Magnetic Properties of Solids [1 ed.] 9781617285349, 9781607415503

Citation preview

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

MATERIALS SCIENCE AND TECHNOLOGIES SERIES

MAGNETIC PROPERTIES OF SOLIDS

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

MATERIALS SCIENCE AND TECHNOLOGIES SERIES Dielectric Materials: Introduction, Research and Applications Ram Naresh Prasad Choudhary and Sunanda Kumari Patri 2009. ISBN 978-1-60741-039-3 Handbook of Zeolites: Structure, Properties and Applications T. W. Wong 2009. ISBN 978-1-60741-046-1

Building Materials: Properties, Performance and Applications Donald N. Cornejo and Jason L. Haro (Editors) 2009. ISBN: 978-1-60741-082-9 Concrete Materials: Properties, Performance and Applications Jeffrey Thomas Sentowski (Editors) 2009. ISBN: 978-1-60741-250-2 Physical Aging of Glasses: The VFT Approach Jacques Rault 2009. ISBN: 978-1-60741-316-5

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Mesoporous Materials: Properties, Preparation and Applications Lynn T. Burness (Editor) 2009. ISBN: 978-1-60741-051-5 Corrosion Protection: Processes, Management and Technologies Teodors Kalniņš and Vilhems Gulbis (Editors) 2009 ISBN: 978-1-60741-837-5 Strength of Materials Gustavo Mendes and Bruno Lago (Editors) 2009. ISBN: 978-1-60741-500-8 Magnetic Properties of Solids Kenneth B. Tamayo (Editor) 2009. ISBN: 978-1-60741-550-3

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

MATERIALS SCIENCE AND TECHNOLOGIES SERIES

MAGNETIC PROPERTIES OF SOLIDS

KENNETH B. TAMAYO

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Magnetic properties of solids / [edited by] Kenneth B. Tamayo. p. cm. Includes index. ISBN 978-1-61728-534-9 (E-Book) 1. Solids--Magnetic properties. I. Tamayo, Kenneth B. QC176.8.M3M35 530.4'12--dc22

Published by Nova Science Publishers, Inc.

2009 2009014697

New York

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

CONTENTS

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Preface

vii

Chapter 1

Non-equilibrium Magnetism of Single-Domain Particles for Characterization of Magnetic Nanomaterials Mikhail A. Chuev and Jürgen Hesse

Chapter 2

Magnetic and Electronic Structure Modifications Induced by Surface Segregation in La0.65Pb0.35MnO3 Thin Films C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

105

Chapter 3

Site Disorder and Finite Size Effects in Rare-Earth Manganites K.F. Wang, S. Dong and J.M. Liu

145

Chapter 4

Processing and Properties of Thin Manganite Films S. Canulescu, C.N. Borca, M. Döbeli, P. Schaaf, T. Lippert and A. Wokaun

179

Chapter 5

Study of the Magnetic Properties of the Semiconductors and the Nanomaterials by Different Theoretical Methods R. Masrour and M. Hamedoun

203

Chapter 6

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review) R. Varga

251

Chapter 7

Substitution- Induced Structural, Ferroelectric and Magnetic Phase Transitions in Bi1-xGdxFeO3 Multiferroics V.A. Khomchenko and A.L. Kholkin

273

Chapter 8

Pinning Effect on Local Magnetization in Ferrimagnets A2FeMoO6 (A = Ba, Ca) Examined by Transmission Electron Microscopy (TEM) X.Z. Yu, T. Asaka, Y. Tomioka, T. Arima, Y. Tokura and Y. Matsui

291

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

1

vi

Contents

Chapter 9

Obtaining Magnetic and Electric Information on Organic Systems Using Electron Spin Resonance L. Walmsley

307

Chapter 10

Orbital Dilution Effect in Mott Insulating System Sumio Ishihara

315

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Index

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

331

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

PREFACE Magnetism is one of the phenomena by which materials exert attractive or repulsive forces on other materials. Some well-known materials that exhibit easily detectable magnetic properties are nickel, iron, cobalt, and their alloys; however, all materials are influenced to greater or lesser degree by the presence of a magnetic field. Materials may be classified by their response to externally applied magnetic fields as diamagnetic, paramagnetic, or ferromagnetic. Magnetic particles of nanometer size are attracting growing fundamental and technological interest because of their unique magnetic properties, which are dominated by superparamagnetism. This book presents a broad range of topics in this growing field, including their use as magnetic agents, receptor particles for microwaves as well as in shielding layers. A great interest of researchers in modern materials containing small magnetic particles or clusters (several nanometers, so that they are single-domain) is primarily due to the wide area of their application in the nanotechnology of magnetic and magneto-optic informationrecording devices, ferrofluids, chemical catalysis, color imaging devices, biotechnology, etc. For this reason, it is necessary to perform systematic investigations of the structural and magnetic properties of these materials by various methods in order to optimize the technology of their growth and to determine the fundamental characteristics of magnetism in an ensemble of magnetic nanoparticles. The most informative techniques to study the non-equilibrium magnetism of nanoparticles seem to be the conventional magnetization measurements and Mössbauer spectroscopy. The principal difference between them is that they can probe magnetic properties of the same material in different frequency ranges: the magnetization measurements are carried out at lower frequencies (of about 1–1000 Hz) while Mössbauer spectroscopy can reveal the magnetic dynamics of nanoparticles at higher frequencies due to the Mössbauer time window (10–11 – 10-6 s for 57Fe nuclei). The only way to extract the reach information from the experimental data is to define a model of the magnetic dynamics in order to describe the whole set of the experimental data for the sample studied. In Chapter 1, magnetic relaxation effects of nanoparticle ensambles revealed in magnetization measurements and Mössbauer spectra are discussed. The basis of this discussion is a recently introduced general model for magnetic dynamics of ensemble of single-domain particles which is based on a generalization of the well-known StonerWohlfarth model within more accurate description of relaxation processes within the Néel’s ideas and corresponding time-dependent hyperfine interactions in such magnetic systems. The generalized model allows one to treat numerically both magnetization curves in alternative

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

viii

Kenneth B. Tamayo

low-frequency magnetic field as well as temperature demagnetization from PHFC (positivehigh-field cooling), NHFC (negative-high-field cooling) and ZFC (zero-field cooling) measurements as well as Mössbauer spectra, including those in static and radiofrequency magnetic field in a self-consistent way within the same set of physical parameters inherent to the system studied. A number of qualitative effects can be explained or predicted within the approach, which include interaction effects, relaxation-stimulated resonances in Mössbauer spectra under radiofrequency field excitation, specific shapes of Mössbauer spectra within precession of particle’s uniform magnetization, and asymptotic high-temperature magnetization and susceptibility behavior different from the classical Langevin’s hightemperature limit for ideal superparamagnetic particles. Corrections within more general models of magnetic dynamics based on the Landau-Lifshitz-Gilbert or Braun kinetic equations also will be discussed. Available experimental results are presented in order to elucidate the merits that result from the generalization of the well-known Stoner-Wohlfarth model combining it with Néel ideas as well as from the general stochastic approaches. Chapter 2 presents an extended study of the La0.65Pb0.35MnO3 thin films, emphasizing the interplay between surface composition and surface electronic structure. We have found that annealing treatments can easily modify the surface composition. A gently annealed La0.65Pb0.35MnO3 surface is ‘soft’, with a surface effective Debye temperature close to Pb single crystal surface, in agreement with the extended Pb segregation in the surface region. The spin-asymmetry of this surface reaches 80% at 0.5 eV above the Fermi level at room temperature. A heavily annealed surface has a reduced Pb segregation and a slightly higher surface Debye temperature, indicating a stiffer lattice, but showing strong evidence of layer restructuring. The polarization of this restructured surface reaches 40% at 0.5 eV above the Fermi level, in the center of the Brillouin zone. We conclude that the surface segregation, consistent with a difference in free enthalpy between the surface and the bulk, is induced by annealing treatments. This surface segregation greatly reduces the spin-polarization near surface. The extreme changes in structure and composition of surface may have a great impact on magnetic and electronic surface structure - properties that are very important in improving the tunnel magnetoresistance and spin valve performance. In Chapter 3, we report our systematic investigation on (1) the site disorders imposed by A-site and B-site doping, and (2) the nanoscale size effect, in various rare-earth manganites. First, we address the effect of A-site size mismatch disorder on colossal magnetoresistance effect (CMR) and electronic phase separation. Given various manganites of different electronic bandwidths, it is demonstrated experimentally that the A-site disorder enhances significantly the instability of ferromagnetic metallic or antiferromagnetic charge-ordered ground states and drives them into spin-cluster-glass state. Second, we pay particular attention to a theoretical understanding of the B-site disorder in antiferromagnetic charge-ordered CE state, induced by nonmagnetic B-site doping. The as-caused charge frustration effect triggers a transition of the CE state into a short range ferromagnetic cluster state with relatively strong ferromagnetic tendency. Our calculation is in good agreement with experiments. Our experimental and theoretical investigations on the finite size effect focus on the spinglass tendency due to the enhanced surface relaxation with reducing sample size. While early works addressed the antiferromagnetic tendency on the surface of a ferromagnetic nanoparticle, we reveal the opposite effect: ferromagnetism appears in some nano-sized manganites with antiferromagnetic charge-ordered CE ground state in the bulk. A possible

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Preface

ix

origin is the development of ferromagnetic correlations at the surface of these small systems. We study the two-orbital double-exchange model near half-doping level n=0.5, using open boundary conditions to simulate the surface of a manganites nanoparticle. We confirm that the enhancement of surface charge density suppresses the charge ordered CE state and produces a weak ferromagnetic signature that could explain experimental observations. The perovskite- type compounds RE1-xMxMnO3 (RE= rare earth, M = Ca, Sr, Ba, Pb) exhibit remarkable electronic and transport properties induced by the presence of mixed Mn3+/Mn4+ valence states. In particular, mixed valence perovskites have been extensively studied due to their spectacular colossal magnetoresistance (CMR) effect. The most important features of the CMR manganites are reviewed. Here we present our findings on La0.7Ca0.3Mn1-xFexO3 (x=0 and 0.2) thin films grown by Pulsed Reactive Crossed Beam Laser Ablation. The structural, electronic and transport properties of the films were determined and correlated with the growth parameters. The influence of the oxygen vacancies on the transport properties of the La0.7Ca0.3MnO3 thin films, and in particular on the colossal magnetoresistance effect is discussed. Changes in the manganese valence states monitored by means of the X-ray photoelectron spectroscopy (as surface sensitive technique) and X- ray absorption spectroscopy technique (as bulk sensitive technique) are correlated with the transport properties of the thin films (bulk resistivity and magnetoresistance). Fe doping (20%) on Mn sites induces changes in the transport properties of La0.7Ca0.3Mn0.8Fe0.2O3 thin films. A considerable decrease in the colossal magnetoresistance value after Fe doping is reported. X- ray absorption spectroscopy measurements for the Fesites in these compounds indicate an increased localization of the Mn eg orbitals upon Fe substitution, which weakens the double exchange interaction. The importance of the double exchange interaction on the magnetoresistance effect is extensively discussed in Chapter 4. In Chapter 5, the magnetic properties of: semiconductors spinels, diluted magnetic semiconductors (DMSs), illuminates, perovskites and nanomaterials are studied by using the mean field theory (MFT), the high temperature series expansion (HTSE), the replica method (RM) and the variational principle. The exchange interactions of a diluted ferrimagnetic spinels materials CoFe2− 2 x Cr2 x O4 , Cox Fe1− x Cr2 S 4 and Lix Zn1− xV2O4 are obtained, by using the probability law and MFT. The HTSE combined with the Padé approximant (PA) and RM are used to determinate the magnetic phase diagrams ( TC versus dilution x ) of the three semiconductors spinels, DMSs Ca1− x Mn x O , illuminates K 2Cu x Mn1− x F4 and perovskites Fex Mn1− xTiO3 materials in the range 0 ≤ x ≤ 1 . The exchanges interactions in the CoRh2 O4 and MnCr2O4 nanomaterials are calculated for different sizes of nanomaterials by using the MFT. The intra-planar and the inter-planar interactions and the exchange energy are deduced for Lix Zn1− xV2O4 and for

CoRh2 O4 nanomaterials with different sizes. The critical exponents associated with the magnetic susceptibility

(γ )

and with the correlation lengths

(ν )

are deduced by using the

HTSE combined with the Padé approximant methods for: spinels the materials, illuminates and DMSs. The effective critical exponent (ν ) of CoRh2 O4 and of MnCr2O4 nanomaterials is deduced for different sizes.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

x

Kenneth B. Tamayo

Domain wall propagation is used in different modern spintronic devices like Race track memory, domain wall logic, domain wall electronics, etc.. The key requirement for the application of thin magnetic wire in such devices is their high speed, which is finally driven by the domain wall propagation velocity. Chapter 6 reviews the results on the fast domain wall dynamics in amorphous glasscoated microwires. Very fast domain wall has been observed that can reach the velocities of up to 20 000 m/s. Such velocities exceed the sound velocities in magnetic microwires (4 700 m/s). Extremely fast domain wall brings new effects the change of the domain structure at around 1 000 m/s or the interaction of the domain wall with phonons when the domain wall approaches sound velocity. Due to the interaction, the domain wall velocity remains constant in a certain range of applied field, which could be used to synchronize the domain walls propagation in spintronic devices. Typically, scientists use materials with low damping in order to observe a fast domain wall. We show, that it is a distribution of the anisotropy that is even more important. Existence of two, perpendicular anisotropy results in their compensation and in the very fast domain wall. Such theory is confirmed by the measurement of the domain wall dynamics of highly magnetostrictive wires in transversal field. In Chapter 7, investigation of room-temperature crystal structure, magnetic and local ferroelectric properties of polycrystalline Bi1-xGdxFeO3 (x= 0.1, 0.2, 0.3) samples has been carried out. Gadolinium substitution has been found to induce a polar- to- polar R3c→Pn21a structural phase transition at x~ 0.1. Increasing content of the substituting element has been shown to suppress the spontaneous polarization in Bi1-xGdxFeO3, resulting in a ferroelectricparaelectric Pn21a→Pnma phase transition at 0.2 kBT

(3)

where kB is the Boltzmann constant and T is the temperature. That is in the absence of an external field the magnetic moment of each particle occupies one of two states with its direction along the easy axis and jumps between these local states are supposed to be very slow according to Eq. (3) so that they can be neglected for the measurement time. When an external field varies, for definiteness say from positive values of h, the particle’s magnetic moment follows the position of and instantaneously changes its direction in accordance with the local energy minimum. Only in a field h > hC(Θ) the particle’s magnetic moment passes immediately into the absolute minimum as can be seen in figure 4. Assuming a random distribution of particles over the easy-axis direction, the projection of m on the field direction represents the relative magnetization value m(H,Θ) of a group of particles with a given Θ value. The total magnetization for an ensemble of the SW particles is naturally the result of averaging over the particles with different orientations:

π

m( H ) = ∫ m(Θ, H ) sin ΘdΘ . 0

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(4)

Non-equilibrium Magnetism of Single-Domain Particles…

Figure 4. The energy density of a SW particle for a fixed angle Θ = 45º versus the angle

9

φ (rotating the

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

particle’s magnetisation vector). The normalized strength h of an external field is indicated as a parameter. The dashed line shows the variation of the angle φ with decreasing h from a positive value above the switching field hC(45º)= 0.5 to negative values.

Figure 5. Angular dependence of the critical field hC.

Of interest are now magnetization curves in an external periodic field. Such magnetization curves calculated as a function of an external periodic field with the amplitude H0 and angular frequency ω0,

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

10

Mikhail A. Chuev and Jürgen Hesse H(t) = H0 sin(ω0t),

(5)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

are shown in figure 6. When an external periodic magnetic field with the normalized amplitude above h = 1 is applied, the time dependence of the particle’s magnetisation takes the form of a hysteresis loop. If the amplitude of the periodic field is lower than the minimum critical field hC(45º) = 0.5, then according to the SW model the particle stays into one of two local energy minima and never leave it for the other. For amplitudes of the external field in the range 0.5 < h0 = H0M0/2K < 1 the magnetization of some particles reverses in the strongfield regime whereas that of the other particles behaves in the weak-field regime (figure 6). The simplicity of the SW model allows one to calculate the magnetic properties of an ensemble of particles and treat the experimental magnetization curves like those shown in figure 1. However, it is clear that the magnetization curves within the SW model do not depend on both the characteristic measurement time (or ω0) and temperature, so that this model can not be applied even qualitatively for analyzing the temperature dependent magnetization curves (Rellinghaus et al [35]) as well as the Mössbauer spectra of nanoparticles collected in a magnetic field (see Sections 4.6 and 5).

Figure 6: Magnetization curves for an ensemble of noninteracting SW particles. From top to bottom: completely textured particles with the angles Θ = 0º , 45º, 90º and a system of randomly oriented easy axes (bottom). h0 = H0M0/2K = 1.5, 0.75 and 0.5 (from left to right). Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

11

1.3. Néel’s Relaxation Approach In accordance with condition (3) thermal excitations into the local energy minima (collective excitations) as well as transitions between these local states (superparamagnetic relaxation) are not taken into consideration in the SW model. A theory of the superparamagnetic relaxation was first considered by Néel [2] who derived the effective relaxation rate between local energy minima separated by an energy barrier U0:

p = p 0 exp(−U 0 / k BT )

(6)

where p0 is the fluctuation rate slightly dependent on temperature T as compared to the exponent (figure 7). The relaxation process in the Néel’s model is related to the magnetostriction and demagnetization fluctuations induced by the lowest energy vibrations accomplished with the Boltzmann factor for describing the probability of precession the particle’s magnetic moment across the energy barrier when the magnetization stays near the barrier (first discussed by Néel and again considered by Jones and Srivastava [36]. This results in Eq. (6) where 1/ 2

p0 =

⎛ 2V ⎞ 2γK ⎟⎟ 3λG + Dp M 02 ⎜⎜ M0 Gk T π B ⎠ ⎝

(7)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

where γ is the gyromagnetic ratio, λ is the magnetostriction, G is the Young’s modulus and Dp is the demagnetization factor.

Figure 7. Schematic of the two-level relaxation model for a single-domain particle.

The Néel’s model has been extended by Brown [37] who has developed a theory based on a Brownian-motion approach for a random walk of the magnetization direction. The standard theory for the stochastic relaxation of the particle’s uniform magnetization vector M for an ensemble of ferromagnetic single-domain particles is based on the following Landau-

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

12

Mikhail A. Chuev and Jürgen Hesse

Lifshitz-Gilbert equation [38, 39] describing particularly the reorientation of the vector M in the presence of a rapidly fluctuating chaotic field h(t) [37]:

dM dM ⎡ ⎤ = γM × ⎢H eff − η + h(t )⎥ dt dt ⎣ ⎦

(8)

where η is the dissipation coefficient,

H eff ≡ H eff (θ ,ϕ ) = −

V ∇E (θ ,ϕ ) , M0

(9)

θ is the angle between the uniform magnetization M and the particle’s easy axis, φ is the azimuth angle. Here, the magnetization direction is arbitrary so that the energy density of the particle is given by a more general expression:

E (θ ,ϕ ) = − K cos 2 θ − HM .

(10)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Under the assumption that the stochastic process h(t) is stationary and isotropic:

hi (t ) = 0 ,

(11a)

hi (t )h j (t + τ ) = μδ ijδ (τ ) ,

(11b)

Brown derived the following differential equation for the probability density (population) W(θ,φ) of states with a given M direction [37]:

∂W = − Pˆ W . ∂t

(12)

⎞ ⎡ ⎤ γ ⎛M V Pˆ = − D ⎢Δ − ∇(∇E (θ ,φ ) )⎥ − ∇⎜⎜ × ∇E (θ ,φ ) ⎟⎟ k BT ⎣ ⎦ M0 ⎝ M0 ⎠

(13)

Here, the diffusion operator

and the diffusion coefficient

D=

γηkBT VM 0

=

γμ 2M 0

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(14)

Non-equilibrium Magnetism of Single-Domain Particles…

13

are introduced in the representation given by Afanas’ev and Sedov in [40]. The diffusion operator defined by Eq. (13) has the form of the Fokker–Planck operator and its terms describe the isotropic diffusion, drift towards the local minima of the anisotropy energy, and precession of the vector M in the effective field Heff, respectively. Braun has also derived the transition rates of jumps between the local energy minima for the axially symmetric case (Θ = 0) in terms of the smallest non-vanishing eigenvalue of the Fokker–Planck equation (12) in the high-energy barrier approximation (3): 1/ 2

γK ⎛ U 0 ⎞ ⎜ ⎟ p12,21 = M 0 ⎜⎝ πkBT ⎟⎠

[

(1 − h 2 )(1 ± h) exp − U 0 (1 + h) 2 / kBT

]

(15)

which in the absence of a field comes to Eq. (6) with 1/ 2

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

γK ⎛ U 0 ⎞ ⎜ ⎟ p0 = M 0 ⎜⎝ πkBT ⎟⎠

.

(16)

A detailed re-examination of the classical Néel’s and Braun’s models has been performed by Jones and Srivastava [36], which shows that the different physical assumptions taken in the models actually lead to a slightly different p dependence on T and V. That is why many researches (including us) prefer to regard p0 in Eq. (6) as a constant in simple treatments when a microscopic relaxation mechanism is not specified explicitly. The Braun’s model can be principally used for analyzing the experimental magnetization curves of an ensemble of nanoparticles, e.g., those shown in figures 1 and 2 as well as the corresponding Mössbauer spectra presented and discussed in Sections 4 and 5. However, its application in practical sense is restricted by only numerical simulations of not only the Mössbauer spectra (Jones and Srivastava [41], Chuev [42]), but even magnetic measurements (Aharoni [43], Schrefl [44], Bauer et al [45], Sun et al [46]) without any analysis of the experimental data taken from real magnetic nanoparticles except for a single study by van Lierop and Ryan [47]. Moreover, almost all the simulations have been performed in the simplest case of axial symmetry (Θ = 0), which is not applicable for analyzing the experimental data taken from real magnetic nanoparticles (see, e.g., figures 1 and 2). Problems of practical techniques for calculating the particle’s magnetic relaxation under nonaxisymmetric potential like that in Eq. (10) have been repeatedly mentioned already in the classic work [37]. However, since this work none has even started to solve the problem of calculations of the temperature- and field-dependent magnetization curves (even of the equilibrium ones) within the non-axisymmetric potential. Instead, researchers prefer to interpret the experimental data taken on real samples within the simplest classical models [1, 2], which often reduces to estimates of one or a finite number of empirical parameters such as the blocking temperature, the coercivity, mean particle’s size and magnetic anisotropy constants (Jönsson et al [48], Wernsdorfer et al [49], Rellinghaus et al [35], Michele et al [33, 50,51], Tronc et al [27], Chuev et al [32], Cador et al [52], Du et al [53], Suzuki K and Cadogan [54], Hupe et al [55], Balogh et al [56], Predoi et al [57], Stankov et al [58], Hendriksen et al [59] , Vasquez-Mansilla et al [60], Miglierini et al [61]).

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

14

Mikhail A. Chuev and Jürgen Hesse

The main aim of this contribution is just to present practical techniques for calculating the magnetization curves and the Mössbauer spectra of an ensemble of nanoparticles within models of the particle’s magnetic relaxation under the non-axisymmetric potentials (1) and (10). In the next section we start with an extension of the original SW model in terms of the Néel’s relaxation of the magnetization in such systems, which is reduced to the derivation of an universal relaxation equation which holds for the populations of the particle’s energy levels defined by the original SW model. Solving this differential equation with proper chosen initial conditions, a number of magnetisation phenomena observed experimentally versus temperature, time and external magnetic fields can be described and understood quantitatively. So, hysteresis loops including also those in high-frequency external magnetic fields can be calculated within this model as a function of temperature as well as demagnetisation curves for arbitrarily heating rates in different external magnetic fields can be simulated. In contrast to a great problem in treating the experimental data within more general stochastic models based on the Landau-Lifshitz-Gilbert equation (8), one can easily fit to a first approximation a wide set of the data taken at the same sample within the extended SW model [3]. Along with qualitative description of experimental magnetization curves the extended SW model results in a nontrivial fundamental pattern: the asymptotic behavior of the magnetization and susceptibility in the high-temperature limit and weak magnetic fields, which is different from the classical Langevin limit for ideal superparamagnetic particles (Bean [16], Bean and Livingston [62], Vonsovski [63]). Moreover, the non-trivial asymptotic magnetization behavior is justified by our experiments on real magnetic nanoparticles. In Section 3 a general way for calculating the temperature dependence of magnetization (including the non-equilibrium one) in terms of the Braun’s differential equation taking into account the precession and diffusion of the particle’s uniform magnetization is outlined. Calculations within the corresponding multi-level relaxation model show that the very realization of the asymptotic high-T magnetization limit is directly related to a nonconventional thermodynamics of each particular particle because its mean magnetization is determined by not only the standard Gibbs distribution leading to the Langevin limit, but also by the ratio of characteristic frequencies of the regular precession and random diffusion of the particle’s uniform magnetization. In Section 4 a theory of relaxation Mössbauer spectra taken on an ensemble of nanoparticles is presented. This theory is principally based on the general equations of stochastic relaxation given in previous sections so that the models of magnetic dynamics can be used for calculating the corresponding hyperfine interaction and for numerical analyzing the Mössbauer spectra including those taken in a weak static magnetic field. As for the later, the theory based on the Brown equation (12) is reduced in the first approximation to a threelevel relaxation model which continues the line of ‘physically oriented’ phenomenological models of the classical two-level relaxation in the absence of a field introduced by Wickman [29] and the extended Stoner-Wohlfarth relaxation in a field, which is discussed in Section 2. In the last section 5 a theory of Mössbauer spectra under radiofrequency magnetic field excitation is described, which is also based on the extended Stoner-Wohlfarth model [64, 65]. Along with a qualitative explanation of the effects observed in the experimental Mössbauer spectra of magnetic nanoparticles under rf field excitation, a principally new shapes of the spectra are predicted, which are originated from the relaxation-stimulated resonance

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

15

processes at the rf-field frequencies coupled by parametric resonance conditions with the Larmor frequencies for the ground and excited nuclear states.

2. The Stoner-Wohlfarth Model Extended within Néel’s Relaxation In contrast to the Néel’s idea, the relaxation procesess were not directly introduced in the original SW model, nevertheless, the relaxation is presented indirectly and plays a rather nontrivial role in this model. Namely, it is assumed that with a magnetic field changing: (i) the particle’s magnetic moment being in a definite energy minimum follows the position of the local energy minimum and instantaneously changes its direction in accordance with the strength of the magnetic field applied; (ii) transitions between the states corresponding to different energy minima are forbidden until the strength of the magnetic field becomes stronger than the critical field. In other words, the relaxation process is both extremely fast and extremely slow. Impossibility of jumps between the states with different energy minima is indirectly substantiated by the presence of high energy barriers Ui hampering the jumps in accordance with condition (3) (see figure 8). However, this assumption appears to be valid in rather wide range of magnetic fields weaker than HC(Θ), but that must be wrong when the magnetic field strenght is in the vicinity of the critical field. Indeed, in the latter case the height of the energy barrier is given by [65]:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

U21(h, Θ) = c(Θ) U0(hc(Θ) − |h|)2

(17)

where c(Θ) is a numerical constant. For instance, if the periodic field amplitude H0 in Eq. (5) is close to HC(Θ), the assumption (3) of a high energy barrier is obviously not valid for a significant while of time (figure 9). A more accurate description of the relaxation process within the SW model has been suggested by Afanas’ev et al [65] where the reorientation of the particle’s magnetic moment is regarded to occur not only in a magnetic field stronger than Hc(Θ), but also in weaker fields when the effective energy barriers U(Θ) are not too high as compared to the thermal energy. Such a generalization of the SW model results in a successful description of remarkable changes in both the magnetic properties and shape of Mössbauer spectra of the SW particles [65]. In accordance with the results and the Néel’s equation (6), we will assume that depending on the instantaneous value of the magnetic field strength in each moment the relaxation process is defined by only two parameters, the constant p0 and the height of the energy barrier U0 in zero field:

p p12 (Θ, H , T ) = 0 ∑ exp[− U1i (Θ, H ) / kBT ] 2 i =1,2

(18a)

p p21(Θ, H , T ) = 0 ∑ exp[− U 2i (Θ, H ) / k BT ] 2 i =1,2

(18b)

and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

16

Mikhail A. Chuev and Jürgen Hesse

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 8. Schematic of the energy levels (local energy minima), the energy barriers and transitions between the local energy minima within the extended SW model.

Figure 9. Time dependence of the energy barrier calculated for SW particles with angle Θ = 45º for different amplitudes h0 of the periodic magnetic field (5). T0 = 2π/ω0.

where

U ij (Θ, H ) = E (max) (Θ, H ) − Ei(min) (Θ, H ) . j

(19)

Here, p12 and p21 make sense of the probabilities of the transitions between the local equilibrium states (figure 8). For each group of particles with different orientations the values

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles… (max)

of Ei

(min)

and Ei

17

can be evaluated by means of not complicated numerical calculations

from Eq. (1). Note that in contrast to Ref. [65, 67] where only the lower energy barrier has been taken into account, we have introduced into Eqs. (18) an average value of relaxation rate of jumps over both the SW barriers in order to better approximate the three-dimensional character of the energy barrier existing in reality (see Section 3). This relaxation model has the advantage that the relaxation process is characterized by only two parameters as a whole: the constant p0 and the height of the energy barrier U0. At the fixed p0 and extremely high U0 just the original SW model is realized, and the higher U0, the smaller the time interval within which the energy barriers can not be regarded as small ones, in the limiting case when U0→∞ this interval is tending to zero. (In the range of small energy barriers more accurate description of the relaxation processes could be given, when p0 is no longer a constant and depends on the strength H of magnetic field (Aharoni [43, 66]). In magnetic fields ⏐h⏐< hc(Θ), each particle can stay only into two states corresponding to the local energy minima between which the relaxation transitions can occur. Then, the relative equilibrium populations of the states are defined by the detailed balance principle:

wi(0) (Θ, H , T ) =

exp[ − Ei(min) (Θ, H ) / kBT ]

exp[ − E1(min) (Θ, H ) / kBT ] + exp[ − E2(min) (Θ, H ) / kBT ]

(20)

and the projection of the equilibrium magnetization of the particle on the direction of the magnetic field is determined by the following expression

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

m(0) (Θ, H , T ) = w1(0) (Θ, H , T )m1(Θ, H ) + w2(0) (Θ, H , T )m2 (Θ, H ) ,

(21)

where mi= Mi(Θ,H)/M0 are the projections of the normalized magnetic moment of the particle on the direction of the magnetic field, corresponding to the local energy minima. Naturally, under the action of an external time-dependent field (which is assumed here to be periodic in time) the true populations of the local states are, in general, not equilibrium ones when the frequency of the field is high enough as compared to the relaxation rate and depend both on the external parameters (H, T, and the rate of changes in one of them) and parameters (K, V, M0, p0 and their distributions) inherent to the system studied. So, at each moment the changes in the non-equilibrium populations of the local states, w1(t) and w2(t), for a group of particles with the given angle Θ in time can be described by two conventional combined differential equations

dw1,2 (t ) dt

= ±[ p21(t ) w2 (t ) − p12 (t ) w1(t )] .

(22)

Taking into account the obvious normalization w1(t) + w2(t) = 1,

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(23)

18

Mikhail A. Chuev and Jürgen Hesse

these equations are reduced to the single equation

~ (t ) dw ~ (t ) − w ~ ( 0 ) (t )] = − p(t )[ w dt

(24)

for the difference of the populations

~ (t ) = w (t ) − w (t ) , w 1 2

(25)

which is a typical relaxation equation. Here we have introduced p(t) = p12(t) + p21(t),

(26)

~ ( 0) (t ) = w( 0) (t ) − w( 0 ) (t ) . w 1 2

(27)

and

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The time-relaxation equation (24) bears the extended Stoner-Wohlfarth model and should be solved together with the initial conditions. Note that we write down the effective relaxation rates pij(t) in Eqs. (22) and (26) assuming that at each moment t the system exhibits energy barriers depending on H(t) and/or T(t), which results in the time dependence p(t) that is generally determined by Eqs. (18). That is why the kinetic equation (24) can principally describe memory effects depending on initial conditions in a specified scheme of measurements. The nonlinear equation (24) should be supplied with the boundary conditions which are determined by the concrete experimental scheme (see Sections 2.1 and 2.2) applied to study ~ (t ) is known, the time evolution of the magnetization of a group of magnetic properties. If w particles with the given angle Θ is determined by

m(t , Θ) = w1(t , Θ) cos φ1( H (t ), Θ) + w2 (t , Θ) cos φ2 ( H (t ), Θ) where

w1,2 (t , Θ) =

~ (t , Θ) 1± w , 2

(28a)

(28b)

the angels φ1 and φ2 correspond to the local energy minima. In order to determine the evolution of the total magnetic moment of an ensemble of the randomly oriented and noninteracting SW particles it is necessary to carry out the sum

M (t ) = M 0 ∫ m(t , Θ) sin ΘdΘ .

(29)

There always exists a distribution of particle’s sizes P(V,σV) with a certain width σV in a real samples, so that one has also to perform an averaging over this distribution

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

19

M (t ) = ∫ M (t ) P(V , σ V )dV .

(30)

The generalization of the relaxation process in the SW model results in essential changes in the non-equilibrium magnetic properties and, first of all, in the shape of hysteresis loops as compared to those in the original SW model [65]. Whereas the differential equation (24) for the differences of populations generally holds, the magnetization behaviour is more complex as can be seen in Eq. (28). Considering the very special case of a system of indentical completely textured SW particles with the easy axes parallel to the external magnetic field (Θ = 0) the form of the kinetic equation (24) also holds for the magnetization [67]:

dm(t ) = − p (t ) ⋅ [m(t ) − m0 (t )] dt

(31)

~ (t ) and m (t ) ≡ w ~ (t ) . where m(t ) ≡ w 0 0

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

2.1. The Extended SW Model: Hysteresis in a Periodic External Magnetic Field The magnetization curve in the original, “static” SW model is determined by only the amplitude of alternating magnetic field (5) and does not depend on the rate of changes in the field strength. In the extended SW model presented above the shape of hysteresis loops becomes already dependent of the magnetic field frequency due to relaxation processes taken into account, i.e., of the ratio ω0/p0 as well as of the value of the effective energy barrier U0/kBT [65]. In order to calculate the magnetization curves of an ensemble of relaxing SW particles in this case one should define the boundary conditions for the differntial equation (24). For the periodic field amplitude H0 < HC(Θ), the periodicity condition

~ (t + 2π / ω ) = w ~ (t ) w 0

(32a)

can be choosen as the bounadry condition. In a periodic magnetic field with the amplitude stronger than the critical one, H0 > HC(Θ), there remains only a single absolute energy minimum for ⏐H(t)⏐ > HC(Θ) in the original SW model, and the particle comes to the state corresponding to this minimum so that the true populations do not depend on time:

~ (t , Θ) ≡ w ~ (0) (t , Θ) = ⎧ 1 , H (t ) > H C (Θ) . w ⎨ ⎩− 1, H (t ) < − H C (Θ)

(32b)

This condition plays the role of a boundary condition for H0 > HC(Θ). Such well-defined states for the partices system can experimentally be achieved also by positive high-field cooling or negative high-field colling described in the next section.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

20

Mikhail A. Chuev and Jürgen Hesse

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Hysteresis loops calculated within the extended SW model in the strong (h0 = 1) periodic magnetic field as a function of the ratio ω0/p0 are shown in figure 10. A simple physical explanation of the observed widenining of hysteresis loop (increase in the effective switching field) with the frequency ω0 increasing can be given: (i) at lower frequencies ω0 the SW particles have more time to come into the equilibrium state and the corresponding switching field appears to be weak, while (ii) at higher frequencies the particles magnetic moments has no time to populate the local energy levels in accordance with their equilibrium population so that a stronger field amplitude H0 is necessary for the particle’s magnetisation reversal. (The latter corresponds to the original SW model.) This fact is directly associated with a large difference between the switching fields for the widely-used soft magnetic Permalloy evaluated from the Mössbauer experiments under radio frequency field excitation (see Section 5), HC is about several oersteds [68], and those from conventional magnetization measurements at low frequencies where HC in Permalloy equals to about only several hundredth oersted [69].

Figure 10. Magnetization curves for an ensemble of particles (U0/kBT = 20) as calculated within the extended SW model in a strong magnetic field with the amplitude h0 = 1 and different frequencies ω0/(2πp0) = 1, 10−2, 10−4, 10−6, 10−8, 10−10 (from the outer hysteresis loop to the inner one).

In a weak periodic field when H0 < HC(Θ) for all particles the hysteresis shape varies in a more complicated way shown in figure 11. In the low-frequency regime (ω0 > KV the relaxation is fast enough that the integral diffusion (70) over all the stochastic states is efficiently realized rather than the differential diffusion (63) between neighboring energy levels is and (ii) changes in the diffusion rate (i.e., in the interaction of the particle with surroundings) in the temperature range of measurements are small. In this case, the calculation scheme is essentially simplified because the equilibrium state of particles with a given Θ value is described by the distribution Wi(E) in accordance with Eqs. (60). Figure 25 shows temperature dependences of the inverse magnetization as a function of the diffusion coefficient D0 (of the ratio D0/Ω0) for a given h value, which are calculated by Eqs. (4), (60), (61), (57), (65), and (66) in the case of homogeneous diffusion (70). As seen in the figure, with the diffusion rate increasing the high-T magnetization asymptotic behavior like that described by Eq. (45) inherent to the slow diffusion limit converts smoothly to the ‘Langevin’ limit (49).

Figure 25. Temperature dependences of the inverse equilibrium magnetization of an ensemble of particles (KV/kB = 300 K) in a magnetic field with h = 0.02 as calculated in the model of homogeneous and constant diffusion given by Eq. (70) with various rate D0. Here the curves for D0/Ω0 = 10 and 100 differ only by their drawing-line width.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

44

Mikhail A. Chuev and Jürgen Hesse

In reality the diffusion rate depends on temperature, for instance, in the Braun’s model this dependence is linear in accordance with Eq. (14) (Brown [37]). Let us write the dependence in the form convenient for our purposes:

pE (θ ,ϕ ) = D0

T , TC

(71)

i.e., assuming again that the diffusion is homogeneous over all the stochastic states. It is easy to see that this model principally differs from the previous ones. In the latter models the mean magnetization M (Θ, E ) for each orbit CE does not depend on temperature while that in the model (71) depends through the instantaneous angular velocity renormalized by diffusion:

~ Ω E (θ ,ϕ ) = Ω 2E (θ ,ϕ ) + D02T 2 / TC2 .

(72)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Just this circumstance allows the interrelation of two ‘conflicting’ limits described by Eqs. (45) and (49) to be made clear.

Figure 26. (Right ordinate) Temperature dependences of the inverse equilibrium magnetization of an ensemble of particles (KV/kB = 300 K) in a magnetic field with h = 0.02 as calculated for various constant D0 of homogeneous and temperature-dependent diffusion specified by Eq. (71). (Left ordinate) Dashed line represents the temperature dependence of the saturation magnetization for bulk nickel that is calculated by Eq. (47) with Na = 1000.

Figure 26 shows typical temperature dependences of the inverse magnetization which are calculated by Eqs. (4), (60), (61), (57), (65), and (72) as a function of the diffusion coefficient D0 within the model (71). (Note that the magnetization in this figure is not normalized by the temperature-dependent saturation magnetization defined by Eq. (47) in contrast to the other

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

45

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

figures.) As seen in the figure, when the diffusion is slow (D0 ≤ 0.1Ω0), at high temperatures lower than the Curie temperature (in the particle’s bulk) the high-T magnetization asymptotic behavior like that given by Eq. (45) is realized. When the diffusion is fast (D0 ≥ 10Ω0), the high-T “Langevin’ limit (49) has time to be realized at T < 0.5TC. At last, in the case of intermediate diffusion (D0 ≈ Ω0) with temperature increasing, but lower enough the asymptote (45) is first developed whereas with further temperature increasing this behavior converts smoothly to the ‘Langevin’ limit (49) which at higher temperatures is ‘killed’ by the temperature-dependent saturation magnetization defined by Eq. (47). That is figure 26 actually illustrates scenario schematically described by Chuev [72]. Note that the extended SW model and the multilevel model presented above can be easily modified to describe the interacting single-domain particles in the framework of the mean field approximation [3], which will significantly expand their application area for analyzing experimental data. It is also clear that a real system containing nanoparticles is always inhomogeneous so that one should inevitably take into account distributions of physical parameters inherent to the system (e.g., particles size distribution) in a treatment of the experimental data within any model of magnetic dynamics. Such a model should also include somehow the temperature dependence of the uniform saturation magnetization M0(T) characterizing each single particle. All the details concerning the new description of nanomagnetism in the frame of the extended SW model and also in the multilevel model for stochastic relaxation presented before will have consequences for each experimental physical method used for investigating the magnetic properties. Very powerful are methods like Mössbauer spectrometry, neutron scattering, electron or muon spin resonances, etc. In the next sections the Mössbauer effect spectrometry, a nuclear method widely used in investigations of magnetic nanoparticles, is discussed. After a short introduction into this method consequently step by step the influence of magnetic relaxation of single-domain particles on the shape of the Mössbauer spectra will be presented.

4. Relaxation Mössbauer Spectra of Magnetic Nanoparticles For 50 years Mössbauer spectroscopy is known as a powerful method for investigating hyperfine interactions in solids. The mechanisms responsible for the hyperfine magnetic structure revealed in the absorption spectra seemed to be reliably established to the present time. The Mössbauer absorption spectra of magnetic materials is usually interpreted within group of lines (partial spectra) forming due to the hyperfine interaction of the nuclear magnetic moment with a static (hyperfine) magnetic field and lines originated by the quadrupolar interaction in the presence of electric field gradient at the nucleus. For instance, as far as the 57Fe isotope widely used in the Mössbauer spectroscopy is concerned, the hyperfine magnetic field splits the energy level of ground nuclear state with spin Ig = 1/2 into two sublevels with different projections mg of nuclear spin onto the direction of the hyperfine field. Whereas the excited nuclear state with energy E0 = 14.4 keV and spin Ie = 3/2 is split into four sublevels with different nuclear spin projections me, in accordance with the Hamiltonians of Zeeman interaction of nuclear magnetic moments with hyperfine magnetic field Hhf (Goldanski and Makarov [76]).

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

46

Mikhail A. Chuev and Jürgen Hesse

Hˆ (g, e) = − g g, e μ N H hf Iˆ (g, e)

(73)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(g, e) where μN is the nuclear magneton, gg,e are the nuclear g-factors and Iˆ are the nuclear spin operators for the ground (g) and excited (e) states. The scheme of splitting of the 57Fe nuclear energy levels is shown in figure 27.

Figure 27. (Top) The scheme of splitting of the 57Fe nuclear energy levels for the excited (e) and ground (g) states in a static hyperfine field and (bottom) the corresponding Mössbauer absorption spectrum, a magnetic sextet. Here and below the spectra are calculated for non-polarized gamma radiation, randomly distributed directions of the hyperfine field and Hhf = 330 kOe. The splitting of the energy levels in this example is of the order of 10-7 eV and should be compared with the 14.4 keV of the resonance transition.

There occur transitions between the nuclear sublevels of the ground and excited states, which are just observed as a set of lines in experimental absorption spectra. Positions and intensities of the lines are determined by the Hamiltonians (73) and multipolarity of the corresponding transition from the ground state to excited one. For the 57Fe nuclei the magnetic dipolar radiation of the M1 type takes place, for which transitions with changes in the nuclear spin projections by more than unity ( mg = ±1 / 2 → me = ∓3 / 2 ) are forbidden. So that the 57Fe absorption spectrum consists of not 8 lines corresponding to the scheme shown in figure 27, but of 6 lines, so-called magnetic sextet (Goldanski and Makarov [76]). Ratios of intensities are defined by the Clebsch-Gordan coefficients and the crystal texture of sample studied, e.g., for iron-based polycrystalline samples or magnetic alloys when the direction of hyperfine field at nucleus takes no preferential orientation and is randomly scattered, the intensities of spectral lines obey the ratios 3:2:1:1:2:3 as shown in figure 27. The majority of 57Fe Mössbauer absorption spectra of magnetic materials are

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

47

analyzed on the basis of this magnetic sextet. An additional important fact is that the hyperfine field at the nucleus is rigidly coupled to the magnetic moment of the Mössbauer atom and in many cases to the magnetisation of the nano particle in question. The absorption spectrum is usually collected as a function of a defined relative velocity between the absorber and a single line source, which results in the Doppler shift of the incident gamma-quanta. In the 57Fe case the velocity value of 1 mm/s corresponds to the energy shift of 4.8 10-8 eV. The Larmor precession frequency in a field of 330 kOe is ωg/2π ≈ 45 MHz and ωe/2π ≈ 25 MHz (see figure 1) for the nucleus in the ground and excited state, respectively. The nuclear life time in the excited state is 1.41 10-7 s which corresponds to an energetic absorption half line width of 4.67 10-12 keV. If the magnetic field at the nucleus starts to fluctuate in times comparable to the Larmor precession time there is a strong influence on the effective life time of the split energy levels and therefore on the line width observed (so called relaxation spectra).

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

4.1. Mössbauer Spectra of Nanoparticles and Hyperfine Field Distribution Figure 28 shows typical Mössbauer absorption spectra of crystalline α-Fe nanosized grains (nanoparticles) embedded in an amorphous magnetic matrix. This kind of spectra often are interpreted as the superposition of the well resolved (at low temperatures) hyperfine magnetic structure (the magnetic sextet for the 57Fe nucleus) corresponding to the nanoparticles and the strongly disperse magnetic component, which is usually attributed to the amorphous matrix (Miglierini and Greneche [77, 78], Miglierini et al [79], Suzuki and Cadogan [54], Hernando [80], Hupe et al [55], Stankov et al [58]). Theoretical models used to analyze such spectra are based on two approaches qualitatively different but identically widespread. The first is based on the introduction of continuous distributions of the hyperfine field values Hhf at the nuclei (Hesse and Rübartsch [81]). The spread of hyperfine fields is associated with the presence of different atomic neighborhoods of the Mössbauer atom, of different magnetic phases, magnetic sub-lattices in ferri- and antiferromagnetic materials as well as with crystal lattice defects. In this model the absorption spectrum σ (ω ) for gamma-quanta with energy Eγ = ω in the presence of the continuous Hhf distribution given by the probability function P(Hhf) is described by the simple expression

σ (ω ) = ∫ L (ω , H hf ) P( H hf )dH hf ,

(74)

where ω is the spectral frequency,

L (ω , H hf ) =

σ a Γ02 4

∑ i

Ai

(ω − ω i ( H hf ) )2 + Γ02 / 4

,

(75)

σa is the effective absorber thickness, Γ0 is the width of the excited nuclear level, Ai is the intensity of i-th hyperfine transition at the resonance frequency

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

48

Mikhail A. Chuev and Jürgen Hesse

ω i ( H hf ) = E0 + (meω e − m g ω g ) ,

(76)

and

ωe,g = ge,gμNHhf

(77)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

are the Larmor frequencies for nuclear spin in the hyperfine field.

Figure 28. (Left panel) 57Fe Mössbauer absorption spectra (vertical dashes) of the nanocrystalline magnetic alloy Fe82CuNb5B13 and (Right panel) the corresponding Gaussian hyperfine field distribution P(Hhf) taken from [55]. Dashed lines are the mean square deviations for P(Hhf).

The basic advantage of this approach is the simplicity of its computer realization, because it does not require a priori information on the form of the desired distributions. The application of this method allows one formally to reconstruct the P(Hhf) function for nanoparticles and its temperature evolution from the experimental Mössbauer spectra which results in qualitative estimates of physical parameters abovementioned. However, the conventional approach does not provide really quantitative estimates of both the distribution found and corresponding parameters. The improved variant of this technique allows one to evaluate the resulting hyperfine field distributions with indication of their mean-square deviations (Hupe et al [55]). In this approach additional Gaussian broadening of lines of a magnetic sextet is introduced, which can be regarded as a good estimate for the distribution of the hyperfine field over different sites in the sample, which are represented by the sextet. The resulting Hhf distribution can be expressed as a sum over all sextets of Gaussian broadened lines (Chuev et al [82]):

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

P ( H hf ) = ∑ i

)

(

⎡ (i ) 2 ⎤ H hf − H hf ⎥ ⎢ exp ⎢− ⎥ 2 2π γ i 2γ i ⎥⎦ ⎢⎣ Ai′

49

(78)

where A'i is the area of i-th sextet, γi is the additional Gaussian linewidth for outer lines of the (i ) sextet, and H hf is the normalized separation between the outer lines of the sextet. The corresponding Hhf distributions evaluated from the Mössbauer spectra shown in figure 28 are displayed on the right panel of the figure. Because the hyperfine field distribution is determined in Eq. (78) by the parameters of lines, which are adjustable in fitting, one can also estimate the mean square deviations of P(Hhf) for each point Hhf (Chuev et al [82])

ΔP( H hf ) =

∑∑ j

k

∂P( H hf ) ∂P ( H hf ) ∂p j

∂p k

(Δp Δp ) j

k

(79)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

where pj are all the parameters involved in Eq. (78), i.e., the areas, positions, and Gaussian widths of lines for all the sextets; (ΔpjΔpk) are the elements of the covariance matrix which is calculated within fitting. The corresponding values of ΔP(Hhf) restricts the resolution of the hyperfine field distribution evaluated by Eq. (78) and defines a spread in P(Hhf) which is also shown by dashed lines in figure 28. The application of this method to determine the quantitative characteristics of desired distributions for analysis of the spectra of magnetic nanomaterials allows the formal reconstruction of the temperature evolution of the form of the distribution of Hhf for nanoparticles and the amorphous phase and the estimation of characteristics of the so-called interface regions between them [54, 55, 58, 77-80].

4.2. Two-Level Relaxation Model Even if the above scheme of analysis within static hyperfine field distributions fit satisfactorily experimental spectra, one should understand that this mathematical approach does not take into account physical processes that occur in real materials incuding those with magnetic nanoparticles and result in shapes of the Mössbauer spectra which can not be described in terms of the static hyperfine structure. That is why more complicated models are involved in order to describe so-called relaxation effects when the strength and/or direction of the hyperfine field can change stochastically in time due to the spin-lattice, spin-spin or other relaxation processes (Wickman [29], Reid et al [83], Mørup [84], Mørup and Tronc [28], Tronc et al [27, 85], Dormann et al [86], Kemény et al [87], Balogh et al [56], Afanas'ev and Chuev [31], Chuev et al [32] , Aksenova and Chuev [88] ). In this case, following the basic Stoner-Wohlfarth ideas, magnetic nanomaterials are treated as an ensemble of uniformly magnetized single-domain particles whose total magnetic moment can relax to the local states of a particle that are determined by the magnetic anisotropy given by Eqs. (1) and (10). Since the particle size in these materials is quite small (5–10 nm), the relaxation time of the

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

50

Mikhail A. Chuev and Jürgen Hesse

magnetic moment of each separate particle can fall in the Mössbauer time window (10–11–10– 6 s for the 57Fe nuclei) and superparamagnetic relaxation can be the decisive factor in realizing a certain shape of the absorption spectrum. The simplest and, correspondingly, most widespread model for describing Mössbauer relaxation spectra is the two-level relaxation model first presented by Wickman [29]. Due to the two-level relaxation the Wickman model seems to be ideal for describing spectra of magnetic nanoparticles following the SW ideas, in which the magnetic moment of a single particle changes its direction to the opposite direction infinitely fast but randomly in time, remaining parallel or antiparallel to the easiest magnetization axis of the particle. In this case, when one may ignore the deviation of the magnetic moment from the direction of the easy magnetization axis, so that the hyperfine field Hhf at the nucleus can only be reversed during the relaxation, the cross-section for absorption of a gamma-ray quantum is usually written as [29]

σ (ω ) = −

σ a Γ0 2

∑ Cα α

2

(

)

−1 Im W ω~ − ωˆα + iPˆ 1

(80)

~ = ω − E / +i Γ 2, α=(m ,m ) labels the hyperfine transitions between the nuclear where ω g e 0 0 states

ωα = meωe− mgωg,

(81)

the coefficients Cα are the intensities of the respective transitions, W = (1 / 2 1 / 2 ) is the

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

state population vector; 1 is a column vector with all components equal to zero,

⎛ω ωˆ α = ⎜⎜ α ⎝ 0

0 ⎞ ⎟ − ωα ⎟⎠

(82)

is the matrix of hyperfine transitions; and

⎛ p − p⎞ Pˆ = ⎜⎜ ⎟⎟ ⎝− p p ⎠

(83)

is the relaxation matrix with the elements p given by Eq. (6). In this case one can easily obtain the following analytic expression for the absorption cross-section

σ (ω ) = −

σ a Γ0 2

∑ Cα α

2

Im

ω + ip ω 2 − ωα2 + p 2

where

ω = ω − E0 / +i (Γ0 2 + p) .

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(84)

Non-equilibrium Magnetism of Single-Domain Particles…

51

Typical absorption spectra calculated by Eq. (84) within the framework of the Wickman two-level model are shown in figure 29. In the slow relaxation limit, p > ωα2 / Γ0 , the spectrum collapses into a single central line (or a quadrupole doublet in the presence of the electric field gradient at the nucleus) according to the approximate expression for the absorption spectrum

σ Γ (Γ + ωα2 / p) σ (ω ) ≈ − a 0 0 4

∑

Cα

2

2 2 2 α ω + (Γ0 + ωα / p ) / 4

.

(87)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

This simple Wickman’s model appeared to be very popular and many researchers have used it also to describe spectra of fine magnetic particles in different materials [27, 28, 56, 8387]. However, only few examples could be found where experimental data are satisfactorily described within this model. Moreover, in a great number of studies lines of asymmetrical shape with sharp outer and strongly extended inward wings are observed in the spectra like those shown in figure 28. To explain such spectra, one usually invokes models in which the particles are scattered in size, i.e., introducing a hyperfine field distribution within the scheme given in the previous section.

4.3. Generalized Two-Level Relaxation Model An alternative explanation for Mössbauer lines of asymmetrical shape with sharp outer and strongly extended inward wings as observed in nanoparticles has recently been suggested by Afanas’ev and Chuev [31] where a generalization of the two-level relaxation model based on the SW particle energy levels has been performed. This new model was successfully applied for evaluation of experimental data by Chuev et al [32], Aksenova and Chuev [88]. The principal postulate of the generalized two-level relaxation model is that the relaxation between the particle’s states with opposite directions of its magnetic moment never occurs as a transition between the states of the same energy because even weak interaction with the environment should inevitably smear out the energy levels as shown in figure 30. In a system like nanostructured ferromagnetic alloys with a great number of degrees of freedom, the energy levels of each particle at a certain time prove to be separated by a certain gap ΔE (figure 30, right) and the average value σ of distribution over ΔE may be rather large and comparable to temperature. Such a separation of the energy levels results in different values of the relaxation rates, p12 (ΔE) and p21 (ΔE), from one state to the other and vice versa. In the

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

53

simplest case of weak interaction (ΔE > ωα / Γ0 ), which physically means Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

2

very fast fluctuations between the energy states of a particle. In this case, the nucleus should ‘feel’ only the stochastically averaged hyperfine field defined by the difference of the equilibrium populations, w1 (ΔE ) and w2(ΔE ), of the states at a given ΔE:

H hf ( ΔE ) = [w1 ( ΔE ) − w2 ( ΔE )]H hf .

(96)

As follows from the detailed balance principle,

w1 ( ΔE ) − w2 ( ΔE ) = Δp / p = tanh( ΔE / k BT ) ,

(97)

Eq. (95) for the absorption cross-section becomes independent of the parameter p and takes the form of a continuous distribution of Lorentzian lines with the natural width Γ0:

σ (ω ) =

σ a Γ02 4

2

∞

1

∑ Cα ∫ [ω − ω tanh x]2 + Γ 2 / 4 g ( x, γ E / kT )dx . α 0 α −∞

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(98)

Non-equilibrium Magnetism of Single-Domain Particles…

55

Figure 31. 57Fe Mössbauer spectra calculated within the generalized two-level relaxation model [31] at

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

p = 30Γ0 (left) and in the fast relaxation limit,

p >> ωα2 / Γ0

(right). Here, ω3/2,1/2 =75Γ0.

We have performed simultaneous fitting for the whole temperature series of the Mössbauer spectra collected on the nanostructured Fe79Cu1Nb7B13 alloy (nanostructured means here iron nanoparticles embedded in a residual amorphous ferromagnetic matrix) within the generalized two-level relaxation model (figure 32). Details of the fitting procedure are given in [32, 88]. The results shown in figure 32 demonstrate a good description of the outermost lines of all the spectra as well as almost complete accordance between the calculated curves and resolved magnetic hyperfine structure (with lines of strongly asymmetrical shape mentioned above) at temperatures higher than the Curie temperature of amorphous phase for both the samples. Remembering that solid lines in figure 32 are calculated for the single hyperfine magnetic component corresponding to nanoparticles of the same size, one can understand that in this case there is no need to introduce a broad distribution of particle’s size at least for this nanoparticles phase even at higher temperatures, because the relaxation of interacting particles can result in the specific broadening of the spectral lines. In any case it is clear that although the generalized two-level relaxation model does not deny principally the particle’s distribution over sizes taking into account the interparticle interaction should strongly modify the shape of the distributions obtained by the conventional technique described in previous section.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

56

Mikhail A. Chuev and Jürgen Hesse

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

It is worth noting that, in addition to the magnetic sextets, the spectra of the magnetic nanoparticles often certainly exhibit line doublets (D1 and D2 marked in figure 32), which are usually attributed to the presence of the quadrupole hyperfine interaction in the presence of an electric field gradient at the nucleus. However, such interpretation often seems very artificial; for example, the doublet D1 with the line splitting Δ1≈0.74 mm/s is realized at lower temperatures, whereas the doublet D2 with the line splitting Δ2≈0.41 mm/s is realized in a quite wide high-temperature range and formally corresponds to the iron nuclei in the amorphous phase above its ordering temperature, but quadrupole splitting is completely absent in both magnetic spectral components for low temperatures. Such behavior suggests that the observed doublets are likely of magnetic rather than electric nature.

Figure 32. Temperature evolution of the Mössbauer spectra (vertical dashes) of the 57Fe nuclei in the nanocrystalline magnetic alloy Fe79CuNb7B13. The partial spectra of nanoparticles (solid lines) as calculated in the generalized two-level relaxation model and the partial contribution from the amorphous phase (points) as calculated with the (left panels) Gaussian Hhf distribution or (right panels) effective line doublet [32, 88].

4.4. Mössbauer Spectra within Precession of the Hyperfine Field Indeed, an additional nontrivial mechanism determining the formation of hyperfine structure in the Mössbauer spectra of magnetic nanomaterials recently has been revealed by

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

57

Afanas’ev and Chuev [74, 75]. It is the qualitative transformation of the absorption spectra in the hyperfine field rotating about a certain axis due to the precession of magnetic moments of nanoparticles in their intrinsic magnetic-anisotropy field. The necessity of inclusion of such a precession in Mössbauer spectroscopy has long been understood (Morup [84], Belozerskii and Pavlov [89], Jones and Srivastava [41]. However, it was suggested earlier that the characteristic precession frequency, Ω, of the particle’s magnetic moment is much higher than the precession frequency of the nuclear spin in the hyperfine field. In this case, the effect of fast rotation about an axis z reduces only to the effective decrease in the hyperfine field [41, 84, 89]:

H hf (θ ) = H hf cos θn z .

(99)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

where nz is the unit vector along the rotation axis and θ is the angle between the Hhf vector and z axis (figure 33, left).

Figure 33. Schematic of the rotating hyperfine field: (left) laboratory- and (right) rotating-frame representations. The quantization axes of the hyperfine-interaction Hamiltonians given by Eqs. (105) for the ground and the excited state of the 57Fe nucleus in nanoparticles with K > 0 (Ω < 0).

According to the results described in Section 3.1, in the absence of an external magnetic field the particle’s magnetic moment inclined by the angle θ to the easy axis stays in the axial magnetic anisotropy field

H an = K cosθ / M 0

(100)

and make a precession about the easy axis with the frequency

Ω = −γH an

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(101)

58

Mikhail A. Chuev and Jürgen Hesse

which can be represented in the following form

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Ω = −Ω0 cosθ .

(102)

Here, the parameter Ω0 is specified by Eq. (55). Note that the sign of precession of the particle’s magnetic moment is defined by the sign of the anisotropy constant K. When K > 0, i.e., in the presence of the easy magnetisation axis, the counterclockwise (Ω < 0) rotation of magnetic moment occurs. In the case of magnetic anisotropy of the ‘easy-plane’ kind (K < 0), the sign of rotation of magnetic moment becomes positive, clockwise rotation (Ω > 0). These situations can be realised when the magnetic anisotropy arises from the shape of particle, for instance, form of the prolate or oblate spheroids. In contrast to the magnetization curves, the sign of rotation of the particle’s magnetic moment cardinally influences the shape of Mössbauer absorption spectra. It is also essential that at angles θ close to zero the frequency of rotation of the particle’s magnetic moment is maximum, Ω ≈ Ω0, and at θ = π/2 the precession frequency goes to zero. Hence, there is always a range of angles θ over which the precession frequency Ω is comparable with the frequencies of precession of nuclear spins in the hyperfine field at nucleus. As simple estimates show, for small particles of size by the order of several nanometers the parameter Ω0 is only several times larger than the Larmor frequencies of nuclear spin precession. So, for the γ-Fe2O3 particles with the average diameter of about 7 nm and the characteristic magnetic anisotropy energy KV of about 1000 K (see, e.g., [27]), one can estimate the value of the parameter Ω0/2π ≈ 0.5 GHz. So that with the angle θ increasing the precession frequency Ω can become comparable or less than the Larmor frequencies of nuclear spin precession. The finiteness of Ω results in a cardinal transformation of the hyperfine magnetic structure of Mössbauer spectra. The direction of the hyperfine field at a nucleus in time t follows that of the magnetic moment, rotating about the magnetic anisotropy axis with the frequency Ω:

[

(

)

]

H hf (t ) = H hf n z cos θ + n x cos(Ωt ) + n y sin(Ωt ) sin θ .

(103)

In this case the Hamiltonians of hyperfine interaction of nuclear magnetic moments with the hyperfine field Hhf(t) becomes time-dependent:

Hˆ (g, e) (t ) = − g g, e μ N H hf (t )Iˆ (g, e)

(104)

To eliminate the time dependence, one can change to a frame of reference rotating about the z axis at the frequency Ω, which yields

(

)

~ˆ H (g, e) = − Ω + ω g, e cos θ Iˆz(g, e) + ω g, e sin θIˆx(g, e) Then, the eigenvalues of the operators can be written as follows

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(105)

Non-equilibrium Magnetism of Single-Domain Particles…

~ ~ ~ Eg, e = λg, e m g, e

59 (106)

~ where m g, e are the projections of nuclear spins onto the quantization axes for which the operators (105) are diagonal, and

~

λg, e =

(− Ω + ω g, e cosθ )2 + ω g,2 e sin 2 θ

(107)

The pressing point here is that the respective quantization axes for the two nuclear states differ essentially in direction which is illustrated by figure 33, right. In order to describe the Mössbauer spectra in this case one can use a general theory of the relaxation Mössbauer absorption spectra for the hyperfine field Hhf(t) depending on time in an arbitrary way (Afanas’ev et al [64, 65, 67]), which is discussed below in Section 5. Using this theory, an analytical expression for the absorption cross-section with averaging over the polarizations η of the incident radiation can be derived (Afanas’ev and Chuev [74, 75]): σ (ω , Ω, θ ) = −

~ m m m ~ m Γ0 g g e e (108) Im ∑ ∑ Vm~(η )m~+ Vm(η )m ~ ~ ~ ~ e g g e ω− λ m 2 λ / 2 − m − Ω m − m + i Γ η mg me e e g g g e 0 ~ m ~ m g e

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

where V

(

) (

)

(η ) are the matrix elements of the operator of interaction of the gamma-quantum me mg

with the nucleus and mg,e are the projections of nuclear spins onto z axis. As follows from Eq. (108), the lines with natural line width Γ0 should be observed in the absorption spectrum in the case of rotating hyperfine field. In general case the number of lines is equal to N = (2Ig +1)2⋅(2Ie +1)2. For the 57Fe isotope N = 64, but the selection rules for the magnetic dipolar M1 transitions reduces the number of allowed lines to 24, each of them being doubly degenerate because the pairs of lines with indices (mg,me) and (mg±1,me±1) have the same energy of transitions. Using Eq. (108) one can calculate the absorption spectrum as a function of the precession frequency Ω and the angle θ. Figure 34 shows the 57Fe Mössbauer spectra calculated in the case of a hyperfine field rotating about an axis under θ = 80° for different values of the frequency Ω, and different signs of rotation. The spectra are actually divided into the central group of lines and side bands, the intensity of side bands being decrease with the rotation frequency increasing and they come out of the spectral range in conventional Mössbauer measurements. In the limiting case of high precession frequency, independent of the sign of rotation, a ‘static’ magnetic sextet is observed, which corresponds to the rotation-average hyperfine field given by Eq. (99). Within an intermediate range of precession frequencies a non-trivial transformation of the spectra is observed. It is clearly seen that instead of the classic sextet of the static hyperfine structure (top spectra in figure 34) there can appear unusual spectra consisting of triplet, quartet and quintuplet of lines for negative values of the precession frequency Ω (K > 0) and a doublet of lines for positive sign of rotation (K < 0). That is a rotation of the hyperfine field drastically changes the Mössbauer line shape.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

60

Mikhail A. Chuev and Jürgen Hesse

The physical nature of the qualitative transformation of Mössbauer spectra can be clarified in the limiting case of high precession frequency,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Ω >> ω g, e .

(109)

Figure 34. 57Fe Mössbauer absorption spectra within the rotating hyperfine field (θ = 80°) for different values of the frequency Ω: (left-hand precession, left panel) Ω < 0 and (right-hand precession, right panel) Ω > 0 [74, 75].

In this case the absorption spectrum consists of the central group of six doubledegenerate lines and side bands. If the condition (18) holds, the side bands go out of the ordinary spectral range and become of minor intensity so that they can be neglected. The ~ = m and major contribution in the spectrum is given by the central lines with indices m g g

~ = m so that Eq. (108) for the absorption cross-section is reduced to the following m e e approximate expression

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

61

2 Γ0 1 (η ) σ (ω , Ω,θ ) = − Im ∑ ∑ Vm m . (110) ~ ~ e g ω − ω e me − ω g mg + iΓ0 / 2 2 η mg me

(

)

~ and ω~ are the effective constants of hyperfine interaction for the ground and Here, ω g e excited nuclear states, which are found from Eqs. (107):

ω~g,e ≡ ω~g,e (θ ) = − g~g,e (θ ) μ N H hf cosθ

(111)

and determined by the renormalized nuclear g-factors for the ground and excited nuclear states:

⎛ ω g, e ⎞ g~g, e (θ ) = g g, e ⎜⎜1 − sin θ tan θ ⎟⎟ . 2Ω ⎝ ⎠

(112)

As follows from Eq. (111), apart from the effective decrease in the value of hyperfine field proportionally to cosθ in accordance with Eq. (99), the rotation transforms qualitatively the shape of Mössbauer spectra by means of the renormalisation of the nuclear g-factors given ~ -factors for the ground and excited by Eq. (112). Moreover, changes in the effective g

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

nuclear states appear to be different because the initial gg- and ge-factors are different. For the 57 Fe isotope the g-factors for the ground and excited nuclear state are different not only by value, but also by sign (gg = 0.18 and ge = −0.10). ~ -factor for the As seen from Eq. (112), if Ω < 0 (left-hand precession), the effective g ground state will decrease in magnitude, whereas that for the excited state will increase. It is ~ for the ground nuclear state can clear that at the angles θ close to π/2 the effective factor g g even change in its sign to opposite one. Just this circumstance is the reason for the qualitative transformation of Mössbauer spectra, including the appearance of the triplet, quartet and quintuplet of lines shown in figure 34. The energy level splitting producing a quartet is illustrated by figure 35, left.

Figure 35. Schematic of the energy level splitting for the excited and ground 57Fe nuclear states in the rotating hyperfine field when (left) (right)

ω~e = 0 , magnetic ‘doublet’.

ω~ g = 0 ,

magnetic ‘quartet’ in the Mössbauer spectrum, and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

62

Mikhail A. Chuev and Jürgen Hesse

~ -factor for the excited state decreases in If Ω > 0 (right-hand precession), the g magnitude, whereas that for the ground state increases as compared with the real g-factors. It ~ for the excited nuclear state can change sign if θ is close is now that the effective factor g e to π/2. And if the angle θ satisfies the following condition

Ω cosθ = ω e / 2 ,

(113)

~ for the excited nuclear state is equal to zero and the absorption the effective factor g e

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

spectrum looks like a ‘magnetic’ doublet of lines. The corresponding scheme of splitting of the energy levels in the hyperfine field and Mössbauer absorption spectrum are shown in figure 35, right. The resulting Mössbauer spectrum of a single-domain fine particle is a superposition of partial spectra corresponding to the energy states with different orientations of the particle’s magnetic moment with respect to the magnetic anisotropy axis. Figure 36 shows the 57Fe Mössbauer absorption spectra calculated in the case of hyperfine field rotating about an axis under different angles θ for the parameter Ω0/2π = 0.5 GHz. One can see that the hyperfine ~ -factors is distinctly observed in the absorption magnetic structure with the renormalized g spectra for the angles θ > 70°. With the angle θ decreasing the qualitative effect of rotation on the shape of Mössbauer spectra become weaker and the hyperfine structure with only the rotation-average hyperfine field given by Eq. (99) remains in the spectra. However, one should mind that appreciable rotation effect can be registered not only if the spectra take nonconventional shape shown in figure 34, but also when the corresponding line shifts are comparable with the natural line width Γ0. For example, for the parameter Ω0/2π = 0.5 GHz one can estimate the rotation effect to be detectable for the angles θ > 30°. In order to observe the qualitative effects discussed above in a distinct form, one should realize a situation when an external perturbation forces the hyperfine field to be rotating about an axis with a given frequency and by a given angle to the axis. For instance, this can be done by means of collecting the Mössbauer absorption spectra of fine magnetic particles under excitation with an external rotating radiofrequency magnetic field strong enough to make the particles magnetic moments follow the changes in the external field, i.e., rotate about the corresponding axis at a given angle (Vagizov et al [91]). Providing the proper characteristics of the external rotating magnetic field, one can realise the conditions necessary to observe the specific shapes of Mössbauer absorption spectra shown in figures 34 and 36. It is also obvious that such nontrivial qualitative transformation of the hyperfine structure must significantly modify the above standard schemes for analyzing experimental Mössbauer spectra of magnetic nanomaterials including those shown in figures 28 and 32. Moreover, this transformation currently requires the development of the general formalism for calculating the spectra of these materials under the condition of a continuous relaxation process with the inclusion of the diffusion and precession of uniform magnetization in the magnetic-anisotropy field of nanoparticles.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Non-equilibrium Magnetism of Single-Domain Particles…

63

Figure 36. 57Fe Mössbauer absorption spectra in the rotating hyperfine field for various values of angle θ and the parameter Ω0/2π = −0.5 GHz.

4.5. Mössbauer Spectra of Nanoparticles within Continuous Diffusion and Precession of Uniform Magnetization According to the consideration in Sections 1.3 and 3.1, the equilibrium state of an ensemble of single-domain particles in the absence of an external magnetic field is described by the Gibbs distribution given by Eq. (50). In the presence of axial magnetic symmetry (10), when the population of states W(θ) and the elements of the matrix Pˆ are independent of the azimuth angle ϕ, the last term in the operator (13) disappears and Eq. (12) becomes onedimensional. In this case, precession can be ignored when calculating the magnetic relaxation time (Brown [37]). However, the magnetization precession with angular frequency defined by Eq. (102) must be taken into account when calculating the time-dependent hyperfine interaction. For this reason, the standard equation (80) is inappropriate for calculating the absorption spectrum in the general case of a continuous relaxation process with the inclusion

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

64

Mikhail A. Chuev and Jürgen Hesse

of the diffusion and precession of uniform magnetization. In principle, Eq. (80) can be appropriately generalized by using the results from Refs. [64, 65, 67], where the theory of Mössbauer spectra was developed for the case of the field Hhf(t) periodically varying with time along an arbitrary trajectory. In this case, in order to calculate the spectra, it is necessary to describe precession by the Liouville hyperfine-interaction superoperators, time ordering operators, and time integration rather than by the matrices ωˆα . In this case, it is not difficult to write the general expression for the absorption spectrum, but the main problems of analysis are associated with the optimization of a calculation procedure, which is still a separate problem. Here, we render the results of Chuev [42] where the simplest case is considered, when only states with high frequencies of uniform magnetization precession are populated in the thermodynamic equilibrium, i.e.,

Ω(θ ) >> ω g ,e ,

(114)

According to Eqs. (55) and (102), such a situation for real nanoparticles whose size is about several nanometers (Ω0 ≈ 1 GHz and KV/k ≈ 1000 K) is realized at quite low temperatures (KV >> kT) in a fairly wide range of angles θ near the easy axis. If condition (114) holds, the absorption spectrum in the presence of continuous diffusion and precession can also be calculated by Eq. (80), because the rotation effect in this case is reduced to the renormalization of nuclear g factors given by Eqs. (111) and (112) the latter being rewritten in the following form:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

⎛ ω g ,e ⎞ g~g ,e (θ ) ≈ g g ,e ⎜⎜1 − tan 2 θ ⎟⎟ . ⎝ 2Ω 0 ⎠

(115)

Then, the elements of the matrix ωˆα can be expressed as

ω~α (θ ) = meω~e (θ ) − m g ω~ g (θ ) ,

(116)

rather than by Eq. (81). A relaxation model in this case can be constructed within the quantum-mechanical description of a nanoparticle with the total spin S and 2S + 1 possible states of the projection Sz. Then, the transition probability between these states per unit of time can be determined under the assumption that relaxation is defined by the transverse components of the random field given by Eq. (11) [41]:

p S z , S z +1 = D S z S − ( S z + 1)

2

= D[S ( S + 1) − S z ( S z + 1)] .

(117)

In this case, the relaxation matrix Pˆ becomes tridiagonal and specified by the following expressions: Pii±1 = − pii±1 f ii±1 , (118a)

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

[

]

65

⎧exp − ( E j − Ei )V / kT , E j > Ei , fij = ⎨ E j < Ei 1, ⎩

(118b)

Pii = − Pii −1 − Pii +1 .

(118c)

Here, i = 1, …, 2S + 1; the transition rates pii + 1 are given by Eq. (117), and the energy density for each state is defined by the magnetic anisotropy (10): 2

⎛ S − i + 1⎞ Ei = − K ⎜ ⎟ . ⎝ S ⎠

(119)

Finally, the absorption spectrum for the given parameters Hhf, KV/kT, D, and Ω0 can be calculated by Eq. (80), where

Wi =

exp(− EiV / kT ) , ∑ exp(− EiV / kT )

(120)

i

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

and the elements of the diagonal matrix ωˆα are given by Eqs. (111), (115) and (116). Since relaxation matrix specified by Eqs. (117)-(119) includes the transitions between all the possible states of the projection Sz or orientations of the particle’s uniform magnetization, there is always a set of states (with small Sz values) for which condition (114) is not valid so that the elements of the diagonal matrix ωˆα for these states are determined by the initial expression (99) rather than by effective constants (111). The choice of such approximation is justified by two circumstances: (i) these states are weakly populated for low temperatures (KV >> kT) and (ii) the density of these states near the flat maximum of magnetic anisotropy energy (10) is quite high, so that even weak relaxation between them leads to effective averaging from Eq. (99). Typical Mössbauer absorption spectra calculated according to the above scheme in the model of the continuous diffusion and precession of uniform magnetization are shown in figure 37. Note that the total spin in real magnetic nanoparticles is about 1000, therefore, the rank N = 2S + 1 of relaxation matrix Pˆ specified by Eqs. (117)-(118) and the resulting matrix to be inversed in Eq. (80) is of the same order of magnitude. However, owing to the special form of these matrices (they are tridiagonal), the calculation of the absorption spectrum by Eq. (80) is reduced to the solution of the combined N linear equations. Such a solution includes about N operations and takes particularly less than 1 s on a PC. As seen in figure 37, the effect of precession with the finite frequency (102) is the most pronounced in the slow relaxation regime, where the shape of the spectra is determined primarily by the equilibrium population of states given by Eq. (50) so that the effective positions of the spectral lines for different precession orbits are determined by the following approximate expression

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

66

Mikhail A. Chuev and Jürgen Hesse

(

ω~α (θ ) ≈ ωα (0) cosθ 1 − δα tan 2 θ

)

(121a)

where

δα =

meω e2 − m g ω g2 . 2ωα (0)Ω 0

(121b)

A simple analysis of the last expression with particular values of the nuclear g factors for the Fe nuclei shows that the effect of the precession with the finite frequency (102) must be maximal for the inner lines ( me = ±1 / 2 → m g = ∓1 / 2 ) of the effective magnetic sextet, 57

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

which is illustrated by figure 37. In this case, for particular values of the model parameters, the shapes of the central part of the spectrum may be very diverse and their difference from the standard shape determined by Eq. (99) increases with decreasing the parameter Ω0. In particular, as seen in the left panel of figure 37, the absorption spectra can display the magnetic component that looks like a doublet D1, which entirely corresponds to the doublet observed in the experimental spectra of a nanocrystalline magnetic alloy for low temperatures (figure 32, left panel).

Figure 37. 57Fe Mössbauer spectra of magnetic nanoparticles (Hhf = 330 kOe, S = 1000) as calculated in the continuous diffusion model with D = (left panels) 0.01 and (right panels) 0.1 mm/s. (Solid lines) precession with the finite frequency Ω0/2π = 0.25 GHz and (Dashed lines) fast-precession limit (|Ω0| >> |ωe|, |ωg|) [42].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

67

As the relaxation rate increases, the qualitative features of the formation of the resulting hyperfine magnetic structure in the presence of the precession of uniform magnetization are significantly dispersed. However, the effect of the precession with finite frequency (102) on the shape of the spectra remains (figure 37, right panel) and must be taken into account in treating experimental data. When analyzing a series of spectra measured at different temperatures (see figure 32), one should take into account the temperature dependence not only on the effective energy barrier KV/kBT but also on the diffusion constant D, so that the fit of the experimental data allows the reconstruction of the temperature dependence of the latter parameter.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

4.6. Mössbauer Spectra of Nanoparticles in a Weak Magnetic Field In contrast to the conventional magnetization measurements that are widely performed as a function of both temperature and magnetic field (see Sections 1 and 2), the Mössbauer spectra of fine magnetic particles are usually collected as a function of temperature only [27, 28, 54-56, 78-80, 83-87]. In some cases, when the spectra are measured in a strong external magnetic field of about several Tesla, which is comparable with the strength of the hyperfine field, one can roughly estimate only the average saturation magnetization of the sample as a whole (Tronc et al [27], Hendriksen et al [59], Vasquez-Mansilla et al [60]). Meanwhile, a substantial influence of weak magnetic fields (of about kilooersted or less) on magnetic characteristics of small particles has been observed not only in magnetization curves [26, 48], but also in the field-dependent shape of Mössbauer spectra measured long ago (Eibschüts and Shtrikman [92], Lindquist et al [93]). Moreover, the Mössbauer spectra of modern nanostructured magnetic alloys display a diverse transformation of their shapes in radiofrequency magnetic field with the amplitude of about only ten Oersteds (Pfeiffer [68], Kopcewicz et al [94, 95], Hesse et al [96]). As an example, figure 38 shows typical 57Fe Mössbauer absorption spectra of nanoparticles of iron oxide in a polymer matrix, which were measured in weak magnetic fields (details of the preparation of the samples and measurements of the Mössbauer spectra can be found in Godovsky et al [97]. The spectra have been analyzed within the ‘universal’ theoretical approach for searching the hyperfine field distribution described in Section 4.1. Even not going into details, one understands that, in spite of a rather good fit with the experimental spectra, this approach describes only an effective magnetization of the studied ensemble of nanoparticles in a magnetic field, which is, of course, formal in character. In this situation, one should develop a theory for describing the magnetic dynamics in the sample studied and corresponding Mössbauer spectra in order to extract something more real from the experimental spectra. In order to develop a physical formalizm for describing the Mössbauer spectra of this kind, i.e. those of nanoparticles in a magnetic field, one can use the general theory of magnetic dynamics decribed in Section 3 and the relaxation approach from previous section, taking into account the most significant points in this changed situation, i.e., the complication of the form of the relaxation matrix specified by Eqs. (13) and the appearance of an inhomogeneous precession in the effective field Heff given by Eq. (9). It is not so difficult to write the general expression for the Mössbauer absorption spectrum in the representation given in Section 3. However, the main problems of analysis are associated with the

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

68

Mikhail A. Chuev and Jürgen Hesse

optimization of a calculation procedure, first of all, for calculation of the Liouville hyperfineinteraction superoperators [64, 65] (see also Section 5.1 and 5.3) under the average magnetization M ( E , Θ) changing continuously both in magnitude and in direction and the corresponding average hyperfine field H hf ( E , Θ) ∝ M ( E , Θ) for different orbits and Θ.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

That is why we consider here a simplified version of such a model, which allows one not only to analyze (in the first approximation) the experimental Mössbauer spectra of an ensemble of nanoparticles in a magnetic field, but also to make qualitative conclusions on specific shapes of the absorption spectra in this case.

Figure 38. (Left panel) 57Fe Mössbauer spectra (vertical dashes) of iron-oxide nanoparticles (mean size of about 4nm [97]) in a magnetic field H. The field direction is perpendicular to the gamma-beam. The resulting and partial spectra (solid lines) are calculated within the corresponding Gaussian hyperfine field distribution P(Hhf) shown on the right panel and an effective doublet of Gaussian-broaden lines (see Section 5.1).

At first glance, this task can be reduced to a formal specification of the generalized Stoner-Wohlfarth model based on Eqs. (18)-(21) to the case of the three-dimensional character of energy barrier (see figure 22) between the local energy minima. That is one can average the uniform magnetization over thermal fluctuations into each local energy minimum by analogy with the well-known approximation (Mørup [84]):

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

M i (T , Θ) =

69

E1(max) ( H ,Θ)

M ( E , Θ)Wi ( E , T , Θ)dE ∫ (min)

(122)

1 exp(− EV / kT ) ∫ dΩ . Zi C

(123a)

( H ,Θ)

Ei

where

Wi ( E , T , Θ) =

E

The integral in Eq. (123a) is taken over the solid angle along the corresponding trajectory CE and the normalization constants Zi are defined by the condition E1(max) ( H ,Θ)

∫ Wi ( E,T , Θ)dE = 1 .

(123b)

Ei(min) ( H ,Θ)

However, as clearly seen in figure 22, the region of stable rotation about the pole (max) corresponding to the absolute energy maximum E2 takes a substansial fraction of the stochastic space (points over the unit sphere), that is increasing with the strength of an external magnetic field. (When H > HC(θ), there remains a single energy minimum

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

characterized by the mean magnetization M1(T , Θ) .) That is why this region must be taken into account in developing the relaxation model for the description both the magnetic dynamics and corresponding Mössbauer spectra of an ensemble of single-domain particles. Thus, we come to the three-level relaxation model within the stochastic states characterized by the mean magnetizations M i (T , Θ) ( M 3 (T , Θ) being defined by Eq. (122) where the (max)

integration limits are E1

(max)

and E2

) and the equilibrium populations determined by the

obvious relation

Wi = Z i /( Z1 + Z 2 + Z 3 ) .

(124)

Assuming that the relaxation between the states occurs as a random walk of the vector M with small step lengths (rotation by a small angle), the transitions between the states corresponding to the local energy minima occur only through the state 3 corresponding to the absolute energy maximum (figure 22), i.e., p12 = p21 = 0 (see also Jones and Srivastava [41]). Then, the other relaxation rates are determined by the detailed balance principle: p3i= p0,

pi3 = p3i

W3 . Wi

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(125a) (125b)

70

Mikhail A. Chuev and Jürgen Hesse

Here, i = 1, 2 and p0 is a phenomenological constant specified by, e.g., the statistical characteristics of the random field h(t) defined by Eqs. (11) by analogy with that given by Brown [37], Jones and Srivastava [41], Chuev [42]). Thus, the three-level relaxation model is completely defined. The calculation of Mössbauer spectra within the three-level relaxation model can be performed in terms of the Anderson’s stochastic approach [98] according to which the absorption spectrum is described by the general expression (Afanas’ev et al [65], Chuev [99]):

(

σ Γ ˆ −1 (ω, Θ) 1 Vˆ + σ(ω, Θ) = − a 0 Im ∑ Sp Vˆη W A η 2 η

)

(126)

where Vˆη is the operator for the interaction of the gamma-quantum with a given polarization η and the nucleus, the superoperator

ˆ (ω , Θ) = ω − Lˆ (Θ) + iPˆ (Θ) + iΓ / 2 A hf 0

(127)

is defined by the Liouville operator of hyperfine interaction that is diagonal over the stochastic states:

i Lˆ hf (Θ) j = Lˆ Hˆ ( M i (Θ))δij

(128)

Pˆ = Pˆ ⊗ 1ˆ n ,

(129a)

Pij = − pij ,

(129b)

Pii = −∑ pij .

(129c)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

and the relaxation matrix

j ≠i

j ≠i

ˆ ˆ ( M (Θ)) acts in the space of (2Ig+1)(2Ie+1) nuclear variables Here, the superoperator L i H (Zwanzig [100]): (e ) (g ) (Lˆ Hˆ ) me mg me′ mg′ = H m δ − Hm δ , m′ mg mg′ m′ me me′ e e

g g

(130)

i.e., is determined by the Hamiltonians of hyperfine interaction for the ground and excited states, which in our case take the forms:

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles… (g,e) M i (Θ ) (g,e) (0) Iˆ ˆ Hi (Θ) = − gg,eμ N H hf M0

71

(131)

where H hf is the hyperfine field at extremely low temperature, and 1ˆ n is the unity superoperator in the space of nuclear variables. The resulting absorption spectrum of an ensemble of single-domain particles is defined by the sum over polarization η according to the scheme proposed by Chuev [99] and the averaging of partial spectra (126) over the chaotic distribution of the anisotropy axes: (0)

σ(ω) = ∫ σ(ω, Θ) sin ΘdΘ .

(132)

Using Eqs. (126)-(132) one can calculate the Mössbauer absorption spectrum in the framework of the three-level relaxation model (122)-(125) for given values of external (H and (0)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

T) and intrinsic ( H hf , K, V, M0 and p0) parameters. Figure 39 shows typical 57Fe Mössbauer absorption spectra calculated within the threelevel relaxation model for various values of the normalized external field h and the effective energy barrier KV/kBT. (Note that the rank of matrices in Eq. (127) in this case is equal to 24.) Along with the obvious effect of the resulting magnetization of the ensemble of nanoparticles in a field, a great diversity of the spectral shapes observed in the figure reflects a non-trivial character of the local and equilibrium magnetization at different values of temperature and field. The spectra in the negligible field (h = 0.01) display a slightly resolved magnetic hyperfine structure (sextet of lines) and a collapsed (due to relaxation) central line. In a stronger, but still weak field (h = 0.1) the magnetic hyperfine structure visually disappears and the spectra look like a doublet of lines with the splitting slightly depending on the effective energy barrier, i.e., on temperature and/or the particle’s size. The spectra of the magnetic nanomaterials often certainly exhibit high-temperature line doublets (see, e.g., the doublet D2 in figure 32), which are usually attributed to the presence of the quadrupole hyperfine interaction in the presence of the electric field gradient on the nucleus. However, such interpretation often seems very artificial; for example, such a ‘doublet’ is often realized at higher temperatures, but quadrupole splitting is completely absent in the spectra at lower temperatures. Such behavior suggests that the observed ‘doublets’ are likely of magnetic rather than electric nature, which is also justified by the present calculations where the quadrupole hyperfine interaction is not taken into account. With an external field increasing (h = 0.5 and 1) there again appears a resolved magnetic hyperfine structure on the background of the ‘doublet’ in the spectra. In a strong magnetic field (h = 2) the magnetic hyperfine structure becomes well resolved. Indeed, such a transformation of the spectral shape with a field changing is directly related to the longstanding and heated debates of two research groups on pages of authoritative journals (see Dormann et al [24], Mørup and Tronc [28], Hansen and Mørup [101], Dormann et al [23] and references therein). One of the groups contended that the presence of interparticle interaction, i.e., a mean (‘molecular’) field at each particle in the ensemble results in fastening the magnetization’s relaxation, whereas the other made the contrary assertion, i.e., ‘an applied field’ is slowing down the relaxation. Not going into details, note that all the arguments of

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

72

Mikhail A. Chuev and Jürgen Hesse

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

both the groups were based on the estimates of average relaxation time (the probabilities of transitions) between two local energy minima. However, the latter are obviously nonequivalent in a magnetic field and the probabilities of transitions from one state to the other are different, which defines local relaxation magnetic properties as a consequence of the asymmetric magnetization fluctuations resulting in specific shapes of Mössbauer spectra [31, 32, 88]. Nevertheless, the spectra shown in figure 39 manifest both the fastening (h = 0.1) and slowing down (h ≥ 0.5) relaxation in a magnetic field (or in the presence of the mean field interaction), which actually try on the conclusions of two groups.

( 0) hf = 500 kOe, p0 = 1 GHz) in a magnetic field, as calculated in the three-level relaxation model as a function of the effective energy barrier KV/kBT (from left to right) and the normalized field strength h (from top to bottom) (Chuev [73]). In order not to mix weak field and polarization effects, the calculations are performed for the case when the field direction composes the ‘magic’ angle (54.70) with the gamma-beam. Figure 39. 57Fe Mössbauer spectra of an ensemble of nanoparticles ( H

A qualitative character of such a behavior of the local magnetization can be clarified if one considers the limiting case of weak magnetic field (h 0) = exp⎜ − 2 ⎟ ⎜ 2σ V ⎠ ⎝

(135)

( 0) hf = 500 kOe, p0 = 3 GHz) calculated in the three-level relaxation model for the Gaussian distribution of the particle’s volume

Figure 41. 57Fe Mössbauer spectra of an ensemble of nanoparticles ( H ( KV

/ kBT = 1

and σ V

/ V = 0.5 ) in a magnetic field of different strength h (from top to bottom).

In order not to mix weak field and polarization effects, the calculations are performed for the case when the field direction composes the ‘magic’ angle (54.70) with the gamma-beam.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

75

The transformation of the spectra with field increasing qualitatively reproduces the evolution of the experimental spectra shown in figure 38. Moreover, an effective ‘doublet’ of lines in a weak magnetic field (h = 0.1) is clearly observed in the volume-averaged spectra due to slight dependence of its splitting on the effective energy barrier (particle’s size and/or temperature) as seen in figure 39. The formalism described above principally allows one to fit the experimental Mössbauer (0)

absorption spectra like those shown in figure 38 and to determine the parameters ( H hf , K,

V , σV, M0 and p0) inherent to the sample under investigation. However, this problem requires a separate, primarily qualitative analysis in order to optimize the corresponding calculation procedure and results will be publish elsewhere. Let us consider here one more qualitative point, namely, the Mössbauer lineshape for an ensemble of nanoparticles in the limiting case of high temperature. First, we consider the limiting case of a weak magnetic field (h > KV) when the regime of fast relaxation between the states 1 and 2 of local energy minima is realized, the hyperfine field Hhf over the nuclear lifetime is to be proportional to the mean magnetization of each particle with a given Θ:

M (Θ) = W1(Θ) M1(Θ) + W2 (Θ) M 2 (Θ) .

(136)

Then, the absorption spectrum (126) in the fast relaxation regime is defined by the mean Liouville operator of hyperfine interaction averaged over the stochastic states (Afanas’ev et al [65]):

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Lˆ hf (Θ) = W1(Θ)Lˆ H (M1(Θ)) + W2 (Θ)Lˆ H ( M 2 (Θ)) = Lˆ H ( M (Θ)) .

(137)

In this case the Mössbauer absorption spectrum of an ensemble of particles is determined by Eqs. (74)-(76) where

P ( H hf ) =

M0

( 0) (∂M (Θ) / ∂ cos Θ) H hf

.

(138)

A routine analysis of Eqs. (122)-(124) in this limiting case allows one to estimate the lower boundary for the mean magnetization as follows

M ( Θ) ≈ M 0

h2 x2 sin 2 Θ + cos 2 Θ , 4 9

(139)

where x is the parameter of the Langevin function [102] specified by Eq. (34). Eq. (139) defines the asymptotic tend of the local magnetization for each group of particles with the given Θ to a constant with temperature increasing in complete analogy with the asymptotic behavior of magnetization and susceptibility in the high-temperature limit given by Eqs. (45)

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

76

Mikhail A. Chuev and Jürgen Hesse

and (46) (Chuev [71]), which is different from the Langevin limit for ideal superparamagnetic particles (Bean [16]). Taking into account Eqs. (137)-(139), the Mössbauer absorption spectrum of an ensemble of particles is defined by Eqs. (74)-(76) with the hyperfine field distribution

P ( H hf ) =

1 ( 0) H hf

2

hhf 2

h /4 − x /9

h

2

2 / 4 − hhf

(140)

where non-zero values of P(Hhf) are realized in the interval

x / 3 < hhf =

H hf

( 0) H hf

< h/2.

(141)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The probability function (140) shows that the effective distribution P(Hhf) for h > 1) a single local energy minimum is realized for a SW particle so that one can estimate from Eqs. (122)-(124) the mean magnetization of each particle at high temperature (kBT >> HM0V):

⎛ x π KV ⎞ sin 2Θ ⎟⎟ . M (Θ) ≈ M 0 ⎜⎜ + ⎝ 3 16 kBT ⎠

(142)

Here, the first term in the brackets in Eq. (142) is the classical high-temperature limit of the Langevin function for ideal superparamagnetic particles, whereas the second term is a small correction for the magnetic anisotropy. The absorption spectrum of an ensemble of particles in this case is again defined by Eqs. (74)-(76) where the effective distribution P(Hhf) is ( 0) concentrated in the vicinity of H hf = H hf x / 3 and defines the shape of spectra shown in figures 39-41 (h = 2).

5. Mössbauer Spectra under Radiofrequency Magnetic Field Excitation Mössbauer spectroscopy under radiofrequency (rf) field excitation proved to be rather informative (Pfeiffer [68], Kopcewicz [94], Kopcewicz and Kotlicki [95], Hesse et al [96]),

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Non-equilibrium Magnetism of Single-Domain Particles…

77

providing a rich transformation of the spectral shape as a function of the rf field amplitude and frequency. Influence of an external rf field on magnetic nanomaterials results in changes in magnetic moments of both single particles and the sample as a whole so that the intrinsic non-equilibrium magnetic dynamics reveals in diverse forms of Mössbauer spectra as seen in figure 42. For a long time efforts of researchers were focused on studies of two major effects revealed in the Mössbauer spectra (Heiman et al [103], Pfeiffer et al [104], Heiman et al [105], Albanese and Asti [106], Baldokhin et al [107], Olariu et al [108], Kopcewicz et al [109, 111], Julian and Daniels [110], Graf et al [112, 113], Dzybulik [114], Vagizov et al [91]): the collapse effect and the sideband one found by Pfeiffer [68]. As for the former that is the collapse of the well-resolved hyperfine magnetic structure in the spectrum with no rf field into to a single central line in a strong enough rf field (see figure 42), there is a simple physical explanation: (i) the magnetization of the sample with soft enough magnetic properties is switched in the direction following the external magnetic rf field; (ii) because the hyperfine field vector at the nuclei is strongly coupled to the magnetic moments of the atoms, its direction also changes in response to the rf field; and (iii) if the rf-field frequency is higher than the nuclear Larmor frequency and the rf-field amplitude is strong enough to drive the magnetization into saturation, the magnetic hyperfine field at the nucleus is averaged to zero and the spectrum looks like a single line pattern. Along with the collapsed spectrum symmetric pairs of sidebands located at ω ≈ ±nωrf (ωrf is the rf-field frequency and n is integer) are observed, which are attributed to the magnetostriction and magnetization reversal effects [94-96, 103-114]. The rf collapse effect as well as the sidebands reveal in a clear form only in rf fields strong enough. In weak or intermediate fields, a very complex transformation of Mössbauer spectra is observed (figure 42), which principally allows one to get a lot of information about the non-equilibrium magnetic dynamics inherent to the sample studied from the spectra on conditions that a proper theory is developed for analyzing the spectra. Such a general theory that describes the transformation of Mössbauer spectra under excitation by rf fields of arbitrary frequency and strength has been recently developed in (Afanas’ev et al [64, 65, 67]). The main idea of the theory is that under the action of the rf field the hyperfine field becomes time-dependent, Hhf(t), as a result of complicated relaxation processes. In the simplest and physically clearest case when the relaxation processes are fast enough Hhf(t) follows the particle’s magnetization M(t) that varies in time under the action of the rf field given by Eq. (5). If a model of magnetic dynamics is given (see, e.g., Sections 2 and 3), M(t) and therefore Hhf(t) are known functions and the Hamiltonians of the system for the ground and excited states of a nucleus takes the following form:

Hˆ = Hˆ 0 + gg,e μ N Iˆ(g,e)H hf (t ) + VˆγN (t0 ) .

(143)

Here, H 0 is the Hamiltonian determining the nuclear energy levels, neglecting the hyperfine interaction and the operator VˆγN (t0 ) describes the interaction of a γ-ray with a nucleus, V(t< t0 ) = 0 according to the general theory of the resonant interaction of radiation with matter (Heitler [115]).

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

78

Mikhail A. Chuev and Jürgen Hesse

When a rf field is strong enough, the direction of magnetization of each particle stays parallel to the direction of the rf field and, assuming extremely fast relaxation, changes instantaneously with the direction of the rf field changing (see the step-like dependence in figure 43). In the simplest case the wave functions of the nucleus in the ground and excited states can be written as

⎤ ⎡ t ⎢ ψ mg (t , t0 ) = exp − ig g μ N mg ∫ H hf (t ′)dt ′⎥ mg ⎥ ⎢ t0 ⎦ ⎣

(144a)

and

⎤ ⎡ t E0 Γ0 ⎢ ψ me (t , t0 ) = exp − ig e μ N me ∫ H hf (t ′)dt ′ + (i − )(t − t0 )⎥ me . (144b) ⎥ ⎢ 2 t0 ⎦ ⎣

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Then, according to the general theory of radiation (Heitler [115]), the amplitude of the absorption of a γ-quantum can be expressed as

Figure 42. 57Fe Mössbauer spectra of the ferromagnetic Fe73.5CuNb3Si13.5B9 alloy in a radiofrequency (ωrf/2π = 60.8 MHz) magnetic field of different strength H0 = 0,…, 20 Oe (from bottom to top): (left panel) the as-quenched alloy, (central panel) the nanocrystalline alloy, (right panel) the alloy in the beginning of microcrystal formation (Hesse et al [96]).

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

79

Figure 43. Time dependence of the hyperfine field (dashed line) in a strong rf field (solid line). ∞

cmg me (ω ) = ∫ ψ m∗ e (t ) VˆγN (t 0 ) exp[iω (t − t 0 )] ψ mg (t ) dt t0

⎡t ⎤ ≡ Vmg me ∫ exp ⎢ ∫ iω mg me (t ′)dt ′ + iω~ (t − t 0 )⎥ dt ⎢⎣t0 ⎥⎦ t0 ∞

.

(145)

The square of the absolute value of the absorption amplitude defines the absorption crosssection that, within averaging over the switch-on time t0, takes the form (Afanas’ev et al [64]): σ Γ2 2 σ (ω ) = a 0 Cα ϕα (ω ) (146) 4 α where

∑

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

2 ϕα (ω ) = Re Trf

∞ ∞ ⎡t ⎤ ~ − ω (t ′)]dt ′⎥ , (147) ⎢ − Γ − dt dt exp[ ( t t )] dt exp i [ ω 0 1 0 ∫ α ∫ 0∫ 1 ⎢∫ ⎥ 0 t0 t1 t 1 ⎣ ⎦

Trf

Trf = 2π/ωrf is the rf-field period. The periodicity of the rf-field results in ωα(t) = ωα(t+Trf) and allows one to derive from Eqs. (146) and (147) an analytic expression for the absorption spectrum in the case when Hhf varies in time as shown in figure 43 (Afanas’ev et al [64]):

ϕα (ω ) = where

⎧ 1 + exp(iω~Trf ) ⎫ 1 Re⎨ [Fα (ω~) + Fα (−ω~)] + [Fα (ω~) − Fα (−ω~)]⎬ (148) ~ 4Γ0 ⎩ (1 − exp(iωTrf )) ⎭

Fα ( ±ω~ ) = G ± (ω~ , ωα ) + G ± (ω~ ,−ωα ) ,

G ± (ω~, ωα ) = ( ~ y / yα + 2) f ± ( ~ y + yα ) , f + ( x) = [1 − cos( x)] / x 2 , f − ( x) = i[x − sin( x)] / x 2 , ~ y = ω~Trf / 2 , yα = ωα Trf / 2 .

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

80

Mikhail A. Chuev and Jürgen Hesse

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Examples of the spectra calculated by Eq. (148) as a function of the radiofrequency are shown in figure 44, which demonstrates the collapse and the sideband effects in accordance with earlier results (Julian and Daniels [110], Dzyublik [114]).

Figure 44. 57Fe Mössbauer spectra (ω3/2,1/2 = 50 MHz) in a strong rf field with ωrf/2π = 21.1, 30, 36.8, 50, and 100 MHz (from bottom to top) [64].

5.1. Mössbauer Spectra under Rf Field Excitation within the StonerWohlfarth Model Another example of the deterministic magnetic dynamics is the SW model described in Section 1.2. In this model the magnetic moment of a particle changes its direction steadily, which results in arbitrary changes in the Hhf(t) direction so that the hyperfine interaction is defined by the Hamiltonians (143) without any restrictions. In this case the wave functions for the ground and excited nuclear states can be sought in the form

ψ (g ) (t , t 0 ) = cmg mg0 (t , t 0 ) mg

(149a)

ψ (e) (t , t0 ) = c~me me0 (t , t0 ) me .

(149b)

and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

81

Here and below summation over repeated indices is assumed. One can easily find from the Schrödinger equation that t ⎡ ⎤ ˆ c mg mg0 (t , t 0 ) = mg T exp ⎢− ig g μ N m ∫ H hf (t ′)dt ′⎥ mg0 . ⎢⎣ ⎥⎦ t0

(150a)

and t ⎡ ⎤ Γ E ~ ˆ cme me0 (t , t 0 ) = me T exp ⎢− ig e μ N me ∫ H hf (t ′)dt ′ + (i 0 − 0 )(t − t 0 )⎥ me0 . (150b) 2 t0 ⎣⎢ ⎦⎥

where Tˆ is the time-ordering operator. Taking into account the averaging over the initial states within the rf-field period, the absorption cross-section for particles with a given orientation (angle Θ) of the easy axis with respect to the rf-field direction can be expressed as

1 σ (ω , Θ) = Trf

Trf

∞

∫ ∑ ∫c 0 mg0 , me0 t0

2 ∗ me0 me

(t , t 0 )Vmemg cmg mg0 (t , t 0 )e

iω ( t −t0 )

dt dt 0 .

(151)

In order to transform this equation one can introduce the Liouville operator L H ( t ) that acts on an ordinary operator A(t ) according to the rule (Zwanzig [100]):

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

i

∂A(t ) = L H (t ) A(t ) = H (t ), A(t ) . ∂t

[

]

(152)

The explicit form of the superoperator is determined by Eq. (130) and for the 57Fe nuclei it is a matrix of rank 8. Using the superoperator (152), Eq. (151) can be reduced to a form similar to Eq. (147): σ (ω , Θ) =

Trf t ∞ ∞ ⎧⎪ ⎡ ⎤ ⎫⎪ 2 Re ∫ dt 0 ∫ dt1 exp[−Γ0 (t1 − t 0 )]∫ Tr ⎨Vˆ ⎢Tˆ exp{∫ i[ω~Iˆ − Lˆ Hˆ (t ′)]dt ′}⎥Vˆ + ⎬dt . (153) Trf t0 t1 t1 0 ⎦⎥ ⎪⎭ ⎩⎪ ⎣⎢

Since the rf field is periodic, averaging over t0 results in the following final expression for the absorption cross-section of particles with a given angle Θ in the SW model (Afanas’ev et al [64]):

σ (ω , Θ) =

Trf t1 +Trf ~ ⎡ ⎤ 2 ˆ −1 (t ) exp[iω (t − t1 )] G ˆ (t )Vˆ + ⎥ Re ∑ ∫ dt1 ∫ dtTr ⎢Vˆη G 1 η ˆ (T ) Γ0Trf Iˆ − exp(iω~Trf )G η 0 t1 rf ⎣⎢ ⎦⎥

where Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(154)

82

Mikhail A. Chuev and Jürgen Hesse t ˆ (t ) ≡ G ˆ (t , Θ) = Tˆ exp ⎧⎨ dt ′[−iLˆ ˆ (t ′, Θ)]⎫⎬ . G ∫ H ⎭ ⎩0

(155)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Averaging over different polarizations η of the incident radiation is also taken into account in Eq. (154). The total absorption cross-section for an ensemble of particles is obviously determined by Eq. (132). If the magnetic dynamics of the system is specified, i.e., M(t,Θ) and therefore Hhf(t,Θ) are known functions (see figure 6), the absorption spectra can be calculated directly by Eqs. (130), (132), (143), (154) and (155). Figure 45 shows typical Mössbauer spectra calculated within the SW model for different values of the rf-field amplitudes and frequencies. Apart from the well-pronounced collapse effect, a much more complicated shapes of the spectra are observed in intermediate fields (h0 ≤ 1): (i) an additional hyperfine structure of the central line and sidebands in a stronger field and (ii) splitting of the individual hyperfine components in a lower field with the resonance frequency ωrf =ωg = 36.8 MHz (figure 45, right) . Note that the latter has been clearly observed in the experiments (Vagizov et al [91]).

Figure 45. 57Fe Mössbauer spectra for an ensemble of SW particles (ω3/2,1/2 = 50 MHz) in a rf field with different frequencies ωrf/2π = 60 (left), 50 (center), and 36.8 (right) MHz and amplitudes h0 = 0.25, 0.5 0.75, 1, and 5 (from bottom to top) [64].

It is obvious that these peculiarities are mainly due to specific shapes of the partial magnetization curves for particles with non-zero angles Θ (figure 6), which result in

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles…

83

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

appearance of the perpendicular components of Hhf. In order to clarify these effects, let us consider two limiting cases.

Figure 46. Schematic of the energy level splitting for the excited and ground 57Fe nuclear states in a static hyperfine field (left part) and in a circularly polarized hyperfine field (right part) corresponding to the SW particles with Θ = 90º and h0 = 1.

When h0 = 1 and Θ = 90º, the hyperfine field Hhf(t) appears to be circularly polarized, i.e., it is described by Eq. (103) where θ = 90º and Ω = ωrf. In this case the absorption is determined by Eq. (108) where

~

2 λg,e = ω rf2 + ω g, e

(156)

~ and m g, e are the projections of nuclear spins onto the quantization axes for which the operators

~ˆ H ( g,e ) = ω rf Iˆz( g,e ) + ω g,e Iˆx( g,e )

(157)

are diagonal. As seen from Eq. (108), a nucleus with spin I under the excitation of a circularly polarized field behaves like a system with energy quasilevels given by Eqs. (106). The number of these quasilevels is equal to (2I+1)2. Taking into account the selection rules for the

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

84

Mikhail A. Chuev and Jürgen Hesse

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

M1 transition, in this case the absorption spectrum is a superposition of 24 doubly-degenerate lines with a Lorentzian shape. For high ωrf the lines separate into a central group consisting of six lines and side groups with a resolved hyperfine structure with 5, 3, and 1 lines with increasing from center. The corresponding scheme of the splitting of energy levels for the 57 Fe ground and excited states is shown in figure 46. This picture of more or less pronounced hyperfine splitting of both the central line and sidebands remains in a wide angular range near Θ = 90º and in a wide range of the rf-field frequencies. This explains the specific shape of the resulting Mössbauer spectra in a strong rf field shown in figure 45.

Figure 47. (Top) Schematic of the energy level splitting for the excited and ground 57Fe nuclear states in a weak rf field, h0 = 0.25, at resonance frequencies ωrf =ωg = 36.8 MHz (left) and ωrf =ωe = 21.1 MHz (right) and the corresponding 57Fe Mössbauer spectra for an ensemble of SW particles (ω3/2,1/2 = 50 MHz) in a rf field with different frequencies in the vicinity of resonances: ωrf =ωg,e + Δω, ωg,e, ωg,e - Δω (from top to bottom); Δω = 5 MHz.

The other qualitative effect, the resonant splitting of the hyperfine components on the right panel of figure 45, can be explained in the limiting case of an extremely weak rf field when h0 HC(Θ), for magnetic fields ⏐H(t)⏐ > HC(Θ) the particle can be found only in one absolute minimum, so that the relaxation is absent. However, one can use the general Eq. (170) to calculate the

ˆ (t , t ′) in the time interval (t, t') where ⏐H(t)⏐ > HC(Θ) if he assumes that the function G elements of the P ( t ) operator are equal to zero and the hyperfine interaction is given by

L(t ) = L H (t ) ⊗ 1e

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(174)

Non-equilibrium Magnetism of Single-Domain Particles…

89

where 1e is the identity operator in the space of energy states. In a time interval (t,t') where H(t) < HC(Θ) and H(t') > HC(Θ), there is a single point in time, tC, at which H(tC) = HC(Θ). According to the original SW model, the particle’s magnetic moment undergoes an instantaneous rearrangement in this point and the relaxation process is prescribed as the final result: in whatever state the particle was found at t < tC, at t > tC it should be found in the state

ˆ (t , t ′) in the time interval can be represented in the form (1). Then, the operator G ˆ (t , t ′) = G ˆ (t , t ) Rˆ G ˆ G C 1 (t C , t )

(175)

where the projection operator is introduced: ⎛ 1 0⎞ R1 = ⎜ ⎟ ⎝ 1 0⎠

(176)

In the region of negative values of the rf field, where H(t) > −HC(Θ) and H(t') < −HC(Θ),

ˆ (t , t ′) can be calculated by the same Eq. (175) in which the operator R must the operator G 1 be replaced by the projection operator ⎛ 0 1⎞ R2 = ⎜ ⎟. ⎝ 0 1⎠

(177)

Using Eqs. (169)-(177) one can calculate the Mössbauer absorption spectrum in the framework of the extended SW model for given values of the rf-field frequency and (0)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

amplitude and parameters ( H hf , K, V, M0 and p0) inherent to the system studied. Figure 48 displays the Mössbauer absorption spectra calculated in the original and extended SW for different rf-field amplitudes. In both the models, the collapse effect is observed. However, the collapse in the extended SW model occurs at considerably smaller rf-field amplitudes as compared to those in the original model, i.e., the relaxation introduced results in an effective decrease in the magnitude of the critical field. Moreover, The spectral lines are observed to be broadened in the extended SW model and their width grows with the rf-field amplitude. It is very important for a quantitative and more accurate description of the experimental spectra like those shown in figure 42.

5.4. Relaxation-Stimulated Resonances in Mössbauer Spectra under Rf Field Excitation As for qualitative effects, they can be expected to appear only in the region of the transition from a resolved hyperfine structure to the collapsed spectrum. Indeed, direct calculations of the spectra within the extended SW model by Eqs. (169)-(177) show that specific spectral shapes as a function of the rf-field frequency are observed in the region.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

90

Mikhail A. Chuev and Jürgen Hesse

Figure 49 shows the corresponding spectra for representative values of the rf-field frequency in the vicinity of

ωrf = ω1= ω3/2,1/2

(178)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

where ω3/2,1/2 is the frequency corresponding to the spectral positions of the outermost lines of the static magnetic sextet. As seen from the figure, when the resonance condition (178) is hold, the outer lines appear to be much narrower than the inner ones and, consequently, their peak intensity is strongly enhanced. With the rf-field frequency detuning from the resonance toward either side, the line widths and ratio of intensities are restored. This effect is preserved in both the partial spectra and those of the entire ensemble of particles (figure 48).

Figure 48. 57Fe Mössbauer spectra for an ensemble of particles (ω3/2,1/2 = 50 MHz) in a rf field (ωrf/2π = 75 MHz) as calculated within the original (left panel) and extended (right panel, U0/kBT = 20 and 2πp0/ωrf = 104) SW models for different values of the rf-field amplitude.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Non-equilibrium Magnetism of Single-Domain Particles…

91

Figure 49. 57Fe Mössbauer spectra of SW particles (ω3/2,1/2 = 50 MHz, U0/kBT = 20 and 2πp0/ωrf = 104) with orientation Θ = 45º (left panel) and an ensemble of particles (right panel) in a rf field with amplitude h0 = 0.2 and frequency ωrf = ω1 + Δω, ω1, ω1 − Δω (from top to bottom). Δω/2π = 5 MHz.

A qualitatively different effect is observed in figure 50 where the absorption spectra of SW particles with orientation Θ = 45º are calculated in the case when the rf-field frequency is located in the vicinity of

ωrf ≈ 2ω1.

(179)

when the resonance condition (179) is valid, the outer lines of the sextet appear to be split whereas the inner lines remain not split and their shape does not essentially depend on the rffield frequency. The resonance splitting of the outer lines is preserved even in the case of small detuning of the rf-field frequency from the resonance one, but becomes asymmetric (figure 50). It is clear that this resonance transformation of the spectral shape is principally different from the resonance effects shown in figure 47. The latter are ‘conventional in the sense that they are realized at the resonance frequencies corresponding to a real splitting between nuclear sublevels whereas the former are realized at the resonance frequencies specified by the resonance condition (179) and being a combination of the frequencies of the hyperfine transitions, which exceed the actual splitting of nuclear levels in the ground and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

92

Mikhail A. Chuev and Jürgen Hesse

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

excited states by ten orders of magnitude. In spite of this circumstance, the resonant rf field sets as if in coherency between the hyperfine sublevels of the nucleus in the ground and excited states. These resonant effects can be observed only when relaxation processes play an essential role in the sample’s magnetization reversal. Due to this very reason the resonances specified by Eqs. (178) and (179) have been called as the relaxation-stimulated resonances (Afanas’ev et al [116, 117]). In order to clarify the physical origin of the non-standard resonance effects one should simplify the model of magnetic dynamics. The simplest model of the kind is the one-way and localized relaxation introduced by Afanas’ev et al [116]). The principal assumptions of the model are the following. (i) When a magnetic field is applied, there are different energy barriers for the downward and upward transitions (figure 8). The energy barrier for the upward transition increases with the field strength increasing and, hence, such transitions can be disregarded. Relaxation processes of the kind can be called as the one-way relaxation. (ii) On the other hand, for the downward transition the energy barrier U21 decreases with the field strength increasing, and the most intense relaxation processes will occur at maximum values of the rf-field strength, i.e., at moments

Figure 50. 57Fe Mössbauer spectra of SW particles (ω3/2,1/2 = 50 MHz, U0/kBT = 20 and 2πp0/ωrf = 104) with orientation Θ = 45º in a rf field with amplitude h0 = 0.2 and frequency ωrf = ω2 + Δω, ω2, ω2 − Δω (from top to bottom). ω2 = 99 MHz and Δω/2π = 5 MHz.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles… tk = (2k +1)Trf/4

93 (180)

where k is integer. Therefore, at high U0 = KV the relaxation processes can be taken into consideretion only in the short time intervals near tk so that the resulting effect of the relaxation can be described completely by integral characteristics of the relaxation process, r and q, defined by (Afanas’ev et al [116]: tk ⎡ ⎤ ⎢ r = exp − 2 ∫ p21 (t )dt ⎥⎥ , q = 1 − r. ⎢ ⎣ t k − Trf / 4 ⎦

(181)

Here r determines the probability for a particle to stay in the same state in passing the point tk and q is the probability to change its state in passing this point. This model actually describes a localized relaxation. (iii) The rf-field amplitude is not large (h0 Γ0 the spectrum is a superposition of narrow and broad lines, which reveals as a sharp increase in the peak intensity for these lines that has been demonstrated by means of numerical calculations (see figures 49 and 51). With ωrf detuning from the exact resonance the width of the first line increases and that of the second line decreases so that at |Δω| = 2γ they becomes equal and with |Δω| increasing further the line widths do not change. Qualitatively different behavior occurs for the odd resonance. In this case the value of λ is always real and the widths of both the lines are equal to each other. However, as seen from Eqs. (186) and (187), there is always a splitting between two lines,

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles… Δ12 = 2γ,

97 (194)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

which clearly reveals in figures 50 and 51. In accordance with the parametric resonance condition (185) the resonance effects described above reveal themselves not only for the outermost spectral lines but also for the resonant frequencies ωrf corresponding to the inner lines as well as for the resonant frequencies of higher orders n, which are shown in figure 52. Moreover, the relaxationstimulated resonances revealed in Mössbauer spectra of nanoparticles under rf field excitation should undergo low-frequency shift upon applying a weak static magnetic field (Afanas’ev et al [117]). The latter extends the possibilities for experimental observation of the predicted effects, because tuning to the resonance may be accomplished not only by changing frequency of the rf field but also by varying the amplitude of the alternating field and the strength of the static field. A natural question arises whether the effects predicted could be observed in real situations, for instance, when a sample consists of particles with random orientation of their easiest magnetisation axes and the hyperfine field at nucleus can change smoothly in its direction. Calculations performed within general Eqs. (169)-(177) show that for such systems the characteristic features of the even resonances remain to a considerable extent whereas the odd ones are essentially smoothed down and in order to observe the latter textured samples is to be prepared. This means that textured samples are required in order to observe the later behaviour.

Figure 52. 57Fe Mössbauer spectra (ω3/2,1/2/2π = 50 MHz) calculated within the model of one-way and localized relaxation for q = 0.5 in a rf magnetic field with the frequency ωrf/2π = 54.5 MHz near the doubled frequency ω1/2,1/2 (a), ωrf = ω1/2,1/2 = 2π×29 MHz (b), ωrf = 2ω3/2,1/2/3 = 2π×33 MHz (c), ωrf = ω3/2,1/2/2 = 2π×25 MHz (d) (Afanas’ev et al [116]).

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

98

Mikhail A. Chuev and Jürgen Hesse

With this topic we will close the part considering Mössbauer spectra collected on magnetic nanoparticles. It was demonstrated that a large variety of different shapes of spectra can appear. Mostly these spectra are relaxation determined. Therefore, the Mössbauer spectroscopists should first check if in their measurements relaxation spectra appear. There exist some ideas how to check objectively the character of spectra measured. An old idea of Hesse and Hagen [118] is based on the fact that if only a hyperfine field distribution is responsible for the shape of the spectrum then the lines of the sextet are similar in shape and only shifted and broadened proportional to their resonance velocity position. A more sophisticated method exploits the idea of Mössbauer line sharpening (Afanas’ev and Tsymbal [119], Tsymbal et al [120]). The topic dedicated towards the evaluation of Mössbauer spectra collected on magnetic nanoparticles should be considered in more detail in future.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

6. Conclusion The extension of the Stoner-Wohlfarth model by means of a more accurate description of relaxation processes within Néel’s ideas results in the derivation of a universal differential equation which holds for the populations of the SW energy levels. Solution of this equation with proper chosen initial conditions allows one to describe quantitatively a number of magnetization phenomena observed experimentally as a function of temperature, time and external magnetic field. So, hysteresis loops including also those in high-frequency external magnetic fields can be calculated within this model as a function of temperature as well as demagnetisation curves for arbitrarily heating rates in different external magnetic fields can be simulated. Various shapes of temperature dependence of magnetization for different cooling regimes, field strengths, and temperature scanning rates obviously contains a large amount of information on the real and significant physical parameters (e.g., the anisotropy energy density K, the saturation magnetization M0, particles volume V, and the relaxation rate p0) and their distribution over the sample studied. To estimate these parameters, it is necessary to know a model of the magnetic dynamics of the entire ensemble for describing the whole body of the experimental data for this sample. The extended SW model just delivers this opportunity instead of a ‘conventional’ way when researchers restrict themselves with qualitative estimates based primarily on the Néel formula (6), which are used to make physical conclusions. Such estimates are rough and artificial, because, for instance, the most popular parameter, blocking temperature Tb, is determined not only by the properties of the sample under investigation, but also by the external parameters of the measurement method (external field strength, temperature scanning rate, prehistory of the sample, etc.). Moreover, even if the external parameters are unchanged, the identity or difference between the Tb values for different samples means nothing, because these values are determined by at least three true physical parameters, K, V, and p0. The new ideas of the extended SW model describing the magnetic behaviour of nanoparticles magnetic systems now must consequently be applied to other experimental methods, i.e. ferromagnetic resonance, neutron scattering etc. In this contribution we focused our attention to Mössbauer spectrometry. So a theory of relaxation Mössbauer spectra taken on an ensemble of nanoparticles is presented. This theory is principally based on the general equations of stochastic relaxation given in previous sections so that the models of magnetic dynamics can be used for calculating the corresponding hyperfine interaction and for

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Non-equilibrium Magnetism of Single-Domain Particles…

99

numerical analyzing the Mössbauer spectra including those taken in a weak static magnetic field. As for the later, the theory based on the Brown equation is reduced in the first approximation to a three-level relaxation model which continues the line of ‘physically oriented’ phenomenological models of the classical two-level relaxation in the absence of a field introduced by Wickman and the extended Stoner-Wohlfarth relaxation in a field. Finaly, a theory of Mössbauer spectra under radiofrequency magnetic field excitation is described, which is also based on the extended Stoner-Wohlfarth model. Along with a qualitative explanation of the effects observed in the experimental Mössbauer spectra of magnetic nanoparticles under rf field excitation, a principally new shapes of the spectra are predicted, which are originated from the relaxation-stimulated resonance processes at the rf-field frequencies coupled by parametric resonance conditions with the Larmor frequencies in the ground and excited nuclear states. Principally, a number of experimental methods probing magnetic dynamics independent of their characteristic time window can be applied to the same sample, and treatment of the whole set of experimental data collected in the different techniques within the same model of magnetic dynamics must give a lot of information about the system studied, even about so complex as a system of magnetic nanoparticles. Note that the extended SW model presented above can be easily modified to describe the interacting single-domain particles in the framework of the mean field approximation, which will significantly expand its application area for analyzing experimental data. It is also clear that a real system containing nanoparticles is always inhomogeneous so that one should inevitably take into account distributions of physical parameters inherent to the system (e.g., particles size distribution) in a treatment of the experimental data within any model of magnetic dynamics. Such a model should also include somehow the temperature dependence of the uniform saturation magnetization M0(T) characterizing each single particle. However, the most serious problem is that in agreement with the basic assumptions of the SW model the continuous diffusion and precession of the particle’s magnetic moment are also not taken into account in the generalized SW model. This will limit the application of the latter in analysing the experimental data and more general models of magnetic dynamics can essentially correct the results of numerical analysis within the generalized SW model. Nevertheless, the main advantage of the latter is its simplicity in numerical calculations so that one can easily use it to fit the experimental data and qualitatively estimate the most principal physical parameters inherent to the sample studied as well as to get a first approximation for further analysis in the framework of more advanced (and much more complicated) models of magnetic dynamics. In this chapter we have also presented the experimental and theoretical evidences for the existence of the asymptotic high-temperature behavior of the magnetization of an ensemble of single-domain particles in weak magnetic fields, which is different from the classical Langevin limit for ideal superparamagnetic particles. It is shown that the physical cause of the constant term in the high-temperature asymptote (45) is the temperature-independent slant of the magnetizations vectors corresponding to the local energy minimain in the field direction. The major contribution to this term is made by the particles with large angles between the easy axis and the field direction whereas the particles with the easy-axis direction close to the field direction mainly contribute to the ‘Langevin’ high-T magnetization. The very existence of the non-trivial high-T asymptotic behavior is established to be directly related to a nonconventional thermodynamics of each particular particle with the equilibrium magnetization

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

100

Mikhail A. Chuev and Jürgen Hesse

determined by not only the Gibbs distribution over the stochastic states of the particle’s uniform magnetization vector, but also by the ratio of characteristic frequencies of the regular precession and random diffusion of the vector. The high-T and low-field magnetization asymptote (45) should radically change the conventional approach for treating the measurements of high-temperature magnetization and Mössbauer spectra of nanoparticles, which is based exceptionally on the ‘Langevin’ limit. In order to describe the magnetic dynamics of an ensemble of single-domain particles in a magnetic field the alternative approach has been proposed within the general theory of stochastic relaxation of the uniform magnetization. In this approach the magnetization precession orbits are considered as stochastic states of each particle, the states (orbits) being characterized by the mean magnetization value determined by the elliptic integrals along the corresponding trajectory. It is shown that within the approach the probabilities of transitions per unit time between the stochastic states can be defined by the statistical characteristic of the random field. This allows a general model of magnetic dynamics to be determined for calculating the magnetic characteristics in various measurement methods and analyzing numerically the experimental data. The technique for calculating the equilibrium magnetization of an ensemble of nanoparticles has been developed on the base of the differential Braun’s equation taking into account inhomogeneous precession and isotropic diffusion of the uniform magnetization. This extends essentially the overview of the magnetism of single-domain particles, which demonstrates different types of high temperature behavior and smooth transfer between them.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Acknowledgments We are grateful to Prof. A.M. Afanas’ev for longstanding and very fruitful combined work. We express our thanks also to Drs. T. Graf, H. Bremers, O. Hupe, O. Michele and V.M. Cherepanov for performing many magnetization and Mössbauer spectra measurements. One of the authors (MCh) thanks the Russian Foundation for Basic Research for financial support (project No. 08-02-00388). The authors are very thankful to Professor Stewart Campbell for accepting the duty of referring this contribution, for many valuable hints and lingual improvements.

References [1] [2] [3] [4] [5]

Stoner, E. C.; Wohlfarth E. P. Phil. Trans. R. Soc. A 1948, 240, 599-642; IEEE Trans. Magn. 1991, 27, 3475-3518. Néel, L. Ann. G´eophys. 1949, 5, 99–136. Chuev, M. A.; Hesse, J. J. Phys.: Condens. Matter 2007, 19, 506201, 1-18. Birringer, R.; Gleiter, H.; Klein, H.-P.; Marquardt, P. Physics Letters A 1984, 102, 365369. Hempelmann, R. special issue of “Zeitschrift für Physikalische Chemie” 2008/2-3, “Progress in Physical Chemistry”, Oldenbourg Wissenschaftsverlag, Vol. 2, pp 1-377.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles… [6]

[7] [8] [9] [10] [11]

[12]

[13] [14] [15] [16] [17] [18]

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[19] [20] [21] [22] [23] [24] [25] [26] [27]

[28] [29] [30]

101

Sumiyama, K.; Hihara, T.; Peng, D. L.; Yamamuro, S. In Encyclopedia of Nanoscience and Nanotechnology; Nalwa, H.S.; Ed.; American Scientific Publishers: New York, 2004; Vol. 10, pp. 471-507. O’Handley, R. C. Modern Magnetic Materials: Principles and applications; Wiley: New York, 2000. Pankhurst, Q. A.; Connolly, J.; Jones, S. K.; Dobson, J. J. Phys. D: Appl. Phys. 2003, 36, R167-R181. Shinkai, M.; Ito, A. Adv. Biochem Engin./Biotechnol. 2004, 91, 191–220. Hergt, R.; Dutz, S.; Müller, R.; Zeisberger, M. J. Phys.: Condens. Matter 2006, 18, S2919-S2934. Hütten, A.; Sudfeld, D.; Ennen, I.; Reiss, G.; Hachmann, W.; Heinzmann, U.; Wojczykowski, K.; Jutzi, P.; Saikaly, W.; Thomas, T. J. Biotechnology 2004, 112, 4763. Behrens, S.; Bönnemann, H.; Matoussevitch, N.; Gorschinski, A.; Dinjus, E.; Habicht, W.; Bolle, J.; Zinoveva, S.; Palina, N.; Hormes, J.; Modrow, H.; Bahr, S.; Kempter, V. J. Phys.: Condens. Matter 2006, 18, S2543-S2561. Salgueirino-Maceira, V.; Correa-Duartec, M. A.; Huchtd, A.; Farle, M. J. Magn. Magn. Mater. 2006, 303, 163-166. Bizdoaca, E. L.; Spasova, M.; Farle, M.; Hilgendorff, M.; Liz-Marzan, L. M.; Carusob, F. J. Vac. Sci. Technol. A. 2003, 21, 1515-1518. Schlachter, A.; Gruner, M. E.; Spasova, M.; Farle, M.; Entel, P. Phase Transitions 2005, 78, 741-750. Bean, C. P. J. Appl. Phys. 1955, 26, 1381-1383. Williams, H. D.; O’Grady, K.; El Hilo, M. J. Magn. Magn. Mater. 1993, 122, 129-133. Wickhorst, F.; Shevchenko, E.; Weller, H.; Kötzler, J. Phys. Rev. B 2003, 67, 224416, 1-11. Weili, L.; Nagel, S. R.; Rosenbaum, T. F.; Rosensweig, R. E. Phys. Rev. Lett. 1991, 67, 2721-2724. El-Hilo, M.; Shatnawy, M.; Al-Rsheed, A. J. Magn. Magn. Mater. 2000, 221, 137-143. Chantrell, R. W.; Walmsley, N.; Gore, J.; Maylin, M. Phys. Rev. B 2000, 63, 024410, 1-14. Vieira, S. R.; Nobre, F. D.; da Costa, F. A. J. Magn. Magn. Mater. 2000, 210, 390-402. Dorman, J. L.; Fioriani, D.; Tronc, E. J.Magn. Magn. Mater 1999, 202, 252-267. Dormann, J. L.; Bessais, L.; Fiorani, D. J. Phys. C: Solid State Phys. 1988, 21, 20152034. Kleemann, W.; Petracic, O.; Binek, Ch.; Kakazei, G. N.; Pogorelov, Yu. G.; Sousa, J. B.; Cardoso, S.; Freitas, P. P. Phys. Rev. B 2001, 63, 134423, 1-5. Hansen, M. F.; Jönsson, P. E.; Nordblad, P.; Svedlindh, P. J. Phys.: Condens. Matter 2002, 14, 4901-4914. Tronc, E.; Ezzir, A.; Cherkaoui, R.; Chanéac, C.; Nougés, M.; Kachkachi, H.; Fiorani, D.; Testa, A. M.; Grenèche, J. M.; Jolivet, J. P. J. Magn. Magn. Mater. 2000, 221, 6379. Mørup, S.; Tronc, E. Phys. Rev. Lett. 1994, 72, 3278-3281. Wickman, H. H. In Mössbauer Effect Methodology; Gruverman, I. J.; Ed.; Plenum Press: New York, 1966; Vol. 2, pp 39-66. Mørup, S. J. Magn. Magn. Mater. 1983, 37, 39-50.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

102

Mikhail A. Chuev and Jürgen Hesse

[31] Afanas’ev, A. M.; Chuev, M. A. JETP Lett. 2001, 74, 107-110. [32] Chuev, M. A.; Hupe, O.; Afanas’ev, A. M.; Bremers, H.; Hesse, J. JETP Lett. 2002, 76, 558-562. [33] Michele, O 2004, PhD Thesis, Technische Universit¨at Braunschweig , (http://opus.tubs.de/opus/volltexte/2005/646) [34] Michele, O.; Hesse, J.; Bremers, H.; Polychroniadis, E. K.; Efthimiadis, K. G.; Ahlers, H. J. Phys.: Condens. Matter 2004, 16, 427-443. [35] Rellinghaus, B.; Stappert, S.; Acet, M.; Wassermann, E. F. J. Magn. Magn. Mater. 2003, 266, 142-154. [36] Jones, D. H.; Srivastava, K. K. P. J. Magn. Magn. Mater. 1989, 78, 320-328. [37] Brown, W. F. Jr. Phys. Rev. 963, 130, 1677-1686. [38] Landau, L. D.; Lifshitz, E. M. Phys. Z. Sowjetunion 1935, 8, 153-169. [39] Gilbert, T. Phys. Rev. 1955, 100, 1243(A). [40] Afanas’ev, A. M.; Sedov, V. E. Sov. Phys. Dokl. 1986, 31, 651-654. [41] Jones, D. H.; Srivastava, K. K. P. Phys. Rev. B 1986, 34, 7542-7548 [42] Chuev, M. A. JETP Lett. 2006, 83, 572-577. [43] Aharoni, A. Phys. Rev. 1969, 177, 793-796. [44] Schrefl, T. J. Magn. Magn. Mat. 1999, 207, 45-65. [45] Bauer, M.; Fassbender, J.; Hillebrands, B.; Stamps, R. L. Phys. Rev. B 2000, 61, 34103416. [46] Sun, Z. Z.; Wang, X. R. Phys. Rev. B 2005, 71, 174430, 1-9. [47] van Lierop, J.; Ryan, D. H. Phys. Rev. B 2001, 63, 064406, 1-8. [48] Jönsson, P.; Hansen, M. F.; Nordblad, P. Phys. Rev. B 2000, 61, 1261-1266. [49] Wernsdorfer, W.; Thirion, C.; Demoncy, N.; Pascard, H.; Mailly, D. J. Magn. Magn. Mater. 2002, 242-245, 132-138. [50] Michele, O.; Hesse, J.; Bremers, H.; Wojczykowski, K.; Jutzi, P.; Sudfeld, D.; Ennen, I.; Hütten, A.; Reiss, G. Phys. Stat. Sol. C1 2004, 12, 3596-3602. [51] Michele, O.; Hesse, J.; Bremers, H. J. Phys.: Condens. Matter 2006, 18, 4921-4934. [52] Cador, O.; Grasset, F.; Haneda, H.; Etournea, J. J. Magn. Magn. Mater. 2004, 268, 232-236. [53] Du, J.; Zhang, B.; Zheng, R. K.; Zhang, X. X. Phys. Rev. B 2007, 75, 014415, 1-7. [54] Suzuki, K.; Cadogan, J. M. Phys. Rev. B 1998, 58, 2730-2739. [55] Hupe, O.; Chuev, M. A.; Bremers, H.; Hesse, J.; Afanas'ev, A. M. Nanostruct. Mater. 1999, 12, 581-584; J. Phys.: Condens. Matter. 1999, 11, 10545-10556. [56] Balogh, J.; Bujdoso, L.; Kaptás, D.; Keméni, T.; Vincze, I.; Szabó, S.; Beke, D. L. Phys. Rev. B 2000, 61, 4109-4116. [57] Predoi, D.; Kuncser, V.; Tronc, E.; Nogues, M.; Russo, U.; Principi, G.; Filoti, G. J. Phys.: Condens. Matter. 2003, 15, 1797-1811. [58] Stankov, S.; Sepiol, B.; Kanuch, T.; Scherjau, D.; Würschum, R.; Miglierini, M. J. Phys.: Condens. Matter. 2005, 17, 3183-3196. [59] Hendriksen, P. V.; Bødker, F.; Linderoth, S.; Wells, S.; Mørup, S. J. Phys.: Condens. Matter. 1994, 6, 3081-3090. [60] Vasquez-Mansilla, M.; Zysler, R. D.; Arciprete, C.; Dimitrijewits, M. I.; Saragovi, C.; Greneche, J. M. J. Magn. Magn. Mater. 1999, 204, 29-35. [61] Miglierini, M.; Schaaf, P.; Skorvanek, I.; Janickovic, D.; Carpene, E.; Wagner, S. J. Phys.: Condens. Matter 2001, 13, 10359-10369.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Non-equilibrium Magnetism of Single-Domain Particles… [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76]

[77] [78] [79]

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94]

103

Bean, C. P.; Livingston, J. D. Suppl. J. Appl. Phys. 1959, 30, S120-S129. Vonsovski, S. V. Magnetism; Nauka: Moscow, 1971; Wiley: New York, 1974. Afanas’ev, A. M.; Chuev, M. A.; Hesse, J. JETP 1998, 86, 983-992. Afanas’ev, A. M.; Chuev, M. A.; Hesse, J. JETP 1999, 89, 533-546. Aharoni, A. Phys. Stat. Sol. 1966, 16, 3-42. Afanas’ev, A. M.; Chuev, M. A.; Hesse, J. Phys. Rev. B 1997, 56, 5489-5499. Pfeiffer, L. J. Appl. Phys. 1971, 42, 1725-1726. Eder F. X. Moderne Messmethoden der Physik; VEB Deztscher Verlag der Wissenschaften: Berlin; 1968. Hesse, J.; Bremers, H.; Hupe, O.; Veith, M.; Fritscher, E. W.; Valtchev, K. J. Magn. Magn. Mater. 2000, 212, 153-167. Chuev, M. A. JETP Lett. 2007, 85, 744-750. Chuev, M. A. JETP Lett. 2008, 87, 707-709. Chuev, M. A. J. Phys.: Condens. Matter 2008, 20, (to be published). Afanas’ev, A. M.; Chuev, M. A. JETP Lett. 2003, 77, 415-419. Afanas’ev, A. M.; Chuev, M. A. J. Phys.: Condens. Matter 2003, 15, 4827-4839. Goldanskii, V. I.; Makarov, E. F. In Chemical applications of Mössbauer spectroscopy; ed Goldanskii, V. I.; Herber, R. H.; Eds.; Academic Press: New York–London; 1968; pp 74-101. Miglierini, M.; Greneche, J.-M. Hyperfine Interact. 1999, 121, 297-301. Miglierini, M.; Grenèche, J.-M. J. Phys.: Condens. Matter. 1997, 9, 2303-2319. Miglierini, M.; Skorvanek, I.; Grenèche, J.-M. J. Phys.: Condens. Matter. 1998, 10, 3159-3176. Hernando, A. J. Phys.: Condens. Matter. 1999, 11, 9455-9482. Hesse, J.; Rübartsch, H. J. Phys. E: Sci. Instr. 1974, 7, 526-532. Chuev, M. A.; Hupe, O.; Bremers, H.; Hesse, J.; Afanas'ev, A. M. Hyperfine Interact. 2000, 126, 407-410. Reid, N. M. K.; Dickson, D. P. E.; Jones, D. H. Hyperfine Interact. 1990, 56, 14871490. Mørup, S. J. Magn. Magn. Mater. 1983, 37, 39-50; Hyperfine Interact. 1994, 90, 171185. Tronc, E.; Prené, P.; Jolivet, J. P.; d’Orazio, F.; Lucari, F.; Fiorani, D.; Godinho, M.; Cherkaoui, R.; Noguès, M.; Dormann, J. L. Hyperfine Interact. 1995, 95, 129-148. Dormann, J. L.; D’Orazio, F.; Lucari, F.; Tronc, E.; Prené, P.; Jolivet, J. P.; Fiorani, D.; Cherkaoui, R.; Noguès, M. Phys. Rev. B 1996, 53, 14291-14297. Kemény, T.; Kaptás, D.; Balogh, J.; Kiss, L. F.; Pusztai, T.; Vincze, I. J. Phys.: Condens. Matter 1999, 11, 2841-2847. Aksenova, N. P.; Chuev, M. A. Russ. Microelectron. 2005, 34, 279-294. Belozerskii, G. N.; Pavlov, B. S. Sov. Phys. Solid State 1983, 25, 974-980. Afanas’ev, A. M.; Chuev, M. A. Dokl. Phys. 2003, 48, 277-282. Vagizov, F. G.; Manapov, R. A.; Sadykov, E. K.; Zakirov, L. L. Hyperfine Interact. 1998, 116, 91-104. Eibschüts, M.; Shtrikman, S. J. Appl. Phys. 1968, 39, 997-998. Lindquist, R. H.; Constabaris, G.; Kündig, W.; Portis, A. M. J. Appl. Phys. 1968, 39, 1001-1003. Kopcewicz, M. Phys. Status Solidi A 1978, 46, 675-685.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

104

Mikhail A. Chuev and Jürgen Hesse

[95] Kopcewicz, M.; Kotlicki, A. J. Phys. Chem. Solids 1980, 41, 631-633. [96] Hesse, J.; Graf, T.; Kopcewicz, M.; Afanas’ev, A. M.; Chuev, M. A. Hyperfine Interact. 1998, 113, 499-506. [97] Godovsky, D. Yu.; Varfolomeev, A. V.; Efremova, G. D.; Cherepanov, V. M.; Kapustin, G. A.; Volkov, A. V.; Moskvina, M. A. Adv. Mater. Opt. Electron. 1999, 9, 87-93. [98] Anderson, P. W. J. Phys. Soc. Jpn. 1954, 9, 316-339. [99] Chuev, M. A. JETP 2006, 103, 243-263. [100] Zwanzig, R. Physica 1964, 30, 1109-1123. [101] Hansen, M. F.; Mørup, S. J. Magn. Magn. Mater. 1998, 184, 262-274. [102] Langevin, P. J. Phys. 1905, 4, 678-693. [103] Heiman, N.; Pfeiffer, L.; Walker, J. C. Phys. Rev. Lett. 1968, 21, 93-96. [104] Heiman, N.; Walker, J. C.; Pfeiffer, L. Phys. Rev. 1969, 184, 281-284. [105] Pfeiffer, L.; Heiman, N.; Walker, J. C. Phys. Rev. B 1972, 6, 74-89. [106] Asti, G.; Albanese G.; Bucci C. Phys. Rev. 1969, 184, 260-263. [107] Baldokhin, Yu. V.; Borshch, S. A.; Klinger, L. M.; Povitsky, V. A. Sov. Phys. JETP 1972, 36, 374-378. [108] Olariu, S.; Popesku, I.; Collins, C. B. Phys. Rev. C 1981, 23, 50-63. [109] Kopcewicz, M.; Gonser, U.; Wagner, H.-G. Nucl. Instrum. Methods Phys. Res. 1982, 199, 163-167. [110] Julian, S. R.; Daniels, J. M. Phys. Rev. B 1988, 38, 4394-4403. [111] Kopcewicz, M.; Jagielski, J.; Graf, T.; Fricke, M.; Hesse, J. Hyperfine Interact. 1994, 94, 2223-2227. [112] Graf, T.; Kopcewicz, M.; Hesse, J. Nanostruct. Mater. 1995, 6, 937-945. [113] Graf, T.; Kopcewicz, M.; Hesse, J. J. Phys.: Condens. Matter 1996, 8, 3897-3901. [114] Dzyublik, A. Yu. Phys. Status Solidi B 1986, 134, 503-513. [115] Heitler, W. Quantum Theory of Radiation; Clarendon: Oxford; 1954; pp 136-203. [116] Afanas’ev, A. M.; Chuev, M. A.; Hesse, J. J. Phys.: Condens. Matter 2000, 12, 623635. [117] Afanas’ev, A. M.; Chuev, M. A.; Hesse, J. JETP Lett. 2001, 73, 519-523. [118] Hesse, J.; Hagen E. Hyperfine Interact. 1986, 28, 475-478. [119] Afanas’ev, A. M.; Tsymbal, E. Yu. Hyperfine Interact. 1990, 62, 325-342. [120] Tsymbal, E. Yu.; Afanas’ev A. M.; Fricke, M.; Hesse, J.; Parak, F. Z. f. Phys. B 1994, 94, 217-222. Refereed by Stewart J Campbell, Professor, Physics Discipline, School of Physical, Environmental and Mathematical Sciences, The University of New South Wales Australian Defence Force Academy, Canberra, ACT 2600, AUSTRALIA

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

In: Magnetic Properties of Solids Editor: Kenneth B. Tamayo, pp. 105-143

ISBN: 978-1-60741-550-3 © 2009 Nova Science Publishers, Inc.

Chapter 2

MAGNETIC AND ELECTRONIC STRUCTURE MODIFICATIONS INDUCED BY SURFACE SEGREGATION IN LA0.65PB0.35MNO3 THIN FILMS C.N. Borca1, Hae-Kyung Jeong2 and Takashi Komesu3 1

Paul Scherrer Institut, Villigen PSI CH-5232, Switzerland 2 Center for Nanotubes and Nanostructured Composites, Sungkyunkwan University, Suwon 440-746, Korea 3 RIKEN Harima Institute, Hyogo 679-5148, Japan

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Abstract This paper presents an extended study of the La0.65Pb0.35MnO3 thin films, emphasizing the interplay between surface composition and surface electronic structure. We have found that annealing treatments can easily modify the surface composition. A gently annealed La0.65Pb0.35MnO3 surface is ‘soft’, with a surface effective Debye temperature close to Pb single crystal surface, in agreement with the extended Pb segregation in the surface region. The spin-asymmetry of this surface reaches 80% at 0.5 eV above the Fermi level at room temperature. A heavily annealed surface has a reduced Pb segregation and a slightly higher surface Debye temperature, indicating a stiffer lattice, but showing strong evidence of layer restructuring. The polarization of this restructured surface reaches 40% at 0.5 eV above the Fermi level, in the center of the Brillouin zone. We conclude that the surface segregation, consistent with a difference in free enthalpy between the surface and the bulk, is induced by annealing treatments. This surface segregation greatly reduces the spin-polarization near surface. The extreme changes in structure and composition of surface may have a great impact on magnetic and electronic surface structure properties that are very important in improving the tunnel magnetoresistance and spin valve performance.

1. Introduction The first studies of lanthanum manganites originate from the early 1950’s [1]. Since then, the interest in compounds which displayed so-called giant magnetoresistance (GMR) has

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

106

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

grown considerably owing to the possibility of technological applications. La1-xPbxMnO3 was first synthesized and characterized by Searle and Wang in 1969 [2]. This ferromagnetic compound crystallizes in a nearly cubic structure, with a rhombohedral distortion which is mainly due to rotations of MnO6 octahedra along each (111) axis, as shown in figure 1. The point group (R3c), one of the most common in perovskite systems [3], is determined in reference [4]. La0.65Pb0.35MnO3 is a ferromagnetic material, with a Curie temperature of 335-350 K [2] and a metal-insulator phase transition occurring at about the same temperature. The interplay among the charge, spin and orbital degree of freedoms, result in fascinating electronic and magnetic properties in all members of lanthanum-perovskite type materials [5]. While the parent compound LaMnO3 is an A-type (layered type) antiferromagnetic insulator and Mn3+ has an ionic configuration t2g3 eg1, the valence of the Mn ions becomes a mixture of Mn3+ and Mn4+ (t2g3 eg0) after doping. The antiferromagnetic phase is weakened as the doping level increases. The ferromagnetic phase emerges in the doping range of 0.2 < x < 0.5 for the Pb- doped compound [2].

Figure 1. Cubic R3c structure of La0.65Pb0.35MnO3. Atoms are shown in red-O, green-La/Pb and in purple-Mn octahedra.

To describe the magnetic and transport properties of the doped perovskites, one usually adopts the “double – exchange” model [6]; the strong on-site coupling, together with the crystal-field, split the spin-up and spin-down electron bands. The t2g electrons are almost localized while eg electrons are itinerant because of hopping. The electronic transport is determined by the hopping of an eg electron of an Mn3+ ion into the unoccupied eg orbital of neighboring Mn4+ ion and the effective hopping parameter is proportional to the inner product of the neighboring Mn moments. Millis et al. [7] argued, however, that the double-exchange model alone may not be enough and the effect associated with Jahn-Teller splitting of the d states could be an important factor. The transport property in doped perovskites can be affected both by the electron-phonon interaction and by strong coupling to the local magnetic moments [8]; this indicates that the spin disorder scattering in these compounds may result in the change of resistivity or even in a metal-nonmetal transition. While there is no specific band structure calculation for Pb-doped perovskite, several groups have calculated the band structure of La0.65Sr0.35MnO3 [9-11], a material that is closely

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Magnetic and Electronic Structure Modifications…

107

similar in electronic and structural properties with La0.65Pb0.35MnO3. The Fermi level lies just above a gap in the minority density-of-states, while there is no gap in the majority electronic states, indicating that the system is very close to being half-metallic (theoretically, 100% spin polarization – that is, metallic for the majority-spin electrons but insulating for the minorityspin electrons). Indeed there is now considerable evidence that La0.65Sr0.35MnO3 is not at all half-metallic even in the bulk [12-13]. Several recent studies are focused on determining the spin-polarization in doped lanthanum perovskites. To our knowledge, there was only one spin-resolved photoemission study performed on La0.7Pb0.3MnO3 [14], which shows a maximum 20% polarization at 10 K. Most of the studies were performed on Sr-doped perovskite and involve spin–polarized tunneling measurements. Using La0.7Sr0.3MnO3 as a spin-polarized electrode, Soulen et al. found 78% polarization at 1.6 K [15], Viret et al. measured 83 % at 4.2 K [16] and more recently, J.-H. Park et al. obtained 100% spin asymmetry at 40 K [17]. There is good reason to doubt that these high values of polarization do not translate to applicable to a device [1820]. Considering the problems outlined by Mazin in reference [21] of defining and calculating the degree of spin polarization in tunneling measurements, the best proof of half metallicity remains the investigation of spin-asymmetry in angle-resolved photoemission and/or inverse photoemission, two surface sensitive techniques. Due to the increased surface sensitivity of these techniques, the expected bulk properties predicted by band calculations may not be observed if the composition is modified in the surface region. Therefore, a thorough investigation of the surface composition is needed before any spin-resolved studies. In the present work, the focus is on characterizing the surface composition and understanding the changes that may be induced by annealing treatments in both physical and electronic structures. The results presented in this study may have a great impact on improving the performances of tunnel magneto-resistive (TMR) devices. Applications of such devices are envisaged for magnetic memories, as they should lead to faster read-write access time. If the junction impedance is low enough, optimized noise properties could allow TMR systems to become competitive with the best giantmagneto-resistance (GMR) devices currently used. We employed several techniques in order to better understand the behavior of the various La0.65Pb0.35MnO3 surfaces: atomic force microscopy, low energy electron diffraction, angle resolved x-ray photoemission, soft X-ray magnetic circular dichroism and spin polarized inverse photoemission spectroscopy. For comparison, magnetoresistivity, X-ray and neutron diffraction were used for investigating the bulk properties.

2. Thin Film Deposition The La0.65Pb0.35MnO3 thin films were deposited on (100) LaAlO3 substrates via the RF magnetron sputtering method. The targets were made from homogeneous powders of La2O3, MnO2 and PbO, prepared by citric acid sol-gel process. The solution was pre-sintered at 900ºC for 12 hours to form a stable phase and then sintered at 1000ºC for 24 hours in order to obtain a dense target. The deposition system with a base pressure of 10-7 torr was filled with a mixture of two gases, Ar and O2 in a 4:1 ratio, up to a pressure of 23 mtorr. The target-tosubstrate distance was around 2 inches. Films with a thickness of 1000 Å have been deposited on-axis, at a power of 15 W for 90 min. The best films have been deposited using a single

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

.

108

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

crystal of LaAlO3, kept at a temperature between 200 to 300ºC. Higher deposition temperature resulted in a random orientation of the films. The (100) LaAlO3 substrate was first cleaned and the surface examined with STM, as shown in figure 2. A slight miscut from the (100) plane creates atomic flat terraces that can be observed across the entire substrate surface, with a maximum roughness of 10 Å.

Figure 2. STM figure of LaAlO3 substrate (color scale goes from 0 to 3nm).

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(a) MFM

(b) AFM

200 nm

Figure 3. (a) MFM and (b) AFM images of a 1 μm thick La0.65Pb0.35MnO3 film. Color range in (b) ranges from 0 to 5nm.

As the freshly prepared films were amorphous, the perovskite phase was formed after annealing first at 650 oC for 10h (in Ar), then at 850 oC for 2h (in Ar), and last, at 650 oC for 10 h (in 5 psi O2). The surface morphology was examined using atomic force microscopy (AFM) combined with magnetic force microscopy (MFM). The AFM image (figure 3b) shows a 5 nm roughness rms for the 1000 Å thick La0.65Pb0.35MnO3 films. The crystallite size ranges from 50-150 nm and is aligned at 45° from the [100] direction of the LaAlO3 substrate.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

109

The MFM scan (figure 3a) shows the presence of complicated ferromagnetic domains structure, in which the domain walls are more likely to be cross-tie walls, rather than Bloch or Neel walls. It has been observed that the domain walls are more stable in thicker films. The magnetic tip was magnetized along the out-of-plane direction (perpendicular to the film surface). The conclusion that can be drawn from figure 3 is that the magnetic domain size is much larger than the crystallite size.

3. Bulk Properties of the Thin Films The post-deposition thermal treatments greatly influence the magnetic properties of the thin film. Figure 4 shows the magnetization curves (in a field of 1 T) for a 1000 Å film treated at different temperatures for 10 hours, in oxygen atmosphere. Both magnetization and Curie temperatures increase with increasing the annealing temperature. Microstructure measurements indicate that the films have not yet crystallized well. The film annealed at 850ºC for 10 hours has the highest magnetization value (78 emu/g) and a Curie temperature as high as 370 K. 80

850 K 800 K 750 K 700 K

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

M (emu/g)

60

40

20

0 0

50

100

150

200

250

300

350

400

Temperature (K)

Figure 4. The curves of magnetization versus annealing temperature for a 1000 Å thick film (annealed at different temperatures for 10 h).

The crystalline structure was studied by means of θ-2θ X-ray diffraction, using Cu Kα radiation. The diffraction spectrum shown in figure 5, contains the (100) and (200) reflections for both the LaAlO3 substrate and the deposited La0.65Pb0.35MnO3 film, with a slight misfit between the lattice constants of about 2.3%. No extra phases can be observed even on logarithmic scale. The two peaks are indexed using the cubic perovskite structure, with a lattice constant of 3.87 Å. The thin film diffraction peaks are very sharp - comparable to those of the LaAlO3 single crystal - indicating a high degree of crystallinity along the (100) direction.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

10

20

200

100

Intensity

LaAlO3

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

LaAlO3

110

30

40

50

60

2 θ (degrees)

Figure 5. X-ray diffraction pattern (θ - 2θ scan) for 1000 Å thick film of La0.65Pb0.35MnO3.

The temperature dependencies of resistivity for the 1000 Å thick La0.65Pb0.35MnO3 film measured in no applied magnetic field, as well as with an applied filed of 5.5 T, are shown in figure 6. The zero field resistivity shows a maximum at 351 K very close to the Curie temperature (354 K). The value of resistivity at 351 K is 4.8 mΩcm close to the value of 5 mΩcm, obtained for a single crystal of La1-xPbxMnO3 with x=0.4 [22].

ρ(m Ω -cm)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

5

0T

0.5

4

5.5 T

0.4

3

0.3

2

0.2

1

0.1

0

0

100

200

Temperature (K)

300

400

Magnetoresistance (%)

0.6

6

0.0

TC

Figure 6. Temperature dependencies of resistivity (dashed lines) measured at 0 T and 5.5 T magnetic fields respectively. The continuous line represents the calculated magneto-resistance (%) on the right vertical axis.

The resistivity ρ extrapolates to a small value (0.2 mΩcm) at T=0 K, which is typical for single crystals or epitaxial films [23]. The peak of the ρ-T curve shifts to higher temperature

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

111

in 5.5 T applied magnetic field, indicating magneto-resistive behavior. The magnetoresistance MR= Δρ/ρ(0,T)=-[ρ(H,T)-ρ(0,T)]/ρ(0,T) on the right vertical axis of figure 6 is plotted as continuous line. The curve displays a sharp peak around 320 K and a negative magnetoresistance of 47% in an applied magnetic field of 5.5 T. In the low temperature region, up to 250 K, the resistivity ρ drops by 90%, as the neighboring spins become aligned and magnetization approaches its saturation value.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

4. Surface Composition Surface studies are essential in explaining physical behavior of heterojunction devices that are based on spin-dependent tunneling mechanism [24]. Such studies are always performed in ultra-high-vacuum (UHV) in order to maintain a consistently well characterized, well defined and clean surface. In the case of thin films of perovskites, one procedure for cleaning the surface in UHV consists of repeated annealing treatments under low energy electron bombardment exposure to stimulate the desorption of light contaminants (such as C and O) [25]. As we will further demonstrate, differences in surface preparation lead to different surface structure and stoichiometric conditions, particularly for surfaces of complex materials. The La0.65Pb0.35MnO3 thin film, of nominal thickness of 1000 Å, has been installed in a UHV system with a base pressure of 2×10-10 torr. In order to obtain a clean surface, we first annealed the samples by resistive heating at 250ºC in an oxygen atmosphere of 1×10-6 torr. At the same time, the surface was exposed to low energy electrons an electron gun, which impinge the surface at an angle of 30 degrees. A standard Omicron electron gun was used for the cleaning the surface contaminants as well as for structural, low energy electron diffraction (LEED) analysis, presented below. The surface contaminants, mainly carbon, were reduced to a minimum, as characterized using core level photoemission. The surface cleanness was also verified using LEED, and a clean surface was seen to exhibit a sharp diffraction pattern at the lowest achievable electron energy (below 50 eV kinetic energy, when the electrons are diffracted by the first surface layers). After 10-15 hours of cleaning treatments, we obtained a very fragile surface, with a four-fold symmetric surface structure and a negligible amount of carbon. Figure 7 shows the influence of oxygen treatments at 250ºC annealing temperature. The LEED pictures were acquired using identical working conditions for the LEED gun (1.1 mA emission current and 7 kV screen accelerating voltage) while the sample was kept in the same position. It is obvious that after the oxygen treatment (right panels in figure 7), the surface order is improved and therefore, sharper and additional diffraction spots can be observed for the images recorded at the incident electron energies of 68 and 106 eV. After careful analysis of this gently annealed surface, the sample was further annealed in UHV at 520ºC for several days, using resistive heating. This second thermal treatment produced a heavily annealed surface with totally different composition and structure from the gently annealed one, as discussed below (vide infra).

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

112

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

Figure 7. LEED images acquired at room temperature, immediately after a thermal treatment of 1 hour at 250ºC performed in vacuum (left images) and in an atmosphere of 1×10-6 torr of oxygen (right images). Three different electron energies were used in order to show that oxygen deficiencies are reduced not only at the surface, but also deeper in the selvedge layers.

Figure 8 shows the comparison of the surface topography for the two types of surfaces, obtained using a scanning tunneling microscope (STM) [26]. The surface of the La0.65Pb0.35MnO3 thin films before the UHV studies shows a roughness of 5 nm for a 1000 Å nominal thickness (panel 8a). We can infer that the as-deposited surface is generally very

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

113

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

smooth for such perovskite structures, with atomic flat terraces, transmitted from the oxidized (100) LaAlO3 substrate (shown in figure 2) to the film’s surface. The heavily annealed surface has a different topography and a surface roughness of 50 nm. The STM image in figure 8b is similar in appearance to a TEM picture obtained on La0.67Sr0.33MnOx films, before post-deposition treatments [27]. The image shown in reference [27] indicates that the surface of the Sr-doped thin film has two phases, a columnar MnO phase embedded into a Ruddlesden-Popper (RP) phase (La,Sr)n+1MnnO3n+1, which is characterized by a bilayer structure. This new phase has n = 1 stacking layer sequence or more exactly, one Mn – O layer followed by two consecutive layers of La/Pb – O, as shown later in figure 14.

Figure 8. STM images of sample surface prior to (a) and after (b) UHV heavy annealing treatment performed on a sample with nominal thickness of 1000 Å. Surface roughness increases 10 times and the atomically flat terraces are not seen in image (b) compared to image (a) (adapted from [26]).

Figure 9 (a) presents the LEED pattern for the gently annealed surface at a temperature of 190 K and at 106 eV incident electron energy. The heavily annealed LEED pattern, acquired at the same temperature and at 107 eV electron energy, is shown in figure 9 (b). The appearance of sharp superlattice spots in the case of the heavily annealed surface may indicate a four-fold modulation in the MnO planes. Temperature dependence of the diffraction pattern in figure 9 (b) showed that the supperlattice spots appear only at temperatures below 250 K. Such a reconstruction could be characteristic of a charge ordering state, with an ordering temperature of about 250 K [28]. An apparent increase of 16% in lattice constant can be observed in figure 9, from approximately 3.8 Å to 4.4 Å. The value of 4.4 Å is characteristic of the cubic MnO phase lattice parameter [27]. At this point we cannot clearly identify whether the apparent reconstruction is characteristic only to the columnar MnO phase, or is also extending over the RP matrix.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

114

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 9. LEED patterns acquired at 190 K for gently (a) and heavily (b) annealed surfaces, using electron beam energies of 106 eV and 107 eV, respectively. The presence of supperlattice spots is evident for the restructured surface, in image (b) (adapted from [26]).

It has been proven that core level X-ray photoemission spectra can provide rich information on the local electronic structure of the atom studied including intrashell electron correlations, charge transfer and covalency or ligand coordinations [29]. However, the information derived from these spectra is not, in general, trivial because of the strong Coulomb interaction between the core-hole and the valence electrons. If the interaction between the core-hole created by the photoelectron and the correlated valence electrons is sufficiently strong, satellites accompanying the main lines appear in the photoemission spectra. Examples of such multiplet splitting of the main photoemission peaks can be found in intermetallic uranium compounds [30] or in late transition metal halides [31]. To excite the core level photoelectrons, we used an Mg anode to generate X-rays from impact of 15 keV electrons. The main Mg-Kα1,2 line obtained is 1253.6 eV, with full width at half maximum of 0.65 eV and a satellite line at 1262.9 eV (Mg-Kα3,4 line, ΔE=9.3 eV). With 9% of the intensity of the Mg-Kα1,2 line, the satellite does not contribute much to the overall line width, and perturbs the spectra only slightly. For monitoring the photoelectron intensities as a function of their binding energies, we have used an hemispherical electron energy analyzer from Physical Electronics (PHI Model 10-360 Precision Energy Analyzer), with 8.5’’ radius, 2’’ working distance, 4×10 mm2 analysis area, acceptance angle of ± 10 degrees and one channel detector. The transmission function of this particular analyzer varies with the square root of the photoelectron kinetic energies, as determined in reference [32]. Core level photoemission data from La-3d, Pb-4f and Mn-2p core levels as a function of emission angle are shown in figure 10 [33]. As stated above, we see enhanced satellite features for the La 3d core levels that indicate a strong Coulomb interaction (or charge transfer) between the La 3d core electrons and La 4f valence electrons. There is a general agreement in the literature that the double peak structure of each spin-orbit split component reflects states with configurations 3d9 4f 0 and 3d9 4f 1 L, where L denotes the oxygen ligand (O – 2p) and underscoring denotes a hole [32]. In other words, one side of the split peak

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

115

corresponds to the La atom in an oxide environment while the other contribution comes from the interaction with the 4f state. However, there has been considerable debate about whether the f 1 configuration corresponds to the higher binding energy peaks, and hence an excitation results in an energy loss, or to a well-screened lower binding energy state. The atomic-like models appear to be too simplistic in order to explain the strong final state mixing [35]. This result arises because the energy difference between the La 4f and O 2p levels is comparable to the Coulomb energy between the La 3d core hole and the 4f electron [34].

La-3d Intensity (arb.units)

3d3/2

3d5/2

Pb-4f

Mn-2p

4f7/2

2p3/2

4f5/2 2p1/2

0

0

0

0

0

20 0 40 0 x2 60

0

0

0

x2

0

20 0 40 0 60 x2

860 850 840 830 148 144 140 136 132

20

0

40

0

60

656 652 648 644 640

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Binding Energy (eV) Figure 10. The X-ray photoemission spectra of the La 3d, Pb 4f and Mn 2p core levels. The spectra were taken at RT as a function of emission angle of the photoelectron, as noted on the side of each panel (adapted from [33]).

Satellites do not occur in the Pb 4f core levels because of negligible interactions between the 4f cores and the higher energy valence electrons. The Pb – 4f cores in figure 10 correspond only to the oxide environment of the Pb atom. The Mn 2p core levels are also not split, as expected for a Mn atom in an octahedral coordination (e.g., Mn in oxides) [29]. The La 3d, Pb-4f and Mn-2p core levels have the same shape and binding energy values for both the gently and heavily annealed surfaces and, in addition, they closely reproduce the spectra obtained from similar CMR perovskites [36]. The independence of the full-width-at-halfmaximum (FWHM) of the three cores with emission angle provides little evidence of surface La, Pb or Mn species that differ substantially from the bulk. The oxygen 1s core level spectra for both types of surfaces are presented in figure 11 as a function of emission angle. All spectra exhibit two dominant peaks, similar to those observed not only in different CMR perovskites [36], but also in high-TC superconductors [37] and in spinel oxides [38]. The binary manganese oxides show just one dominant O 1s peak, characteristic of the lattice O2- ions and comparable to the lower binding energy peak of the perovskite materials. Thus, the lower binding energy O 1s state corresponds to a well-ordered

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

116

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

oxygen lattice. The higher binding energy O 1s peak, found at ~ 532 eV in perovskites, is related to either a different surface oxide or to oxygen associated with defects in the perovskite structure. The exact nature of the oxygen species giving rise to this 532 eV feature has not yet been determined. It has been proposed, however, that this 532 eV oxygen 1s core level feature, caused by metal vacancies and/or interstitials, is associated in general with lattice defects in oxides [37-38]. When the near surface distribution of the two different O 1s species was probed by changing the XPS emission angle (figure 11a), the higher binding energy species is seen to emerge from the outermost surface layers mainly for the gently annealed surface. While both types of surfaces are influenced by Pb segregation (as later presented), the gently annealed surface seems to be terminated in “PbO-like” layers while the heavily annealed surface appears to be mainly MnO terminated.

Intensity (arbitrary units)

0

0

20

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

60

532

530

528

0

0

40

534

Emission Angle (degrees)

(b)

(a)

20

0

x2

0

526534

0

532

530

528

0

40

0

60

0

526

Binding Energy (eV) Figure 11. Oxygen 1s core level as a function of emission angle, with respect to the surface normal, for the gently annealed (a) and for the heavily annealed (b) surfaces.

A shift in binding energy with emission angle in the case of core levels of heavy elements, as Pb for example, is a clear evidence of a valence state modification in the surface atoms compared to the bulk atoms. The binding energy of the Pb 4f7/2 core level is plotted in figure 12 as a function of emission angle for both gently and heavily annealed surfaces. The Pb 4f7/2 core for both types of surfaces changes with emission angle, increasing binding energy as the Pb atoms are found closer to the surface. These binding energies are greater than those observed for the Pb2+ state seen in PbTe (137.3 eV [39]), PbS (137.5 eV [39]) and PbO (137.4 eV [40]). The higher binding energies for the Pb 4f7/2 at greater emission angles indicate an effectively higher oxidation state of Pb in the surface region consistent with a concomitant lower oxidation state of Mn in the surface region. Semiempirical calculations based on the angle - resolved XPS data can be successfully applied for miscible substitutional binary alloys [41]. Quantitative estimation of surface

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

117

segregation has been already performed in Sr-doped [42] and Ca-doped [43] perovskites. The terminal layer identification procedure presented in [42-43], is not directly applicable to LPMO structure as surface stoichiometry was altered in the post-deposition thermal treatments. Therefore, we consider Pb segregation for both the gently and the heavily annealed surfaces separately.

gentle annealed heavily annealed

Binding Energy Pb 4f 7/2 (eV)

138.4

138.2

138.0

137.8

137.6

0

10 20 30 40 50 Emission Angle (degrees)

60

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 12. The Pb 4f7/2 binding energy plotted as a function of emission angle at 300 K for gently (z) and heavily () annealed surfaces.

In the simplest model we can assume an exponential segregation profile [41]. Let A and B denote the two constituents of a binary alloy, with A chosen as the element that segregates to the surface. The atomic fraction of element A, fj(A), for the j-th layer can be written as:

⎛ jd ⎞ ⎟ f j = b + δ exp⎜⎜ − ⎟ ⎝ G ⎠,

(1)

where b is the bulk fraction for element A. The parameters δ and G represent the segregation at the top most layer and the segregation depth expressed in units of the distance d between components, respectively. These two quantities are also the fitting parameters when comparing the model with the experimental values. From the knowledge of the profile form fj, one can calculate the apparent surface concentration (or relative intensity) of element A for a particular core level. The experimental core level intensities for the two components are acquired at several emission angles, usually from θ = 0º (normal emission) to 60º (off-normal emission). Then, a linear background contribution is systematically subtracted from each raw spectrum. The

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

118

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

peak intensities are further normalized by the corresponding differential cross section for emission and by the analyzer transmission function. The normalized intensity ratio is thus given by:

R

exp

p ⎡ I ( A) / σ A ⎤ E kin ( A) − C (θ ) = ⎢ ⎥ p ⎣ I ( B ) / σ B ⎦ E kin ( B) − C ,

(2)

where θ is the emission angle with respect to the surface normal, σA, σB are the cross sections (using calculations by Scofield in reference [44]) and the term (Ekinp− C) corrects for the transmission of the electron energy analyzer at the kinetic energy of the core level A. Based on the measured transmission functions for a PHI 10-360 hemispherical analyzer [45] we have set p = 0.5 and C = 0. Using the simple model outlined above, the theoretical ratio can be written as: ∞

∑

R theor (θ ) =

j =0 ∞

∑

j =0

jd ⎞ ⎛ ⎟ f j ( A) exp⎜⎜ − ⎟ ⎝ λ A cos θ ⎠ jd ⎛ ⎞ ⎟ f j ( B) exp⎜⎜ − ⎟ λ cos θ B ⎝ ⎠,

(3)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

f ( B ) = 1 − f ( A) = 1 − b − δ exp(− jd / G )

j j is the atomic fraction for where constituent B, and λA, λB are the mean free paths of the photoelectrons corresponding to the monitored core levels. Photon absorption processes that might influence the intensity of emitted electrons are neglected, as the absorption length of the photons is much bigger than the electron’s mean free path. The theoretical fits Rtheor(θ ) to the experimental ratios Rexp(θ ) performed as a function of emission angle often have no unique solutions due to distortions introduced by the forward scattering processes [25]. In order to improve the fits we need to restrict the validity of the model to layered systems with more than one atomic species per layer. In this case we can construct more than one experimental intensity ratio and still use only two fitting parameters (δ and G). The theoretical values of λ have been tabulated in reference [46] as a function of core level kinetic energy (Ekin), but can also be estimated. Assuming that the chemical composition of the host material is uniform in the surface region, the mean free path - λ - due to core electron excitation is given by:

λ =

E kin

a (ln E kin + b )

,

(4)

where a, b depend on the electron concentration of the host material as well as on the core levels of the constituent host atoms and Ekin represents the kinetic energy of the emitted

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

119

photoelectrons. Unfortunately, the quantities a and b depend on the dielectric function ε of the host, which is not known for non-free-electron materials like the CMR compounds. As best approximation, we used an average value of λ, obtained by considering different host materials for the same impurity atom, as calculated in Table 1. For example, an Mn 2p photoelectron, with 611 eV kinetic energy in an Mn host material has a mean free path of 8.75 Å, in a La host the mean free path becomes 13.4 Å and in a Pb host is 10.41 Å. The Mnphotoelectron mean free path we used therefore is the average value of λMnavg = 10.85 Å. Table 1. Approximation of the theoretical escape depths for the Mn, La and Pb, obtained by taking the average of the respective impurity mean-free-paths in different pure host materials Element Mn La Pb

Ekin(eV) 611 416 1115

a 19.7 9.47 13.2

λ in Mn(Å) 8.75 6.682 13.65

b -2.87 -1.6 -1.97

λ in La(Å) 13.4 9.916 21.73

λ in Pb(Å) 10.41 7.762 16.73

λavg (Å) 10.85 8.12 17.37

The Pb/La and Mn/(La+Pb) ratios thus obtained have been plotted as a function of emission angle for both gently and heavily annealed surfaces, as seen in figure 13 [26]. The intensity ratio of the Pb to the La core level increases as a function of the increasing emission angle. Qualitatively, this indicates that the surface region is Pb-rich. At each emission angle, each spectrum was analyzed by first subtracting a linear background, then dividing by the correspondent cross section value, adapted from Scofield calculations [44] (σLa = 44.74, σPb = 22.88, σMn = 13.62 barns). 3.5

1.8

(a) Pb / La

(b) Mn / (La+Pb)

Ratio I(Pb)/I(La)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

1.4

2.5

gently annealed heavily annealed

2.0

1.2 1.0

1.5

0.8

1.0

0.6

0.5 0.0

Ratio I(Mn) / [I(La)+I(Pb)]

1.6

3.0

0.4 0

10

20

30

40

50

60

0

10

20

30

40

50

60

Emission Angle (degrees)

Figure 13. Panel (a) shows the relative XPS normalized intensity of Pb(4f) to that of La(3d) and the best fits (continuos lines) to the data as a function of emission angle, for the gently („) and heavily (▲) annealed sample. Similarly, panel (b) shows the experimental ratio of (Mn/[La+Pb]) and the corresponding fits for both types of surfaces (adapted from [26]).

In order to fit the experimental points shown in figure 13, we consider different stacking scenarios of the naturally layered La0.65Pb0.35MnO3 material, as schematically shown in figure 14. The layers are at a distance of d = 2.1 Å apart from each other, which also represents the Mn-O bond length determined by neutron scattering experiments [4]. Figure 14(a) presents the n = ∞ sequence, specific to La1-xDxMnO3 compounds (D = Ca, Ba, Sr, Pb), while figure

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

120

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

14(b) shows the stacking sequence (n = 1) characteristic to a Ruddlesden – Popper phase (La1xPbx)2MnO4.

Figure 14. Assumed stacking sequence of the La/Pb-O and MnO planes for (a) the unreconstructed and (b) the reconstructed surfaces. The rhombohedral shapes represent the MnO octahedrons. The model fits indicate that surface (a) is La/Pb-O terminated while surface (b) has a MnO terminal layer.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

For the gently annealed surfaces, the best fits indicate that La/Pb-O is the terminal layer and Pb segregation is highly increased compared to the bulk doping level (35%). In the n = ∞ [47] assumed stacking sequence (figure 14), the La/Pb atoms occupy even (2j) layers, while the Mn atoms are assumed to occupy odd layers (2j+1). The Pb-atomic fraction in the surface can be written as:

⎛ jd ⎞ ⎟ , and f 2 j + 1 ( Mn) = 1 ; f 2 j ( La ) = 1 − f 2 j ( Pb) , (5) f j ( Pb) = b + δ exp⎜⎜ − ⎟ G ⎝ ⎠ The Pb/La ratio is given by:

I ( Pb) I ( La )

=

1− 1−

⎛

⎞ ⎟ ⎟ λ θ cos j =0 Pb ⎝ ⎠ = = ∞ 2 jd ⎛ ⎞ ⎟ ∑ (1 − f 2 j ) exp⎜⎜ − ⎟ j =0 ⎝ λ La cos θ ⎠ b + exp(−2d /[λ Pb cos θ ]) 1 − exp(−2d / G 1−b − exp(−2d /[λ La cos θ ]) 1 − exp(−2d / G ∞

∑ f 2 j exp⎜⎜ −

2 jd

δ − 2d /[λ Pb cos θ ])

δ

− 2d /[λ La cos θ ])

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

,

(6)

Magnetic and Electronic Structure Modifications…

121

The Mn/(La+Pb) ratio is given by: ∞

I ( Mn) = I ( Pb) + I ( La )

⎛

∑ exp⎜⎜ − λ

2 j +1 ⎞ d ⎟⎟ Mn cos θ ⎠

⎝ ⎛ ⎛ 2 jd ⎞ ∞ 2 jd ⎜ − exp f ∑ 2j ⎜ λ cos θ ⎟⎟ + ∑ (1 − f 2 j ) exp⎜⎜ − λ cos θ j =0 La Pb ⎝ ⎠ j =0 ⎝ exp(− d /[λ Mn cos θ ]) 1 − exp(− 2d /[λ Mn cos θ ]) , (7 ) = I ( Pb) + I ( La ) j =0

∞

⎞ ⎟⎟ ⎠

=

For the gently annealed surfaces, the results of the best fits (in figure 13) give a value of b = 0.35 for the bulk Pb concentration, a Pb-segregation of δ =0.6 in the most top layer and a segregation depth of G = d, where d is the distance between two consecutive layers. For the heavy annealed surface, the best fits indicate that MnO is the terminal layer and Pb segregation is still enhanced compared to the bulk doping level (35%), but not as much as in the gently annealed surface. In the assumed stacking sequence (figure 14a) with n = 1 [47], the La/Pb atoms occupy (3j+1, 3j-1) layers, while the Mn atoms occupy (3j) layers. The elemental relative intensities can be calculated with the following expressions:

3j + 1

∞ 3j − 1 ⎞ ⎛ ⎞ d ⎟⎟ + ∑ f 3 j −1 exp⎜⎜ − d ⎟⎟ = ⎠ j =1 ⎝ λ Pb cos θ ⎠ b exp(−d /[λ Pb cos θ ])(1 + exp(−d /[λ Pb cos θ ])) = + 1 − exp(−3d /[λ Pb cos θ ]) Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

I ( Pb) =

+

⎛

∞

∑ f 3 j +1 exp⎜⎜ − j =0 ⎝ λ Pb cos θ

δ exp(−d / G − d /[λ Pb cos θ ])(1 + exp(−d / G − d /[λ Pb cos θ ])) 1 − exp(−3d / G − 3d /[λ Pb cos θ ])

∞ 3j + 1 3j − 1 ⎛ ⎞ ⎛ ⎞ f 3 j +1 ) exp⎜⎜ − d ⎟⎟ + ∑ (1 − f 3 j −1 ) exp⎜⎜ − d ⎟⎟ = j =0 ⎝ λ La cos θ ⎠ j =1 ⎝ λ La cos θ ⎠ (1 − b) exp(−d /[λ La cos θ ])(1 + exp(− d /[λ La cos θ ])) − = 1 − exp(−3d /[λ La cos θ ])

I ( La ) =

−

∞

∑ (1 −

δ exp(−d / G − d /[λ La cos θ ])(1 + exp(−d / G − d /[λ La cos θ ]))

I ( Mn) =

1 − exp(−3d / G − 3d /[λ La cos θ ]) ∞

⎛

j =0

⎝

∑ exp⎜⎜ −

1 ⎞ , ⎟ = ⎟ 1 − exp( −3d /[λ Mn cos θ ]) cos θ ⎠

3 jd

λ Mn

(8)

Using the above elemental intensities we can construct the relative ratios Pb/ La and Mn/(La+Pb) for the heavily annealed surface, adjusting not only δ and G, but also the bulk Pb concentration b. The best fits indicate that the value of b decreased to 0.2, the Pb-segregation

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

122

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

in the top most layer became δ =0.35, and the segregation depth increased to G=2.12 d for the heavily annealed surface. Note that b is the Pb atomic concentration in the surface and selvedge regions, and not the one corresponding to the entire bulk which can not be determined using photoemission spectroscopy. As derived from the model fits to the experimental angle resolved XPS data where the model are given by the equations (6-8), the Pb atomic fraction can be plotted as a function of layer number, as shown in figure 15 [26]. In the case of heavily annealed surface, the best fits for both Pb/La and Mn/(La+Pb) ratios have been obtained for a bulk Pb doping value of 20%, instead of 35%. The reconciliation with the bulk magnetization and resistivity measurements presented in figure 6, as well as bulk X-ray fluorescence data and XRD allot which are consistent with a doping level of 35%, can be made if we consider the possibility of Pbevaporation from the surface during the vacuum thermal treatments at 520ºC. The melting temperature of bulk Pb is around 330ºC and is 40 degrees lower for (100) and (110) surfaces of Pb single crystals [48]. Another plausible explanation for the anomalous value of Pb doping level in the heavily annealed surface can be found if we consider that the mean-freepath of the electrons increases as a result of enhanced insulator properties, allowing us to probe deeper in the selvedge region, where the Pb concentration is reduced compared to the gently annealed surface.

Pb atomic fraction (%)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

100 gently annealed heavily annealed

80 60 40

b = 35%

bulk

b = 20%

b = 35%

20 0

0 2 4 6 8 10 12 14 16 18 20 22 24 Layer Number

Figure 15. The concentration of Pb in the near surface region of La0.65Pb0.35MnO3 thin film. The (■) symbols denote the gently annealed surface segregation profile, while the (▲) stand for the Pb concentration in the heavily annealed surface. The hatched area indicates the presence of a transition region that occurs only for the (▲) profile (adapted from [26]).

Below the selvedge region (with b = 20%) of the heavily annealed surface, we assume the existence of a transition region (hatched area in figure 15) in which the doping level is restored to the bulk value of 35%. In the case of the gently annealed surface, the fits indicate no change in the bulk value of 35%, but a highly Pb segregated surface. This clearly demonstrates that annealing treatments can easily modify the surface composition and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

123

structure. Such differences in the surface composition are expected to have a strong effect on the magnetic and electronic properties of the different types of surfaces (as shown in next section). A stable Ruddlesden-Popper phase (n = 1 stacking sequence) has also been observed to occur at the surface region of the related La0.65Sr0.35MnO3 structure [42]. Surface segregation is a strong indication that the surface enthalpy differs significantly from the bulk, in the context of standard statistical models. In a simple statistical-mechanical model of segregation [49], the total free energy, F, for the system is written as:

F =

∑ nib g ib +nis g is − k B T ln Ω i

,

(9)

where nib and nis are the number of the bulk and surface atoms of type i, with individual free energies gis and gis, respectively; kB is the Boltzman constant, T is the temperature of the system, and Ω is the entropy due to the mixing of the components. For a two-component system, an Arrhenius expression can be written as:

n As n Bs

=

H ⎞ ⎛ ⎜− ⎟ exp ⎜ k T⎟ n Bb B ⎝ ⎠, n bA

(10)

0.00

-0.04

-0.08

ΔH (eV)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

where H is the enthalpy of the segregation. Equation (9) implies that the surface segregation is a competition to minimize the total free energy by a maximization of entropy by evenly mixing the two elements. Using the bulk concentration values for the A and B components, one can easily extract the dependence of the segregation enthalpy of the atomic fraction.

-0.12 gently annealed heavily annealed

-0.16 0

2

4

6

8

10

12

Layer number Figure 16. The enthalpy difference between the surface and the bulk based on the extensive Pb segregation profile shown in Figure 15. The filled (open) symbols indicate the change in energy that occurs for the gently (heavily) annealed surfaces.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

124

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

Using equations (9-10) we can calculate the respective dependence of the enthalpy on the layer number, for both gently and heavily annealed surfaces, as shown in figure 16. The driving force toward equilibrium segregation (enthalpy change) is much larger for the heavily annealed surface compared to the gentle annealed surface. One explanation can be the possibility that the terminal layer is a polar surface – a surface likely to be unstable toward surface reconstruction. In conclusion, these differences at the surface suggest that under equilibrium conditions, the surface will not have a composition and/or a structure representative of the bulk.

5. Surface Debye Temperature It is well known that the Debye temperature is an important characteristic structure parameter of a solid, which is directly related to the binding force between atoms comprising the material, and can indicate the level of electron-phonon coupling across a transition (in case of electron scattering [50]). In the Debye model, the phonon spectra of a crystalline lattice has an upper limit, given by the vibrational frequency limit of the atoms. The Debye temperature is related to this frequency by:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

νD =

kB h

θD

,

(11)

where kB is the Boltzmann constant and h is the Planck constant. Any surface sensitive technique that involves diffraction of electrons or atoms can be used to extract the value of the effective surface Debye temperature (Ref. [51] and the references therein). We have calculated the effective Debye temperature for the two different surfaces of La0.65Pb0.35MnO3 using LEED and core level photoemission (XPS). Generally, in electron spectroscopy techniques it is assumed, in the absence of a surface phase transition, that the emerging electron beam intensity depends exponentially on the sample temperature. This dependence is applicable to core level intensities obtained from X-ray photoemission at normal emission angle and LEED intensities at normal incidence. Within the kinematic approximation, the photoelectron intensities can be written as:

I = I 0 exp[−2W (T )] , with

2W =

3= 2 (Δk ) 2 T mk Bθ D2

,

(12)

where W is the Debye-Waller factor, = is the Planck constant, T is the sample temperature, =(Δk) is the electron momentum transfer, m is the scattering center mass, kB is the Boltzmann constant, and θD is the effective surface Debye temperature. This surface θD is dominated by the atoms’dynamic motions parallel to the scattering vector and typically does not contain significant in-plane or anharmonic contributions to the true surface Debye temperature. We regard this parameter (θD) as the measure of the dynamic motions of specific atoms along the direction of the surface normal.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

125

Using LEED, we have extracted the value of the effective surface Debye temperature using equation (12), in which the scattering center mass m was considered as an average value between La and Pb atomic masses ( m = [ M La + M Pb ] / 2 = 173.05a.u ). The masses of Mn and O atoms are small compared to La and Pb and therefore, we assume that electrons scatter mainly from the surface La – Pb planes. Note that the La – Pb planes in the restructured surface are staked differently than in the as-deposited surface (see figure 14), fact that might influence the value of the surface effective Debye temperature. The temperature dependence of LEED intensities is shown in semi-log plots in figure 17 [26], for both gently and heavily annealed surfaces.

(a)

0.0

(b)

ln[(I-Ibkg)/I0]

-0.2 -0.4

E = 109 eV θD=227K

-0.6

E = 32.1 eV θD=71K

-0.8 -1.0

E = 29 eV θD=77K

-1.2 -1.4

200

300

400

500

600200

300

400

500

600

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Temperature (K) Figure 17. Natural logarithm of the diffracted electron spots obtained in LEED as a function of temperature for the gently and heavily annealed surfaces, panels (a) and (b), respectively. Using different electron energies we probed the near surface (32 and 27 eV) or deeper in the selvedge layers (109 eV). A linear background was subtracted for each spectrum and normalization was achieved by dividing with the intensity value (I0) at the lowest temperature (adapted from [26]).

The values of the effective surface Debye temperature obtained for the gently annealed surface are 71± 2 K (at 32 eV incident electron energy) and 77 ± 2 K (at 29 eV) for the heavily annealed surface. Increasing the incident electron energy up to 109 eV we have obtained a “bulk-like” effective Debye temperature of 227 ± 7 K for the restructured surface after heavy annealing. In case of XPS (emission angle θ = 0 degrees), we used the same equation (12), as in the case of LEED, in order to extract the effective surface Debye temperature for the two surfaces. We have monitored the change in photoelectron intensity in XPS from the 4f7/2 subshell of Pb atoms and appropriately corrected for the scattering center atomic mass mPb. The temperature dependence of the XPS intensities is shown on semi-log plots in figure 18 (a) and (b) for the as-deposited and the restructured surfaces, respectively [26]. The value of the effective surface Debye temperature obtained for the gently annealed surface is 310 ± 16

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

126

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

K, while for the heavily annealed surface we have determined an effective Debye temperature of 352 ± 19 K. All XPS measurements were conducted using the main Mg-Kα1,2 line at 1253.6 eV, which extracts the photoelectrons from the 4f7/2 Pb core with a kinetic energy of 1115 eV.

0.0

(b)

(a)

ln[(I-Ibkg)/I0]

-0.2

-0.4

θD=310K

θD=352K

-0.6

-0.8 200 250 300 350 400 450200 250 300 350 400 450 Temperature (K)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 18. Natural logarithm of the XPS intensity of the Pb-4f7/2 core level vs. temperature for the gently (a) and heavily (b) annealed surfaces. A linear background was subtracted for each spectrum and normalization was achieved by dividing with the intensity value (I0) at lowest temperature (adapted from [26]).

There is a big discrepancy between the values of θD obtained from LEED and XPS for the same surface. This difference can be understood considering that the probing depths are not the same in the two techniques, XPS having a larger electron mean free path, more bulklike, while LEED (depending on incident electron energies) provides a more surface sensitive measurement. Almost independent of surface orientation, the expected surface θD is 70% of the bulk Debye temperature value [52]. Still, the gently annealed surface exhibits a much lower effective Debye temperature (71 K) in the surface region compared to the more bulklike value (310 K). We attribute this drop to the presence of Pb surface segregation as shown in figure 18. Indeed, the value of surface Debye temperature for Pb single crystals obtained from similar RHEED studies, is around 50 K while the bulk Pb Debye temperature is 90 K [53]. Moreover, we can compare the bulk θD value of 310 K for La0.65Pb0.35MnO3 thin film with the bulk Debye temperature of 312 K found for NiMnSb [54] or with the value of 350380 K found for Y1Ba2Cu3Ox [55]. We can conclude that the gently annealed samples of La0.65Pb0.35MnO3 have a very soft surface, with a high Pb concentration that decreases rapidly towards the bulk where the lattice is much stiffer. The heavily annealed surfaces exhibit a similar behavior to the gently annealed surfaces, with an overall increase of the Debye temperature in the surface layers and in the selvedge region. Depending on the energy of the incident electrons in LEED, a value of 77 K for 29

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

127

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

eV, and 227 K for 109 eV is extracted for the first surface layers; and from XPS we obtained a θD value of 352 K for the selvedge region. The difference in Debye temperatures obtained for the same surface can be understood if we recall that the mass of the scattering center was different in LEED and XPS. Generally higher values for the effective Debye temperature, and hence lower mean-square displacements of the scattering centers, provide a further indication that extensive annealing in UHV creates a surface different in composition and structure than the gently annealed surface. Nevertheless, we cannot rule out the possibility that the observed Debye temperature increases as a result of greater probing depths (electron mean free path increase), which in turn results from enhanced insulating properties of the new heavily annealed surface. As a quick summary of the measurements reported above on the surface of La0.65Pb0.35MnO3 thin film, we have found that annealing treatments can easily modify the surface composition. We have studied the composition of two types of thermal treated surfaces and we found that gentle annealing (up to 250ºC) of the sample induces Pb segregation to the surface, while a heavy annealing (520ºC) completely alters the surface and selvedge structure and induces Pb evaporation. The well-annealed surface goes through a restructuring transition characterized by the formation of a Ruddlesden-Popper phase with a presumably embedded MnO columnar phase. The gently annealed La0.65Pb0.35MnO3 surface is soft, with a surface effective Debye temperature close to Pb single crystal surface, in agreement with the extended Pb segregation in the surface region. The heavily annealed sample has a reduced Pb segregation in the surface region and a slightly higher surface Debye temperature, indicating a stiffer lattice. These extreme changes in surface structure and composition may have a great impact on magnetic and electronic surface structure properties that are very important in improving the tunnel magneto-resistive junctions and spin valve performance.

6. Surface Electronic Structure The understanding of the electronic structure is the key to insight in most properties of the solid, including the dielectric, magnetic and bonding behavior. Fundamental elements of magnetism, such as the exchange coupling, the spin-orbit coupling, the magnetic moment, and the Curie temperature are determined by the electronic structure. The valence and conduction band structure can be experimentally investigated using photoemission, inverse photoemission and X-ray absorption spectroscopies. After a brief introduction to these three techniques, this section reveals the similarities and differences between the band structures corresponding to the two types of perovskite surfaces. Photoelectron spectroscopy involves measuring the kinetic energies Ekin of ejected photoelectrons from a molecule or a surface atom due to the excitation caused by monoenergetic radiation. Depending on the energy of the incident radiation, it is possible to extract electrons from the valence band (using low energy incident radiation) or from the core levels of the solid (using high-energy photons). Throughout this study, He II (40.8 eV) lines were used as the incident radiations for the ultraviolet photoemission spectroscopy. The probing depth is up to 5 atomic layers from the surface, which gives a high level of surface sensitivity The UPS spectra are normalized by the corresponding integrated intensity of each

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

128

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

curve, after subtracting an exponential background, which originates from the emerging secondary electrons. Inverse photoemission spectroscopy (IPES) is used to study the unoccupied electronic structure of solids, molecules or atoms. The inverse photoemission process is time reversed as compared to the photoemission process. An incident electron undergoes a radiative transition between an initial unoccupied state above the vacuum level to a final unoccupied state above the Fermi level, mapping the unfilled electronic states between the Fermi level and the vacuum level. The technique requires two key components – a spatially resolved, variable energy low electron source (of preferential spin for spin-polarized inverse photoemission SPIPES) and an isochromat photon detector which, in our case is based on a Geiger-Muller isochromat. More details about the setup can be found in reference [56]. The basic process of X-ray absorption (XAS) is the excitation of electrons from deep core levels of a selected atom by absorption of a photon. For an ordered magnetic material, if we can orient the magnetic moments (by applying a magnetic field to the sample), we obtain local magnetic information, using the circular polarization of the X-rays (MCD). The XAS and XMCD spectra were measured at the NRL-NSLS Magnetic Circular Dichroism Facility U4B located at the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory. The details of the U4B beam line have been published elsewhere [57]. At a quick glance, the IPES and XAS processes might give the same results due to the fact that both have same final state in the unoccupied band. An in depth analysis reveals three main differences between XAS and IPES techniques. First one refers to the fact that the absorption process in XAS is site selective. Tuning the X-ray photon energy allows us to select each element in a compound material and to collect data site by site. This gives us information about the site symmetry, spin state, oxidation state and nature of chemical bonds for the absorbing atom. IPES is not site selective, but measures the combined electronic unoccupied electronic structure of all atoms located in the surface region, having a probing depth much smaller than in the case of XAS. The second difference can be attributed to the cross sections involved in the two processes. In the case of IPES, the dipole selection rules and the one-electron approximation are similar to XAS, but we must introduce another requirement for the initial state in IPES – namely to be fully symmetric (or even) with respect to the mirror plane (the plane that contains the normal to the surface and the detection direction). This difference in the cross sections for XAS and IPES plays a very important role in the case of polarization dependent incident (XAS) or emitted (IPES) light measurements. The third distinction that must be made between the XAS and IPES spectra refers to the reciprocal space probed in the two techniques. In IPES, one can probe only one k-point in the Brillouin zone for each incidence angle. Therefore, mapping of the unoccupied electronic states near EF, as a function of wave-vector k, can be achieved successfully in IPES. In the case of XAS, using the total electron yield technique, the signal is given by the charge carriers collected throughout the entire Brillouin zone. In the XAS case, band mapping as a function of k-vector is not possible. The occupied (UPS) and unoccupied (IPES) bands of gently and heavily annealed La0.65Pb0.35MnO3 surfaces are shown in figure 19 and 20, respectively [58]. According to the photoionization cross sections [59], the emission intensity is due mostly to O 2p states in the valence band. The valence-band spectra of the two types of surfaces present two distinct photoemission structures, located at 3-5 eV and ~ 6 eV below the Fermi level. The first peak is emission due

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

129

to the Mn 3d- t2g↑ states and the second peak is entirely O 2p in origin, consistent with the results obtained on the Sr-perovskite series [60-61]. Although very low, the emission intensity between the Fermi level and ~ 2eV below it corresponds to the occupied eg↑ electronic band. For the restructured surface (after heavy annealing) we find that at the spectral feature at about 4 eV binding energy decreases in intensity compared to the same band corresponding to the gently annealed surface. Such spectral weight transfer is directly related to the increased hole doping levels of the surface region [60-61].

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 19. Valence band (left) and conduction band (right) at 300 K and 180 K, for the gently annealed surface of La0.65Pb0.35MnO3 (adapted from [58]).

The experimental valence band structure obtained from the gently annealed surface (figure 19) corresponds to a high Pb doping level, while the restructured surface (figure 20) displays a valence band indicative of low Pb content. The Pb atomic fraction in the two types of surfaces is related with the intensity ratios of the O 2p and Mn 3d (t2g) photoemission bands, as previously described in literature [60-61]. These results are in agreement with the Pb-concentration determined using XPS in the surface region of gently and heavily annealed thin films of La0.65Pb0.35MnO3. Temperature dependence of the valence band structure shows an approximate 0.8 eV binding energy shift in case of the gently annealed surface (figure 19), while the restructured surface remains largely unchanged between 300 K and 180 K (figure 20). For the lightly annealed sample, the presence of an energy shift towards the Fermi level indicates an increasingly more metallic state at low temperatures, consistent with the decrease in resistivity with decreasing with temperature, shown in figure 6. The lack of a shift in binding energy in case of the heavily annealed surface reveals an anomalous behavior of the CMRperovskites and provides the first indication of the surface electronic phase transition. As it will be shown later in this section, the heavily annealed surface resembles a p-type semiconductor at low temperatures.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

130

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

Figure 20. Valence band (left) and conduction band (right) at 300 K and 180 K, for the heavily annealed surface of La0.65Pb0.35MnO3 (adapted from [58]).

The influence of surface composition on the electronic structure can be observed in temperature dependence of the core level binding energies for the heavily annealed MnO rich surface. Figure 21 shows the XPS O 1s core level as a function of temperature and X-ray source power, for the restructured (heavily annealed) sample. It is to be noted here that in case of the gently annealed surface, there was no evidence of energy shift with temperature, regardless of the source power used. s

TC

Oxygen 1s binding energy (eV)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

530.4

X-ray power 300 Watts 200 Watts 125 Watts

530.3 530.2 530.1 530.0 529.9 529.8 529.7 529.6 529.5 529.4 529.3 175

200

225

250

275

300

325

350

375

400

Temperature (K) Figure 21. The binding energy of the oxygen 1s peak as a function of temperature and X-ray source power (flux) for the heavily annealed MnO rich surface.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Magnetic and Electronic Structure Modifications…

131

In case of the restructured surface, the extent of the binding energy shift is observed to depend strongly upon the X-ray source power (flux). The greater the incident X-ray flux becomes, the larger the core level shift (toward higher binding energies) for temperatures below 240 K. The fact that the core level binding energies depend upon the X-ray flux indicate that surface charging of the sample occurs below 240 K. From valence band photoemission (UPS), we have shown above the effect of doping on the electronic structure. Using the same measurements of the valence band it is possible to investigate the spin disorder effects on the electronic structure. Such manifestations of the spin-disorder effects have been carried out for La1-xSrxMnO3 and LaNi1-xMnxO3 compounds in reference [62]. The author demonstrates that the effective strength of the Coulomb interaction between the correlated electrons (U) decreases with decreasing temperature. Also, it is shown that the width of the effective d-band (W) decreases with lowering temperature. With increasing ferromagnetic order, or as the temperature is lowered below TC, both these effects lead to a pronounced decrease of the effective U / W, which can be observed as a decrease of the DOS at EF in the UPS spectra. In the case of La0.65Pb0.35MnO3, the gently annealed surface shows a reduced spectral weight at EF compared to the heavily annealed surface, at all temperatures, as shown in figures 19 and 20. This effect could be the manifestation of the spin-disorder on the electronic structure, which specifically demonstrates that the gently annealed surface is characterized with a higher degree of spin disorder than in case of the restructured surface. Also, the ratio U / W is presumably higher for the gently annealed surface compared with the heavily annealed surface. The doped electron or hole states arising from different surface compositions can be investigated by studying the unoccupied electronic band structure of the surface. To accomplish this, we have carried out a detailed inverse photoemission investigation. The right sides of figures 19 and 20 show the inverse photoemission spectra acquired using normal incident electrons for gently and heavily annealed La0.65Pb0.35MnO3 surfaces. The IPES spectra are normalized by integrated area of each curve after subtracting a linear background. The position of the Fermi level was determined from the Ta IPES spectrum, which represents the material used for fabricating the sample holder. In figures 19 and 20, both a broad unoccupied band from approximately 5 to 8 eV above EF and a shoulder structure centered on 3.6 eV above EF are observed. The former is mostly due to the La 5d band and the latter originates from the nonbonding or antibonding like states with Mn 3d – O 2p hybridized character [63,64]. In comparison with other CMR-manganite materials, the Mn 3d – O 2p antibonding-like states are positioned at approximately 2 eV above the Fermi level for both La0.65Ca0.35MnO3 and La0.65Ba0.35MnO3, while the same states are found at about 4 eV above Fermi level for the La0.65Sr0.35MnO3 [65]. The feature at 6.5 eV, corresponding to the La 5d states, is clearly observed in the conduction band of La0.65Ca0.35MnO3, while the La 5d state for La0.65Sr0.35MnO3 are substantially broader and less apparent, as in case of La0.65Ba0.35MnO3. In general, the similarity of the spectra for all doped manganites argues in favor of the model proposed by Taguchi and Shimada [66] for La1xCaxMnO2.97, proving that the bonding is generally covalent and not ionic for the moderately doped manganite species. In the case of the restructured surface (after heavy annealing) of La0.65Pb0.35MnO3, the unoccupied peak corresponding to La 5d states is enhanced, while the Mn 3d - O 2p shoulder moves towards the Fermi level (to 2.4 eV) at 300 K, as shown in figure 20. These changes are

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

132

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

consistent with Pb-depletion of the surface layers, shown in figure 15. Similar results were obtained in references [63,64] for La0.84Sr0.16MnO3.

La(Pb) 5d

Normalized intensity (arb. units)

Mn 3d

300 K 268 K 240 K 200 K 180 K

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

8 4 2 6 10 EF Energy above Fermi level EF (eV) Figure 22. Inverse photoemission (IPES) spectra acquired at normal emission as a function of temperature, for the restructured (heavily annealed) MnO rich surface.

The temperature dependence of the unoccupied levels is quite pronounced for the restructured surface, in comparison with the lightly annealed surface. Figure 19 shows no change in the unoccupied bands between 300 K and 180 K, while figure 20 presents a 2.5 eV energy shift of both bands at 180 K compared to the spectrum acquired at 300 K. This anomalous thermal behavior of the restructured surface has been encountered only for the similar La0.65Sr0.35MnO3 structure [42]. In both Sr- and Pb-doped perovskite structures, the dramatic decrease in density of states at EF with lowering temperature indicates that the surface becomes insulating at low temperatures, consistent with the O-1s core level shift shown in figure 21. This insulating character appears below approximately 240 K, while above this temperature the reconstructed surface is metallic or semi-metallic. This metal to insulator transition is shown clearly in figure 22. If one considers the low spectral density of the valence band at temperatures shown in figures 19 and 20, it appears that La0.65Pb0.35MnO3 exhibits a gap or a pseudo-gap with very low density of states near EF. This is consistent with the results obtained for the La1-xSrxMnO3 with x = 0 – 0.4, which indicates that the conductivity gap of about 0.2 eV exists for all compositions at T > 300 K [61]. Though of great debate in the literature, the metallicity of both Pb – and Sr – doped manganates can

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

133

originate in several ways. One possibility is that the small gap found at high temperatures is closed by a very low density of states at lower temperatures due to the tailing of the states near EF; this density of states would have to be low enough to be undetectable by photoemission and inverse photoemission experiments reported here. Another possibility is that the single-particle density of states continues to have a gap even at lower temperatures, which is consistent with the results of figures 19 and 20, while the multi-electron processes, which are expected to be of increasing importance for conductivity with decreasing temperature [68], introduce the metallic behavior observed in figure 6. The spin-polarized unoccupied bands for the gently and heavily annealed surfaces are presented in figure 23 and 24 respectively, at two different temperatures. To our knowledge, these results are the first obtained for a CMR-material. The spin-polarized bands are acquired for normal incidence of the electron beam, which represents the center ( Γ ) of the Brillouin zone. Several sets of data are averaged together in the spectra shown in figure 24. The spin asymmetry, shown at the bottom of figure 23, is obtained from P = (N↑-N↓)/(N↑+N↓), where N↑ (N↓) are the spin up (down) density of states, respectively.

Spin Asymmetry (%)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Intensity (counts)

(a) T = 300K spin up spin down

(b) T = 175K spin up spin down

EF

EF

100

100

50

50

0

0

-50 -100

-50

0

2

4

6

8

10 0

2

4

6

8

-100 10

Energy above Fermi level EF (eV) Figure 23. Spin-polarized inverse photoemission spectra acquired on a gently annealed LPMO surface at normal incidence, and at a temperature of (a) 300K and (b) 175 K. The spin asymmetry is shown at the bottom for each spectrum.

After a linear background subtraction, the maximum polarization of the gently annealed surface reaches almost 80%, while the heavily annealed surface shows a maximum of 40% at energy of 0.5 eV above the Fermi level. At the exact position of the Fermi level, the spin

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

134

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

asymmetry is approximately zero, or has a negative value, for the gently and heavily annealed surfaces respectively. This result does not agree with the requirement for half-metallic behavior of the surfaces studied, which postulates the existence of 100% spin asymmetry at the Fermi level across the entire Brillouin zone. The negative value of the spin-polarization at EF suggests that the surface layer is antiferromagnetic or antiferromagnetically aligned with the layers below. This magnetic coupling might be induced by the presence of an MnO2 phase, as found previously, in section 5, for the heavily annealed surface. Another explanation could relate to the occurrence of a spin standing wave (or RKKY-type interaction between adjacent spins) in the restructured Ruddlesden-Popper phase, as demonstrated for the semi-metal Sb [68]. The energy dependence of the spin-up contribution to the spin asymmetry in the gently annealed surface shows an enhanced ferromagnetic order compared to the heavily annealed surface at room temperature. However, at low temperatures (180 K), the spin asymmetry of both surfaces drops unexpectedly, especially in the case of the restructured surface. This result suggests that, while polarization at the Brillouin zone center (k⎜⎜ = 0) does corresponds to long range magnetic order, polarization cannot be easily related to magnetization, as it is well known that the ferromagnetic moment should increase with decreasing temperature. Another possibility would be the occurrence of a spin-reorientation transition at low temperatures, from in-plane to out-of-plane ferromagnetic order. If the total magnetic moment develops an out-of-plane component, the geometry of our SPIPES measurement will not be able to detect it. This reorientation transition has been already proven to exist for La0.7Sr0.3MnO3 thin films, deposited on LaAlO3 under compressive stress [69].

Intensity (counts) Spin Asymmetry (%)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(a) T = 300K spin up spin down

(b) T = 180K spin up spin down

EF

EF

100

100

50

50

0

0

-50

-50

-100 0

2

4

6

8

10

0

2

4

6

8

-100 10

Energy above Fermi level EF (eV)

Figure 24. Spin-polarized inverse photoemission spectra acquired on the reconstructed LPMO surface at normal incidence, and at a temperature of (a) 300 K and (b) 175 K. The spin asymmetry is shown at the bottom for each spectrum.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

135

The main conclusion obtained from the spin-polarized inverse photoemission measurement is that the surface region (maximum 10 Å from the surface) of the gently annealed sample, which preserves the n = ∞ layered structure, shows a maximum spinasymmetry in the center of the Brillouin zone at room temperature. The study of the unoccupied bands of La0.65Pb0.35MnO3 can be extended using X-ray absorption spectroscopy. Next, we continue our study by looking at the elemental specific energy levels lying above the Fermi level using in X-ray absorption spectroscopy. Using the total yield detection method, the absorption spectra corresponding to Mn 2p → 3d transitions are shown in figure 25, for both the gently and heavily annealed surfaces of La0.65Pb0.35MnO3 at 206 K. The spectra present two broad multiplets separated, in a first approximation, by the spin orbit splitting of the Mn 2p core hole, of magnitude 11.25 eV.

Absorption (arb. units)

Mn 2p -> 3d gently annealed heavily annealed at T = 206 K LIII

LII

0.20

MCD signal

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

0.15 0.10 0.05 0.00 -0.05 -0.10 635

640

645

650

655

660

Photon Energy (eV) Figure 25. Mn L2,3 XAS and MCD spectra for gently (solid lines) and heavily (dashed lines) annealed LPMO surfaces.

To identify the ground state of the Mn ion, we have compared our spectra with the theoretical calculation performed by Abbate et al. in reference [70]. Since this is the one of the few spectroscopic studies of Sr-doped manganite and, keeping in mind the the Sr- and Pbdoped manganite have similar crystal structure and transport properties, we consider legitimate the following comparison between our results on Pb-manganite and the results published in reference [70] on the Sr-manganite series. The shape of the Mn spectrum in

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

136

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

LaMnO3 corresponds to a 3d4 (t2g3 eg1) ground state, while the Mn spectrum for the heavily Sr doped manganite La0.1Sr0.9MnO3 has a 3d3 (t2g3) ground state, as shown in reference [69]. The change of shape, specifically the development of a second peak in the low energy part of the LIII and LII edges, is caused by holes (Sr2+) induced by substitution in the La1-xSrxMnO3 compound. Comparing our results on Pb-doped perovskite, with the results in reference [70], we suggest that the ground state for low Pb doping becomes Mn3+ while for high Pb doping is Mn4+. However, the Mn spectrum characteristic of high Pb doping is quite similar to the Mn 2p results in MnO2 obtained from Electron Energy Loss Spectroscopy (EELS), with Mn4+ in octahedral symmetry [71] and therefore, we need additional data to clearly identify the Mn oxide, as shown later in this section. Similar results, showing peak doubling in the Mn 2p spectrum, have been published in references [72,73]. S

B

TC

TC

Relative magnetic moment (arb. units)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

1.0

0.5

(a) gentle annealed MCD on Mn SQUID

0.0 1.0

0.5

(b) heavily annealed MCD on Mn SQUID

0.0 50

100

150

200

250

300

350

400

Temperature (K) Figure 26. Relative magnetic moment obtained from SQUID measurements (filled symbols) and MnMCD (open symbols) for gently (top) and heavily (bottom) annealed surfaces (adapted from 58).

In order to extract magnetic information from our two types of surfaces, we have monitored the MCD signal from the Mn 2p core hole as a function of temperature. Because it is not straightforward to apply the sum-rules [74] for such a complex material, we have limited our expertise of the magnitude of magnetic moment to the LIII integrated MCD

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

137

signal, without determining its absolute values. As can be seen in figure 26 (upper panel), the MCD signal for the gently annealed surface is greatly enhanced compared to the one obtained from the heavily annealed surface [58]. Figure 26 (lower panel) shows the temperature dependence of the LIII – MCD integrated signal. For the gently annealed surface, the Curie temperature remains close to the bulk value of 340 K, while for the restructured surface, the ordering temperature drops to approximately 240 K. The shift in TC may be caused by the tilting of the MnO6 octahedra, a structural distortion induced as the vacancy rate increases [75]. Another possibility that would explain the shift in TC is the occurrence of the reordering transition below 240 K, as expected for a double layer structure [76]. The total magnetic moment, extracted from the SQUID measurements at lowest temperature (5 K), is 10% reduced in absolute value in the heavily annealed (restructured) sample relative to the gently (as-deposited) thin film. The shape of the temperature dependence of the magnetic moment obtained from SQUID is similar for the two types of samples, with no evidence of a reduced surface Curie temperature. Note that the inplane temperature dependence of magnetization was measured in an applied external field of 1000 Oe, case in which the out-of-plane contribution of the magnetic moment cannot be detected. Figure 27 shows the O K-edge X-ray absorption measurements for the gently and heavily annealed surfaces measured at room temperature. The part of the spectrum between 530 and 535 eV is, in general, interpreted as originating from unoccupied O 2p states of the ground state covalently mixed with Mn 3d states. These features represent the crystal field and Coulomb interaction multiplet structure of the Mn 3d electron addition states with eg and t2g symmetry [70].

La(Pb) 5d

XAS signal (arb. units)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

eg t2g eg

O 1s -> 2p transition gently heavily annealed at T = 300 K

528

530

532

534

536

538

540

Photon energy (eV) Figure 27. O 1-s X-ray absorption spectra of gently (solid line) and heavily (dashed line) annealed LPMO surfaces at 300 K (adapted from [58]).

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

138

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

The three peaks at photon energies of 531, 532 and 534 eV have in the past been interpreted as being due to transition into the unoccupied eg↑, t2g↓ and eg↓ states (↑ and ↓ describing majority and minority spin), respectively, resulting in the peak assignment given in figure 27. The actual assignment of these pre-edges is still a matter of controversy [60]. It is important to note that the pre-peak at 531 eV has in all cases been interpreted as being of Mn 3d eg↑ majority character depleted by Pb doping. The broad structure centered at 538 eV is attributed to bands of La 5d mixed to Pb 5d character [70]. The gently annealed surface shows an O K-edge similar with that for x = 0.4 hole doping level in reference [70]. Accidental superposition of majority eg and minority t2g bands due to similar magnitudes of the exchange splitting and crystal field splitting requires enhanced resolution of the measurements in order to separate the two bands. However, the minority eg band is greatly enhanced in the case of heavily annealed surface, a unique feature compared to the Sr-manganite studied in reference [70]. The eg↓ band is hidden in the low-energy tail of the La 5d band for the gently annealed surface. Heavy population of the eg↓ in the case of heavily annealed surface indicates that the high-spin d4 state is lost, in the favor of a d3 state, which is characteristic to MnO2, with Mn4+ in octahedral symmetry [70].

IPES intensity (counts)

Mn 3d - O 2p Mn 3d - O 2p

(c) Absorption (arb. units)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(b)

(a)

0

EF

2

4

6

8

10

(d)

0

2

EF

4

La(Pb) 5d

6

eg

8

10

La(Pb) 5d

t2g eg

eg t2g eg

528

530

532

534

536

538

540528

530

532

534

536

538

540

Energy above Fermi level EF (eV) Figure 28. Unoccupied bands assignment is done comparing the XAS (c and d) spectra for O 1s core hole with the inverse photoemission spectra (a and b), for the as-deposited sample (a and c) as well as for the restructured (b and d) LPMO surface (adapted from [58]).

Both Mn 2p and O 1s absorption spectra indicate that the restructured surface is dominated by a columnar MnO2 phase, in agreement with the STM and LEED measurements Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

139

presented in sections 2 and 3. Pb segregation cannot be the cause of the changes in the shape of the absorption spectra, as they differ substantially from the changes induced by Sr-doping described in reference [70]. Our XPS surface measurements indicate that the Pb segregation is limited to the first 20-30 Å from the surface into the bulk, while the probing depth of XAS is around 50-100 Å from the surface. In the case of the CMR manganites, it is quite probable that the XAS and MCD techniques are not strictly surface sensitive, as the results considerably differ with surface preparation procedures. Once the core level binding energy has been taken into account for O 1s core level, the XAS and inverse photoemission spectra can be roughly compared, as shown in reference [77]. Figure 28 presents the inverse photoemission spectra and oxygen K edge absorption at room temperature for the gently and heavily annealed La0.65Pb0.35MnO3 surfaces. In both inverse-photoemission and XAS, the joint density of states just above EF is largely eg↑ (at 531 eV in O 1s XAS) plus t2g↓ (at 532 eV in O 1s XAS) in origin, while the enhanced shoulder at 3 eV above EF (534 eV in O 1s XAS) has an eg↓ provenance and originates from the nonbonding or antibonding like states with Mn 3d – O 2p hybridized character. The feature at 6.5 eV above EF corresponds to the La(Pb) 5d - O 2p states (at 538 eV in O 1s XAS), as mentioned earlier in this section. The comparison between the density of states of the gently and heavily annealed surfaces reveals that the eg↓ minority band is enhanced in case of the restructured surface, which indicates the predominant antiferromagnetic coupling between the magnetic moments on the O and Mn ions, situated in different layers. This band assignment is consistent with the energy distribution of the spinasymmetry presented in figure 24, which shows a distinct polarization reduction of the (eg↓) unoccupied band.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

7. Summary High quality La0.65Pb0.35MnO3 thin films have been fabricated using RF magnetron sputtering with subsequent annealing cycles in order to obtain the optimal oxygen content. The LEED, STM and X-ray diffraction pattern show that the films have an ordered (100) orientation with an approximate 5 to 10 nm surface roughness and a low number of surface defects. The magnetic and transport properties of the films are comparable to those of bulk single-crystals, with a Curie temperature of 354K and a magnetoresistance as high as 47% at ambient temperature. We have shown that the surface of La0.65Pb0.35MnO3 thin films is not stable, and that the surface can be modified by annealing treatments. The composition of two types of thermal treated surfaces was measured layer-by-layer, revealing that gentle annealing (up to 250ºC) induces Pb segregation to the surface, while a heavy annealing (520ºC) completely alters the surface and selvedge structure and induces Pb evaporation. The well-annealed surface goes through a restructuring transition characterized by the formation of a Ruddlesden-Popper phase with a presumably embedded MnO columnar phase. The gently annealed La0.65Pb0.35MnO3 surface is soft, with a surface effective Debye temperature close to Pb single crystal surface, in agreement with the extended Pb segregation in the surface region. The heavily annealed sample has a reduced Pb segregation in the surface region and a slightly higher surface Debye temperature, indicating a stiffer lattice.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

140

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

The as-deposited (gently annealed) La0.65Pb0.35MnO3 surface, although already Pb-rich in the outermost layers, shows a maximum spin-asymmetry of 80% at 0.5 eV above the Fermi level and at room temperature. As the sample is heavily annealed, the Pb-doping level drops due to temperature induced sublimation, while the surface region goes through a major restructuring transition characterized by formation of a new MnO2 phase embedded into a Ruddlesden-Popper phase, as evidenced by angle-resolved photoemission and STM studies. The maximum polarization of this restructured surface is 40% at a value of 0.5 eV above the Fermi level, in k⎢⎢ = 0 and at room temperature. The electronic bands closest to the Fermi level are assigned almost entirely to the Mn 3d bands. The first peak in the conduction band has a strongly mixed Mn 3d – O 2p character. Half metallic character requires integer values of the number of electrons in the initial states of Mn. The as-deposited surface of LPMO shows a clear Mn d4 initial state, while the heavily annealed perovskite surface has a preferential Mn d3 ground state. The half metal character expected from the as-deposited LPMO is lost by the extensive Pb-segregation proven to exist in the outermost layers, which explains the low spin-asymmetry value, measured at EF in spin-polarized inverse photoemission. The results obtained here are of great importance not only in explaining lower than expected polarization values found in literature, but also because they underscore the importance of characterizing the interface in any effort to understand the properties of spinelectronics junctions. We present clear evidence that changes in surface composition greatly affect the electronic structure of the surface. Throughout this work, the photoemission, using a He I source (21.3 eV), and inverse photoemission spectra were undertaken at normal

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

emission or incidence, corresponding to the center of the surface Brillouin zone ( Γ ). The surface Brillouin zone is the k-point representative of long-range magnetic order in realspace, but is not a test of half-metallic character by itself.

References [1] Jonker G.H., van Santen J.H., Physica. 1950, 16, 337 [2] Searle C.W., Wang S.T., Can. J. Phys. 1969, 47, 2703; Searle C.W., Wang S.T., Can. J. Phys. 1970, 48, 2023 [3] Glazer A.M., Acta Crystallogr. 1972, B 28, 3384 [4] Borca C.N., Adenwalla S., Liou S.-H., Xu Q.-L., Robertson J.L., Dowben P.A., Mat. Lett. 2002, 57, 325 [5] Ramirez A.P., J. Phys.: Condens. Matter. 1997, 9, 8171 [6] Zener C., Phys. Rev. 1951, 81, 440; de Gennes P.G., Phys. Rev. 1960, 118, 141 [7] Millis A.J., Littlewood P.B., Shraiman B.I., Phys. Rev. Lett. 1995, 74, 5144 [8] Lynn J.W., Erwin R.W., Borchers J.A., Huang Q., Santoro A., Penn J.-L., Li Z. Y., Phys. Rev. Lett. 1996, 76, 4046; Martinez B., et al., Phys. Rev. B 1996, 54, 10001 [9] Youn S.J., Min B.I., J. Korean Phys. Soc. 1998, 32, 576 [10] Pickett W.E., Singh D.J., Phys. Rev. B 1997, 55, R8642 [11] Livesay E.A., West R.N., Dugdale S.B., Santi G., Jarlborg T., J. Phys.: Condens. Matter 1999, 11, L279 [12] Nadgorny B., J. Phys. Cond. Matter. 2007, 19, 315209

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Magnetic and Electronic Structure Modifications…

141

[13] Nadgorny B., Mazin I.I., Osofsky M., Soulen R.J. Jr., Broussard P., Stroud R.M., Singh D.J., Harris V.G., Arsenou A., Mukovskii Ya., Phys. Rev. B 2001, 63, 184433 [14] Alvarado S.F., et al., Phys. Rev. B 1976, 13, 4918 [15] Soulen R.J. Jr., Osofsky M.S., Nadgorny B., Ambrose T., Cheng S.F., Broussard P.R., Tanaka C.T., Nowak J., Moodera J.S., Barry A., Coey J.M.D., Science. 1998, 85, 282 [16] Viret M., Nassar J., Drouet M., Contour J.P., Fermon C., Fert A., J. Magn. Magn. Mater. 1999, 1, 198; De Teresa J.M., et al., Phys. Rev. Lett. 1998, 82, 4288 [17] Park J.-H., Vescovo E., Kim H-J., Kwon C., Ramesh R., Venkatesan T., Nature. 1998, 392, 794 [18] Dowben P.A., Skomski R., J. Appl. Phys. 2004, 95, 7453 [19] Dowben P.A., Jenkins S., Frontiers in Magnetic Materials; Narlikar A.; Ed.; Springer Verlag. 2005, 295 [20] Katsnelson M.I., Irkhin V.Y., Chioscel L, Lichtenstein A.I., de Groot R.A., Rev. Mod. Phys. 2008, 80, 315; Irkhin V.Y., Katsnelson M.I., Lichtenstein A.I., J. Phys. Cond. Matter. 2007, 19, 315201 [21] Mazin I.I., Phys. Rev. Lett. 1999, 83, 1427 [22] Searle C.W., Wang S.T., Canadian J. Phys. 1970, 48, 2023 [23] Jia Y.X., Lu Li, Khazeni K., Crespi V.H., Zettl A., Cohen M.L., Phys. Rev. B 1995, 52, 9147 [24] Velev J.P., Dowben P.A., Tsymbal E.Y., Jenkins S.J., Caruso A.N., Surf. Sci. Rep. 2008, 63, 400 [25] Kono S., Goldberg S.M., Hall N.F.T., Fadley C.S., Phys. Rev. B 1980, 22, 6085 [26] Borca C.N., Xu B., Komesu T., Jeong H.-K., Liu M.T., Liou S.-H., Dowben P.A., Suf. Sci. 2002, 512, L346 [27] Zhao Y.G., Li Y.H., Ogale S.B., Rajeswari M., Smolyaniova V., Wu T., Biswas A., Salamanca-Riba L., Greene R.L., Ramesh R., Venkatesan T., Phys. Rev. B 2000, 61, 4141 [28] Fujishiro H., Fukase T., Ikebe M., J. Phys. Soc. Japn. 1998, 67, 2582; Fujishiro H., Ikebe M., Komo K., J. Phys. Soc. Japn. 1998, 67, 1799 [29] Nelson A.J., Reynolds J.G., Ross J.W., J.Vac. Technol. A 2000, 18, 1072 [30] Okada K., J. Phys. Soc. Jap. 1999, 68, 752 [31] Okada K., Kotani A., Thole B.T., J. Electr. Spec. Rel. Phenom. 1992, 58, 325 [32] Seah M.P., Jones M.E., Anthony M.T., Surf. Interface Anal. 1984, 6, 242; Seah M.P., Surf. Interface Anal. 1993, 20, 243 [33] Borca C.N., Ristoiu D., Xu Q.L., Liou S.-H., Adenwalla S., Dowben P.A., J. Appl. Phys. 2000, 87, 6104 [34] Suga S., Imada S., Muro T., Fukawa T., Shishidou T., J. El. Spectrosc. Rel. Phenom. 1996, 78, 283 [35] Schneider W.D., Delley B., Wuilloud E., Immer J.M., Baer Y., Phys. Rev. B 1985, 32, 6819 [36] Choi J., Dulli H., Liou S.H., Dowben P.A., Langell M.A., Phys. Stat. Sol. (b) 1999, 214, 45 [37] Steiner P., Courthis R., Kisinger V., Sander I., Siegwart B., Hufner S., Politis C., Appl. Phys. A 1987, 44, 75 [38] Carson G.A., Nassir M.H., Langell M.A., J. Vacuum Sci. Technol. 1996, 14, 1637; Oku M., Sato Y., Appl. Surf. Sci. 1992, 55, 37

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

142

C.N. Borca, Hae-Kyung Jeong and Takashi Komesu

[39] Shalvoy R.B. et al., Phys.Rev. B 1977, 15, 1680; Penderson R.L. et al., J. El. Spectrosc. Rel. Phenom. 1982, 28, 203 [40] O’Leary J., Winograd N., Anal. Chem. 1973, 45, 2214 [41] Surface Segregation Phenomena, Dowben P.A.and Miller A.; Ed., CRC Press, Boston, 1990, 145 [42] Dulli H., Plummer E.W., Dowben P.A., Choi J., Liou S.-H., Appl. Phys. Lett. 2000, 77, 570 [43] Choi J., Zhang J., Liou S.-H., Dowben P.A., Plummer E.W., Phys. Rev. B 1999, 59, 13453 [44] Scofield J.H., J. El. Spectrosc. Rel. Phenom. 1976, 8, 129 [45] Seah M.P., Jones M.E., Anthony M.T., Surf. Interface Anal. 1984, 6, 242; Seah M.P., Surf. Interface Anal. 1993, 20, 243 [46] Penn David R., J. El. Spectrosc. Rel. Phenom. 1976, 9, 29 [47] Morimoto Y., Asamitsu A., Kuwahara H., Tokura Y., Nature. 1996, 380, 141 [48] Frenken Joost W.M., et.al., Phys.Rev. B 1986, 34, 7506 [49] Liu S.Y., Kung H.H., Surf. Sci. 1981, 110, 504 [50] Dai P., Zhang J., Mook H.A., Liou S.-H., Dowben P.A., Plummer E.W., Phys. Rev. B 1996, 54, R3694 [51] Waldfried C., McIlroy D.N., Zang J., Dowben P.A., Katrich G.A., Plummer E.W., Surf. Sci. 1996, 363, 296 [52] van Hove M.A., Weinberg W.H., Chan C.M., Low-Energy Electron Diffraction, Springer Series in Surface Science. 1986, 6, 134 [53] Elsayed-Ali H.E., J. Appl. Phys. 1996, 79, 6853 [54] Borca C.N., Ristoiu D., Komesu T., Jeong H.-K., Hordequin Ch., Pierre J., Nozieres J.P., Dowben P.A., Appl. Phys. Lett. 2000, 77, 88 [55] Lin S., Lei M., Ledbetter H., Mat. Lett. 1993, 16, 165 [56] Komesu T., Waldfried C., Jeong H.-K., Pappas D.P., Rammer T., Johnston M.E., Gay T.G., Dowben P.A., “Apparatus for Spin - Polarized Inverse Photoemission and Spin Scattering”, in: Laser Diodes and LEDs in Industrial, Measurement, Imaging and Sensor Applications II: Testing, Packaging, and Reliability of Semiconductor Lasers V; Burnham G.T., He X., Linden K.J., Wang S.C.;Ed.; Proceedings of the SPIE – The International Society for Optical Engineering. 2000, 3945, 6 [57] Chen C.T., Nucl. Instrum. Methods Phys. Res. A 1987, 256, 592 [58] Borca C.N., Xu B., Komesu T., H.-K. Jeong, Liu M.T., S.-H. Liou, Stadler S., Idzerda Y., Dowben P.A., Europhys. Lett. 2001, 56, 722 [59] Yeh J.J., Landau I., At. Data Nucl. Data Tables 1985 32, 1 [60] Saito T., Bocquet A.E., Mizokawa T., Namatame H., Fujimori A., Abbate M., Takeda Y., Takano M., Phys. Rev. B 1995 51, 13942 [61] Chainani A., Mathew M., Sarma D.D., Phys. Rev. B 1993, 47, 15397 [62] Sarma D.D., J. El. Spectros. Rel. Phenom. 1996, 78, 37 [63] Kuwata Y., Suga S., Imada S., Sekiyama A., Ueda S., Iwasaki T., Harada H., Muro T., Fukawa T., Ashida K., Yoshioka H., Terauchi T., Sameshima J., Kuwahara H., Morimoto Y., Tokura Y., J. El. Spectros. Rel. Phenom. 1998, 88-91, 281 [64] Suga S., et al., J. El. Spectros. Rel. Phenom. 1996, 78, 283 [65] Choi J., Dulli H., Liou S.-H., Dowben P.A., Langell M.A., Phys. Stat. Sol. (b) 1999, 214, 45

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Magnetic and Electronic Structure Modifications…

143

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[66] Taguchi H., Shimada M., J. Solid St. Chem. 1987, 67, 37 [67] Pollack M., Kotek M.L., J. Non-Cryst. Solids. 1979, 32, 141 [68] Skomski R., Komesu T., Borca C.N., Jeong H.-K., Dowben P.A., Ristoiu D., Noziere J.P., J. Appl. Phys. 2001, 89, 7275 [69] Kwon C., Robson M.C., Kim K.C., Gu J.Y., Lofland S.E., Bhagat S.M., Trajanovic Z., Rajeswari M., Venkatesan T., Kratz A.R., Gomez R.D., Ramesh R., J. Magn. Magn. Mater. 1997, 172, 229 [70] Abbate M., de Groot F.M.F., Fuggle J.C., Fujimori A., Strebel O., Lopez F., Domke M., Kaindl G., Sawatzki G.A., Takano M., Takeda Y., Eisaki H., Uchida S., Phys. Rev. B 1992, 46, 4511 [71] Patterson J.H., Krivanek O.L., Ultramicroscopy. 1990, 32, 319 [72] Pellegrin E., et al., J. El. Spectros. Rel. Phenom. 1997, 86, 115 [73] Stadler S., Idzerda Y.U., Chen Z., Ogale S.B., Venkatesan T., Appl. Phys. Lett. 1999, 75, 3384 [74] Thole B.T. et al., Phys. Rev. Lett. 1992, 68, 1943; Carra P. et al., Phys. Rev. Lett. 1993, 70, 649 [75] Abdelmoula N., et al., Phys. Stat. Sol. (a) 2000, 180, 533 [76] Moritomo Y., Tomioka Y., Asamitsu A., Tokura Y., Phys. Rev. B 1995, 51, 3297 [77] Borca C.N., Cheng R.H., Xu Q.L., Liou S.-H., Stadler S., Idzerda Y.U., Dowben P.A., J. Appl. Phys. 2000, 87, 5606.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

In: Magnetic Properties of Solids Editor: Kenneth B. Tamayo, pp. 145-177

ISBN: 978-1-60741-550-3 © 2009 Nova Science Publishers, Inc.

Chapter 3

SITE DISORDER AND FINITE SIZE EFFECTS IN RARE-EARTH MANGANITES K.F. Wang1, S. Dong1 and J.M. Liu1,2* 1

Nanjing National Laboratory of Microstructure, Nanjing University, Nanjing 210093, China 2 School of Physics, South China Normal University, Guangzhou 510006, China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China

Abstract Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

In this chapter, we report our systematic investigation on (1) the site disorders imposed by Asite and B-site doping, and (2) the nanoscale size effect, in various rare-earth manganites. First, we address the effect of A-site size mismatch disorder on colossal magnetoresistance effect (CMR) and electronic phase separation. Given various manganites of different electronic bandwidths, it is demonstrated experimentally that the A-site disorder enhances significantly the instability of ferromagnetic metallic or antiferromagnetic charge-ordered ground states and drives them into spin-cluster-glass state. Second, we pay particular attention to a theoretical understanding of the B-site disorder in antiferromagnetic charge-ordered CE state, induced by nonmagnetic B-site doping. The as-caused charge frustration effect triggers a transition of the CE state into a short range ferromagnetic cluster state with relatively strong ferromagnetic tendency. Our calculation is in good agreement with experiments. Our experimental and theoretical investigations on the finite size effect focus on the spinglass tendency due to the enhanced surface relaxation with reducing sample size. While early works addressed the antiferromagnetic tendency on the surface of a ferromagnetic nanoparticle, we reveal the opposite effect: ferromagnetism appears in some nano-sized manganites with antiferromagnetic charge-ordered CE ground state in the bulk. A possible origin is the development of ferromagnetic correlations at the surface of these small systems. We study the two-orbital double-exchange model near half-doping level n=0.5, using open boundary conditions to simulate the surface of a manganites nanoparticle. We confirm that the enhancement of surface charge density suppresses the charge ordered CE state and produces a weak ferromagnetic signature that could explain experimental observations. *

E-mail address: [email protected] (To whom any correspondence should be addressed)

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

146

K.F. Wang, S. Dong and J.M. Liu

I. Introduction 1.1. Outline of Manganites Rare-earth manganites, which have a universal formula RE1-xAExMnO3 (where RE=La3+, Pr , Nd3+, etc, and AE=Ca2+, Sr2+, Ba2+, etc) and similar perovskite structure, are typical representatives of strongly correlated electronic systems and have raised general interest in modern condensed matters for two famous phenomena: colossal magnetoresistance (CMR) effect and electronic phase separation (PS) effect [1-5]. Some manganites exhibit metalinsulator transition (MIT) at critical point Tm, accompanied by a ferromagnetic (FM) to paramagnetic (PM) transition at Curie temperature (TC). The resistivity near TC (Tm) can change several orders of magnitude upon an external magnetic field H of a few Telsa. Moreover, related to the CMR effect, manganites exhibit rich phase diagrams, offering a variety of phases with unusual spin, charge, lattice, and orbital orders [5]. The CMR effect can be understood in the framework of double exchange (DE) model [68]. When the rare earth site (RE, A-site) is doped with a divalent ion (AE), a proportional number of Mn3+ ions are converted into Mn4+ ions and mobile eg electrons are generated, mediating the FM interaction between Mn3+ and Mn4+ according to the DE interaction. The hopping of eg electrons between two partially filled d-orbitals of neighboring Mn3+ and Mn4+ ions via the orbital overlap eg-O(2pσ)-eg, and the strong Hund coupling between the t2g core spins and mobile eg electron’ spins, cause the FM interaction between Mn3+ and Mn4+ ions. On the other hand, the rich ground states are considered to be determined by the transfer interaction of the eg conduction electrons or the effective one electron bandwidth (W) of the eg-band. The bandwidth W, dependent of the perovskite structure distortion, can be described as: 3+

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

1 3.5 W = cos( π − β ) d Mn −O , 2

(1)

where is the average angle of the Mn-O-Mn bonds and dMn-O is the average length of Mn-O bonds modulated by the average radius of A-site ions, . Since the DE interaction responsible for the FM metallic (FMM) state is scaled by W [8], the FM state becomes destabilized in a distorted perovskite with a small , and often is replaced by competing phases, such as charge ordered (CO) and orbital ordered (OO) antiferromagnetic (AFM) insulating phase. However, it has been recently recognized that the phase diagram of CMR manganites, as well as many strongly correlated electron systems, is multicritical, involving competing spin, charge or orbital, and lattice orders [1, 5]. The competition between these interactions and orders inherent in manganites, i.e. phase competing at the boundaries between those phases, generates interesting phenomena. For instance, the competition between the DE-FM phase and superexhange AFM phase, or that between the CO-OO phase and FMM phase, will produce a multicritical state [5, 9]. In relation to the competition between the CO-OO phase and FMM phase, a scenario of electronic phase separation has recently been arisen as a generic feature of CMR manganites. It has been accepted that given a temperature T and field H, the electronic and magnetic ground state of manganites can be inhomogeneous due to the coexistence of FMM phase and CO-AFM insulating (AFI) phase [1, 10, 11].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Site Disorder and Finite Size Effects in Rare-Earth Manganites

147

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

1.2. Disorder Effect Moreover, the physical properties of real materials are strongly influenced by randomness and electronic correlation. In particular, dealing with the localization and delocalization of carriers, the electron Coulomb correlation and disorder can be both driving forces for MIT process [12]. The interplay between the correlation and disorder represents one of the most fundamental, but least understood problems in contemporary condensed matter physics [1316]. For example, inhomogeneity has been observed in underdoped high-TC superconducting cuprates, where spin glass and stripe phase are the features of disorder effect near the MIT [16]. The disorder effect in manganites was already identified in the very beginning of CMR researches [17], but it has not attracted sufficient attention until very recently. For example, early studies on the transport property of RE0.7AE0.3MnO3 series revealed that the residual resistivity vary significantly upon different (RE, AE) combinations, as shown in figure 1 [17]. This difference must be attributed to the increasing carrier scattering and shortened quasiparticle life induced by existing disorder, because this series of samples have the same carrier density and these carries have the same effective mass, as proven by specific heat experiments. The MIT critical point Tm can also change remarkably upon different (RE, AE) combinations, while in the simple DE model Tm is merely controlled by effective bandwidth W. From combination (La, Sr) to (Nd, Sr), W only varies 2%, but Tm falls down about 30%, as shown in figure 1. These experiments revealed that the disorder effect suppresses the hopping of eg electrons. Moreover, those samples with higher residual resistivity usually have higher resistivity and sharper resistivity variation over Tm, as schematically shown in figure 1. The response of resistivity near Tm to H is very significant; ultimately resulting larger magnetoresistive ratio. This also evidences the important role of disorder effect in modulating the CMR effect. In parallel to researches on the transport behaviors, investigation on spin wave excitations in manganites revealed that the magnon dispersion relationship for manganites of low TC exhibit softening and extending effect at the boundary of Brillouin zone, which is absent in manganites of high TC and can’t be interpreted by simple DE model [18]. This softening and extending effect can be reasonably explained if disorder effect is involved in the DE model [19]. The disorder effect can be illustrated by some specific experiments and one example is Ln0.5Ba0.5MnO3, where the site occupation of Ln and Ba ions can be either ordered or disordered [20]. For the ordered case or “clean limit”, where Ln and Ba ions form a periodic layered structure, the phase diagram shows a multicritical behavior where the FMM state and CO-OO state compete with each other. For the disordered case, the phase diagram becomes very asymmetric: the FMM phase is partially suppressed but still survives at finite T, while the CO-OO state is replaced by a glassy-like phase in low T region. Enhanced CMR effect is identified in the region with the FMM state rather than the CO-OO state. This site occupation disorder not only produces the glassy state but also enhances the fluctuation of the multiphase competition near the original bicritical point. Such large fluctuation is amenable to an external H, favoring the FMM phase and making the disorder effect one of the most essential ingredients of the CMR physics [21-23].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

148

K.F. Wang, S. Dong and J.M. Liu

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 1. A sketch relationship between resistivity and temperature in manganites RE0.7AE0.3MnO3.

It is well developed that site doping, generating either ionic size mismatch, charge disproportion, or spin fluctuation, is the mainstream to introduce disorder into manganites. In fact, the disorder generated by Mn-site doping, which would result in localization of carries and holes at eg-orbits, was studied carefully [24-29]. For manganites of the CO-OO ground state, the Cr-doping at Mn-site gives rise to the FM clusters embedded in the CO-OO background. The doping in La2/3Ca1/3MnO3 with non-magnetic Ga ions makes the first-order transition be continuous, allowing the coexistence of long-range FM phase with short-range magnetic correlations at low doping level, which seems to support an argument that the firstorder transition in pure La2/3Ca1/3MnO3 is induced by fluctuations from the competition. Theoretical researches also predict that the disorder in the critical region may produce phase separation of two competing ordered phases on various time and spatial scales, although the mechanism remains unclear. Besides the B-site doping, A-site cationic size disorder has been an issue of special interest. Hereafter, the A-site cation size disorder (in shorts the A-site disorder) is specifically designated to the disorder induced by A-site ionic radii mismatch. It seems that the variance of the A-site ionic radii, σ2=∑i(xiri2-2), where xi and ri are the atomic fraction and ionic radii of i-type ions at A-site, respectively, governs the magnetic and transport properties of manganites, given the constant A-site cationic mean radius [30,31]. σ2 can then be viewed as a parameter to scale the A-site disorder degree. Our motivation to perform the present study is that previous studies mainly dealt with the relevance between Tm and σ2, and no sufficient data on PS behaviors associated with the A-site disorder are available before our work.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Site Disorder and Finite Size Effects in Rare-Earth Manganites

149

1.3. Finite Size Effect

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

In addition to the disorder effect highlighted above, here we have a brief outline of the surface and finite size effects in nanostructure, which in fact can be considered as one type of disorder effects, too. Quite a few experimental and theoretical studies focusing on the size effect of perovskite manganites were recently reported [32-39] and interesting effects associated with the materials downsizing to tens of nanometers were revealed. For example, some experiments found the spin glass ground state in nanosized manganites [32], but the enhanced ferromagnetism was also observed in nanosized La0.7Ca0.3MnO3, which was attributed to the contraction of lattice volume [38]. Nevertheless, neither nanosize effect in the CO state nor much theoretical analysis on the size effects is available before our recent work. In this chapter, we are not going to cover every aspect of the disorder effects, including the finite size effect, which seems to be an impossible task to us. Instead, we pay our attention to those aspects relevant to our own work on the disorder effect and size effect, in various rare-earth manganites. First, we address the effect of A-site disorder on the CMR and electronic phase separation. For various manganites of different W, it will be demonstrated that the A-site disorder enhances significantly the instability of the FMM or CO-AFM ground states and drives them into cluster spin glass state. Second, we pay particular attention to a theoretical understanding of the B-site disorder in the CE state, induced by nonmagnetic ions doping. The charge frustration effect, caused by the B-site doping, can trig a transition of the CE state into a short range FM cluster state with relatively strong FM tendency. Finally, we describe our experimental and theoretical investigations on the finite size effect, focusing on the spin glass tendency due to the enhanced surface relaxation with reducing sample size. The highly stable CO state can be significantly suppressed upon reduction of the grain size down to nanometer scale, while the ferromagnetism is enhanced.

II. A-Site Disorder Effects One of the key parameters to determine the ground state in manganites is the transfer interaction of the eg-state conduction electrons between neighboring Mn sites or the effective one electron bandwidth (W), which is controlled by the size of A-site ions. In ideal cubic structure with ideal radius of A-site ions, rA0, no lattice distortion exists and the Mn-O-Mn bond angle is 180o. The ground state is FMM state, as shown in figure 2(a) and (b). When rA0 decreases, the Mn-O-Mn bond would be bended and the MnO6 octahedra would exhibit collective distortion, as shown in figure 2(d). In this case, the DE interaction becomes no longer dominant and then the FMM state will be replaced by a CO state. Alternatively, if the A-site ions consist of two ions with different radius rA0+σ and rA0+σ, although the A-site ionic mean radius remains identical to rA0, the size mismatch between the two neighboring A-site ions would induce local bending of the Mn-O-Mn bonds, resulting in the local distortion of the MnO6 octahedra and lattice around them, as shown in figure 2(c). Even so the local lattice can be seriously distorted, the macroscopic structure remains unchanged as long as =rA0 is fixed. In details, the size difference between the neighboring A-site RE3+ and AE2+ ions around one oxygen ion may enable the random oxygen displacement and consequently the local distortion of MnO6 octahedra [30, 31]. This is the so-called A-site size

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

150

K.F. Wang, S. Dong and J.M. Liu

mismatch disorder. The strength of the local distortion and density of distorted MnO6 octahedra can be measured by variance σ2 defined above.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 2. Models for local oxygen displacements in ABO3 perovskites. Ideal cubic structure with rA0 is shown in schematically in (a) and as spherical ions in (b). Cation size disorder in (c) gives rise to random oxygen displacements. A reduction in the A site radius in (d) leads to ordered oxygen displacement. (Reproduced from Ref.[31]).

Compared with B-site doping, the A-site disorder can be tuned over a broad range and then does the local lattice distortion to a large extent. However, different from B-site doping, this disorder doesn’t cause significant change of the crystallographic structure and lattice constants, in the sense of X-ray diffraction over a macroscopic volume. Also no magnetic impurity and variation of the Mn3+/Mn4+ ratio will be generated by this disorder. Therefore, an investigation on the A-site disorder would be a more direct and purified roadmap to disclose the physical mechanism underlying the disorder effects in manganites. Subsequently, we will carefully illustrate the effect of the A-site disorder on the CMR and electronic phase separation in three types of manganites, after a brief introduction of previous results on the Asite disorder effects.

2.1. Early Studies on the A-Site Disorder Significant A-site disorder effects were firstly found in manganites (RE0.7AE0.3)MnO3, usually of FMM ground state [30, 31]. It was revealed that Tm varies as Tm=Tm(0)-pQ2 due to strain fields resulting from the disordered oxygen displacements Q that are parameterized by variance σ2. The value of p is related to the Mn-O force constant showing that the Mn3+ JahnTeller distortions assist electron localization at Tm [31]. Neutron diffraction revealed that the

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Site Disorder and Finite Size Effects in Rare-Earth Manganites

151

lattice structure of a series of samples with the same but increasing σ2 show an increasing local radial distortion associated with distorted MnO6 octahedra. The decrease in Tm across this series may also reflect the increasing strength of the distortion and the number of the distorted units [31]. While the samples with large σ2 becomes insulating over the whole T-range, the nature of electronic conduction changes gradually from thermal-activated type to variable range hopping type with increasing σ2 [40]. When the ground state is the CO phase, the impact of the A-site disorder was also studied. The resistivity and ac-susceptibility measurements revealed that the A-site disorder influences greatly the transport and magnetic properties which are associated with CMR. For example, Pr1-xCaxMnO3 does not exhibit any MIT and remains insulating over the whole Trange because of the CO state at high temperature. The mismatch generated by replacing Pr3+ ions with larger La3+ and smaller Y3+ ions, given a constant ~1.18Å, suppresses the CO state. Eventually, the MIT at Tm=130K and 60 K occurs, respectively, in the samples with x~0.25 and 0.30. For 0.300.016. In such a case, the external H favors the FM phase which expands at the expense of the AFI phase. Further expansion of the FM phase requires field H to be higher than a critical value, in order for the Zeeman energy to overcome the strain energy and enforce continuous expansion of the FM regions. At this critical field, a sharp step in M is observable, as shown in figure 5. Such a stepwise like behavior illustrates the coexistence of the FM regions and the CO-OO AFI regions in these samples [49-55]. We demonstrate that the A-site disorder will cause enhanced CMR effect. External H enforces the ground state of the samples with σ2=0.016, 0.018 and 0.020 from the insulator to metal, accompanied with the MR ratio approaching to 100%. The reason is that H destroys the CO-OO regions by growth of the FMM regions, as described above. When σ2 reaches up to 0.025, the long-range FM ground state is completely melted into the short-range magnetically ordered regions. The system can no longer develop sufficient symmetrical domains to induce the FM ordered regions. Consequently, M remains very low (~1.12μB for sample with σ2=0.025 under H=7.0T). Very similar to typical cluster-glass systems, the short-range magnetically ordered regions may frustrate upon decreasing T down

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Site Disorder and Finite Size Effects in Rare-Earth Manganites

157

to a frustrating point at which these regions begin to freeze due to the inter-cluster frustration. Then a cluster glass transition is activated.

2.3. Intermediate Bandwidth Manganites For manganites of intermediate bandwidth W, we choose La0.7Ca0.3MnO3 with =1.20Å as the parent compound of our investigation [1]. Its bandwidth is larger than Pr0.7Ca0.3MnO3 but smaller than La0.7Sr0.3MnO3 [56, 57]. In shorts, the increasing σ2 results in the metal-insulator transition from the insulating ground states, as listed in Table II, which is similar to the case in Sec.2.2. However, because W takes an intermediate value, La0.7Ca0.3MnO3, which is considered as typical PS system and located at the phase boundary between the FMM state and CO state [1, 58, 59], is more sensitive than La0.7Sr0.3MnO3 against the variation of σ2. The long-range FM order preferred at σ20.009. At σ2~0.008, the coexistence of both states is expected because of the competition between the DE interaction and the AFM superexchange interaction. For the cases of intermediate disorder, these two states coexist and the samples of highly disorder will change from the metallic state to the insulating state. Our experiments seem to coincide with predictions of Dagotto et al [1, 60-62] and reveal that the two competing interactions plus the quenched disorder give rise to a cluster-glass like behavior. This effect may be induced by the enhanced quantum fluctuations between the competing interactions as the consequence of the quantum phase transition, an issue worthy of further address.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

2.4. Small Bandwidth Manganites We finally address the magnetic and transport behaviors of manganites of small W. We prepare a series of manganites RE0.55AE0.45MnO3 which have the same =1.18Å but with different σ2 [62], as listed in Table III, noting that base system of this series is Pr0.55Ca0.45MnO3 with =1.18Å, which usually offers the CO-OO ground state at low T. We focus on the effect of the A-site disorder on the stability of the CO-OO state as the ground state of manganites with small W. Based on all of the transport and magnetic results [62], as shown in figure 8 and figure 9, we conclude that upon increasing σ2 from 0.00002Å2 to 0.0025Å2, the CO/OO ground state is gradually suppressed and the ground state changes to a FMM state. Nevertheless, further increasing of σ2 (>0.0025Å2) suppresses the FMM ground state either, and ultimately leads to the cluster-glass insulating ground state. To understand the physics underlying the collapse of the CO-OO state, one may consider the PS scenario in response to the A-site disorder and magnetic field. This series we study here have the constant = 1.18Å, which is much smaller than the ideal value. Therefore, the sample with the smallest σ2=0.00002Å2 must have the CO-OO ground state. The increasing σ2 will result in destabilization. First, a moderate A-site disorder will introduce larger A-site cations into the neighbors of seriously distorted MnO6 ocathedra and may help to minimize the crystal distortion in these regions. This effect ultimately makes these regions transit back to the FMM state. However, in the other regions, the disorder will enhance the A-

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

158

K.F. Wang, S. Dong and J.M. Liu

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

site size mismatch and the local lattice distortion, which favors the CO-OO AFI state. Second, it is argued that Pr0.55Ca0.45MnO3 locates near the multicritical points and the competition among the FMM phase and CO-OO phase is very significant, and consequently the coexistence of the FMM and CO-OO phases becomes preferred [63,64]. Here, the A-site disorder will enhance the competition.

Figure 8. Measured ρ-T relations for the samples with =1.18Å and σ2=0.00002 Å2, 0.0004 Å2, 0.0025 Å2, 0.0057 Å2 and 0.0078 Å2; (b) σ2-dependences of M measured at T=3K under H=3T, and σ2dependences of zero-field ρ measured at T=50K. (Reproduced from Ref. [62]).

Therefore, a slight increasing of σ2, referring to Pr0.55Ca0.45MnO3, would suppress the long-range CO-OO state but favor the FMM phase. The FMM phase increases and becomes predominant with increasing σ2. Nevertheless, in this condition, the CO-OO state still survives, thus the magnetization of the sample with σ2=0.0004 exhibits a weak cusp at T~214.8K, corresponding to the weak CO-OO transition and then the FMM transition at T~109.8K, as shown in figure 9(b) and the inset. Moreover, the M-H curve of this sample shows an evident step-wise behavior, corresponding to the field-driven destroy of the CO-OO regions and the growth of the FMM phase. These effects are popular in PS manganites, demonstrating the coexistence of the CO-OO and FMM phases. Increasing of σ2 to 0.003 allows further growth of the FMM phase and no longer CO-OO transition is observable. Correspondingly, both TC and M increase.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Site Disorder and Finite Size Effects in Rare-Earth Manganites

159

Figure 9. Measured M-T relations under ZFC and FC conditions for the samples with =1.18Å and (a) σ2=0.00002 Å2, (b) 0.0004 Å2, (c) 0.0025 Å2, and (d) 0.0078 Å2, respectively. The arrows in (d) indicate the cluster-glass transition point. The inset in (b) is the part of the CO-OO transition as described in text. (Reproduced from Ref. [62]).

When the A-site disorder is enhanced up to some threshold, the strength of the random lattice distortion increases, ultimately destroying the FMM phase, too. For the samples of σ2=0.0025Å2 to 0.0057Å2, the FMM phase becomes destablized, characterized by the decreasing TC and M. At σ2>0.0057Å2, the long-range FMM phase and CO-OO phase are

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

160

K.F. Wang, S. Dong and J.M. Liu

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

completely melted into short-range ordered regions. This again corresponds to the so-called cluster-glass state. The CMR effect in manganites of intermediate W is quite sensitive to the A-site disorder. Pr0.55Ca0.45MnO3 scarcely exhibits MR effect, as shown in figure 10. However, for the samples with σ2=0.0004Å2, 0.0025Å2 and 0.0057Å2, very big CMR effect is offered and the MR ratio approaches almost to 100%. The reason is that the ground state of these samples is the coexisting FMM phase and short-range CO-OO phase. External magnetic field can destroy the CO-OO phase and enforce remarkable growth of the FMM phase, resulting in significant CMR effect. We have demonstrated that the A-site disorder has significant impact on the PS microstructure. Because of the competitions and fluctuations between various spin/charge ordered phases in manganites, the A-site disorder enhances the competition allowing the PS state. Moreover, the disorder enhances the fluctuations of the competing ordered phases, allowing sensitive response of the phase separated state to external stimuli.

Figure 10. Magnetoresistance ratio defined as MR=-Δρ/ρ0 a function of T for samples with different σ2.

III. B-Site Disorder Effects 3.1. Introduction In Sec.II we addressed the effects of the A-site disorder. In fact, major attentions from researchers went to detailed investigation on the effect of B-site disorder by doping. Differently from the A-site disorder, the B-site doping usually generates disorder locally in the Mn-O network and it mainly modulates parameter like eg electron-density, n. Therefore, the B-site disorder can have even stronger impact on the physical properties of manganites than the A-site disorder.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Site Disorder and Finite Size Effects in Rare-Earth Manganites

161

For example, in the FMM ground state, Cr-doping on the B-site gives rise to the FM clusters in the CO-OO background [65, 66]. The doping with non-magnetic Ga ions makes the first-order transition in La2/3Ca1/3MnO3 be continuous, and results in long-range FM phase in coexistence with the short-range magnetic correlations at low doping level, so it can be concluded that the first-order transition in pure La2/3Ca1/3MnO3 is induced by fluctuations from competition [25, 26]. The effect of the B-site doping on the CO ground state is more interesting. Experiments indicated that a low level B-site doping at only a few percentage, such as chemical substitution of Mn by Al/Ga in the half-doped CE state of manganites, can activate phase separation with strong FM tendencies [67-71]. For instance, 2.5% Al substitution in Pr0.5Ca0.5MnO3 is sufficient to convert the CE insulating state into a state of FMM characters. These experimental results are very surprising since (1) the substituting ions are nonmagnetic, and (2) Al/Ga doping leads to the reduction of the eg electron density, both of which are disadvantageous for the FM tendency [72]. In fact, such a reduction in the eg electron density is expected to favor an AFM insulating state. In this section, we will carefully study the B-site nonmagnetic doping effect in the CE manganites using the double orbital model defined on a two-dimensional L×L square lattice (L=8 with periodic boundary conditions) incorporated with Monte Carlo simulation.

3.2. Theoretical Model We start from the double orbital model Hamiltonian which has been extensively employed to investigate the ground state and dynamics of phase separation. This mode reads: αβ

H = − ∑ trαβ Ωij ci+α c jβ + J AF ∑ Si ⋅ S j + ∑ (ε i − μ )ni Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

< ij >

< ij >

i

2 1 2 2 + λ ∑ (Q1i ni + Q2iτ xi + Q3iτ zi ) + ∑ (2Q1i + Q2i + Q3i ) 2 i i

,

(4)

where the first term is the two-orbital double exchange interaction, α and β denote the two Mn +

eg-orbitals a ( d x 2 − y 2 ) and b ( d 3 z 2 − r 2 ), operator c ja (cia ) annihilates (creates) an eg electron in orbital a of site i with its spin parallel to the localized t2g spin Si. The hopping direction is denoted by vector r, and the hopping amplitudes can be found in literature. The infinite Hund coupling generates the factor

Ω ij = cos( θ i / 2 ) cos( θ j / 2 ) + sin( θ i / 2 ) sin( θ j / 2 ) exp[ −i( π i − π j )],

(5)

where θ and π are the angles of the t2g spins in spherical coordinates. The second term in Eq.(4) is the superexchange interaction between the nearest-neighbor (NN) t2g spins. In the third term, μ is the chemical potential, εi corresponds to a site dependent Coulomb potential, and ni is the eg charge-density at site i. The fourth term stands for the electron-phonon coupling, where Qs are the phonon modes corresponding to the Jahn-Teller modes (Q2 and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

162

K.F. Wang, S. Dong and J.M. Liu

Q3) and breathing mode (Q1), τ is the orbital pseudospin operator given by and

τ x = ca+cb + cb+ca

τ z = ca+ ca − cb+ cb . The last term is the elastic energy of phonons. For simplicity, we

assume that both the t2g spins and the phononic degrees of freedom are classical variables. To dope a non-magnetic impurity (without 3d electrons) at the B-site, we assume that the impurity has no contribution to the electron conductivity and, thus, consider it as a lattice defect. Given this assumption, the double exchange, superexhange, and Jahn-Teller coupling around the impurity can be ignored, retaining only the elastic energy of local phonons. These localized defects distinguish the B-site substitution models from the A-site disorder models in which the disorder effects were applied to all sites. Therefore, the topological structure of the lattice defects, which is absent in the A-site disorder cases, is especially important in patterning the electron configuration of the original CE/CO state. Our model Hamiltonian is studied via a combination of exact diagonalization and Monte Carlo (MC) technique: the classical t2g spins and phonons evolve following the MC procedure, and at each MC step, the fermionic sector of the Hamiltonian is numerically exactly diagonalized. The first 104 MC steps are used for thermal equilibrium and another 103 MC steps are used for measurements. More details about this extensively employed two-orbital Hamiltonian and the MC algorithm can be found in Ref. [1].

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

3.3. Results and Discussion Now let us investigate the effect of the B-site nonmagnetic substitution in manganites of form RE0.5AE0.5Mn1−yByO3. Such a substitution will lead to the appearance of lattice defects and, simultaneously, a variation of the eg electron density. To clarify their respective roles, here we first address the effect of the lattice defects. For such purpose, the substituting cations are assumed to be +3.5 in charge to keep the average eg charge density for the remaining Mnsites (nr) invariant, i.e. nr=0.5. Our calculations show that for an appropriate doping level y, the CE/CO state will turn into a state with strong FM tendency. Figure 11(a) shows the structure factor S(q) for several typical spin structures evolving with y at the point (JAF=0.1, λ=1.4). The FM order at q=(0, 0) emerges at y>0.031 and it is enhanced quickly up to S(q)~0.12, demonstrating strong FM tendency. On the other hand, the E-type AFM order at q=(π/2, π/2) and C-type AFM order at q=(0, π) are rapidly suppressed when y reaches up to 0.063, indicating that the CE spin order is destroyed by the lattice defects. This CE destruction can be further identified by observing the MC snapshot of a spin configuration at y=0.094 substitution. As shown in figure 11(b), there is no trace of any CE chains, and the CE phase is converted into a state consisting of small FM clusters with various orientations, similar to the results of the A-site disorder efforts revealed earlier. The fundamental reason for the CE/CO destabilization can be understood based on the breaking of the zigzag FM chains and concomitant charge frustration. Let us consider the magnetic order first. The CE phase consists of zigzag FM chains that are easily cut down by lattice defects, leading to a substantial increase in the kinetic energy. However, the competing FM phase has a two-dimensional (three-dimensional in real case) character, which is much more robust than the CE phase against the lattice defects. For a pure system, the Jahn-Teller coupling favors the long-range staggered CO pattern. Upon the doping, the CE phase has two different B-site locations: as a bridge site (B1) with higher eg density and as a corner site (B2)

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Site Disorder and Finite Size Effects in Rare-Earth Manganites

163

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

with lower eg density, as shown in figure 11(c). In real cases, the B-site substitution should be randomly distributed and the two types of occupancy are available. This randomness can break the original CO state, causing the charge frustration which spreads over the whole lattice and leads to the collapse of the long-range CO phase. We show two typical MC snapshots of the charge redistribution in figure 11(d) and (e) with an intermediate and an large λ, respectively. For the intermediate case (here λ=1.3), the charge density distribution is homogeneous except some regions around the defects, while for the large λ case (here λ=1.6) the charge disproportionation is obvious although it occurs without a long-range ordered pattern.

Figure 11. (a) spin structure factors S(q) evolving with y at the point (JAF=0.1, λ=1.4) (b) Typical MC snapshot of the spin configuration showing the short range FM domains (y=0.094, JAF=0.1, λ=1.4), where the substitution ions are represented by filled circles. (c-e) Typical MC snapshot of the charge distribution. Here the circle radius is proportional to the local charge density. High and low-density sites (comparing with 0.5) are colored by blue and red, respectively. (c) The staggered CO pattern in the clean limit (JAF=0.1, λ=1.3). (d) CO pattern at y=0.094, JAF=0.1, and λ=1.3. (e) CO pattern at y=0.094, JAF=0.1, and λ=1.6, in the spin-glass regime (R1). The locations of the substituting ions are the same in (b), (d) and (e). (f) Energy difference between the FM and CE ordered states as a function of λ, for various values of y.

The eg electronic density of states (DOS) also provides insight in the effect of lattice defects. Figures 12(a)-(c) present the calculated DOS at several values of y and λ, for a fixed JAF=0.1. The DOS at small λ (=0.5) is qualitatively not modified by the lattice defects, showing the anticipated robustness of the metallic state. As for a large electron-phonon coupling λ=1.6, the DOS shows a large energy gap at the Fermi level in the clean limit y=0, corresponding to the long range CE/CO phase. A substitution of y=0.094 clearly shrinks this

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

164

K.F. Wang, S. Dong and J.M. Liu

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

gap, but yet no states at the Fermi level (although there are some states close to it) are available. However, for an intermediate coupling λ=1.2, the most exotic features in the DOS are obtained. In the clean limit, the gap is wide and obvious, but it completely vanishes at y=0.094, suggesting ensuing of a finite DOS at the Fermi level and, if Anderson localization is not considered, metallic behavior in the electronic transport. This lattice-defects induced insulator to metal transition is similar to that induced by the A-site disorder, in terms of the transport properties. Finally, we discuss the effect of electron density reduction, which should be included for the case of +3 impurity doping. Comparing with above results, the FM metallic regime shrinks while the FM cluster regime is relatively extended. Thus, the reduction of the eg electron-density suppresses both the long-range CE and FM orders and enhances the C-type AFM order. Therefore, the total combined effect of the lattice defects and electron density reduction over the clean-limit CE state is an inhomogeneous state with coexistence of shortrange FM and C-type AFM orders.

Figure 12. Calculated DOS at several values of y and λ, for a fixed JAF=0.1.

IV. Finite Size Effects in Nanostructure Manganites Recently, quite a few experimental and theoretical studies focusing on the size effect of perovskite manganites were reported, in which some fascinating effects associated with the downsizing of the materials to tens of nanometers were revealed [32-39]. One of these effects is the distabilization of the robust CO state as the ground state of bulk manganites. In consequence, a transition of the AFM order to a weak FM state was observed in both nanowires and nanoparticles [33, 34], where the key role of the surface effect was argued. The surface disordering of manganite nanoparticles would prevent the establishment of longrange CO state, while the FM order could still survive due to the short-range nature of the exchange interactions. Similar prediction was made by considering the surface PS sequence,

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Site Disorder and Finite Size Effects in Rare-Earth Manganites

165

although more relevant experimental evidence is required. In fact, it is surprising to observe such significant size effect when the materials size is only reduced down to tens of nanometers at which the conventional quantum fluctuations are far from considerable. Therefore, it would be of interest to investigate whether those manganites of the high CO stability have such significant size effect or not.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

4.1. Experimental Results In this work, we study the stability of the CO state and related magnetic behaviors of electron-doped La1-xCaxMnO3 (LCMO) nanoparticles at x~0.6, noting that earlier work was mainly on LCMO at x~0.5 where the CO state is associated with the CE phase [74]. A choice of LCMO at x=0.6 was based on the fact that LCMO is of intermediate bandwidth and shows tremendous CMR effect. For x>0.5, LCMO is electron-doped and the stability of the CO ground state increases with x and the maximal CO transition point is at x~0.6. Thus, it is believed that the energy difference between the CO state and FM state is smaller than that of the narrow bandwidth manganites. The La0.4Ca0.6MnO3 (LCMO) nanoparticles with size of 20nm (LCMO-20) and 60nm (LCMO-60) were prepared by facile sol-gel method. The detected magnetization (M) over the whole T-range is very different from one and another for these samples, no matter how the magnetic ordering is, as shown in figure 13. When the LCMO bulk sample shows its M ~10-2 emu/g, this value for the LCMO-60 and LCMO-20 is roughly two orders of magnitude larger. This tremendous difference refers to the fact that the dominant magnetic order in the LCMObulk sample is AFM-type, while it is not for the LCMO-60 and LCMO-20. For the nanoparticle samples, no clear CO transition is identified from the M-T curves. Instead, a clear PM-FM transition can be identified and the Curie point Tc for both the LCMO-60 and LCMO-20 is ~220 K, determined by the inflection point defined by the minimum of dM/dT, as shown in the insets in figure 13(b) and (c), respectively. Furthermore, for both the LCMO60 and LCMO-20, the tremendous difference between the zero-field cooled (ZFC) curve and field-cooling (FC) curve over a broad T-range indicates the nature of spin-glass-like behaviors, although we have no intention to identify it as spin glass in the strict sense. In particular, a cusp-like peak in the ZFC curve for the LCMO-20 sample can be observed, and the spin-glass-like transition temperature T=Tf ~120 K, indicated by the arrow at lower temperature in the inset of figure 13(c). However, typical FM hysteresis can be observed for the LCMO-60 and LCMO-20 samples, shown in figure 14(b) and (c), and the saturated M as large as 25 emu/g was recorded at H~0.3T. It is known that if all spins at Mn sites aligns in the FM order, the maximum spin-only ordered moment would be 3.4μB per formula unit. The measured M at T=3K and H=3T is ~1.0μB per formula unit for both LCMO-60 and LCMO-20. A general argument is that the nanoparticles or nanowires have the surface spin-disordered magnetic state composed of noncollinear spin arrangement due to the reduced coordination and broken exchange bonds between surface spins. Due to this magnetic frustration behavior, the spin-glass like surface layer would contribute to the large difference between the ZFC and FC curves as well as the cusp-like peak for the ZFC case.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

166

K.F. Wang, S. Dong and J.M. Liu

Figure 13. Measured M as function of T under condition of ZFC and FC for (a)LCMO bulk, (b) 60nm and (c) 20nm nanoparticles. The insert in (b) and (c) is the first derivative of M with respect to T. (Reproduced from Ref. [74]).

The properties of nanoparticles of another charge-ordered manganite, Pr0.57Ca0.43MnO3 are carefully studied too, and the low temperature CO AFM transition in bulk is completely suppressed in nanoperticles [75]. Besides, we also study the properties of nanostructures of other charge-ordered oxides. For example, our experiments show that the CO phase observed in bulk La1/3Sr2/3FeO3 is suppressed in nanoparticles [76]. An earlier explanation on the size effects in nanosized FM La0.7Ca0.3MnO3 attributes the enhancement of ferromagnetism to the contraction of lattice volume [77]. However, in our and some other experiments, only slight expansion of the lattice volume were observed, which can’t induce the significant enhancement of ferromagnetism. An alternative understanding seems to be required.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Site Disorder and Finite Size Effects in Rare-Earth Manganites

167

Figure 14. Measured M as function of H for (a) LCMO bulk at 3K (main panel) and at 50K (the inset), (b) 60nm and (c) 20nm nanoparticles at 3K.

4.2. Theoretical Approach Let’s start with a phenomenological model. For simplicity, the CO phase is specified as the CE-type AFM-CO phase. In the CE ground state, Mn cations form FM zigzag chains in X-Y plane, with AFM couplings between the neighboring chains in X-Y plane and neighboring sites along Z axis. For this spin structure, each Mn has four AFM bonds and two FM bonds with its nearest neighbors (NN). The NN superexchange contribution is JAFSiSj, i.e. (2-4)JAF/2=-JAF per cell in the CE phase, where JAF is the superexchange interaction and spin S of t2g electrons is simplified as a classical unit. In contrast, the spins in the FM phase are all

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

168

K.F. Wang, S. Dong and J.M. Liu

parallel, which contribute 6JAF/2=3JAF per cell to the energy. The energy per cell for the CO and FM phases in bulk form can be written as respectively:

ECO = EkCO − J AF EFM = EkFM + 3J AF

,

(6)

where Ekx accounts the energy terms from all other interactions in phase x (x=FM or CO), including the double exchange, Jahn-Teller distortion, intra-site Coulomb repulsion and so on. When the grain size is down to nanoscale, the surface relaxation becomes significant due to increased surface/volume ratio. For manganites, those Mn cations on the surface layer have only five neighbors rather than six, as shown in figure 15, i.e. the AFM superexchange is relaxed on the surface layer. The energy of each surface cell will be raised JAF/6 for the CE phase while lowered JAF/2 for the FM phase. This effect reduces the energy gap between the CO and FM phases, and may destabilize a pure CO phase into a phase-separated state. The grain surface may favor FM state rather than CO state. For simplification, a phase-separated state in a core-shell structure: a CO core wrapped by a FM shell, will be discussed, as shown in figure 15. In the sphere approximation, the core radius is rc while the particle radius is r (0 TC ) , the presence of the field magnetization S

z

h is necessary for deduced the

. We shall explicitly consider the case of a sufficiently small field, and

defined the magnetic susceptibility as:

⎡ Sz ⎤ ⎥ χ (T ) = lim ⎢ h →0 ⎢ g μ h ⎥ B ⎣ ⎦ In the limit h → 0 , the quantity

gμ B h S

z

(9)

will approach the inverse susceptibility

χ −1

will

approach the inverse susceptibility; whereas the correlation function approaches its isotropic value

S (S + 1) , therefore the implicit equation for χ (T ) in the paramagnetic phase may be 3

evaluated in the RPA [65,66] as follows:

χ −1 (T ) = τ

with τ =

(

1 1 + χϕ

(10)

G q

)

3k BT et ϕ qG = J 0 − J qG . S (S + 1)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

T is the absolute temperature and J q is the Fourier transform of the exchange integrals J ij defined by: J qG = The symbol ...

G q

1 N

∑ J ij exp[iq (Ri − R j )] G G

G

(11)

G

is the average value when the wave vector q runs over its N allowed

values in the first Brillouin zone. At high-temperature, the magnetic susceptibility goes to zero; we can then expand the relation (10) as:

χ −1 (T ) = τ

1 1 + χϕ

G q

= τ 1 − ϕχ − ...

G q

(12)

To zero-order approximation, we have:

χ (T ) = τ −1

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(13)

Study of the Magnetic Properties of the Semiconductors…

211

By replacing (13) into (10) and expanding this latter to the first-order, we obtain:

(χτ )−1 =

1−

ϕ − ... τ

G q

= 1−

ϕ

G q

− ...

τ

(14)

we replace again (14) into (10) and using the same procedure, we get :

(χτ )

−1

= (1 − ϕ

) 1−

(ϕ −

ϕ

) (ϕ −

τ

−

ϕ

)2

τ2

− ...

(15)

After some algebra we arrive at the following expression of the magnetic susceptibility:

χ where: A1 =

−1

∞ ⎡ ( − 1) p A p ⎤ = τ ⎢1 + ∑ ⎥ p ⎣⎢ p =1 τ ⎦⎥

ϕ A2 = ϕ 2 − ϕ

2

A3 = ϕ 3 − 2 ϕ 2 ϕ + ϕ

3

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

A4 = ϕ 4 − 3 ϕ 3 ϕ + 3 ϕ 2 ϕ For

2

+ ϕ

4

p ≥ 1 , a possible expression of Ap can be generalised in the form A p = (− 1)

where

(16)

p −1

ϕ

p

p −2

+ ∑ C kp −1 (− 1) ϕ p −k ϕ k

(17)

k =0

C nm are binomial coefficient defined by: C nm =

If we take into account all the

k

n! , for n ≥ m . m!(n − m )!

nn and nnn exchange interactions J 1 and J 2 , ϕ qG

takes the form :

G

G

ϕ qG = z1 J1 (1 − γ 1 (q )) + z 2 J 2 (1 − γ 2 (q )) G

(GG)

γ i (q ) = ( zi )−1 ∑ exp iqδ i δi

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(18) (19)

212

δi

R. Masrour and M. Hamedoun is the vector connecting ith nearest-neighbours, the total number of such neighbours, to a

given ion, being

G

z i . The factor γ i (q ) depends on the lattice geometry. In the case of spinel

structure the method should be developed for four interpenetrating sub-lattices. Unfortunately, mathematical difficulties arise and it becomes progressively more difficult to obtain the analytic solution of the problem, especially when one include the nnn exchange coupling. The inverse of the 4× 4 matrices of the susceptibility is more complicated and no mathematical approximation can be used because of the translational invariance of the green functions. However, one may approximate the spinel structure by a simple cubic lattice which has the same first and second coordinate numbers as the spinel the factor structure [67]. G γ i q is defined by:

( )

G

γ 1 (q ) = G

γ 2 (q ) =

[

]

( )

1 cos(q x a ) + cos q y a + cos(q z a ) 3

[ ((

))

((

(20)

) )]

1 ∑ cos q x + εq y a + cos((q x + εq z )a ) + cos q z + εq y a 6 ε =±1

(21)

By replacing the summation by an integral over the three-dimensional Brillouin zone, we can write that:

G

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

ϕ (q )

G q

=

1 N

G

G

∑G ϕ (q ) → 8π 3 ∫∫∫ϕ (q )dq x dq y dq z V

(22)

q

The summation is to be taken for q x , q y , q z running independently between −

+

π a

.

π a

and

V volume of the unit cell and a is the lattice parameter. γn =

The relationship between the reduced correlation function inverse magnetic susceptibility

γn =

S (S + 1)

and the

χ −1 is given by [64, 65]: G G S 0 .S n

S (S + 1)

=τ

Using the previously evaluated series of correlation functions; with

G G S 0 .S n

γ aa , γ ab

and

[ (

G G G exp iq. Rn − R0

ϕ+χ

−1

)]

(23) G q

χ −1 , we may readily calculated the three first

γ ac

:

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

213

γ aa = 2γ 1 + 4γ 3 is the in plane correlation. γ ab = 4γ 1 + 8γ 2 is the correlation between neighbouring planes. γ ac = 4γ 2 + 8γ 3 is the correlation between the second-neighbour planes. We have computed these correlation functions as a function of temperature T and for different composition x to order 6 in

β=

1 . The sums over the brillouin zones were k BT

performed using Gauss approximation quadrature method similar to that given in the appendix of reference [65]. In recent work [68, 69, 44], a relation between the correlation length ξ (T ) and the correlation functions is given in the case of the case of diluted magnetic

G

B-spinel lattice with a particular ordering vector q (0,0, q ) . 2 2 1 ⎛⎜ 1 γ ab ⎛ξ ⎞ = − + γ ⎜ ⎟ G ac 16 γ ac 8S (q0 ) ⎜⎝ ⎝a⎠

G

G

⎞ ⎟ ⎟ ⎠

(24)

G

G

G

with S (q 0 ) is the structure factor at q = q 0 . In the ferromagnetic case we get q 0 = 0 . The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility

χ is that the neighbour hood of the critical point the above two

functions exhibit an asymptotic behaviour:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

χ (T ) ∝ (T − TC ) −

γ

(25)

ξ 2 (T ) ∝ (TC − T ) −2ν

(26)

The [M,N] P. A to the series F (β ) is a relational fraction polynomials, of degree M and N in β, satisfying: F (β ) =

PM with PM and Q N , QN

)

(

PM + θ β M + N +1 . The QN

sequence of [M,N] P. A to both the Log (χ (T )) and Log

(ξ

2

(T )) ,

was found to be

convergent. The simple pole correspond to the residues to the critical exponents

γ and ν .

II.4. High-Temperature Series Expansions (Classical Method) In order to deduce the expression of the susceptibility of the system with two sub-lattices (ferrimagnetic), the Hamiltonian of the Heisenberg with extern field hex may be put in the form:

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

214

R. Masrour and M. Hamedoun GG H = −2 J AA ∑ Si Si ' − 2 J BB i ,i

G

where S and

`'

GG

G G

⎛

⎞

∑ σ jσ j − 2 J AB ∑ Siσ j − μ B hex ⎜⎜ g A ∑ Siz − g B ∑ σ zj ⎟⎟ '

j, j

'

i, j

⎝

i

G

G

(27)

⎠

j

G

σ are spin vectors of magnitudes S 2 = S (S + 1) and σ 2 = σ (σ + 1) in sub-

lattice A and B respectively. g A and g B are the corresponding gyromagnetic factors and μ B is the Bohr magneton. hex is an external magnetic field (z direction) introduced in order to provide an easy determination of the magnetic susceptibility. The first summation is over all spin pairs nearest-neighbours in sub-lattice A, the second is over all spin pairs nearest-neighbours in sub-lattice B and the third is between all spin pair nearest-neighbours in A an B . J AA , J BB and J AB are the intra and the inter-sub-lattice exchange interactions between neighbouring spins. The magnetisation of the ferrimagnetic spinels materials is:

⎛ M = μ B ⎜⎜ g A ∑ S iz − g B ∑ σ zj i j ⎝

⎞ ⎟ ⎟ ⎠

(28)

⎛ ∂M ⎞

⎟⎟ After computing the first derivative of magnetisation χ = ⎜⎜ , we obtained h ∂ ⎝ ex ⎠ hex →0

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

the general expression of susceptibility for the collinear normal ferrimagnetic spinel as follows:

χ=

μ B2 ⎛

⎞ ⎟ (29) ⎟ ⎠

GG GG G G 2 2 2 2 2 2 ⎜ N A g A S + N B g Bσ − g A ∑ Si Si \ − g B ∑ σ jσ j \ − 2 g A g B ∑ Siσ j 3k BT ⎜⎝ i, j i ≠i \ j≠ j\

where N A and N B are respectively the number of ion and the spin value of each type of spin. Finally, we obtain simple form

χ=

μ B2

(N 3k T

2 2 A g AS

+ N B g B2σ 2 − N A g A2 γ AA − N B g B2 γ BB − 2 N B g A g Bγ BA

)

(30)

B

Following the procedure in [70-76, 51], we compute the expressions of spin correlation functions γ AA , γ BB and γ AB in terms of powers of β and mixed powers of J BB , J AB and

J AA . The correlations functions γ AA ,

γ BB and γ AB may be expressed as follows:

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

K

K

7

K

K

∑ ∑ ∑ a ( m, n, p, q )J

m=0

n=0

p =0

7

q

q −m

q −m−n

q =1

K

7

∑ ∑ ∑ b ( m, n, p, q ) J

m=0

γ AB = 2m+ n + p σ 2 m+ n +1S m+ 2 p +1 ∑ q =1

with S =

q −m−n

q =1

γ BB = 2m + n + p σ 2 m + n + 2 S n + 2 p ∑ K

q −m

q

γ AA = 2m + n + p σ 2 m + n S m+ 2 p + 2 ∑

n=0

p =0

q−m

q

n AB

m p J BB J AA βq

n AB

m p βq J BB J AA

q−m−n

∑ ∑ ∑ c ( m, n, p, q )J

m=0

n=0

215

p=0

n AB

m p J BB J AA βq

S( S + 1 ) , σ = σ ( σ + 1 ) .

Table 1. Nonzero coefficients c(m , n , p , q ) of the correlation function

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(m,n, p,q)

c (m,n, p,q)

(m,n, p,q)

(1,0,0,1)

2

(0,1,1,2)

8/3

(1,1,0,2)

γ BA .

c (m,n, p,q)

(m,n, p,q)

(0,1,4,5)

104/45

(1,3,3,7)

872398/3645

(0,3,2,5)

87622/2025

(3,1,3,7)

226472/18225

34/9

(2,1,2,5)

80/9

(0,1,6,7)

81584/42525

(1,1,1,3)

16/3

(0,3,3,6)

1019216/18225

(0,5,2,7)

171526118/382725

(2,1,0,3)

172/27

(2,3,1,6)

221953/729

(0,6,1,7)

16/567

(0,1,2,3)

8/3

(3,1,2,6)

1792/135

(1,6,0,7)

68/2835

(0,3,0,3)

494/45

(2,1,3,6)

10112/1215

(5,2,0,7)

8/315

(0,3,1,4)

746/27

(1,5,0,6)

2149376/6075

(0,3,4,7)

126088366/1913625

(3,1,0,4)

3868/405

(0,5,1,6)

3860318/18225

(2,5,0,7)

483802024/382725

(2,1,1,4)

80/9

(3,3,0,6)

4452341/18225

(6,1,0,7)

2038348/76545

(1,1,2,4)

16/3

(4,1,1,6)

848/45

(3,3,1,7)

35844608/54675

c (m,n, p,q)

(0,1,3,4)

112/45

(0,1,5,6)

2032/945

(4,1,2,7)

848/45

(1,3,0,4)

1195/27

(5,1,0,6)

808316/42525

(2,1,4,7)

9424/1215

(2,3,0,5)

701276/6075

(1,3,2,6)

3332276/18225

(6,0,1,7)

16/2835

(3,1,1,5)

1792/135

(1,1,4,6)

1888/405

(1,1,5,7)

1744/405

(4,1,0,5)

5492/405

(4,3,0,7)

41712593/91125

(0,7,0,7)

192541946/637875

(0,5,0,5)

33316/567

(5,1,1,7)

482752/18225

(0,2,5,7)

8/567

(1,3,1,5)

27803/243

(2,3,2,7)

135323753/273375

(1,1,3,5)

2024/405

(1,5,1,7)

2506351409/1913625

Nonzero coefficients a ( m, n, p, q ) , b ( m, n, p, q ) on

(31)

and c ( m, n, p, q ) up to order 7

β are given in Ref [51] (tables 1, 2 and 3). We use the powerful P.A method [48], to

estimate the critical temperature. In this method, the Curie point is determined by locating the singularities in the P.A method to the HTSEs of the magnetic susceptibility. The excellent reviews of these methods are available in Refs [69, 70]. In the ferromagnetic (or antiferromagnetic) and in the spin glass phases, we have used the results of the magnetic susceptibility χ(T ) and the correlation length obtained by Ref [43]:

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

216

R. Masrour and M. Hamedoun 6

n

χ (T ) =

∑ ∑ a(m, n) y mτ n

(32)

m = − n n =1

ξ 2 (T ) =

6

n

∑ ∑ b(m, n) y mτ n

(33)

m = − n n =1

where

τ=

J 2 S ( S + 1) J1 and y = 2 . The series coefficients a (m, n ) and b( m, n) are k BT J1

given in Ref [43].

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Table 2. Nonzero coefficients b(m , n , p , q ) of the correlation function γ BB . (m,n, p,q)

b (m,n, p,q)

(m,n, p,q)

b (m,n, p,q)

(m,n, p,q)

b (m,n, p,q)

(1,0,0,1)

2

(1,4,0,5)

1416524/6075

(1,6,0,7)

630901223/382725

(2,0,0,2)

10/3

(0,2,3,5)

4024/405

(0,2,5,7)

218516/25515

(0,2,0,2)

22/3

(5,0,0,5)

15616/1575

(0,4,3,7)

11784542/54675

(3,0,0,3)

224/45

(1,2,2,5)

9856/243

(0,6,1,7)

99633071/127575

(1,2,0,3)

739/27

(6,0,0,6)

592664/42525

(4,2,1,7)

148310524/382725

(0,2,1,3)

92/9

(0,2,4,6)

11216/1215

(2,2,3,7)

1849297/18225

(0,4,0,4)

5528/135

(2,4,0,6)

325121/405

(1,4,2,7)

268887851/273375

(0,2,2,4)

284/27

(1,2,3,6)

47072/1215

(1,2,4,7)

131708/3645

(4,0,0,4)

106/15

(1,4,1,6)

11051807/18225

(3,4,0,7)

4108034257/1913625

(1,2,1,4)

1049/27

(2,2,2,6)

127876/1215

(5,2,0,7)

24520799/54675

(2,2,0,4)

3097/45

(3,2,1,6)

758651/3645

(7,0,0,7)

12441284/637875

(2,2,1,5)

40231/405

(0,6,0,6)

1097084/5103

(3,2,2,7)

489181/2187

(3,2,0,5)

172586/1215

(0,4,2,6)

2998372/18225

(2,4,1,7)

4669868/2187

(0,4,1,5)

125788/1215

(4,2,0,6)

3702346/14175

In the spin-glass (SG) region, critical behaviour near the freezing temperature J SG is expected in the nonlinear susceptibility χ s =

χ − χ 0 rather than in the linear part χ 0 of the

dc susceptibility χ . This is due to the fact that the order parameter q in the SG state is not the magnetization but the quantity q =

[ Si N∑ 1

2

i

]

.

As was suggested by Edwards and

av

Anderson, [77] leading to an associated susceptibility χ s = the correlation length of the correlation function

[SS ] 2

i

j

1

⎡ ss i j 3 ∑⎢ ⎣ NT ij

2⎤

⎥⎦ av

, where

possibly diverges at T = TSG . We

use the powerful P.A method [48] to estimate the critical temperature. In this method, the Curie point is determined by locating the singularities in the P.A method to the HTSEs of the Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

217

magnetic susceptibility. The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility χ (T ) is that the neighbourhood of the critical point the above the following functions exhibit the asymptotic behaviour:

χ (T ) ∝ (T − TN ) −

γ

(34)

The usual approach is to compute the series for the logarithmic derivative of χ (T ) ,

d −γ log ⎡⎣ χ (T ) ⎤⎦ ≈ dT T − TN

(35)

as this function has a simple pole TN and should be well represented by Padé approximants [ M , N ]. The exponent

γ is then re-estimated from the approximates to

(T − TN )

d log ⎡⎣ χ (T ) ⎤⎦ dT

(36)

evaluated at T = TFerriM .

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Table 3. Nonzero coefficients a (m , n , p , q ) of the correlation function

γ AA .

(m,n, p,q)

a (m,n, p,q)

(m,n, p,q)

a (m,n, p,q)

(m,n, p,q)

a (m,n, p,q)

(0,0,1,1)

4/3

(0,4,1,5)

706946/6075

(3,2,3,7)

915829/6075

(0,2,0,2)

53/9

(1,4,0,5)

167294/1215

(5,2,0,7)

6620456/382725

(0,0,2,2)

4/3

(0,0,5,5)

1016/945

(2,4,1,7)

386566502/273375

(1,2,0,3)

116/9

(0,2,3,5)

88/3

(1,6,0,7)

675001052/637875

(0,2,1,3)

404/27

(1,2,3,6)

9434/135

(0,2,5,7)

4803382/127575

(0,0,3,3)

56/45

(0,2,4,6)

867712/25515

(0,4,3,7)

84221632/212625

(1,2,1,4)

2788/81

(2,4,0,6)

2254517/6075

(0,6,1,7)

301410296/382725

(2,2,0,4)

1802/81

(0,0,0,6)

41152/42525

(4,2,1,7)

1496656/10935

(0,2,2,4)

1028/45

(3,2,1,6)

340796/3645

(2,2,3,7)

50792/405

(0,4,0,4)

4364/135

(1,4,1,6)

9330116/18225

(1,4,2,7)

59782103/54675

(0,0,4,4)

52/45

(2,2,2,6)

7750/81

(1,2,4,7)

6224368/76545

(2,2,1,5)

14620/243

(0,6,0,6)

860921/5103

(3,4,0,7)

44023634/54675

(3,2,0,5)

41386/1215

(0,4,2,6)

885472/3645

(0,0,7,7)

21856/25515

(1,2,2,5)

7267/135

(4,2,0,6)

179194/3645

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

218

R. Masrour and M. Hamedoun

II.4. Replica Method To study the critical phenomena in quenched random spin systems Sarbach established in 1980 [78] a method based on evaluating the mean field free energy. He applied the replica method to Ising simple-cubic systems with quenched random interactions. The free energy taking into account the randomness must be averaged over all configurations of the disorder and in a straightforward way is deduced by using the variational principle. This gives the parameters one needs to characterise the magnetic phase diagram. In this work, we have applied this procedure to the diluted diamagnetic ions spinellattices Ax A'1− x B2 X 4 with competitive interactions. We have used the XY model of the Hamiltonian of the systems of interest in this paper. The Hamiltonian has the form:

H =−

G G G G 1 J ijm S i ⋅ S j + ∑ H j ⋅ S j ∑∑ 2 ij m j

(37)

The sum over ij includes all the n.n and the n.n.n pair interactions J 1 ( x ) and K ( x )

respectively. If i and j are n.n J ij = J 1 ( x ) and if i and j are n.n.n J ij = K ( x ) .

G G S m is the spin operator at site m with two components Sm = (Smx , Smy ) . We redefine the G G G G G spin S m by the spin vector unity U such that: S m = S mU where S m is the module of S m . G G G G The magnetic field H j is defined by a vector unity V j such that H j = hV j , where h is G G the module of H j . H j is introduced to provide an easy derivation of the magnetic Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

susceptibility with probability distribution

P( H j ) = δ ( H j − H )

(38)

A kind of calculating the average free energy is the replica method. The problem is reduced to calculate a partition function of n identical replica of the original system, Λ = {1,2,....n} then:

Z (n) = Z n = Tr exp(− H eff ) ,

(39)

Nn

where H eff is the effective Hamiltonian. Using the following properties k n G G ⎛ n G G ⎞ ⎜ ∑U i ⋅U j ⎟ = ∑ a kr (n ) ∑ U i ⋅U j R =r r =0 ⎝ α =1 ⎠

(

)

R

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(40)

Study of the Magnetic Properties of the Semiconductors… k n G G ⎛ n G G ⎞ ⎜ ∑Vi ⋅U j ⎟ = ∑ bkr (n ) ∑ Vi ⋅U j r =0 R =r ⎝ α =1 ⎠

(

219

)

R

(41)

the effective Hamiltonian H eff can be written as, R being a subset of Λ of r elements :

H eff = −

G G G G 1 J ij (r , n)(U i ⋅U j ) R − ∑ ∑ H l (r , n) Vl ⋅U l ∑ ∑ 2 ij R⊂Λ l R⊂Λ

(

)

R

(42)

∞

1 c J ij (k , n)akr (n) , k =1 k!

(43)

1 c H l (r , n)bkr (n) . k =1 k!

(44)

J ij (r , n) = ∑ H l ( r , n) = ∑ c

c

where J ij (k) and H l (k) are the k-th cumulants as defined as usual, see e.g. [79]. In our case where r = 1 corresponds to the long range order and r = 2 to the S.G phase we have obtained the coefficients akr (n) and bkr (n ) as [52 (see annex 1):

⎤ ⎡⎛ I (λ ) ⎞ r ⎟⎟ (I 0 (λ ))n ⎥ a kr (n) = bkr (n ) = Dλ ⎢⎜⎜ 2 1 ⎥⎦ ⎢⎣⎝ I 0 (λ ) ⎠ k

(45)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

k

where I0, I1 are the Bessel functions and Dλ designs the differential operator at the point

λ =0. To compute the free energy of the mean field theory F, we have defined the trial operator density ρ T ,

F = lim min φ N (n, σ (k )) n →0

ρ

(46)

T

where

φ N (n, {σ (k )}) = and the n order parameters

1 [Tr ρ T H eff + Tr ρ T Logρ T ] Nn Nn Nn

(47)

σ (k ) obey the variational principle dφ N (n, {σ (k )}) = 0, dσ (k )

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(48)

220

R. Masrour and M. Hamedoun

for all

σ (k ) (k=1, 2, …., n.). The field H (k , n) is coupled to each σ (k ) in (48) and

consequently “n” susceptibility can be derived to be

χ ( k ,n ) =

∂σ (k) ∂H(k,n)

In the lim , and limiting ourselves to the order 2 in n→0

where all

(49)

σ ( k ) near the critical transition

σ (k ) are very small, the magnetic susceptibility has the form χ ( q, k ) =

e( k ) 1 − d (k ) J q (k )

(50)

where

GB G G ⎡ 1 G σ Tr ⎢ ∑ ∑ (b) U j j V jU j σ σ j (k) nj ⎣ K⊂ ΛΛ B⊂ΛΛ (2π )n e( k ) = GBG G K⎤ ⎡ 2 1 G Tr σ (b) U σ (k) U ∑ ⎢ j j j j ⎥ σ 2j (k ) nj ⎣B⊂Λ (2π )n ⎦

(

1

)

K

⎤ ⎥ ⎦

GBG G A G G K⎤ ⎡ G 1 1 σ (b) U σ (a) U Tr ⎢ ∑ ∑ ∑ G j i i j (U i U j ) ⎥ σ σ i (k)σ j (k) nij ⎣ K ⊂ ΛΛ B⊂ ΛΛ A ⊂ ΛΛ (2ππ )2n ⎦ d (k ) = GBG G K⎤ ⎡ 2 1 G Tr ⎢ ∑ σ j (b)U j σ j (k)U j ⎥ σ 2j (k ) nj ⎣ B⊂ Λ (2π )n ⎦

(51)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

G

(52)

in the case of k=1 and k=2 we have e(k=1)=1/2

d(k=1)=1/4

e(k=2)=1/2

d(k=2)=1/2.

In (15) J q (k ) is the Fourier transform of exchange integrals for a magnetic structure

G

G

with a wave vector q . The order phase corresponds to q = (0, 0, q 0 ) . At the temperature

Tq (k ) where the denominator [ 1 − d(k)J q (k) ] vanishes in (50) the paramagnetic phase becomes unstable. As usual in mean field theory, the largest temperature T0 is the relevant critical temperature i.e:

T0 = max max Tq (k ) k

q

The numeric program uses to calculate the critical temperature is given in annex 2.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(53)

Study of the Magnetic Properties of the Semiconductors…

221

III. Applications for Different Semiconductors Magnetic Materials III. 1. Semiconductors Spinels Material III.1.A. Mean Field Theory and Probability Law -Semiconductors materials Lix Zn1− xV2O4 , Cox Fe1− x Cr2 S 4 and CoFe2− 2 x Cr2 x O4

To determine the exchange integrals J 1 ( x ) and J 2 ( x ) in the antiferromagnetic phase, we have used the Eqs (4) and (5), with using the experimental values of TN and

θ p obtained by

magnetic measurement [18] for the spinels systems Lix Zn1− xV2O4 . From these values, we have derived the variation of the intra-plane coupling and the coupling between nearest and next-nearest plane in the spinels Lix Zn1− xV2O4 materials with 0.95 ≤ x ≤ 1 . The obtained values of J 1 ( x ) and J 2 ( x ) are given in table 4 for this material.

For determinate the exchange interactions in the all dilutions, we have used the probability law, given by Eq(7). Lix Zn1− xV2O4 is diluted spinel LiV2O4 material with nn exchange interactions J1 = 4.37 K and nnn super-exchange interactions J 2 = −1.78 K and ZnV2O4 with J1 = 3.33 K and J 2 = −1.66 K . The obtained values of J 1 and J 2 are given in the table 4, for 0 ≤ x ≤ 1 . By mean field theory and probability law, the values of the corresponding classical

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

exchange energy

E [61] for the magnetic structure are given (see table 4). The values of kB S 2

the intra-plane and inter-plane interactions J aa = 2J 1 , J ab = 4 J 1 + 8 J 2 , J ac = 4J 2 , respectively and the ratio of inter to intraplanar interactions

J int er J + J ac for the first = ab J int ra J aa

method (MFT) for x = 0 , 1 and for the second method (PL) in 0 ≤ x ≤ 1 are obtained. The results obtained are presented in table 4. For calculated the exchange interactions J AB ( x) of the Cox Fe1− x Cr2 S 4 and

CoFe2−2 x Cr2 x O4 materials, we have used the Eq(8). The obtained results are given in tables 5 and 6. CoFe2− 2 x Cr2 x O4 is diluted spinels materials of CoCr2O4 and CoFe2O4 with the values of the exchange integrals:

J Co −Cr = −51.47 K

for

CoCr2O4

J Co − Fe = −22.7 K for CoFe2O4 [81].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

[80] and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Table 4. The Néel temperature TN (K ) (or the freezing temperature TSG ( K ) ), the Curie-Weiss temperature θ p , the values of the first and the second exchange interactions obtained by mean filed theory (MFT) and probability law (PL)of Lix Zn1− xV2O4 as a function of dilution x

TN (K )

x

(K)

Or TSG [18]

θ (K ) [18]

J aa kB

J ab KB

J ac kB

J1 (K) kB

J2 (K ) kB

J1 (K ) kB

J2 ( K) kB

(MFT)

(MFT)

(PL)

(PL)

(K )

(K )

(K )

Jab +Jac Jaa

E kBS2

( K)

1

17

500

4.37

-1.78

4.37

-1.78

8.74

3.24

-7.12

0.44

4.86

0.9

-

-

-

-

4.25

-0.12

8.50

16.04

-0.48

1.83

24.06

0.8

6

-

-

-

4.14

0.82

8.28

23.12

3.28

3.18

34.68

0.7

9.37

-

-

-

4.03

1.32

8.06

26.69

5.28

3.96

40.3

0.6

11.63

-

-

-

3.92

1.55

7.85

28.11

6.20

4.37

42.17

0.5

-

-

-

-

3.82

1.61

7.64

28.16

6.44

4.52

42.24

0.4

13.77

-

-

-

3.71

1.53

7.42

27.08

6.12

4.47

40.62

0.3

-

-

-

-

3.61

1.30

7.23

24.88

5.20

4.16

37.32

0.2

14

-

-

-

3.51

0.81

7.02

20.52

3.24

3.38

30.78

0.1

14

-

-

-

3.42

-0.09

6.84

12.96

-0.36

1.84

19.44

0.05

25

-

-

-

3.37

-0.77

6.74

7.32

-3.08

0.63

10.98

0

40

-420

3.33

-1.66

3.33

-1.66

6.66

0.04

-6.64

-0.99

0.06

Study of the Magnetic Properties of the Semiconductors…

223

Cox Fe1− x Cr2 S4 is diluted spinels materials of CoCr2 S4 and FeCr2 S4 with the value of the exchange integrals J Co−Cr = −17.5 K for CoCr2 S 4 and J Fe−Cr = −10 K for

FeCr2 S 4 , and given by Ref[82]. The interactions energies Ei (i = 1 − 3) are deduced by using the expressions E1 = E2 = 5 J AA − 3 J AB and E3 = 3 J BB − 3 J AB given by [83]. In CoCr ( Fe )2 O4 and Fe ( CO ) Cr2 S 4 materials, the exchange interactions J Fe− Fe and

J Co −Co are negligible. Table 5. the absolute value of the exchange interactions J AB ( x) for the spinels CoFe2− 2 x Cr2 x O4 materials.

x

J AB ( x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

22.7 24.62 26.72 29.02 31.52 34.24 37.18 40.37 43.80 47.50 51.47

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Table 6. the absolute value of exchange interaction J AB ( x) and the exchange energies Ei (i = 1 − 3) for the materials Cox Fe1− x Cr2 S 4 .

x

J AB ( x)

E1 ( K )(= E2 ( K ))

E3 ( K )

0

10

30

27.15

0.1

10.54

31.63

31.74

0.3

11.75

35.27

38.36

0.5

13.14

39.44

43.64

0.7

14.73

44.18

47.92

0.8

15.60

46.79

49.01

0.9

16.52

49.56

48.57

1

17.5

52.5

45.6

III.1.B. High Temperature Series Expansions To determine the magnetic phase diagrams of the spinels materials Lix Zn1− xV2O4 ,

Cox Fe1− x Cr2 S4 and CoFe2−2 x Cr2 x O4 , we have used the HTSE combined the P.A Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

224

R. Masrour and M. Hamedoun

approximant. For the first material, we have used the HTSEs quantum and classical methods

⎛

of the magnetic susceptibility and correlation length to order sixth in β ⎜ β =

⎝

1 ⎞ ⎟ . The k BT ⎠

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

P.A approximant applied to the Eqs (25), (32) and to the Eqs (26) and (33). In figure 1, we have presented the magnetic phase diagram of the Lix Zn1− xV2O4 material.

Figure 1. Magnetic phase diagram of (PM), phase

antiferromagnetic

( 0.1 < x ≤ 0.8)

(AFM)

Lix Zn1− xV2O4 . The various phases are the paramagnetic phase phase ( 0 ≤ x ≤ 0.1 , 0.8 < x ≤ 0.9 ) and spin glass

. The solid circles are the present results (given by HTSE method). The solid

squares represent the experimental points deduced by magnetic measurements [18] and the solid triangle represents the result obtained by Replica method.

For the ferrimagnetic spinels material CoFe2− 2 x Cr2 x O4 , the P.A approximant has applied to the magnetic susceptibility given by Eq (30), for determinate the ferrimagnetic (ferriM) temperature TN . In figures 2 and 3, we have given the magnetic phase diagram of the

CoFe2−2 x Cr2 x O4 and Cox Fe1− x Cr2 S4 materials, respectively.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

225

Figure 2. The magnetic phase diagram of the systems CoFe2− 2 x Cr2 x O4 . The circles and the squares

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

represent the results given by: the HTSE method and magnetic measurements [25], respectively.

Figure 3. The magnetic phase diagram of the

Cox Fe1− x Cr2 S4

systems. The circles and the squares

represents the results given by: the results given by HTSE method and magnetic measurements [18], respectively.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

226

R. Masrour and M. Hamedoun In other hand, we have calculated the values of critical exponent γ as function of intra-

sub-lattices exchanges integrals J AA and J BB , and for arbitrary values of spins for the series up to order 7. This procedure was repeated for series up to order 6. The obtained value of γ is similar to that obtained above for the order 7. For short series n ≤ 5 the accuracy in the calculation is not expected to be high. The gyromagnetic factors g A and g B was assumed to be equal to 2. For all the cases, J AB = −1 K . It is necessary to point out here that we will not take into account the stability of the spin configuration. First, we analyse the case where J AA is weaker (i.e. J AA = 0 ). The behaviour of

γ

with J BB is reported in figure 4. From this figure we can see that the critical exponent (i) decreases rapidly with increasing antiferromagnetic J BB value until a minimum (γ = 1.2433) , (ii) increases and tends to be constant

(γ

= 1.3838) for large

ferromagnetic value of J BB . To examine this variation, we display in the same figure the behaviour

of

the

ratio

R of inter-plane correlation to the intra-plane G G ⎛ ∑ S0 Si ⎞⎟ ⎜ i∈ int erplan correlations ⎜ R = G G ⎟ . We remark that for large ferromagnetic value of J BB S ∑ ⎜ 0S j ⎟ j∈ int raplan ⎝ ⎠ and weak values of R , γ is close to one of 3D Heisenberg model [84, 85]. There is no interaction between the spins in sublattice A and B. The system can be considered as isotropic. When the effect of inter-plane anisotropy is more pronounced, R increases, γ takes the value of Ising-like system [84]. For the large values of R ( J BB < −1 K ), the Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

frustration becomes very important and will be responsible of a net divergence of γ .

Figure 4. The critical exponent

γ

(Solid line) and R , representing the ratio of inter to intraplanar

correlations, (dashed line) as function of the intra sub-lattice

J BB

for

J AA = 0

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

227

In figure 5, we present the case when J AA is equal to J BB . The dependence of the critical exponent γ on the intra-sub-lattice exchange coupling is similar to this of figure 1. The minimum of

γ is 1.3259 . This value is somewhat similar to that of known XY model

[80]. From the plot of ratio R we note that the intra-plane sub-lattice correlations are important in the antiferromagnetic exchange integral J BB region and as consequence, the γ

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

takes the value of the planar model. For J BB < −0.25 , γ diverges as a consequence of the strong frustration between different interactions in sub-lattices.

Figure 5. As in Fig. 3 but with J AA

= J BB .

We have also examined the case where J AA = − J BB . Figure 6 illustrates the critical exponent versus the intra-plane exchange integral J BB . It can be seen that the curve has two minimums ( γ = 1.2747 and 1.2938 ) and diverges for large value of J BB when the frustration is strong. The major frustrations among spins arise from the competition between the ferromagnetic and antiferromagnetic interactions within and between spins in sub-lattice A and B. The ratio R governs the behaviour of γ . Finally, we apply this model to magnetic spinel semiconductors FeCr2 S 4 and

CoCr2 S 4 . Both systems are normal spinel ferrite with collinear configuration. The physical parameters are taken from reference [87] and are given by: (i) for FeCr2 S 4 , the exchange coupling are J Fe−Cr = −10 K , J Cr −Cr = −0.95 K , the gyromagnetic factors are

g Cr = 1.98 and g Fe = 2.1 . The system presents the

ferrimagnetic order below the critical temperature Tc = 177 K . (ii) for CoCr2 S 4 ,

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

228

R. Masrour and M. Hamedoun

Tc = 240 K , J Co−Cr = −17.5 K , J Cr −Cr = −2.3 K and g Co = 2.3 . In the two systems the interaction between Fe-Fe and Co-Co is negligible.

Figure 6. As in Fig. 3 but with J AA

= − J BB .

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The sequences of [ M , N ] Padé approximants to the series have been evaluated and are presented in tables 7 and 8. The estimated critical temperatures are in good agreement with the experimental values found in FeCr2 S 4 and CoCr2 S 4 . Table 7. The critical temperature and critical exponent

γ for

magnetic susceptibility of FeCr2 S 4 . [M,N]

[3,2]

[4,2]

[3,3]

[4,3]

[1,4]

[2,4]

[1,5]

[2,5]

Tc

176.788

176.824

178.556

178.797

176.725

178.587

176.889

178.730

γ

1.307

1.309

1.321

1.307

1.329

1.326

1.322

1.322

γ for

Table 8. The critical temperature and critical exponent magnetic susceptibility of CoCr2 S 4 . [M,N]

[3,2]

[4,2]

[3,3]

[4,3]

[1,4]

[2,4]

[1,5]

[2,5]

Tc

239.393

241.349

240.303

239.725

241.421

240.267

240.447

239.429

γ

1.300

1.312

1.326

1.300

1.330

1.327

1.326

1.330

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

229

III.1.C. Replica Method In this subsection, we have estimated the phase diagrams of the diluted spinel B-site lattice Ax A'1− x B2 X 4 by using the replica method and the variational principle. Adopting the

XY model as the simplest one to describe order and disorder phases of the spinel lattice, the equation for calculating the magnetic susceptibility (Eq (50)) has been given. In the above system, the disorder in the distribution of magnetic interactions is due to the substitution between diamagnetic ions (A ↔ A’) on the tetrahedral sites. We have applied the obtained results to the particular material Lix Zn1− xV2O4 . In figure 1, we have presented the magnetic phase diagram of material Lix Zn1− xV2O4 in the range 0 ≤ x ≤ 1 .

III. 2. Diluted Magnetic Semiconductors (DMS) Using the experimental values of TN or TSG obtained by magnetic measurement for the diluted magnetic semiconductors Ca1− x Mn x O [88]. We have deduced the values of

exchange integrals J 1 ( x ) and J 2 ( x ) in the range 0 ≤ x ≤ 1 . From these values, we have

derived also the variation of the intra-plane coupling and the coupling between nearest and next-nearest plane with the concentration x in the Ca1− x Mn x O system with 0 ≤ x ≤ 1 . The obtained values of the nearest and next nearest neighbour interactions J 1 ( x ) and J 2 ( x ) ,

respectively are given in table 9. The values of corresponding classical exchange energy for the magnetic structure [55], are given in the table 1 for the Ca1− x Mn x O materials with

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

0 ≤ x ≤ 1. Table 9. The Curie-Weiss temperature θ p (K ) , the Néel temperature TN (K ) , the values of the first, the second exchange integrals and the energy of Ca1− x Mn x O as a function of dilution x

x

θ P (K ) [88]

TN (K ) [88]

J1 (K ) KB

J2 (K ) KB

E KBS 2

(K )

1

-536

120

-6.85

-3.42

82.14

0.9

-463

99

-5.65

-2.82

67.74

0.8

-372

77

-4.40

-2.20

52.80

0.75

-315

67

-3.82

-1.91

45.84

0.7

-276

57

-3.25

-1.62

38.94

0.65

-248

48

-2.74

-1.37

32.88

0.6

-206

38

-2.17

-1.08

25.98

0.5

-162

21

-1.20

-0.60

14.40

0.45

-153

17

-0.97

-0.48

11.58

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

230

R. Masrour and M. Hamedoun

The HTSEs combined with the P.A method has been applied, for get the magnetic phase diagrams of the Ca1− x Mn x O systems. Figure 7 shows magnetic phase diagram of material Ca1− x Mn x O . In this figure, we have included, for comparison, the experimental results obtained by magnetic measurement. We can see the good agreement between the magnetic phase diagram obtained by the HTSEs technique and the experimental ones, in particular in the case of the last systems of which the phase diagrams have been established well by different methods [88,89]. In the other hand, the values of critical exponents γ and

ν associated with the magnetic susceptibility χ (T ) and with the correlation length ξ (T ) ,

have been estimated in the range of the composition 0.5 ≤ x ≤ 1 . The sequence of [M, N] PA

χ (T ) and ξ (T ) has been evaluated. By examining the behaviour of these PA, the convergence was found to be quite rapid. The simple pole corresponds to TN and the residues to the critical exponents γ andν . The obtained central values are γ = 1.36 ± 0.02 and ν = 0.68 ± 0.01 .

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

to series of

Figure 7. Magnetic phase diagram of DMS

Ca1− x Mn x O . The various phases are the paramagnetic

phase

(AFM)

(PM),

antiferromagnetic

phase

(0.5 ≤ x ≤ 1)

and

spin

glass

phase

(SG)

(0.25 ≤ x ≤ 0.4) . The solid circles are the present results obtained by HTSEs method. The squares are deduced by magnetic measurement [88]. The solid line is a guide to the eye.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

231

III.3. Illuminates and Perovskites Materials The magnetic phase diagrams of the systems K 2Cu x Mn1− x F4 are obtained by the Hightemperature series expansions and the replica methods used in subsections III.4 and III.5, respectively. For the Fex Mn1− xTiO3 we have used replica method for determinate the their magnetic phase diagrams.

III.4. Magnetic Properties and Finite Size Scaling in Nanomaterials The

exchange

interactions

and

magnetic

energy

of

the

MnCr2O4

and

CoRh2O4 nanomaterials are obtained for different sizes L ( nm ) [89], by using of the Eq (5).

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The results obtained are given in tables 9 and 10, respectively. The Eqs (4) and (5) are used to determine the exchange interactions of the first, the second, the intra-plane, the inter-plane exchange integrals and the energy of CoRh2 O4 for different sizes (see table 11). The critical

Figure 7. Log-log plot of the shift of the Néel temperature

MnCr2O4

TN ( ∞ ) TN ( L )

−1

versus diameter

nanomaterials.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

L ( nm )

of

232

R. Masrour and M. Hamedoun

(ν )

exponent associated to the correlation lengths

is deduced for different sizes of

MnCr2O4 [57] and CoRh2 O4 [91] nanomaterials. In figures 8 and 9, we exhibit the dependence of the shift determine the exponent

δT =

T N (∞ ) − T N ( L ) whit L(nm ) in a Log- Log scale to TN ( L)

λ , by using the equation (1) for the MnCr2O4 and

CoRh2O4 nanomaterials. Table 10. The different sizes L ( nm ) , the Néel temperature TN (K ) , the values of the

nearest neighbour (nn ) interactions J1 ( L) and the energy of the magnetic structure of

MnCr2O4 nanomaterials E (K ) kB S 2

J1 (K ) kB

L ( nm )

TN (K ) [57]

11

52

3.9

15.6

13

49

3.675

14.7

16

47

3.525

14.1

19

46

3.45

13.8

bulk

45

3.375

13.5

−

Table 11. The Curie-Weiss temperature θ P (K ) , the Néel temperature TN (K ) , the

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

values of the first, the second, intra-plane, inter-plane exchange integrals and the energy of CoRh2 O4 nanomaterials

J1 (K ) L(nm) θ P (K ) TN (K ) kB [91] [91]

J2 (K) kB

Jaa (K ) kB

Jab (K) kB

Jac (K) kB

kB

E

(K)

Bulk

-44.23

27±0.5

-1.474

-0.737

-2.948

-11.792

-2.948

17.688

70

-42.80

27±0.5

-1.426

-0.713

-2.852

-11.408

-2.852

17.112

50

-42.05

27±0.5

-1.401

-0.700

-2.802

-11.204

-2.802

16.806

32

-41.84

27±0.5

-1.394

-0.697

-2.788

-11.152

-2.788

16.728

IV. Discussions and Conclusions The J 1 ( x) and J 2 ( x) exchange interactions have been determined through mean field theory (MFT), using the experimental data of TN and

θ p [18] for each dilution. We have

used the probability distribution law (PDL) adapted of the nature of dilution problem in material lattice, to determine the first two exchange integrals, the nearest neighbour and nextnearest neighbour exchange interactions J 1 and J 2 , respectively, of the Lix Zn1− xV2O4

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

233

materials. These results are given in table 4 for 0 ≤ x ≤ 1 . From the value obtained by MFT method and PDL, we have deduced the values of the intra-plane and the inter-plane interactions J aa , J bb and J ac , respectively, the energy of the magnetic structure and the ratio of inter to intr-aplanar interactions

J int er . The obtained values for Lix Zn1− xV2O4 are J int ra

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

given in table 1.

Figure 9. Log-log plot of the shift of reduced critical temperature

L(nm ) of CoRh2 O4

TN (∞ ) −1 TN (L )

versus diameter

nanomaterials.

The MFT has been applied to the ferrimagnetic materials Cox Fe1− x Cr2 S 4 and

CoFe2−2 x Cr2 x O4 for calculate the exchange interactions J AB ( x ) for each dilution x (see tables 5 and 6). In ordered phase the values of exchange interactions obtained by two methods are comparable. From table 4, the sign of super-exchange interactions J 2 ( x) are negative in the all range of dilution. From the same table, one can see that J 1 ( x) decreases with increases of x . The decrease of J 1 ( x) is due to the change in the average value of the Cr-Cr distance. J 2 ( x) represents the strength of the V-O-A-O-V super-exchange, we expect its value to depend on the nature of the A-site cation. The ratio

(J + J ac ) and the J int er = ab J int ra J aa

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

234

R. Masrour and M. Hamedoun

energy of the magnetic structures increase when x increases. The collapse of the long-range antiferromagnetic order and the appearance of the spin-glass behavior in this material are essentially related to site disorder and fluctuating competing super-exchange (nn and nnn) interactions which appear on substitution. From the tables 5 and 6, one can see that the J AB ( x) increases in absolute value with x increases for Cox Fe1− x Cr2 S4 and

CoFe2−2 x Cr2 x O4 . In figure 10, we have presented the magnetic exchange interactions Ei (i = 1 − 3) and the difference δ E = E1 − E3 for Cox Fe1− x Cr2 S4 , versus the dilution x . The values of J 1 ( x) and J 2 ( x) have been determined from mean field theory, using the experimental data of TN given in Ref [88], for each dilution (see table.9) for the DMS

Ca1− x Mn x O . From these values, we have deduced the values of the intra-plane and interplane interactions J aa , J bb and J ac , respectively, and the energy of the magnetic structure (see table. 9). The values of J 1 ( x) and J 2 ( x) decreases with the absolute value when x decreases. The sign of J 1 ( x) and J 2 ( x) are negative in the whole range of concentration. We have used the MFT to determine the values of exchange interactions with different sizes of CoRh2 O4 (see table 11) and of MnCr2O4 nanomaterials (see table 10). For the

MnCr2O4 nanomaterial, on can see that J 1 ( x) and the energy of the magnetic structure increases with the absolute value of when L ( nm ) decreases. For CoRh2 O4 , on can see that

J 1 ( x) and the energy of the magnetic structure decreases with the absolute value of when

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

L ( nm ) decreases. The sign of J 1 ( x) and J 2 ( x) are negative in the whole range of dilution for two nanomaterials. The HTSE extrapolated with Padé approximants method is shown to be a convenient method to provide valid estimations of the critical temperatures for real system. By applying this method to the magnetic susceptibility χ (T ) we have estimated the Néel temperature TN (or the freezing temperature TSG ) for each dilution x . We have used the HTSEs quantum and Replica methods to determine the magnetic phase diagrams of the systems Lix Zn1− xV2O4 (see figure 1). Several thermodynamic phases may appear including the paramagnetic (PM), antiferromagnetic (AFM) 0 ≤ x ≤ 0.1 and spin glass phase 0.1 < x < 0.9 for

Lix Zn1− xV2O4 . In 0.8 < x ≤ 1 , no such tendency is observed, which may be consistent with the fact that these compounds are metallic. In this figure we have included, for comparison, the experimental results obtained by magnetic measurement. In addition, we have determined the region spin glass while using the expression of the nonlinear susceptibility. The classical site percolation threshold in the system Lix Zn1− xV2O4 is x p ≈ 0.8 and comparable with given by Ref [1]. In the other hand, the value of critical exponents γ and ν associated with

the magnetic susceptibility χ (T ) and with the correlation length ξ (T ) , have been estimated

in the ordering phase. The sequence of [M, N] PA to series of

χ (T ) has been evaluated. The

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

235

HTSEs has been applied to the materials Cox Fe1− x Cr2 S 4 and CoFe2− 2 x Cr2 x O4 for obtained the magnetic phase diagrams (see figures 2 and 3). The several thermodynamic phases may appear including the paramagnetic (PM), ferrimagnetic (FerriM) 0 ≤ x ≤ 1 . From these figures one can see good agreement between the theoretical phase diagram and experimental results. The application of the present theories to particular methods spinels Cox Fe1− x Cr2 S 4 ,

CoFe2−2 x Cr2 x O4 and Ca1− x Mn x O gives the estimates values of critical temperature TC and critical exponent

γ . In figure 7, we have presented the magnetic phase of DMS

Ca1− x Mn x O . The several phases are appearing including the paramagnetic phase (PM), antiferromagnetic phase (AFM) ( 0.5 ≤ x ≤ 1) and spin glass phase (SG) ( 0.25 ≤ x ≤ 0.4 ) . The HTSE extrapolated with Padé approximants method is shown to be a convenient method to provide valid estimations of the critical temperatures for real system. By applying this method to the magnetic susceptibility χ (T ) , we have estimated the critical temperature

TC ( TN ) and the

TSG

(in

spin

glass

region)

for

each

dilution x

in

the

K 2Cu x Mn1− x F4 systems. In figures 10 to 15, we have presented the magnetic phases diagrams of K 2Cu x Mn1− x F4 , with different values of J AA , J AB and J BB . Several thermodynamic phases may appear including the paramagnetic (PM), the ferromagnetic (FM), the antiferromagnetic (AFM) and the spin glass phases (SG). For J AA > 0 and

J BB < 0 alls phases mentioned above may appear (see figures 11 and 12). The spin glass phase moves towards higher values of x when J AB decreases. For J AA < 0 , J BB < 0 and Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

small J AB , only an antiferromagnetic takes place at low temperature (see figure 13), and spin *

glass phase (SG) appears above a critical value of the exchange integral J AB (see figure 14, *

in the case of J AA = −1 and J BB = −0.66 , J AB is closer to 0.179 ). These results are with the approximated mean filed analysis [92]. It known that the occurrence of the SG phase depends on kind of quenched randomness, besides symmetry of spin and the range of interactions [93]. Figure 15 confirms that the range of concentration x , where SG exists, increases with J AB . Theses results are in rough qualitative agreements with those deduced from phase diagrams of a number of real systems; the competition between different interactions contributes to the appearance of SG phase. The range of concentrations, where this phase appears, depends on the degree of frustration. For fixed values of J AA and J BB the nature of the phase diagrams depends on the spin number and on the sign and the strength of J AB . It is worth noting that if J AA > 0 and J BB > 0 , the AFM and FM phase’s exchanges *

their role and SG phase appears for a negative threshold − J AB . It can be observed that in our model we deal with classical spins. The results obtained by HTSE method are comparable with those given by Replica method [46,52]. In figure 16, we presented the magnetic phase diagram of the planar mixed system K 2Cu x Mn1− x F4 . In this material, we cross the way

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

236

R. Masrour and M. Hamedoun

between the opposite pure compounds K 2 MnF4 and K 2CuF4 . We took values for the exchange interactions ( J Cu −Cu = 11.4 K , J Mn − Mn = −4.2 K and J Cu − Mn = −5.05 K )[94-96]

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 10. Variation of the energies interaction

Figure 11. Magnetic phase diagram with J AA

Ei (i = 1 − 3)

versus

= 1K , J BB = −0.66

x

for

and J AB

Cox Fe1− x Cr2 S4 .

= 0.81K . The circles

represent the results given by HTSE method and the squares represent the results given by Replica method [52].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors… *

237 *

The spin glass phase limits are situated on Cu-concentrations xc ≈ 0.5 and xc ≈ 0.85 . These values are predicted by magnetic measurements [97] and by replica method [52].

Figure 12. Magnetic phase diagram with J AA

= 1K , J BB = −0.66

and J AB

= −0.81K .

The

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

circles represent the results given by HTSE method and the squares represent the results given by Replica method [52].

Figure 13. Magnetic phase diagram with J AA

= −1K

,

J BB = −0.66

and J AB

= 0.17 K .

The

circles represent the results given by HTSE method and the squares represent the results given by Replica method [52].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

238

R. Masrour and M. Hamedoun

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 14. Magnetic phase diagram with J AA = −1K , J BB = −0.66 and J AB = 0.24 K . The circles represent the results given by HTSE method and the squares represent the results given by Replica method [52].

Figure 15. Magnetic phase diagram with J AA

= −1K

,

J BB = −0.66

and J AB

= 0.81K .

The

circles represent the results given by HTSE method and the squares represent the results given by Replica method [52].

Figure 16 gives the phase diagram of the mixed compound Fex Mn1− xTiO3 . This system is considered to be an Ising system [98, 99]. However, the results of the Mössbauer measurements indicate the existence of the transverse spin component [100,101]. In this figure, we have included for comparison, the experimental results obtained by magnetic measurements given in Ref. [98]. We can see the good concordance between the experimental

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

239

results and the theoretical one. The frustration within c-layers leads to SG behaviour for intermediate Fe dilution x . The percolation values between the spin glass and the continuous *

**

antiferromagnetic phases are estimated to be at xc = 0.40 and xc = 0.60 .These values are compatible with those obtained by experimental measurements. The sequence of [ M , N ] PA to the series have been evaluated. By examining the behaviour of these PA , the convergence was found to be quite rapid and we expect the result to be accurate to within 1%. Estimates of the critical exponents associated with susceptibility are found to be γ = 1.317 ± 0.001 for FeCr2 S 4 and γ = 1.328 ± 0.001 for CoCr2 S 4 . The critical exponents associated with magnetic susceptibility and with the correlation γ = 1.312 ± 0.002 andν = 0.672 ± 0.004 for the length are found to be

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

systems Lix Zn1− xV2O4 .

Figure 16. the magnetic phase diagram of the

K 2Cu x Mn1− x F4

systems. The circle, triangle, square

and the stars represents: the results given by replica method [52], experiment results [13,106], the results given by Monte Carlo simulations [107] and the results given by HTSE method, respectively.

These values are insensitive to dilution x in ordered phase. The values obtained of

γ and

ν are nearest to the one of 2 D XY model. These values are comparable with given by [49]. To conclude, it would be interesting to compare the critical exponents γ with other theoretical values. A lot of methods of extracting critical exponents have been given in the literature. We have selected many of the methods, and summarised our findings below. The obtained central values of the critical exponent of Ca1− x Mn x O are γ = 1.36 ± 0.02 and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

240

R. Masrour and M. Hamedoun

ν = 0.68 ± 0.01 . In the magnetic order the values of γ and ν are nearest to the one of Heisenberg model and insensitive to the dilution. the critical exponents associated with magnetic susceptibility for the K 2Cu x Mn1− x F4 systems are found to be γ = 1.32 ± 0.02 . This value is comparable with those given by [103] and may be compared with other theoretical results based on XY model. Zarek [102] has found experimentally by magnetic balance for CdCr2 Se4 is γ = 1.29 ± 0.02 ; for HgCr2 Se4 γ = 1.30 ± 0.02 and for

CuCr2 Se4 is γ = 1.32 ± 0.02 . The effective critical exponents ν associated to correlation length of the CoRh2 O4 and the MnCr2O4 nanomaterials are obtained:

ν b = 0.90 ± 0.01 and ν b = 0.283 ± 0.020 ,

respectively. The first value is qualitative accordance with the universality class hypothesis [104, 105, 58].

Annex 1 I. Calculation of the Coefficients a kr (n ) The coefficients a kr (n ) are defined by:

G G ⎛ n G G ⎞ ⎜ ∑ U iU j ⎟ = ∑ a kr (n ) U iU j ⎠ R⊂Λ ⎝ α =1

(

)

R

(1)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

To calculate these coefficients, we use the following equation:

G G ⎞ G G ⎛ exp ⎜ λ ∑ U iα U αj ⎟ = ∑ g (λ , r ) U iU j ⎝ α ⎠ R⊂Λ

(

)

R

(2)

(

G G

One calculation draws it of the equation (2) after have multiply by the term U iU j

G G Tr U iU j nij

(

)

A

G G ⎞ G G ⎛ exp ⎜ λ ∑ U iα U αj ⎟ = Tr U iU j ⎝ α ⎠ nij

(

G G

) ∑ g (λ ,r )(U U ) G G G G = ∑ g (λ , r )Tr (U U ) (U U ) A

i

R⊂ Λ

G G Tr U iU j nij

(

)

A

i

nij

j

A

R

j

A

R⊂Λ

), (3)

R

i

j

G G ⎞ G G A G G ⎛ exp ⎜ λ ∑ U iα U αj ⎟ = Tr U iU j ∏ exp λ U iα U αj α ⎝ α ⎠ nij G G G G a G G n−a (4) = ⎡Tr U iα U αj exp λ U iα U αj ⎤ ⎡Tr exp λ U iβ U βj ⎤ ⎥⎦ ⎢⎣ βij ⎥⎦ ⎣⎢ αij = T1a T2n − a

(

)

(

(

) (

)

)

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(

)

Study of the Magnetic Properties of the Semiconductors…

(

Gα Gα

Gα Gα Gβ Gβ et T2 = Tr exp λ U i U j i Uj

) (λ U

where T1 = Tr U i U j exp αij

)

(

βij

241

)

we have

T1 = ∫

2π 0

cos(θ i − θ j )exp (λ (θ i − θ j )) dθ i dθ j

Using the change of variables following:

θi −θ j = θ θi +θ j = ϕ We obtained T1 =

2π

2π

0

0

∫ ∫

J ac cos(θ ) exp (λ cos(θ )) dθ dϕ

(5)

J ac is the Jacobian being worth 2 in this case. After integration, we get: π 2π T1 = 4 π ⎛⎜ [sin (θ ) exp (λ cos (θ ))] 0 + λ ∫ sin 2 (θ ) exp (λ cos (θ )) dθ ⎞⎟ 0 ⎝ ⎠ π

= 4 π λ ∫ sin 2 (θ ) exp (λ cos (θ )) dθ

(6)

0

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

= 4 π 2 I 1 (λ ) I 1 is the Bessel function of the first order. T2 = ∫

2π 0

exp (λ cos (θ i − θ j ))dθ i dθ j = 4 π

∫

2π 0

exp (λ cos (θ )) dθ = 4 π 2 I 0 (λ )

(7)

I 0 is the Bessel function of the zero order. G G Tr U iU j nij

(

)

A

G G ⎞ ⎛ I (λ ) ⎞ ⎛ ⎟⎟ 4 π 2 exp ⎜ λ ∑ U iα U αj ⎟ = ⎜⎜ 1 ( ) λ I ⎝ α ⎠ ⎝ 0 ⎠ a

(

G G

Of another quoted, to calculate Tr U iU j nij

G G

) (U U ) A

i

R

j

(

) (I (λ )) n

n

0

, we have two cases:

* First case: A = R and A = a (here, A is the cardinal of A)

A = {α 1 ,α 2 ," ,α a }

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(8)

242

R. Masrour and M. Hamedoun

G G Tr U iU j nij

(

=∫

G G

) (U U ) A

i

R

j

G G = Tr U iU j nij

(

2π

(

)

G G

) (U U ) A

A

i

j

(

G G = Tr U iU j

(

) (U U )

(

)

aij

)

G G

A

i

A

j

Tr 1

( n − a )ij

cos 2 θ iα1 − θ jα1 cos 2 θ iα 2 − θ jα 2 " cos 2 θ iα a − θ jα a dθ iα1 dθ jα1 " dθ iα a dθ jα a

0

= (4 π )

a

π

∫

0

( )

( )

( )

(9)

cos θ α1 cos θ α 2 " cos θ α a dθ α1 dθ α 2 " dθ α a Tr 1 2

2

2

(2 π ) 2 (n − a ) a⎛π ⎞ = (4 π ) ⎜ ⎟ (2 π ) = 2a ⎝2⎠ a

** Second case: A ≠ R ,

G G Tr U iU j

A

n −α ij

i

α ∈ A and α ∉ R with A1 = A − {α }

G G

G G

G G

Gα Gα

) (U U ) = ( Tr) (U U ) (U U ) Tr (U U ) G G G G = Tr (U U ) (U U ) ∫ cos(θ − θ )dθ dθ ( ) nij

(

( n − a )ij

2n

i

R

j

A1

j

i

n −α ij

j

i

R

j

αij

i

j

2π

R

i

A1

j

iα

0

jα

iα

jα

(10)

=0 Taking into account the two cases us gets

G G Tr U iU j nij

(

G G

) (U U ) A

i

R

=

j

(2 π )2 n δ 2a

finally,

∑

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

R⊂Λ

G G g (λ , r )Tr U iU j nij

(

G G

) (U U ) A

i

R

j

= p (a )

AR

(2 π )2 π g (λ , a ) a 2

(11)

where p (a ) is the permutation of a. Introducing (11) and (8) in (3), ones find: a ⎤ 2 r ⎡⎛ I 1 (λ ) ⎞ ⎟⎟ (I 0 (λ ))n ⎥ ⎢⎜⎜ g (λ , r ) = p (r ) ⎢⎝ I 0 (λ ) ⎠ ⎥⎦ ⎣

(12)

Using the following equality: k ⎛ n Gα Gα ⎞ ⎛ n G G ⎞ ⎜ ∑ U i U j ⎟ = Dλk exp ⎜ λ ∑ U iα U αj ⎟ ⎝ α =1 ⎠ ⎝ α =1 ⎠

dk where : Dλ f (λ ) = f (λ ) dλ k k

we get, λ =0

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(13)

Study of the Magnetic Properties of the Semiconductors… k G G ⎛ n Gα Gα ⎞ ⎜ ∑ U i U j ⎟ = ∑ Dλk g (λ , r ) U iU j R⊂Λ ⎝ α =1 ⎠

(

) =∑a

G G

R

R⊂Λ

kr

243

(n ) (U iU j )

R

(14)

finally, one finds:

a kr (n ) = Dλk g (λ , r ) In this calculation, r = 1 corresponds to the order to long range whereas r = 2 corresponds to the phase glass of spin. One has

2r = 2 , for the two previous cases. Eventually, we find: p (r ) ⎡⎛ I (λ ) ⎞ a ⎤ ⎟⎟ (I 0 (λ ))n ⎥ a kr ( n ) = 2 ⎢⎜⎜ 1 ⎢⎣⎝ I 0 (λ ) ⎠ ⎥⎦

(15)

II. Calculation of the Coefficients e (k ) and d (k ) , for k = 1, 2 From the formula of e (k ) given in the text (Eq. 51), one can write e (k ) =

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

N (k ) =

G G G ⎡ 1 G Tr ⎢ ∑ ∑ σ j (b )U Bj V j U j n σ j (k ) nj ⎣ K ⊂ Λ B ⊂ Λ (2 π )

(

1

)

K

⎤ ⎥ ⎦

N (k ) , with: M (k ) (16)

and

M (k ) =

GB G GK⎤ ⎡ 1 G ( ) ( ) Tr σ b U σ k U ⎢ ∑ j j j j ⎥ σ 2j (k ) nj ⎣ B ⊂ Λ (2 π )n ⎦

2

(18)

* For k = 1 , one has:

(

)

1 1 2π σ x V x cos 2 (θ 1 ) + σ y V y sin 2 (θ 1 ) dθ 1 ∫ 0 σ (1) 2 π G G 1 σ (1)V 1 = = 2 σ (1) 2

N (1) =

M (1) =

2

1 σ (1) 2 π 2

π ∫ (σ 2

0

2 x

)

cos 2 (θ 1 ) + σ y2 sin 2 (θ 1 ) dθ 1 = 1

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(19)

(20)

244

R. Masrour and M. Hamedoun

1 . 2

therefore, e (k ) =

** For k = 2 , one has : N (2) = =

4

2π

(cos θ (2 π ) ∫ 2

4

(2 π )2

0

(π

2

1

cos θ 2 + sin θ 1 sin θ 2 ) (V x cos θ 1 + V y sin θ 1 )(V x cos θ 2 + V y sin θ 2 )dθ 1 dθ 2

V x2 + π 2 V y2

)

=1

and

M (2 ) =

4

π (cos θ ∫ (2 π ) 2

2

0

cos θ 2 + sin θ 1 sin θ 2 ) dθ 1 dθ 2 = 2 2

1

1 . 2

finally, one has, e (2 ) =

N (k ) , M (k )

From the formula of d (k ) , given in the text (Eq. 52), one can write d (k ) = with :

G G G G G ⎡ G 1 Tr ⎢ ∑ ∑ ∑ σ i (b )U iB σ j (a )U jA U iU j n 2 σ i (k )σ j (k ) nij ⎣ K ⊂ Λ B ⊂ Λ A⊂ Λ (2 π )

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(

1 G

N (k ) = G

)

K

⎤ ⎥ (21) ⎦

and M (k ) guard the same definition that before. For k = 1 , one has: N (1) = G

1 G

1

σ i (1)σ j (1) (2 π )2

∫ (σ

x i

= G

1 G

2π

0

)(

)

cos θ i + σ iy sin θ i σ xj cos θ j + σ jy sin θ j (cos θ i cos θ j + sin θ i sin θ j )dθ i dθ j 1

σ i (1)σ j (1) (2 π )

=

×

2

(π

2

σ ix (1)σ xj (1) + π 2 σ iy (1)σ jy (1))

1 4

** For k = 2 : Calculate it is the same and one finds N (2 ) = 1, d (2 ) =

1 1 et d (2 ) = . 4 2

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(22)

Study of the Magnetic Properties of the Semiconductors…

245

Annex 2

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The numeric program uses to calculate the critical temperature by replica method. Programme implicit real*8(a-h,o-z) dimension t(3) x=0.00 open(unit=2, file='text.dat') open (unit=4,file='stru.out') v=(2*s*(s+1)/(3*s*s)) write(*,*)'val1,val2,s,w' read(*,*)val1,val2 50 continue xj1=val1*v*w xj2=val2*v*3*w alfa=6. x1j1=(x**alfa)*xj1 x1j2=(x**alfa)*xj2 x2j1=(x**alfa)*(xj1**2) x2j2=(x**alfa)*(xj2**2) xja1=2.*x1j1 xja2=4.*x1j1+8*x1j2 xja3=4*x1j2 t(1)=s*s*(xja1+xja2+xja3)/8 t(2)=s*s*(xja1-xja2+xja3)/8 xb=(1./8)*(6*(x2j1-x1j1**2)+12*(x2j2-x1j1**2)) if(xb.ge.0)then t(3)=s*s*(xb/2)**0.5 else t(3)=-50.1 endif write(2,5)x, t(1), t(2),t(3) write(4,5)x, t(1), t(2),t(3) 5 format(6x,f6.2,6x,f6.2,6x,f6.2,6x,f6.2) x=x+0.002 if(x.le.1.0) goto 50 end

References [1] [2]

K. Binder, A.P. Young, Rev. Mod. Phys. 58, (1986) 801. V.Y. Kalinnikov, T.G. Aminov, V. M. Novotortsev, Neorg. Mater. 39, (2003) 997; V.Y. Kalinnikov, T.G. Aminov, V.M. Novotortsev, Inorg. Mater. Engl.Trans. 39, (2003) 1159.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

246 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

R. Masrour and M. Hamedoun G. Güner, F. Yildiz, B. Rameev, B. Atkac¸, J. Phys.: Condens. Matter. 17, (2005) 3943. Wiedemann, M. Hamedoun, J. Rossat-Mignod, J. Phys. C: Solid State Phys. 18, (1985) 2549. M. Hamedoun, A. Wiedemann, J.L. Dormann, M. Nogues, J. Rossat-Mignod, J. Phys. C: Solid State Phys. 19, (1986) 1801. M. Nogues, M. Hamedoun, J.L. Dormann, A. Saifi, J. Magn. Magn. Mater. 54-57, (1986) 85. S.K. Sampath, D.G. Kanhere, R. Pandey, J. Phys: Condens. Matter. 11, (1999) 3635. T. Grón, J. Kopyczok, I. Okónska-Kozłowska, J. Warczewski, J. Magn. Magn. Mater. 111, (1991) 53. J. Krok-Kowalski, J. Warczewski, T. Mydlarz, A. Pacyna, I. Okónska- Kozłowska, J. Magn. Magn. Mater. 111, (1991) 50. Okónska-Kozłowska, E. Malicka, A. Waskowska, T. Mydlarz, J. Solid State Chem. 148, (1999) 215. N. Godenfeld: Lectures in Phase Transitions and the Renormalisation Group (AddisonWesley, Reading, MA, 1992) Y. Ueda, N. Fujiwara, H. Yasuoka, J. Phys. Soc. Jpn. 66, (1997) 778 and references therein. D B. Rogers, J. L. Gillson, T. E. Gier, Solid State Commun. 5, (1967) 263. Fujimori, K. Kawakami, N. Tsuda, Phys. Rev. B. 38, (1988) 7889. K. Kawakami, Y. Sakai, N. Tsuda, J. Phys. Soc. Jpn. 55, (1986) 3174. H. Mamiya, M. Onoda, Solid State Commun. 95, (1995) 217. H. Mamiya, M. Onoda, T. Furubayashi, J. Tang, I. Nakatani, J. Appl. Phys. 81, (1997) 5289. Yu. Izyumov, V. E. Naish, S. B. Petrov, J. Magn. Magn. Mater. 13, (1979) 267. C.S. Kim, M. Y. Ha, H.M. Ko et al. J. Appl. Phys. 75 (10) (1994) 6078-6080. H. Mohan, I.A. Shaikh, R.G. Kulkarni, Physica B. 217, (1996) 292-298. J. K. Furdyna, J. Appl. Phys. 53, (1982) 637. K. Ramdas, J. Appl. Phys. 53, (1982) 7649. J. Mycielski, Prog. Cryst. Growth Charact. 10, (1984) 101. R. R. Galazka, J. Cryst Growth. 72, (1985) 364. J. K. Furdyna, N. Samarth, J. Appl. Phys. 61, (1987) 3526. G. Shull, W.A. Strausser, E.O. Wollan., Phys. Rev. 83, (1951) 333; W.L. Roth, Phys. Rev. 110, (1958) 1333. R.J. Birgeneau, et al, Phys. Rev. B. 3, (1971) 1736-1749. R. J. Birgeneau, H. J. Guggenheim, G. Shirane, Phys. Rev. B. 8, (1973) 304-311. K. Hirakawa, H. Ikeda, J. Phys. Soc. Jpn. 35 (1973) 1328. Yamada, J. Phys. Soc. Jpn. 33, (1972) 979. R. Masrour, Hamedoun, K. Bouslykhane, A. Hourmatallah, N. Benzakour, J. Chinese. Phys, 47, (2008) 1. S. Krupika, P. Novak, in Ferromagnetic materials, edited by E.P. Wolfarth (NorthHolland, Amsterdam). 3, (1982) 189. D.G. Wickham, J.B. Goodenough, Phys. Rev. 115, (1959) 1156. V. Tsurkan, M. Mcksch, et al. Phys. Rev. B. 68, (2003) 134434. J.M. Hasting, L.M. Corliss, Phys. Rev. 126, (1962) 556. S. Mukherjee, R. Ranganathan, S.B. Roy, Phys. Rev. B. 50, (1994) 6866.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors… [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]

247

K. Tomiyasu, J. Fukunaga, H. Suzuki, Phys. Rev. B. 70, (2004) 214434. J. M. Hastings, L. M. Corliss, Phys. Rev. 126, (1962) 556 – 565. R. Seshadri, private communication. W. Schiessl, W. Potzel, et al., Phys. Rev. B. 53, (1996) 9413. G. Blasse, D. J. Schipper, Phys. Lett. 5, (1963) 300. M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, Physica B. 270, (1999) 384. M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, J. Phys: Condens. Mater. 10, (1998) 3611. M. Hamedoun, R. Masrour, K. Bouslykhane, A. Hourmatallah, N. Benzakour, J. Magn. Magn. Mater. 320, (2008) 1431-1435. M. Hamedoun, R. Masrour, K. Bouslykhane, A. Hourmatallah, N. Benzakour, A Filali, Chin, Phys. Lett. 24, (2007) 2077. M. Hamedoun, R. Masrour, K. Bouslykhane, A. Hourmatallah, N. Benzakour, J. Trans world Research Network (India), Chapter 5, No 56/2008-2009. M. Hamedoun, R. Masrour, K. Bouslykhane, A. Hourmatallah, N. Benzakour, J. Alloys Compds. 462, (2008) 125-128. “Padé Approximants”, edited by G.A. Baker and P. Graves-Morris (Addison-Wesley, London, 1981). R. Navaro, Magnetic Properties of Layered Transition Metal Compounds, Ed. L.J.DE Jonsgh, Daventa: Kluwer. 105, (1990). M.C. Moron, J. Phys: Condensed Matter. 8, (1996) 11141. M. Hamedoun, H. Bakrim, K. Bouslykhane, A. Hourmatallah, N. Benzakour, A. Filali, R. Masrour, J. Phys.: Condens. Matter. 20, (2008) 125216. M. Hamedoun, M. Hachimi, A. Hourmatallah, K. Afif, A. Benyousef, J. Magn. Magn. Mater. 247 (2), (2002) 127. M. Henkel, S. Andrieu, P. Bauer, M. Piecuch, Phys. Rev. Lett. 80, (1998) 4783. Saber, A. Ainane, F. Dujardin, N. El Aouad, M. Saber, B. Stébé, J. Phys.: Condens. Matter. 12, (2000) 43. R. Zhang, R. F.Wills. Phys. Rev. Lett. 86, (2001) 2665. J. Cabral Neta, J. Ricardo de Sousa, J. A. Plascak, Phys. Rev. B. 66, (2002) 064417. R. N. Bhowmik, R. Ranganathan, R. Nagarajan, Phys. Rev. B. 73, (2006) 144413. R. Masrour, M. Hamedoun, A. Hourmatallah, K. Bouslykhane, N. Benzakour, Phys. Scr, 78, (2008) 025702. Domb, J. Phys. A. 6, (1973) 1296. W.E. Holland, H. A. Brown, Phys-Stat. Sol. (a) 10, (1972) 249. M. Hamedoun, A. Wiedenmann, J. L. Dormann. M. Nogues, J. Rossat-Mignod, J. Phys. C : Solid State Phys. 19, (1986) 1783-1800. M. Hachimi, M. Hamedoun, H. Bakrim, A. Hourmatallah, Z. Elachen, L. Ajana, J. Magn. Magn. Mater. 256, (2003) 412. Tawardowski, H. J. M. Swagten, W. J. M. de Jonge, M. Demianiuk, Phys. Rev. B. 36 (1987) 1. R. Masrour, M. Hamedoun, A. Hourmatallah, K. Bouslykhane, N. Benzakour, Can. J. Phys, 86 (2008) 1287. A.P. Young, B. S. Shastry, J. Phys. C15, (1982) 4547. M.E. Lines, Phys. Rev. 139, (1965) A 1304.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

248

R. Masrour and M. Hamedoun

[67] J.M. Weisselinowa, Phys. Stat. Sol. 120, (1983) 585; J. M. Weisselinowa, A. T. Apostolov, J. Phys: Condens. Matter. 8, (1996) 473. [68] M. Hamedoun, M. Houssa, N. Y. Cherriet and F. Z. Bakkali, Phys. Stat. Sol. (b). 214, (1999) 403. [69] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, Physica. B. 270, (1999) 384. [70] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah: J. Phys: Condens. Matter. 10, (1998) 3611. [71] M. Hamedoun, H. Bakrim, A. Filali, A. Hourmatallah, N. Benzakour, V. Sagredo: J. Alloys Compd. 369, (2004) 70. [72] M. Hamedoun, H. Bakrim, A. Hourmatallah, N. Benzakour, Surf. Sci. 539 (2003) 159. [73] M. Hamedoun, H. Bakrim, A. Hourmatallah, N. Benzakour, Superlattices Microstruct. 33, (2003) 131. [74] M. Hamedoun, M. Houssa, Y. Cherriet F. Z. Bakkali: Phys. Status Solidi B. 214, (1999) 403. [75] M.C. Moron: J. Phys.: Condens. Matter. 8, (1996) 11141. [76] H.E. Stanley: Phys. Rev. 158, (1967) 537. [77] S.F. Edwards, P. W. Anderson, J. Phys. F5, (1975) 965. [78] S. Sarbach, J. Phys. C. 13, (1980) 5033 [79] E.J. S. Lage, J. Phys. C. 10, (1977) 701 [80] Ederer: arXiv: cond-mat/0611502 v1 19 Nov 2006 [81] C.M. Srivastava, G. Srivasan, N.G. Nanadikar, Phys. Rev. B. 19, (1979) 1. [82] P. Gibart, J.L. Dormann, Y. Pellerin, Phys. Status. Solidi. 36, (1969) 187. [83] M. Nougues, M. Mejai, L. Goldstein, J. Phys. Chem. Solids. 40, (1979) 375-379. [84] J. C. LE. Guillou, J. Zinn- Justin, Phys. Rev. Lett. 2, (1977) 95. [85] For a review, see M. F. Collins, Magnetic Critical Scattering (Oxford: Oxford University Press, 1989). [86] K. Afif, A. Benyoussef, M. Hamedoun, Chin. Phys. Lett. 19, (2002) 1187 and references therein. [87] P. Gibart, J.L. Dormann, Y. Pellerin, Phys. Status. Solidi. 36, (1969) 187. [88] S. Kolesnik; B. Dabrowski, J. Supercond. Incorporating Novel Magnetism. 16, (2003) 501-505(5). [89] M. Alba, J. Hammann, M. Nougues, J. Phys. C. 15, (1982) 5441. [90] K. Afif, A. Benyoussef, M. Hamedoun, A. Hourmatallah, Phys. Stat. Sol. (b). 219, (2000) 383. [91] R.N. Bhowmika, R. Nagarajanb, R. Ranganathana: cond-mat/0210678 v1 (2002). [92] F. Matsubara, Prog. Theo. Phys. 52, (1974) 1124. [93] K. Hukushima, Y. Nonmura, Y. Ozeki, H. Takayama, J. Phys. Soc. Jpn. 66, (1997) 215. [94] M. Itoh, M. Yamada, T. Sakakibara and T. Goto, J. Phys. Soc. Jpn. 58, (1989) 684. [95] S. Funahashi, F. Moussa, M. Steiner, Solidd State Commun. 18, (1976) 433. [96] H.W. deWijn, L.R. Walsted, Phys. Rev. B. 8, (1973) 433. [97] Y. Kimishima, H. Ikeda, A. Furukawa, H. Nagano, J. Phys. Soc. Jpn. 55, (1986) 3574 [98] H. Yoshizawa, S. Mitsuda, H. Aruga, A. Ito, J. Phys. Soc. Jpn. 58, (1989) 1416; H. Aruga, A. Ito, J. Phys. Soc. Jpn. 62, (1993) 4488. [99] Ito, H. Aruga, M. Kikuchi, Y. Syono, H. Takei, Solid State Commun. 66, (1988) 475. [100] Ito, S. Morimoto, H. Aruga, Hperfine Interactions. 54, (1990) 567.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Study of the Magnetic Properties of the Semiconductors…

249

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[101] Ito, E. Torikar, S. Morimoto, H. Aruga, M. Kikuchi,Y. Syono, H. Takei, J. Phys. Soc. Jpn. 59, (1990) 829. [102] W. Zarek, Acta. Phys. Polon. A 52, (1977) 657. [103] J. A.Nielsen, Phase transitions and critical phenomena, 5a (ed. C. Domb and M. S. Green). Academic Press, New York. (1976). [104] K. K. Pan, Phys. Rev. 71, (2005) 134524. [105] P. Butera, M. Comi, Phys. Rev. B. 56, (1997) 8212. [106] Y. Todate, Y. Ishikawa, K. Tajima, S. Tomiyoshi, H. Takei, J. Phys. Soc. Jpn. 55, (1986) 4464. [107] K. Affif, A. Benyoussef, M. Hamedoun, Chin. Phys. Lett. 19, (2002) 1187.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

In: Magnetic Properties of Solids Editor: Kenneth B. Tamayo, pp. 251-272

ISBN: 978-1-60741-550-3 © 2009 Nova Science Publishers, Inc.

Chapter 6

FAST DOMAIN WALL DYNAMICS IN THIN MAGNETIC WIRES (REVIEW) R. Varga Inst. Phys., Fac. Sci., UPJS, Park Angelinum 9, 041 54 Kosice, Slovakia

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Abstract Domain wall propagation is used in different modern spintronic devices like Race track memory, domain wall logic, domain wall electronics, etc.. The key requirement for the application of thin magnetic wire in such devices is their high speed, which is finally driven by the domain wall propagation velocity. In the given contribution I review our results on the fast domain wall dynamics in amorphous glass-coated microwires. Very fast domain wall has been observed that can reach the velocities of up to 20 000 m/s. Such velocities exceed the sound velocities in magnetic microwires (4 700 m/s). Extremely fast domain wall brings new effects the change of the domain structure at around 1 000 m/s or the interaction of the domain wall with phonons when the domain wall approaches sound velocity. Due to the interaction, the domain wall velocity remains constant in a certain range of applied field, which could be used to synchronize the domain walls propagation in spintronic devices. Typically, scientists use materials with low damping in order to observe a fast domain wall. We show, that it is a distribution of the anisotropy that is even more important. Existence of two, perpendicular anisotropy results in their compensation and in the very fast domain wall. Such theory is confirmed by the measurement of the domain wall dynamics of highly magnetostrictive wires in transversal field.

Introduction Domain wall dynamics in thin magnetic wire is used in many modern devices. Either it is a Race track memory developed by the IBM in last years [1,2], Field Driven Domain Wall Motion Memory (FDDWMM) [3,4], domain wall logic [5,6], domain wall electronics [7,8], or many other sensors [9,10]. Although, the domain wall propagation has been studied more than 70 years [11,12], there are still some problems than avoid full application of such

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

252

R. Varga

devices in praxis. Moreover, new phenomenon appears as the dimensions of the materials decreases and new materials are developed for such applications. The most important problems that appear in applications of the above mentioned devices are: i) the propagation of the domain wall through the real material that contains defects [13], ii) the pinning and depining of the domain wall at desired position [14], and iii) the speed of such devices [15]. The domain wall propagation is driven either by magnetic field or current. However, the speed of these devices depends on the domain wall propagation velocity, which is mostly dependent on the material. Theoretically, the fastest domain wall velocity in thin magnetic wires could be calculated up to 1000 m/s [16,17]. However, much faster domain walls have been observed experimentally [18,19,20]. In this chapter, I will give an overview on our experimental results obtained during the study of the single domain wall propagation in glass-coated microwires. Very fast domain wall could be obtained in these materials under special conditions [20] that sometimes could reach even supersonic speed [21].

Theory of Domain Wall Propagation The domain wall propagation in real material can be described similarly as the linear harmonic oscillator under external force F(t) in a viscous medium [22]. Its time (t) dependent oscillation is described:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

d 2x dx m 2 +β + ξx = F (t ) , dt dt

(1)

where m is the effective mass of the domain wall, β is the damping coefficient that characterizes the viscous medium, ξ is the stiffness coefficient, x is the displacement of the domain wall from its equilibrium position. In the case of domain wall, the force F is represented by the force, acting on the domain wall b the applied magnetic field H and it is expressed as: F= bμ0MsH,

(2)

where b is the constant that depends on the domain wall configuration and is equal 2 for 180o domain wall or √2 for 90o domain wall, μ0 is the permeability of vacuum and Ms is the saturation magnetization. Assuming the domain wall propagation at constant velocity (d2x/dt2→0), linear dependence of the domain wall velocity v on the applied magnetic field H can be simply obtained [23]: v=S(H-H0).

(3)

Here S is the so-called domain wall mobility and H0 is the critical field, below which the domain wall propagation cannot be observed, neither theoretically. Comparing the eq.(1) and

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

253

eq.(3), it can be shown that domain wall mobility is inversely proportional to the domain wall damping parameter:

S=

bμ 0 M s

β

.

(4)

The critical propagation field H0 is a more mystic parameter. Sometimes, it is taken to be a meaningless constant [23]. However, according to eq.(1) and (3), it is expressed:

H0 =

αx . bμ 0 M s

(5)

which is very similar to expression for the coercive field [22]. Therefore, some authors take it simply as coercivity [24]. Another called it dynamic coercivity [25], since it differs from the static coercivity measured from the hysteresis loops. In the following sections, we will show some interesting aspect that are connected to the critical propagation field and that strongly affects the domain wall velocity. However, the most important parameter that controls the domain wall velocity is the domain wall damping. At the beginning, the only domain wall damping was assumed to arise from the eddy current [25]. In the case of magnetic wire, the domain wall damping due to eddy current βe is expressed as [27]:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

βe =

4 μ 02 M s2 r0 ⎛ r0 8 ⎞ ⎜⎜ ln + 2 ⎟⎟ ρ ⎝ rb π ⎠

(6)

where ρ(Τ) is the resistivity and r0 and rb the radii of the wire and that of the inner domain core, respectively. However, the strong damping for highly insulated materials, such as ferrites, cannot be explained by the eddy current damping. Therefore, another damping mechanism, based on Landau-Lifshitz equation has been introduced [28], which is still basis for many theoretical approaches to solve the domain wall propagation [29,30]. The idea starts from famous Landau - Lifshitz model [31] of the domain wall that assumes the homogeneous distribution of magnetization within the domain wall. In such case, the spin dynamics can be described by the famous Landau-Lifshitz equation [24]:

λ dM = γ (M × H ) − 2 (M × M × H ) dt MS

(7)

where γ is the gyromagnetic ratio, and factor λ defines the damping action of the magnetic moment and is called relaxation frequency. The first term on the right side represents precession motion of magnetization, whereas the second term describes the damping of

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

254

R. Varga

precession motion given in first term. In practice, it is more favorable to introduce the parameter α:

α=

λ MS

,

(8)

which is also called Gilbert damping parameter. The switching motion of magnetization is more viscous as the Gilbert damping parameter becomes large. However, the switching time becomes very large, too, as the Gilbert damping parameter becomes small, since the magnetization performs too many precession rotations [26]. Hence, the fastest switching is attained for an intermediate value of Gilbert damping parameter α, called critical damping. It is possible to obtain the relation between the domain wall damping β and Gilbert dampig parameter α [24, 26]. Such damping, called relaxation βr, is inversely proportional to the domain wall width δw:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

βr ≈

μ M α ≈ 0 s γ δw π

K μ0 M s ≈ A π

3λ sσ 2A

(9)

where K is the magnetic anisotropy energy density (in the case of amorphous microwires magnetoelastic), A the exchange stiffness constant, λs the magnetostriction and σ is the mechanical stress. Anyway, neither eddy currents, nor magnetic relaxation could explain a strong variation of domain wall damping with the temperature for some magnetic materials [32, 33]. Therefore, another damping was suggested in [32] that arises from the structural relaxation of the mobile defects on atomic scale. Such contribution, called structural relaxation damping βs, was firstly approved in amorphous microwires [33]:

βs ∝ τ(c0/kT)G(T,t)

(10)

where τ is the relaxation time of the defects, εeff is the interaction energy of the domain wall with the defects, c0 is number of the defects, k is Boltzman constant and G(T,t) is so-called relaxation function [34]. The structural relaxation damping was approved also by the thermal treatment [35], and it is specially important at low temperature, when the atomic mobility decreases, leading to the increase of a local anisotropy and therefore to the increase of the structural relaxation damping [33,36]. Now, the situation should be easy if one wants to achieve the fast domain wall. According to eq.(3) and (4), one should simply keep domain wall damping, β, low (in order to obtain high domain wall mobility S) and critical field H0 also low (even negative, if possible). However, the situation is much more complicated in practice. Generally, the viscous domain wall motion occurs only up to a certain limit. For very high magnetic field, the magnetization precession inside the domain wall becomes non-uniform and the conditions of validity of the above-described model (constant velocity of the domain wall, homogeneous distribution of the magnetization inside the domain wall) are not fulfilled anymore. This situation occurs at

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

255

the so-called Walker limiting velocity vw [37], which is mostly taken as the maximum domain wall velocity for its propagation in viscous regime. Walker limiting velocity can be estimated as [24]:

vw =

γ μ0 M Sδ w 2

(11)

However, the field range in which the domain wall velocity is proportional to the applied magnetic field depends on the presence of the transverse anisotropy. The Walker limiting velocity appears at the so-called Walker field, Hw, which is proportional to the transverse anisotropy field Hk [16]: Hw~ αHk/2.

(12)

Above this Walker field, the domain wall propagation is not uniform and the domain wall velocity decreases and its mobility is negative [25]. Although nice, the Walker model cannot describe fast domain walls observed experimentally in some Garnet films when a perpendicular field, H ⊥ , was applied [38]. These velocities exceed the Walker limiting velocity more than ten times. The maximum velocity, vm, can be related to the Walker limiting velocity, vw, by the following expression [24]:

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

v m 2 H ⊥ (Q + 1) = , vw M SQ

(13)

being Q dimensionless parameter representing the anisotropy of material (Q=2Ku/μ0Ms2). Ku is the anisotropy constant. Finally, very fast domain walls have been observed in ortoferrites [39-41]. Such domain wall velocity exceeds the sound velocity with a maximum ~ 20 000 m/s. v Quasi –relativistic behaviour was found with the maximum velocity equal to the spin waves velocity [40,41]. A theory for such quasirelativistic behaviour was worked out by Zvezdin [42] or Baryakhtar [43]. This is an overview of the physical models mostly used currently for description of the domain wall dynamics. However, new materials and new measuring conditions bring new, surprising results that do not fits to the above given model. Surely, there are much more parameters that defines the domain wall velocity, therefore new experiments on new materials are necessary in order to understand the thing. In the following section, the new material, almost ideal for study the domain wall dynamics, will be presented.

Amorphous Glass-Coated Microwires Amorphous glass coated microwires are novel materials that are not only perspective for technical applications, but they are also ideal material for study the single domain wall

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

256

R. Varga

magnetization processes [44-47]. They consist of metallic nucleus (of diameter 1-30 μm) that is covered by the glass-coating of thickness 2-20 μm (see figure 1). They are prepared by Taylor-Ulitovski technique [48,49] by a rapid quenching of molted master alloy and drawing.

30 μm

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 1. Amorphous glass-coated microwire.

Figure 2. Schematic domain structure of the amorphous glass-coated microwire with a positive magnetostriction.

Due to their amorphous nature, the most important interaction that defines their magnetic properties is magnetoelastic one. It arises from the stresses introduced into the microwire during preparation. The stresses are induced by the quenching, drawing as well as by the different thermal expansion coefficients of metallic nucleus and a glass coating. Three types of stresses exists in glass coated microwires: i) axial one (originates from drawing), ii) radial one (originates from quenching and different thermal expansion coefficient) and iii) circular one. It was shown that axial stress dominates in the center of microwire [50,51]. However, radial stress prevails just below the surface of the metallic nucleus [52]. As a result of magnetoelastic interaction between the magnetic moments and the given stresses, the domain structure of amorphous microwire with positive magnetostriction consists of single axial

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

257

domain, which is surrounded by the radial domain structure as given schematically in figure 2. Moreover, small closure domain appears at the end of microwire in order to decrease the stray fields. Hence, the peculiar domain structure of microwires with positive magnetostriction results in the magnetization process, which runs through the depining and subsequent propagation of single closure domain wall along entire microwire. Such process is characterized by the bistable hysteresis loop, where the magnetization can achieve only two values Ms and –Ms (figure 3).

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 3. Bistable hysteresis loop of amorphous glass coated FeNiMoB microwire with positive magnetostriction.

Besides, the magnetostatic interaction is the second one most important in the case of microwire. There exists so-called critical length, below which the bistable behaviour is lost. In the case of glass coated microwires, it was found to be 2 mm [47].

Single Domain Wall Propagation in Amorphous Glass- Coated Microwires Due to their special domain structure as well as special magnetization process, amorphous glass- coated microwires are ideal material to study the single domain wall behaviour. Either it is the study of single domain wall potential and its particular contributions [53-56], or study of the single domain wall dynamics either for the wall between the axially magnetized domains (in the case of microwires with positive magnetostriction) [33, 57, 58, 59] or the single domain wall dynamics for the domain wall between circularly magnetized domains (for microwires with negative magnetostriction and hence, the circular easy axis) [60,61]. The dimensions of microwires allow us to study the single domain wall propagation at a distance of few cm. There are not a lot of similar materials, in which the experiments can be performed in such a pure manner (e.g. the single domain wall dynamics can be studied without interaction with another domain structure). The

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

258

R. Varga

domain wall velocity v is constant during the domain wall propagation (see figure 4), hence it fulfill perfectly the necessary condition for eq.3, e.g. v is perfectly proportional to the applied magnetic field H in a wide range of temperature (see figure 5). Moreover, planar shape of the domain wall can be estimated from a perfect symmetric shape of emf pulse. Hence, different parameters, that determine the domain wall dynamics (domain wall damping, etc..), can be studied with very small error. Firstly, new contribution to the domain wall damping that arises from the structural relaxation was firstly calculated (figure 6) [33]. Later, it was confirmed in [35] by annealing the microwire at different temperatures.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 4. emf pulse, dM/dt, induced in three equidistant pick-up coils confirms the constant domain wall velocity during its propagation along entire microwire [62].

Figure 5. Domain wall velocity v dependence on the applied magnetic field H for amorphous glass coated FeSiB microwire. Temperature as a parameter [33].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

259

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 6. The temperature dependence of the domain wall damping for amorphous glass coated FeSiB microwire [33].

However, the most interesting fact was a negative critical propagation field H0 that has been found for each temperature. The negative critical propagation field has been found in amorphous glass coated microwires by many authors, however, without mention it [57, 58, 63]. Theoretically, it is very interesting fact, since it points to a possible domain wall propagation even without applied external magnetic field. Such possibility (domain wall propagation without applied magnetic field) was also introduced by Chikazumi [26] for a special domain wall configuration. Negative critical propagation field has also been observed in ferrites [64] and the reason for such phenomenon was estimated to be a dramatic changes in the domain wall structure before it can begin to move [24]. In order to explain the role of negative critical propagation field, we have studied the domain wall dynamics in a wide temperature range [33, 36]. It was found that the amplitude of the negative H0 decreases with temperature and is proportional to the static switching field of the domain wall. This means that both fields are governed by the same mechanism [33]. Moreover, it was found that the critical propagation field H0 is proportional to the effective domain wall mass (as it was suggested theoretically in [25]) and also it is inversely proportional to the domain wall mobility S [36]. In order to avoid all speculations on the negative critical propagation field H0, we have performed detailed domain wall dynamics analysis at low applied magnetic fields using a nucleation coil to force the beginning of the domain wall propagation [65]. It is surely not surprising that no propagation has been observed without applied magnetic field. Instead, different kind of the domain wall dynamics has been found (see figure 7). At higher fields, the domain wall dynamics fulfill the model for the domain wall propagation in a viscous medium given by eq.3. Alternatively, the low-field domain wall dynamics can be described by a power law: v=S’(H-H’0)q

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(14)

260

R. Varga

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

where S’ is an effective domain wall mobility parameter, H’0 is the dynamic coercive field and q is the power exponent. Such a power law results from the interaction of the propagating wall with the defects of the material, which different sources in the actual amorphous microwire have been described before [33,56]. As it was shown in ref.56 and 66, the domain wall potential E consists of two terms: long-range magnetoelastic one and the short-range terms arising from the pinning of the domain wall on the defects in the amorphous medium. The pinning centres are randomly distributed along the amorphous microwire and the pinning field. Therefore, the domain wall potential fluctuates as the domain wall moves through the material and the restoring force ξx acting on the domain wall due to the gradient of the internal potential is given by ξ=dE/dx. When the domain wall passes the region with a local maximum of the restoring force ξx, a local jump occurs until the wall reaches a new site with the restoring force ξx greater than the force 2MSH acting on the domain wall.

Figure 7. Dependence of the domain wall velocity v on the applied field H measured at 298K. Full lines correspond to the fit according to eq.3 and 14 [65].

Therefore, the domain wall motion in the low field limit is adiabatic. Under the action of a small force 2MSH, the domain wall moves slowly close to some local minimum in its potential that arises from the elastic interactions within the domain wall and from the impurities pinning. At some point, the local minimum disappears and the domain wall moves forward rapidly to another local minimum. Such motion is characterized by intermittent jumps from defect to defect. Hence, the local domain wall dynamics description is governed by the generalized eq. (3): v=S(H-(Hdm+Hp)),

(15)

where Hdm includes all long range contribution ( such as the geometry dependent demagnetizing field and the magnetoelastic contribution) and is separated from a random component, Hp, that includes all short-range counterfield contributions. The pinning field, Hp, is assumed to exhibit statistical properties governed by details of the local pinning potentials

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

261

that inhibit domain wall motion. In the first approximation, the distribution of the pinning field takes Gaussian shape with the width of R. Then the magnetization changes during the domain wall jump is given by the power law [67]:

ΔM~ ((R-Rc)/Rc)q

(16)

where Rc is a critical distribution width, below which the small intermittent domain wall jumps do not appear. As a result, the domain wall moves with average velocity (v=ΔM/Δt) given by eq.(14) [68,69].

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 8. emf recorded waveform measured in viscous and adiabatic regime (at very low domain wall vecocity).

Figure 9. Domain wall velocity as a function of magnetic field amplitude for a range of indicated measuring temperatures. Full lines represents the linear fit [66].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

262

R. Varga

The different shape of the induced peaks at the pick-up coils (obtained by a “single shot” acquisition) also confirms the presence of the two regimes (figure 8). Perfect symmetric shape measured in viscous regime confirms the planar shape of the domain wall, which propagates at constant velocity. The emf waveform measured in adiabatic regime (at very low domain wall velocity - 50 m/s) suggest the fluctuation of the domain wall velocity during its motion across the randomly distributed defects. The power law at low fields was confirmed also in a wide temperature range, as proved in log-log plot in figure 9 and in a variety of microwires of different compositions [65,70,71].

Fast Domain Wall Propagation in Amorphous Glass Coated Microwires

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Amorphous glass coated microwires are ideal material to study the fast domain wall propagation. Due to their amorphous structure, they show high resistivity. Hence, the eddy current contribution to the domain wall damping (given by eq.6) is small. Moreover, their amorphous nature results in the lack of magnetocrystalline anisotropy. Magnetoelastic or magnetostatatic (shape) anisotropy, the ones that are responsible for the magnetic properties in the case of amorphous microwires, are much smaller than magnetocrystalline one. Hence, the relaxation contribution to the domain wall damping (given by eq.9) is also small. Finally, the structural relaxation contribution is also small at room temperature as shown in ref.[33,36]. Therefore, it is quite easy to observe very fast domain wall (up to 2000 m/s) even at low field (figure 10) [20,72]. Moreover, negative critical propagation field (although the reason is still questionable) helps to keep the domain wall velocity high even in the case of small domain wall mobility [20].

Figure 10. Dependence of the domain wall velocity v in amorphous Fe62Ni15.5Si7.5B15 microwire on the applied magnetic field H. Full line corresponds to the fit according to eq.2. [72].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

263

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 11. Dependence of domain wall velocity on applied magnetic field for amorphous FeCuNbSiBbased microwire. Temperature as parameter [73].

Figure 12. Schematic structure of different 180o domain wall configuration in thin magnetic wires. a) transversal. b) vortex.

The domain wall damping is not the only parameter that drives the domain wall velocity. Few different regimes of the domain wall dynamics has been found in magnetic microwires under very similar measuring conditions (see also figure 11) [36,72,73]. The first one, characterized by low domain wall mobility, appears at low applied magnetic field. The second one, characterized by high mobility, appears at higher applied field. One possible explanation is the change of the domain wall structure. It was shown theoretically that two types of the domain wall structure could appear in thin magnetic wires [74,75]. Firstly, it is a transversal-type domain wall (figure 12a). Such a domain wall configuration can be expected due to the internal stress distribution in microwires with strong axial anisotropy in the center

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

264

R. Varga

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

and strong radial anisotropy below the surface [52]. The transversal configuration has lower energy and therefore it appears at low fields. Moreover, it creates the free poles at its surface that interact with the radial domain structure resulting in the suppression of the domain wall motion. As it was shown in [74], the transversal domain wall has lower mobility On the other hand, vortex configuration of the domain wall can appear in thin magnetic wires (figure 12b) [74,75]. It has higher domain wall energy and therefore it appears at higher fields. As it was shown theoretically it appears at thicker wires and it has higher mobility.

Figure 13. Up: Temperature dependence of the critical propagation field for both transversal (H01) and vortex (H02) domain wall in amorphous FeCuNbSiB microwire. Down: Temperature dependence of the domain wall damping for both transversal (β1) and vortex (β2) domain wall in amorphous FeCuNbSiB microwire [73].

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

265

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The two domain walls have different dynamics in the microwires [73]. The critical field of the transversal domain wall (H01) is negative (figure 13 up). This can be explained by the fact that the closure domain wall already exists even without applied field and its nucleation field is negative. The absolute value of the critical propagation field H01 increases with increasing temperature as a result of the stabilization of the domain structure through the structural relaxation. On the other hand, the domain wall at high field has positive critical propagation field H02. It does not appear at low temperature and when it appears, it decreases with the temperature in contrary to H01 since the stabilization has no effect at higher fields. The domain wall damping has also different behaviour for different domain wall structure (figure 13 down). The domain wall damping of transversal domain wall β1 varies with temperature according to the variation of anisotropies and equivalent contribution of the domain wall damping. Typically, the domain wall damping in magnetic materials decreases with the temperature as a result of the higher thermal activation of magnetic moments. It can explain smooth increase of the domain wall damping β1 at 77K. An increase of domain wall damping β1 at higher temperatures in figure 13 could be associated to the stabilization of the domain structure through the structural relaxation and subsequent increase of appropriate domain wall damping contribution (see eq.10). In contrary to the transversal domain wall, the domain wall damping of such domain wall β2 almost does not depend on the temperature (see figure 13 down) and is very low giving rise to a very fast domain wall that can reach the velocity of up to 2000 m/s.

Figure 14. Domain wall velocity v as a function of the magnetic field amplitude H for Co68Mn7Si10B15 microwire with the applied tensile stress as a parameter[21].

Having material with very low anisotropy (as one of Co68Mn7Si10B15, which has the lowest possible anisotropy in order to still observe bistable behaviour) one can reach very high domain wall velocities (figure 14) [21]. Such domain wall can approach 20 000 m/s, and is much faster than sound speed in amorphous microwires (that was measured to be somewhere around 4500 m/s [76]). In this case, the domain wall is supersonic and its velocity is 4 MACH. Application of the stress results in an increase the maximum velocity. Also, very high domain wall mobilities have been observed in the microwires with almost zero magnetostriction (hence, almost zero anisotropy, too) [77, 78]. Except domain wall damping,

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

266

R. Varga

there is also another very important fact that influences the domain wall velocity. It is the distribution of anisotropies. In the case of amorphous glass-coated microwires the most important are two perpendicular anisotropies: axial and radial one. Having two perpendicular anisotropies, it is easier to rotate the magnetic moment from one direction into another. The fact is confirmed by the measurement on the highly-magnetostrictive Fe36Co40Si11.1B12.9 microwire (figure 15), where also supersonic wall has been observed [21]. However, the maximum velocity here is lower (8000 m/s) and the domain wall reaches the Walker limit in this case. Application of the axial mechanical stress decreases the maximum velocity and enhances the region in which the transversal domain wall is observed.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 15. Domain wall velocity v as a function of the magnetic field amplitude H for Fe36Co40Si11.1B12.9 microwire with the applied tensile stress as a parameter[21].

Figure 16. Domain wall velocity v as a function of the axial magnetic field amplitude H for Fe77.5Si7.5B15 microwire. The transversal magnetic field as a parameter. Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

267

The importance of the anisotropy distribution is confirmed also by measuring the domain wall dynamics under the influence of both axial as well as transversal magnetic field (figure 16). The increase of the domain wall velocity has been found after application of the transversal magnetic field. Although, the domain wall mobility decreases, the velocity remains constant due to increase of the amplitude of the negative critical propagation field H0. The domain wall velocity increases almost twice at low field (300 A/m) after application of the transversal magnetic field of amplitude 1280 A/m. Finally, very important fact that increases the domain wall velocity is its shielding from the surface. Radial domain structure presented in the microwires shields the propagating domain wall from its pinning on the surface. The surface domain wall pinning is the second most important pinning in amorphous materials [79] that obstruct its propagation along the material. It was also calculated in [16] that introducing the defects just below the surface results in the increase of the domain wall velocity. The effect of the defects is that they invade the easy magnetization axis and decreases the uniaxial anisotropy below the surface, where the domain wall can be pinned. Hence the domain wall propagation is enhanced in such a case. In the case of microwires, the effect of the defects is played by the radial domain structure. New exciting measurements always bring new effects. This is also the case of supersonic domain wall in microwires. Since they have positive magnetostriction (this is the key condition for observing the monodomain structure in amorphous microwires), the domain wall propagation results in microscopic changes of their structure. Such changes generate the elastic waves (phonons). Typically, these elastic waves dissipate on the thermal phonons very quickly. However, when the domain wall approaches the sound speed limit, the dissipation in the elastic subsystem of a microwire grows substantially and the DW transfers part of its magnetic energy to the elastic subsystem [80]. As a result, the domain wall velocity remains constant. This effect could be seen in a magnetic microwires that have higher magnetostriction (figures 17, 18) at various levels (v~ 3100, 4000, 5500 m/s). The magnetic energy E transferred to the elastic subsystem can be calculated from the constant part of the domain wall velocity dependence v(H) [41]: E = 2μ0MsΔH,

(17)

where ΔH is the width of the constant part in v(H) dependence. The transferred magnetic energy increases the amplitude of the elastic waves until the elastic waves damping becomes equal to 2μ0MsΔH. Finally, the domain wall exceeded the sound speed when the driving field becomes greater than the force of dynamic breaking of the domain wall by the elastic waves. Then, domain wall velocity steeply jumps to higher values. Such effect is already well known in ferrites [39-41,80] that have much smaller magnetization comparing to the amorphous Febased microwires. Generally, the sound velocity s in solids is given by: s = (E/ω)1/2,

(18)

where E is Young modulus and ω is mass density. In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, it is possible to generate sound waves with different velocities dependent on the deformation mode.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

268

R. Varga

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 17. Dependence of the domain wall velocity v in amorphous Fe49.6Ni27.9Si7.5B15 microwire on the applied magnetic field H. Full line corresponds to the fit according to eq.3.[72].

Figure 18. Dependence of the domain wall velocity v in amorphous Fe38.7Ni38.8Si7.5B15 microwire on the applied magnetic field H. Full line corresponds to the fit according to eq.2.[72].

Assuming the above mentioned stress distribution in amorphous glass coated microwires (with axial, radial and circular stresses), different types of the elastic waves could be expected (figure 19). Firstly, it is longitudinal elastic wave that arises from the axial stresses. Secondly, transversal elastic wave could be expected as a result of the radial stresses. Finally extensional (Lamb) wave arises from the circular stresses. At least two of them (longitudinal and transversal) can be recognized from the domain wall dynamics measurement in amorphous glass coated FeNiSiB microwires [72], comparing their wave velocities with that measured for stainless steel rod [81] (see also table 1). The last one (Extensional) is

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

269

questionable maybe due to the complex stress distribution in amorphous glass coated microwire. Also, another type of elastic waves could exist, but it is hardly to resolve this question at the moment.

Figure 19: Different types of the elastic waves as could be expected according to the stress distribution in amorphous glass coated microwires.

Table 1. Sound velocities as measured for stainless steel rod [81] and for amorphous glass-coated Fe-Ni-Si-B microwires [72] LONGITUDINAL V (M/S)

Transversal v (m/s)

Extensional v (m/s)

Stainless Steel rod [81]

5790

3100

5000

Fe49.6Ni27.9Si7.5B15

5430

3130

4070

Fe38.7Ni38.8Si7.5B15

5860

3200

4120

Phonon wave

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The interaction of the domain wall with the phonons is very interesting effect, which allows us to study the interaction of magnetic subsystem with mechanical one. From the practical point of view, it hinders the domain wall to increase its velocity. However, the interaction of the domain wall with the phonons can be efficiently applied to synchronize the multidomain movement in applications mentioned above (Race-track memory, FDDWMM, Domain wall logic, etc..). Moreover, the synchronization occurs at very high velocities of the domain wall.

Acknowledgment I wish to thank all my colleagues and students who have greatly contributed to the study of the domain wall dynamics in magnetic microwires. Mainly, it is: Prof. A. Zhukov, Prof. M. Vazquez, Prof. Vojtanik, Prof. J. Gonzalez, Dr. K. L. Garcia, Dr. V. Zhukova, K. Richter, Y. Kostyk, J. Olivera. The work was supported in part by the Slovak Ministry of Education under the projects: APVT-20-007804, VEGA 1/3035/06 and MVTS 6RP/Manunet/UPJS/08.

References [1] Parkin S.S, ‘‘Shiftable Magnetic Shift Register and Method of Using the Same.’’ U.S. Patent 6834005, 2004; ‘‘System and Method for Writing to a Magnetic Shift Register.’’ U.S. Patent 6898132, 2005. [2] Parkin S.S.; Int. J. Modern Phys. B, 2008, vol. 22, 117-118. [3] You C.Y.; Appl. Phys. Lett., 2008, vol.92, 152507 [4] You C.Y.; Appl. Phys. Lett., 2008, vol.92, 192514

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

270

R. Varga

[5] Allwood D. A.; Xiong G.; Faulkner C. C.; Atkinson D.; Petit D.;, Cowburn R. P.; Science, 2005, vol.309, 1688-1692. [6] Allwood D. A.; Xiong G.; Cowburn R. P.; J. Appl. Phys., 2006, vol. 100, 123908. [7] Allwood D. A.; Xiong G.; Cowburn R. P.; Appl. Phys. Lett., 2004, vol. 85, 2848-2850. [8] Bryan M. T.; Schrefl T.; Allwood D. A.; Appl. Phys. Lett., 2007, vol. 91, 142502. [9] Vazquez M.; Physica B, 2001, vol.299, 302-313. [10] Vazquez M.; Badini G.; Pirota K.; Torrejon J.; Zhukov A.; Torcunov A.; Pfuetzner H.; Rohn M.; Merlo A.; Marquardt B.; Meydan, T.; Int. J. Appl. Electromagnetics and Mechanics, 2007, vol. 25, 441-446. [11] Sixtus K.J. and Tonks L., Phys. Rev., 1932, vol.42, 419. [12] O’Handley R.C., J. Appl. Phys., 1975, vol. 46, 4996. [13] Atkinson D.; Faulkner C.C.; Allwood D.A.; Cowburn RP.; Topics Appl. Physics, 2006, vol.101, 207–223. [14] Hayashi M.; Thomas L.; Rettner C.; Moriya R.; Parkin S. S. P.; Appl. Phys. Lett.,2008, vol. 92, 162503. [15] Cowburn R.P.; Science, 2007, vol.311, 183-184. [16] Nakatani Y.; Thiaville A.; Miltat J.; Nature Mater., , 2003, vol. 2, 521. [17] Kunz A.; IEEE Trans. Magn., 2006, vol. 42, 3219. [18] Atkinson D.; Allwood D.A.; Xiong G.; Cooke M.D.; Faulkner C.C.; Cowburn R.P.; 2003, Nature Mater., 2, 85. [19] Glathe S.; Mattheis R. Mikolajick T.; presented as AB-11 at INTERMAG 2008, Madrid, Spain. [20] Varga R.; Zhukov A.; Blanco J. M.; Ipatov M.; Zhukova V.; Gonzalez J.; Vojtaník P.; Phys. Rev. B, 2006, vol. 74, 212405. [21] Varga R.; Zhukov A.; Zhukova V.; Blanco J.M.; Gonzalez J.; Phys. Rev. B, 2007, vol.76, 132406. [22] Chen C.W., Magnetism and metallurgy of soft magnetic materials, Dover Publications Inc., New York., 1986, pp.154. [23] Cullity B.D., Introduction to magnetic materials, Addison-Wesley,1972, pp.446. [24] O’Dell T.H., Ferromagnetodynamics, The MacMillan Press Ltd., 1981, pp. 5. [25] Hubert A. and Schafer R., Magnetic Domains., Springer-Verlag, Berlin, 1998, pp. 271. [26] Chikazumi S., Physics of Ferromagnetism., Claredon Press, Oxford, 1997, pp. 562. [27] Chen D.X., Dempsey N.M., Vazquez M., Hernando A., IEEE Trans. Magn., 1995, vol. 31, 781. [28] Williams H.J., Shockley W., Kittel C., Phys. Rev., 1950, vol.80, 1090. [29] Thiaville A., Garcia J.M., Miltat J., J. Magn. Magn. Mater., 2002, vol.242-245, 1061. [30] Forster H., Schrefl T., Scholz W., Suess D., Tsiantos V., Fidler J., J. Magn. Magn. Mater., 2002, vol.249, 181. [31] Landau L., Lifshitz E., Phys. Z. Sowjetunion, 1935, vol.8, 153. [32] Kittel C., Galt J.K., Solid State Phys., 1956, vol. 3, 437. [33] Varga R., Garcia K.L., Vázquez M., Vojtanik P., Phys. Rev. Lett., 2005, vol. 94, 017201. [34] Kronmuller H., Fahnle M., Micromagnetism and the microstructure of the ferromagnetic solids, Cambridge Univ. Press., Cambridge, (2003), pp.274. [35] Novak R. L., Sinnecker J. P., Chiriac H., J. Phys. D: Appl. Phys., 2008, vol. 41, 095005.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Fast Domain Wall Dynamics in Thin Magnetic Wires (Review)

271

[36] Varga R., Zhukov A., Usov N., Blanco J.M., Gonzalez J., Zhukova V., Vojtanik P., J. Magn. Magn. Mater., 2007, vol. 316, 337. [37] Schryer N.L., Walker L.R., J. Appl. Phys., 1974, vol. 45, 5406. [38] de Leeuw F.H., IEEE Trans. Magn., 1973, vol. 9, 614. [39] Tsang C. H., White R.L., White R.M., J. Appl. Phys., 1978, vol.49, 6052. [40] Chetkin M. V. , Kurbatova Yu. N., Shapaeva T. B., Borschegovsky O. A., Physics Letters A, 2005, vol.337, 235. [41] Chetkin M. V., Kurbatova Yu. N., Shapaeva T. B., Borshchegovskii O. A., J. Exp. Theor. Phys, 2006, vol. 103, 159. [42] Zvezdin A. K., JETP Lett., 1979, vol. 29, 553. [43] Baryakhtar V. G., Ivanov B. A., Sukstanskioe A. L., Sov. Phys. JETP, 1980, vol. 51, 757. [44] Vázquez M., “Advanced magnetic microwires”in Handbook of Magnetism and Advanced Magnetic Materials ed. Kronmuller H. and Parkin S., John Wiley & Sons (2007), pp.221-222. [45] Zhukov A., Gonzalez J., Vazquez M., Larin V., Torcunov A., Nanocrystalline and amorphous magnetic microwires, in: H.S. Nalwa (Ed.), Encyclopedy of Nanoscience and Nanotechnology, American Scientific Publishers, 2004 p.23 (Chapter 62). [46] Chiriac H., Ovari T.A., Progress Mater. Science, 1996, vol.40, 333. [47] Vazquez M., Physica B, 2001, vol.299, 302. [48] Taylor G.F., Phys. Rev., 1924, vol.23,655. [49] A.V. Ulitovski, method of continuous fabrication of microwires coated by glass, authors certifications (USSR patent), No 128427, 3.9.1950 [50] Chiriac H., Óvári T.-A., Corodeanu S., Ababei G., Phys. Rev. B, 2007, vol.76, 214433. [51] Antonov A. S., Borisov V. T., Borisov O. V., Prokoshin A. F., Usov N. A., J. Phys. D: Appl. Phys., 2000, vol.33, 1161. [52] Chiriac H., Ovari T. A., Pop G., Phys. Rev. B, 1995, vol.52, 10104. [53] Zhukov A., Vazquez M., Velazquez J., Garcia C., Valenzuela R. and Ponomarev B., Mater. Sci. Eng. A, 1997, vol. 226-228, 753. [54] Varga R., Garcia K.L., Zhukov A., Vazquez M., Vojtanik P., Appl. Phys. Lett., 2003, vol. 83, 2620. [55] Aragoneses P., Blanco J.M., Dominguez L., Zhukov A., Gonzalez J., Kulakowski K., J. Phys. D, 1998, vol. 31, 494. [56] Varga R., García K.L., Vázquez M., Zhukov A., Vojtanik P., Phys. Rev. B, 2004, vol.70, 024402. [57] Zhukov A., Appl. Phys. Lett., 2001, vol. 78, 3106. [58] Neagu M., Chiriac H., Hristoforou E., Darie I., Vinai F., J. Magn. Magn. Mater., 2001, vol. 226-230, 1516. [59] Puerta S., Cortina D., Garcia-Miquel H., Chen D.X., Vázquez M., J. Non-Cryst.Solids, 2001, vol. 287, 370. [60] Antonov A.S., Buznikov N.A., Granovsky A.B., Joura A. V., Rakhmanov A.L., Yakunin A. M., J. Magn. Magn. Mat., 2001, vol. 249, 95. [61] Ziman J., Kladivová M., Zagyi B., J. Magn. Magn. Mat., 2001, vol. 234, 529. [62] Varga R., Garcia K.L., Zhukov A., Vazquez M., Ipatov M., Gonzalez J., Zhukova V., Vojtanik P., J. Magn. Magn. Mater., 2006, vol. 300, e305.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

272

R. Varga

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[63] Chiriac H., Hristoforou E., Neagu M., Darie I., Mater. Sci. Eng. A, 2001, vol. 304-306, 1011. [64] Tsang C.H., White R.L., White R.M., J. Appl. Phys., 1978, vol. 49, 1838. [65] Kostyk Y., Varga R., Vazquez M., Vojtanik P., Physica B, 2008, vol. 403, 386. [66] Varga R.; Zhukov A.; Ipatov M.; Blanco J. M.; Gonzalez J.; Zhukova V.; Vojtaník P.; Phys. Rev. B, 2006, vol. 73, 053605. [67] Sethna J.P., Dahmen K.A. and Perkovic O., “Random Field Ising Models of Hysteresis”, in “The Science of Hysteresis”, (2006) ed. G. Bertotti, I. Mayergoyz, Academic Press, 107. [68] Durin G. and Zapperi S., “Barkhausen Effect”, in “The Science of Hysteresis”, (2006) ed. G. Bertotti and I. Mayergoyz, Academic Press, p. 181. [69] Yang S. and Erskine J.L., Phys. Rev. B, 2005, vol.72, 064433. [70] Varga R., Kostyk Y., Zhukov A., Vazquez M., “Single domain wall dynamics in thin magnetic wires”, J. Non-Cryst. Solids, 2008, vol.354, 5101. [71] Varga R., Torrejon J., Kostyk Y., Garcia K. L., Infantes G., Badini G. and Vazquez M., J. Phys. Cond. Mater., 2008, vol. 20, 445215. [72] Varga R., Richter K., Zhukov A., IEEE Trans. Magn., 2008, vol. 44, 3946. [73] Olivera J., Varga R., Vojtanik P., Prida V.M., Sanchez M.L., Hernando B., Zhukov A., J. Magn. Magn, Mater., 2008, vol. 320, 2534. [74] McMichael R. D. and Donahue M.J., IEEE Trans. Magn., 1997, vol. 33, 4167. [75] Forster H. Schrefl T, Scholz W, Suess D, Tsiantos V, Fidler J., J. Magn. Magn. Mater., 2002, vol. 249, 181. [76] Chiriac H., Hristoforou E., Neagu M., Darie I., Moga A.E., J. Non-Cryst. Solids, 2001, vol. 287, 413. [77] Chiriac H., Tibu M., Ovari T.A., Presented as F-021 at SMM 18, 2007, Cardiff, United Kingdom. [78] Chiriac H., Ovari T.A., Tibu M., IEEE Trans. Magn., 2008, vol. 44, 3931.

[79] Kronmüller H., J. Magn. Magn. Mater.,1981, vol. 24, 159. [80] Demokritov S.O., Kirilyuk A., Kreines N.M., Kudinov V., Smirnov V.B., Chetkin M.V., J. Magn. Magn. Mater., 1991, vol.102, 339. [81] Weast, R.C., M.J. Astle and W.H. Beyer, eds. 1984. CRC Handbook of Chemistry and Physics, 65th edition. Chemical Rubber Company Press Inc., Boca Raton, Florida.

Magnetic Properties of Solids, edited by Kenneth B. Tamayo, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

In: Magnetic Properties of Solids Editor: Kenneth B. Tamayo, pp. 273-289

ISBN: 978-1-60741-550-3 © 2009 Nova Science Publishers, Inc.

Chapter 7

SUBSTITUTION- INDUCED STRUCTURAL, FERROELECTRIC AND MAGNETIC PHASE TRANSITIONS IN BI1-XGDXFEO3 MULTIFERROICS V.A. Khomchenko and A.L. Kholkin Department of Ceramics and Glass Engineering & CICECO, University of Aveiro, 3810-193 Aveiro, Portugal

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Abstract Investigation of room-temperature crystal structure, magnetic and local ferroelectric properties of polycrystalline Bi1-xGdxFeO3 (x= 0.1, 0.2, 0.3) samples has been carried out. Gadolinium substitution has been found to induce a polar- to- polar R3c→Pn21a structural phase transition at x~ 0.1. Increasing content of the substituting element has been shown to suppress the spontaneous polarization in Bi1-xGdxFeO3, resulting in a ferroelectric- paraelectric Pn21a→Pnma phase transition at 0.2