrugation amplitude, [1 + (3q 2 /2κ2 )]2 , could be substantial. For example, on most close-packed metal surfaces, a ≈ 0.25 nm, which implies q ≈ 12.5 nm−1 . An enhancement of 11.2 is expected. Most of the commonly used tip materials are d-band metals, for example, W, Pt, and Ir. In all first-principles computations of the tip electronic states, as shown in Section 7.4, the dz2 tip state often dominates the DOS near the Fermi level. It could enhance the corrugation amplitude by more than one order of magnitude. A straightforward calculation using the tunneling matrix elements listed in Chapter 3 shows that the dxz state results in a large but inverted corrugation amplitude on metal surfaces, because the tunneling matrix element ¯ point vanishes. The role of this state for the sample wavefunction at the Γ and the dx2 state in the inverted corrugation will be discussed in Section 7.3.3. 7.3.2

Surfaces with hexagonal symmetry

Probably the most commonly encountered surfaces in STM experiments are of trigonal symmetry. The close-packed metal surfaces and most cleaved surfaces of layered materials belong to this category. In Fig. 7.13, the structure of a close-packed metal surface is shown. The large dots represents the atoms in the top layer. The circles represent the atoms in the second layer. The small dots are those in the third layer. However, experimentally, it was found that only the atoms in the first layer are observed. Therefore, the surface has an approximately hexagonal symmetry, p6mm, which is the highest symmetry in all plane groups (Appendix D). The high symmetry makes the treatment much simpler, since the basic features of the images

7.3 Analytic treatments

201

with the lowest non-trivial Fourier components are determined by symmetry only. In this case, the charge density should have a hexagonal symmetry, i.e., invariant with respect to plane group p6mm (Fig. 7.13). Up to the lowest nontrivial Fourier components, the most general form of surface charge density with hexagonal symmetry is: ρ(r) =

EF 

|ψ(r)|

2

≈ a0 (z) + a1 (z) φ(6) (kx),

(7.24)

EF −ΔE

√ where x = (x, y), and k = 4π/ 3a is the length of a primitive reciprocal lattice vector. A hexagonal cosine function is defined for convenience, 1 2  + cos ω n · x, (7.25) 3 9 n=0 √ √ where ω 0 = (0, 1), ω 1 = (− 12 , 12 3), and ω 2 = ( 12 3, − 12 ), respectively. By plotting it directly, it is clear that the function φ(6) (kx) has a maximum value 1 at each atomic site, and nearly 0 in the space between atoms. The function [1 − φ(6) (kx)] has a minimum value 0 at each atomic site, and nearly 1 in the space between atoms, which describes an inverted corrugation (Fig. 7.14). Similar to the one-dimensional case, the a0 (z) term in Eq. 7.24 comes ¯ mainly from Bloch functions near Γ, 2

φ(6) (x) ≡

a0 (z) ∝ e−2κz .

(7.26)

Fig. 7.13. Geometrical structure of a close-packed metal surface. Left, the second-layer atoms (circles) and third-layer atoms (small dots) have little influence on the surface charge density, which is dominated by the top-layer atoms (large dots). The top layer exhibits a six-fold symmetry, which is invariant with respect to the plane group p6mm (that is, point group C6v together with the translational symmetry). Right, the corresponding surface Brillouin zone. The lowest non-trivial Fourier components of the ¯ and K ¯ points. (The symbols for plane LDOS arise from Bloch functions near the Γ groups are explained in Appendix D. Reproduced with permission from [14]. Copyright 1990 American Physical Society.)

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Atomic-Scale Imaging

Fig. 7.14. The hexagonal cosine function and its complementary function. (a) The hexagonal cosine function defined by Eq. 7.25, φ(6) (kx), has a maximum value 1 at each atomic site, and nearly 0 in the space between atoms. The function [1−φ(6) (kx)] has a minimum value 0 at each atomic site, and nearly 1 in the space between atoms, which describes an inverted corrugation. (Reproduced with permission from [122]. Copyright 1992 American Physical Society.)

¯ points have the longest decay length, which The Bloch functions near the K are the dominating contribution to the second term in Eq. 7.24. In general, a surface Bloch function at that point has the form: ψK ¯ =



aG e−

√

κ2 +|k1 +G|2 z

ei(k+G)·x ,

(7.27)

G

√ with | k1 | ≡ q = k/ 3. By inspecting Eq. 7.27 and Fig. 7.13, one finds that the only slow-decaying symmetric Fourier sums of the Bloch functions ¯ points are: near K 1

ψ1 = B e− 2 γz

2 

cos(qτn · x),

(7.28)

sin(qτn · x),

(7.29)

n=0

and

ψ2 = B e

− 12 γz

2  n=0

√ √ where τ0 = (1, 0), τ1 = (− 12 , 12 3), τ2 = (− 12 , − 12 3); B is a real constant;  and γ = 2 κ2 + q 2 is the corresponding decay constant. The symmetric ¯ point, Eqs 7.28 and 7.29, are invariant under a wavefunctions at the K rotation 2mπ/3, individually. Under a rotation of π, ψ1 is invariant, whereas ψ2 changes sign. These two wavefunctions are degenerate because, while shifting the origin by one lattice constant, these two wavefunctions mix with each other by the matrix

7.3 Analytic treatments

203

Fig. 7.15. Charge-density contour plot of Al(111) film. Results of a firstprinciples computation of a nine-layer Al(111) film. The contours are in steps of 0.27 electrons per atom. (Reproduced with permission from [234]. Copyright 1981 American Physical Society.)

⎡ ⎣ ∓

1 2 √

±

3 2

√

3 2

⎤ ⎦.

(7.30)

1 2

This can be easily shown by direct calculation. The charge density is the sum of Eq. 7.26 and the sum of the charge density proportional to |ψ1∗ ψ1 | + |ψ2∗ ψ2 |. A straightforward calculation gives ρ(r) ∝

EF 

|ψ(r)|

2

EF −ΔE



= ΔE C0 e−2κz + C1 e−γz

 φ(6) (kx) ,

(7.31)

where C0 , C1 are constants. The corrugation charge-density contour, Δz, as a function of z, can be obtained from Eq. 7.31 Δz(x) =

C1 −βz (6) e φ (kx). 2κC0

(7.32)

Similarly, β = γ − 2κ. The ratio (C1 /C0 ) can be determined by comparing Eq. 7.32 with the corrugation amplitudes of the charge-density contours obtained from first-principles calculations. For example, from Fig. 7.15, averaged from five contours ranging from three contours of the thinnest densities, we find (C1 /C0 ) ≈ 5.7 ± 1.0. Following the procedure for the one-dimensional case, the STM image for the pz tip state is   q2 C1 1 + 2 e−βz φ(6) (kx), (7.33) Δz(x) = 2κC0 κ

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Atomic-Scale Imaging

Fig. 7.16. Interpretation of atom-resolved STM images on Al(111). The predicted corrugation amplitude with a dz2 tip state (solid curve) agrees well with the experimental data from [88] (circles with error bars). The parameters of the theoretical curve are taken from Fig. 7.15. The corrugation from an s-wave tip state (dashed curve), that is, the corrugation of the Fermi-level LDOS contour, is included for comparison. (Reproduced with permission from [14]. Copyright 1990 American Physical Society.)

and the STM image for the dz2 tip state is Δz(x) =

C1 2κC0

 1+

3q 2 2κ2

2

e−βz φ(6) (kx).

(7.34)

A comparison of the theory with experiments is shown in Fig. 7.16. For Al(111), a = 0.288 nm, φ = 3.5 eV, it follows that κ = 9.6 nm−1 , γ = 34.8 nm−1 . The slope of the ln Δz ∼ z curve from Eqs 7.32 through 7.34 fits well with experimental data. The absolute tip–sample distance is obtained from curve fitting, which gives the shortest average tip–sample distance at I = 40 nA (with bias 50 mV) to be about 0.29 nm, consistent with the measured tip–sample distance by D¨ urig et al, [235, 20], about 0.1 nm before a mechanical contact. It is also consistent with the formula linking tunneling conductance and tip–sample distance based on the Landauer theory, Eq. 1.24 and Table 1.2. 7.3.3

Corrugation inversion

In this section, we discuss the effect of m = 0 tip states. We show that those tip states will create inverted images, where the sites of surface atoms are minima rather than maxima in the topographic images [48]. Qualitatively, the explanation of the effect of m = 0 tip states is as follows. For simplicity, we assume that the tip has an axial symmetry (actually, a trigonal symmetry suffices). In other words, the two m = 1 states, xz and yz, are degenerate. Similarly, the two m = 2 states, xy

7.3 Analytic treatments

205

and x2 − y 2 , are also degenerate. The LDOS of those tip states are shown in Fig. 7.17. Consider a simple metal, for example, Au(111). Each Au atom has only s-wave states near the Fermi level, and the tunneling current from each sample atom is additive. Thus, the tunneling current distribution between a single Au atom and the tip is the tip-state LDOS, measured at the center of that Au atom. For a dz2 tip state, it has a sharp peak centered at the atom site. The total current distribution is the sum of the tunneling current for all the Au atoms at the surface. The sharpness of the tunneling current distribution for the dz2 -tip state, compared with that of the s-wave tip state (Fig. 7.17), again illustrates why the dz2 -tip state enhances image corrugation. The m = 1 and 2 tip states exhibit a ring-shaped LDOS, as shown in Fig. 7.17. The tunneling current distribution for a single Au atom should be proportional to the tip LDOS, which is ring-shaped (Fig. 7.17). The total current distribution is the sum of the tunneling current for all the Au atoms at the surface. Therefore, with an m = 0 tip state, an inverted STM image should be expected. In other words, with an m = 0 tip state, every Au atom site at the surface should appear as a depression rather than a protrusion in the STM image. The general expression for the tunneling current can be obtained using the explicit forms of tunneling matrix elements listed in Table 3.2. To put the five d-states on an equal footing, normalized spherical harmonics,

Fig. 7.17. LDOS of several tip electronic states. Evaluated and normalized on a plane z0 = 0.3 nm from the nucleus of the apex atom. Axial symmetry is assumed. (a) s state. (b) l = 2, m = 0 state (d3z2 −r2 ). (c) l = 2, m = 1 states (dxz and dyz ). (d) l = 2, m = 2 states (dx2 −y2 and dzy ). (Reproduced with permission from [122]. Copyright 1992 American Physical Society.)

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as listed in Appendix A, are used. The wavefunctions and the tunneling matrix elements are listed in Table 7.1. Up to a constant, the tunneling current is I = 4 |D0 B0 | e−2κz 2

2  2 +9 |D0 B1 | e−2κ1 z 3(κ1 /κ)2 − 1 φ(6) (kx)  2   2 1 − φ(6) (kx) +54 |D1 B1 | e−2κ1 z qκ1 /κ2  2 4 +(27/2) |D2 B1 | e−2κ1 z (q/κ) 1 − φ(6) (kx) .

(7.35)

The first term in Eq. 7.35 represents the uncorrugated tunneling current, which decays much more slowly than the corrugated terms. Therefore, if D0 is not too small, the corrugation of the topographic image is !  2 2 3κ21 1 3  D1   qκ1 2 Δz = − − 2κ2 2 2  D0  κ2 (7.36) "  2 3  D2   q 4 −  Δz0 , 8 D0  κ where

Table 7.1: Tunneling matrix elements for d-tip states. The tip is assumed to have an axial symmetry. For brevity, the common factor in the normalization constant of the spherical harmonics and a common factor (2π2 /κme ) in the expressions for the tunneling matrices is omitted. State

Tip wavefunction

3z 2 − r2

  D0 k2 (κr) 3 cos2 θ − 1

xz

√ D1 k2 (κr) 3 sin 2θ cos φ

yz

√ D1 k2 (κr) 3 sin 2θ sin φ

x2 − y 2

√ D2 k2 (κr) 3 sin2 θ cos 2φ

xy

√ D2 k2 (κr) 3 sin2 θ sin 2φ

Tunneling matrix element

3 ∂2 D0 − 1 ψ(r0 ) κ2 ∂z 2 ! √ " 2 3 ∂2 ψ(r0 ) D1 κ2 ∂x∂z " ! √ 2 3 ∂2 ψ(r0 ) D1 κ2 ∂y∂z !√  " ∂2 3 ∂2 ψ(r0 ) D2 − 2 κ2 ∂x2 ∂y " ! √ 2 3 ∂2 ψ(r0 ) D2 κ2 ∂x∂y

7.3 Analytic treatments

207

Fig. 7.18. Enhancement factor for different tip states. The shaded area near E = 0 is the area where the corrugation amplitude is within the limit of the Fermi-level LDOS contours. In the hatched area near the bottom, the theoretical amplitude of the negative corrugation shows a spurious divergence. (Reproduced with permission from [122]. Copyright 1992 American Physical Society.)

 2 9  B1  −2(κ1 −κ)z (6) Δz0 = e φ (kx) 2κ  B0 

(7.37)

is the corrugation of the Fermi-level LDOS of the sample. The ratio | B1 / B0 | can be determined independently by first-principles calculations or independent experimental measurements, such as helium atom scattering. For Au(111), a = 0.287 nm, q = 14.6 nm−1 , κ = 9.6 nm−1 , and κ1 = 1.74 nm−1 . From Eq. 7.36, we obtain       D 1 2  D 2 2     Δz = 19.6 − 11.4  − 2.0  (7.38) Δz0 . D0  D0  The enhancement factor E, that is, the quantity in the parenthesis in this equation, is displayed in Fig. 7.18. Because the corrugation amplitude depends only on the relative intensities of different components, we normalize it through | D0 |2 + | D1 |2 + | D2 |2 = 1.

(7.39)

Naturally, the results can be represented by a diagram similar to a threecomponent phase diagram, as shown in Fig. 7.18. Several interesting features are worth noting. First, when the m = 0 or dz2 state dominates, a large, positive enhancement is expected. The condition for a substantial enhancement is quite broad, for example, when the condition | D0 |2 > 1.2 | D1 |2 + 0.2 | D2 |2 is satisfied, the positive enhancement should be greater than 10, or a full order of magnitude. It is about 15% of the total

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Atomic-Scale Imaging

phase space. To have an enhancement of more than 5, one third of the total phase space is available. Therefore, the experimental observation of large positive corrugation enhancement should be frequent. Second, when m = 0 states dominate, an inverted corrugation should be observed. Again, the probability of a negative image to occuring is large. Actually, when the condition | D0 |2 < 0.58 | D1 |2 + 0.1 | D2 |2 is fulfilled, the image corrugation is inverted. This is about 43% of the total phase space. To have negative corrugations with an enhancement factor of 5 or more, 14% of the total phase space is available. Third, from Eq. 7.38 and Fig. 7.18, it is apparent that the effect of m = 1 states in generating inverted corrugation is much stronger than that of m = 2 states. This is expected from Fig. 7.17. The m=1 states have a much sharper rim than the m=2 states. Finally, there is a small region in which an almost complete cancellation of the positive enhancement and the negative enhancement can occur, as indicated by the shaded area near zero corrugation. In this case, the image is similar to the prediction of the s-wave model. The observed image corrugation in this case should be equal to or smaller than the corrugation of the Fermi-level LDOS. From Eq. 7.38 and Fig. 7.18, the available phase space is about 2.8% of the total phase space. Therefore, the probability is small. Practically, when this situation occurs, an almost flat image is observed. The experimentalist explains it as a bad tip. A tip-sharpening procedure is then conducted until a large corrugation is observed, which is explained as having a good tip. Experimental observations of the corrugation inversion during a scan have been repeatedly reported, for example, Figs 1.16, 7.3, and 7.4. Owing to a sudden change of the tip state, the image switched from positive corrugation to negative corrugation. One interpretation is that, before the tip restructuring, an m = 0 tip state dominates; after the tip restructuring, an m = 0 tip state dominates. This could result from a spontaneous change of the tip structure or the chemical identity of the apex atom. The corrugation inversion due to m = 0 tip states is a universal phenomenon in the STM imaging of low-Miller-index metal surfaces. For most metals (except alkali and alkaline earth metals, rarely imaged by STM), the nearest-neighbor atomic distance a ≈ 0.25nm. Consequently, the numerical coefficients on Eq. 7.38 are very close to those for Au(111). 7.3.4

Profiles of atomic states as seen by STM

In this section, we study the images of atomic states. Starting with an analytic expression for the tunneling conductivity distribution, we compare the theoretical predictions and experimental findings of a measurable quantity: the radius of the STM image of an atomic state near its peak. To determine the image, the first step is to determine the distribution of tunneling current as a function of the position of the nucleus of the apex

7.3 Analytic treatments

209

atom of the tip. We set the center of the coordinate system at the nucleus of a sample atom. The tunneling matrix element as a function of the position r of the nucleus of the apex atom can be evaluated by applying the derivative rule to the Slater wavefunctions. The tunneling conductance as a function of r, g(r), is proportional to the square of the tunneling matrix element: g(r) ∝ | M |2 .

(7.40)

Because the equal-conductance contour is independent of an overall constant, we neglect it for convenience. In Table 7.2, the results for three types of m = 0 tip states and three types of m = 0 sample states are listed. A convenient quantity for comparing with the STM image profile for a single atomic state is the apparent radius of the image near its peak, R, or the apparent curvature, K = 1/R. Considering the m = 0 state only, these quantities are related to the tunneling conductance distribution by K ≡

1 = R



∂g(r) ∂z

−1

∂ 2 g(r) , ∂x2

(7.41)

which is evaluated at (0, 0, z). To illustrate the meaning of Eq. 7.41, we make an explicit calculation for a spherical tunneling-conductance distribution, g(r) ≡ g(r), where r2 = x2 + y 2 + z 2 . Using the following relations,    z ∂g = g = g , ∂z z=r r z=r  2    2 2 ∂ g g g  x  x − g , = g + = ∂x2 x=0 r2 r 3r3 x=0 r

(7.42)

(7.43)

(7.44)

we find that the definition of the radius, Eq. 7.41, coincides with the nominal distance between the two nuclei, as shown in Fig. 7.19. Intuitively, we expect that for pz or dz2 states on the tip and on the sample, the images should be sharper, i.e., the apparent radius should be smaller. This is indeed true. For example, for a dz2 tip state and an s-wave sample state, the tunneling conductance distribution is Fig. 7.19. Apparent radius for a spherical conductance distribution. For a spherical tunneling conductance distribution, the apparent radius equals the nominal tip–sample distance, regardless of the specific functional form of the distribution.

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Atomic-Scale Imaging

Table 7.2: Apparent curvature of individual atomic states. Conductance distribution and apparent curvature of the STM images of individual atomic states. Tip

Sample

Conductance distribution

Apparent curvature

s

s

e−2κr

1 z  1 1+ z  1 1+ z  1 1+ z  1 1+ z  1 1+ z  1 1+ z  1 1+ z  1 1+ z

2

−2κr

s

pz

s

dz 2

pz

s

cos2 θ e−2κr

pz

pz

cos4 θ e−2κr

pz

dz 2

dz 2

s

dz 2

pz

dz 2

dz 2

cos θ e 

2

3 cos2 θ − 1



2

3 cos2 θ − 1



2



2



4

3 cos2 θ − 1

3 cos2 θ − 1 3 cos2 θ − 1

 g(r) =

e−2κr

cos2 θ e−2κr e−2κr cos2 θ e−2κr e−2κr

3 z2 1 − 2 r2 2

2

e−2κr .

 1 κz  3 κz  1 κz  2 κz  4 κz  3 κz  4 κz  6 κz

(7.45)

Using Eq. 7.41, we find 1 1 = K ≡ R z

  3 1+ . κz

In general, the apparent curvature is   1 h S + hT 1 = 1+ , K ≡ R z κz

(7.46)

(7.47)

where hS is determined by the atomic state at the sample surface under probing: for an s state, hS = 0; for a pz state, hS = 1; and for a dz2 state, hS = 3. The same holds for the tip states. The results are listed in Table 7.2, and are illustrated in Fig. 7.20. Clearly, the apparent radius is reduced

7.3 Analytic treatments

211

Fig. 7.20. Apparent radius as a function of tip–sample distance. For different combinations of tip states and sample states, the apparent radii are given by Eq. 7.47. The shaded area indicates the condition for achieving atomic resolution (that is, having an apparent radius comparable to the actual radius of an atom). The minimum tip– sample distance is limited by the mechanical contact (that is, about 0.25 nm. Therefore, most of the images with true atomic resolution are obtained with pz or dz2 tip states.

substantially for p and d states on the tip as well as on the sample. In other words, with p and d states, the images of individual atoms at surfaces look much sharper than those with s states, which is expected. The images for mixed states, for example, sp3 states, can be treated using the same method. For states with n > 1, the enhancement is more pronounced. As we have shown, the apparent radius for a spherical distribution does not depend on the specific functional form of the distribution. Therefore, the algebraic factor in the radial Slater functions results in a small correction to the apparent radius for l = 0 states. It is easy to show that asymptotically the correction term is proportional to n − 1 and to z −3 . A glance in Table 7.2 reveals an interesting fact. Not only does the radius for s-states depend on the tip–sample distance z, but it also depends on the differences between different types of state. At large distances, for which κz  1, the apparent sizes of images with different types of atomic state differ by a constant. The relative difference is reduced with increasing distance. At short distances, where κz ≈ 1, the apparent sizes for different types of atomic state can differ by a factor of 2, 4, or even more. This is in accordance with the result in Chapter 5 that at large tip–sample distances, that is, κz  1, and for large features, where κa  1, the difference between different tip states is reduced. Naturally, we observed the reciprocity principle: by interchanging the tip state and the sample state, the conductance distribution, and consequently, the apparent radius of the image, are unchanged.

212

7.3.5

Atomic-Scale Imaging

Independent-orbital approximation

The previous sections presented a method for calculating the STM images based on leading Bloch functions. However, this method can only treat simple surfaces. For more complex surfaces, a few special points on the surface Brillouin zone are not sufficient. Besides, the choice of special points on the surface Brillouin zone becomes ambiguous. In the theory of gas– surface interactions [236, 237], the summation of pairwise interactions has been a powerful method for treating the problem of a single atom with surfaces. In this section, we present an alternative method for predicting STM images for crystalline surfaces utilizing the pairwise summation concept. It is based on the assumption that the total tunneling conductance can be approximated as the sum of the tunneling conductance from individual atomic states on the sample surface. In other words, the atomic orbitals are assumed to be independent. This assumption is not always valid. For example, it does not work for solids exhibiting charge-density waves. However, for many metal and semiconductor surfaces, it provides an adequate description. We discuss the general method using the example of a crystalline surface with square lattice and tetragonal symmetry, that is, belonging to the plane group p4mm. General expressions for the Fourier coefficients are given. Other applications, for example, Si(111)-2×1 and close-packed metal surfaces, will be left as Problems. A tetragonal lattice can be considered as a combination of two perpendicular one-dimensional lattices with the same lattice constant and having a reflectional symmetry, as shown in Fig. 7.21. Therefore, we start with a one-dimensional case. The tunneling conductance from the n-th atom is g(x − na, z). The total tunneling conductance from all the atoms is G(x, z) =

∞ 

g(x − na, z).

(7.48)

n=−∞

Apparently, it is a periodic function with periodicity a. Therefore, it can be expressed as a Fourier series, G(x, z) =

∞ 

˜ j (z) eijbx , G

(7.49)

j=−∞

where b = 2π/a is the length of the primitive vector of the reciprocal lattice. The Fourier coefficient is ˜ j (z) = 1 G a 1 = a

 

a

∞ 

dx e−ijbx

0

g(x − na, z)

n=−∞ ∞

dx e −∞

−ijbx

g(x, z).

(7.50)

7.3 Analytic treatments

213

Fig. 7.21. Close-packed surface with tetragonal symmetry (a) The square lattice in real space. There is an atom on each lattice point. (b) The reciprocal space.

As shown, owing to the infinite sum in Eq. 7.48, the integration extends over the entire x axis. To construct an image including the lowest non-trivial Fourier components, only three terms in the Fourier series are significant. Those are: ˜ 0 (z), G ˜ −1 (z), and G ˜ 1 (z). Because of the reflectional symmetry of the conG ductance function g(x, z), the last two Fourier coefficients are equal, and ˜ 1 (z). Up to this term, are denoted as G ˜ 1 (z) cos(bx). ˜ 0 (z) + 2G G(x, z) = G

(7.51)

Following the general relation between current images and topographic images, Eq. 6.6, the topographic image is Δz(x) = 

˜ 1 (z0 ) 2G ˜ 0 (z0 ) dG

cos bx.

(7.52)

dz0 Therefore, the problem of calculating the STM images reduces to the problem of evaluating the Fourier coefficients for the tunneling conductance distribution of a single atomic state, Eq. 7.50. This treatment can be extended immediately to the case of surfaces with tetragonal symmetry. The Fourier coefficients for the tunneling conductance of a tetragonal lattice are   ˜ nm (z) = 1 G dx dy e−i(nx+my) g(r). (7.53) a2 ˜ −1,0 = G ˜ 0,1 = G ˜ 0,−1 . We ˜ 1,0 = G Because of the axial symmetry of g(r), G ˜ denote them as G1 . The tunneling current distribution, up to the lowest non-trivial Fourier components, is ˜ 1 (z) (cos bx + cos by). ˜ 0 (z) + 2G G(r) = G

(7.54)

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Atomic-Scale Imaging

The maximum value of the function (cos bx + cos by) is 2, and the minimum ˜ 1 , it is convenient to ˜ 0 is much larger than G value of it is –2. Because G define a tetragonal cosine function φ(4) (X) ≡

1 1 + (cos X + cos Y ) . 2 4

(7.55)

The function φ(4) (kx) has a value 1 at each lattice point, and 0 at the center of four neighboring lattice points. Eq. 7.54 thus becomes ˜ 1 (z) φ(4) (bx). ˜ 0 (z) + 8G G(r) = G

(7.56)

The topographic image is Δz(x) = 

˜ 1 (z0 ) 8G ˜ 0 (z0 ) dG

φ(4) (bx).

(7.57)

dz0 In order to obtain the final form of the theoretical images, we need to calculate the Fourier coefficients for the functions f (r) = rn−1 cosm θ e−2κr . We start with the mathematical identity:   2π −γz dx dy −2κr+ipx+iqy e e = , I≡ r γ

(7.58)

(7.59)

where r2 = x2 +y 2 +z 2 and γ 2 = 4κ2 +p2 +q 2 . A proof of this mathematical identity can be found in [236]. By taking derivatives with respect to z, a factor x/r = cos θ is generated. Within the same approximation of the Slater atomic wavefunctions, the differentiation acts on the exponential factor only. Therefore,   dx dy rn−1 cosm θ e−2κr+ipx+iqy n  m −γz  e ∂ ∂ − . = 2π − 2∂κ 2κ∂z γ

(7.60)

Using these equations and the conductance distribution functions listed in Table 7.2, the corrugation amplitudes for a tetragonal close-packed surface with different tip states and sample states can be obtained. For example, for a 1s state, using Eqs 7.59 and 7.60, we have ˜ 0 (z) = πz e−2κz , G κ and

(7.61)

7.4 First-principles studies: tip electronic states

215

Table 7.3: The independent-orbital model. The theoretical corrugation amplitudes for surfaces with a tetragonal symmetry. Tip

Sample

s

s

s

pz

s

dz 2

pz

s

pz

pz

pz

dz 2

dz 2

s

dz 2

pz

dz 2

dz 2

Corrugation amplitude 16κ −βz e γ2  γ 2 16κ e−βz 2κ γ2 2  3 γ2 16κ −βz 1 − e 2 4κ2 2 γ2  γ 2 16κ e−βz 2κ γ2  γ 4 16κ e−βz 2κ γ2 2  γ 2  3 γ 2 16κ −βz 1 − e 2 2κ 2 4κ 2 γ2 2  3 γ2 16κ −βz 1 − e 2 4κ2 2 γ2 2  γ 2  3 γ 2 16κ −βz 1 − e 2κ 2 4κ2 2 γ2   4 3 γ2 16κ −βz 1 − e 2 2 4κ 2 γ2

˜ 1 (z) = 4πκz e−γz . G γ2 From Eq. 7.57, the corrugation amplitude is Δz ≡ − 

˜ 1 (z0 ) 8G ˜ 0 (z0 ) dG

=

16κ −βz e . γ2

(7.62)

(7.63)

dz0 Here β = γ − 2κ. For p and d states, using Eq. 7.60, similar results can be obtained. A list of formulas is shown in Table 7.3.

7.4

First-principles studies: tip electronic states

It is clear that the atomic and electronic structures of nanotips, and their interactions with the samples, are essential to the understanding of the STM

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Atomic-Scale Imaging

images. Between the late 1980s and early 2000s, a number of extensive firstprinciples numerical studies are devoted to that problem [92, 98, 99, 158, 238, 239, 240, 241, 242, 243]. Here we discuss the common conclusions: First, the tips used in STM and AFM for atomic-resolution studies are made of either transition metals, for example, W, Ir, Pt, etc., or semiconductors, notably Si. All first-principles studies on transition-metal tips showed that, near the Fermi level, the d states dominate the tip DOS. The DOS of s states is of the order of a few percent, and the DOS of p states is of the order of some 10%. More than 80% are d states. Furthermore, different d states appear at different energy levels. All first-principles studies on semiconductor tips, notably Si tips, showed that on many types of tip structure, the p states, especially the sp3 dangling-bond states dominate the Fermi-level DOS. Second, the p and d tip states, especially the dz2 states and sp3 states, protrude deeply into the vacuum, and show the strongest interactions with the sample surface. Third, the computed atomic corrugations of the Fermi-level LDOS on low-Miller-index metal surfaces at a reasonable tip–sample separations—0.1 to 0.3 nm from the mechanical contact, z = ze —are often more than one order of magnitude smaller than the experimentally observed values. In most cases, the atomic corrugations of the Fermi-level LDOS are close to or less than one picometer, way below the detection limit of STM instruments. However, with dz2 -tip states and sp3 -tip states, the corrugation of tunneling conductance could be enhanced up to more than one order of magnitude, which agrees with experimental observations.

7.4.1

W clusters as STM tip models

To investigate the STM imaging mechanism, Ohnishi and Tsukada [92] made a first-principles computation of the electronic states of a number of W clusters. Their main results are summarized in Fig. 7.22. The electron configuration near the tip apex depends on the structure of the cluster, as well as the energy level. As shown, for cluster W5 , the HOMO is basically a dxy state at the tip apex. The state just below HOMO is a dz2 state. For cluster W4 , the HOMO is a dz2 state, whereas the state just below HOMO is a dxy state. Their results indicate that, by changing the bias voltage, different dstates at the tip apex can be selected. It is also affected by the atomic structure near the tip apex. Using Green’s function methods, they also found that about 90% of the tunneling current is contributed by those dstates. Ohnishi and Tsukada [92] also proposed that the dz2 orbitals would be advantageous for sharp STM images.

7.4 First-principles studies: tip electronic states

217

Fig. 7.22. Electronic states of W clusters near the Fermi level Charge-density contours for several states on W clusters. (a) and (c) The highest occupied molecular orbitals (HOMO) of W4 and W5 . (b) and (d) The eigenstates just below HOMO. At low bias, these d-like tip states contribute more than 90% of the tunneling current. (Reproduced with permission from Ohnishi and Tsukada [92]. Copyright 1989 Elsevier.)

7.4.2

DFT study of a W–Cu STM junction

To study the electronic states and interactions of an STM system, Lamare et al. [241] made a first-principles study of a coupled system, consisting of a W cluster and a Cu cluster, as shown in Fig. 7.23(a). DFT with localdensity approximation (LDA) is used in the computation. Electron density plots are obtained and analyzed. The main results are as follows: First, as shown in Fig. 7.23(b), the total DOS of the W cluster is dominated by d-electrons. The sp-component is insignificant. Second, in the STM configuration, the main contributions to the tip– sample chemical bonding are those of the W-5d states. Among them, the strongest contribution is because of the W-5dz2 atomic tip states. The W-5dxz and W-5dx2 −y2 being less important. These results suggest the major role played by the 5dz2 states in the electronic structure of STM tungsten tips. 7.4.3

Transition-metal pyramidal tips

Another first-principles study of the electronic structures of transition-metal nanotips, especially the dependence of the tip electronic states with tip configuration, was conducted by Ness and Gautier [239]. The tip is modeled as a pyramid of four layers, supported by a semiinfinite bcc crystal. Both (001) and (111) orientations are considered. The pyramidal tips are computed using a tight-bonding method, typically with eight recursions. It is found that, at the apex atom, various d states dominate the DOS from a few eV below the Fermi level to a few eV above.

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Fig. 7.23. A model STM system: a W tip and a Cu sample. (a) A firstprinciples study of a model STM system with a W cluster as the tip, and the flat surface of a Cu cluster as the sample. Density-functional theory is used in the computation. (b) Total and partial DOS of a tungsten-tip cluster. As shown, the DOS is dominated by the d-states. (Reproduced with permission from [241]. Copyright 1999 Elsevier.)

Special attention was paid to the influence of tip-atom relaxation upon the electronic structure, as shown in Fig. 7.24. The relaxation parameter da is defined as the distance of the apex atom from the immediate layer of the pyramid versus the corresponding distance in the bulk. As shown, the partial density of states for different components of the d states is well separated on the energy scale, and sensitive to da . 7.4.4

Transition-metal atoms adsorbed on W slabs

Hofer et al. [242, 243] made two large-scale first-principles studies of the electronic states of STM tips, modeled as a transition-metal atom adsorbed on a W slab. Their motivation was to settle a long-term question as to why the contamination of a tip with a single atom should play a key role in STM imaging: although the Tersoff–Hamann model [115, 116] is widely used, a more realistic model of the tip should include the p- and d-tip states. A firstprinciple study of the electronic states of a single atom on a W tip would provide answers. The relaxation effect was taken in full. Taking advantages of recent advances in computing power and models, they expected to gain a better understanding of the STM images. They modeled the tip as a five layer p(2×2) supercell with a single apex atom. They demonstrated that the electronic structure of a realistic tip is determined by the mixture of s, p, and d states. Contrary to frequent assumptions, the DOS around the Fermi level is not constant, but highly dependent on the energy and the contaminant atom. Typical results from their studies are shown in Fig. 7.25. For all 5d tran-

7.4 First-principles studies: tip electronic states

219

Fig. 7.24. Electronic states of a W pyramid. The STM tip is modeled as a fourlayer W pyramid, supported by a semi-infinite bcc crystal. The effect of atom relaxation is also shown. The relaxation parameter da is the ratio of the apex atom to the next layer of the pyramid versus the distance in the bulk. (a), (b) and (c) show the effect of a slight change of the relaxation to the distribution of the electronic states. As shown, the partial density of states for different components of the d-states is well separated on the energy scale, and varies with the relaxation parameter da . (Reproduced with permission from [239]. Copyright 1995 IOP Publishing Limited.)

sition metal atoms, the contributions of the s-band were fairly insignificant. It is dominated by the d states [242]. Furthermore, they also found that for all 3d and 4d metal atoms [243], there is a dominance of d states in the apex atom for the energy range considered (–2 eV to +2 ev). In most cases they found a composition of pz - and dz2 -like states dangling out of the tip atom and decaying exponentially for larger vertical distances. The result suggests that STM tips contaminated with a 3d, 4d, or 5d transition-metal atom will measure currents and corrugations much higher than that from the Fermi-level DOS contours. To summarize, the first-principles studies of STM tips have reached the following general conclusions: (1) If the tip has a transition-metal apex atom, adsorbed on a typical tip stem, for example made of W, then near the Fermi level (–2 eV to +2 eV), the d states dominate the density of states. (2) On all the transition-metal tips studied to date, only a few percent of the DOS near the Fermi level are from s states. Therefore, the contributions of s-wave tip states is fairly insignificant.

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Atomic-Scale Imaging

Fig. 7.25. Electronic states of a transition metal atom adsorbed on a W slab. The tip is simulated by a five-layer p(2×2) supercell with a single apex atom, which is a transition-metal atom. For all transition metal atoms, including 3d, 4d, and 5d, the contributions of the s-band were fairly insignificant. It is dominated by the d-states. In most cases there is a composition of pz - and dz2 -like states dangling out of the tip atom and decaying exponentially for larger vertical distances. (Reproduced with permission from Hofer et al. [242]. Copyright 2000 Elsevier.)

(3) The different types of d states, especially the m = 0, |m| = 1, and |m| = 2 states, are in general shifted from each other on the energy scale. At a narrow interval of energy, one of the d-states would dominate. (4) In most cases, it was found that, there was a composition of pz - and dz2 -like states dangling out of the tip atom and decaying exponentially for larger vertical distances.

7.5 7.5.1

First-principles studies: the images Transition-metal surfaces

Strong atomic corrugations are observed on W(110), Ta(110), and Fe(110) surfaces with STM [90]. A first-principles study shows that the atomic corrugation predicted by the Tersoff–Hamann model is one order of magnitude too small; and the existence of p-tip states, especially the d-tip states, provides a satisfying explanation [90], see Fig. 7.26. The tip–sample distance z0 is estimated using the universal relation between tunneling conductance and tip–sample distance, Eq. 1.24, G = G0 e−2κ(z−ze ) ≈ 77.48 e−2κ(z−ze ) [μS].

(7.64)

From the tunneling conditions, It =10 nA, V = 40 mV, or G = 0.25 μS, Take κ ≈ 11 nm−1 , one finds z − ze ≈ 2.4 nm. Because the lattice plane separation in tungsten is 0.16 to 0.25 nm, the absolute tip–sample distance is 0.40–0.46 nm. From Fig. 7.26(b), one finds that for an s-tip state, the

7.5 First-principles studies: the images

221

Fig. 7.26. Experimental and theoretical corrugations on W(110). (a) Atomic corrugation observed on W(110) surface with STM. The corrugation amplitude is about 15 pm. (2) Theoretical predictions of atomic corrugations, comparing with experimental observations. Squares, circles, and diamonds represent the numerical results with s-, pz and dz2 -tip states, respectively. The arrow marks the tip–sample distance z0 inferred from tunneling conditions. The corrugation from a s-wave tip state is one order of magnitude too small. The corrugation from a dz2 -tip state fits well with the experimental observation. (Adapted with permission from Heinze et al. [90]. Copyright 1998 American Physical Society.)

corrugation amplitude is about 1 pm. For a dz2 -tip state, the corrugation amplitude is about 10 pm, which agrees with the experimental value. To verify the enhancement factors due to the pz - tip state and the dz2 -tip states, Heinze et al. [90] performed a first-principles numerical computation. As shown in Fig. 7.26(b) the results of the numerical computations, marked in squares, circles, and diamonds, agree well with the analytical expressions, Eqs 7.22 and 7.23. The first-principles computation confirmed the analytical treatments of the corrugation enhancement due to pz - and dz2 -tip states. Another topic discussed there is the bias-voltage-dependent corrugation inversion observed on transition-metal surfaces [90], see Fig. 7.27. At the W(110) surface, the d states dominate to charge density. However, the different types of d states are peaked at different energy levels. At one energy Fig. 7.27. Corrugation inversion on W(110) surface. At different energy levels around the Fermi level, different types of d-states dominate the DOS. If an m = 0 or dz2 state dominates, the corrugation of DOS should be noninverted. If an m = 0, or dxz , dyz , dxy , or dx2 −y2 state dominates, the corrugation of DOS should be inverted. This is similar to the case of tip states, see Figs 7.17, 7.22, and 7.24. (Adapted with permission from Heinze et al. [90]. Copyright 1998 American Physical Society.)

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Atomic-Scale Imaging

level, the dz2 state dominates. A non-inverted DOS corrugation could be observed. At some other energy levels, the m = 0 d states could dominate the sample DOS. An inverted corrugation in DOS should be observed. According to the principle of reciprocoty, it is similar to the case of tip states, where different types of d tip states shift in energy level. At a specific energy level, one of those d tip states could dominate. See Section 7.4. 7.5.2

Atomic corrugation and surface waves

In Section 6.2.6, we presented the theory and experimental observations of scattered surface states in terms of the Tersoff–Hamann model [115, 116]. Because the typical periodicity of the surface waves is 1 to 2 nm, the STM images of the surface waves are independent of the tip structure and tip chemistry. However, under favorable tip conditions, the underlying atomic structure can also be observed [227, 228]. A study for the simultaneous observation of scattered waves and atomic corrugation on the Be(10¯ 10) surface has been conducted [244], as shown in Fig. 7.28. In Fig. 7.28(a), two experimental curves, in thin lines, show the combined effect of the atomic corrugation with a periodicity of 0.3 nm, and the scattered surface wave with a much longer periodicity, but with amplitude decaying with the lateral distance. A Fourier analysis of the experimental corrugation, Fig. 7.28(a), showed that the observed corrugations can be decomposed into three components, as shown in Fig. 7.28(b). The first component (3) is the atomic corrugation. The second component (4) is the free-electron component. The third component (5) is the long-period oscillation. The Fermi-level LDOS, obtained from first-principles numerical computations, shows a very small atomic corrugation. To resolve the discrepancy, Briner et al. [244] assumed that the tip has a dz2 state. A satisfactory overall agreement with experiments is achieved. Instead of representing the tunneling current with LDOS,  I(r|| , z0 ) = const ×

EF +eV

ρ(r|| , z0 , E) dE,

(7.65)

EF

Briner et al. use the expression of the tunneling matrix for a dz2 -tip state, following the derivative rule presented in Section 3.3,  I(r|| , z0 ) = const ×

EF +eV EF

 2  ∂ψ(r|| , z0 ) 1  2   ψ(r − , z ) || 0  d k|| .  κ2 ∂z 2 3

(7.66)

Here the lateral wave vector k|| is the parameter of the wavefunctions ψ. Tunneling images are obtained by numerically evaluating Eq. 7.66. The theoretical curves are shown in Fig. 7.27(a), in bold curves. As shown, for

7.5 First-principles studies: the images

223

¯ Fig. 7.28. Atomic corrugation and surface wave on Be(1010). (a) Theoretical and experimental STM images of two cases. The theoretical curves are derived by assuming a dZ 2 -tip state. (b) Three components of the observed corrugation, obtained by Fourier analysis: (3), short-period atomic corrugation. (4), free-electron component. (5), long-period oscillation. (Reproduced with permission from Briner et al. [244]. Copyright 1998 American Physical Society.)

both cases, except for the region immediately adjacent to the scattering center, the theoretical curves agree well with the experimental curves. According to [244], the underlying physics is as follows. The s-wave approximation predicts a strong suppression of the short-period Bloch components in STM images. With a dz2 -tip state, the STM image of the longperiod oscillations stays almost identical to that of the s-wave tip model. However, the dz2 -tip state makes a one-order-of-magnitude enhancement for atomic corrugations. Therefore, with the assumption of a dz2 -tip state, the entire image is well explained. 7.5.3

Atom-resolved AFM images

The observation of atomic resolution in AFM, Fig. 1.22, highly suggests the critical role of tip electronic states. The atomic corrugation changed three times in a single scan, owing to spontaneous tip restructuring. Only for one of the imaging modes was a clear atomic resolution observed. In addition, stable imaging with true atomic resolution is only possible over a restricted range of tip–sample distances. Because both the sample and the tip are made of silicon, using the growing power of computers, one could assume many different types of tip structure, and make a first-principles computation of the force distribution, to clarify the imaging mechanism. Careful and convincing numerical simulations were published by P´erez et al. [98, 99, 240]. In their studies, the sample surfaces are various members of the Takayanagi reconstructions, including Si(111)-3×3, Si(111)-5×5, and Si(111)-7×7. The Si(111)-5×5 surface contains most of the features of the

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Atomic-Scale Imaging

Fig. 7.29. Silicon clusters as models for non-contact AFM tips. Three types of clusters are considered, based on the fact that most likely the tips have the natural cleavage plains, Si(111). The simplest one, Si4 , does not have saturated dangling bonds in the base. The two with hydrogen termination, Si4 H9 and Si10 H15 , yield very similar results. (Adapted with permission from P´ erez et al. [98]. Copyright 1997 American Physical Society.)

Si(111)-7×7 version, but computationally more tractable. Three types of Si tips were tried, as shown in Fig. 7.29. From Si4 to Si4 H9 , the hydrogen termination makes a significant difference. However, the difference between the Si4 H9 and the Si10 H15 is insignificant, indicating that the Si10 H15 is quite close to a bulk tip, defined by three natural (111) cleavage planes. The interaction energies and atomic forces are calculated with DFT, in its plane-wave pseudopotential formalism. Because the long-range van der Waals interaction cannot be described correctly by the standard DFT method, a Hamaker summation method is used to treat it, see Section 4.1.3. The Hamaker constant for the Si-Si interaction is A = 1.16 eV [99]. Assuming the radius of curvature of the Si tip is Rc ≈ 4 nm, from Eq. 4.17, the van der Waals interaction energy is U =−

ARc 0.776 ≈ [eV]. 6R R(nm)

(7.67)

Fig. 7.30. Van de Waals force between a Si tip and a Si surface. The van der Waals force between a tip with local radius of curvature R and a flat sample surface. See also Section 4.1.3. (Adapted with permission from P´erez et al. [99]. Copyright 1998 American Physical Society.)

7.5 First-principles studies: the images

225

Table 7.4: Parameters of the Morse functions Atom Adatom Rest atom Two tips

U0 (eV) 2.273 2.636 3.094

a (nm−1 ) 14.97 15.26 13.28

Rc (nm) 0.235 0.235 0.229

Since 1eV/1nm = 0.16 nN, the normal van der Waals force is F ≡−

∂U 0.124 ≈− 2 [nN]. ∂R (R(nm))

(7.68)

In the non-contact AFM experiments, the physical quantity directly measured is the shift of the resonance frequency of the cantilever. Details are presented in Chapter 15. For small vibrational amplitudes, the frequency shift is directly proportional to the force gradient. For large vibrational amplitudes, the force gradient can be inferred from the frequency shift, although non-trivial. Theoretically, the force gradient can be obtained by differentiating the Morse function, fitted to the results of numerical computations. Fig. 7.31(b) shows the force gradients over the same sites. The force gradient of the van der Waals force is also shown. From the figure, it is clear that the force gradient on the adatom is much stronger than the van der Waals force gradient.

Fig. 7.31. Fitting of the chemical-bond forces by Morse functions. The chemical-bond force between a single-atom tip and various positions of the Si(111)-5×5 surface can be fitted accurately with a Morse function. (a) The force on various locations as a function of tip–sample distance. (b) The force gradient on various locations as a function of tip–sample distance. The case of the force between two tips and the van der Waals force are also shown. (Adapted with permission from P´ erez et al. [99]. Copyright 1998 American Physical Society.)

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Atomic-Scale Imaging

The covalent chemical forces between the tip and the sample are calculated using DFT method. The force as a function of tip–sample distance can be fitted with a Morse function, Eq. 4.84:   E (R) = U0 e−2a(R−Re ) − 2e−a(R−Re ) ,

(7.69)

where U0 is the dissociation energy, Re is the equilibrium internuclear distance, and a is a constant. The force curves for three different cases are shown in Fig. 7.31(a). The parameters of the Morse functions are listed in Table 7.4. The adatom shows a strong chemical-bond force even as far as 0.4 nm. On the other hand, the rest atom would show a significant chemicalbond force only when the tip–sample distance reaches 0.2 nm. The force between two Si4 H9 tips is also shown for comparison. The lateral distribution of force at a tip–sample distance of 0.5 nm is shown in Fig. 7.32. Fig. 7.32(a) is a normal projection of the Si(111)-5×5 surface. The force along two lateral scan lines, A–A and B–B, is shown in Fig. 7.32(b). Apparently, only the chemical-bond force shows a strong corrugation. From the calculated force distribution, it is clear that the covalent chemical bond interaction dominates the force gradients, which provides a mechanism to achieve atomic resolution [98, 99, 240].

Fig. 7.32. Chemical-bond force between a Si tip and a Si surface. A map of the normal force between a single-Si-atom tip and a Si(111)5×5 surface. Two cross sections are shown: along a symmetric axis of the Si(111)5×5 structure, A–A, and the off-diagonal adatoms, B–B. (Adapted with permission from P´ erez et al. [99]. Copyright 1998 American Physical Society.)

7.6 Spin-polarized STM

7.6

227

Spin-polarized STM

Another important case for which the tip cannot be considered as a geometrical point is spin-polarized STM. Here the spin polarization of the tip electronic states plays a central role. The direction of the tip spin polarization is determined by two angle parameters, for example, the azimuth (or longitude), and the zenith (or polar angle). The degree of the tip spin polarization is characterized by a polarization parameter. All those parameters, including the direction and degree of spin polarization, often depend on energy level, or bias voltage. Therefore, the property of the tip cannot be taken out of the problem, and the STM images are not related to a property of the surface alone. Section 2.4 presents a general theory of tunneling with electron spin, together with an application to the spin-valve effect [149]. Following Heinze [245], we present a simple theory of the spin-polarized STM based on the independent-orbital approximation, see Section 7.3.5. To treat the tunneling problem with spin polarization, similar to Section 2.4, the electronic states of both the tip and the sample are represented by two-component spinors. To simplify derivation, we use the coordinate systems in which the z-component of the Pauli matrix is diagonized, and assume that the spatial wavefunction identical. For the tip,     1 0 , ΨT ↓ = χ(r) . (7.70) ΨT ↑ = χ(r) 0 1 For the n-th surface atom on the sample,     1 0 , Ψn↓ = ψn (r) . Ψn↑ = ψn (r) 0 1

(7.71)

Spin polarization is characterized by the difference in density of states of the spin-up and spin-down states at a given energy level, ρT ↑ = ρT ↓ , ρn↑ = ρn↓ .

(7.72)

In general, the direction of polarization of the tip eT is different from that of an individual atom on the sample en , as shown in Fig. 7.33. Denoting the angle between eT and en as θn , in the coordinate system of the tip, the wavefunctions of the n-th surface atom are [18]   cos(θn /2) i sin(θn /2) Ψn↑ = ψn (r) , Ψn↓ = ψn (r) . (7.73) i sin(θn /2) cos(θn /2)

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Atomic-Scale Imaging

Fig. 7.33. Spin-polarized STM in independent orbital approximation. The tip, centered at rT , has a spin polarization direction eT . The degree of spin polarization, or the relative difference of the spin-up DOS and spin-down DOS, is schematically shown by the two arrows. On each atom of the sample surface, the direction en and the polarization could be different. By assuming that the total tunneling conductance is the sum of the tunneling conductance of each surface atom, a simple mathematical model is established, which fits well with experimental data.

The tunneling matrix elements are   M M cos(θn /2) i sin(θn /2) n↑↑ n↑↓ ˆn = M = Mn0 Mn↓↑ Mn↓↓ i sin(θn /2) cos(θn /2)

.

(7.74)

For simplicity, we assume the spatial factor of the tunneling matrix element  2 [ψn ∇χ∗ − χ∗ ∇ψn ] · dS (7.75) Mn0 = 2m Σ is independent of the spin. Similar to Eq. 2.122, for the n-th atom of the sample surface, the tunneling conductance is  Gn = 2π 2 G0 ρT ↑ ρn↑ |Mn↑↑ |2 + ρT ↑ ρn↓ |Mn↑↓ |2  (7.76) +ρT ↓ ρn↑ |Mn↓↑ |2 +ρT ↓ ρn↓ |Mn↓↓ |2 . As usual, one defines the spin-averaged density of states ρT = ρT ↑ + ρT ↓ ,

ρn = ρn↑ + ρn↓ ;

(7.77)

and the spin-polarized density of states mT = ρT ↑ − ρT ↓ ,

mn = ρn↑ − ρn↓ .

(7.78)

7.6 Spin-polarized STM

229

Substituting Eqs 7.74, 7.77, and 7.78 into Eq. 7.76, using the identities cos2

θn θn + sin2 = 1, 2 2

(7.79)

θn θn − sin2 = cos θn , (7.80) 2 2 we obtain an expression for the tunneling conductance between the tip and the n-th sample atom, cos2

Gn = π 2 G0 |Mn0 |2 ( ρT ρn + mT mn cos θn ) .

(7.81)

The value of the tunneling matrix element |Mn0 | can be estimated using the Landauer theory of tunneling, see Sections 1.2.3 and 2.2.10. For simplicity, the spatial wavefunction of the tip is modeled as being spherically symmetrical, similar to the Tersoff–Hamann model [115, 116]. Furthermore, the spatial wavefunctions of the sample atoms are also modeled as spherically symmetrical. Therefore, in addition to the spin polarization parameters, we only need a distance to describe the interaction. The minimal distance between the tip atom and a sample atom is the equilibrium distance re , typically 0.25 nm, see Eq. 1.24. It is convenient to use the off-equilibrium distance between the tip nucleus and the nucleus of the n-th atom of the sample surface, (7.82) rn ≡ |rT − rn | − re . Using Eq. 2.75, one finds 4π 2 |Mn0 |2 ρT ρn = e−2κrn ,

(7.83) √ where κ = 2mφ/ is the decay constant, and φ is the work function. As usual, we define the polarization parameter of the tip PT and that of the n-th sample atom Pn as PT ≡

mT , ρT

Pn ≡

mn . ρn

(7.84)

Using Eqs 7.83 and 7.84, the tunneling conductance between the tip and the n-th atom, Eq. 7.81, becomes Gn =

1 G0 (1 + PT Pn cos θn ) e−2κrn . 2

(7.85)

In the spirit of independent orbital approximation, see Section 7.3.5, the total tunneling conductance is the sum of the partial tunneling conductances for each atom on the sample surface, as shown in Fig. 7.33,   1 G= Gn = G0 (1 + PT Pn cos θn ) e−2κrn . (7.86) 2 n n

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Atomic-Scale Imaging

In spite of the simplicity in concept and mathematics, in many cases the calculated images are in excellent agreement with experimental data, and with those obtained from computationally much more demanding ab initio calculations [62, 245]. For many practical structures, using the method in Section 7.3.5, especially Eqs 7.59 and 7.60, the sum in Eq. 7.86 can be evaluated analytically to generate close-form expressions.

7.7

Chemical identification of surface atoms

The Tersoff–Hamann model [115, 116] descbibes the STM image as determined by a single physical quantity of the sample surface, the Fermi-level LDOS at the center of curvature of the tip, ρS (EF , r0 ). The information provided by this quantity is not sufficient to distinguish the chemical identity of surface atoms. Even by extending the Tersoff–Hamann model to finite bias voltages, the information is still too sparse for that purpose. In order to discern the chemical identity of surface atoms, the role of tip electronic structures is essential. And for the interpretation, it is better to rely on the individual orbital model rather than the continuum model. As an example, Ruan et al. [246] performed experiments by reversibly manipulating the chemical identity of the apex atom of the tip. Clear and reproducible information for the chemical identification of surface atoms is found (Fig. 7.34). The experiments were performed with an UHV STM, at room temperature, on Cu(110) and Ni(110) surfaces. After exposing a Cu(110) surface to 4 langmuir of oxygen, a (2×1)O structure is formed, which covers about 20% of the surface. Fig. 7.34(a) is an atomistic model. The small black dots represent the O atoms, and the large hollow circles are added rows of Cu atoms. The gray circles are the original Cu atoms of the (110) surface. Two types of tip are used, both based on the standard electrochemically etched tungsten tip. The first is made by cleaning the tip using field emission, i.e. by ramping the bias to 5–10 V in 1 msec. This generally causes the transfer of material from the tip to the sample. A clean W tip is expected. The second is by letting tiny amounts of oxygen into the chamber (0.05 L), or by just scanning the (2×1)O area for a while. A sudden tip restructuring would occur at some time, resulting in a vastly different image. It is expected that an O atom is attached to the tip apex. By doing field emission again, the clean W tip and the original image are recovered. Therefore, the entire process is reversible. An image from the ‘clean W’ tip is shown in Fig. 7.34(b). On the (2×1)O regions, the protrusions are aligned with the Cu(110) rows. Comparing with Fig. 7.34(a), the positions of the protrusions coincide with those of the O atoms. Another image, with the ‘O contaminated’ tip, Fig. 7.34(c), shows that the protrusions are out of registry with the Cu rows. Comparing with

7.8 The principle of reciprocity

231

Fig. 7.34. Chemical identification of surface atoms. (a) An atomistic model of the oxygen-induced (2×1)O structure on the Cu(110) surface. The small black dots represent the O atoms. The large circles are added rows of Cu atoms. The gray circles are underlying Cu atoms of the (110) surface. (b) The STM image with a clean W tip. The protrusions are aligned with the Cu(110) rows, pointing to the O atoms. (c) The STM image with an oxygen-contaminated tip. The protrusions are aligned with the Cu(110) valleys, pointing to the added Cu atoms. (Adapted with permission from Ruan et al. [246]. Copyright 1993 American Physical Society.)

Fig. 7.34(a), the positions of those protrusions coincide with the positions of the added Cu atoms. To interpret those images, neither the Fermi-level LDOS nor a perturbation of the Fermi-level LDOS are appropriate. Ruan et al. [246] proposed the following explanations in terms of a partial chemical bond between the apex atom of the tip and the surface atoms. For a clean W tip, the apex W atom may form a partial chemical bond with the O atom on the surface. For an O contaminated tip, the O atom may form a partial chemical bond with the added Cu atoms, and the chemical bond reveals itself in tunneling. These experiments clearly showed that chemical identification of surface atoms can be practiced by intentionally changing the tip structure. Different types of atoms are highlighted under different tips. And the experiments could be reproducible and reversible. In view of the equivalence between the covalent chemical bond and the tunneling conductance we discussed in Chapter 5, their explanation is plausible.

7.8

The principle of reciprocity

At atomic scale, the imaging mechanism of STM and AFM is a convolution of tip electronic states and sample electronic states. That fundamental symmetry has a consequence: if the electronic state of the tip and that of the sample under observation are interchanged, the image should be the same. In other words, at atomic resolution, an STM or AFM image can be interpreted as either imaging the sample state with a tip state, or imaging the tip state with a sample state. This is the principle of reciprocity of STM and AFM at an atomic scale.

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Atomic-Scale Imaging

Practically, this principle has two consequences. First, if the sample is well characterized, for example, by first-principles computations or independent measurements, especially when the features on the sample surface are well separated, then the STM or AFM is imaging the tip states using the known sample. Second, to obtain a complete and reproducible information with STM and AFM, a two-step experiment is desirable: first, by characterizing the tip with a known sample; and second, by using the already characterized tip to image an unknown sample. The observations of the electronic states of the tip using STM [248] and AFM [247] were reported. In both experiments, the sample is Si(111)-7×7 surface, and the sp3 dangling bond on each adatom was used as the ‘probe’. Because the lateral distance between adjacent adatoms is about 0.7 nm, and the diameter of an atomic orbital is about 0.3 nm, each image of the tip apex atom is independent. In other words, the overlap would be negligible, and the individual image of the apex atom of the tip is clearly recognizable. In the first experiment, a refined frequency-modulation method is used to detect the attractive force between the tip and the sample, and it is used to generate a constant-force topographical image. The details of that technique are presented in Chapter 15. The sample is the Si(111)-7×7 surface, prepared using the standard method of annealing at 1300◦ C for 30

Fig. 7.35. Observation of a tip state by AFM. (a) A constant-force topographical image observed on a Si(111)7×7 surface. The image of each dangling bond on top of a adatom on the Si(111)7×7 surface splits into two crescent-like features with an average distance of less than 0.2 nm. This distance is smaller than the interatomic distance of the tip, either W or Si. Therefore, it must be a feature of a single atom at the tip apex. (b) The line scan of the image. (Reproduced with permission from Giessibl et al. [247]. Copyright 2000 the American Association for the Advancement of Science.)

7.8 The principle of reciprocity

233

seconds. During the scan, the features of the image often undergo spontaneous changes. In one instance, the shape of the image of a dangling bond splits into two crescent shaped features, one strong and one weak (Fig. 7.35). From the shape and the distance, which is less than 0.2 nm, it is unlikely to be caused by a double-tip effect. The seam between the two crescents is likely to be a nodal surface between two lobes of opposite signs in a wavefunction. In the second experiment [248], a dynamic-mode STM is applied – see Chapter 15 for details. This technique allows the tip to approach closer to the sample, thus to improve the quality of the signal. A Co6 Fe3 Sm tip is used. The image of each dangling bond on top of an adatom on the Si(111)-7×7 surface appears as a pattern of straw-hat shape (Fig. 7.36). The measured radius of curvature of the central feature is 0.12 to 0.15 nm along two directions – see Fig. 7.37(b). Apparently, the lateral resolution should be close to that radius. To interpret the observed radius of curvature, we recall first the basic concepts of the Tersoff–Hamann model. If both the tip and the sample atom are spherically symmetric, keeping a constant tunneling conductance means keeping the distance between the center of the tip and the center of the sample atom constant. Also, the tip could move around the sample atom. Therefore, the observed radius of curvature equals the vertical distance between the center of the tip and the center of the sample atom. In Tersoff and Hamann’s original paper, see Section 6.2 and Fig. 6.1, the distance between the center of the tip and the sample surface is 1.5 nm. Following that data, the radius of curvature of the constant-current image

Fig. 7.36. Observation of a tip state by STM. A constant-current topographical image observed on a Si(111)-7×7 surface. The image of each dangling bond on top of an adatom on the Si(111)-7×7 surface appears as a pattern of straw-hat shapes, which may enable a very high spatial resolution. The arrow points to a defect of the Si(111)-7×7 surface, a missing center atom. (Adapted with permission from [248]. Copyright 2003 American Physical Society.)

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Fig. 7.37. Quantitative profile of a tip state observed by STM. (a) Details of a single image of an adatom on the Si(111)-7×7 surface. (b) The radius of curvature of the A − A cross section and the B − B cross section. By using a local parabolic fit, the radius of curvature is found to be 0.12 nm and 0.15 nm. (Adapted with permission from [248]. Copyright 2003 American Physical Society.)

should be 1.5 nm, which is 10 times greater than the observed value. Assuming that the tip and the sample are still spherically symmetric, but the distance is much closer, then the discrepancy is reduced. However, since the closest distance between the center of the tip atom and that of a sample atom is 0.25 nm, even the tip atom is in firm contact with the sample atom, and the radius of curvature of the constant current image should still be 0.25 nm. Realistically, the shortest distance should be 0.35 nm, and the radius of curvature should be greater than 0.35 nm. Apparently, for the actual circumstances of their experiment, the assumption that the tip is spherically symmetric must not be correct. In fact, according to Section 7.3.5, for different types of tip states and sample states, the radius of curvature is different. For example, for a dz2 tip state and a pz sample state, the apparent radius of curvature is, see Table 7.2,   1 1 4 = 1+ . (7.87) R z κz Since κ = 11.4 nm−1 , with distance z ≈ 0.35 nm, the apparent radius should be 0.15 nm. However, the estimated distance z is probably too small. Herz et al. [248] proposed that the tip state is a 4fz3 state of a Sm atom, for which the apparent radius of curvature is   1 1 7 = 1+ . (7.88) R z κz

Chapter 8 Imaging Wavefunctions The concept of wavefunction was introduced in the first 1926 paper by Erwin Schr¨ odinger entitled “Quantisierung als Eigenwertproblem (erste Mittelung)” [160] as the central object of the atomic world and the cornerstone of quantum mechanics. It is a mathematical representation of de Broglie’s postulate that the electron is a material wave. In that historical paper, wavefunction ψ(r) was defined as “everywhere real, single-valued, finite, and continuously differentiable up to the second order.” Schr¨ odinger defined the quantity −e|ψ(r)|2 as the charge density distribution of an electron extended in real space. The normalization condition  d3 r |ψ(r)|2 = 1 (8.1) is required because the total charge of an electron is always the elementary charge −e. The meaning of −e|ψ(r)|2 as the charge density distribution of a single electron is the basis for quantum-mechanical treatments of manyelectron atoms, molecules, and solid state. The wavefunction of a manyelectron system is a Slater determinant of the wavefunctions of individual electron states. The wavefunction of each individual electron state, also known as an orbital, is the solution of the Schr¨odinger equation in a potential field formed by the nuclei and the electrical charge densities of all other electrons. It is the basis of the self-consistent field theory, including the Hartree method and the Hartree-Fock method, especially the DFT [249, 250]. It is documented in standard quantum mechanics textbooks [251, 252] as well as standard quantum chemistry textbooks [253, 254]. Nevertheless, for many decades since its introduction as a physical concept, wavefunction has not been characterized as an observable for legitimate reasons. First, it is too small. The typical size of a wavefunction is a fraction of a nanometer. Second, it is too fragile. The typical bonding energy of a wavefunction is a few electron volts. Using an optical microscope is out of the question: the wavelength of visible light is thousands of times greater than the size of a wavefunction. Heisenberg [255] suggested using a gamma ray microscope to observe electrons in a subatomic scale. However, the energy of such gamma ray photons is thousands of times greater than the bonding energy of the wavefunctions, which would severely disturb the electrons under observation. This was an argument for his uncertainty principle. An electron microscope has a much better resolution than an optical *OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

236

Imaging Wavefunctions

microscope. However, to achieve spatial resolution smaller than a fraction of a nanometer, the energy of the electrons is still too high. According to de Broglie, the wavelength of an electron with energy E in eV is  1.504 h h = λ= = √ nm. (8.2) p E 2me E Table 8.1 shows some typical values of the wavelength λ of an electron. Therefore, wavefunction is not observable by an electron microscope. The advancement of STM and AFM has made wavefunctions observable. First, the accuracy of position determination can be a fraction of a picometer. Second, both STM and AFM are nondestructive, which leaves the object of observation undisturbed. The importance of nondisturbance in quantum mechanical measurement process is discussed in Section 8.4. The meaning of expression “imaging wavefunctions” needs to be clarified. To take advantage of the Euler formula e−ix = cos x + i sin x, many quantum mechanics textbooks define wavefunctions as complex. However, it contradicts the original definition of wavefunctions by Schr¨odinger, and it is often unnecessary. According to Wigner’s theorem, if the Hamiltonian is time-reversal invariant, all wavefunctions can be real, see Appendix B. For applications in STM and AFM, it is always fine to follow the original definition of Schr¨odinger that all wavefunctions of bound states are real. In most cases, the wavefunction has multiple lobes of opposite signs. By multiplying the wavefunction with −1, it is still a good solution of the Schr¨odinger equation. Nevertheless, the signs of all lobes are reversed. The field quantities directly observed and mapped by STM and AFM are derived from the local values of wavefunctions, as shown by the following examples. First, the square of the wavefunction, which is proportional to the charge density distribution of the electron at that location, ρ(r) = −e|ψ(r)|2 . Second, the square of the lateral derivatives of the wavefunction, as measured using a p-type tip wavefunction. Third, the absolute value of the wavefunction, as measured through the chemical-bond interaction energy between an s-type tip and the sample. Lastly, the absolute value of the lateral derivatives of the wavefunction by AFM using a p-type tip. If the wavefunctions are degenerate (in the tip or in the sample), then a linear superposition of the degenerate wavefunctions can change the observable quantity from an individual wavefunction. However, because the degenerate wavefunctions are orthogonal and normalized, the observable quantity arising from all de-

Table 8.1: Wavelength and electron energy Energy (eV) Wavelength (nm)

25

50

100

200

0.245

0.173

0.123

0.087

8.1 Use of ultrathin insulating barriers

237

generate wavefunctions at the same energy level is invariant under a linear superposition, for example, due to a coordinate rotation. For all those cases, a field quantity representing the local values of wavefunctions is observed and mapped [3]. An umbrella term “imaging wavefunctions” is thus used. For more details, see Section 8.4.

8.1

Use of ultrathin insulating barriers

Atoms and molecules adsorbed on metal surfaces have been observed even in the early years of STM. However, in those early experiments, the wavefunctions of the adsorbed atoms and molecules are seriously perturbed by the metal substrate. The images are often interpreted as a disturbance of the LDOS of the metal surface at the Fermi level caused by the adsorbed atoms or molecules. By placing an ultrathin insulating barrier between the atom or molecule under observation and the metal substrate, atoms and molecules, including the amplitudes and the derivatives of the wavefunctions, can be investigated in pristine condition. A well-verified system consists of two atomic layers of NaCl on a Cu(111) substrate [256, 257, 258]. The perturbations of the metal substrate to the wavefunctions become negligible. The substrate with an ultrathin NaCl insulating film on Cu(111) is relatively easy to prepare and well studied. The underlying substrate, Cu(111) surface, can be prepared with repeated sputtering and annealing. NaCl is then evaporated thermally, with the sample temperature kept at about 320 K. NaCl forms (100)-terminated islands up to several microns in size, starting with a double layer [257]. Islands of three-layer film also exist, which

Fig. 8.1. Ultrathin insulating barrier for imaging wavefunctions. (A) An STM image of a NaCl film on Cu(111) substrate. (B) A schematic diagram of a top view, showing a (100) structure of NaCl on top of the Cu(111) surface. (C) A lateral view. The average thickness of the double NaCl film is about 564 pm. The interference to the atoms and molecules from the metal substrate is substantially reduced.

238

Imaging Wavefunctions

can be identified by STM, see Fig. 8.1(A). The structure, with a lattice parameter of 0.564 nm, is shown in Fig. 8.1(B). The energy gap of NaCl is 8.5 eV, which has been shown to exist already for a bilayer [3]. It is equivalent to a vacuum gap of 0.564 nm, see Fig. 8.1(C). The work function of Cu(111) is 4.94 eV. The gap causes a decay constant √ κ = 5.1 × 4.94 ≈ 11.3 nm−1 . (8.3) The reduction ratio for the tunneling current is R = e−0.564×11.3 ≈ 1.7 × 10−3 .

(8.4)

The tunneling current reduction can be compensated by increasing amplification, whereas the perturbation to the wavefunctions under observation is reduced by about three orders of magnitude.

8.2 8.2.1

Imaging wavefunctions with STM Imaging atomic wavefunctions

One of the most well-known wavefunctions, described in all quantum mechanics textbooks, is the ground state of a hydrogen atom. It is 1 |1s =  3 e−r/a0 , πa0

(8.5)

where

4π 0 2 ≈ 52.9 pm me e 2 is the Bohr radius. The energy level is a0 ≡

E=−

me e4 ≈ −13.6 eV. 32 20 π 2 2

(8.6)

(8.7)

Fig. 8.2. Wavefunction of ground-state hydrogen atom. (A) Amplitude plot of the ground-state wavefunction of hydrogen. (B) The amplitude profile.

8.2 Imaging wavefunctions with STM

239

Table 8.2: Wavefunction radii and energy levels. Atom

Configuration

H Li Na K Cu Ag Au

1s 1s2 2s ...2s2 2p6 3s ...3s2 3p6 4s ...3p6 3d10 4s ...4p6 4d10 5s ...5p6 5d10 6s

Radius (pm)

Energy level (eV)

53 167 190 243 145 165 174

-13.60 -5.39 -5.14 -4.34 -7.72 -7.57 -9.22

A density plot and an amplitude contour of the wavefunction of ground-state hydrogen atom are shown in Fig. 8.2. As shown, the equal-amplitude contours of the wavefunction are spheres. If it is possible to map the amplitude contours, the results should be spheres of different radii. However, the size of the wavefunction of the ground-state hydrogen atom is very small, and the energy level is much too deep with respect to the work function of the metal substrate, typically 5 eV. Other atoms with a single outer electron are the alkali metals and noble metals. The size of the wavefunctions and the energy levels are shown in Table 8.2. As shown, for alkali metal atoms (Li. Na, K), the energy level is very close to the Fermi level and roughly equals the work function of the metal substrate. Therefore, the wavefunction of those atoms tends to spread over to the surface. The three group-11 elements (Cu, Ag, Au), the so-called noble metals, have a single electron moving in the potential field of a spherical charge distribution formed by the nucleus and other electrons. The energy

Fig. 8.3. STM images of Au atoms on NaCl film. The images of Au atoms on a NaCl insulating film appear as protrusions of apparent height of 0.20 nm to 0.25 nm, as part of a spherical electrical charge distribution. The charge state of the Au atom can be switched by applying electrical pulses. (A) By applying an electrical pulse of V ≥0.6V near the center of the Au atom, the charge state of the Au atom is apparently changed, as the diameter and height of the image is changed, see (B) By applying another electrical pulse of -1 V, the charge state of the Au atom is switched back, see (C).

240

Imaging Wavefunctions

Fig. 8.4. Explanation of the charge-state switching of an Au atom. Calculated electronic and geometrical properties of the neutral (A and C) and negatively charged (D and F) Au atom. The geometry of the atomic arrangement and the energy levels are changed due to the state switching After Repp et al. [1].

levels are a few eV below the Fermi level. Those metal atoms are convenient for wavefunction observation and mapping. Figure 8.3 shows examples of experimental observations of Au atoms on a NaCl insulating film. As expected, the images of the Au atoms appear as round-shaped protrusions of 0.20 to 0.25 nm, the top part of the contours of the spherical electrical change distribution. An interesting effect is, the charge state of the Au atoms can be switched using electrical pulses. As shown in Fig. 8.3(A), by applying a pulse of V ≥ 0.6 V on top of an Au atom, the image is dramatically changed, see Fig. 8.3(B). By applying a pulse of higher voltage and opposite polarity, the state can be switched back, see Fig. 8.3(C). Another interesting fact is, before and after state change, not only the shape of the image is changed, but also peak bias voltage to observe the images is changed. It is because the energy levels of different states are different, and the bias voltage realigns the energy levels. Figure 8.4 is an explanation of the process. The geometry of the atomic arrangement and the energy levels are changed due to the state switching. 8.2.2

Imaging molecular wavefunctions

The STM imaging of the wavefunctions of metal atoms on an insulating barrier showed definitively its observability, as its charge density distribution can be mapped in real space [1]. Much richer images can be observed on organic molecules on top of the ultrathin insulating films. With an swave tip state of a typical metallic apex atom, representing the square of wavefunctions at different energy levels [13] 2

Is (x, y) ∝ |ψ(x, y, z0 )| ,

(8.8)

are mapped in real space [258, 259]. Not only are the structures of the wavefunctions more complex, for many organic molecules, but also both

8.2 Imaging wavefunctions with STM

241

Fig. 8.5. Charge density and STM image of naphthalocynine. (A) The molecular structure of naphthalocynine. (B) The theoretical charge density distributions and STM images of naphthalocynine, for both HOMO and LUMO. The theoretical charge densities contours are computed using density-functional theory. The STM images closely resemble the theoretical charge density contours, represented by the square of the wavefunctions. [261].

the HOMO (highest occupied molecular orbitals) and LUMO (lowest unoccupied molecular orbitals) states have energy levels within the reach of STM by using reasonable bias voltages. In 2005, the wavefunctions of both HOMO states and LUMO states of pentacene were observed by STM [258]. The images and the energy levels fit well with the wavefunctions and energy eigenvalues of the free molecules from first-principle numerical computations using DFT based on quantum mechanics [260], see plate 14. Figure 8.5 shows another example of imaging naphthalocynine molecule on a NaCl insulating film. (A) is the molecular structure. (B) shows the theoretical and experimental images. The HOMO state is observed with a bias of -1.75 V, and the LUMO state is observed with a bias of +0.6 V. The charge density contours, represented by the square of the wavefunctions, are computed using DFT methods. The STM images are shown to closely resemble the charge density contours, or the square of wavefunctions. Those STM images show clearly that the electron charge density distributions, or the square of the wavefunctions, are observable field quantities. 8.2.3

Imaging nodal structures

In 2011, using a tip functionalized by a CO-molecule to image organic molecules adsorbed on a NaCl thin film, completely different STM images were observed [3]. The process of controlled transfer of a CO atom between the STM tip and the sample was discovered in 1997 [206]. The physics of the process and the electronic states of the CO molecule is well understood [207], see Section 14.7.2. The carbon atom is directly attached to the metal base. The oxygen atom has a pair of degenerate px and py states, that dominate the tunneling process. According to the derivative rule of tunneling

242

Imaging Wavefunctions

Fig. 8.6. STM images of HOMO wavefunctions of pentacene, s-wave-tip and p-wave-tip. (a) The theoretical and observed images of HOMO wavefunctions by an s-wave tip, from [258]. (b) The theoretical and observed images of HOMO wavefunctions by an p-wave tip, from [3].

theory [13, 15, 262], the tunneling matrix elements are proportional to the derivatives of the sample wavefunction ψ, which are Mpx ∝

∂ψ ∂ψ and Mpy ∝ , ∂x ∂y

(8.9)

respectively. The tunneling current Ip (x, y) is proportional to the sum of the squared tunneling matrix elements. On a plane z = z0 , it is      ∂ψ(x, y, z0 ) 2  ∂ψ(x, y, z0 ) 2  +  . Ip (x, y) ∝     ∂x ∂y

(8.10)

Fig. 8.7. STM images of LUMO wavefunctions of pentacene, s-wave tip and p-wave-tip. (a) The theoretical and observed images of LUMO wavefunctions by an s-wave tip, from [258]. (b) The theoretical and observed images of LUMO wavefunctions by an p-wave tip, from [3].

8.2 Imaging wavefunctions with STM

243

Fig. 8.8. Mechanism of STM imaging with a p-wave tip. (a) The tip wavefunction is dominated by the px and py orbitals of the O atom. At the center of a lobe of the sample wavefunction, the contributions to the tunneling current from the two lobes of the p-orbital cancel each other. The net tunneling amplitude is zero. (b) At a nodal plane of the sample wavefunction, the contributions from the two lobes of the p-orbital of the tip wavefunction are additive, and the tunneling amplitude reaches maximum. (c) At a site of four-fold asymmetry, the contributions of different lobes of the p-orbital of the tip again cancels each other.

The predicted and observed images from such a p-type tip for the HOMO wavefunction of pentacene are shown in Figure 8.6(b). Those for the LUMO wavefunction are in Figure 8.7(b). Those images resemble the squares of the lateral derivatives of wavefunctions. A combination of images with an s-type tip and a p-type tip enables the experimental determination of complete wavefunctions up to a global sign. Typically, in many places where the tunneling current from a metal tip Is (x, y) (see Eq. 8.8) vanishes, the current from a p-type tip Ip (x, y) (see Eq. 8.10) is at its maximum. As a consequence of Schr¨ odinger’s equation, the derivatives of the wavefunction should be continuous. A sign change in the wavefunction is observed. Consequently, the phase contrast of different lobes of wavefunctions is observable. Therefore, the entire wavefunction can be mapped experimentally up to a global sign. Figure 8.8 is an intuitive explanation of the observed contrast. The CO molecule has two degenerate πx and πy orbitals near the Fermi level, that dominate the tunneling current. As shown in Fig. 8.8(a), on the O atom of the tip, the wavefunctions are basically a px and a py atomic orbitals, each has two lobes. Here the polarity (positive or negative) of the lobes are marked by grey or white. If the tip is located on top of the center of a lobe of the sample wavefunction, the tunneling current from the positive lobe and

244

Imaging Wavefunctions

Fig. 8.9. Imaging naphthalocynine with a Cu tip and a CO tip. (a) The topographical image of naphthalocynine HOMO wavefunction using a Cu tip. The gross features are resolved. (b) and (c) Images of the same, using a CO tip. The fine features of the eight-fold patterns are clearly resolved. (d) Theoretical images of the square of the HOMO wavefunction, showing similar features as (a). (e) and (f) Theoretical images of the square of lateral derivatives of the HOMO wavefunction of naphthalocynine, showing great details of the eight-fold features. The theoretical image based on a mixture s-state and p-state in (f) fits perfectly with the observed image.

the negative lobe of the p-orbital cancel each other. The tunneling matrix element |M | is zero. In Fig. 8.8(b), the tip axis is located at a nodal plane of the molecular orbital. The tunneling amplitudes from the two lobes of the p-orbital of the O atom to the two sides of the sample orbital have the same sign, thus are added together, similar to a constructive interference. The tunneling conductance reaches a maximum. Figure 8.8(c) shows a third case. At a sample site of a four-fold symmetry, the tunneling conductance from different lobes of the p-orbital again cancel each other, and shows a zero tunneling current. Figures 8.9 and 8.10 show the images of another molecule, naphthalocynine, with more details on the effects of the combinations of different tip wavefunctions. Figure 8.9 shows the theoretical results and the observed images of the HOMO wavefunction of naphthalocynine with a Cu tip and a CO tip. As shown in (b) and (c), by using a CO tip, many more details of the wavefunction are resolved than the images with a Cu tip, as shown in Fig. 8.9(c) . Furthermore, the theoretical image based on a mixture of s-tip wavefunction and p-tip wavefunction fits accurately to the observed image with a CO tip. It indicates that the actual tip wavefunction is a mixture s-tip wavefunction and p-tip wavefunction.

8.3 Imaging wavefunctions with AFM

245

Fig. 8.10. Imaging naphthalocynine with a Cu tip and a CO tip. (a) The topographical image of naphthalocynine LUMO wavefunction using a Cu tip. The gross features are resolved. (b) and (c) Images of the same, using a CO tip. The fine features of the eight-fold patterns are clearly resolved. (d) Theoretical images of the square of the LUMO wavefunction, showing similar features as (a). (e) and (f) Theoretical images of the square of lateral derivatives of the HOMO wavefunction of naphthalocynine, showing great details of the eight-fold features. The theoretical image based on a mixture s-tip wavefunction and p-tip wavefunction in (f) fits perfectly with the observed image.

In Fig. 8.10, images of the LUMO wavefunction of naphthalocynine are presented. The images with a CO tip, Fig. 8.10(b) and (c), show more details of the internal structure of the LUMO wavefunction than with a Cu tip, Fig. 8.10(a). Again, the theoretical images based on a mixture of s-tip wavefunction and p-tip wavefunction makes a perfect match to the experimental images using a CO tip. The interpretation of the STM images with a CO-functionalized tip as associated with lateral derivatives of sample wavefunctions has been extended and verified by further experiments. For example, in 2015, Corso et al. reported the observation of enhanced resolution of internal structures of acetylene by a CO-functionalized tip [263]. In 2018, Shiotari et al. reported the observation of lateral derivatives of molecules with a NO-functionalized tip, which also has 2pπ ∗ states at the apex [264].

8.3

Imaging wavefunctions with AFM

As discussed in Chapter 4, the chemical bond energy can be represented by the Bardeen tunneling matrix elements. Therefore, the main component of

246

Imaging Wavefunctions

Fig. 8.11. AFM images of a pentacene molecule with different tips. The nature of the AFM image depends drastically on the tip. (A) Ag tip. (B) CO tip. (C) Cl tip. (D) Pentacene tip. See Gross et al. [210].

the attractive atomic force, the chemical bond energy, is capable of imaging the sample wavefunction. A straightforward inference from the STM imaging of wavefunctions in the previous sections is that the observed AFM images should dramatically depend on the atomic structure of the tip. A properly functionalized tip could improve the resolution. One complication of AFM compared to STM is, the Pauli repulsive force is an unavoidable component. The repulsion between atomic cores would eventually appear at shorter tip-sample distances. In 2009, both the effects of tip functionalization and the Pauli repulsive force were observed with pentacene molecules on a Cu(111) substrate buffered by two atomic layers of NaCl, see Gross et al. [210]. Figure 8.11 shows four AFM images of a pentacene molecule with different tips. (A) is from an Ag tip. As shown, only a single broad peak is observed. (B) is from a CO tip. A lot of details, including the atomic skeleton, are apparent. (C) is from a Cl tip. The resolution is much higher than the Ag tip, but the image is somewhat different from that with a CO tip. (D) is from a pentacene tip. The image is very different from that with a CO tip. Apparently, using different tips, different features related to the molecular wavefunction are imaged. Furthermore, by measuring the AFM frequency shift (Δf ) versus voltage dependence to obtain the local contact potential difference (LCPD) at each point in the space, three-dimensional images of the charge distribution within a single molecule are obtained [265]. The experiment was done on naphthalocyanine on a Cu(111) substrate, with a two atomic buffer layer of NaCl, similar to the conditions in Sections 8.2.2 and 8.2.3. A plot of the three-dimensional electrical charge density inside the molecule is shown

8.4 Meaning of wavefunction observation

247

Fig. 8.12. Three-dimensional electrical charge map inside a molecule. The vertical height in nm on each cross-sectional plot is in reference to the STM set point. The scale bar in each graph is 0.5 nm. See Mohn et al. [265].

in Fig. 8.12. The vertical height in nm on each cross-sectional plot is in reference to the STM set point (I = 2 pa, V = 0.2 V). The scale bar in each graph is 0.5 nm. As shown, the closer the tip is to the molecule base plane, the more details of the charge distribution discovered. Therefore, the three-dimensional electrical charge distribution of the electron, or the square of the wavefunction, can be measured and mapped in real space without significant disturbance to the system under observation.

8.4

Meaning of wavefunction observation

The concept of wavefunction was introduced in the first 1926 paper by Schr¨ odinger [160] as the mathematical representation of de Broglie’s material wave of an electron, which is “everywhere real, single-valued, finite, and continuously differentiable up to the second order.” Schr¨odinger defined the quantity −e|ψ(r)|2 as the charge density distribution of an electron. It is the basis for quantum-mechanical treatments of many-electron atoms, molecules, and solid states [249, 250, 251, 252, 253, 254]. The experimental observation and mapping of wavefunction through local field quantities by STM and AFM, including the square of wavefunction and the square of the lateral derivative as well as their absolute values, implied that the wavefunction is a physical field. Schr¨odinger’s original interpretation is confirmed. The statistical interpretation of the wavefunction, although valid and indispensible, has to be properly defined.

248

8.4.1

Imaging Wavefunctions

Interpretations of wavefunctions

Many quantum mechanics textbooks [266, 267, 268] present the following interpretation of wavefunction. Wavefunction as one of the representations of an abstract state is not observable. Its absolute square |ψ(r)|2 as a real function in space is also not observable. As a material point, the position of an electron in real space is an observable. Each time a measurement of the position of an electron is conducted, values of its coordinates are returned. In general, the result of each position measurement is different. The only meaning of the wavefunction is: the probability of finding an electron in an elementary volume d3 r around r is P (r) d3 r = |ψ(r)|2 d3 r.

(8.11)

Some textbooks [266, 267] further state that after a position measurement, the wavefunction collapses to a position eigenstate as a delta function ψ(r) =⇒ δ(r − r0 ), where r0 is the result of measurement. Max Born was awarded a Nobel prize in physics in 1954 for his statistical interpretation of the wavefunction. Nevertheless, according to his Nobel Lecture, his statistical interpretation has a well-defined meaning. He did not mention anything related to wavefunction collapse [269]: Wave mechanics enjoyed a very great deal more popularity than the G¨ottingen or Cambridge version of quantum mechanics. It operates with a wave function ψ, which in the case of one particle at least, can be pictured in space, and it uses the mathematical methods of partial differential equations which are in current use by physicists. Schr¨odinger thought that his wave theory made it possible to return to deterministic classical physics. He proposed (and he has recently emphasized his proposal anew), to dispense with the particle representation entirely, and instead of speaking of electrons as particles, to consider them as a continuous density distribution |ψ|2 (or electric density −e|ψ|2 ). To us in G¨ottingen this interpretation seemed unacceptable in face of well established experimental facts. At that time it was already possible to count particles by means of scintillations or with a Geiger counter, and to photograph their tracks with the aid of a Wilson cloud chamber. Born referred to the matrix mechanics of Heisenberg as the G¨ottingen version and the q-number formulation of Dirac as the Cambridge version of quantum mechanics. According to Born, Schr¨odinger’s version is a great deal more popular because of its intuitiveness and easier mathematics. According to Born, Schr¨ odinger insisted that the electron is an extended physical field. And the quantity −e|ψ|2 represents the charge density distribution of the electron as a continuous field. In other words, Schr¨odinger

8.4 Meaning of wavefunction observation

249

disagreed with the view that the electron can be represented by a geometrical point of ultimate sharpness and arbitrary precision. Later developments favored Schr¨odinger’s definition of electron charge density distribution. It is the basis of the DFT, now a standard method for computational quantum mechanics [249, 250]. Accordingly, the total electron density in a many-electron system is the sum of the density distributions of individual electron wavefunctions, n(r) =

N 

|ψj (r)|2 .

(8.12)

j=1

8.4.2

Wavefunction as a physical field

The experiments described in this Chapter indicate that the wavefunction is a physical field. Especially, −e|ψ(r)|2 is a charge density distribution. The nature of wavefunction is similar to the Maxwellian electromagnetic fields, enabling a realistic interpretation of quantum mechanics. In an article for the centenary of Maxwell’s birth, Maxwell’s Influence on the Development of the Conception of Physical Reality, Einstein wrote [270]: Before Maxwell, people thought of physical reality—in so far represented events in nature—as material points, whose changes consist only in motions which are subject to total differential equations. After Maxwell, they thought of physical reality as represented by continuous fields, not mechanically explicable and subject to partial differential equations. This change in the conception of reality is the most profound and the most fruitful that physics has experienced since Newton.

Fig. 8.13. Observing and mapping electrical fields. Using an electrical probe, the electrical field can be observed and mapped. Note the similarity to the STM mapping of wavefunctions by using another wavefunction as the probe.

250

Imaging Wavefunctions

Fig. 8.14. Observing and mapping magnetic fields. Using iron filings, the magnetic field can be observed and mapped. Note the similarity to the STM mapping of wavefunctions by using another wavefunction as the probe.

For centuries after Newton, the forces between material points, including gravitational and electrical, were considered as acting over a distance. Maxwell introduced electromagnetic fields that exist outside the material points, governed by partial differential equations. The electromagnetic fields can be observed independently of the particles, and have energy density in the space; for example, the black-body energy of standing electromagnetic waves in a cavity, and the mass and momentum of gamma ray. Figure 8.13 shows an apparatus for observing and mapping the electrical field. Note that the probe for observing and mapping is an electrical device, which is also governed by the same Maxwell equations. Figure 8.14 shows an experiment for observing and mapping the magnetic field. Also note that the probes, iron filings, are magnetic devices, and the mapping process is governed by the same Maxwell’s equations. The observation and mapping of the wavefunctions by STM and AFM is similar to the observation and mapping of Maxwell’s electromagnetic fields. In the same article about Maxwell’s fields, Einstein made a criticism about the probabilistic interpretation of quantum mechanics [270]: Nevertheless, I am inclined to think that physicists will not be satisfied in the long run with this kind of indirect description of reality, even if an adaptation of the theory to the demand of general relativity can be achieved in a satisfactory way. Then they must surely be brought back to the attempt to realize the program which may suitably be designated as Maxwellian: a description of physical reality in terms of fields which satisfy partial differential equations in a way that is free from singularities. Einstein’s prophecy may have been realized as the wavefunction becomes an observable physical reality because of STM and AFM.

8.4 Meaning of wavefunction observation

8.4.3

251

Born’s statistical interpretation

Although the experimental observations of wavefunctions showed the similarity of the wavefunction to the electrical field and magnetic field, as soon as quantization is brought up, including energy, charge, and mass, Born’s statistical interpretation is valid and indispensable. The statistical interpretation is necessary even for electromagnetic waves. Take an example of the double-slit experiment, see Fig. 8.15. A number of single-photon detectors of width δ are on the detector side, D. Each detector is made of thousands of atoms and has many energy levels. An electromagnetic wave with circular frequency ω causes a transition between two quantum states with energy difference ω of a detector. The transition rate is proportional to the square of the field intensity at the location of that detector. Therefore, the probability of photon detection is proportional to the square of the field intensity at the location of a detector. In certain sense, this is Einstein’s statistical interpretation of electromagnetic waves. Note that in the entire space between the light sourse and the detectors, the only valid description of light is an electromagnetic wave. There is no point-like photons moving around. Energy is quantized only when light is produced or converted into other forms. Born was the first scientist to publish a paper about the relation of the probability of particle detection with the wavefunction in 1926 [271]. As Born stated in his Nobel Lecture [269], his inspiration of statistical interpretation came from Einstein’s idea that the probability density of the occurrence of light quanta is proportional to the square of field intensities. Born said, “This concept could at once be carried over to the ψ-function: |ψ|2

Fig. 8.15. Double-slit experiment with single-photon detectors. A plane wave of wavelength λ falls on a screen with two slits S1 and S2 of distance a. An interference pattern is formed on an array of single-photon detectors δ. The probability of photon detection is proportional to the square of field intensity at a detector.

252

Imaging Wavefunctions

ought to represent the probability density for electrons [269].” Therefore, Born’s statistical interpretation of wavefunctions is an analogy to Einstein’s statistical interpretation of electromagnetic waves. Note that Born’s statistical interpretation is valid only on a macroscopic scale, such as the double-slit experiment and the particle detection in scattering experiments, where each individual position detector (such as detector δ in Fig. 8.15, photographic plates, scintillators, Geiger counters, and tracks in Wilson chambers [269]) is composed of thousands of atoms. On the other hand, STM and AFM experiments show that wavefunction is an extended physical field on a subatomic scale. Although on a macroscopic scale the electrons appear as individual entities, on a subatomic scale, an electron is a continuous field. Both Born’s statistical interpretation and Schr¨ odinger’s field interpretation are valid and indispensable.

Chapter 9 Nanomechanical Effects As shown in the previous chapters, under normal STM operating conditions, there is a weak attractive force between the tip and the sample. Although the force is only a few nN, because the contact area is much smaller than 1 nm2 , the local stress could be high enough to generate a large strain. Furthermore, since the tunneling conductance is very sensitive to tip-sample distance, a measurable effect can be created. As early as 1978, a few years before the invention of STM, the effect of the weak attractive force on tunneling measurements was already discovered by Teague in a metal–vacuum– metal tunneling experiment [272]. An important issue related to the effect of force in STM and AFM is the mechanical stability of the tip-sample junction. If the tip-sample junction is too soft, under the influence of that force, the tip might jump into contact with the sample. The approaching and retracting processes become nonreversible. A hysteresis takes place. Nevertheless, experiments repeatedly showed that for well-prepared tips and rigid samples, there is no jump into contact. The hysteresis occurs only when the tip condition is pathological. Conversely, for soft samples, the phenomenon of jumping into contact always occurs. In this chapter, a simple mathematical model is presented to clarify the condition of mechanical stability of the STM junction. One of the mysteries in the earlier years of STM is that the corrugations observed on graphite could often be as large as a good fraction of a nanometer [273]. Soler et al. [198] interpreted the giant corrugation observed on graphite and other soft materials as due to the amplification of the corrugation by the deformation of the sample surface. Their theory [198] was confirmed and extended by Mamin et al. [274] and Mate et al. [275] to cases with a contamination layer between the tip and the sample. However, for hard materials, including most metal surfaces, the effect of force and deformation is to reduce the apparent corrugation, instead of amplifying it. This fact was pointed out by Ciraci, Baratoff, and Batra in 1990 [276], and confirmed by the first fully ab initio density-functional computation of a combined tip-sample system by Di Ventra and Pantelides in 1999 [277]. The effect of force and deformation can be determined quantitatively by measuring the apparent barrier height using a lock-in amplifier [199]. That method has been extended to the measurement of the force and deformation between a tip and a single molecule on the sample surface [278], which is a useful method for studying the mechanical properties of single molecules. *OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

254

9.1

Nanomechanical Effects

Mechanical stability of the tip-sample junction

Because of the existence of the attractive atomic force, under certain circumstances, the STM gap might become mechanically unstable. However, the stability of the tip-sample junctions is a necessary condition for successful STM and AFM experiments. The phenomenon of instability is also important for understanding the imaging mechanism of STM and AFM. Under certain conditions, the instability of the junction causes large observed corrugation. Consequently, it is important to know the conditions under which the tip-sample junction is mechanically stable. 9.1.1

Experimental observations

In the earlier years of STM, the instruments were not rigid enough, and the atomic structures of the tip were unknown and out of control. The instability or hysteresis of the tip-sample junction has been discussed [279, 280]. In recent years, precision design of instruments combined with careful preparation and characterization of tips result in a better understanding of the tip-sample junction. Especially, a series of experiments performed with a combined STM and FIM repeatedly showed that if the tip is well prepared, there is no jump-into-contact syndrome. The force versus tipsample distance curve shows no or only modest hysteresis, and the FIM images reveal an atomically unchanged tip apex after repeated approaching and retracting [281, 282]. Hooke’s law is valid even when the tip is in mechanical contact with the sample surface [281, 282]. A schematics of the instrument is shown in Fig. 9.1. It is a combination of STM and FIM. The sample is mounted on a cantilever. The deflection of the cantilever is detected by an optical interferometer. The sample stage with the interferometer can be shifted to the side, to expose the tip to the FIM field and screen. The experiments were performed in ultrahigh vacuum conditions (p < 2×1011 mbar) and at a temperature of 150 K. Au(111) thin films evaporated on mica substrates served as samples. At the beginning of the experiments, a tip made from single crystal W(111) wire is cleaned and sharpened in situ. The tip apex was tailored to its final trimer shape by means of field evaporation under visual control using FIM. Using those trimer tips, remarkably √ stable tunneling current was observed, and imaging of Au(111) 22× 3 reconstruction was readily achieved, which proves that the tip-sample distance could be reliably controlled on a 0.01-nm level. A typical STM experiment takes about 2.5 hours, with about 100 cycles of approaching and retracting. The tip structure is stable for hours. Two FIM images of the initial trimer tip and that afterwards are shown in Fig. 9.1 (a) and (b). As shown, only two residual-gas atoms are found to be adsorbed on the shaft of the tip. The trimer structure remains intact.

9.1 Mechanical stability of the tip-sample junction

255

Fig. 9.1. A combined STM and FIM. The sample is mounted on a cantilever. The defection of the cantilever is measured by an optical interferometer. The tip is attached to the piezo scanner. The sample stage with the interferometer can be shifted to the side, to expose the tip to the FIM field and screen. The tip is engineered and imaged by FIM. After the tunneling experiment, the atomic structure of the tip apex can be investigated again using AFM. (a) The FIM image of the tip before tunneling experiments. (b) The FIM image of the tip apex after tunneling experiments. The trimer structure (marked by stars) remain unchanged. Two additional atoms, probably from the rest gas in the vacuum system, show up in the FIM image. (Reproduced with permission from [281]. Copyright 1998 American Physical Society.)

To make the study statistically meaningful, three sets of approaching and retracting cycles were executed [281, 282]. Each set has 32 approaching– retracting cycles, see Fig. 9.2. For each set, the tip was retracted 0.5 nm from the 100 MΩ set point, then programmed to approach the sample by 1.0 nm, 1.5 nm, and 2.2 nm, respectively. Both strong attractive force and strong repulsive forces were observed, indicating that, each time, the tip atoms strongly interact with the sample. However, for 94/96 cycles, the process was completely reversible. There is no spontaneous jump-intocontact. The repulsive branch of the force curve is essentially linear and reversible, indicating that the Hooke’s law is still valid, see Fig. 9.2. In summary, systematic experiments have shown that if the instrument is well-designed, and the atomic structure of the junction—especially the structure of the tip apex—is well-defined, the approaching and retracting process is completely continuous and reversible. This is true even if the tip mildly touches the sample surface. It has two consequences to the understanding of the imaging mechanism of STM and AFM.

256

Nanomechanical Effects

First, most of the high-resolution images are obtained at small tipsample distances, typically 0.05 to 0.25 nm from a mechanical contact. In terms of absolute tip-sample distance (the distance from the nucleus of the apex atom of the tip to the nearest nucleus of the sample atoms) it is 0.3 to 0.5 nm. In that range of distances, the atomic corrugations are the strongest. These systematic experiments of repeated approaching and retracting showed that in such a distance range, the force curve is completely continuous and smooth. Even a mild collision would only occasionally cause instability and inelastic deformation, depending on the condition of the tip and the sample. It is consistent with the fact that the range of stable observation of true atomic resolution on close-packed metal surfaces could be more than 0.2 nm [88, 89]. Fig. 9.2. Records of three sets of approaching and retracting curves. Three approaching and retracting cycles were executed and recorded. Each set has 32 cycles. The reference point was determined by the tunneling condition: 1 nA at 100 mV bias voltage, or a tunneling resistance of 100 MΩ. Other values of tunneling resistance were estimated using Eq. 1.7 assuming the work function is about 5 eV. The point of zero force, or a mechanical contact, marked as M, corresponding to a tunneling resistance of about 10 kΩ, agrees well with the theoretical value of G0 = 77.48 μS. For each of the three sets, the tip was retracted 0.5 nm from the 100 MΩ set point, and then the tip was programmed to approach the sample by 1.0 nm, 1.5 nm, and 2.2 nm, respectively. (a) The tip is almost touching the sample at the closest approaching point. The approaching and retracting curves are completely reversible. (b) The tip is mildly touching the sample. The approaching and retracting curves are still completely reversible; only a minor hysteresis is observed. (c) The tip is pressing into the sample by more than 0.5 nm. Still, in most cases, the process is reversible. Only for two cases was substantial inelastic deformation observed. The repulsive branch of the force curve is essentially linear and reversible. In all cases, there is no evidence of spontaneous jump-into-contact. (Reproduced with permission from [281, 282]. Copyright 1998 American Physical Society.)

9.1 Mechanical stability of the tip-sample junction

257

Second, in the early years of STM, there were arguments that the atomic resolution images observed on metal surfaces were due to mechanical effects, such as the inelastic deformation of the tip and the sample, in analogy to the case of graphite. Numerous experimental evidences, including the above systematic study, have repeatedly shown that in the normal operation of STM and AFM with metal surfaces, the instability is an exception rather than a rule. The instability is often related to unfavorable instrument designs and unfavorable conditions of the tip and the sample, unless it is intentionally induced, for example, to make atom manipulation. In other words, the instability and inelastic deformation cannot be responsible for the atomic resolution routinely observed on metal surfaces under well-defined tip and sample conditions. 9.1.2

Condition of mechanical stability

Carefully designed experiments have shown that, under a wide range of conditions, the tip-sample junction is stable, the approaching and retracting process is reversible, and there is no jump-into-contact syndrome from a large separation up to the presence of a repulsive force [281, 282]. The observed stability is also verified by carefully executed first-principles computations of combined tip-sample systems. For example, in the case of a Si tip and a Si sample, even after full consideration of coordinate relaxation, the final force curve can be well described by a Morse function, which is completely continuous and reversible [98, 99, 240]. On the other hand, jump-into-contact syndrome and hysteresis have been observed in some other cases, especially on soft samples such as graphite [274, 275]. However, there are still questions raised in the literature from pure theoretical point of view. The first question is, is the jump-into-contact syndrome inevitable for any tip-sample junction? In other words, could any tip-sample junction be stable on the entire range of tip-sample distances [280]? If the answer is positive, then the second question is, what is the condition of mechanical stability of a tip-sample junction [283]? The experimentally observed stability of tip-sample junctions on the entire range of distances can be explained using a simple model based on the concept of coordination number [137]: the number of nearest neighbor atoms for an apex atom of the tip or a surface atom on the sample. A well-structured tip can be described as a pyramid, cut from a bulk crystal. For a (111)-oriented tip cut from a diamond structure, the coordination number is 3. For a (100)-oriented tip cut from a body-centered structure or a face-centered structure, the coordination number is 4. Assuming that the bond strength, or the spring constant with each nearest neighbor is roughly equal, the spring constant for the first layer of the tip pyramid is 3 or 4 times the spring constant of each bond. The coordination number of atoms in the sample is higher than those in the tip. The interaction force between

258

Nanomechanical Effects

Fig. 9.3. Stiffness of a W(100) tip. (a) The equilibrium configuration of the top layer of a tungsten tip, pointing to the (100) direction. (b) Under the action of a force F , a strain  is generated. The strain can be estimated from the classical elasticity theory: the force is acting on a surface area of a2 , which generates a strain  = F/a2 E, where E is the Young’s modulus, see Appendix E.

the apex atom of the tip and a sample atom is roughly one such bond. Therefore, the tip-sample interaction force is not sufficient to break all the bonds on the first atom either on the tip or on the sample. Here is a quantitative treatment for the condition of stability. By using experimental values of the materials properties, it is shown that a large class of tip-sample junctions can be stable over the entire distance range. If the deformation is not too large, then Hooke’s law should apply. The displacement of the tip-sample distance δζ due to an external force F is δζ =

F , kJ

(9.1)

where the stiffness of the tunneling junction, kJ , is related to the stiffness of the tip, kT , and that of the sample, kS , via the relation 1 1 1 = + . kJ kT kS

(9.2)

Consider first the stiffness of the tip. Many important tip materials, such as tungsten, are of bcc structure. Figure 9.3 shows the top layer of a (100)oriented tip, made of a metal of bcc structure. Reacting to a force F , the height of the top layer changes from h = a/2 to h→

a (1 + ), 2

(9.3)

where a is the lattice constant, and is the strain induced by that force. It can be estimated using Young’s modulus E of the material: =

F . a2 E

(9.4)

9.1 Mechanical stability of the tip-sample junction

259

Table 9.1: Stiffness of various pyramidal tips Item

Unit

W

Mo

Lattice constant a Young’s module E Tip stiffness kT

nm GPa N/m

0.316 408 157

0.315 310 119

Cr 0.288 248 87

It is clear from the geometry of the bcc lattice that, the top layer of the pyramid has 4 bonds, the second layer has 16 bonds, and the n-th layer has 4×n2 bonds. Assuming the stiffness of those bonds is approximately equal, the displacement δζ of the apex atom under a force F is δζ =

∞ F  1 F π2 . = 2aE n=1 n2 2aE 6

(9.5)

Therefore, the stiffness of the tip is kT =

12aE F = ≈ 1.216 aE. δζ π2

(9.6)

Table 9.1 shows the stiffness of three (100)-oriented pyramidal tips with commonly used bcc metals, W, Mo, and Cr. The data of Young’s moduli are taken from a standard reference book [284]. As shown in Chapter 4 and Chapter 5, for both the covalent bond and the metallic bond, the force and the interaction energy between a tip and a sample surface can be represented by a Morse function [169]:   F = −2κU0 e−κ(ζ−ζe ) − e−2κ(ζ−ζe ) , (9.7) where U0 is the binding energy, and ζe is the equilibrium distance, i.e., the distance at which the net force is zero. Here, we use ζ for the actual distance between the nuclei of the atoms, and use z for the tip-sample distance measured through the piezo. For electronic states near the Fermi level, the constant κ in the √ expression of the Morse curve is determined by the work function, κ = 2mφ/. Typically, κ = 11nm−1 . The maximum force occurs at κ(ζ − ζe ) = ln 2, with the value Fmax =

1 κ U0 . 2

(9.8)

The observed z-piezo displacement is then the sum of δz = F/kJ and the true tip-sample displacement ζ. Using Eqs 9.1 and 9.7, we find

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Nanomechanical Effects

z=ζ+

 2κU0  −κ(ζ−ζe ) e − e−2κ(ζ−ζe ) . kJ

(9.9)

Taking derivatives with respect to ζ on both sides, it becomes   dz = 1 − 8γ e−κ(ζ−ζe ) − 2e−2κ(ζ−ζe ) . dζ

(9.10)

Here a dimensionless quantity γ = κ2 U0 /4kJ is introduced as a measure of the stiffness of the STM or AFM system with respect to the force gradient. In order to have a stable tip-sample junction, the quantity dz/dζ must be positive. In other words,   8γ e−κ(ζ−ζe ) − 2e−2κ(ζ−ζe ) < 1. (9.11) The above function has the value –8γ at ζ = ζe , and 0 at ζ = ∞. The maximum is at   8γκ e−κ(ζ−ζe ) − 4e−2κ(ζ−ζe ) = 0. (9.12) It implies eκ(ζ−ζe ) = 4.

(9.13)

Therefore, the condition of stability is γ < 1.

(9.14)

Using Eq. 9.8, the condition can be written in a more convenient form, 1 κFmax < kJ . (9.15) 2 Equation 9.15 has an intuitive meaning. The left-hand side is the maximum force gradient. In order to have a mechanically stable tunneling junction, the stiffness of the junction, kJ , must be greater than the maximum gradient of the interaction force. The typical value of the decay constant κ is 11 nm−1 , and the typical value of maximum attractive force is 3 nN. The left-hand side is 15.5 nN/nm. If the sample is made of a similar material, its stiffness should be higher than the tip. The stiffness of the junction is greater than 0.5 kT . From Table 9.1, it is clear that condition 9.15 is always satisfied. The tip-sample junction is always stable. In STM experiments, various types of samples are used, such as gold, aluminum, graphite, etc. Some of them are soft. To make a quantitative estimation of the deformation, consider the case that the tip apex is on top of a hollow site, i.e., the center of a triangle of three adjacent surface atoms. From the geometry of the bcc structure, it is straightforward to show that the surface area S of the three atoms is

9.1 Mechanical stability of the tip-sample junction

261

√ 3 3 2 a , 4 and the thickness of the first layer is S=

(9.16)

1 h = √ a. 3

(9.17)

Thus, the deformation due to a force F on top of the three atoms is δζ =

hF . SE

(9.18)

The stiffness of the sample surface with respect to the tip is kS ≡

SE 9 F = = aE. δζ h 4

(9.19)

Equation 9.19 slightly overestimates the actual relaxation, because the force from the neighboring surface atoms is neglected. However, it should be a good order-of-magnitude estimate. It is much better than the case where the tip is on top of a surface atom, for which the effect of the six neighboring surface atoms could be substantial. Table 9.2 shows the stiffness of four close-packed metal surfaces frequently encountered in STM and AFM. The data of Young’s moduli are taken from a standard reference book [284]. The case of graphite is also included, where the lattice constant is the nearestneighbor distance, the surface area is that of a honeycomb, and the interlayer distance is obtained directly from the data sheet. Here is some numerical results. The typical maximum force Fmax of a covalent bond is 3 nN. The typical decay constant is 11 nm−1 . Therefore, the typical maximum force gradient, the left-hand side of Eq. 9.15, is 16.5 N/m. Taking an example of a W(100) tip and a Au(111) surface, the stiffness of the junction is −1  1 1 + ≈ 49.1 N/m. (9.20) kJ = 157 71.6

Table 9.2: Stiffness of some close-packed metal surfaces Item

Unit

Al

Au

Cu

graphite

Lattice constant a Young’s modulus E Surface area S Layer thickness h Surface stiffness kS

nm GPa nm2 nm N/m

0.405 62 0.213 0.233 56.5

0.408 78 0.216 0.236 71.6

0.361 128 0.169 0.208 104

0.142* 20 0.105 0.34* 6.18

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Nanomechanical Effects

Fig. 9.4. Relaxation of a W–Au STM junction. (a) The geometrical configuration of a W–Au STM junction. The apex atom of the W tip is on top of the hollow site, the center of the triangle of three Au atoms. The force between the tip and the sample causes a displacement δζT of the tip apex atom, and a displacement δζS of the three sample atoms. (b) Variation of the relaxation displacements, δζT and δζS , with the tip-sample distance as measured by the piezo, z. If the tip and the sample are well-structured, the junction is completely stable over the entire range of tip-sample distance, with no jump-into-contact syndrome, and no hysteresis.

It is about three times the maximum force gradient of the typical value of a chemical bond, or γ = 0.33. Therefore, a well-structured W-Au STM junction is completely stable over the entire range of tip-sample distance. Figure 9.4 shows the dependence of the relaxation displacements of a W(100)tip and a Au(111) sample over a range of tip-sample distance from 0.25 nm to 0.6 nm. The maximum total relaxation, the sum of the relaxation of the apex atom of the tip and the surface atoms, could be as large as 50 pm, which occurs at a tip-sample distance of z = 0.33 nm. At z = 0.45 nm, the tip apex relaxes 10 pm, and the sample atom relaxes 20 pm. Although the displacement is substantial, the entire process is continuous and reversible. There is no discontinuity, no inelastic deformation, and no hysteresis. It explains the experimental observations [281, 282]. From Tables 9.1 and 9.2, it is clear that for metals commonly used in STM and AFM experiments, if the tip is well-structured, the junction should be stable over the entire range of tip-sample distance. On the other hand, for soft samples such as graphite, from Table 9.2, we find  kJ =

1 1 + 158 6.18

−1 ≈ 5.94 N/m,

(9.21)

which is about three times smaller than the typical maximum force gradient, or γ = 2.8. Therefore, the tip-sample junction is intrinsically unstable. The approaching and retracting process is shown by the solid curve in Fig. 9.5. At about z = 0.35 nm, dz/dζ becomes negative. Hysteresis takes place. As the tip approaches the sample surface, at point A, dζ/dz becomes infinite.

9.1 Mechanical stability of the tip-sample junction

263

Fig. 9.5. Stability of STM and the rigidity of surfaces. Three cases are shown. The dotted curve represents the case of a perfectly rigid junction, γ = 0. The dashed curve represents the typical case of a metal tip and a metal sample surface, γ = 0.3. The solid curve represents the case of a soft sample surface, such as graphite, γ = 1.5. For this case, a jump to contact, or a hysteresis, should be observed. During approaching, the true tip-sample distance would jump from point A to point B while the observed zdisplacement through the piezo does not change. During retracting, the true tip-sample distance would jump from point C to point D while the observed z-displacement through the piezo remains constant.

The sample surface (and the tip) expands to make a jump of about 0.2 nm. The tip touches the sample at point B. By retracting the tip from mechanical contact with the sample surface, at point C, dζ/dz again becomes infinity. The sample surface (and the tip) restores to make an increase of separation of about 0.2 nm, to reach point D. 9.1.3

Relaxation and the apparent G ∼ z relation

In Section 1.2, the elementary theory of tunneling gives an exponential relation between tunneling conductance G and tip-sample distance ζ, G = G0 e−2κ(ζ−ζe ) .

(9.22)

As a result of relaxation, the observed relation between the tunneling conductance G and the apparent tip-sample distance z measured through the z-piezo would deviate from an exponential relation. To make a quantitative analysis, a dimensionless quantity Γ is introduced: Γ ≡ ln

G , G0

(9.23)

264

Nanomechanical Effects

which is positive only under a firm tip-sample contact, and always negative under tunneling conditions. Equation 9.22 now becomes Γ = −2κ(ζ − ζe ).

(9.24)

From Eq. 9.8, the maximum z–relaxation is δzmax =

κ U0 . 2 kJ

Substituting Eqs 9.23 and 9.25 into Eq. 9.9 gives   Γ Γ + 4 δzmax exp − exp Γ . z − ze = − 2κ 2

(9.25)

(9.26)

Compared to Eq. 9.24, in addition to the linear term, there is a relaxation term with a maximum z-relaxation δzmax . Under tunneling conditions, only for small values of |ζ − ζe |, the relaxation is significant. Figure 9.6 shows the relation between the tunneling conductance and the observed tip-sample distance with a work function 5 eV and a maximum relaxation δzmax ≈ 60 pm, typical for metal–metal systems. If the sample is clean and the tip is well structured, the approaching and retracting process is reversible and free from jump-into-contact syndrome. Experimental observations of such relaxation effect have been repeatedly reported. For example, Fig. 3 and Fig. 4a in from Limot et al. [285]

Fig. 9.6. The effect of relaxation on the apparent G ∼ z relation. The tunneling conductance G has an exponential dependence on the true tip-sample distance ζ. The apparent tip-sample distance z, measured through the z-piezo, is calculated using Eq. 9.26. Parameters used: work function 5 eV, maximum relaxation δzmax ≈ 60 pm. At small tip-sample distances, because of the relaxation, the apparent G ∼ z relation noticeably deviates from an exponential dependence.

9.2 Mechanical effects on observed corrugations

265

show a good agreement with Fig. 9.6 both in experimental data and numerical simulation. A statistical study would further clarify the conditions of the observations: by doing computer-automated experiments of repeated approaching and retracting a well-structured tip onto a clean metal sample surface down to 0.1 nm before a mechanical contact (corresponding to a tunneling resistance of about 100 kΩ), each time randomly changing to a different location on the sample surface. Limot et al.’s relaxation curves in Fig. 3 and Fig. 4a in [285] should be statistically prevalent.

9.2 9.2.1

Mechanical effects on observed corrugations Soft surfaces

Binnig et al. [273] observed that the corrugation amplitude on graphite with STM can be as large as a good fraction of one nanometer, whereas the lateral dimensions of the corrugations are as expected from the crystallographic data. The observed giant corrugation was interpreted by Soler et al. [198] as originating from the deformation of the graphite surface, which amplifies the electronically based corrugations. The interpretation of Soler et al. [198] was based on the experimental fact that the corrugation amplitude depends dramatically on tunneling conductance [273, 198, 274], as shown in Fig. 9.8. Qualitatively, their interpretation is as follows: during the experiment, the tip actually comes into mechanical contact with the graphite surface. In other words, there is a repulsive force between the tip and the sample, which generates a compression of the graphite surface. By moving the z piezo, the tip moves relative to the back side of the sample. Because graphite is very soft, the graphite surface in the vicinity of the tip deforms. The change of the true gap width, d, is much smaller than the observed z displacement of the piezo, Δz, which is the sum of the gap displacement, Δd, and the deformation of the graphite surface, Δu, as shown in Fig. 9.8. Because of the softness of graphite, the corrugation observed from the displacement of the z piezo is much larger than the true corrugation, that is, the displacement of the true gap width. In the attractive-force regime, the situation could be the opposite. As shown in Fig. 9.8, the attractive force generates an expansion of the sample surface. The retracting of the tip results in a reduction of the attractive force. The expansion of the graphite surface is reduced. The displacement of the true gap is then greater than the observed z-piezo displacement. In other words, in the attractive-force regime, the observed corrugation amplitude is reduced by the elastic deformation. Quantitatively, Soler et al. [198] expressed the force between the tip (a carbon cluster) and the graphite surface with a Morse function,

266

Nanomechanical Effects

Fig. 9.7. Amplification of corrugation amplitude by deformation Due to the softness of the graphite surface, the observed mechanical displacement of the z piezo, as the sum of the displacement of the gap width and the displacement of the sample surface, can be orders of magnitude greater than the displacement of the gap width. (Reproduced with permission from [198]. Copyright 1986 American Physical Society.)

 F (z) = 2κU0

 e−2κ(z−z0 ) − e−κ(z−z0 ) ,

(9.27)

and the constants are obtained from first-principles computations of the forces between graphite planes. The theory of Soler et al. [198] was confirmed and extended by Mamin et al. [274]. While explaining the large corrugations in terms of an amplification of the tip motion arising from surface deformation, they proposed that while graphite is imaged in air, there is a contamination layer between the tip and the sample. The force, which is the origin of the corrugation amplification, is mediated by that contamination layer. Tunneling proceeds through a miniature tip protruding through the contamination. Also, a systematic study of the dependence of apparent corrugation amplitude on tunneling resistance is conducted. A sharp dependence of the corrugation amplitude on tunneling conductance is found, as shown in Fig. 9.8. The existence and value of the force between the tip and the graphite surface

9.2 Mechanical effects on observed corrugations

267

Fig. 9.8. Dependence of corrugation on tunneling conductance A systematic study shows that the corrugation amplitude is determined by tunneling conductance but has no direct connection with the bias. It indicates that the tunneling characteristics of graphite are metal-like. (Reproduced with permission from [274]. Copyright 1986 American Physical Society.)

during tunneling experiments were directly measured [275]. The force can be as large as a few hundred nN. The same phenomenon is observed on other layered materials, for which the deformation perpendicular to the cleavage surface is relatively easy. When the large corrugation amplitudes are observed, the apparent barrier height becomes very low, indicating a nearly synchronous motion of the sample surface with the tip. 9.2.2

Hard surfaces

The effect of the repulsive atomic force in the STM imaging process is well established experimentally and theoretically. It is natural to inquire about the effect of the corrugation of attractive forces in the STM imaging process [88]. The problem was studied using a first-principles numerical method by Ciraci, Baratoff, and Batra [286]. The major conclusions are as follows: (1) At the operating tip-sample distances, the attractive force prevails. (2) The magnitude of the attractive force at the hollow site (H) in Fig. 9.9 is larger than that at the top site (T ). In other words, there is a reversed corrugation for the attractive force. (3) The deformation of the tip would reduce the observed corrugation of the constant-current topographic image.

268

Nanomechanical Effects

Fig. 9.9. Atomic force between an Al tip and an Al sample. The magnitude of the attractive force on top of an Al atom (T site) is smaller than that on top of a hole (H site). (1 eV/1 nm= 0.16 nN.)

(4) For most metals, the effect of deformation on the corrugation in the topographic image is much smaller than the observed values. The conclusions of Ciraci et al. [286] can be understood with a simple independent-atom model, that is, by considering the tip as a single atom, and the total force as the sum of the forces on every atom on the sample surface. By representing the force between individual pairs of atoms as a Morse function, the z component of the force to the inth atom at a point in space, r, is:   fz = 2κU0 e−κ(r−r0 ) − e−2κ(r−r0 ) cos θ,

(9.28)

where r =| r − rn | is the distance from the point in space to the center of the nth atom on the sample surface, and θ is the angle between the norm of the surface and the line (rr0 ). For a crystalline surface which has one atom at each lattice point with primitive vectors a1 and a2 , the total force at point r, F (r), is F (r) =

∞ 

fz (r + n a1 + m a2 )

n,m=−∞

=

∞  j,k=−∞

F˜jk (z) ei( j b1 +k b2 )·x .

(9.29)

9.2 Mechanical effects on observed corrugations

269

The Fourier coefficients are F˜jk (z) =

1 l |a1 × a2 |



d2 x F (r) e−i( j b1 +k b2 )·x ,

(9.30)

where b1 = 2π(a2 × k)/[(a1 × a2 ) · k] and b2 = 2π(k × a1 )/[(a1 × a2 ) · k] are primitive vectors of the reciprocal lattice, and the integration extends over the entire surface. Consider now the lowest nontrivial Fourier components only. For elementary crystalline surfaces with hexagonal symmetry, similar to the arguments leading to Eq. 7.56, the force distribution is ˜ 1 Φ(6) (bx), F (r) = F˜0 + 9G

(9.31)

where the function Φ(X) has a maximum value of 2/3 at the T sites and a value of 1/3 at the H sites. The Fourier coefficients can be evaluated using a mathematical identity, Eqs 7.59 and 7.60. The results are:  4πzU0  −κ(z−r0 ) − e−2κ(z−r0 ) , 2e F˜0 (z) = √ 3a2   4πzκU0 2 −β1 z+κr0 1 −β2 z+2κr0 ˜ , e − e F1 (z) = − √ β1 β2 3a2

(9.32) (9.33)

where  κ2 + b 2 ,

(9.34)

 (2κ)2 + b2 .

(9.35)

β1 = β2 = At a T site, the force is

FT (z) = F˜0 (z) + 6F˜1 (z),

(9.36)

and at an H site, the force is FH (z) = F˜0 (z) − 3F˜1 (z).

(9.37)

An example of an Al(111) surface is shown in Fig. 9.9. From the measured work function, φ=3.5 eV, we find κ = 9.6 nm−1 . The atomic distance is a = 0.288 nm, which also equals the parameter r0 in the Morse formula. An estimation of the parameter U0 in the Morse formula, or the binding energy per pair of Al atoms, can be made as follows: The evaporation heat of aluminum is 293 kJ/mol, which is 3.0 eV per atom. Aluminum is an fcc crystal, where each atom has 12 nearest neighbors. Therefore, the binding energy per pair of Al atoms is about 0.5 eV. Substitute these numbers into Eqs. 9.36 and 9.37, we find the forces at the T site and the H sites which reproduce the result of first-principles computations by Ciraci et al. [286].

270

9.3

Nanomechanical Effects

Force in tunneling-barrier measurements

For a large number of materials, the stability condition, Eq. 9.14, is satisfied. In addition, in most cases, the STM images are taken under attractive-force conditions. The amplification effect does not occur. However, the effect of force and deformation is still observable. By measuring the apparent barrier height using an AC method, based on Eq. 1.10,  φapp ≡ 95

d ln I dz

2 ,

(9.38)

even a slight variation of gap distances can be detected. In fact, if the apparent barrier height in terms of the actual displacement of tip-sample distance dζ is a constant,  φ0 ≡ 95

d ln I dζ

2 = const.,

(9.39)

but due to the force and deformation, dz = 1, dζ

(9.40)

an apparent variation of the measured barrier height, Eq. 9.38, should be observed. In the strong repulsive-force regime, (dz/dζ)  1, a very small apparent barrier height should be observed. This phenomenon was first described and analyzed by Coombs and Pethica [287], and can be easily observed on graphite and other layered materials, such as the case of VSe2 [288]. In the attractive-force regime, (dz/dζ) < 1, an increase of the measured value of apparent barrier height should be observed. Actually, this was first reported by Teague [272] in an MIM tunneling experiment, and then reported by Binnig et al. using STM [289]: at a very short tip-sample distance, where the actual barrier collapses, the current becomes even higher than what was expected from an exponential dependence on the distance. A systematic study was conducted on a clean Si(111) surface with a clean W tip [199]. The entire curve of the dependence of the measured apparent barrier height, Eq. 9.38, with z-piezo displacement, was recorded. The experiment was performed under the condition that a clear 7×7 pattern was observed, which indicated that both the tip (near the apex atom) and the sample were clean. By carefully moving the tip back and forth, so as not to press into the sample surface too deeply, the entire process is completely reversible. The experimental barrier height measurements were performed using an ac modulation method, by applying a small 5 pm modulation to the z piezo at a frequency ωmod ≈2 kHz. The AC method provides better accuracy than the DC method. The data points in Fig. 9.10 are the experimentally measured apparent barrier height as a function of tip-sample

9.3 Force in tunneling-barrier measurements

271

Fig. 9.10. Variation of the measured apparent barrier height with distance. Circles are data points. The solid curve is derived from Eq. 9.14. The dashed curve is the actual gap displacement as a function of the measured z-piezo displacement. The dotted curve, the fictitious gap displacement in the absence of force, is included for comparison. (Reproduced with permission from [199]. Copyright 1991, AVS The Science and Technology Institute.)

separation obtained in this manner on a clean Si(111)-7×7 sample with a sample bias of –1 V with respect to the tip. As a function of tip-sample separation, the barrier height versus distance can be separated into four distinct regimes: (1) At large separations the barrier height is approximately 3.5 eV, which is roughly equal to the average of the work functions of tungsten and silicon. (2) As the tip-sample separation is decreased, the barrier height first exhibits a small increase to about 4.8 eV. (3) Further decreasing the tip-sample separation causes the barrier height to plummet by more than a factor of ten with only 0.1 nm change in tip-sample separation. (4) Pushing the tip toward the sample even further produces only a small modulation of the current dI/dz, leading to an apparent barrier height of near zero, a phenomenon reported first by Coombs and Pethica [287]. In both cases, the observed behavior of the apparent barrier height is continuous and reversible if the tip is not pushed too deeply. The observed variation of apparent barrier height can be understood quantitatively by assuming that the force follows a Morse function. Thus, the relation between the z-piezo reading and the true gap displacement ζ follows Eq. 9.10,   dz = 1 − γ e−κ(ζ−ζe ) − 2e−2κ(ζ−ζe ) . dζ

(9.41)

272

Nanomechanical Effects

The solid curve in Fig. 9.10 is drawn with γ=0.95 and assume the actual apparent barrier height is 3.5 eV throughout the entire region. The accurate fit indicates that the model is reasonable. Using Eq. 9.1, the characteristic radius of the end of the tip can be estimated. Assuming Ue = 5 eV, with the elastic constants of tungsten, E = 340 GPa and ν = 0.26, we find a0 ≈ 0.5 nm. This is a reasonable value for tips that exhibit atomic resolution. The normal tip-sample distance in STM experiments can be obtained accurately from this experiment. In Fig. 9.10, the equilibrium distance, where the net force is zero, is taken as the origin of z. As shown for the case of aluminum, because the attractive force has a longer range than the repulsive force, the absolute equilibrium distance between the apex atom and the counterpart on the sample surface is slightly less than the sum of the atomic radii of both atoms, which is about 0.2 nm. The normal topographic images on Si(111) are usually taken at I = 1 nA, corresponding to a distance of ≈ 0.3 nm from the equilibrium point, or ≈ 0.5 nm from nucleus to nucleus. The method presented here is extended to the study of the mechanical properties of individual molecules. In addition to the lock-in signal for the first harmonics, that of the second harmonics is also utilized [278, 290].

Part II Instrumentation

Part II: Instrumentation I suppose that when the bees crowd round the flowers it is for the sake of honey that they do so, never thinking that it is the dust which they are carrying from flower to flower which is to render possible a more splendid array of flowers, and a busier crowd of bees, in the years to come. We cannot, therefore, do better than improving the shining hour in helping forward the cross-fertilization of the sciences. Lecture given in Cambridge, May 14, 1878 while directing a demonstration of the telephone James Clerk Maxwell Distinguished Professor of Experimental Physics, Cambridge University

In view of the extreme simplicity of the STM as an instrument, one may wonder why it was not invented many decades ago. Probably the reason is that the STM is a hybrid of several different branches of science and technology that generally have very little communication with each other. These parent areas are classical tunneling experiments (a branch of lowtemperature physics), surface science (a branch of vacuum physics), and microscopy. The technological implementation of STM required skills and knowledge in different disciplines such as mechanical, electronic, and control engineering. The invention of the STM was a result of the cross-fertilization of different branches of science (Binnig and Rohrer [32]), much as advocated by Maxwell more than a century ago. In Part II, we discuss the essential elements of STM instrumentation. Except for a few cross-references, all the chapters can be read independently. The chapter on piezodrives starts with an introduction to piezoelectricity and piezoelectric ceramics at the general physics level. Three major types of piezodrives, the tripod, the bimorph, and the tube, are analyzed in detail. The chapter on vibration isolation starts with general concepts and vibration measurements. Various vibration isolation devices used in STM are described and analyzed. The chapter on electronics and control is presented at the college electronics course level. The analysis of the transient responses of feedback circuits requires some knowledge of the Laplace transform, but the presentation is aimed at general scientists rather than specialists in this field. The actual mechanical design of the STM largely depends on the coarse-positioning mechanism, which is discussed in the same

276

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chapter. Tip preparation and characterization is a crucial, but still not well understood, technique in STM. This topic also has a dedicated chapter. It is followed by a chapter on scanning tunneling spectroscopy, where tip treatment and tip-electronic-structure characterization are essential. The chapter on atomic-force microscopy only touches the repulsive-force mode, which provides atomic resolution, as STM’s next of kin. The applications of STM are so broad that a number of books are required to describe them. In the final chapter of this book, a number of illustrative applications are presented. The theme of this book is atomic-scale imaging through tunneling. For it to be realized, all the instruments must be well orchestrated. The limited size of this book prevents the author from including all the variations. After proper instrumentation, may you play and develop the theme in your sphere: be it physics, chemistry, electrochemistry, biology, materials science, or any engineering science.

Chapter 10 Piezoelectric Scanner The heart of STM is a piezoelectric scanner, sometimes called a piezodrive or simply a piezo. In this chapter, we provide a brief summary of the basic physics of piezoelectricity as well as piezoelectric materials including quartz and piezoelectric ceramics. Both are relevant to the applications in STM and AFM. Major types of piezodrives, the tripod, the bimorph, the tube scanner, and the sheer piezo, are analyzed.

10.1 10.1.1

Piezoelectricity Piezoelectric effect

The piezoelectric effect was discovered by Pierre Curie and Jacques Curie, about 100 years before the invention of the STM [291]. A sketch of their experiment is shown in Fig. 10.1. A long, thin quartz plate, cut from a single crystal, was sandwiched between two tin foils. While one tin foil was grounded, another tin foil was connected to an electrometer. By applying a weight to generate a stress, an electric charge was detected.

Fig. 10.1. Piezoelectric effect. (a) A quartz plate, cut from a single crystal. (b) By stressing the quartz plate, an electric charge is generated; after [292].

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278

10.1.2

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Inverse piezoelectric effect

A few months later, Lippmann [293] predicted the existence of the inverse piezoelectric effect: by applying a voltage to a quartz plate, a deformation should be observed. This effect was soon confirmed by the Curie brothers [294], who designed a clever experiment to measure the tiny displacement, as shown in Fig. 10.2. Here, a light-weight lever with an arm of about 1:100 amplifies the displacement by two orders of magnitude. An optical microscope further amplifies it by two orders of magnitude. The displacement is then measured by an eyepiece with a scale. The definitions of the parameters for describing piezoelectric effect are shown in Fig. 10.3. A voltage V is applied to a rectangular piece of piezoelectric material. Inside it, the electric field intensity is E3 =

V . z

(10.1)

In the standard convention, the directions x, y, and z are labeled as 1, 2, and 3, respectively. As a result of the electric field, a strain is generated. The xx component of the strain tensor is (see Fig. 10.3.) S1 ≡

δx , x

(10.2)

S3 ≡

δz . z

(10.3)

and the zz component is

A piezoelectric constant is defined as the ratio of a strain component over a component of the applied electrical field intensity, for example,

Fig. 10.2. The inverse piezoelectric effect. A thin and long quartz plate, QQ, is sandwiched between two tin foils. By applying a voltage to the tin foils, the quartz plate elongates or contracts according to the polarity of the applied voltage. To measure the very small displacement, Curie used a lever ABD with a small mirror v attached at its end, the displacement of which is measured with an optical microscope; after [295].

10.1 Piezoelectricity

279

Fig. 10.3. Definition of piezoelectric constants. A rectangular piece of piezoelectric material, with a voltage V applied across its thickness, causes a strain in the x as well as the z direction. A piezoelectric constant is defined as the ratio of a component of the strain with respect to a component of the electric field intensity.

d31 ≡

S1 E3

(10.4)

d33 ≡

S3 . E3

(10.5)

and

Because strain is a dimensionless quantity, the piezoelectric constants have dimensions of meters/volt or coulomb/newton in SI units. In the literature, the unit pm/V, or 10−12 m/V is commonly used. Using primitive means as shown in Fig. 10.2, the Curie brothers [294] obtained a value of 2.1 pm/V for the parallel piezoelectric constant for quartz, which matches favorably the results of modern measurements [296]. From the very beginning of their experiments, the Curie brothers realized that the linear piezoelectric effect only exists in anisotropic crystals, such as quartz, tourmaline, and Rochelle salt [291]. In fact, as seen from Fig. 10.3, if the +z and −z directions of the plate are equivalent, by reversing the direction of the electric field E3 , the strain should be the same. In such cases, the strain should be proportional to |E3 |2 instead of E3 . In other words, there should be no linear piezoelectric effect. Lippmann’s reasoning [293] for the inverse piezoelectric effect is based on thermodynamics. Consider the experiment shown in Fig. 10.1. The increment of the total energy due to an applied voltage V is dE = T dS + F d(δx) + Q dV,

(10.6)

where F is the force in the vertical direction, x, and Q is the electric charge on the surface. The reversibility of the process implies that the Gibbs free energy G is also a function of the state, that is, dG = −S dT − (δx) dF + Q dV, from which we obtain immediately the Maxwell relation

(10.7)

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Piezoelectric Scanner



∂Q ∂F



 =

T,V

∂(δx) ∂V

 .

(10.8)

T,F

Using Eq. 10.1 and Eq. 10.2, note that the polarization in the z direction is P3 = Q/xy and the stress in the x direction is σ1 = F/yz, where y is the width of the quartz plate, Eq. 10.8 can be rewritten as 

∂P3 ∂σ1



 = T,E3

∂S1 ∂E3

 .

(10.9)

T,σ1

The left-hand side of the equation is the forward piezoelectric constant, in units of coulomb/newton. The right-hand side is the reverse piezoelectric constant, in units of meters/volt. They are equal. The coexistence of forward and inverse piezoelectric effects provides a simple method to test the piezodrive used in STM, which is discussed in Section 10.4.2. The piezoelectric effect was one of the crucial elements for Pierre Curie and Marie Skladowska Curie in their discovery of radioactivity. However, there was no technological application for the following thirty years, until it was used in radio transmitters and ultrasonics on the 1910s. Currently, the largest application of piezoelectric devices is time keeping, such as in wrist watches, cellular phones, computers, etc. The original piezoelectric material discovered by the Curie brothers, quartz, is still the material of choice. To date, the number of the time-keeping quartz devices is well over the number of human population, and the unit price is below one dime.

Fig. 10.4. A photograph of an artificial quartz crystal. Grown within a liquidphase process, the artificial quartz crystal has replaced the natural quartz as the material of choice for piezoelectric devices. It is remarkably uniform, free from defects, much larger than most of the natural crystals, and the axes are easily identified. (By courtesy of Shenzhen Crystal Science and Technology Ltd.)

10.2 Piezoelectric materials in STM and AFM

10.2 10.2.1

281

Piezoelectric materials in STM and AFM Quartz

Crystalline quartz is the most abundant crystals found in Nature. In the first half of the twentieth century, Brazil was the largest supplier of natural quartz crystals for the radio technology. After the invention of the liquid-phase crystal growth process, artificial quartz crystals became the raw material of choice for quartz-based piezoelectric devices. Currently, quartz is the second most manufactured single crystal only next to silicon. Figure 10.4 is a photograph of a typical artificial quartz crystal, which is much bigger than the typical natural quartz crystal, uniform, defect-free, and can be easily cut to desired form. Figure 10.5 shows the definition of axis with regard to the artificial quartz crystal. The x-axis is called the electrical axis. An application of an electrical field along it would cause a strain along the y-axis, the mechanical axis. By shining a polarized light through the z-axis, the optical axis, a rotation of the polarization angle is observed. Figure 10.5 also shows various cuts of quartz devices used in the industry. The odd angles are designed for minimizing temperature dependence of the resonant frequencies. Table 10.1 lists some most important constants of quartz. The negative sign indicates that by applying a positive electric field in the x-direction, a contraction in the y-direction occurs.

Fig. 10.5. Axes and various cuts of quartz crystal. The three axis of a quartz crystal are different. The x-axis is called the electrical axis, where an electric field is applied, which causes a strain along the y-axis, called the mechanical axis. By shining a polarized light through the z-axis, or the optical axis, a rotation of the polarization angle is observed. Various cuts are applied in the quartz device industry. The angles are designed to minimize temperature dependence of the resonant frequency.

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Table 10.1: Physical constants of quartz Item

10.2.2

Unit

Value

Density Young’s module

3

g/cm GPa

2.65 97.2

Piezo constant d11

pm/V

–2.25

Lead zirconate titanate ceramics

The piezoelectric materials used in STM are various kinds of lead zirconate titanate ceramics (PZT).1 The mechanism of piezoelectricity in PZT is somewhat different from single-crystal piezoelectric materials such as quartz. Pure PbZrO3 and PbTiO3 and their solid solutions are ferroelectric or antiferroelectric, and exhibit a permanent electric dipole even in the absence of external electric field. PZTs have never been used in single-crystal form. Those are useful only in ceramic form. The as-made ceramics do not exhibit a piezoelectric effect. Macroscopically, they are isotropic owing to the random arrangement of the electrical dipoles. A poling process is then applied to produce a permanent electric polarization, similar to the process of making a permanent magnet from a piece of hard ferromagnetic material. After poling, the dipoles are aligned with the poling field. Macroscopically, the piece of material becomes anisotropic. A strong piezoelectric effect is generated. One of the advantages of the piezoelectric ceramic over single-crystal materials is that it can be shaped easily and poled in a desired direction. However, inherently, its piezoelectric parameters are not as stable and reproducible as single crystals such as quartz. The first practical material in this category is barium titanate, BaTiO3 , which had been extensively used in ultrasonic transducers in the 1940s and 1950s [297]. One of the disadvantages of BaTiO3 is that its Curie temperature is too low. Once has been heated above this temperature, its piezoelectricity is permanently lost. In the 1950s, Jaffe et al. [298] discovered that ceramics based on a mixture of PbZrO3 and PbTiO3 , after a similar poling process, exhibit excellent piezoelectric properties. The Curie temperatures of such ceramics range from 200◦ C to 400◦ C, and are therefore much more stable than barium titanate ceramics. The PZT ceramics are made by firing a mixture of PbZrO3 and PbTiO3 together with a small amount of additives at about 1350◦ C under strictly controlled conditions. The result is a solid solution. Macroscopically, it is isotropic. Microscopically, it consists of small ferroelectric crystals in 1 PZT is the trade name of the lead zirconate titanate piezoelectric ceramics of one of its largest producers, Morgan Electro Ceramics, Inc. It is also commonly used in the scientific literature as a standard acronym.

10.2 Piezoelectric materials in STM and AFM

283

Fig. 10.6. PZT: piezoelectric properties and composition. Dependence of piezoelectric properties on composition. The zirconate-rich phase is rhombohedral, whereas the titanate-rich phase is tetrahedral. The piezoelectric constants reach a maximum near the morphotropic phase boundary. (After [298].)

random orientations. After machining and metallization, a high electric field is applied for a sufficiently long period of time, for example, 60 kV/cm for 1 hour. A strong piezoelectricity is generated. As a convention, the direction of the poling field is labeled as the 3 direction, or the positive z direction. The crystallographic and piezoelectric properties of the ceramics depend dramatically on composition. As shown in Fig. 10.6, the zirconate-rich phase is rhombohedral, and the titanate-rich phase is tetragonal. Near the morphotropic phase boundary, the piezoelectric constant reaches its maximum. Various commercial PZT ceramics are made from a solid solution with a zirconate-titanate ratio near this point, plus a few percent of various additives to fine tune the properties for different applications. In addition to the parameters discussed in the previous section, that is, the piezoelectric constants d31 , d33 , and the velocity of sound, c, there are several other parameters that are important for applications in STM. Curie point As a ferroelectric material, each piezoelectric ceramic is characterized by a Curie point or Curie temperature, Tc [299]. Above this temperature, the ferroelectricity is lost. An irreversible degradation of the piezoelectric property occurs if the ceramic is heated above such a temperature. At

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Fig. 10.7. Variation of piezoelectric constant with temperature. (By courtesy of Morgan Electro Ceramics, Inc., Bedford, Ohio.)

temperatures close to but still much lower than the Curie temperature, serious degradation may occur. Therefore, each piezoelectric material has a well-defined maximum operating temperature, which is much lower than its Curie point. Temperature dependence of piezoelectric constants While designing or using STM at low or high temperatures, the variation of piezoelectric constants with temperature has to be considered seriously. The variation differs for different PZT materials. Figure 10.7 shows measured variations of d31 for several commonly used PZT materials with temperature. Depoling field As mentioned, the piezoelectric properties of ceramics are generated by a poling process. Apparently, if a strong electric field in a direction other than the poling direction is applied, the piezoelectric property is altered or lost. The safe value of an AC field to avoid causing depoling, Ed , can be found in the product specifications. Mechanical quality number This quantity, usually denoted QM , is a measure of the internal mechanical energy loss of the material. Roughly speaking, it is the number of vibrations it can sustain without substantial amplitude reduction. The larger the number QM , the smaller the internal loss.

10.2 Piezoelectric materials in STM and AFM

285

Coupling constants Probably the best measure of the effectiveness of a piezoelectric material is its electromechanical coupling constant, k, defined as

k=

electrical energy converted to mechanical energy . input electrical energy

(10.10)

This is the efficiency of energy conversion between mechanical and electrical forms. For PZTs, it ranges from 0.5 to 0.7, which are the most efficient of all known piezoelectric materials, see Table 10.2. For quartz, the coupling constant is about 0.1. Aging In contrast to piezoelectric single crystals, such as quartz, the piezoelectricity of PZT ceramics decays with time, owing to relaxation. Experimentally it is found that on a large time scale (for example, months and years), the aging process can be accurately described by a logarithmic law. For example, the coupling constant k varies with time t as k(t) = k(0)(1 + δ log1 0 t),

(10.11)

where δ is the relative variation of the coupling constant per time decade. The zero point of time is the completion of poling.

Table 10.2: Properties of PZT ceramics. Electro Ceramics Inc., Bedford, Ohio. Item d31

Unit pm/V

d33 d15 Y ρ c Tc Ed QM kp Aging

pm/V pm/V 1010 N/m2 g/cm3 km/sec ◦ C kV/cm (rms) – – kp /time decade

By courtesy of Morgan

PZT-4D –135

PZT-5H –262

PZT-7A –60

PZT-8 –95

315 – 7.5 7.6 3.3 320 10 600 –0.60 –1.7%

583 730 6.1 7.5 2.8 195 4 65 –0.64 –0.2%

150 360 9.2 7.6 2.9 350 15 600 –0.52 –0.006%

220 330 8.7 7.6 3.4 300 15 960 –0.52 –2.3%

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Piezoelectric Scanner

Table 10.2 shows selected parameters of several PZT ceramics commonly used in STM. PZT-5H has by far the highest sensitivity (that is, the largest piezoelectric constant d31 ). However, its Curie temperature is low and its internal friction is high, which means it has a serious hysteresis problem. Also, its properties sharply depend on temperature. On the other hand, PZT-7D, with a lower sensitivity, exhibits very low hysteresis and a very small aging effect. The properties of PZT-4D have very low temperature variation near room temperature. At cryogenic temperatures, PZT-8 has the lowest temperature variations. Also, PZT-8 has a high depoling field and a high Curie temperature, which is suitable for high-temperature applications.

10.3 10.3.1

Piezoelectric devices in STM and AFM Tripod scanner

In the early years of STM instrumentation, tripod piezoelectric scanners were the predominant choice, see Fig. 10.8. The displacements along the x, y, and z directions are actuated by three independent PZT transducers. Each of them is made of a rectangular piece of PZT, metallized on two sides. Those three PZT transducers are often called x piezo, y piezo, and z piezo, respectively. By applying a voltage between the two metallized surfaces of a piezo, for example the x piezo, the displacement is L , (10.12) h where V is the applied voltage, L the length, and h the thickness of the piezo. The quantity δx = d31 V

L dx = d31 (10.13) dV h is called a piezo constant. For example, with L = 20 mm, h = 2 mm, and using PZT-4D, d31 = –135 pm/V, Eq. 10.13 gives K = 1.35 nm/V. The accurate calculation of the resonant frequencies of a tripod scanner is a complicated problem. The flexing modes are effectively coupled with the stretching modes. An evaluation of the lowest resonant frequency of the flexing mode provides an order-of-magnitude estimation of the lowest K≡

Fig. 10.8. Tripod scanner. Three PZT bars, Px , Py , and Pz , to control the x, y, and z displacements, respectively. The tip is mounted at the vertex of the tripod.

10.3 Piezoelectric devices in STM and AFM

287

Fig. 10.9. Bimorph. Although the word bimorph is a trade name of Morgan Electro Ceramics, Inc., it was described and analyzed by Curie [295]. The most common mode of connection is to ground the two outer electrodes and to apply a voltage to the center electrode. The operation is similar to the bimetal thermometer. (By courtesy of Morgan Electro Ceramics, Inc., Bedford, Ohio.)

resonant frequency of the tripod scanner. For a piezo made √ of PZT-5A, 20 mm long and 2 mm thick, the radius of gyration is 2 mm/ 12 = 0.577mm. The speed of sound is about 2.8 km/sec. Using Eq. 10.37, the resonant frequency is found to be 3.3 kHz, which is close to the values often observed experimentally. Because the piezo constant is proportional to L/h, whereas the resonant frequency is proportional to h/L2 , it is clear that, by reducing the length and thickness in proportion, the piezo constant remains the same, and the resonant frequency will increase. This is in general true. The natural limit of such a reduction is the depoling field of the material. 10.3.2

Bimorph

If larger displacements are required, an arrangement as shown in Fig. 10.9, the bimorph,2 can be applied. The principle is similar to the bimetal thermometer. Two thin plates of piezoelectric material are glued together. By applying a voltage, one plate expands and the other one contracts. The composite flexes. A common arrangement of a PZT bimorph is to pole both plates in the same direction (normal to the plane). When it is used, both outer electrodes are grounded. The driving voltage is applied to the center electrode. Such an arrangement reduces the stray field of the applied voltage to a minimum, and simplifies mounting. The following is the treatment of bimorphs presented by Curie [295]. As shown in Fig. 10.10(a), immediately after the application of a voltage V , a strain d31 V (2/h) is generated. It in turn generates a stress σ1 in the x direction: 2 Bimorph is a trade name of Morgan Electro Ceramics, Inc., for such flexing-type piezoelectric elements. Historically, it was developed by the Curie brothers in the 1880s [295]. No specific term was suggested by the Curie brothers. Currently, the term bimorph is used frequently in the literature for the flexing-type piezoelectric element.

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Fig. 10.10. Deflection of a bimorph. Two long, thin plates of piezoelectric material are glued together, with a metal film sandwiched in between. Two more metal films cover the outer surfaces. Both piezoelectric plates are poled along the same direction, perpendicular to the large surface, labeled z (a) By applying a voltage, stress of opposite sign is developed in both plates, which generates a torque. (b) The bimorph flexes to generate a stress to compensate the torque. The neutral plane, where the stress is zero, lies at h/3 from the central plane.

σ1 = ±Y d31 V

2 , h

(10.14)

where Y is Young’s modulus. A torque occurs that flexes the element. As shown in Fig. 10.10(b), the bimorph bends to produce a distribution of stress to compensate the torque. It then reaches equilibrium. The additional stress is assumed to be linear in z, σ(z) = σ1 − αz.

(10.15)

The constant α is determined by the condition of equilibrium, that is, with zero torque M :  M = z σ(z) dz = 0. (10.16) Substituting Eq. 10.15 for Eq. 10.16, we obtain α=

3σ1 . h

(10.17)

Substituting Eq. 10.16 for Eq. 10.15, the position of the neutral plane (where σ(z) = 0) is found to be at z = h/3. Using elementary geometry and Eq. 10.35, we obtain the radius of curvature of the bimorph R R=

h2 . 6 d31 V

(10.18)

Another application of elementary geometry gives the deflection δz at the end of the bimorph, Δz = 3 d31 V

L2 . h2

(10.19)

10.4 The tube scanner

289

As an example, a bimorph with the same material and dimensions as used in the last section, PZT-4D, 20 mm long and 2 mm thick, has a sensitivity of 40.5 nm/V. This is substantially higher than the stretching mode, 1.35 nm/V. However, it is very difficult to construct a three-dimensional scanner based on bimorphs. An example of such a design is reported by Muralt et al. [300]. It is much more complicated than the tripod scanner.

10.4

The tube scanner

Invented by Binnig and Smith in 1986 [301], the tube scanner immediately became the predominant STM scanner. It has high piezo constants as well as high resonant frequencies. Moreover, it is much simpler than both the tripod scanners and the bimorph-based scanners. The original design of the tube scanner [301] is shown in Fig. 10.11. A tube made of PZT, metallized on the outer and inner surfaces, is poled in the radial direction. The outside metal coating is sectioned into four quadrants. In their original arrangement, the inner metal coating is connected to the z voltage, and two neighboring quadrants are connected to the varying x and y voltages, respectively. The remaining two quadrants are connected to a certain DC voltage to improve linearity. The tip is attached to the center of one of the DC quadrants. The first tube scanner was made by EBL Product Inc., East Hartford, using PZT-5H as the piezoelectric material. The tube, 12.7 mm long, 6.35 mm in diameter, and 0.51 mm thick, has a measured piezo constant of 5 nm/V in the x and y directions. The resonant frequency in the x and y directions is 8 kHz, and in the z direction, 40 kHz. Thus, the overall performance is much better than that of the tripod scanners. One of the problems with this design is that the motions driven by the x, y, and z voltages are nonlinear and not precisely orthogonal. This can be taken care of by proper programming in the control system. Substantial

Fig. 10.11. The tube scanner. In the initial design, the two quadrants are used for DC, and another two quadrants are used for AC. The tip is placed at the edge of the tube. (Reproduced with permission from [301]. Copyright 1986 American Institute of Physics.)

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Piezoelectric Scanner

improvement can be achieved by using bipolar, symmetric x and y voltages, and by placing the tip at the center of the tube. In this case, +Vx and −Vx voltages are applied on the opposite x quadrants, while +Vy and −Vy voltages are applied on the opposite y quadrants. Further improvements are discussed later in this chapter. 10.4.1

Deflection

Here we present a treatment of the deflection of a piezoelectric tube with quartered electrodes in the symmetric-voltage mode, see Fig. 10.12. The wall thickness of the piezoelectrics is usually much smaller than the diameter. Therefore, the variation of strain and stress over the wall thickness can be neglected. As shown in Fig. 10.12, two voltages, equal in magnitude and opposite in sign, are applied to the two y quadrants. The inner metal coating and the two x quadrants are grounded. Immediately after the onset of y voltages, a strain in the z direction, S3 = d31 V /h, is generated. It in turn creates a stress σ3 = Y S3 in the z direction, where Y is Young’s modulus. The torque of this pair of forces causes the tube to bend. The bending of the tube generates a torque in the opposite direction. At equilibrium, the total torque in any cross section should be zero. By virtue of the symmetry of the problem, it is sufficient to consider one quarter of the circle. Assuming that the stress generated by the bending is linear with respect to y, the total stress σ(θ) in the piezoelectrics, as a function of angle θ, is π , 4 π π ω0 , the characteristic equation Eq. 11.7 has two negative real roots. The viscous resistance is so large that the mass creeps to its equilibrium position from any initial condition. It is convenient to define a dimensionless number to characterize the “quality” of the oscillator, a number used extensively in mechanical engineering and electrical engineering: the Q factor or Q number, Q ≡

ω0 . 2γ

(11.12)

Roughly speaking, the Q factor is the number of oscillations the oscillator can sustain after an initial push. The stronger the damping, the smaller the Q factor. The general solution of Eq. 11.5 with an external force is a superposition of the general solution, Eq. 11.10, plus a term representing the response of the mass to the external force, 1 x(t) = ωd



t

f (τ )e−γ(t−τ ) sin ωd (t − τ ) dτ,

(11.13)

0

which can be verified by direct substitution. For a sinusoidal vibration, X(t) = X0 eiωt ,

(11.14)

at the steady state, the motion of the mass should also be sinusoidal, x(t) = x0 eiωt .

(11.15)

Substituting Eq. 11.14 and Eq. 11.15 into Eq. 11.5 we obtain x0 ω 2 + 2iγω . = 2 0 2 X0 ω0 − ω + 2iγω

(11.16)

302

Vibration Isolation

The ratio of the amplitudes is the transfer function or the response function. of the vibration isolation system:  #   x0  ω02 + 4γ 2 ω 2 = . (11.17) K(ω) ≡  2  X0 (ω0 − ω 2 )2 + 4γ 2 ω 2 In the engineering literature, the decibel (dB) unit is frequently used. The transfer function in terms of decibels is Z = 20 log10 K(ω) (dB).

(11.18)

Fig. 11.1 shows the frequency dependence of the transfer function for the one-stage system. An efficient vibration isolation means a small K(ω). The qualitative features of such a vibration isolation system can be visualized by considering the following special cases: 1. At high frequencies, if the damping is negligible, then the transfer function is inversely proportional to the excitation frequency:  ω 2  f 2 0 0 K(ω) ≈ = . (11.19) ω f 2. If the excitation frequency is close to the natural frequency of the system, then the transfer function can be greater than unity, that is, the vibration is worsened. Actually, at ω = ω0 , # ω2 ω0 = Q. (11.20) K(ω0 ) = 1 + 02 ≈ 4γ 2γ If the Q factor is too large, the external vibration at ω0 would be amplified tremendously. To avoid such resonance excitation, appropriate damping must be applied. 3. Damping worsens the vibration isolation at higher frequencies. When ω > Qω0 , Eq. 11.17 becomes K(ω) ≈

1 ω0 1 f0 = . Q ω Q f

(11.21)

The dependence of the transfer function on the ratio (f0 /f ) becomes linear instead of quadratic. Therefore, a compromise between the suppression of resonance and the suppression of high-frequency vibrations has to be made. A Q value of 3–10 is usually chosen. It is clear that the lower the natural frequency ω0 , the better the vibration isolation. The natural frequency is the prime parameter of a vibration isolation system. In a realistic STM system, the reduction of natural frequency is not unlimited. If a suspension spring is used, the elongation of the suspension spring with stiffness k due to the weight of the mass M is ΔL =

Mg , k

(11.22)

11.2 Environmental vibration

303

where g ≈ 9.8m/sec2 is the gravitational acceleration. The natural fre quency of the system is f0 = 2π k/M , or  g 5.0 f0 = 2π ≈ . (11.23) ΔL ΔL(cm) To make a suspension-spring system with a natural frequency of 1 Hz, the weight of the mass should stretch the spring by 25 cm. Notice that Eq. 11.23 is exactly the formula for the natural frequency of a simple pendulum with length ΔL. To isolate the horizontal vibration, a pendulum is the natural choice. The suspension spring then acts as the isolation device for both vertical and horizontal environmental vibrations. There is another incarnation for the model in Fig. 11.1 by interpreting the frame as the base plate (with the sample) of the STM, and the mass as the tip assembly; the model describes the influence of external vibration on the relative displacement of the tip versus the sample, which is the quantity we want to reduce. A good STM design means a high resonance frequency. When the excitation frequency is much lower than the natural frequency of the STM, then the tip assembly moves closely with the frame. In fact, when f f > f I , the overall transfer function is  2  2  2 fI fI f = . K(f ) = fS f fS

(11.25)

(11.26)

For example, if the natural frequency of the STM is 2 kHz, and a suspensionspring with natural frequency 2 Hz is used for vibration isolation, the overall transfer function for intermediate frequencies is a constant, 10−6 , or 120 dB. Therefore, the rigidity of the STM unit is the most important factor in vibration isolation.

11.2

Environmental vibration

As described, the purpose of vibration isolation is to reduce the effect of environmental vibration. In this section, we present the method of character-

304

Vibration Isolation

Fig. 11.2. Schematic and working mechanism of the Hall–Sears seismometer, HS-10-1. A metal cylinder, typically 1 kg in weight, is hung from the case with a set of springs. The springs are highly anisotropic; that is, it only allows the cylinder to be mobile in the vertical direction. Two coils are arranged on the mobile cylinder. A permanent magnet in the center generates strong radial magnetic fields. The instantaneous vertical velocity of the metal cylinder generates an instantaneous electromotive force, which can be measured by an oscilloscope or a frequency analyzer. The metal cylinder is also the vibration damper through the eddy current.

izing the environmental vibration, and the way to reduce the environmental vibration at the foundation level. 11.2.1

Measurement method

Before a vibration isolation device is designed, the vibration characteristics of the laboratory has to be measured. A typical instrument for measuring vibration is shown in Fig. 11.2. The simple mechanical system analyzed in the previous section that can be used as an instrument for this measurement is a seismometer. Fig. 11.2. is the schematic of a typical seismometer, the Hall–Sears Geophone.1 The function of the seismometer can be understood in terms of the simple model of the previous section. As shown in Fig. 11.2, a metal cylinder with two coils is suspended by a set of springs. The springs allow the cylinder to vibrate in the vertical direction with a typical natural frequency of 1 Hz. Therefore, for environmental vibrations with frequency higher than 1 Hz, the vibration amplitude of the metal cylinder is negligibly small. The relative vertical motion of the cylinder versus the case is virtually equal to the environmental vibration. A permanent magnet and the pole piece (a soft steel ring) create a strong radial magnetic field. The motion of the coil then generates an electromotive force (emf) that can be detected by an oscilloscope or a frequency analyzer. The quantity measured by the seismometer is the instantaneous velocity. The sensitivity of the seismometer depends on the model. It ranges from 1 The Hall-Sears Geophone is currently available from Geospace Corporation, 7334 North Gessner Road, Houston, TX 77040.

11.2 Environmental vibration

305

Fig. 11.3. Vibration spectra of laboratory floors Four locations in the Electrotechnical Laboratory, Ibaraki, Japan. (a) Basement, (b) first floor, (c) first floor, another location, and (d) third floor. (Reproduced with permission from [309]. Copyright 1987, AVS The Science and Technology Institute.)

1.2 to 25 V/(cm/sec). The typical frequency of environmental vibration ranges from 10 to 200 Hz (which are also the most harmful frequencies for STM operation). The actual amplitude and frequency distribution of floor vibration depends on the structure of the building, the location of the room inside the building, and the source of vibration. Pohl [308] observed that the peak frequency of floor vibration in the IBM Zurich laboratory is about 17 Hz, which is probably the resonance frequency of the building. Secondary peaks at 50, 75, and 100 Hz are also observed. Okano et al. [309] observed that in the Electrotechnical Laboratory of Japan, the peak frequency of the floor vibration is at 180 Hz, see Fig. 11.3. To achieve atomic resolution, an overall transfer function of 10−6 or better is needed. 11.2.2

Vibration isolation of the foundation

Since the source of the vibration in from the environment, the first step to reduce the vibrational noise is to make the instrument cradle quiet. This can be achieved by constructing a separate concrete foundation block for the instrument, which is vibrationally isolated from the rest of the laboratory

306

Vibration Isolation

Fig. 11.4. Vibration isolation of the foundation. A concrete sink box is connected to the laboratory floor. A 5 cm-thick elastomer barrier is placed between the sink box and the concrete instrument cradle to reduce the amplitude of environmental vibration. (By courtesy of N. Dix of Hamburg University.)

floor. An example of such construction is shown in Fig. 11.4. As shown, a 5 cm-thick elastomer barrier is placed between the concrete sink box which is connected to the laboratory floor and the concrete cradle for the instrument. The elastomer barrier functions both as an elastic medium and as a damping medium. Because of the huge mass of the concrete cradle, the resonance frequency of the system is low. The concrete cradle has two layers, separated by a plastic basket, for additional vibration isolation, and a moisture barrier. Figure 11.5 shows two photographs of the system. Fig. 11.5(a) shows the instrument cradle. Fig. 11.5(b) shows a general view of the laboratory.

Fig. 11.5. A photograph of the foundation with vibration isolation. (a) The concrete instrument cradle. (b) A view of the laboratory, showing the opening on the ceiling, and the acoustic absorption panels on the wall. (1) The instrument cradle. (2) Sound-absorbing panels. (3) Opening to the upper floor. (Photographs taken at the Institute for Applied Physics of Hamburg University by courtesy of R. Wiesendanger.)

11.3 Vibrational immunity of STM

307

There is an opening in the ceiling, which is used for the hoist of the instrument. The reduction of acoustic noise is achieved by placing a number of absorption panels on the wall. An alternative approach could be shielding the entire instrument in a sound-proof box. However, this would result in some inconvenience in operation. The sound absorption panels are quite effective and do not introduce any inconvenience to the operation of the instrument.

11.3

Vibrational immunity of STM

Additionally, the rigidity of the STM will finally determine how much the environmental vibration will affect the STM measurements. The most sensitive displacement to the STM measurement is the variation of the tip-sample distance, because the tunneling current changes about 2% per picometer. The early STMs had an “open frame” as shown in Fig. 11.6(a). The bending mode of the frame is most likely the weakest link. Later STMs have a “closed frame” as shown in Fig. 11.6(b). An environmental vibrational displacement z0 (t) = a0 cos(ωt) will propagate along the frame with the speed of sound of the frame, c1 , as well as along the piezo tube with the speed of sound of the frame, c2 . If the length of the piezo tube assembly is L, the change of the tip-sample distance is



  

L L − cos ω t − δz ≈ a0 cos ω t − c1 c2   1 1 ≈ a0 ωL − sin ωt. c1 c2

(11.27)

Fig. 11.6. Vibrational immunity of STM. (a) The open-frame design of STM. The bending mode of the frame is usually the weakest link. (b) The closed-frame design of STM. The vibration is propagated with the velocity of sound. The effect of external vibration can be minimized by choosing a frame material with a matched speed of sound, c1 ≈ c2 , and with a short length L.

308

Vibration Isolation

For a numerical example, if the vibration frequency is 100 Hz, the length is L=20 mm, the speed of sound in PZT-8 is 3.4 km/sec, the speed of sound in steel is 5.1 km/sec, the amplitude of the external vibration is 1 nm, the vibrational amplitude of the tip-sample distance is 1.23 pm. According to Eq. 11.27, in order to improve the vibrational immunity of STM, a shorter piezo tube and a frame material of a speed of sound that matches the speed of sound of the piezo is preferred. In Chapter 13, we show how this concept is applied to the mechanical design of STM.

11.4

Suspension-spring systems

In this section, we will discuss vibration isolation systems based on suspension springs with eddy-current damping. To date, it is probably the most efficient vibration isolation system. The design and choice of springs are also discussed. An elementary theory of helical springs, sufficient for all the applications in STM and AFM, is presented in Appendix E. If the STM is rigid enough, a single-stage suspension spring stage, as described in Section 11.1, is sufficient. Hansma et al. [95] described a vibration isolation system that consists of a concrete block hung from the ceiling with rubber tubes. The rubber tubes are at the same time springs and dampers. The STM, a very rigid unit (the nanoscope, see Section 13.3) is placed directly on the concrete block.

11.4.1

Analysis of two-stage systems

If the STM is not rigid enough, or in UHV and cryogenic environments, more sophisticated vibration isolation systems are needed. To date, vibration isolation systems using a two-stage suspension spring with eddy-current damping are probably the most efficient. The following analysis is based on the study of Okano et al. [309]. Fig. 11.7. A two-stage suspension-spring vibration isolation system. Two masses are hung from the frame via two springs and two damping mechanisms. The ratio between the vibration amplitudes of the frame and of the second mass (the transfer function) is calculated. The efficiency of its vibration isolation is much better than the single-stage system. Analysis shows that one damping mechanism alone is sufficient.

11.4 Suspension-spring systems

309

Fig. 11.8. Transfer functions for two-stage vibration isolation system. Parameters for both (a) and (b): M1 =2.4 Kg, M2 =2.9 Kg, k1 =800 N/m, k2 =700 N/m. In (a), the damping stages are equally arranged, c1 = c2 = c. I, c=0. II, c=10 Ns/m. III, c=20 Ns/m. IV. c=50 Ns/m. In (b), the effect of different arrangements of damping is illustrated. I, c1 = c2 =20 Ns/m. II, c1 =20 Ns/m, c2 =0. III, c1 = 0, c2 =20 Ns/m. IV, c1 = c2 =0. (Reproduced with permission from [309]. Copyright 1987, AVS.)

For the two-stage system shown in Fig. 11.8, Newton’s equations for the two masses are: M1 x ¨1 + c1 x˙ 1 + k1 x1 + c2 (x˙ 1 − x˙ 2 ) + k2 (x1 − x2 ) = c1 x˙ + k1 X, M2 x ¨2 + c2 (x˙ 2 − x˙ 1 ) + k2 (x2 − x1 ) = 0.

(11.28)

For a sinusoidal external excitation, X = X0 eiωt ,

(11.29)

the equations can be brought into matrix form, [A] x = X, where [A] =

!

"

k1 + k2 − M1 ω 2

−k2

−k2

k 2 − M2 ω 2

(11.30) !

+ iω

 x1 , x= x2   X . X= 0

c1 + c2 −c2 −c2 c2

" ,

(11.31)



The transfer function, in units of decibels, is x   2 Z ≡ 20 log10  . X

(11.32) (11.33)

(11.34)

310

Vibration Isolation

It is straightforward to obtain an analytic formula for the transfer functions from the matrix equation, Eq. 11.30. Using a desktop computer, the curves can be easily displayed. Typical results are shown in Fig. 11.8. In general, the efficiency of vibration isolation using a two-stage system is much better than a single-stage system. As in the case of a single-stage system, in the absence of viscous damping, the vibration isolation for higher excitation frequencies is optimized. However, the resonances at the natural frequencies of the isolation system are huge. By introducing various amount of viscous damping, the resonance is suppressed. The efficiency of vibration isolation at higher frequencies is affected. An important fact observed from Fig. 11.8(b) is that a single viscous damping mechanism is enough to suppress the resonance. This is much simpler than having two damping mechanisms from a design point of view. 11.4.2

Choice of springs

A proper choice of the parameters for the extension springs can provide the desired stiffness and endurance, and minimize the dimensions of the instrument. A simple theory of the deflection and maximum stress of springs is presented in Appendix F. The relation between the axial load P and the deflection F of a spring with coil diameter D, number of coils n, and wire diameter d is: Gd4 P = F, (11.35) 8nD3 where G is the shear modulus of elasticity of the wire. The quantity (Gd4 /8nD3 ) is often called the stiffness or rate of a spring. The maximum shear stress in the wire is 8D P. (11.36) πd3 In choosing a spring, the maximum shear stress must not exceed the endurance limit of the material. Table 11.1 lists these two parameters for commonly used spring materials. From Eq. 11.35 we see that, to make a soft spring, it is not necessary to use thin wires. The stress in the thin wire might exceed its yield strength, and permanent deformation might occur. A larger coil diameter and a τmax =

Table 11.1: Properties of common spring materials Material Music wire Stainless steel

Modulus of rigidity G 2

8.0 × 1010 N/m 2 6.7 × 1010 N/m

Yield strength τmax 2

3.9 × 104 N/m 2 5.8 × 104 N/m

11.4 Suspension-spring systems

311

larger number of coils also make a softer spring. According to Eq. 11.36, the maximum shear stress in a relatively thick wire is much smaller. We have shown in Eq. 11.23 that the natural frequency of a spring– mass system is only linked with the stretch ΔL under gravity. In any STM design, it is better to reduce both the total length of the spring and the resonance frequency. This can be achieved by choosing extension springs that are preloaded, or with an initial tension. The extension spring will not be stretched unless a weight greater than the initial tension is applied. In the actual design work, the springs are chosen from catalogs of manufacturers, e.g., Lee Spring Company, 1462 62nd Street, Brooklyn, New York 11219. 11.4.3

Eddy-current damper

When a conductor moves in a magnetic field, damping forces are generated by eddy currents induced in the conductor. The magnetic damper, with its reliability and thermal stability, has been utilized in various branches of engineering. Nagaya and Kojima analyzed it [315, 316]. A schematic of an eddy-current damper is shown in Fig. 11.9. A copper block with resistivity ρ is under the influence of the magnetic field B of a permanent magnet. The relative motion of the magnet and the copper block in the x direction causes a force in the −x direction. By solving Maxwell’s equation numerically, Nagaya [315] calculated the force to be  2 2  B πa t Fx = −cVx = −C0 Vx . (11.37) ρ This equation shows the general characteristics of the damping: the force is proportional to the area πa2 , thickness t, velocity Vx , conductivity of the copper block 1/ρ, and square of the magnetic field intensity. Therefore, by

Fig. 11.9. Eddy-current damper. A magnet is placed against a metal plate of resistivity ρ. When there is a relative velocity between them, a viscous force occurs. The force constant depends on the strength of the magnetic field and the dimensions of the plate and the magnet. (Reproduced with permission from [309]. Copyright 1987, AVS The Science and Technology Institute.)

312

Vibration Isolation

Fig. 11.10. Dimensionless constant in the calculation of the damping constant. While using Eq. 11.37 to calculate the damping constant, the dimensionless parameter C0 can be chosen from this chart (Reproduced with permission from [309]. Copyright 1987, AVS The Science and Technology Institute.)

using strong permanent magnets, for example, Co5 Sm magnets, the damping force can be made very large. If the size of the copper block is much larger than the diameter of the magnet, C0 = 0.5. If the sizes are comparable, the dimensionless constant C0 depends on the geometry. Nagaya [315] gave the values of C0 as a function if the ratios 2a/b and d/b, as shown in Fig. 11.10 In a design by Okano et al. [309], B= 0.2 T, a=2 cm, b = 6 cm, d = 6 cm, t = 2 cm, and for copper, ρ = 1.56 × 10−8 Ω/m. From Fig. 11.10, one finds C0 = 0.31. Eq. 11.37 gives c ≈ 20 Ns/m. It fits reasonably well with their measurements.

11.5

Pneumatic systems

Numerous pneumatic vibration isolation systems are commercially available. The prime market of these systems is for optical benches. The typical natural frequency is 1–2 Hz. For vibrations with frequencies larger than 10 Hz, a transfer function of 0.1 can be achieved. Some systems provide effective vibration isolation only in the vertical direction, whereas others are effective for horizontal directions as well. All those systems are fairly bulky. If the STM cannot be isolated from the chamber in which it resides, the entire chamber has to be vibration isolated. In this case, the commercial pneumatic system is the choice.

Chapter 12 Electronics and Control 12.1

Current amplifier

The tunneling current occurring in STM is very small, typically from 0.01 to 50 nA. The current amplifier is thus an essential element of an STM, which amplifies the tiny tunneling current and converts it into a voltage.1 The performance of the current amplifier, to a great extent, influences the performance of the STM. There are natural limits for the overall performance of current amplifiers, as determined by the thermal noise, stray capacitance, and the characteristics of the electronic components. In this section, we present these issues by analyzing several typical current amplifier circuits, which can be easily made and used in actual STMs. Figure 12.1 shows two basic types of current amplifiers. The first is called a feedback picoammeter [317]. The circuit consists of two components, an operational amplifier A, and a feedback resistor RFB . An operational amplifier (op-amp) has a very high input impedance, a very high voltage gain, and a very low output impedance (see Appendix H for details). To a very good approximation, the output voltage should provide a feedback current through the feedback resistance RFB to compensate the input current such that the net current entering the inverting input of the op-amp is zero. The noninverting input is grounded, and the voltage at the inverting input should be equal to ground. This implies VOUT = −IIN RFB .

(12.1)

The minus sign indicates that the phase is reversed. For RFB = 100 MΩ, one nanoampere of input current results in an output voltage of 100 mV. In some cases, the sample must be grounded, then the noninverting input of the op-amp may be connected to a fixed DC voltage as the bias. The output voltage is the sum of the fixed DC voltage and the voltage corresponding to the current. The fixed DC voltage is then subtracted from the output. There are several factors which impose natural limits for the performance of that current amplifier, such as the stray capacitance parallel to the feed1 Strictly speaking, it is a transimpedance amplifier, because the output is voltage instead of current. However, we will use the term current amplifier when no ambiguity occurs.

*OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

314

Electronics and Control

Fig. 12.1. Two basic types of current amplifiers. (a) Feedback picoammeter. It consists of two components, an operational amplifier (op-amp) A, and a feedback resistor RFB . A typical value of the feedback resistor used in STM is 108 Ω. The stray capacitance CFB is an inevitable parasitic element in the circuit. In a careful design, CFB ≈ 0.5pF. The input capacitance CIN is also an inevitable parasitic element in the circuit. Those parasitic capacitors, the thermal noise of the feedback resistor, and the characteristics of the op-amp are the limiting factors to the performance of the picoammeter. (b) An electrometer used as a current amplifier (the shunt current amplifier). The voltage at the input resistance RIN is amplified by the circuit, which consists of an op-amp and a pair of resistors R1 and R2 . The parasitic input capacitance CIN limits the frequency response, and the Johnson noise on RIN is the major source of noise. Also, the input resistance for this arrangement is large.

back resistance, and the stray capacitance parallel to the input terminals, which is discussed in the following subsections. Another possible circuit is a voltage amplifier, with a shunt resistor to convert the input current to input voltage, see Figure 12.1(b). This type of current amplifier has more disadvantages than the picoammeter: the input capacitance is always much larger than the stray capacitance across the feedback resistance, which seriously affects the frequency response; the input impedance of that current amplifier is very large, which makes the actual bias different from the applied bias. Therefore, we concentrate on the picoammeter. 12.1.1

Johnson noise and shot noise

The thermal motion of electrons in a resistor results in noise that is independent of the nature of the resistance. The power of such thermal noise, within a frequency interval B, is P = 4kB T B.

(12.2)

It is often called the Johnson noise, after its discoverer (see, for example, [318]). From Eq. 12.2, we find that the rms Johnson voltage noise in a resistor R is  (12.3) E = 4kB T R B,

12.1 Current amplifier

and the rms Johnson current noise through a resistor R is  4kB T B . I= R

315

(12.4)

The peak-to-peak noise value is approximately 8 times the rms value. For example, at room temperature, over a frequency interval of 3 kHz, on a 100 MΩ resistor, the Johnson current noise is  4 × 1.38 × 10−23 × 3 × 103 I ≈ 8× ≈ 0.3pA. (12.5) 108 The larger the feedback resistor, the smaller the current noise. By using a 1 MΩ feedback resistor, the theoretical noise becomes 2 pA, a tangible value when very small tunneling current is measured. The noise of an actual resistance is always higher than the theoretical limit. While for metal resistors the noise level is close to the theoretical limit, the noise level in carbon resistors is much higher. The resistance of the tunneling junction, which is parallel to the feedback resistor, should be taken into account when its value is comparable to that of the feedback resistor. In addition to the thermal noise, the discrete nature of the current results in shot noise, whose rms value over a frequency interval B is of the form [318]:   ¯ Ishot = 2eI¯ B ≈ 5.66 × 10−10 I, (12.6) where e is the electron charge and I¯ is the average DC current, in units of amperes. Except for an extremely low current ( I¯ 1,

(12.45)

the system becomes absolutely unstable. To improve the accuracy of imaging, a high open-loop gain G is desirable. From Eq. 12.45 it is obvious that the system would soon become unstable with a moderately high G. This situation can be partially improved by decreasing the Q of the piezo, for example, by filling the piezo tube with an elastomer. A common method of achieving a high gain and at the same time preventing instability is to introduce a compensation circuit, for example a large RC time constant in the feedback circuit. Sometimes, it can be as long as 1 sec. In this case, the pole given by the large RC constant dominates the response of the entire system. In other words, the 1/RC pole becomes the dominant pole. In this case, Eq. 12.35 becomes ZT (s) =

1 1 . RC s s 1+ G

(12.46)

The inverse Laplace transform gives zT (t) = 1 − e−Gt/RC .

(12.47)

If the gain G is sufficiently large, the response of the tip can still be sufficiently fast. Nevertheless, the other factors will show up, and oscillation will occur. Practically, the gain G is limited by the amplifiers in the circuit. The optimum condition is chosen in between, either by a more careful analysis or by measuring the actual time response of the system. An example of the measured time response is shown in Fig. 12.7. The insertion of an RC circuit in the feedback electronics right before the high-voltage amplifier for the z piezo has some other advantages. First, much of the high-frequency noise is efficiently filtered out. Second, it facilitates the realization of the electronics for spectroscopic study, which we will discuss in the following section.

12.3

Computer interface

Computer control has been an essential part of STM ever since the very beginning. The different variations of computer interface and software are virtually unlimited. In this section, we describe the essential elements of computer interfaces. Although in the early years the x, y scanning was made by function generators, most of the laboratory STMs and commercial STMs use software and D/A converters to generate raster scan voltages. An important point

12.3 Computer interface

327

Fig. 12.8. The essential elements of a computer-controlled STM. The feedback electronics are replaced by a single-CPU computer. A Motorola 68020 microprocessor and a 68881 math coprocessor are used to perform the feedback control. A commercial VME crate is applied. The versatility of the software-controlled system facilitates the optimization of the transient response of the STM. (Reproduced with permission from [320]. Copyright 1989 American Institute of Physics.)

is that D/A converters have finite step sizes. A typical D/A converter has a full range of ±10 V, with 12-bit accuracy. Each step is 20 V/4096=4.88 mV. By using it directly to drive the x, y piezo with a typical piezo constant of 6 nm/V, each step is about 0.03 nm. If amplification is installed, the step size becomes larger. The reading of z piezo voltage is taken to the computer with an A/D converter. Using a 16-bit A/D converter, each step is 0.305 mV. For a z piezo of 2 nm/V, each step is 0.6 pm. The whole range is 40 nm. If a range of 400 nm is required, each step becomes 6 pm. To expand the range of z piezo and at the same time retain a high reading accuracy, a DC offset for the z piezo reading can be implemented. The advantage of using a computer and A/D and D/A converters for piezo control and reading is that, for tube piezos, especially when the tip is mounted at the edge of the tube instead of at the center, the x, y scan is not linear, and there is substantial crosstalk between x, y, and z. This nonlinearity and interference can be corrected by software. The feedback electronics, an analog circuit described in the previous section, can be executed with a dedicated computer or, more precisely, by a dedicated microprocessor. Figure 12.8 shows an example of such a system, described by Piner and Reifenberger [320]. The transfer function of the

328

Electronics and Control

feedback system is then software controlled. It is more versatile than the analog circuits. The local tunneling spectroscopy can be easily implemented on such a digital feedback system by simply holding the z piezo voltage and ramping the bias. Some of the commercial STMs use such a digital feedback system. The bias can be implemented from the computer through a D/A converter. The output from a typical D/A converter, ±10 V in range and 4.88 mV per step, is ideal for bias control and for local tunneling spectroscopy. The speed of output from a D/A converter and the speed of reading by an A/D converter are typically 30 kHz, which matches the speed of the current amplifier. With an additional A/D conversion for the tunneling current (the output of the current amplifier), the local tunneling spectroscopy can be implemented by the computer without additional analog electronics. 12.3.1

Automatic approaching

In the early years of STM operation, coarse approaching was conducted manually. It is a painfully difficult process. With an electrically controlled coarse advance mechanism, the coarse positioning can be performed automatically. The process is as follows: 1. Disable the feedback loop. 2. Withdraw the z piezo to the limit. 3. Take one step forward with the coarse positioner. 4. Activate the feedback loop. 5. If a tunneling current is detected and the tip is stabilized, then stop. Otherwise repeat the loop. The automatic approaching procedure is used in many home-made STMs and almost all commercial STMs.

Chapter 13 Mechanical Design We have discussed the common elements of STM in the previous chapters, including piezodrives, vibration isolation, and electronics. An important element of an STM is the coarse positioner, which moves the relative position of the tip versus the sample in steps exceeding the range of the piezodrive (typically a fraction of a micrometer). The initial success of the STM was partly because of the invention of a novel piezoelectric stepper, nicknamed the louse. Later, various other stepping mechanisms were introduced. The coarse positioner largely determines the mechanical design of the STM. In this chapter, we discuss several important examples of the coarse positioners and the mechanical design of STM.

13.1

The louse

The piezoelectric stepper, nicknamed the louse, was the first successful nanostepper used in UHV STM [205]. A schematic of the louse is shown in Fig. 13.1. As shown, the actuating element of the louse is a piezoelectric plate, which can be expanded or contracted by applying an actuation voltage (±100 to ±1000 V). It is resting on three metal feet, separated by high-dielectric-constant insulators from the three metal ground plates. The feet are clamped electrostatically to the ground plates by applying a voltage. The motion of the louse resembles the walking of a creature with three legs. Each step consists of the following six substeps. First, loosen one of the three feet by eliminating the clamping voltage on that foot. Second, expand or contract the piezo plate to move the loose foot to a new position. Third, apply clamping voltage on that foot. Fourth, loosen another foot. Fig. 13.1. The piezoelectric stepper: The louse. It consists of a piezoelectric plate, standing on three metal feet, separated by high-dielectric-constant insulators from three metal ground plates. The feet can be clamped electrostatically to the ground plate by applying a voltage. By alternatively activating the clamping voltage and the voltage on the piezo plate, the louse crawls like a creature with three legs. (After Binnig and Rohrer [32].)

*OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

330

Mechanical Design

Five, release the voltage on the piezoelectric plate. Six, apply clamping voltage on the other foot. The louse is then moved in a new position. By alternatively activating the three feet, the louse can move in any direction on the ground plate. The size of a step ranges from 10 nm to 1 μm, up to 30 steps per second. To make a louse work properly, the feet and the ground plate have to be polished and cleaned carefully. However, once working, the louse is fairly reliable.

13.2

The pocket-size STM

By virtue of the small size of the louse, a pocket-size STM was developed and incorporated in an UHV scanning electron microscope chamber [321], see Fig. 13.2. The actual dimensions are 10×6×4 cm3 . Many of the early spectacular STM images were obtained using the pocket-size STM, or slightly improved versions of it, for example, Feenstra, Thomson, and Fein [322], Chiang, Wilson, Gerber, and Hallmark [87]. The excellent performance of the pocket-size STM is largely due to the smallness and the rigidity of the piezoelectric stepper (louse). The efficiency of vibration isolation by the stack of metal plates is not as good as the two-

Fig. 13.2. The pocket-size STM Vibration isolation and damping are achieved by a stack of stainless-steel plates separated by Viton O rings (not shown) in between. On the top metal plate are the louse and the piezoelectric tripod. (1) A metal plate with the sample holder, with only one of the three screws tightened. (2) Piezoelectric plate. (3) Anodized aluminum feet. (4) Tip holder. (5) The current lead. Viton pieces on the edges of the metal plates are used for isolating the vibration transmitted through the wires. (Reproduced with permission from Gerber et al. [321]. Copyright 1986 American Institute of Physics.)

13.3 The single-tube STM

331

stage suspension-spring system. To reduce the influence of low-frequency vibration further, an additional suspension-spring stage is often applied. Feenstra, Thomson, and Fein [322] used a two-stage suspension-spring vibration isolation system, whereas Chiang, Wilson, Gerber, and Hallmark [87] used very long springs (longer than 25 cm) to isolate their pocket-size STM from the vibrations of their vacuum chamber. Another advantage of the louse is that two-dimensional coarse motions can be achieved. By placing the pocket-size STM in a vacuum chamber with a scanning electron microscope (SEM), it is possible to locate interesting features on the sample surface with the tip. This enabled several authors to locate the interface of a device, for example, Muralt et al. [300], Salmink et al. [323], and Albrektsen et al. [324].

13.3

The single-tube STM

After the tube scanner was invented (Binnig and Smith [301]), it soon became the primary choice of piezo scanners in STM. Its small size and high natural resonance frequency make the mechanical design and vibration isolation much easier. Currently, most of the commercial STMs, as well as home-made STMs, use tube scanners. Figure 13.3 shows an STM designed by Drake et al. [325]. The tube piezo scanner is adhered at the center of a heavy and sturdy metal cylinder. The tip is mounted on the edge of the piezo tube, similar to the original arrangement of Binnig and Smith [301]. The preamplifier, mounted on Fig. 13.3. Single-tube STM. The tube piezo scanner is adhered inside a sturdy metal cylinder, which sits on three screws on the base plate. The two front screws make the coarse approaching. The rear screw, actuated by a stepping motor, makes fine approaching by using the two front screws as the pivot axis. The preamplifier (not shown) is mounted directly on top of the metal cylinder to eliminate the microphone effect, see Section 12.1.3. The entire unit is rigid enough that a mediocre vibration isolation device can provide atomic resolution. (Reproduced with permission from [95]. Copyright 1988 American Association for the Advancement of Science.)

332

Mechanical Design

a small printed circuit board of less than 25 mm in diameter, is directly connected to the tip through a hole in the metal cup. This entire assembly constitutes the scanning head. The head is seated on three screws with polished spherical ends, which are bolted through the base plate on which the sample is mounted. The two screws near the front are coarse advance devices. By using the front screws, the tip is advanced to be very close to the sample surface. Then, fine approaching is executed through the screw at the rear. The two front screws now become the pivot axis. Because the line connecting the two front screws is very close to the end of the tip, there is a huge reduction in the mechanical motion of the rear screw. The rear screw is usually actuated by a stepping motor. Automatic approaching is easily implemented. As expected from its structure, such an STM is very rigid. To improve its rigidity further, the polished tops of the three screws are made of permanent magnets, and the cup is made of a soft ferromagnetic material. The magnetic force adds to the gravitational force to increase the natural resonance frequency of the entire unit. The lowest natural resonance frequency of the entire unit can easily be greater than 10 kHz. From the previous discussion of vibration isolation, if a single-stage vibration isolation system with a natural frequency of 10 Hz is used, the overall transfer function in the frequency range of some 10 Hz to 10 kHz is 10−6 . Remember that a pendulum of natural frequency 10 Hz means a length of 2.5 mm; so the entire STM unit can be made very small. The simple single-tube STM is preferentially operated in air. With an air bag, it can be operated in controlled ambient conditions. It is also convenient to operate in liquid. A refined version of this STM has become a commercial product, Nanoscope, which is probably still the most popular STM in the open market.

13.4

The Besocke-type STM: the beetle

Ever since the first publication by Besocke in 1987 [326], the design has been adapted by numerous laboratories worldwide as the basic structure of homemade STM and AFM. It is characterized by simple mechanical structure, effective temperature compensation, and immunity from vibration. As shown in Fig. 13.4, the design uses four identical tube scanners. The center tube is the xyz-scanner. The three outer tubes, the carriers, with a ball bearing attached at the end of each, are used to support the sample ring, perform tip approaching and retrieving, as well as horizontal course positioning of the sample relative to the tip. The approaching and retrieving of the tip as well as the coarse lateral motion relative to the sample are performed by a slip-stick motion including three steps – see Fig. 13.5. First, the outer tubes are extended to lift the

13.4 The Besocke-type STM: the beetle

333

Fig. 13.4. The Besocke-type STM. The basic building blocks are four identical tube piezo scanners. The center tube is the xyz-scanner. The sample disc is sitting on the three carriers through the ball bearings by gravity. The three carriers are performing tip approaching and retrieving, as well as horizontal coarse positioning. (Reproduced with permission from [326]. Copyright 1987 Elsevier.)

microscope ring. Second, the tubes are contracted, moved sidewise, then extended back quickly to reach a new position. Third, the tubes move slowly sidewise to complete a motion. The tip approaching and retrieval are processed through a ramp – see Fig. 13.6. In this figure, the overall arrangement is reversed: the four tube scanners are mounted on the microscope ring, and the tip is pointing downwards towards the sample ring. Because of the ramp, as a result of a rotation, the distance between the tip and the sample is changed. The design has a nickname of the “beetle”. Because the four tube scanners are identical, the system is thermally compensated. Furthermore, because the four piezo tubes have the same speed of sound, it is fairly immune to external vibration. To improve stability, the resonance frequency of the Besocke-type STM can be made very high by choosing a small dimension. For example, in the

Fig. 13.5. Working principle of the Besocke design. The basic building blocks are four identical tube piezo scanners. First, the outer tubes are extended to lift the microscope ring. Second, the tubes are contracted, move sidewise, then extended back quickly to reach a new position. Third, the tubes move slowly to complete a move. (Reproduced with permission from [326]. Copyright 1987 Elsevier.)

334

Mechanical Design

Fig. 13.6. Tip approaching and retrieving of the Besocke design. The three carriers and the scanner are mounted on the microscope ring. The sample ring has three ramps, typically having a slope of 2◦ . By stepping the three carriers to perform a rotation about the center, the distance between the microscope ring and the sample is changed. The shape resembles a beetle. (Reproduced with permission from [327]. Copyright 1989 American Institute of Physics.)

original design, the piezo tubes have outer diameter of 2 mm, length 10 mm, and the size of the rings is about 10 mm. An important variation of the Besocke design is the variable-temperature STM developed by Wilson Ho’s group [328], see Fig. 13.7. The design enables the temperature to vary over a wide range for the study of temperaturedependent phenomena on the atomic scale. The temperature is varied by using a continuous-flow cryostat, see the upper half of Fig. 13.7. The temperature of the cold tip is regulated by adjusting the cryogen flow rate with a needle valve and with the use of a heater wrapped around the cold tip. The STM is vibrationally isolated with three suspension springs, with a resonance frequency of approximately 2 Hz. Eddy current damping is provide by three samarium-cobalt magnets and the copper piece under the STM. The three outer tubes with high-purity tungsten balls at the ends are used for xy scanning and the coarse approach (note that this is different from the original Besocke design). The central piezo tube, with the tip assembly at its end, is for the z motion only. Coarse approach is accomplished by inducing the sample holder to rotate by repeated slip-stick motions. The tungsten balls are in contact with the ramps on the sample holder sloped at 2.5◦ . Guard rails are attached to the outside of the ramps to prevent the sample holder from falling off the STM. Because the Besocke design provides a natural temperature compensation, that STM has been able to image the same area of the surface while changing the temperature from 80K to 8K. For other examples of Besocke-type STM and AFM designs, see [329, 330, 331, 332, 333].

13.5 The walker

335

Fig. 13.7. A Besocke-type variable-temperature STM. The STM is cooled by a continuous-flow cryostat. The first-stage vibration isolation is accomplished by bolting the UHV chamber to an optical table equipped with pneumatic legs. The second-stage vibration isolation is accomplished by hanging the STM on three springs, together with eddycurrent damping. Atomic resolution is achieved even if the temperature is varied during the scanning. (Reproduced with permission from [328]. Copyright 1999 American Institute of Physics.)

13.5

The walker

In case very high rigidity and stability are required, especially under cryogenic temperature and UHV environments, the walker, developed by S. H. Pan and coworkers [334], has been found to be suitable. An STM with the walker designed for millikelvin temperature and high magnetic field is shown

Fig. 13.8. STM with a walker. The moving part is a sapphire prism, containing a piezo tube. The stepping is actuated by six shear piezo stacks, see Section 10.5. Four of the shear piezos are placed on the interior of a macor body. Another two are mounted on a macor plate, pressed against the sapphire prism using a spring plate through a ruby ball. Therefore, the pressures of the six shear piezo stacks are approximately equalized. (Reproduced with permission from [334]. Copyright 1999 American Institute of Physics.)

336

Mechanical Design

Fig. 13.9. Working principle of the walker. (1) through (4) The shear piezo stacks are first moved individually in the same direction by applying a voltage with high ramp speed on each one consecutively. (5) Then, the applied voltages on all the shear piezos are reversed slowly but simultaneously. The sapphire prism then follows the synchronized move of all the shear piezo stacks. (Reproduced with permission from [334]. Copyright 1999 American Institute of Physics.)

in Fig. 13.8. As shown, a tube scanner is placed inside a sapphire prism. The sapphire prism is mounted by six pieces of shear piezo stacks. Four of the shear piezo stacks are placed on the interior of a macor body. Another two are mounted on a macor plate, pressed against the sapphire prism using a spring plate through a ruby ball. Therefore, the pressures of the six shear piezo stacks on the sapphire prism are approximately equalized. The functioning mechanism of the walker is shown in Fig. 13.9. For clarity, only four shear piezo stacks are shown here. As shown in Fig. 13.9(a) 1 through 4, the shear piezo stacks are first moved individually in the same direction by applying a voltage with high ramp speed on each one consecutively. Because there are five others (here only three are shown) that stand still, the sapphire prism does not move with that single shear piezo. After all the individual shear piezo stacks have been moved to the new position, the applied voltages on all the shear piezos are reversed slowly but simultaneously. The sapphire prism then follows the synchronized move of all the shear piezo stacks. An important issue of the design is the material for the prism and the surfaces of the shear piezo stacks. Because the piezo and the prism are in firm contact, during the first phase of the move, the friction is high. Therefore, the materials must be very hard to resist wear and tear. The best material for the prism is sapphire, and the surfaces of the shear piezo stacks must be covered with alumina, the ceramic form of sapphire.

13.6

The kangaroo

The walker has superb stability and repeatability. However, the electronics, consisting of six independent functional generators and six power amplifiers, is fairly complicated. Because of the sequential actions, the speed is low. Another design, based on the stick-slip action, is much simpler and much faster [335, 336]. The mechanical design, however, is identical to the walker, see Fig. 13.8.

13.6 The kangaroo

337

Fig. 13.10. The stick-slip stepper. (a) The driving waveform. The six shear piezo stacks, Fig. 13.8, are connected together. From t0 to t1 , the driving voltage follows a quadratic curve. The force is a constant. From t1 to t2 , the piezo stacks move with a constant deceleration. The acceleration is designed such that the force does not to exceed the static friction threshold. From t2 to t3 , the voltage descends rapidly. Due to inertia, the prism does not follow the motion of the piezo stacks. (b) The design of the shear-piezo stack. The four shear piezo plates are arranged to have reversed polarization directions for adjacent ones. The dashed lines show the motion, exaggerated. An alumina ceramic pad is glued on top of the stack to provide friction and improve the life of the device. (Image curtsey of O. Pietzsch, Hamburg University.)

Instead of moving the six shear piezo stacks individually and sequentially, they are connected and thus move together. Therefore, only one set of electronics is required. To achieve maximum speed, the driving waveform is carefully designed, see Fig. 13.10(a). From t0 to t1 , the driving voltage ascends with a quadratic curve. Consequently, the six piezo stacks move synchronously with a constant acceleration. The force between the piezo stacks and the prism is a constant. From t1 to t2 , the piezo stacks move with a constant deceleration. The force between the piezo stacks and the prism is still a constant but of opposite direction. The magnitude of the force is designed such that it is below the static-friction threshold. Therefore, the prism moves with the shear piezo stacks, which is the stick phase. From t2 to t3 , the voltage descends rapidly. Due to inertia, the prism does not follow the move of the shear piezo stacks, which is the slip phase. The magnitude of the force could be well above the static friction threshold. The action is similar to a jumping kangaroo rather than a sneaking cat. To make the actions more efficient, the shear-piezo stacks are carefully designed. Figure 13.10(b) shows the details. Four pieces of shear piezo plates are stacked such that the orientation of polarization is as shown in Fig. 13.10(b) with half arrows. The four shear piezo plates are cut into shapes such that the electrodes can be accessed easily. On top of the shear piezo stack is a pad made of alumina ceramics. It makes a secure and durable contact with the sapphire prism. For a quantitative comparison of the inertial and frictional configurations, see [337].

338

Mechanical Design

Fig. 13.11. The Inchworm. It consists of an alumina shaft inside a piezoelectric tube, which consists of three sections. Top, a photograph of the MicroInchworm Motor, actual size. Left, walking mechanism: (1) Clamp the lefthand side by applying a voltage to section 1. (2) Extend section 2. (3) Clamp the right-hand end. (4) Unclamp section 1. (5) Contract the center section. (6) Clamp section 1. (7) Unclamp section 3. (By courtesy of Burleigh Instruments, Inc.)

13.7

The Inchworm

Translational motions of very small steps can be utilized by a commercial device, the Inchworm.1 Figure 13.11 shows the principle of the walking mechanism of the Inchworm. An alumina shaft is slid inside a PZT sleeve, which has three sections. The three sections of the PZT sleeve are numbered 1 through 3, respectively. By sequentially activating the three sections, axial motions in small steps are made. The outer two PZT elements, numbered 1 and 3, act as clamps. When a voltage is applied to one of them, its diameter shrinks to grip the shaft tightly. The center section does not contact the shaft, but the length can change according to the applied voltage. The function of the Inchworm is illustrated in Fig. 13.11. 1 Inchworm is a registered trademark of Burleigh Instruments, Inc., Burleigh Park, Fishers, New York 14453.

13.8 The match

339

Fig. 13.12. An STM with an Inchworm as coarse positioner. (By courtesy of Burleigh Instruments, Inc.)

Figure 13.12 shows an STM design utilizing an Inchworm. The entire STM, including a fixed sample mounting block and a tube scanner attached to the center shaft of an Inchworm, is on a thick base plate. The base plate is vibration isolated with four suspension springs from the flange. For sample loading and tip loading, the base plate can be locked down to the frame, which is attached to the flange.

13.8

The match

The Inchworm has some problems requiring attention. First, the clamping mechanism is based on the radial contraction of the piezo tube. Even by using the PZT material with a high piezoelectric constant, PZT-5H (see

340

Mechanical Design

Fig. 13.13. The Aarhus STM design. The sample (1) is placed in a tantalum holder (2), which is firmly pressed against a metal block by springs (3). The sample assembly is thermally and electrically insulated from the STM body by three quartz balls (10). The tip (4) is attached to the top of the tube scanner (6), which is glued to a φ 2.5 mm SiC rod (7). The rod is driven by a small Inchworm motor (9), mounted to the STM body through a Macor ring (8). The Zener diode (11) is used for heating the STM body to maintain a constant temperature. (By courtesy of E. Laegsgaard for the original high-resolution figure. See [338, 339] for details.)

Table 10.2), at room temperature, with a maximum applied voltage 300 V on a h = 1 mm thick tube, the contraction of the radius R is ΔR = d31 V

274 × 10−12 × 300 R ≈ R ≈ 8 × 10−5 R. h 10−3

(13.1)

It is extremely small. Since the typical thermal expansion coefficient is 5 × 10−6 , if the thermal expansion coefficient of other components does not match that of PZT precisely, the temperature window of operation is only 10 to 20 degrees. Second, the piezoelectric constant of PZT decreases with lower temperature. For example, at cryogenic temperature, the piezoelectric constant of PZT-5H would drop to 50 pm/V, see Fig. 9.7. Third, because of the friction between the piezo tube and the rod, the lifetime of the Inchworm is limited. According to the data sheet from the manufacturer, the life is about 2000 meters of motion. Fourth, the rotation of the rod in the piezo tube is out of control. To limit the free rotation, other devices must be attached. However, the advantages of the Inchworm—simplicity, rigidity, and small size—are attractive. The principle of the Inchworm has been successfully applied to variable temperature STM and AFM operable from 20K to above room temperature by a group in Aarhus University with innovations: The temperature of the Inchworm and the temperature of the sample are controlled independently, and the matching of the diameters is accomplished by using the solidification of a low-melting-point metal [338, 339]. Using a very small size, the instrument is highly immune to environmental vibration. As discussed in Section 11.3, an effective approach to reducing the effect of environmental vibration is to reduce the size of the scanner.

13.8 The match

341

Fig. 13.14. Photographs of an Aarhus STM. The size of the ZrO2 rod resembles that of a match. The small size is favorable for vibration isolation and fast scanning for video-rate STM to observe dynamic processes [339]. Courtesy of F. Besenbacher and E. Laegsgaard for allowing me to take those pictures.

Figure 13.13 shows the design of the Aarhus STM. It uses a very small tube scanner, 3 mm in diameter and 4 mm long. Coarse positioning is accomplished by a home-made Inchworm motor, which consists of a φ 2.5 mm zirconia (ZrO2 ) rod and a matched piezo tube. The STM and the sample are thermally isolated using three quartz balls, which have a very low thermal conductivity but a high mechanical strength. Because the sample and the STM are thermally isolated, the temperature of the sample can be regulated without affecting the temperature of the STM. Such an arrangement has two advantages. First, since the thermal capacitance of the sample assembly is small, heating and cooling can be fast. Second, the STM assembly can be maintained at a fairly constant temperature to stabilize the operation of the Inchworm. To maintain a stable temperature, the STM assembly is heated by a 70 V Zener diode. Using a high-voltage Zener diode as the heating element has an advantage: since the current required is small, the wires can be quite thin, which would not affect vibration isolation. Figure 13.14(a) shows a photograph of the Aarhus STM, to provide a feeling of the actual size and shape. Figure 13.14(b) is a photograph of the central part of the STM, the SiC rod and the piezo tube. As shown, the size and the shape resemble that of a typical match.2 The smallness of the tube scanner provides two advantages. First, it is fairly immune to external vibration. Second, the resonance frequency could be very high. Typically, the x, y resonance frequency is 8 kHz, and the z resonance frequency is 90 kHz. A high scanning speed could be achieved, which enables video-rate imaging for the observation of dynamic processes [339].

2 As narrated by Hans Christian Andersen in Den lille Pige med Svovlstikkerne, when the tip of a match scratches a surface, it generates a hallucination of Heaven, the kingdom of atoms. Also see Plate 5, Stairway to Heaven to touch atoms.

Chapter 14 Tip Treatment 14.1

Introduction

The importance of tip treatment in STM was recognized by Binnig and Rohrer from the very beginning of their experimentation [205]. In order to understand the observed resolution, they realized that if the tip was to be considered as a piece of continuous metal, the radius must be smaller than 1 nm. The number does not appear very meaningful for a tip radius, but it is nevertheless clear that conventional field-emission tips of radius ≈ 10–100 nm would not provide the resolution they observed. In the first set of STM experiments, the tips were mechanically ground from Mo or W wire of about 1 mm in diameter. Scanning electron micrographs showed an overall tip radius of < 1μm, but the rough grinding process created a few rather sharp minitips. The extreme sensitivity of the tunneling current versus distance then selects the minitip closest to the sample surface for tunneling. Several in situ tip-sharpening procedures were reported even in the very early reports [205]: by gently touching the tip onto the sample surface, the resolution was often improved, and the tips thus formed were quite stable. By exposing the tip to high electric fields, of the order of 108 V/cm, the tips are often sharpened. To date, those alchemy-style tip treatment procedures are still practiced in STM laboratories worldwide. In the STM literature, there are many scattered discussions on the procedures of tip treatment. A brief summary of the facts found in the STM literature is as follows: 1. A tungsten tip, prepared by electrochemical etching, with a perfectly smooth end of very small radius observed by SEM or TEM, would not provide atomic resolution immediately. 2. Atomic resolution might happen spontaneously by repeated tunneling and scanning for an unpredictable time duration. 3. A crashed tip often recovers to resume atomic resolution, unexpectedly and spontaneously. 4. During a single scan, the tip often undergoes unexpected and spontaneous changes that may dramatically alter the look and the resolution from one half of an STM image to another half. *OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

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Tip Treatment

5. Various in situ and ex situ tip sharpening procedures were demonstrated. These procedures might provide atomic resolution, but often make the tip end looks even worse under SEM or TEM. 6. The tip that provides atomic resolution often gives unpredictable and non-reproducible tunneling I/V curves. 7. A mechanically cut Pt–Ir tip often generates beautiful images even if it looks like a warthog under SEM or TEM. In this chapter, we present various methods of tip preparation, treatment and characterization. Most of them are found empirically. A thorough understanding of these procedures is still incomplete. Some of the explanations are tentative. Tip preparation and characterization is one of the central experimental problems in STM, and we can certainly expect that more development and understanding will be achieved in the future.

14.2

Electrochemical tip etching

The art of making sharp tips using electrochemical etching was developed in the 1950s for preparing samples for FIM and field electron spectroscopy (FES). A description of various tip-etching procedures can be found in Section 3.1.2 in Tsong [341]. The preferred method for preparing STM tips is the DC drop-off method. The basic setup is shown in Fig. 14.1. It consists of a beaker containing an electrolyte, typically an aqueous solution of NaOH, 1 M to 2 M. A piece of

Fig. 14.1. Electrochemical etching of tungsten tips. (a) A tungsten wire, typically 0.5 mm in diameter, is vertically inserted in an aqueous solution of NaOH, typically 1M to 2M. A counterelectrode, usually a piece of platinum or stainless steel, is kept at a negative potential relative to the tungsten wire. (b) A schematic illustration of the etching mechanism, showing the “flow” of the tungstate anion down the sides of the wire in solution. (Reproduced with permission from [340]. Copyright 1990, AVS The Science and Technology Society.)

14.2 Electrochemical tip etching

345

W wire, mounted on a micrometer, is placed near the center of the beaker. The height of the W wire relative to the surface of the electrolyte can then be adjusted. The cathode, or counterelectrode, is a piece of stainless steel or platinum placed in the beaker. The shape and location of the cathode has very little effect on the etching process, so it can be chosen for convenience. A positive voltage, 4 V to 12 V, is applied to the wire, which is the anode. Etching occurs at the air–electrolyte interface. The overall electrochemical reaction is [340]:

The etching takes a few minutes. When the neck of the wire near the interface becomes thin enough, the weight of the wire in electrolyte fractures the neck. The lower half of the wire drops off. Actually, this procedures makes two tips at the same time. The sudden rupture leaves a ragged edge of work-hardened asperities at the very end of the tip, the detailed structure of which is usually irregular, see Fig. 14.3. To remove the residual NaOH from the tip surface, a thorough rinsing with deionized water and pure alcohol is necessary. Several parameters affecting the etching process have been studied by Ibe et al. [340]. First, regarding the cell potential, or the voltage between the electrodes, it was found experimentally that the etching rate is virtually flat for a voltage of 4–12 V. Below 4 V, the etching rate drops significantly. Although any voltage between 4 and 12 V is adequate, a voltage close to 4 V may reduce the oxide thickness. Second, the shape of the meniscus affects the aspect ratio and the overall shape of the tip. If the position of the meniscus drops too much during the etching process, a readjustment of the wire height is helpful in obtaining the desired tip shape. Third, the lower the electrolyte concentration, the slower the etching process. Because the OH− is consumed in the reaction, the NaOH solution should be changed after a few etching processes. Fourth, the length of the wire in solution affects the radius of curvature of the tip end. Since the drop-off occurs when the narrow neck no longer supports the weight of the wire in solution, a shorter wire in the liquid is favorable for generating sharper tips. For the 0.25 mm diameter W wire, the optimum wire length in the liquid is found to be 1–3 mm. The most important parameter that affects the final shape of the tip end is the time for the etching current to cut off after the lower part drops off. The shorter the cutoff time of the etching current, the sharper the tip end, see Fig. 14.2. To shorten the cutoff time, a simple electronic circuit is helpful. When the lower part of the wire drops off, the etching current

346

Tip Treatment

Fig. 14.2. Dependence of tip radius of curvature with cutoff time Scanning electron micrographs of tips with different etching-current cutoff time. (a) 600 ns, with an average radius of curvature 32 nm. (b) 140 ms, with an average radius of curvature 58 nm. (c) 640 ms, with an average radius of curvature 100nm. (Reproduced with permission from [340]. Copyright 1990, AVS The Science and Technology Society.)

suddenly drops. The electronic circuit senses the drop of the etching current and turns off the current completely. An example of such a circuit is given in detail in [340]. Various improved methods for making STM tips linked with SEM studies have been discussed in the literature, for example [342, 343] and references therein. However, since the STM resolution does not have a direct correlation with the look of the tip under SEM, the simple DC dropoff method, as described here, is usually sufficient. From the experience of the author, two simple improvements can be helpful. The first is to install an insulating piece between the cathode and anode across the liquid surface to prevent the hydrogen bubbling on the cathode from perturbing the meniscus near the anode. The second is to save the dropped piece as the tip, which might be better than the upper one. The exposure of the tip to the electrolyte and air results in the formation of a surface oxide (WO3 ). This oxide coating has to be removed before tunneling can occur. We describe various methods for removing the oxide in the following. To etch Pt–Ir tips, a solution containing 3 M NaCN and 1 M NaOH is used [344]. A circular Ni foil is used as the counterelectrode. The etching current, which depends on the area of immersed wire and the applied voltage, is adjusted to an initial value of 0.5 A using a bias of 20 V ac by varying the immersed depth of the Pt–Ir wire with a micrometer. The etching proceeds faster at the air–solution interface, and a narrow neck is formed. The part of the wire in the liquid breaks off, and two tips are formed. The tips should be thoroughly rinsed, similar to the case of W tips.

14.3 Ex situ tip treatments

14.3

347

Ex situ tip treatments

The W tips generated by electrochemical etching are seldom applicable immediately. First, immediately after etching, the tip is covered with a dense oxide layer, and often contaminated with sodium compounds from the etchant, as well as organic molecules. Therefore, procedures to remove the oxides and various contaminants must be executed before beginning STM experiments. Next, the arrangement of the apex atoms may not generate the electronic states required to generate atomic resolution, or to generate reproducible tunneling spectra. Figure 14.3 shows a typical FIM image of an as-etched W tip from a single-crystal W wire with (111) orientation. Although, on a large scale, the threefold feature is preserved, locally, serious disorder is always present. This is because, immediately before the lower part of the W wire drops off, the narrowest part of the wire experiences an enormously large stress. A plastic deformation occurs, which eventually causes the wire to rupture. The very end of the tip is often seriously distorted. The distortion does not mean that atomic resolution is impossible. If by accident the arrangement of the last few atoms is a favorable configuration to provide a protruding local orbital, atomic resolution is immediate. However, the chances are slim. Tip sharpening procedures are often needed. Many tip-sharpening procedures have been reported. Some of them are in situ, that is, executed under nearly tunneling conditions, which we discuss later. In this section, several ex situ tip-treatment procedures are described. 14.3.1

Annealing

The purpose of tip annealing is to remove the contaminants and oxides without causing blunting. Because tungsten has a very high melting temperature (3410◦ C), the temperature-and-time process window is wide.

Fig. 14.3. FIM image of a W tip immediately after etching. The tip is etched from a single-crystal W wire with (111) orientation. The threefold symmetry is visible on a large scale. Locally, severe dislocations are observed. (By courtesy of U. Staufer.)

348

Tip Treatment

Fig. 14.4. The phase diagram of the W–O system. At lower temperatures, the W surface is always covered with some kind of oxide. Above 725◦ C, at relatively low oxygen pressure, only W and WO2 are present. Because WO2 is volatile, by heating a piece of tungsten in vacuum above this temperature, a metallic W surface is generated. (After [345].)

The removal of tungsten oxide is based on the following mechanism. On tungsten surfaces, the stable oxide is WO3 . At high temperature, the following reaction takes place: 2WO3 + W → 3WO2 ↑.

(14.1)

While WO3 has a low vapor pressure at its melting point (1473◦ C), WO2 sublimes at about 800◦ C. Figure 14.4 is the phase diagram of the W– O system. As shown, with a high oxygen content, at low temperature, the mixture of tungsten oxides are stable. At T > 725◦ C, only metal tungsten and WO2 are present. Because WO2 is volatile, as well as the organic substances, a clean metal surface is generated. Several methods for annealing the tips are shown in Fig. 14.5. 14.3.2

Field evaporation and controlled deposition

Fink [346] conducted a systematic study of the methods for making tips with a well-defined atomic arrangement near the apex. The motivation is twofold. First, the lateral resolution of STM is determined by the atomic

Fig. 14.5. Tip annealing methods (a) Electron bombardment. A filament, biased negatively with regard to the tip, emits electrons to heat up the tip. (b) Resistive heating by a W filament. The tip is spotwelded to the filament. After heating, the tip is removed from the chamber and detached from the filament, and put into the vacuum chamber quickly. (c) Using the tip shank as the heating element. The tip is brought into touch with a thicker W wire, which is connected to a power supply. Current flows through the tip shank to the ground.

14.4 In situ tip treatments

349

Fig. 14.6. Tip formation by field evaporation. Top left, FIM image of a (111)oriented W tip; the (111) apex plane contains 18 atoms. Top right, the field evaporation process. Bottom, tip with pyramidal apex with one, three, and seven W atoms at the apex plane. (Reproduced with permission from [346]. Copyright 1986 Wiley-Blackwell.)

arrangement and its electronic structure. Second, this study is essential for understanding the tunneling between two microscopic objects—the tip and the sample. The atomic structure of both microscopic objects will eventually be known. The methods used by Fink [346] originated from the field evaporation procedure of FIM. (For details, see M¨ uller and Tsong [347], and Tsong [341].) Tips made of single-crystal W(111) wires are chosen as the starting material. By controlling the field intensity at the tip apex, the most protruding W atoms are stripped off, leaving a well-defined tip shape. By carefully controlling the field intensity, a well-defined pyramid is formed (Fig. 14.6). A single W atom is located at the apex. By carefully controlling the conditions, the first atom can be removed. Using well-controlled vacuum evaporation, a tip atom of different and known chemical identity can be put on top of the three-atom platform. The electronic configuration of the tip can be optimized by choosing the right tip atom, to provide the highest lateral resolution.

14.4

In situ tip treatments

The tip treatment can be done during actual tunneling. Often, these in situ tip treatments take a few seconds to complete. The effect of the tip treatment process can be verified by actual imaging immediately. If one action is not successful, another action can proceed immediately – it takes a few more seconds. As mentioned at the beginning of this chapter, these methods was already being used at the birth of the STM by the inventors in their first set of experiments [205].

350

14.4.1

Tip Treatment

High-field treatment

A detailed description of the high-field treatment was described by Wintterlin et al. [88]. It starts with a clean W tip made through electrochemical etching, and a clean Al(111) surface. The tunneling conditions are –500 mV at the sample, with a current set point 1 nA. The images produced with this tip do not show atomic resolution immediately. The tip sharpening is performed by suddenly raising the bias to –7.5 V (at the sample), and leaving it at this voltage for approximately four scan lines. The tip responds to the voltage jump by a sudden withdrawal by ≈3 nm. This is much more than the 0.2–0.4 nm that could be expected from the constant-current condition. While the bias is kept at –7.5 V, the scan lines are strongly disturbed. Subsequently the bias voltage is reduced to its initial value of –500 mV again. Now the tip does not return to its former z position but remains displaced from that by about 2.5 nm. It is obvious that the tip actually gets longer by about 2.5 nm. This process turns out to be completely reproducible and in most cases results in tips achieving atomic resolution. At the beginning, the mechanism of such tip-sharpening process was not well understood. Two hypotheses were proposed: it is either a restructuring of the tip itself or a transfer of material from the sample to the tip. If the latter is the correct mechanism, its result should depend on the sample material. Later, the same tip sharpening procedure was successfully applied on Ru(0001), Ni(100), NiAi(111), and Au(111), indicating that this phenomenon is not specific to certain surfaces [85]. Therefore, there must be a restructuring of the tip; that is, the W atoms move from the shank surface to the apex, see Fig. 14.7. The magnitude and the direction of the tip-treatment voltage is consistent with the tip-restructuring hypothesis. Actually, the directional walk of W atoms on W surfaces under a nonuniform electric field was observed using FIM by Tsong and Kellogg [348], studied in detail by Wang and Tsong [349], as summarized in Tsong’s book [341]. Near the end of a W tip, the electrical field is highly nonuniform. The W atoms on the surface, polarized

Fig. 14.7. Mechanism of tip sharpening by an electric field. (a) W atoms on the tip shank walk to the tip apex owing to the nonuniform electric field. (b) A nanotip is formed. (Reproduced with permission from [19]. Copyright 1991, AVS The Science and Technology Society.)

14.5 Tip treatment for spin-polarized STM

351

Fig. 14.8. Mechanism of tip sharpening by controlled collision. (a) The W tip picks up a Si cluster from the Si surface. (b) A Si cap forms at the apex of the tip, providing a pz dangling bond. (Reproduced with permission from [19]. Copyright 1991, AVS The Science and Technology Society.)

by the high field, are attracted to the apex, where the field intensity is the highest. The topmost atoms are then ionized and stripped off by the electric field. The experimental demonstration of the atomic metallic ion emission, as discussed in the previous section, further confirms the hypothesis of tip restructuring [350, 351]. 14.4.2

Controlled collision

Tip sharpening by controlled collision (Fig. 14.8) was also used by Binnig and Rohrer in their very first experiments with Si(111)-7×7 [32, 205]. Demuth et al. [94] provided experimental evidence that during a mild collision of a W tip with a Si surface, the W tip picks up a Si cluster. The tip then provides atomic resolution, and a crater is left on the Si surface. The pz dangling-bond state on the Si cluster is apparently the origin of the observed atomic resolution.

14.5

Tip treatment for spin-polarized STM

As shown in Section 7.6, the operation of spin-polarized STM (SP-STM) requires that both the tip and the sample surface are spin-polarized. Therefore, the preparation of the spin-polarized tip is a crucial step of SP-STM. In this section, we describe two types of method: by coating the (nonmagnetic) tip with a ferromagnetic or antiferromagnetic film; and controlled collision with a ferromagnetic or antiferromagnetic surface. 14.5.1

Coating the tip with ferromagnetic materials

Tungsten tips coated with Fe and Gd are fabricated and studied extensively with samples with known noncollinear magnetism [352]. The raw tip is an etched polycrystalline W tip, flashed in vacuum at T > 2200 K to remove oxide layers. As revealed by scanning electron microscopy, this results in

352

Tip Treatment

Fig. 14.9. The spin-density orientation of Gd- and Fe-coated tips. (a) and (b) are images taken with a Gd-coated tip. (a) The topographic image. (b) The dynamic conductance (dI/dU ) image, shows a contrast of domains. (c) and (d) are images taken with an Fe-coated tip. (c) The topographic image. (d) The dynamic conductance (dI/dU ) image, shows a contrast of domain walls. Size of the images: 200 nm × 200 nm. (Reproduced with permission from [352]. Copyright 2003 IOP Publishing Limited.)

a blunt tip with a diameter of about 1 nm. The tips were magnetically coated with 7 ± 1 monolayers of Gd or 5 ± 1 monolayers of Fe while held at 300K, subsequently annealed at T ≈ 500K for 4 minutes. Then the tip is transferred into the cryogenic STM. The sample for the study is a 1.5 monolayer of Fe epitaxially grown on stepped W(110). The Fe overlayer forms a self-organized grating of stripes with thickness alternating between one and two monolayers with domain walls between spin up (+z) and spin down (−z) magnetic domains [353]. Such a magnetic pattern is ideal for characterizing the spin polarization of the tip. The study shows the following: 1. The Gd-coated tip has a spin-density orientation along z, or an out-of-plane spin-density orientation. It is sensitive to the domains. In other words, there is a tunneling conductance contrast between the spin-up domain and the spin-down domain, as shown in Fig. 14.9(b). 2. The Fe-coated tip has a spin-density orientation along x and y, or an in-plane spin-density orientation. It is sensitive to the domain walls. In other words, there is a tunneling conductance contrast at each domain wall, but not between the spin-up domain and spin-down domain, see Fig. 14.9(d). The azimuth in the x, y plane depends on the local crystallographic orientation of the W tip, and thus cannot be predicted or controlled.

14.5 Tip treatment for spin-polarized STM

353

Fig. 14.10. Bias-dependence of spin-density orientation of tips. (a) The topographic image. (b) Dynamic-conductance (dI/dU ) image taken at bias U = −0.1 V, showing pure in-plane spin-density orientation. (c) Dynamic-conductance (dI/dU ) image taken at bias U = +0.1 V, showing a substantial out-of-plane spin-density orientation. (Reproduced with permission from [352]. Copyright 2003 IOP Publishing.)

3. The spin-density orientation of a given tip often also depends on the bias voltage, as shown in Fig. 14.10(b) and 14.10(c). The tips coated with ferromagnetic films are typically macroscopic or mesoscopic, which can resolve magnetic structures down to a few nanometers. The images can be interpreted in terms of the spin-valve effect – see Section 2.4.2. 14.5.2

Coating the tip with antiferromagnetic materials

The tips coated with ferromagnetic materials usually does not provide atomic or nearly-atomic resolution because, near the apex, there is only one magnetic domain, which generates stray magnetic field into the gap. Furthermore, the stray magnetic field of the tip could alter the magnetic structure of the sample. By coating the tip with an antiferromagnetic material, such as chromium, typically there is only one magnetic atom protruding into the gap, and the stray magnetic field is negligible [354]. 14.5.3

Controlled collision with magnetic surfaces

The process of coating the tip with ferromagnetic or antiferromagnetic materials is a slow process, which must be conducted outside the STM chamber. If the sample surface has a (partial) magnetic layer then, by colliding the tip with the surface, the tip may pick up a cluster of magnetic atoms, and a spin-polarized tip could be formed. The success rate of the process is low. However, the process can be repeated many times until a satisfactory result is accomplished. A successful example of the controlled collision method is in the dissertation by L. Berbil-Bautista at Hamburg University [58]. The sample is a Dy film deposited on W(110). By gently touching the tip with the Dy

354

Tip Treatment

surface, the tip often becomes magnetically sensitive. This tip preparation method is highly flexible but uncontrolled. However, it is always possible to repeat the tip preparation process to obtain a different tip magnetization orientation or polarization. Stable, highly polarized tips are prepared and used at temperatures as high as T = 65K.

14.6

Tip preparation for electrochemistry STM

For STM experiments in an electrolyte solution, the Faradaic current between the tip shank and the sample becomes a background signal. To reduce it, the tip shank should be coated with an insulating film, and only the apex of the tip is exposed. The allowed area of exposure can be estimated as follows. The typical Faradaic current density is 1μA/cm2 , or 0.01 pA per μm2 [111]. The typical tunneling current is 0.01 nA to 1 nA. For reducing the Faradaic current to 1 pa, the exposed area of the STM tip should be smaller than 100 μm2 . Because the radius of curvature of a typical electrochemically etched tungsten tip is smaller than 1 μm, that condition can be fulfilled without excess difficulties, and the result can be assessed using an optical microscope. The coating of the tip shank can be made with various insulating materials, such as glass, plastics, Apiezon wax, or even nail polish. Coated tips for

Fig. 14.11. Tip preparation for electrochemistry studies. (a) A home-made apparatus for tip coating. The temperature of the Apiezon wax film, supported by a copper ring heated by a soldering iron, is controlled by a variac. The tip is forwarded through the Apiezon wax film using a gear. (b) The tip, after coating, under an optical microscope. The apex of the tip, with a radius smaller than one micron, is exposed. (Original photographs by courtesy of J. Ulstrup’s group at Technical University of Denmark.)

14.7 Tip functionalization

355

electrochemistry STM are commercially available. However, good-quality coated tips are easily produced in a chemistry laboratory. Figure 14.11(a) shows a simple home-made apparatus for tip coating. A copper ring in place of the tip of the soldering iron, holds a film of Apiezon wax. A variac, connected to the soldering iron, is used to control the temperature of the wax film. After the molten Apiezon wax film reaches a suitable viscosity, the tip is forwarded from below by a gear to pass through the wax film. After cooling down, owing to surface tension, the apex of the W tip is exposed. The area of exposure can be controlled by the temperature. The right condition can be determined by trial. Figure 14.11(b) shows a W tip coated with Apiezon wax under an optical microscope.

14.7

Tip functionalization

As presented in Chapter 6 and Chapter 7, for imaging nanometer-scale features, the exact structure of the tip is not critical. The tip can be modeled as a locally spherical potential well, as in the Tersoff–Hamann theory. For resolving atomic-scale and subatomic-scale features, tip structure plays a decisive role. For imaging wavefunctions, the tip structure becomes even more significant. The importance of functionalize tip is known in the early years of STM: in order to improve the resolution on silicon surfaces, a gentle touch of the tip with a silicon surface often helps. It is explained by picking up a Si atom from the sample, and the dangling bond state of a Si atom provides a better resolution than a pure metal tip [94]. Later, tip functionalization has been practiced intentionally by reversible vertical transfer of atoms or molecules between the sample and tip. Here we offer examples of Xe atom and CO molecule. 14.7.1

Tip functionalization with Xe atom

Reproducible and reversible tip functionalization with Xe atom was reported by Eigler et al. in 1991 [355]. The experiment was performed on a STM operating at 4K under UHV condition. The sample is the (110) surface of a single crystal of nickel. The tip is made from polycrystalline tungsten wire. Through a leak valve, the Ni(110) surface is dosed with a small fraction of monolayer of Xe, which can be imaged. To prepare for a transfer, the tip is positioned on top of a Xe atom with a junction resistance of 1.5 MΩ at −0.02 V on the tip with respect to the sample. During the transfer, the feedback loop is disabled. See Fig. 14.12(3), situation A. By applying a positive electrical pulse to the tip, for example, +0.8 V for 64 msec (see Fig. 14.12(3), situation B), an Xe atom is transferred to the tip, also shown in Fig. 14.12(1). The transfer is signified by a dramatic change of the tunneling current, see Fig. 14.12(3), situation C.

356

Tip Treatment

Fig. 14.12. Transferring a Xe atom to and from the tip. (1) By applying a positive electrical pulse on the tip, a Xe atom is transferred from the Ni(110) sample to the tip. (2) By applying a negative electrical pulse on the tip, a Xe atom originally on the tip is transferred back to the Ni(110) surface. (3) The Xe atom transfer is signified by a change of tunneling conductance. After Eigler et al. [355].

To switch the Xe atom back from the tip to the sample, a negative electrical pulse to the tip is applied, for example, −0.8 V for 64 msec, see Fig. 14.12(3), situation D, also shown in Fig. 14.12(2). After that, the tunneling current recovers to the original value. The transfer process is reproducible and reversible for many cycles. As expected, the higher the voltage of the electrical pulse, the shorter time it needs to complete the transfer. In addition to the change of tunneling conductance, the STM resolution becomes better after the tip is functionalized with a Xe atom. 14.7.2

Tip functionalization with CO molecule

Functionalizing the tip with a CO molecule has many merits, and it has been practiced frequently. First, the procedure of transferring a CO molecule from the sample surface to the tip and back is well understood and fairly reproducible. Second, the characteristic electronic structure, a pair of px and py states generate special electronic effect to enhance the resolution. Third, it is easy to verify the status once it is formed. As reported by Bartels et al. [206], the experiment was performed on a home-made STM under UHV condition operating at 15 K. The starting system is a Cu(111) sample surface and an electrochemically etched tungsten tip. The Cu(111) surface is cleaned by sputtering and annealing. Due to frequent touching of the sample, the tip is expected to be covered by copper atoms. Ultralow coverage of CO molecules is generated by controlled leaking the gas to the chamber. It is well known that the adsorbed CO molecule stands upright on the surface with the C atom bonding to metal atoms. With a metal tip, regardless of the bias voltage, the CO molecule always appears as a depression. The typical starting condition is to position the tip to the CO molecule with a sample bias of +2 V and a tunneling current of 1 nA. Then, the feedback loop is turned off, and the sample voltage is increased to +3 V. Subsequently the voltage is slowly decreased to +0 V, which simultaneously decrease the gap distance by 0.1 nm to compensate the decrease in the current. During the process the current stays below 10

14.7 Tip functionalization

357

Fig. 14.13. Picking up and putting down a CO molecule. Note that the CO molecule stands upright on the surface with the C atom bonding to a metal atom. The position is flipped during the transfer process. After Bartels et al. [206].

nA. A CO molecule is transferred to the tip. The whole process takes a few seconds with a yield over 50 %. To transfer the CO molecule from the tip to the sample, position the tip at a bias of −2.5 V, then ramp the voltage from −4.5 V to −0.5 V while again decreasing the gap distance. The process also takes a few seconds, and the yield is slightly smaller than the picking up process. Figure 14.13 schematically shows the transfer process. Figure 14.14 shows STM images of an area of the sample with a metal tip (a) and after picking up a CO molecule (b). The item with a white arrow in (a) is a CO molecule, obviously being pickup up by the tip. With a CO tip, shown in (b), the CO molecules appear as protrusions. Some of the depressions observed by a metal tip do not change with a CO functionalized tip, for example, the feature indicated by a black arrow in Fig. 14.14(a) and (b). Those features cannot be transferred to the tip, and therefore are not CO molecules. To identify the nature of those features, the sample is exposed to 1 L of oxygen, and the coverage of O atoms can be estimated. From the frequency of occurance, the depression does not change with a CO tip is identified as an oxygen atom. A later experiment was done to show the reproducibility of image inversion of CO molecules by different tips [208]. The substrate Cu(111) is dosed with about 1/3 monolayer of CO. A larger number of CO molecules are

Fig. 14.14. Images before and after picking up a CO molecule. (a) before picking up, with a metal tip, the CO molecules appear as depressions. The item with a white arrow, a CO molecule, is picked up by the tip. (b) after the tip picks up a CO molecule, the images of CO molecules appear as protrusions. However, certain features such as the depression marked by a black arrow, did not change with a CO tip. It is identified as an oxygen atom. After Bartels et al. [206].

358

Tip Treatment

Fig. 14.15. Images of CO molecules with a higher dosage. (a) With a metal tip, the CO molecules appear as depressions. (b) With a CO tip, the CO molecules appear as protrusions. A statistially significant number of cases shows the reproducibility of image inversion of CO molecules by different tips [208].

observed. Figure 14.15 shows STM images with a metal tip and a CO tip. As shown in (a), with a metal tip, the CO molecules appear as depressions. With a CO tip, as shown in (b), the CO molecules become protrusions. The event that the images of CO molecules change from depressions with a metal tip into protrutions with a CO tip is highly reproducible and the number of occurances are statistically overwhelming [208].

Part III Related Methods

Part III: Related Methods By searching “SPM” (scanning probe microscopy) or “STM and related methods” on Google, you will find dozens of entries. The most relevant are STS (scanning tunneling spectroscopy) and AFM (atomic force microscopy). This Part presents those two related methods. One reason only those two are presented is personal: those are my research projects at IBM and at Hamburg University. My very first publications in STM are Theory of scanning tunneling spectroscopy in 1988 [13], and Topography and local conductance images of YBa2 Cu3 O7 crystals fractured in ultra high vacuum in 1989 [356]. The two papers became the raison d’ˆetre of my research projects to build a low-temperature cross-sectional STM for the study of the basic physics of high Tc superconductors, as approved by the management of IBM Research Center. The results of that research, the formulas for the design, test, and calibration of tube scanner under cryogenic temperatures, as well as the quantum mechanics for the interpretation of STM images, became the basis of the first edition of this book. But I have no personal experience on other related methods. A section on the STS study of high Tc superconductors is added to Chapter 15. Although I was not involved in the design and operation of AFM in early years, my background in molecular spectroscopy and quantum chemistry during my graduate study motivated me to study the relation between tunneling and atomic forces, resulting in two early publications: Attractive Atomic Force as a Tunneling Phenomenon in 1991 [123], and a chapter in Volume III of Scanning Tunneling Microscopy, edited by Wiesendanger and G¨ untherrodt, entitled Unified Perturbation Theory for STM and SFM in 1993 [194]. The theoretical predictions of those publications are well verified experimentally, as presented in Chapters 4 and 5. During 2004 to 2006, when I was a Guest Scientist at Hamburg University, I did a serious experimental reaearch project in AFM, trying to find the optimal design of quartz force detectors. A paper entitled Three-electrode self-actuating self-sensing quartz cantilever: Design, analysis, and experimental verification [357] was published three years after I joined Columbia University. According to ResearchGate, it is one of my most read publications. Although the principle of its operation is well verified, and the theory of its design is detailed, certain technical problems have to be resolved for its practical manufacturing and operation. It is not an exception in the history of science that it takes many years to convert a verified principle into a widely used product. Hopefully this will happen soon.

Chapter 15 Scanning Tunneling Spectroscopy By interrupting the feedback loop to keep the tunneling gap constant and applying a voltage ramp to the tunneling junction, the tunneling current as a function of bias is observed. It is convolution of the tip DOS and the sample DOS, and an extension of the tunneling junction experiment [120, 121, 127, 132]. However, it has a richer information content: while the tip is scanning over the sample surface, a map of tunneling spectra is generated. Actually, the original idea of building the STM was to perform tunneling spectroscopy (STS) locally on an area less than 10 nm in diameter [32]. The first successful demonstrations of STS were made in 1986 [54, 358, 359, 360, 361].

15.1

Electronics for scanning tunneling spectroscopy

In this section, we discuss a typical circuit and the methods for such measurements. Although a commercial sample-and-hold device can be used to hold the z voltage, the accuracy and holding time of those commercial integrated circuits are not good enough. In the feedback circuit shown in Chapter 11, there is a natural point where a custom sample-and-hold device can be easily inserted, as shown in Fig. 15.1. The integration capacitor C is usually of the order of 10 μF and is located right before the amplifier for the z piezo. An op-amp with FET input stage has a typical leakage current I0 of 1–10 pA. Assuming I0 = 10−11 A and C = 10−5 F, the time for the voltage on the capacitor to slip by 1 mV is Δt = 0.001

C ≈ 103 sec. I0

(15.1)

Therefore, leakage due to the amplifier after the capacitance is not a problem. The problem is the switching device before the capacitor. Feenstra et al. [54] used a reed relay as the switching device. The open-circuit resistance of a reed relay is larger than 1012 Ω. The leakage current is even smaller than the input of the FET op-amp. (Experience has shown that the printed circuit board containing those parts must be cleaned meticulously after soldering to avoid any unexpected leakage.) Therefore, if the mechanical structure of the STM is sufficiently stable, the position can be held for several seconds at virtually the same voltage. However, the reed relay has a *OUSPEVDUJPO UP 4DBOOJOH 5VOOFMJOH .JDSPTDPQZ 5IJSE &EJUJPO $ +VMJBO $IFO 0YGPSE 6OJWFSTJUZ 1SFTT   ª $ +VMJBO $IFO %0* PTP

364

Scanning Tunneling Spectroscopy

Fig. 15.1. Electronics for scanning tunneling spectroscopy. By using an op-amp with FET input stage as the isolation amplifier to the high-voltage amplifier for the z piezo, the holding time on the capacitor can be as long as 100 sec. The values of R and C show typical ranges.

switching time of a few milliseconds. To obtain one I/V curve, at least 10 msec is required. Therefore, this circuit is suitable for measuring tunneling spectra at selected points with a long holding time (up to a few seconds). Another choice is to use an electronic switch. The switching time is of the order of a few microseconds. However, the leakage current for a typical analog switch (AD7510) is 1–10 nA. An estimation similar to Eq. 15.1 gives a holding time of 1 sec. In most cases this is enough. The reason is as follows. The typical sampling rate for I/V data can be estimated from the sampling frequency of the A/D converter and the time delay of the current amplifier, which is about 50 kHz. At room temperature, the thermal broadening is 4kB T ≈ 0.1 V. A 100-point spectrum contains a sufficient amount of information. Even if a few repeated ramps have to be executed for making an average, a few milliseconds at a point is sufficient. Therefore, if tunneling spectra are taken from every point of the image, or a substantial number of points, the electronic switch is the appropriate choice as the input switching device for the feedback capacitor.

15.2

Nature of the observed tunneling spectra

Chapter 6 presents an elementary theory of STS based on the Tersoff– Hamann model. If both the tip DOS and the tunneling matrix elements are independent of the energy level, then the dynamic tunneling conductance at bias V is proportional to the local density of states of the sample at the center of curvature of the tip at energy EF + eV , Eq. 6.41:  G(V ) ≡

dI dU

 ∝ ρS (EF + eV, r0 ). U =V

(15.2)

15.2 Nature of the observed tunneling spectra

365

Although this elementary model can be used to explain some experiments, such as the scattered surface waves discussed in Section 6.2.6, in many cases, the dependence of the tip DOS and the tunneling matrix element on energy level cannot be overlooked. In planning and interpreting STS experiments, it is essential to understand the nature of the local tunneling spectra obtained by the STM. In Sections 2.2.3 and 2.2.4, using the Bardeen tunneling theory, an expression for the tunneling current with a bias V is obtained, 4πe |M (0)|2     12 eV κ0 s 1 1 d . × ρS (EF + eV + ) ρT (EF − eV + ) exp 2 2 φ¯ − 12 eV (15.3) Here the first line is a constant, and the second line gives the dependence of tunneling current on the tip DOS, the sample DOS, and the variation of the tunneling matrix element on energy. Because of the exponential factor, the values of DOS with positive always play a greater role in determining the tunneling current than the values of the DOS with negative . As a result of Eq. 15.3, if the tip-sample distance s and the absolute value of the bias voltage is not too small, for a positive bias V > 0, the electrons mainly tunnel into the empty states of the sample at E ≈ EF + eV from the tip states near the Fermi level,   dI ≈ ρS (EF + eV ) ρT (EF ). (15.4) dU U =V I =

And for V < 0, the electrons mainly tunnel from the sample states near the Fermi level into the empty states of the tip at E ≈ EF + e|V | 

dI dU

 ≈ ρS (EF ) ρT (EF + e|V |).

(15.5)

U =V

Figure 15.2 shows examples of experimental observations [362]. The experiments aimed to study the magnetic nanostructures using spin-polarized scanning tunneling spectroscopy. Figure 15.2(a) shows a constant-current topographic image of a vicinal W(110) surface covered by 1.4 monolayers of Fe. On top of a nearly complete single-layer surface, double-layer Fe islands appeared. The double-layer Fe islands are ferromagnetic, which are magnetized up or down. Using a spin-polarized tip, the tunneling spectra on the differently magnetized Fe islands are unlike. The most prominent contrast appears at a sample bias voltage near 0.75 V. The features appear regardless of the tip situation. On the other hand, the strong peaks at −0.25 and −0.54 V in one of the curves are solely due to tip state effects, as it comes

366

Scanning Tunneling Spectroscopy

Fig. 15.2. Nature of the observed tunneling spectrum. (a) Constant-current topographic image of a vicinal W(110) surface covered by 1.4 monolayers of Fe. On the background of a nearly complete monolayer of Fe, islands of Fe double layers are apparent. Tunneling spectra were taken on the double-layer sites (A) and (B), as well as a monolayer site (C). (b) Three sets of tunneling spectra observed on the three locations. With a positive bias, the spectroscopic features of the sample dominates. Especially, the prominent feature at +0.75 V shows a strong dependence on spin polarization. On the other hand, at a negative bias, the features originate from the tip spectrum. Upon a change of tip, the features change dramatically. (Adapted with permission from [362]. Copyright 2001 American Physical Society.)

and goes with a tip change. Other features, such as the peak at –0.48 V, also comes and goes with different tips. The observed phenomenon also indicates the importance of choosing a proper stabilization voltage to fix the tip-sample distance during the measurement of tunneling spectrum. The rule of thumb is: the stabilization voltage should be positive and slightly higher than the spectroscopic features of interest. For example, all the STS data in Fig. 15.2 were taken with a stabilization voltage of +1.0 or +1.2 V.

15.3

Tip treatment for spectroscopy studies

As we showed in the previous section, in order to obtain reproducible tunneling spectra, the STM tip must have reproducible DOS, preferably a flat DOS, that is, with a free-electron-metal behavior. However, the tips freshly made by mechanical or electrochemical methods, especially those providing atomic resolution, often show nonreproducible tunneling spectra. The DOS of such tips is often highly structured. To obtain reproducible STS data, a special and reproducible tip treatment procedure is required. 15.3.1

Annealing

Feenstra et al. [54] developed an in situ tip DOS smoothing procedure. This procedure uses field emission current to locally heat up the end of the tip,

15.3 Tip treatment for spectroscopy studies

367

Fig. 15.3. Tip treatment for tunneling spectroscopy. (a) By applying a relatively large positive bias to the sample, a sharp tip generates a field-emission current. (b) When the field-emission current is very high, the tip end melts. (c) The tip end recrystallizes to form facets with low surface energy. In the case of tungsten, the W(110) facets are preferred. Its surface DOS resembles a free-electron metal. (Reproduced with permission from [363]. Copyright 1992 Elsevier.)

which eventually causes a local melting and recrystallization. Details of this process are as follows: 1. Start with an electrochemically etched W tip. The cleaning or sharpening is not critical. 2. The sample, or the anode, can be any conducting material. Because the process will cause local damage to the anode, it is better to use a scratch sample or an unused area on the sample. 3. Apply a relatively high voltage (for example +10 volts) to the sample, then use the coarse positioning mechanism and z piezo to bring the tip and sample into tunneling distance. The actual value of the tunneling current is not important. The use of a relatively high bias is to avoid tip crashing due to the oxide layer on the tip surface. 4. Pull the tip back by about 100 nm. 5. Increase the bias slowly up to some 100 V until a field emission current is observed. 6. Either by further increasing the bias voltage or by reducing the tip– sample distance, carefully ramp up the field-emission current. 7. At a current of the order of 1 to 2 μA, a catastrophic decrease of the field-emission current is observed. In a small fraction of a second, the field emission current drops to zero. Schematically, the steps in this process are shown in Fig. 15.3. At the beginning, the local radius of the tip end is small. Field emission can be

368

Scanning Tunneling Spectroscopy

Fig. 15.4. Structure of some low-Miller-index W surfaces. W(110), 1.41 × 1015 atoms per cm2 . W(111), 1.15 × 1015 atoms per cm2 . W(100), 9.98 × 1014 atoms per cm2 .

established. A high current through the tip end then causes local melting. The local curvature at the end of the tip suddenly decreases. The field emission current is then reduced dramatically. The tip end recrystallizes to have a relatively large radius. Feenstra et al. [54] observed that the tips prepared in this way always provide reproducible tunneling spectra, although atomic resolution is generally not observed. A possible interpretation of this process is as follows. As the tungsten tip recrystallizes, it always tends to reduce the surface energy by forming facets with the highest density of surface atoms. For tungsten, a bcc metal, the (110) facet has by far the highest density of atoms, 1.41×1015 cm−2 , followed by the (111) facet, 1.15 × 1015 cm−2 . The (100) facet, with a surface density of atoms 9.98 × 1014 cm−2 , is unlikely to be formed when recrystallized, see Fig. 15.4. The (112) facet is nearly impossible to form spontaneously. Both field-emission spectroscopy [364] and photoemission spectroscopy [365, 366, 367] show that near the Fermi level, the DOS of the W(110) and W(111) surfaces is similar to free-electron metals, in contrast to the W(100) and W(112) surfaces, where localized dz2 surface states exhibit highly structured surface DOS near the Fermi level.

15.4

Inelastic scanning tunneling spectroscopy

The macroscopic inelastic electron tunneling spectroscopy (IETS) [144, 145, 143, 133] succeeded in detecting the vibrational spectra of molecules in a tunneling junction. Using the STM tip as one electrode, the vibrational spectra of single molecules can be observed [368, 369, 370]. A schematic of STM-IETS is shown in Fig. 15.5. Here, (a) shows an STM tunneling junction with a molecule of vibrational frequency ω. (b) is the I/V curve. (c) is the dI/dV plot. In both (b) and (c), the signal is too small. In (d) we see the d2 I/dV 2 plot, where the IETS signal is easier to identify [370].

15.4 Inelastic scanning tunneling spectroscopy

369

Fig. 15.5. A schematic of STM-IETS. (a) An STM tunneling junction with a molecule of a vibrational frequency ω. (b) The I/V curve. The change in tunneling current due to vibration excitation is too small to be detected directly. (c) In the dI/dV plot, a change in dynamic conductance can be detected, but still too small. (d) In the d2 I/dV 2 plot, the vibration excitations appear as peaks, which is easier to identify and detect. (Reproduced with permission from Ho [370]. Copyright 2002 American Institute of Physics.)

15.4.1

Instrumentation

The instrumentation of STM-IETS is identical to the classical junction IETS. Figure 15.6 is a schematic of circuitry for measuring the second derivative, reproduced from a classical paper [145]. Today, using a digital lock-in amplifier, the circuit diagram is simplified. However, the original circuit diagram shows more clearly the principle of operation. Typically, an oscillation frequency of 50 kHz is used. The second harmonic is 100 kHz. The typical modulation amplitude is 1–5 mV. Although a larger modulation voltage would generate a larger output signal, it would result in a broadening of the spectrum. 15.4.2

Tip treatment for STM-IETS

In the experiments of STM-IETS [368, 369, 370], the substrate is often a nearly-free-electron metal surface, such as Cu(100). The target molecules only cover a small proportion of the surface. A proven recipe to treat the tip is to make a controlled collision of the tip with the bare Cu(100) surface. The raw tip is made of tungsten using the standard procedure. After a controlled collision, the last atoms on the tip are likely to be copper rather than tungsten. This can be verified by making a spectroscopy on the bare

370

Scanning Tunneling Spectroscopy

Fig. 15.6. Instrumentation of inelastic electron tunneling spectroscopy. The oscillator supplies a signal of a typical frequency of 50 kHz, superimposed on a DC voltage, and applied on the sample. A frequency doubler generates a signal of typically 100 kHz. The second harmonic signal from the tunneling current is then detected using a lock-in amplifier. (Reproduced with permission from [145]. Copyright 1968 American Physical Society.)

Cu(100) surface, where a successful tip treatment would result in a flat tunneling spectrum [368]. The best substrate materials for STM-IETS are copper, silver, and gold, which are nearly free-electron metals. To make a tip with free-electron metal characteristics, the following procedures are often followed: 1. Move the tip to a clean area of the sample surface. 2. Stop feedback. 3. Drive the tip towards the sample to make a mechanical contact. 4. Keep the tip in contact with the sample for a fraction of a second. 5. Withdraw the tip away from the sample. 6. Turn on feedback. 7. Test the tip on a clean area of the sample.

15.4 Inelastic scanning tunneling spectroscopy

15.4.3

371

Effect of finite modulation voltage

The application of a modulation signal causes the broadening of the observed peaks. If the original I∼V relation is I(V), after adding an ac modulation voltage u = U cos ωt,

(15.6)

I (V + U cos ωt).

(15.7)

it becomes

The second harmonic of the current from the lock-in amplifier is then

I2ω

π/ω  I (V + U cos ωt) cos 2ωt dt.

2ω = π

(15.8)

t=0

Because cosine is an even function, the integral only extends to one half of the period. The factor before the integral is a normalization constant, as the average value of the square of cosine function is 0.5. Using Eq. 15.6, two partial integrations give

I2ω

1 = π

π/ω  I (V + U cos ωt) d sin 2ωt t=0

8 = 3πU 4

U −U

(15.9) 3 d2 I (V + u)  2 U − u2 2 du. 2 dV

Fig. 15.7. Broadening of the spectral peak owing to a finite modulation amplitude. A plot of Eq. 15.10. The observed width of the peaks is determined by the convolution of the intrinsic width, the thermal width, and the instrumental width.

372

Scanning Tunneling Spectroscopy

It can be interpreted as the convolution of d2 I/dV 2 with an instrument function 3 8  2 φ (u) = U − u2 2 for |u| < U ; 4 3πU (15.10) φ (u) = 0 for |u| ≥ U. The observed line width W is the convolution of the intrinsic width WI , the temperature broadening WB , and the modulation broadening WM , 2 . W 2 = WI2 + WB2 + WM

15.4.4

(15.11)

Experimental observations

The possibility of observing single-molecule IETS was analyzed by Persson and Baratoff in 1987 [371]. The predicted cross sections are quite small, indicating difficulties in experimentation. Since its successful demonstration in 1988 [368], STM-IETS has continues to expand [370]. Figure 15.8 shows some experimental results [369]. The molecules chosen for that study are acetylene and its variations with deuterium, C2 H2 , C2 D2 , and C2 HD. The topographic STM images of those molecules look identical. However, the positions of peaks in the d2 I/dV 2 plot are totally different. Three types of d2 I/dV 2 curves are observed. The first type has a peak at 358 mV, which agrees well with the vibrational energy of the C-H stretching mode observed by electron energy loss spectroscopy (EELS), which gives the averaged value over a large number of C2√ H2 molecules. The second type with a peak at 266 mV, which is about 2 times smaller than the first, corresponding to the mass ratio of hydrogen and deuterium, also agrees well with the EELS measurements. The third type has two features at 358 mV and 266 mV, respectively. It is from a C2 HD molecule.

Fig. 15.8. Observed STM-IETS, showing isotope effect. Molecules under study are C2 H2 , C2 D2 , and C2 HD. (a) Topographical images. The three types of molecules look identical. (b) d2 I/dV 2 plots. The peaks agree well with the measurements of electron energy loss spectroscopy, and reflects mass ratio of hydrogen and deuterium. (Reproduced with permission from [369]. Copyright 1999 American Physical Society.)

15.5 High-Tc superconductors

15.5

373

High-Tc superconductors

In 1986, the year Binnig and Rohrer of IBM Zurich Research Laboratory won their Nobel Prize for the invention of STM, Bednorz and M¨ uller of the same laboratory published a paper Possible high Tc superconductivity in the Ba-La-Cu-O system in Zeitschrift f¨ ur Physik B [372]. Within several months, their experimental discovery was verified by many research institutions in the world, and they won another Nobel Prize for IBM in 1987. The search of high critical temperature (high-Tc ) superconductivity in ceramic materials became a perfect storm. A number of ceramic materials were discovered with critical temperature above the boiling point of nitrogen [373, 374, 375]. The inquiry about the mechanism of the high-Tc superconductivity [376] and the search for new high-Tc materials [377] is still ongoing today. As presented in Section 2.1, the tunneling experiments of Giaever [121, 127, 132] provided key evidence for the BCS theory of superconductivity [131]. The success was largely owing to a gift from Nature: a thin insulating film of Al2 O3 which is reproducibly generated by heating metallic aluminum in air. An Al2 O3 film of thickness of 3 nm thus created shows a very high electrical resistance, and often free of pinholes. A superconductor, such as Pb, is evaporated on top of the thin Al2 O3 film to become a good tunneling junction. Aluminum itself becomes superconducting at 1.75 K. A tunneling junction with two superconductors can be formed. Undoubtedly, STM experiments could provide vital clues for the understanding of high-Tc superconductors. In 1988, collaborating with C. C. Tsuei, using a homemade STM capable of cleaving a crystal in ultrahigh vacuum in situ then observe scanning tunneling spectroscopy, I conducted an early experiment and published a report Topography and local conductance images of YBa2 Cu3 O7 crystals fractured in ultra high vacuum [356]. Since 1989, I was in charge of designing a low-temperature cross-sectional STM for studying high-Tc superconductors. One research project was the design of a tube scanner as well as its testing and calibration under cryogenic temperature. The results are summarized in Section 10.4. Another research project was to improve the understanding of the imaging mechanism of STM, as summarized in Part I. According to a recent review by Fischer et al. [378], the early STM experiments on high-Tc superconductors were often irreproducible, partly due to the inhomogeneities of the materials. It took many years to find the best STM experiments with the high-Tc superconductors, and what can be learn from those experiments. The most productive experiments include tunneling spectroscopy with a controllable vacuum gap, and the direct observation of the Abrikosov flux lattice with the mapping of local electronic structures in the vortex cores. STM is the unique instrument for those spectroscopic measurements.

374

15.5.1

Scanning Tunneling Spectroscopy

Measuring the energy gap

Because of the inhomogeneneities of the high-Tc superconductors, the traditional method of using a solid-state tunneling barrier such as Al2 O3 is not practical. STM as an instrument is unique in providing a controllable vacuum tunneling barrier for tunneling spectroscopy, although scanning is not required. Even for low-Tc superconductors, the controllable vacuum gap of STM enabled more reliable spectroscopic measurements. Figure 15.9 shows some representative results. Figure 15.9(a) shows the result for Nb, a low-Tc superconductor by Pan et al. [379]. To obtain the DOS of the superconductor, the counterelectrode should have a flat DOS. Atomic resolution is not necessary, and sometimes even harmful because of the variations in tip DOS. Actually, that STS experiment was performed with a Nb tip on an Au sample surface, well known to have a flat DOS. The circles are experimental data, and the solid curve is a BCS fit. As shown, the curve is fully describable by the BCS theory, and the observed value of the energy gap Δ agrees with an s-wave pairing Δ=

7 k B Tc . 2

(15.12)

As shown in Fig. 15.9(b)—(d), the tunneling spectra observed with high-Tc superconductors exhibit some similarities with those of low-Tc superconductors, such as the existence of an energy gap, but with noticeable differences.

Fig. 15.9. Tunneling spectroscopy of superconductors. (a) The tunneling spectrum of Nb, obtained by an STM using a Nb tip and Au sample [379]. Dots represent experimental data, solid curve is a BCS fit. (b)—(d), tunneling spectra of high-Tc superconductors. For details of the graphs, see Fischer et al. [378].

15.5 High-Tc superconductors

375

First, the observed energy gap is greater than the value expected from an s-wave paring symmetry. It is an indication of a d-wave pairing symmetry [380]. Second, instead of a flat and zero conductance within the energy gap, a V-shaped curve is observed, indicating the existence of nodes in the gap. For details, see Fischer et al. [378]. 15.5.2

The Abrikosov flux lattice

As predicted by Abrikosov in 1957, by applying a magnetic field perpendicular to a flat surface of type-II superconductor, a flux lattice with hexegonal symmetry shows up, see page 368 of Kittel [223]. Before the invention of STM, it could only be observed indirectly, for example, by neutron diffraction [382]. In 1989, using scanning tunneling spectroscopy, Hess et al. [381] directly imaged the Abrikosov flux lattice in real space and revealed the details of the DOS near and inside a fluxoid. The experiment of Hess et al. [381] was performed on NbSe2 , because of its excellent surface quality. NbSe2 is a layered material which can be cleaved to expose an atomically flat surface. At 33 K, it undergoes a chargedensity wave (CDW) transition. Individual atoms and the CDW patterns can be easily observed with STM, which provides an accurate calibration of length scale. For similar CDW and atomic images see Plate 5 and Fig. 1.16.

Fig. 15.10. Abrikosov flux lattice of NbSe2 imaged by STM. By applying a magnetic field of 1 T, and taking an image of the dynamic conductance dI/dV around 1.3 meV, the Abrikosov flux lattice is imaged. The gray scale ranges from 1 × 10−8 mho to 1.5 × 10−9 mho. After Hess et al. [381].

376

Scanning Tunneling Spectroscopy

NbSe2 is a type-II superconductor with a Tc of 7.2 K. The gap measured by STS is 1.12 meV. At a bias of 1.3 meV, the differential conductance reaches maximum, a convenient point for vortex imaging. The theoretical vortex spacing of an hexagonal flux lattice is # h 48.9 a = √ nm. (15.13) ≈ 3eH H(T) By applying a magnetic field of 1.04 T, the expected spacing is about 48 nm. An STS imaging of the dynamic conductance is performed at 1.8 K. The result is shown in Fig. 15.10. The regularly spaced hexagonal pattern of vortices is clearly shown. The measured vortex spacing is 48 nm, which agrees with the theoretical expectation. By changing the magnetic field intensity, the spacing of the flux lattice changes. By reducing the magnetic field intensity to 0.02 T, the vortex spacing is expanded to 346 nm. Different locations inside each vortex can easily be accessed. Complete I/V vs V curves at various points in and out of the vortex are acquired, see Fig. 15.11. As shown, at the center of the vortex, the differential conductance is not flat. At about 7.5 nm from the center of the vortex, the dynamic conductance is roughly flat. For high-Tc superconductors, imaging Abrikosov flux lattice is difficult. It is nontrivial to prepare surfaces with homogeneous electronic characteristics. Typically, the vortex pattern is more disordered. Nevertheless, with carefully prepared samples, precious information can be acquired, for example, the experiments of Matsuba et al. [383], see Fig. 15.12.

Fig. 15.11. Tunneling spectra at different points of the vortex. dI/dV vs V for NbSe2 at 1.85 K and a field of 0.02 T. (a) Differential conductance at the center of a vortex. (b) Differential conductance at about 7.5 nm from the center. (c) Differential conductance at about 200 nm from the center. After Hess et al. [381].

15.5 High-Tc superconductors

377

Fig. 15.12. Vortex lattice and scanning spectroscopy of Bi2 Sr2 CaCu2 Ox . (a) The Abrikosov vortex lattice, showing a nearly square symmetry. (b) Tunneling spectra in a vortex core showing localized states at ±9 meV (solid circles) and outside a vortex core (open circles). After Matsuba et al. [383].

Figure 15.12(a) is an image of Abrikosov vortex lattice observed at 4.2 K with a 8 T magnetic field on Bi2 Sr2 CaCu2 Ox , showing a nearly square symmetry almost aligned with the crystallographic ab direction. It is an indication of a dx2 −y2 pairing symmetry aligned with the crystallographic orientation. The lattice constant of the fluxoids is  h 45.5 a = ≈ nm. (15.14) 2eH H(T) With a magnetic field of 8 T, the lattice constant is about 16 nm. Figure 15.12(b) shows dynamic conductance curves obtained at different points in the vortex core. Near the center of the vortex, localized states at ±9 meV are observed. Outside the vortex, the typical dynamic conductance curve of a high-Tc superconductor is observed [383]. In a summary, for the measurement of superconducting gaps on a macroscopic scale, the scanning capability of the STM is not used. However, the ability of maintaining a controllable vacuum gap at a chosen location is a valuable function of STM. Even for traditional superconductors, using STM, the tunneling experiment is more reliable. The observation of Abrikosov flux lattice is a unique ability of STM. Because the typical scale is much greater than a nanometer, as shown by Eqs. 15.13 and 15.14, atomic resolution is unwanted. The best tip is a piece of free-electron metal with a local radius of a few nanometers, having a flat DOS with no atomic details, as defined by the Tersoff-Hamann model in Chapter 6.

Chapter 16 Atomic Force Microscopy Motivated by the observation of atomic forces in STM experiments and its visible influence on the images, Binnig, Quate and Gerber [31] created the atomic force microscope (AFM). Its basic idea of scanning and feedback is identical to STM. Instead of using the tunneling current, the force between a sharp tip and the sample surface, detected by a sensitive cantilever, is used as the signal to drive the feedback system. Their original ideas are shown in Fig. 16.1. In the first paper on AFM [31], four methods were proposed for the detection of force. Three of them require the sample assembly or the cantilever (carrying the tip) to oscillate at or near its resonance frequency. The influence of force is then detected by the change of amplitude, phase, or frequency. The fourth method uses the tunneling current between the back of the cantilever D and a tip C to measure the minuscule static deformation of the cantilever. In their first set of experiments, the static force detection method was shown to be superior to the dynamic methods, which they attribute to the existence of a water layer on the sample surface. Those preliminary experiments, performed in air, have already demonstrated a lateral resolution of 3 nm and a vertical

Fig. 16.1. Schematic of the AFM. A figure from the first publication about the atomic force microscope. The xyz piezo drives and the feedback system are identical to those of an STM. The force between the tip, B, and the sample, A, is detected by two types of method. The first, the dynamic mode, utilizes a modulating piezo E to oscillate the cantilever D at or near its resonance frequency. The interaction force between the tip B and the sample A causes various effects to the vibration of the cantilever, such as amplitude, frequency, and phase changes. Those changes are then detected to become the signal. The second, the static mode, detects the deflection of the cantilever directly through the change of the tunneling gap between C and D. The change of tunneling conductance is then detected as the signal. (Adapted with permission from Binnig, Quate, and Gerber [31]. Copyright 1986 American Physical Society.)

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380

Atomic Force Microscopy

resolution better than 100 pm, with a force of 1 nN. This is several orders of magnitude better than the stylus profilometer, which has a typical lateral resolution of one micron at a force of 10−2 to 10−5 newton. The results clearly showed that, by using the method of STM, forces at atomic scale could be detected and mapped. The original goal of AFM was to detect and measure the force between two individual atoms, then use the interatomic force as a signal to drive the feedback, and to see individual atoms even on insulating surfaces. However, at that time, it was not immediately clear that the AFM is capable of detecting the force between individual atoms as they proposed.

16.1

Static mode and dynamic mode

The AFM, in general, has two modes of operation, the static mode and the dynamic mode. The static-mode AFM is essentially a refined stylus profiler, usually operating in the contact mode. During scanning, the tip is simply dragging across the surface. The repulsive or attractive force causes a deflection of the cantilever. The deflection is then detected to become the signal. In the dynamic mode AFM, the cantilever is allowed to or made to oscillate at or near its resonance frequency. Under the influence of a force between the tip and the sample, a rich body of physical phenomena is generated, which provides information about the force. According to the type of physical phenomenon used as the signal, there are several versions of the dynamic mode AFM. The following are the most important ones: 1. The amplitude-modulation mode, or the tapping mode. The cantilever is actuated by a constant external AC voltage, usually through a bimorph piezo. Typically, the frequency is 50 kHz to 500 kHz. When the cantilever is free, the amplitude is typically greater then 10 nm. Once the tip starts to make contact with the sample surface, even only during a small fraction of a period, the amplitude starts to decrease. The reduction of the amplitude is used as the signal. In the static mode, the sustaining vertical and shear forces can damage the sample and the tip by compressing, tearing, or even removing parts of it. The tapping mode only makes vertical contact sparsely, with virtually no shear force. It is currently the most popular mode of AFM operation. 2. The force-modulation mode. By keeping the tip always in contact with the sample surface, the amplitude of oscillation under constant actuation reflects the local elastic property of the sample. Usually, it is operated with a small oscillation amplitude. 3. The frequency modulation mode. Technically, frequency measurement can be made with extraordinary accuracy, such as a small fraction of 1 Hz on a basic frequency of several hundred kHz. For very weak forces, while the change of amplitude is below the detection limit, the change of resonance

16.2 Cantilevers

381

frequency of a high-quality-factor cantilever can be detected comfortably. Therefore, by using a circuit to allow the cantilever to oscillate at its resonance frequency while keeping the amplitude constant, and then measuring the frequency shift, atomic forces well below a fraction of a nN can be detected. This method is especially useful in noncontact AFM to detect the van der Waals force and the covalent-bonding force. Although initially operated under ultrahigh vacuum, it has been shown to work under ambient conditions and even in liquids. 4. The higher-harmonics mode. Because the force gradient is always nonuniform, even if the cantilever is excited by a pure sinusoidal voltage, the oscillation contains higher harmonics of the resonance frequency of the cantilever. The signal, properly deciphered, is a direct measurement of the force at the closest approach point. 5. The dissipation-signal mode. While operating frequency modulation mode in ultrahigh vacuum, the power needed to keep the cantilever oscillating is related to the dissipative components of the force. While plenty of experimental data has been collected, a detailed understanding of the source of the signal is still not achieved.

16.2

Cantilevers

The AFM has a number of elements common to STM: the piezoelectric scanner for actuating the raster scan and z positioning, the feedback electronics, vibration isolation system, coarse positioning mechanism, and the computer control system. The major difference is that the tunneling tip is replaced by a mechanical tip, and the detection of the minute tunneling current is replaced by the detection of the minute deflection of the cantilever. 16.2.1

Basic requirements

In order to achieve sufficient sensitivity for atomic resolution, the cantilever has to satisfy several requirements [384]. First, the cantilever must be flexible yet resilient, with a force constant from 10−2 to 102 N/m. Therefore, a change of force of a small fraction of a nN can be detected. Second, the resonance frequency of the cantilever must be high enough to follow the contour of the surface. In a typical application, the frequency of the corrugation signal during a scan is up to a few kHz. Therefore, the natural frequency of the cantilever must be greater than 10 kHz. Third, as a direct consequence, the cantilever must be very small. Actually, from Eq. E.32, the resonance frequency f of a cantilever is related to its length L, cross-sectional area S, and density ρ as

382

Atomic Force Microscopy

f2 =

0.314EI . L4 ρS

(16.1)

On the other hand, from Eq. E.23, the force constant K of a beam is K=

3EI . L3

(16.2)

Eliminating Young’s modulus E and moment of inertia I from Eqs 16.1 and 16.2, we find K = 9.57ρLSf 2 = 9.57M f 2 ,

(16.3)

where M is the mass of the cantilever. Equation 16.3 is very similar to the equation of a spring–mass oscillator, except that the factor (2π)2 is replaced by 9.57. If we need K < 1 N/m, and f > 104 Hz, then the mass of the cantilever must be much smaller than 1 μg. In other words, the dimensions of the cantilever must be in the micrometer range. Fourth, in the vertical and horizontal directions, the stiffness should be very different. When the AFM is operated in the repulsive-force mode, frictional forces can cause appreciable image artifacts. Choosing an appropriate geometry for the shape of the lever can yield substantial lateral stiffness, thus minimizing the hard-to-handle artifacts. Fifth, when optical beam deflection is used to measure cantilever deflection, the sensitivity is inversely proportional to the length of the cantilever. If the length of the cantilever is of the order of 100 μm, the length of the ‘optical lever’ can be as short as 1 cm for atomic resolution with an inexpensive position-sensitive detector. Finally, a sharp protruding tip must be formed at the end of the cantilever to provide a well-defined interaction with the sample surface, presumably with a single atom at the apex. The slope of the tip should be as steep as possible, and as smooth as possible. In other words, a high aspect ratio is preferred. The following section describes a typical method of fabricating the cantilevers and their typical characteristics. 16.2.2

Fabrication

In the beginning of AFM operation, the cantilevers were cut from a metal foil, and the tips were made from crushed diamond particles, picked up by a piece of eyebrow hair, and painstakingly glued manually onto the cantilevers. This situation has changed completely since the methods for mass production of cantilevers with integrated tips have progressed. A review of various methods for making cantilevers using standard microfabrication techniques was published by Albrecht et al. [384], and an improved method

16.2 Cantilevers

383

is described by Akamine et al. [385]. Those AFM cantilevers with integrated tips are now available commercially. In this section, the method of making cantilevers with integrated tips of Si3 N4 is briefly described [384]. The process starts with a Si(100) wafer with a thermally grown SiO2 layer, as shown in Fig. 16.2. The steps are as follows: 1. Using a photolithographic method, etch a square opening on the SiO2 film. 2. Use KOH solution to etch the part of the silicon wafer exposed through the square opening. The etch self-terminates at the Si(111) planes, and a pyramidal pit is formed. 3. Remove the SiO2 protection layer. 4. Deposit Si3 N4 on the wafer, to form the shape of the cantilever, using a lithographic method. 5. Attach a piece of glass as the carrying substrate using anodic bonding. The area with the cantilever is protected by a Cr layer on the glass. 6. Remove the unwanted part of glass and all the remaining Si.

Fig. 16.2. Fabrication of silicon nitride microcantilevers with integrated tips. (a) A pyramidal pit is etched in the surface of a Si(100) wafer using anisotropic etching. (b) A Si3 N4 film is deposited over the surface and conforms to the shape of the pyramidal pit, and patterned into the shape of a cantilever. (c) A glass plate is prepared with a saw cut and a Cr bond-inhibiting region. The glass is then anodically bonded to the annealed nitride surface. (d) A second saw cut releases the bond-inhibited part of the glass plate, exposing the cantilever. (e) All Si is etched away, leaving the Si3 N4 microcantilever attached to the edge of a glass block. The back side of the cantilever is coated with metal (Au) for the deflection detector. (Reproduced with permission from [384]. Copyright 1990, AVS The Science and Technology Society.)

384

Atomic Force Microscopy

Fig. 16.3. Photographs of microcantilevers. (a) A glass substrate with four cantilevers. (b) One of the cantilevers. (c) Close-up view of the tip. (Reproduced with permission from [384]. Copyright 1990, AVS The Science and Technology Society.)

7. The wafer is diced into pieces. Each piece is a small glass block with several cantilevers attached to its edges. Figure 16.3 is a micrograph of a tip made with this process. The tips made through a similar process have been used commercially for the following reasons. First, the cantilever material, silicon nitride, is robust and inert. Second, the tip sidewalls are extremely smooth and have a slope of 55◦ , which facilitates low friction sliding over rough surfaces. In addition, since the shape of the tip is well defined, the effects of tip morphology on the image can be understood and taken into account. The typical force constant of such mass-produced cantilevers is 0.0006 to 2 N/m, and the typical resonance frequency is 3 kHz to 120 kHz [384].

16.3

Static force detection

In the earlier years of AFM, the contact mode was the norm, and the static force detection method was sufficient. Later, the dynamic modes dominate. However, the static detection methods are still used. First, for force-image mode, static force detection is required. Second, for deflection detection, or measuring the vibrational amplitude, the same methods are often used. 16.3.1

Optical beam deflection

The simplest deflection detection method is the optical beam deflection method [386], as shown in Fig. 16.4. In the following, we analyze the detection limit of the optical beam deflection method. First, the relation between the variation of tip height Δz and the deflection angle θ at the end of the cantilever with length l is, according to Eq. E.24, θ=

3 Δz . 2 l

(16.4)

16.3 Static force detection

385

Fig. 16.4. Detection of cantilever deflection by optical beam deflection. A light beam, typically from a solid-state laser, is reflected by the top surface of the cantilever. The cantilever is vacuum deposited with gold, which reflects red light almost perfectly. The deflection of the mechanical cantilever deflects the optical beam, thus changing the proportion of light falling on the two halves of the split photodiode. The difference of the signals from the two halves of the photodiode is detected. (Adapted with permission from [386]. Copyright 1988 American Institute of Physics.)

The dimension of the mirror w should be a fraction of its length l. It imposes a diffraction limit on the spot size D of the laser beam at the detector, which is at a distance L from the mirror: D≈

λL , w

(16.5)

where λ is the wavelength of the light. The differential signal detected by the photo detector is S=

3 2w P Δz, 2 l λ

(16.6)

where P is the total power incident on the photodiode. At low temperature, the shot noise is the major source of uncertainty. According to Eq. ??, over a bandwidth B, it is Ishot =

 2eI¯ B,

(16.7)

where I¯ is the average signal power, and e is the elementary charge. The minimum detectable cantilever displacement is # 2lλ S Δz = 3w N

Be . 2πηP

(16.8)

386

Atomic Force Microscopy

Here S/N is the signal-to-noise ratio, η is the spectral responsivity. For a measurement bandwidth 1 kHz, using a red light of λ=0.67 μm with power P = 1 mW, with η = 0.4, and w/l ≈ 0.3, at a signal-to-noise ratio of S/N =10, the detection limit is Δz ≈ 3 pm. Notice that the detection limit is inversely proportional to the power of the laser beam. To achieve better resolution, a more powerful laser beam is required. If the sample is sensitive to photons or the experiments are running at cryogenic temperatures, the possible harmful effect of the high power has to be considered. 16.3.2

Optical interferometry

Another method, optical interferometry, is also widely used to detect minute displacements of the cantilever due to the action of force [388, 387]. A typical experimental setup is shown in Fig. 16.5. A coherent light beam, typically from a HeNe laser, is redirected by a polarizing splitter, then focused on the back side of the cantilever through a microscope objective,

Fig. 16.5. Detection of cantilever deflection using an optical interferometer. A laser beam is redirected by a polarizing splitter. The beam reflected from the optical flat and the beam reflected from the back of the cantilever are directed to a photodiode. An interference fringe is formed. The sensitivity of the displacement of the cantilever can be better than 1 pm. This method can be used to detect static as well as dynamic displacements of the cantilever. Using a signal generator to vibrate the cantilever and a lock-in amplifier to the signal from the optical interferometer, the change of vibrational amplitude due to the force can be detected. (Adapted with permission from [387]. Copyright 1988, AVS The Science and Technology Institute.)

16.4 Tapping-mode AFM

387

then directed through the same lens upon the beam splitter. The reference beam is reflected by an optical flat, and then projected together with the light reflected from the cantilever onto the photodiode. An interference fringe is formed. The optical flat is connected to the microscope body via piezoelectric tubes to allow for adjustment of the phase between the two interfering beams. To improve optical isolation and to prevent instability, a λ/4 plate is installed in the light path. An adjustable aperture selects an area of the fringe to maximize the sensibility. In the optimal situation, the apparatus can detect a displacement smaller than 1 pm. The optical interferometer can be used in static mode as well as dynamic mode of AFM, see Fig. 16.5 [388].

16.4

Tapping-mode AFM

In the early years of AFM, the contact mode was the norm. The probe tip was simply dragged across the surface. The signal is the repulsive force between the tip and the sample. While this technique was successfully applied to many cases, the dragging motion, combined with the adhesive forces between the tip and the surface, can cause substantial damage to both sample and tip, and can create artifacts in the images. The problem is more severe when the AFM is operating in air, see Fig. 16.6(a). The invention of the tapping mode or intermittent contact mode AFM resolved many of the problems. The principle is schematically shown in Fig. 16.6(b). The cantilever is actuated by an external piezoelectric device, and vibrational amplitude is detected by a measuring mechanism, typically those in Sections 16.3.1 and 16.3.2. Once the tip is starting to touch the surface, the vibrational amplitude is reduced. The change of amplitude is

Fig. 16.6. Principle of tapping-mode AFM. (a) In contact-mode AFM, the dragging motion of the tip and the surface can cause substantial damage to both sample and tip, and can create artifacts in the images. (b) In tapping mode, the tip only touches the sample surface intermittently. It is also called intermittent contact mode AFM. The change of vibrational amplitude is taken as the signal. The damage of the tip and the sample is substantially reduced. (Adapted with permission from C. Prater et al. [389] by courtesy of Veeco Instruments.)

388

Atomic Force Microscopy

Fig. 16.7. Instrumentation of tapping-mode AFM. The cantilever is vibrated by an oscillator through a piezo actuator. The vibrational amplitude is detected by the laser-beam deflection method. When the tip starts to touch the sample surface, the vibrational amplitude is reduced. The change of vibrational amplitude is taken as the signal. (Adapted with permission from C. Prater et al. [389] by courtesy of Veeco Instruments.)

taken as the signal. A block diagram of tapping-mode AFM is shown in Fig. 16.7. The cantilever is vibrated by a high-resolution oscillator through a piezo actuator. Typically, the frequency ranges from some ten kilohertz to hundreds of kilohertz. The amplitude ranges from a fraction of a nanometer to tens of nanometers. The deflection of the cantilever is typically detected by an optical beam deflection mechanism. When there is no force between the tip and the sample, the cantilever oscillates with a constant amplitude. With the presence of a force between the tip and the sample, the amplitude of the cantilever oscillation is reduced. The amount of amplitude reduction is a measure of the magnitude of the force. By setting a reference amplitude, and use the difference signal for feedback, a constant-force topographic image is obtained.1 16.4.1

Acoustic actuation in liquids

Compared to AFM in vacuum or gas, liquid operation has several advantages. First, the capillary force and van der Waals forces can be eliminated. Second, there are many technologically important processes in liquids. Third, most of the biological objects are only active in liquids, especially for the study of “live” processes. The tapping mode AFM can also be operated in liquids. By submerging the cantilever and sample in Fig. 16.7 in a liquid, it could be operational. However, submersion in a liquid significantly changes the oscillatory behav1 The combined word TappingMode is a trademark of Veeco Instruments Inc., 112 Robin Hill Road, Santa Barbara, CA 93117.

16.4 Tapping-mode AFM

389

Fig. 16.8. Excitation spectrum of the cantilever in liquid. (a) the excitation spectrum of the liquid cell. (b) the thermal excitation spectrum of the cantilever. (c) The observed excitation spectrum is the product of the two. (Reproduced with permission from [390]. Copyright 1996 American Institute of Physics.)

ior of the cantilevers [390]. The resonance frequencies in water are significantly lower than in air. The quality factor in water is of order 1, compared with 10-100 in air. The reduced resonance frequencies can be explained by the effect of fluid loading that increases the effective mass of the cantilever, while the low quality factors are due to increased hydrodynamic damping. A more dramatic effect of the liquid is the occurrence of a forest of vibrational peaks. Those peaks are much stronger and sharper than the peaks due to the resonance frequencies of the cantilever, and depend on the configuration of the cantilever holder and the structure of the liquid cell. In fact, the piezodrive actuates the cantilever indirectly. Directly, it excites the acoustic modes of the liquid cell. The cantilever is then excited through the acoustic vibrations of the liquid cell [390]. Despite the not-so-clean situation, the acoustic actuation method works for most cases. Regardless of the process of excitation, when there is a force between the tip and the sample, the oscillation amplitude of the cantilever is reduced. 16.4.2

Magnetic actuation in liquids

An alternative method of actuating the cantilever is through an AC magnetic field. A tiny permanent magnet is glued on the cantilever, or the cantilever is coated with a ferromagnetic film. An electromagnet, with an AC current passing through it, actuates the cantilever, see Fig. 16.9. In this setup, an actuating piezo is also included for comparison of the two methods. The basic results are shown in Fig. 16.10. (a) shows the response spectrum of acoustic actuation. (b) shows the response spectrum of magnetic actuation. While the cantilever amplitude spectrum from the acoustic actuation exhibits a forest of peaks, mostly unrelated to the resonance frequency of the cantilever, the response from the magnetic actuation follows the simple harmonic oscillator model, with resonance frequency 29.8 kHz and Q=6.2.

390

Atomic Force Microscopy

Fig. 16.9. Tapping-mode AFM in liquid-phase using magnetic actuation. (a) The front view, showing the optical-beam system for deflection detection, the placement of the electromagnet and the cantilever with a permanent magnet. The electromagnet coil is cast in epoxy to reduce the “microphone effect”, that is, the acoustic excitation of the fluid cell. An actuating piezo is also installed to make a comparison of the two actuation methods. (b) Top view. (Reproduced with permission from [391]. Copyright 1996 American Institute of Physics.)

While theoretically, the magnetic actuation method generates a clean response spectrum, practically, the AFM image from both methods could be almost identical [391]. Conversely, the magnetic actuation method may cause problems in certain cases. First, the AC current in the magnetic coil generates heat. By reducing the current to reduce the heat, the sensitivity suffers. Second, some samples may be sensitive to the reactive magnetic ions from the permanent magnet on the cantilever. Therefore, the choice of actuation methods should be studied on a case-by-case basis [391]. Fig. 16.10. Cantilever response spectra from acoustic and magnetic actuation methods. (a) The cantilever response spectrum from an acoustic actuation system. A tree of resonance peaks are observed. (b) The cantilever response spectrum from a magnetic actuation system. The dots (and the thick curve formed by coalescing dots) represent the data. The solid curve is based on a simple harmonic oscillator model, with resonance frequency 29.8 kHz and Q=6.2. The experimental data can be well represented by the simple harmonic oscillator model. (Reproduced with permission from [391]. Copyright 1996 American Institute of Physics.)

16.5 Noncontact AFM

16.5

391

Noncontact AFM

The tapping mode reduces the possible damage of the tip and the sample by only allowing the tip to contact the sample a small fraction of time and by eliminating the lateral force. However, during the short time of contact, the impact could still be high. Furthermore, the tapping mode AFM could only achieve a resolution of a few nm, not true atomic resolution. True atomic resolution with AFM was achieved using the frequency modulation mode, see Section 1.4.1. A schematic is shown in Fig. 1.21. The frequency modulation mode of AFM was first proposed and demonstrated in 1991 [392] for magnetic force microscopy. In frequency modulation mode, the change of resonance frequency of the cantilever due to the influence of the force is detected and used as the signal. The cantilever is allowed to oscillate at its resonance frequency while keeping the amplitude of the cantilever oscillation constant. The phase information is used to keep the cantilever on track with its resonance frequency quickly and accurately. The sensitivity and resolution could be much higher than the amplitudemodulation mode. Typically, it is operating in UHV, sometimes also at cryogenic temperature. It detects, in most cases, the weak attractive force to achieve true atomic resolution. Therefore, it is called noncontact AFM (NC-AFM). In addition to frequency shift, the dissipation signal and the higher-harmonics signal can also be utilized. Since the first demonstrations of true atomic resolution by frequency modulation AFM [97, 393], NC-AFM has grown into a stable and rapidly expanding field. Since 1998, the International Conference of NC-AFM has been held each year. A multi-author book was published in 2002 [394]. An account of the history and the basics can be found in a review article of Giessibl [96]. Several authors have analyzed the operational principle of frequency modulation AFM. Using the Hamilton–Jacobi equation, Giessibl [202, 203, 395] derived general formulas relating frequency shift with force. Using the least-action principle and Chebyshev polynomials, D¨ urig [396, 397] explored more details. Here we present an elementary analysis using Newton’s equation and Fourier series. The results are equivalent to those derived using more sophisticated mathematics. For practical cases where the force can be written as a sum of exponential functions of the tip–sample distance, such as the Morse function, simple analytic formulas are derived. 16.5.1

Case of small amplitude

Consider a mechanical oscillator with spring constant k and mass m. The force to be detected, F (z), is expanded in power series up to the quadratic term. The equation of motion is

392

Atomic Force Microscopy

Fig. 16.11. Effect of force on the resonance frequency of a cantilever. The tip position with regard to sample surface as a function of time, z(t), is expanded into a Fourier series. The zero-order coefficient, a0 , is the equilibrium position of the tip. Up to the first order, the distance of closest approach to the sample is a0 − a1 = zmin .

d2 z F  (0) 2  z . + kz = F (0) + F (0) z + dt2 2 Without external force, the solution is m

z = a0 + a1 cos(ω0 t + φ),

(16.9)

(16.10)

where a0 is the equilibrium position, a1 is the amplitude, and ω0 is the resonance circular frequency  k , (16.11) ω0 = 2πf0 = m and φ is the phase. In the absence of dissipative force, one can always choose φ = 0. With external force, a natural Ansatz is z = a0 + a1 cos ωt + a2 cos 2ωt.

(16.12)

Insert Eq. 16.12 into Eq. 16.9, using the trigonomical identity 1 (cos 2ωt + 1), 2 and noticing that the force is small, the constant terms are cos2 ωt =

(16.13)

a2 1 F (0) + 1 F  (0). (16.14) k 4k Therefore, the force gradient has no effect on the equilibrium position, but the second-order term generates a shift of the equilibrium position of the cantilever. The terms with cos(ωt) generate  k F  (0) ∼ ω0  ω= − F (0). (16.15) = ω0 − m m 2k a0 =

16.5 Noncontact AFM

393

To directly relate this to experimental data, Eq. 16.15 can be written in terms of frequency shift, Δf ≡

ω − ω0 f0  =− F (0). 2π 2k

(16.16)

Since ω ≈ ω0 , the terms with cos(2ωt) generate a2 a2 ∼ = 1 F  (0). 12k

(16.17)

To summarize, the constant term of force generates a shift of the equilibrium position, which is in principle not observable. The force gradient generates a frequency shift. The magnitude of the observable frequency shift is proportional to the force constant F  , the resonance frequency, and inversely proportional to the stiffness k of the cantilever. The quadratic term generates a non-observable shift of the equilibrium position, and a secondharmonic term which is observable. The magnitude of the amplitude of the second-harmonic term is also inversely proportional to the spring constant. Those features will be preserved in the case of finite amplitude, see Section 16.5.2. In AFM, the oscillator is a cantilever. Assume that the cantilever is rectangular and uniform, clamped at one end, and the mass of the tip is negligible; with Young’s modulus E, length L, width b, thickness h, and density ρ, the spring constant is (Appendix E) Ebh3 , 4L3 and the lowest resonance frequency is k=

(16.18) #

h ω0 f0 = = 0.162 2 2π L

E . ρ

(16.19)

The equation of motion of the cantilever at the lowest resonance frequency can be written in the form of Eq. 16.9 with an equivalent mass m related to the actual mass ρLbh by m = 0.243ρLbh. 16.5.2

(16.20)

Case of finite amplitude

The atomic forces have a rather short range, with a typical decay length of 0.1 nm. The amplitude of cantilever oscillation is often comparable to or greater than 0.1 nm. Therefore, often, the small-amplitude approximation does not provide an adequate description of the sensing process. In general, the equation of motion is

394

Atomic Force Microscopy

d2 z + kz = F (z). dt2 Substituting Eq. 16.12 into Eq. 16.21, we obtain m

ma0 + (k − mω 2 ) a1 cos ωt + (k − 4mω 2 ) a2 cos 2ωt = F (a0 + a1 cos ωt).

(16.21)

(16.22)

The second-harmonic term is small. Inside the expression of force, it can be neglected. Introducing a variable proportional to time ϕ = ωt,

(16.23)

multiplying the left-hand side of Eq. 16.22 by cos ϕ and integrate over a period, 2π, it becomes  πa1 m  2  πa1  k − mω 2 = ω0 − ω 2 ∼ (16.24) = 4π 2 a1 mΔf. ω ω Here, Eqs 16.11 and 16.16 are used. Multiplying the right-hand side of Eq. 16.22 with cos ϕ and integrating over 2π, the frequency shift is  2π f0 Δf = cos ϕ F (a0 + a1 cos ϕ) dϕ. (16.25) 2πa1 k 0 Integrating by parts, gives Δf =

f0 2πk



2π

sin2 ϕ F  (a0 + a1 cos ϕ) dϕ.

(16.26)

0

There is a simple explanation of the result: the frequency shift is proportional to a time-average of the force constant with a weighting function sin2 ϕ, which has an average value of 1/2. If F  is a constant, Eq. 16.16 is recovered. Notice that a0 − a1 is the position of the tip at the closest approach to the sample, denoted as zmin , see Fig. 16.11. Taking it as the origin, and changing the variable for integration to ζ = a1 (1 + cos ϕ),

(16.27)

Eq. 16.25 can be written as Δf =

f0 πa21 k



2a1 0

(a1 − ζ)dζ . F (zmin + ζ)  ζ (2a1 − ζ)

(16.28)

If the amplitude of cantilever oscillation is comparable to or larger than the range of force, then the force is significant only for small values of ζ. The integral can be approximated to  ∞ dζ f0 F (zmin + ζ) √ . (16.29) Δf = √ 3/2 ζ 2πa1 k 0

16.5 Noncontact AFM

395

Some quantities in the above expression depend only on the force to be detected, and others depend only on the instrument. Separating the two groups of quantities, gives Δf =

f0 γ 3/2 a1 k

(zmin ),

(16.30)

where the normalized frequency shift [202] depends only on the force curve,  ∞ 1 dζ (16.31) γ (z) ≡ √ F (z + ζ) √ . ζ 2π 0 From Eq. 16.30, we see that the frequency shift depends on three instrumental parameters. First, it is proportional to the resonance frequency of the cantilever, f0 . Therefore, if possible, it is preferable to use a cantilever of higher resonance frequency. Second, it is inversely proportional to the 3/2 power of the oscillation amplitude. Therefore, it is preferable to use a smaller oscillation amplitude. Third, the frequency shift is inversely proportional to the stiffness of the cantilever. Therefore, it is preferable to use a softer cantilever, or a cantilever with smaller spring constant, within the limits of mechanical stability. In other words, if the cantilever is too soft, then the tip could spontaneously jump into contact with the sample, and that must be avoided. However, within the limits of mechanical stability, the softer the cantilever, the more sensitive it is. 16.5.3

Response function for frequency shift

For the Morse function, a closed form of the frequency shift can be obtained:   (16.32) F (z) = 2κU0 e−2κ(z−z0 ) − e−κ(z−z0 ) , where U0 is the dissociation energy, κ is the decay constant of the attractive atomic force, and z0 is the equilibrium internuclear distance. Using the integral form of the modified Bessel function I1 (u),  2π 2π I1 (u), sin2 ϕ e±u cos ϕ dϕ = (16.33) u 0 the general expression of the frequency shift is then  f0  2 R1 (κa1 ) κ2 U0 e−κ(zmin −z0 ) − R1 (2κa1 ) (2κ) U0 e−2κ(zmin −z0 ) . Δf = k (16.34) Here the response function R1 (u) is defined as R1 (u) ≡

2 −u e I1 (u). u

(16.35)

396

Atomic Force Microscopy

Fig. 16.12. Dependence of frequency shift on vibrational amplitude. When the vibrational amplitude is nearly zero, the frequency shift approaches the value in Eq. 16.16. As the amplitude increases, the frequency shift is decreased. The asymptotic behavior is proportional to the –3/2 power of the amplitude.

The quantity κ2 U0 exp{−κ(zmin −z0 )} is the gradient of the attractive force, and the quantity −κ2 U0 exp{−2κ(zmin −z0 )} is the gradient of the repulsive force. For small amplitude, R1 (U ) ≈ 1. The value of the frequency shift approaches Eq. 16.16. As the amplitude becomes larger, the frequency shift is gradually reduced. As κa1 >5, approximately,  2 −3/2 u , (16.36) R1 (u) ≈ π the form of Eq. 16.30 is recovered, see Fig. 16.12. 16.5.4

Second harmonics

Now we turn our attention to the second-harmonic signal. Multiplying both sides of Eq. 16.22 by cos 2ϕ and integrating over a period, and using the condition ω ≈ ω0 , a general expression for the amplitude of the secondharmonic signal is generated, a2 =

1 3πk



2π

cos 2ϕ F (a0 + a1 cos ϕ) dϕ.

(16.37)

0

Integrating by parts twice, gives a2 =

a21 9πk



2π 0

sin4 ϕ F  (a0 + a1 cos ϕ) dϕ.

(16.38)

16.5 Noncontact AFM

397

Again, there is a simple explanation of the result: a2 is proportional to a time-average of F  with a weighting function sin4 ϕ, which has an average value of 3/8. If F  is a constant, Eq. 16.17 is recovered. For a finite amplitude, using Eq. 16.27, the expression of a2 becomes 2 a2 = 3πka21



2a1 0

2ζ 2 − 4a1 ζ + a21  F (zmin + ζ)dζ. 2ζa1 − ζ 2

(16.39)

If the amplitude is much greater than the decay length of force, approximately a2 =



2 √

3πk 2a1

∞ 0

F (zmin + ζ) 2 √ dζ ≡ √ γ(zmin ). 3k a1 ζ

(16.40)

Here the normalized frequency shift γ(zmin ) is defined in Eq. 16.31. From Eqs 16.17 and 16.40 we see that, for very small and very large amplitudes of cantilever oscillation, the second harmonic is small. There must be a maximum in the middle. For the Morse function, Eq. 16.32, using the integral form of the modified Bessel function I2 (u), 

2π 0

sin4 ϕ e±u cos ϕ dϕ =

6π I2 (u), u2

(16.41)

Fig. 16.13. Second harmonics in the dynamic mode AFM. For small amplitudes, the amplitude of the second harmonics is proportional to the square of the vibrational amplitude. For large vibrational amplitudes, the amplitude of the second harmonic decreases. In the intermediate range of vibrational amplitudes, the second-harmonic signal reaches a broad maximum.

398

Atomic Force Microscopy

the general expression of the amplitude of the second-harmonic signal is a2 =

 2κU0  R2 (2κa1 )e−2κ(zmin −z0 ) − R2 (κa1 )e−κ(zmin −z0 ) . 3k

(16.42)

Here the response function R2 (u) is defined as R2 (u) ≡ 2e−u I2 (u).

(16.43)

As seen from the chart, when κa1 is between 3 and 15, the response function is essentially a constant, approximately 1.0. Under such a condition, approximately, the amplitude of the second harmonic is directly related to the force at the closest approach point, 1 F (zmin ). a2 ∼ = 3k 16.5.5

(16.44)

Average tunneling current

The dynamic mode AFM can also be operated as a dynamic mode STM. The measured tunneling current is the time average of the instantaneous tunneling current. If the z-dependence of the tunneling current follows a exponential law, similar to Eq. 1.24, I(x, y, z) = I(x, y) e−2κz ,

(16.45)

Fig. 16.14. Average current in dynamic mode AFM-STM. The ratio of the measured (average) current to the current at the closest approach position. When the vibrational amplitude is nearly zero, the average current equals the maximum current. The average current is reduced along with the vibrational amplitude of the cantilever.

16.5 Noncontact AFM

then the average tunneling current is  2π ¯ y, z) = I(x, y) e−2κ(a0 +a1 cos ϕ) dϕ. I(x, 2π 0

399

(16.46)

Again, by introducing a response function R0 (u) ≡ e−u I0 (u),

(16.47)

the average tunneling current is ¯ y, z) = I(x, y) e−2κ(a0 −a1 ) R0 (2κa1 ). I(x,

(16.48)

The factor before the response function is the tunneling current at the closest approaching point. For small a1 , R0 (2κa1 ) ∼ = 1. For large a1 , the response function is always smaller than 1, and approaches [203] R0 (u) ∼ =√

1 . 2πu

(16.49)

The average current is approximately [203] I(x, y) −2κ(zmin ) I(x, y) −2κ(a0 −a1 ) ¯ y, z) = √ e =√ e , I(x, 4πκa1 4πκa1

(16.50)

where zmin is the distance of closest approach of the tip to the sample. From Fig. 16.14, it is clear that the approximation, Eq. 16.50, is fairly accurate over the entire range except in the immediate vicinity of u = 0. 16.5.6

Implementation

The typical implementation of NC-AFM is shown in Fig. 16.15. There are two feedback loops. The function of the first one, including the AFM cantilever and the controller, Fig. 16.15(a) and (b), is to maintain vibration of the cantilever at its resonance frequency with a given amplitude. The frequency shift, detected and converted to an analog signal by the phaselocked-loop controller, is sent to the SPM feedback circuit, similar to the tunneling current signal. That feedback system (not shown in Fig. 16.15) then drives the z-piezo of the scanner. The frequency modulation AFM has made substantial progress. The optimization of cantilever design is an ongoing research project. The initial demonstration of atomic resolution of the frequency modulation AFM was based on the conventional silicon-based cantilevers, which are massproduced for contact-mode and tapping-mode AFM [97]. However, the spring constant of those cantilevers is small (10 to 50 N/m). To prevent jump-to-contact phenomena, large oscillation amplitudes are applied, typically 10 to 20 nm. According to Eq. 16.30, the frequency shift is inversely

400

Atomic Force Microscopy

Fig. 16.15. A schematic of the frequency-modulation AFM. The cantilever is actuated by an external piezo, and the vibration is detected by an external deflection sensor. With a controller unit, the cantilever is set to oscillate at its resonance frequency with a predetermined amplitude by a positive-feedback circuit. The interaction between the tip and the sample causes a shift of the resonance frequency. The frequency shift is then converted into an analog signal by the phase-locked loop (PLL) detector, which becomes the force signal. (Adapted with permission from Giessibl [96]. Copyright 2003 American Physical Society.)

proportional to the 3/2 power of the vibrational amplitude. Therefore, the sensitivity is low. A solution is to use cantilevers of high spring constant, for example, the qPlus sensor made from a quartz tuning fork [398]. Small vibrational amplitudes, typically 0.25 nm, can be used, which substantially improves sensitivity.

Appendix A Green’s Functions The concept of Green’s functions can be illustrated by the example of the electrostatic problem. The potential U (r) for a given charge distribution ρ(r) is determined by the Poisson equation ∇2 U (r) = −4πρ(r).

(A.1)

It is the inhomogeneous counterpart of the Laplace equation, ∇2 U (r) = 0, which describes the potential in the vacuum. A convenient method to solve Eq. A.1 is to introduce a Green’s function, denoted as G(r, r0 ), which is the potential of a unit point charge located at r0 : ∇2 G(r, r0 ) = −4πδ(r − r0 ). The solution of the Poisson equation, Eq. A.1, is then  U (r) = G(r, r0 )ρ(r0 ) d3 r0 .

(A.2)

(A.3)

By directly solving Eq. A.2 in spherical coordinates, which leads to an ordinary differential equation, we find the Green’s function is simply the Coulomb potential of a unit charge, G(r, r0 ) =

1 . |r − r0 |

Therefore, the general solution of the Poisson equation is  ρ(r0 ) 3 d r0 . U (r) = |r − r0 |

(A.4)

(A.5)

Eq. A.4, the Green’s function of the Poisson equation, has an obvious physical meaning. By labeling the point charge at r0 as the cause, the potential it creates at a point r is its effect, and the influence of the point charge at r0 to the potential at r is described by the Green’s function G(r, r0 ), which is the Coulomb potential of a point charge. The Schr¨ odinger equation in vacuum with a negative energy eigenvalue E is ∇2 ψ(r) − κ2 ψ(r) = 0,

(A.6)

402

Green’s Functions

 where κ = 2me |E|/. It is a homogeneous differential equation that is often called the modified Helmholtz equation. The Green’s function of the modified Helmholtz equation is defined in analogy to Eq. A.2, ∇2 G(r, r0 ) − κ2 G(r, r0 ) = −4π δ(r − r0 ).

(A.7)

Similarly, the Green’s function should depend only on r = |r − r0 |; that is, 1 d2 [rG(r)] − κ2 G(r) = −4π δ(r − r0 ). (A.8) r dr2 The general solution of Eq. A.8, except for the point r = r0 , can be obtained by direct integration. It is A κr B −κr e + e . (A.9) r r For large r, G(r, r0 ) must vanish, which requires that A = 0. For small distances, where κr 0 represents a counterclockwise circular motion of the particle around the z axis. A solution with m < 0 represents a clockwise circular motion of the particle around the z axis. In the presence of an external magnetic field, those states might have different energy levels. In the absence of an external magnetic field, that is, when the system exhibits time reversal symmetry, two states with equal and opposite quantum number m are at the same energy level. Therefore, all those wavefunctions can be written in forms of sin mφ or cos mφ, which represent standing waves with respect to the azimuth φ. The spherical harmonics in real form have explicit nodal lines on the unit sphere. Morse and Feshbach [159] have given a detailed description of those real spherical harmonics, and gave them special names. Here we list those real spherical harmonics in normalized form. In other words, we require  2

|Ylm (θ, φ)| cos θ dθ dφ = 1.

(B.10)

The first of the real spherical harmonics is a constant, which does not have any nodal line: 1 Y00 (θ, φ) = √ . 4π

(B.11)

The ones with l = 0 and m = 0 have nodal lines dividing the sphere into horizontal zones, which are called zonal harmonics. The first two are  Y10 (θ, φ) =

3 cos θ, 4π

(B.12)

Real Spherical Harmonics  Y20 (θ, φ) =

5 (3 cos2 θ − 1). 16π

405

(B.13)

The ones with m = l divide the sphere into sections with vertical nodal lines, which are called sectoral harmonics. Those are:

Fig. B.1. Real spherical harmonics The first one, Y00 , is a constant. The two zonal harmonics, Y10 and Y20 , section the unit sphere into vertical zones. The unshaded area indicates a positive value for the harmonics, and the shaded area indicates a negative value. The four sectoral harmonics are sectioned horizontally. The two tesseral harmonics have both vertical and horizontal nodal lines on the unit sphere. The corresponding “chemists notations,” such as (3z 2 − r2 ), are also marked.

406

Real Spherical Harmonics  e Y11 (θ, φ)

3 sin θ cos φ, 4π

(B.14)

3 sin θ sin φ, 4π

(B.15)

15 sin2 θ cos 2φ, 16π

(B.16)

= 

o Y11 (θ, φ)

= 

e Y22 (θ, φ)

= 

15 sin2 θ sin 2φ. (B.17) 16π The rest of the real spherical harmonics are called tesseral harmonics, which have both vertical and horizontal nodal lines on the unit sphere:  15 e Y21 (θ, φ) = sin 2θ cos φ, (B.18) 16π  15 o Y21 (θ, φ) = sin 2θ sin φ. (B.19) 16π These real spherical harmonics are graphically shown in Figure B.1. o Y22 (θ, φ)

=

Appendix C Spherical Modified Bessel Functions The solution of the Schr¨odinger equation in vacuum with a negative energy eigenvalue, in terms of spherical coordinates, is in the form of spherical modified Bessel functions. Therefore, in treating tunneling problems in three-dimensional space, these functions are of fundamental importance. Although considered as a kind of special function, the spherical modified Bessel functions are actually elementary functions, that is, simple combinations of exponential functions and power functions. Here is a list of important formulas of those functions following the definition and notation of Arfken [140]. The radial part of the Schr¨ odinger equation in spherical coordinates is d dz

 z

2 df (z)

dz

 − [z 2 + n(n + 1)] f (z) = 0,

(C.1)

where z = κr. By making a substitution 1 f (z) = √ g(z), z

(C.2)

the differential equation for g(z) is ! 2 "  g dg 1 2 − r + n+ +r g = 0. r dr2 dr 2 2d

2

(C.3)

This is the modified Bessel equation of order ν = n + 1/2. The solutions of Eq. C.3 are modified Bessel functions of the first kind, which is defined through the Bessel function Jν (Z) as Iν (z) ≡ e−2νπi Jν (iz),

(C.4)

and the modified Bessel function of the second kind, which is defined through (1) the Hankel function Hν (x) as Kν (z) ≡

π 2(ν+1)πi (1) e Hν (iz). 2

(C.5)

The solutions of Eq. C.1 are defined through the modified Bessel functions. Those are spherical modified Bessel functions of the first kind

408

Fig. C.1.

Spherical Modified Bessel Functions

Spherical modified Bessel functions

 in (z) =

π In+1/2 (z), 2z

(C.6)

and of the second kind, 

2 Kn+1/2 (z); (C.7) πz see Fig. C.1. These two functions are linearly independent. The function in (z) is the only solution of Eq. C.1 that is regular at z = 0, and the function kn (z) is the only solution which is regular at z = ∞. These socalled special functions are actually elementary functions, with the following general expression n  d sinh z in (z) = z n , (C.8) zdz z n  d exp(−z) . (C.9) kn (z) = (−1)n z n zdz z kn (z) =

The first three pairs of these functions are i0 (z) =

sinh z z

sinh z cosh z + 2 z z   3 3 1 sinh z − 2 cosh z + i2 (z) = 3 z z z i1 (z) =

(C.10) (C.11) (C.12)

Spherical Modified Bessel Functions

409

1 −z e z   1 1 + 2 e−z k1 (z) = z z   1 3 3 + 2 + 3 e−z . k2 (z) = z z z k0 (z) =

(C.13) (C.14) (C.15)

Actually, by starting with Eqs C.8 and C.9 as the definitions, all the properties of the spherical modified Bessel functions can be obtained, without tracing back to the formal definition, Eqs C.6 and C.7. Following the standard series expansion of Bessel functions, the powerseries expansion of the function in (z) near z = 0 has the following form: in (z) = z

∞ 

n

k=0

1 k!(2n + 2k + 1)!!



z2 2

k .

(C.16)

The first term is proportional to z n : in (z) ≈

zn . (2n + 1)!!

(C.17)

An in (z) of even order only has even powers of z, and an in (z) of odd order only has odd powers of z. These properties are essential in the derivation of the tunneling matrix elements. On the other hand, following the standard asymptotic expansion of Bessel functions, it is easy to prove that the functions kn (z) have the following exact general expression, kn (z) =

n 1 e−z  (n + k)! . z k! (n − k)! (2z)k

(C.18)

k=0

Also, following the standard recursion relations of Bessel functions, the recursion relations for both in and kn are: (2n + 1) fn (z) = z fn−1 (z) − z fn+1 (z),

(C.19)

d n fn (z) = fn+1 (z) + fn (z). dz z Finally, the Wronskian of the pair is W (in (z), kn (z)) ≡ in (z) k  n (z) − i n (z) kn (z) = −

(C.20)

1 . z2

(C.21)

Appendix D Plane Groups and Invariant Functions For most of the surfaces of interest, in addition to the two-dimensional translational symmetry, there are additional symmetry operations that leave the lattice invariant. If the tip has axial symmetry, then the STM images and the AFM images should exhibit the same symmetry as that of the surface. The existence of those symmetry elements may greatly reduce the number of independent parameters required to describe the images. The collection of all symmetry operations that leave a crystalline lattice invariant forms a space group. Each type of crystal lattice has its specific space group. The problem of enumerating and describing all possible space groups, both two-dimensional and three-dimensional, is a pure mathematical problem. It was completely resolved in the mid-nineteenth century. A contemporary tabulation of the properties of all space groups can be found in [400]. For a pedagogical treatment, see [401]. In three-dimensional space, there are 14 Bravais lattices and 230 space groups. By comparison, in two-dimensional space, there are only five Bravais lattices and 17 different groups. The five Bravais lattices are shown in Table D.1. As shown in Table D.1, there is only one centered lattice, oc. It is easy to show that, for monoclinic, orthorhombic, and hexagonal cases, the centered lattice reduces to primitive lattices with halved unit cells.

D.1

A brief summary of plane groups

The symbols for plane groups, the Hermann–Mauguin symbol, are standard in crystallography. The first place indicates the type of lattice, p indicates primitive, and c indicates centered. The second place indicates the axial Table D.1: Two-dimensianal Bravais lattices

412

Plane Groups and Invariant Functions

symmetry, which has only five possible vales, 1-, 2-, 3-, 4-, and 6-fold. For the rest, the letter m indicates a symmetry under a mirror reflection, and the letter g indicates a symmetry with respect to a glide line, that is, one-half of the unit vector translation followed by a mirror reflection. For example, the plane group p4mm means that the surface has fourfold symmetry and mirror reflection symmetries through both x and y axes. Figures D.1 and D.2 show various types of plane groups. The 17 plane groups are not mutually unrelated. Some of them are subgroups of other plane groups, as shown in Fig. D.3. The order of the factor group, that is, the number of different symmetry operations other than translational symmetry, is also shown for each plane group.

Fig. D.1.

The plane groups I. In monoclinic, orthorhombic, and tetragonal families.

D.2 Invariant functions

Fig. D.2.

D.2

413

The plane groups II. Plane groups in the hexagonal family.

Invariant functions

As we have discussed previously, any function with two-dimensional periodicity can be expanded into a two-dimensional Fourier series. If a function has additional symmetry other than translational, then some of the terms in the Fourier expansion vanish, and some nonvanishing Fourier coefficients equal each other. The number of independent parameters is then reduced. In general, the form of a quantity periodic in x and y would be Q(r) = Q0 (z) + Q1 (z)f1 (x, y) + Q2 (z)f2 (x, y) + . . . ,

(D.1)

414

Plane Groups and Invariant Functions

Fig. D.3. Relations among plane groups. In this figure, the plane groups are shown in their degrees of symmetry, as indicated by the order of the factor groups. A plane group with high symmetry always has one or several subgroup(s). The chart shows such relations within the same lattice.

where f1 (x, y), f2 (x, y), are symmetrized sums of sinusoidal functions. At a given distance z, the first term Q0 (z) is a constant. We call the function f1 (x, y) etc. an invariant function of that group, which is normalized to be max [f1 (x, y)] − min [f1 (x, y)] = 1.

(D.2)

The quantity Q1 (z) is the corrugation amplitude of the quantity Q(r) with f1 (x, y) describing the way it varies with x and y. The invariant functions for several important plane groups are derived and listed here. Plane group pm An example is the (110) plane of III–V semiconductors, such as GaAs(110). The only nontrivial symmetry operation is a mirror reflection through a line connecting two Ga (or As) nuclei in the [001] direction, which we labeled as the x axis. The Bravais lattice is orthorhombic primitive (op). In terms of real Fourier components, the possible corrugation functions are f1 (x, y) =

2nπx 2mπy 1 cos cos , 2 a b

(D.3)

f2 (x, y) =

2nπx 2mπy 1 sin cos , 2 a b

(D.4)

D.2 Invariant functions

f3 (x, y) =

2nπx 2mπy 1 cos sin , 2 a b

415

(D.5)

2nπx 2mπy 1 sin sin . (D.6) 2 a b The mirror-image symmetry with respect to the x axis eliminates f3 and f4 . Also, the functions f1 and f2 should have the same z dependence. Thus, we only need one type of invariant function:   2nπx 2mπy 1 fnm (x, y) = cos + φnm cos , (D.7) 2 a b where φnm are phase constants. f4 (x, y) =

Plane group p2gm By replacing Ga and As atoms with the same species, such as Si or Ge, the symmetry becomes higher. In Fig. D.4, the Si(110) plane is shown as an example. The additional gliding symmetry operation means that, by letting y → y + b/2 and x → −x simultaneously, the function should not change. The only Fourier components satisfying this condition are f1 (x, y) =

2nπx 4mπy 1 cos cos , 2 a b

(D.8)

and 2nπx (4m + 2)πy 1 sin cos . (D.9) 2 a b The general form of a corrugation function that contains the first four terms (in order of the absolute value of g) is: f2 (x, y) =

Q(r) = Q0 (z) +

Fig. D.4.

2πx Q2 (z) 4πx Q1 (z) cos + cos 2 a 2 a

2πx 2πy Q3 (z) sin cos . 2 a b

The Si(110) surface. An example of plane group p2gm.

(D.10)

416

Plane Groups and Invariant Functions

The term Q0 is a constant. The next two terms describe the corrugation in the [001] direction. The last term describes the corrugation in the [11¯10] direction, which should be the smallest among all the terms in Eq. D.10. Plane group p2mm A number of surfaces of interest exhibit p2mm symmetry, for example, Si(111)2 × 1, Si(100)2 × 1, etc. The symmetric function is similar to that of plane group p2gm. We leave the derivation of it as an exercise. Plane group p4mm By setting the origin of the coordinate system at the intersection of the two mirror reflection lines, it is easy to see that only Eq. D.3 of the four corrugation functions is invariant under the mirror reflection operation. The four-fold rotational symmetry further requires n = m, and a = b. To the lowest nontrivial corrugation component, the general form of the corrugation function is 2πy 1 2πx cos . (D.11) Q(r) = Q0 (z) + Q1 (z) cos 2 a a This function describes a simple tetragonal surface with atoms located at (na, ma), where n and m are arbitrary integers. Plane group p6mm The general form is listed Eqs 7.24 and 7.25: (D.12) Q(r) = Q0 (z) + Q1 (z) φ(6) (kx), √ where k = 4π/ 3a is the length of a primitive reciprocal lattice vector, and φ(6) (X) ≡

2 1 2  + cos ω n · X 3 9 n=0

(D.13)

√ is the hexagonal cosine function, where ω 0 = (0, 1), ω 1 = (− 12 3, − 12 ), √ and ω 2 = ( 12 3, − 12 ), respectively, see Fig. 7.14.

Appendix E Elementary Elasticity Theory Excellent treatises and textbooks on elasticity are abundant, for example, Landau and Lifshitz [402], Timoshenko and Goodier [403], and Timoshenko and Young [404]. It takes a lot of time to read and find useful information. This appendix contains an elementary treatment of elasticity relating STM and AFM.

E.1

Stress and strain

Imagine that a small bar along the z direction is isolated from a solid body, as shown in Fig. E.1. The force in the x direction per unit area on the x facet is denoted as σx , a normal stress: σx ≡

Fx . Sx

(E.1)

A body under stress deforms. In other words, a strain is generated. With a normal stress σx , the bar elongates in the x direction. The standard notation to describe strain is by introducing displacements, u, v, and w, in the x, y, and z directions, respectively, as shown in Fig. E.1. The dimensionless quantity x ≡

∂u ∂x

(E.2)

Fig. E.1. Normal stress and normal strain (a) The normal force per unit area in the x direction is the x component of the stress tensor. (b) The normal stress causes an elongation in the x direction and a contraction in the y and z directions.

418

Elementary Elasticity Theory

Fig. E.2. Shear stress and shear strain (a) The shear force per unit area is a component of the stress tensor. (b) The shear stress causes a shear strain.

is called the unit elongation, which is a component of the strain tensor. Hooke’s law says that the unit elongation is proportional to the normal stress in the same direction: x ≡

σx , E

(E.3)

where the quantity E is the Young’s modulus. Under the same stress σx , the y and z dimensions of the bar contract: y ≡

σx ∂v = −ν , ∂y E

(E.4)

z ≡

σx ∂w = −ν . ∂z E

(E.5)

This effect was discovered by Poisson, and the dimensionless constant ν is called Poisson’s ratio. For most materials, ν ≈ 0.25. The x component of the force per unit area on a facet in the y direction is denoted as τxy , and is called a component of the shear stress: τxy ≡

Fx . Sy

(E.6)

The condition of equilibrium, that is, the absence of a net torque on a small volume element, requires that τxy = τyx . A shear stress causes a shear strain, defined as (see Fig. E.2 ) γxy ≡

∂u ∂v + . ∂x ∂y

(E.7)

The relation between shear stress and shear strain can be established based on the relation between normal stress and normal strain, in Eqs E.3 and E.4. Actually, by rotating the coordinate system 45◦ , it becomes a problem of normal stress and normal strain. Using geometrical arguments, it can be shown that (see, for example, [403]):

E.2 Small deflection of beams

τxy =

E γxy ≡ Gγxy . 2(1 + ν)

419

(E.8)

The quantity G ≡ E/2(1 + ν) is called the shear modulus.

E.2

Small deflection of beams

The deflection of a beam under a vertical distribution of force, see Fig. E.3, is a very basic problem in the theory of elasticity. Under the condition that the deflection is small and there is no force parallel to the main axis, the deflection u as a function of distance z satisfies a simple differential equation, as we show in this section. To start with, we show two simple relations in statics; see Fig. E.3. First, consider the equilibrium of the net force in the vertical direction. At each cross section of the beam, there is a vertical force V (z), which should compensate the external force F (z): dV (z) ≡ V (z + dz) − V (z) = −F (z) dz.

(E.9)

Second, at each cross section of the beam, the normal stress forms a torque M (z). The equilibrium condition of a small section of the beam dz with respect to rotation requires (see Fig. E.3 ) dM (z) ≡ M (z + dz) − M (z) = V (z)dz.

(E.10)

Combining Eqs E.9 and E.10, we have d2 M (z) = −F (z). dz 2

(E.11)

Under the influence of a torque, the beam deforms. In other words, the slope tan θ changes with z, as shown in Eq. E.4. For small deflections, tan θ ≈ θ. In other words, du(z) = θ. dz

(E.12)

Consider a small section Δz of the beam. The total horizontal force must be zero. For symmetrical cross sections, such as rectangles, circular bars, and tubes, symmetry conditions require that the neutral line of force must be in the median plane, denoted as x = 0, see Fig. E.4. The distribution of normal strain is then z ≡

xdθ d2 u Δw = z = x 2. Δz d dz

(E.13)

420

Elementary Elasticity Theory

Fig. E.3. Bending of a beam Under the action of a vertical force F distributed along the axis of the beam, the beam bends. The deflection u as a function of position z is determined by the force F (z). (a) The equilibrium of a small section dz of the beam respect to the force in the vertical direction. (b) The equilibrium of a small section dz of the beam with respect to the torque.

Using Eq. E.3 and neglecting the contraction of the width A(x), the total torque acting on a cross section is d2 u M = 2E dz

 A(x)x2 dx ≡

d2 u EI. dz 2

(E.14)

The quantity I is the moment of inertia of the cross section. For simple shapes, the integral can be evaluated easily, which is left as an exercise. For a rectangular bar of width b and height h, I=

1 3 bh . 12

(E.15)

I=

π 4 D . 64

(E.16)

For a rod of diameter D, it is

For a tube with outer radius D and inner radius d, I=

π (D4 − d4 ). 64

(E.17)

Combining Eqs E.10 and E.14, we obtain a differential equation for the deflection u(z), EI

d4 u(z) = −F (z). dz 4

(E.18)

E.2 Small deflection of beams

421

Fig. E.4. Deformation of a segment of a beam Under the influence of a torque acting on a cross section, a beam bends. For small deformations, the slope is θ = du(z)/dz. The change of slope with distance dθ/dz is connected with a strain distribution in the beam, Δw/Δz. The strain is connected with a distribution of normal stress σz in the beam. The total torque is obtained by integration over the cross section of the beam.

We give an example of a concentrated force W acting at a point on the beam, z = L, which is clamped at the origin, z = 0. Thus, F (z) = W δ(z − L). Integrating Eq. E.18 once, we find  d3 u(z) W : 0 < z < L, EI = (E.19) 0 : z > L or z < 0. dz 3 Integrating it again and using the condition that, at z = L, the torque is zero, d2 u(z)/dz 2 = 0, we get d2 u(z) = W (z − L). (E.20) dz 2 Integrating twice and using the condition that, at z = 0, u = 0 and du/dz = 0, we obtain EI

EI

1 du(z) = W z 2 − W Lz, dz 2

(E.21)

1 1 W z 3 − W Lz 2 . 6 2

(E.22)

EIu(z) = The deflection at z = L is

u(L) =

W L3 . 3EI

(E.23)

The angle at z = L is  θ=

du(z) dz

 = z=L

3 u(L) . 2 L

(E.24)

422

Elementary Elasticity Theory

Fig. E.5. Vibration of a beam. One end of the beam is clamped; the displacement and the slope are zero. Another end of the beam is free; the force and the torque are zero. The lowest resonance frequency, which corresponds to a vibration without a node, is determined by Eq. E.32.

E.3

Vibration of beams

If a deflection exists in a beam in the absence of external force, the elastic force will cause an acceleration in the vertical direction. From Eq. E.18, using Newton’s second law, the equation of motion of the beam is EI

d2 u(z, t) d4 u(z, t) = −ρS , dz 4 dt2

(E.25)

where ρ is the density of the material, and S is the cross-sectional area of the beam. For a sinusoidal vibration, u(z, t) = u(z) cos(ωt + α), Eq. E.25 becomes d4 u(z) = ρSω 2 u(z). (E.26) EI dz 4 Denoting κ4 =

ω 2 ρS , EI

(E.27)

the general solution of Eq. E.26 is u(z) = A cos κz + B sin κz + C cosh κz + D sinh κz,

(E.28)

which can be verified by direct substitution. The most important case in STM and AFM is the vibration of a beam with one end clamped and another end free; see Fig. E.5. At the clamped end, z = 0, the boundary conditions are u = 0;

du = 0. dz

(E.29)

At the free end, z = L, the vertical force V and the torque M vanish. The boundary conditions are

E.4 Torsion

423

d2 u d3 u = 0; = 0. (E.30) 2 dz dz 3 By going through simple algebra (we leave it as an exercise for the reader), the vibration frequency is found to be determined by the equation cos κL cosh κL + 1 = 0.

(E.31)

Denoting the solutions of Eq. E.31 as κL = rn , where n = 0, 1, 2, 3, ..., the first three are r0 = 1.875, r1 = 4.694, r2 = 7.854.

(E.32)

The solutions with n > 2 are almost exactly 2n + 1 rn ∼ π. = 2 The resonance frequencies of the cantilever is determined by # E rn2 h . fn = 2π L2 ρ

(E.33)

(E.34)

The lowest resonance frequency is # h f0 = 0.56 2 L

E.4

E . ρ

(E.35)

Torsion

An important issue in the theory of elasticity related to STM and AFM is the shearing stress and the angles of twist of circular bars and tubes subjected to twisting torque T , see Fig. E.6. Consider a small section of a circular bar with length L and radius r. If the twist angle θ per unit length is not too large, every cross section perpendicular to the axis remains unchanged. A small, initially rectangular volume element at radius ρ will have a shear strain θρ . L The shear stress is, according to Eq. E.8, γ=

τ =G

θρ . L

(E.36)

(E.37)

424

Elementary Elasticity Theory

Fig. E.6. Torsion of a circular bar. (a) Under the action of a torque T , a bar of length L is twisted through an angle θ. (b) The twist results in a distribution of shear strain γ in a cross section, which generates a distribution of shear stress τ .

The torque acting on a ring of radius ρ and width dρ is 2πGθ 3 ρ dr. L

dT +

(E.38)

By integrating over the entire area, the total torque is T =

2πGθ L



r

ρ3 dρ = 0

πGθ 4 r . 2L

(E.39)

In terms of diameter d = 2r, the torque is T =

πGθ 4 d . 32L

(E.40)

Another important case is the torsion of a long thin plate, such as a rectangular cantilever, see Fig. E.7. If the thickness h is much smaller than the width b, the torsional rigidity is, C≡

1 T = Gh3 b. θ 3

(E.41)

Similar to the discussion in the previous section, the frequency of torsional vibration of a rectangular cantilever can be determined as follows. The moment of inertia around the z axis is, see Fig. E.7, 

b/2

x2 hdx =

I = 2ρ 0

1 3 ρb h. 12

(E.42)

The velocity of the elastic wave is  v=

# C 2h = I b

G . ρ

(E.43)

E.5 Helical springs

425

Fig. E.7. Torsion and torsional vibration of a rectangular cantilever (a) By applying a torque on both ends of a rectangular cantilever, a twist of angle θ is generated. (b) Torsional vibration of a rectangular cantilever, and the evaluation of its moment of inertia.

If one end of the cantilever is clamped and the other end is free, the wavelength of the standing wave is n + 14 times the length of the cantilever, L. The resonance frequencies are  fn =

E.5

1 n+ 4



v = L

#   1 2h G n+ . 4 Lb ρ

(E.44)

Helical springs

Helical springs are common elements of many mechanical devices. The important parameters of a helical spring are: wire diameter r, coil diameter (from an axis of the wire to another axis) D, number of coils n, and the modulus of elasticity of shear of the material G. To provide an understanding of the helical springs, we give a simple derivation of the total stretch f and the maximum shear stress τmax as a function of the axial load P acting on the spring. The stretch f of a spring is the increase of the pitch h times the number of coils n. To simplify the derivation, we imagine that every coil in the spring is a flat ring at rest; that is, the pitch at rest is zero. Because all the quantities are linear in the pitch, an incremental value of pitch results in an incremental value of axial load and maximum shear stress. When the pitch is increased by h = f /n, the cross section of the wire is twisted by an angle φ = f /2nD. For every one half of a coil, the length of wire is L=

πD , 2

(E.45)

which has a total twist angle, see Fig. E.8 θ = 2φ =

f . nD

(E.46)

426

Elementary Elasticity Theory

Fig. E.8. Stiffness of a helical spring (a) A fictitious position with a zero pitch. (b) When the pitch h of a spring increases, the twisting angle of the wire increases. The torque also increases. Using the formula for torsion, the stiffness of a helical spring is obtained.

On the other hand, to generate a torque T in a wire, the axial load at the center of the spring should be 2T . D Substituting these values into Eq. E.40, we obtain P =

P

Gd4 f. 8nD3

(E.47)

(E.48)

The quantity (Gd4 /8nD3 ) is often called the stiffness of a spring. In designing a spring, the maximum stress must not exceed the allowable stress or working stress of the material. From Eq. E.37, we find the maximum shear stress in the wire is τmax =

E.6

Gd Gθr 8D = f= P. L nπD2 πd3

(E.49)

Contact stress: The Hertz formulas

By pressing a sphere upon a planar surface, a deformation occurs near the original point of contact, see Fig. E.9. This problem was first solved by Hertz in 1881 ([402, 403]). The derivations are complicated. We state the results without proof. Consider a sphere of diameter D of material 1 in contact with a planar surface of material 2. The reaction of the load P is determined by an effective Young’s modulus E ∗ , which is defined as 1 1 − ν12 1 − ν22 = + , E∗ E1 E2

(E.50)

E.6 Contact stress: The Hertz formulas

427

Fig. E.9. Contact stress and contact deformation When a sphere (a) is pressed upon a flat surface with a vertical load P , deformation occurs (b). The yield y, the maximum stress σmax , and the diameter of the contact area d were analyzed by Hertz in 1881.

in terms of the Young’s modulus and Poisson’s ratios of material 1 and material 2. The diameter of contact is  d=

3P D E∗

1/3 .

(E.51)

The total yield, or total relative motion, is y=

1 2



9P 2 DE ∗2

1/3 .

(E.52)

The average normal stress in the contact area is 4 4P = σ ¯= πd2 π



P E ∗2 9D2

1/3 .

(E.53)

The maximum normal stress at the point of contact, according to Hertz, is 1.5 times the average normal stress: σmax =

2 π



3P E ∗2 D2

1/3 .

(E.54)

These formulas are important in the discussions of forces in AFM, tip and sample deformation, as well as some mechanical design problems of STM and AFM.

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Index Aarhus STM design, 339–341 Abrikosov flux lattice, 375 AFM, see Atomic force microscopy Apparent barrier height, 5, 10, 138, 168, 267, 270–272 attractive force, and, 270 definition, 5 repulsive force, and, 272 Apparent radius of an atomic state, 208–211, 234 Atom charge superposition, 175, 195, 228 Atom manipulation, 15, 38–39, 133, 154–157, 257 threshold conductance, 133, 154 threshold resistance, 133, 154 Atomic force microscope, 22–25, 93, 379–381, 399 Atomic force microscopy, 379–400 cantilever, 22–25, 379–400, 423–425 dynamic mode, 380 frequency modulation, 22 frequency modulation mode, 381, 391 static mode, 380, 384 tapping mode, 380, 387 Atomic forces, 93–130, 379, 393 Au(111), 17, 19, 33, 36–254 and cysteine, 32 and MoS2 , 36 atomic resolution, 19, 187 in liquid, 31 interaction with a W tip, 254 self-assembled monolayer, 33 surface states, 179 Automatic approaching, 328, 331 Bardeen approximation, see Bardeen tunneling theory Bardeen integral, 86, 87, 106, 111, 128–131 Bardeen tunneling theory, 45, 48–64, 77, 161, 365 asymmetry in polarity, 55 error estimation, 59 one-dimensional case, 48 three-dimensional case, 57 transfer-Hamiltonian formalism, 61 tunneling spectroscopy, 53 wavefunction correction, 60 Beam, 381–384, 419 moment of inertia, 420 torsion, 423–425 vibration, 421

Beetle, see Besocke design of STM Besocke design of STM, 332–334 Bessel function, 78 modified, 395, 397 spherical modified, 78–83, 88–89, 407–409 Black-ball model of atoms, 113–116 Bogoliubov transformation, 133, 161 Bohr radius, 96, 107 Born’s statistical interpretation, 248, 251 Bravais lattices, 411–412 relation chart, 412 table, 411 Canonical transformation, 133, 159, 161 Cantilever, 22–25, 132, 138–143, 254, 379–399, 423–425 fabrication, 382 requirements, 381 Catalysis research, 34–38 hydrodesulphurization, 36 steam reforming, 34 Chemical bond, see Covalent bond Coarse positioner, 1, 329–341 beetle, 332 Inchworm, 338 kangaroo, 336 lever and screw, 331 louse, 329 walker, 335 Computer interface, 326–328 Contact stress, 426–427 Controlled collision, 18, 351, 353–354, 369 Corrugation, 167–169 Corrugation amplitude, 17–18, 24 enhancement, 192 Fermi-level LDOS, 173 general features, 175, 191 theoretical, 196–215 Corrugation inversion, 188, 204–208 qualitative interpretation, 204 quantitative interpretation, 205 topographic images, in, 188, 204 Covalent bond, 25–26, 100–130 and Bardeen integral, 102–104, 123 and resonance, 123, 126 concept, 100 homonuclear diatomic molecules, 118–123 many-electron atoms, 111 perturbation theory, 126

448

Index Current amplifier, 2, 313, 318 frequency response, 315 microphone effect, 317 Cysteine, 32 DAS model, 17 DB, see decibel DC drop-off method, 344, 345 Decay constant, 3–5, 12, 54–57, 78, 98, 132, 137, 173, 174, 182, 191, 195, 202, 229, 259–261, 321, 395 definition, 4 Decibel, 302, 309 Deflection detection method, 384–387 optical beam deflection, 384 optical interferometry, 386 tunneling, 380 Density functional theory, 217–224, 226, 235 Density of states, 48, 53, 56, 69, 143, 146, 363, 364 in classic tunneling junctions, 46 in STS, 53 Derivative rule, 77–92, 196, 199, 209, 222 derivation, 83–89 lateral effects, 91–92 DFT, see Density functional theory Dissociation energy, 98, 120, 226, 395 Dominant pole, 326 DOS, see Density of states Eddy-current damper, 308, 311–312 Elastic tunneling, 50–51, 63 Elasticity theory, 258, 417–427 Electrochemical tip etching, 344–346 Electrochemistry STM, 30–33 Au(111), 31 self-assembled monolayers, 33 the four-electrode system, 30 tip treatment, 354 voltammogram, 32 Environmental vibration, 2, 303–305, 307 Equivalence principle, 131–137, 140, 231 early experiments, 138 general arguments, 157–163 metal tip and metal sample, 140–143 silicon tip and silicon sample, 143–145 Feedback circuit, 1, 23, 320, 323, 326, 328, 363, 399 Fermi distribution, 53, 67–69 Fermi level, 4, 7, 8, 17, 19, 54, 56, 67, 69, 72, 134, 145, 159, 170, 171, 180, 197, 216, 218, 219, 222, 368 definition, 4

449

FES, see Field emission spectroscopy Feynman invitation, xxix, 38 Field emission spectroscopy, 344 Field evaporation, 348 Field ion microscopy, 254, 344, 346–348, 350 FIM, see Field ion microscopy First-principles computation, 80, 173, 174, 176, 196, 215–220 tip electronic states, 215, 218 First-principles simulation, 19, 196, 220–226 AFM images, 223 STM images, 220 Graphite, 253, 257, 260, 262, 265–267 Green’s function, 60, 63, 81–85, 173, 216, 401–402 definition, 401 modified Helmholtz equation, 402 tip wavefunction, and, 81 Yukawa potential, 402 Hamaker constant, 97, 138, 165 Hard-core repulsion, 93, 98, 132 Helical spring, 308, 425–426 allowable stress, 426 stiffness, 426 Herring-Landau theory of covalent bonds, xxvi, 100–111 Hertz formulas, see Contact stress Hexagonal symmetry, 200–208, 269 High-Tc superconductors, 373–377 Hofer–Fisher theory, 163–166 comments and reply, 166 experimental verification, 165 Hooke’s law, 254, 258 Hydrogen molecular ion, 46, 104–111 resonance interaction, 126 van der Waals force, 106 Imaging wavefunctions, xxv, 235–252 atoms, 238 CO-functionalized tip, 241 general concept, 236 meaning, xxviii, 247–252 molecules, 240 nodal structures, 241 ultrathin insulating barriers, 237 with AFM, 245 Inchworm, 338–340 Independent-orbital approximation, 212, 214, 227 Inelastic tunneling, 46, 51, 64–69, 368–370 experimental facts, 65 frequency condition, 66

450

Index

instrumental broadening, 371 temperature effect, 67 Invariant functions, 413–416 Ionic bond, 98 Johnson noise, 313–316 Junction magnetoresistance, 77 Landauer theory, 6–10, 132, 157, 204, 229 and Bardeen theory, 64 and conductance quantum, 10 derivation, 7 Laplace transform, 322–324, 326 Lateral resolution, 14–15, 167, 233, 349 LDA, see Local-density approximation LDOS, see Local density of states Lead zirconate titanate ceramics, 282–286, 338 Curie point, 284 in Inchworm, 340 piezoelectric constants, 283 properties, 285 LEED, see Low energy electron diffraction Lennard-Jones potential, 98, 99 Local density of states, 17, 172, 173, 175, 176, 183, 187, 196, 203, 205, 207, 216, 221, 222, 230 Local-density approximation, 217 Logarithmic amplifier, 318, 320, 321 Louse, 329–331 Low energy electron diffraction, 15 Mapping wavefunctions with AFM, 147–153 p-wave tip, 149 s-wave tip, 147 and reciprocity, 151 CO-functionalized tip, 149 experimental observations, 245–247 intuitive explanation, 152 Mechanical design, 329–341 Mica, 32 Microphone effect, 317 Modified Helmholtz equation, 402 Morse function, 98, 120, 121, 138, 141, 142, 225, 226, 257, 259, 265, 271, 391, 395, 396 Muffin-tin potential, 113 Nanomechanical effects, 253–272 Operational amplifiers, 313 Optical interferometry, 254, 386–387 Pauli equation, 69–72

Pauli repulsion, 132 Pauli repulsion force, 138 Picoammeter, 313–314 Piezodrive, 1, 277–297 bimorph, 287–289, 380 tripod scanner, 286, 296 tube scanner, 289–297, 331–334, 338, 341 Piezoelectric constant, 279–283, 339 definition, 279 in-situ measurement, 294 lead zirconate titanate ceramics, 282, 283 quartz, 279, 281 temperature dependence, 284 Piezoelectricity, 277–280, 282, 285 Plane groups, 200, 212, 411–416 Platinum–iridium, 1, 200, 216, 344, 346 Poisson’s ratio, 418, 427 Potential barrier, 3, 6, 52, 55, 157 Principle of equivalence, 131–137, 140, 231 early experiments, 138 general arguments, 157–163 metal tip and metal sample, 140–143 silicon tip and silicon sample, 143–145 Principle of reciprocity, 52, 195, 211, 231–234 Pt–Ir, see Platinum–Iridium PZT, see Lead zirconate titanate ceramics Q-factor, see Quality factor Quality factor, 301, 302, 389 Quantum mechanics, 248–252 double-slit experiment, 251 measurement, 248 Reciprocity principle, 52, 195, 211, 231–234 Repulsive atomic force, 6, 96, 98, 99, 126, 137, 138, 145, 199, 255, 257, 267, 270, 272, 380, 382, 387 Resonance, 126 in classical mechanics, 125 in quantum mechanics, 126–129 Resonance frequency, 380, 381, 384, 387, 389, 392, 393, 399, 422 SAM, see Self-assembled monolayer Scanning tunneling microscope, 1–2, 39 pocket-size, 330 schematic diagram, 1, 331 single-tube, 331 Scanning tunneling spectroscopy, 363–377 Abrikosov flux lattice, 375 asymmetry in polarity, 55

Index electronic circuits, 363 inelastic, 65–69, 368–372 superconductors, 372–377 tip treatment, 366–368 Self-assembled molecules, 26–29 bias voltage dependence, 28 role of solvents, 27 Self-assembled monolayer, 33 Shear modulus, 310, 419 Shear piezo, 297, 335–337 and the kangaroo, 336 and the walker, 335 Shot noise, 314, 315, 385 current amplifier, in, 315 deflection detection, in, 385 Shunt current amplifier, 313 Single-atom tip, 10, 225 Space groups, 411 Spherical harmonics, 77, 79, 80, 88, 89, 205, 403–406 real form, 80, 403 Spherical modified Bessel functions, 78, 79, 81, 83, 87–89, 407–409 differential equation, 407 explicit forms, 408 first kind, 407 recursion relations, 409 second kind, 408 Taylor expansion, 409 Spin-polarized STM independent orbital approximation, 227–230 tip treatment, 351–354 Spin-polarized tunneling, 69–76, 227–230 Spin-valve effect, 75 Spinors, 69–72 Spontaneous tip restructuring, 18, 189, 208, 223 Springs, 299, 302, 304, 308–311, 425–426 allowable stress, 426 stiffness, 426 Stability of STM junction, 254–263 experimental facts, 254 theoretical analysis, 257 Stiffness, 258–259, 299, 310, 426 junction, 261 spring, 299, 310, 426 surface, 261, 262 tip, 258, 259, 261, 262 STM, see Scanning tunneling microscopy STS, see Scanning tunneling spectroscopy Suspension springs, 302, 303, 308–311, 331, 334, 338 Takayanagi model, see DAS model

451

Tersoff–Hamann model, 17, 83, 167, 169–185, 192, 198, 218, 220, 222, 229, 230, 233 concept, 169 derivation, 83, 170 extension to finite voltages, 176 heterogeneous surfaces, 184 limitations, 184 surface states, 178 surface structure, 173 Tip functionalization, 21–22, 355–358 STM experiment with CO, 21–22 with CO, 358 with Xe, 355 Tip preparation, 343–346 cutting, 344 electrochemical etching, 344–346 mechanical grounding, 344 Tip treatment, 20, 25, 193, 346–358 annealing, 366 controlled collision, 351, 353, 369 controlled deposition, 353 field evaporation, 348 for electrochemistry STM, 354 for scanning tunneling spectroscopy, 366–368 high-field treatment, 350 spin-polarized STM, 351–354 Tip wavefunction, 1, 60, 61, 77–80, 85, 88, 89, 169, 170, 182, 193, 194 s-wave, 172 as spinors, 227 explicit form, 79 Green’s function, and, 81 Tip–sample distance, 4, 6–12, 64, 191, 204, 208, 221, 226 and corrugation, 18, 191 and tunneling conductance, 10, 64, 204 definition, 6 Tip–sample interaction, 15, 97, 191 and atom manipulation, 15 and STM corrugation, 191 van der Waals, 97 Tip-sample distance, 132, 137, 173, 254, 256, 258, 262, 267, 270–272, 307 and corrugation, 175 and tunneling conductance, 132, 137 Tip-sample interaction, 145, 167, 253–272 and mechanical stability of STM, 258 and Tersoff-Hamann model, 167 in NC-AFM, 145 Tip-sample junction, 138, 145 Tips with axial symmetry, 89 Transfer Hamiltonian, 45, 61–63, 160–161

452

Index

Transient response, 319, 322, 324, 326 Transmission coefficient, 4, 6, 64, 159 Tube scanner, 22, 289–297, 331, 332, 338, 341 deflection formula, 290 in-situ testing and calibration, 292 resonance frequency, 295 Tungsten, 1, 5, 19, 195, 217–220, 230, 231, 262, 270, 345, 347, 351, 366 Tunneling, 1, 3–12, 45–77 concept, 3–12 controllable vacuum gap, through, 10 elastic, 51, 54, 63 inelastic, 46, 51, 64–69, 368–372 spin-polarized, 69–76 Tunneling conductance, 1, 6–12, 39, 51, 68, 131–137, 161, 209, 210 and covalent-bond force, 39, 138–166 and tip–sample distance, 10, 64 differential, 176–183 spin dependence, 69–76 Tunneling matrix element, 45, 50, 59, 62, 72, 77–92, 95, 106, 127, 128, 132, 160–166, 171, 176, 177, 182, 194–196, 198–200, 206, 209, 227, 229, 409 Bardeen integral, 52 energy dependence, 54, 56 table, 85 Van der Waals force, 93–98, 106, 137, 141, 142, 223–225, 381, 388 between tip and sample, 96–98, 137 hydrogen molecular ion, 106 quantum-mechanical origin, 95 van der Waals equation, 93 Vibration isolation, 2, 299–312, 331, 335, 341 environmental vibration, 305 measurement, 304 quality factor, Q, 302 Vibrational frequency, 25 cantilever, 25, 395 diatomic molecules, 119–120 molecules, 368, 372 of molecules, 65 W, see Tungsten Wavefunction, 21–22, 235–252 and electron charge density, 235, 249 atoms, 239 collapse, 248 definition, 235, 237, 247, 249 derivatives, 241 experimental observation, xxv, 21–22, 235–252

molecules, 240 nodal structures, 241 Wigner theorem, 128, 181, 403 Work function, 4, 11, 55, 56, 78, 137, 170, 182, 197, 229, 256, 259, 269, 270 definition, 4 typical values, 5 Young’s modulus, 258, 288, 290, 292, 295, 382, 393, 417, 426 definition, 417 typical values, 259, 261 Yukawa potential, 402