Electron Theory in Alloy Design 9781907625596, 9780901716170

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ELECTRON THEORY IN ALLOY DESIGN

ELECTRON THEORY IN ALLOY DESIGN Edited by

D.G. PETTIFOR and

A.H. COTTRELL

THE INSTITUTE OF MATERIALS 1992

Book No. 534 Published in 1992 by The Institute of Materials 1 Carlton House Terrace London SW 1Y 5DB

© 1992 The Institute of Materials All rights reserved

British Library Cataloguing-in-Publication Electron Theory in Alloy Design I. Pettifor, D.G. I I. Cottrell, Sir Alan 530.41 ISBN 0-901716-17-0

Typeset in Great Britain by Alden Multimedia Ltd, Northampton Printed in Great Britain by The Alden Press Ltd, Oxford

Data

Contents Preface

v

Note on the Choice of Units

Vll

1. Introduction ALAN COTTRELL

10

2. Modern Electron Theory M.W. FINNIS

3. Nearly-Free-Electron Approximation: or a Tool for "Research? JURGEN

Text-Book Physics 44

HAFNER

4. The Tight Binding Approximation:

Concepts and Predictions

81

D.G. PETTIFOR

5. Order and Phase Stability in Alloys

122

FRANCOIS DUCASTELLE

158

6. Point, Line and Planar Defects A.T. PAXTON

Properties of Metals at the Nanometre Scale

7. Mechanical A.P.

SUTTON, J.B.

8. Permanent

PETHICA,

H. RAFII-TABAR

AND J.A.

191

NIEMINEN

Magnets

234

R. COEHOORN

9. Magnetic Layered Structures

264

D.M. EDWARDS

284

10. Future Directions ALAN COTTRELL

Appendix: A.l. A.2 A.3 A.4 Index

Green functions Dirac notation Quantum mechanical Ising model

methods for periodic systems

294 297 299 301

Preface The past decade has witnessed a revolution in the ability of electron theory to predict numbers and provide concepts that are of direct import to understanding the structural, mechanical and magnetic properties of metals and alloys. As recently as 1962 William Hume-Rothery was stressing 'the extreme difficulty of producing any really quantitative electron theory'. The breakthrough came, in fact, two years later when Pierre Hohenberg, Walter Kohn and Liu Sham showed that it was possible to transform the complicated many-electron problem into an effective one-electron problem which could, in principle, be solved within the so-called Local Density Approximation. The advent of fast computers and improved numerical algorithms now allows the routine solution of these equations and reliable prediction of ground state properties such as the heat of formation, planar fault-energies, shear moduli and magnetic anisotropy. In addition, this new found ability for making accurate predictions has been accompanied by the development of simple, yet .reliable Nearly Free Electron or Tight Binding models which provide direct physical and chemical insight into the origin of bonding and structure at the atomistic level. The purpose of this book is to present these new developments in electron theory which are now starting to make an impact on the search for novel alloys with improved mechanical or magnetic properties. It was produced as an accompanying volume to a two day short course at Imperial College from 14-15 September 1992. We wish to thank Keith Wakelam and Peter Danckwerts of The Institute of Materials for their unfaltering help and good cheer in keeping to the very tight publishing schedule. We also thank the European Office of the US Office of Naval Research for financial assistance. Finally we wish to thank Professor David West and Professor Harvey Flower of the Department of Materials at Imperial College, the former for suggesting the timeliness of such a course, the latter for helping to put the course together on the ground. D.G. Pettifor A.H. Cottrell June 1992.

Note on the Choice of Units The energy and length scales which are appropriate to electron theory are those set by the ionisation potential and first Bohr radius of the hydrogen atom. In SI units the energy and radius of the nth Bohr stationary orbit are given by (1)

and (2) where m is the electronic mass, e is the magnitude of the electronic mass, eo is the permittivity of free space and h is Planck's constant divided by 2n. Substituting in the values m = 9·1096 x 10-31 kg, e = 1·6022 x 10-19 C, 4neoc2 = 107,c = 2·9979 x 108ms-1andh = 1·0546 x 10-34js,wehave (3) and (4)

Therefore the ground state of the hydrogen atom, which corresponds to n = 1, has an energy of - 2·18 x 10-18 J and an orbital Bohr radius of 0·529 x 10-lO m or O·529A. Because of the small value of the energy and radius in SI units, it is customary for electron theorists to work in atomic units where the unit of energy is the Rydberg (Ry) and the unit of length is the atomic unit (au). The former is the ionisation potential of the hydrogen atom, the latter if the first Bohr radius. Thus, in atomic units we have (5) and (6) where

1 Ry

=

2·18

X

10-18

J

13·6 eV

and

1 au

=

5·29 x 10-11 m

=

O·529A. It follows from equations (1), (2), (5) and (6) that 112/2 m = 1 and e2j4nBo = 2 in atomic units. An alternative choice of energy unit is the Hartree which is obtained by setting m = 1, e = 1, h = 1 and 4nBo = 1. It follows from equation (1) that 1 Hartree = 2 Rydbergs. The total energy of the bulk metal will usually be given in either Ry/atom or eV/atom. Conversion to other units may be achieved by using I mRy/atom = 1·32kjmol-1 = Oo314kcalmol-1•

1

Introduction ALAN COTTRELL Department of Materials Science and Metallurgy, University of Cambridge, Cambridge CB2 3QZ, U.K.

1.1 FREE ELECTRONS The electron theory of metals is an exact twentieth century creation for it was in 1900 that Drude first perceived that a metal contains a gas of charged particles, the mobility of which is responsible for its superb electrical conductivity. Soon afterwards, Lorentz identified these particles as electrons and so thefree electron theory of metals was born. From the start the theory was very successful. It not only explained the high electrical (a) and thermal (K) conductivities of metals, but also the Wiedemann-Franz ratio (1.1) in good agreement with experiment (k = Boltzmann's constant, e = electronic charge, T = temperature). It also explained the characteristic optical properties of metals. Somewhat later, people realised that the metallurgical properties also come under its spell. Because the atoms in a metal are held together, not by chemical bonds, but by their electrostatic attraction to the free electrons moving about, among them, it is easy for different kinds of such atoms to join together, even in arbitrary proportions as in alloy solid solutions; also for large clean pieces of metal to adhere simply by contact, as in welding; and for the atoms in a metallic crystal to slip over one another easily, as in plastic deformation. Pauli's application of Fermi-Dirac quantum statistics to the electron gas, in 1926, enabled Sommerfeld a year later to explain why the specific heat of this gas fell far short of that of a classical one. Quantisation of this gas becomes easy if we push the idea of free electrons to its limit by postulating a completely constant potential energy for a free electron inside the metal, at an energy level, 1'0, which is sufficiently far below the constant level in the vacuum outside (e.g. about 15eV below, for AI) as to keep all such electrons in the metal, except at

2

Electron Theory in Alloy Design

very high temperatures. We can make Va the reference energy zero, so that Schrodinger's equation, inside this model of the metal, then reduces to that of a free-electron gas. However, the sinusoidal waves which are its solution must then have nodes at the surfaces of the metal, to fit the zero amplitudes in the vacuum outside. The wavelengths }L, are thus quantised. It is customary to represent these plane wave solutions in the form ( 1.2) where r is position, A is the normalisation amplitude, and k is the wave vector (k = 2n/A), i.e. the quantum number of the wave t/I, with components ( 1.3) along three axes in the metal, i = x,y, Z of lengths Li; and ± n, = 1, 2, 3, etc. The allowed quantum states of the free electrons, indicated by the various differen t sets of ni values, are then represented as a lattice of poin ts in k-space (or reciprocal space). Since k is equivalent to momentum, the kinetic energy of an electron (and hence total energy E, relative to the chosen flo = 0 at the bottom of this free electron energy band) increases as k2, up to the Fermi energy (1.4 ) at which all the free electrons are accommodated, with two of opposite spins in each state, according to the Pauli principle. Here kF is the Fermi wave number and n is the number of free electrons per unit volume. The surface of constant energy EF, bounding the occupied region of k-space is the Fermi surface, in this case a sphere. The Fermi energy EF is also identified as the Fermi level, sometimes positive when measured from flo, sometimes negative when measured from the external vacuum level. The smallness of the electronic specific heat then follows quite easily from the fact that only a small fraction of the electrons, near the Fermi surface, can absorb thermal energy. Those deeper inside the Fermi distribution cannot do so because there are no empty quantum states, at nearby energy levels, into which they could be excited by thermal energy.

1.2 NEARLY

FREE ELECTRONS

Practically all the a justification-of began with Bloch .was into how the

theory from this point onwards is a probing-and hopefully the audacities of the free electron theory. One enquiry, which in 1928 and resumed with the pseudopotential theory of 1959, electrons could move so freely among the strong electrical

Introduction

3

fields of the positive ions. Bloch's great contribution was to replace the constant Vo of the free electron theory by a periodic lattice field; and to find the general solution of the Schrodinger equation for this. His solution, the Bloch wavefunction t/I ( r)

=

u ( r )ei k . r

(1.5)

replaces the plane wave function of equation (1.2) by the simplest generalisation, i.e. a plane wave that is modulated at each lattice site by a function u(r) which has the periodicity of the lattice and looks rather like an atomic wave function at each site. Alternatively, we can regard it as a sequence of repetitive atomic-like orbitals, with phase differences, strung together along the bow of a sine wave. This latter view particularly fits the tight-binding variant of the theory, in which the orbitals really are atomic ones, slightly perturbed by the presence of weakly overlapping orbitals on neighbouring atoms. The Bloch theory introduced two major effects: (1) The Bloch wave, like its plane wave predecessor, repeats exactly, ad infinitum, throughout the entire domain of the perfect lattice. Then, despite the periodic lattice field which is responsible for u(r), a Bloch electron is just as free to move throughout the lattice, without impediment, as is a free electron. The high conductivities of pure, defect-free, metal crystals at low temperatures, which are indicative of extremely long mean free paths, are thus explained. (2) At critical wavelengths and angles of incidence, the Bloch electrons suffer Bragg reflection. This is the equivalent, inside the crystal, of electron diffraction. Consider a set of Bragg reflecting planes (hkl) whose spacing a, and orientation (normal to an x axis), are represented by a reciprocal lattice vector g (g = 2n/a). Let the Bloch wave of equation (1.5) represent an electron approaching these planes from one direction. If it is reflected, then the component k of k, normal to the planes, changes sign. This Bragg condition, in terms of wave vectors, is then that k is replaced by k', where k'

=

k

+ g

(1.6)

Obviously, the reflected wave can be re-reflected and so on. The two waves, k and k', thus have equal status and so can be superposed to give more significant solutions for the x-component of the wave, k, ~(ikx

±

e-ikx)

=

cos kx

or

isin kx

(1.7)

These are standing waves. The cosine function has its maxima at the lattice planes and so represents a lower energy state than the corresponding free electron one, since the standing wave of electron density is then greatest in the vicinity of the positively charged cores. Similarly, the sine wave represents a higher energy state. The energy band of the Fermi distribution is thus

4

Electron Theory in Alloy Design split, by the energy gaps between the cosine and sine solutions, into allowed energy bands separated by others which are 'forbidden' in the sense that no Bloch states exist within them.

In three-dimensional k-space this behaviour is conveniently represented by Brillouin zones. These are planes, each related to one of the reflecting crystallographic planes, and each defined in k-space by its ~g value. Bragg reflection and the energy gap are thus located by those Bloch wave functions whose k values lie on one of these Brillouin zone boundaries. These boundaries divide k-space up into polyhedral Brillouin zones, each of which contains N Bloch states for a crystal of N unit cells. While this theory applies in principle to all electrons, in practice only the valency electrons are significantly perturbed by the presence of neighbouring atoms. Thus, in for example sodium, with one electron per atom (epa), the first Brillouin zone is half-filled. In aluminium, with 3epa, the Fermi surface overlaps slightly into the third zone, the first zone (i.e. lowest energy) being completely full, the second being empty in some regions of k-space.

1.3 APPLICATIONS OF THE BRILLOUIN ZONE THEORY There are two famous applications of this theory. First, to the distinction between metals and insulators. In both of the metals just mentioned all or a large part of the Fermi surface is free-electron-like in the sense that, where it crosses the interior of a zone, it separates filled and empty electron states which have almost identical energy levels. This is the basic requirement which allows an arbitrarily small applied voltage to produce a flow of electrical current, i.e. a slight off-centering of the Fermi distribution in k-space. These elements are thus metallic conductors; sodium because the first zone is only half full, aluminium because the smallness of the energy gaps allow a new zone to start filling before its predecessor is completely full. By contrast, in materials where a strongly varying 1'0 produces a corresponding u(r) which differs greatly from the free electron A (equations (1.5) and (1.2)) and so produces large band gaps, it is possible for some zones to become completely full while all others, at higher energies, remain completely empty. The above condition for electrical conductivity, at the Fermi surface, no longer holds. Such materials are insulators (e.g. diamond). Second, the Jones theory of alloy phases. Hume-Rothery had proved experimentally that in alloys of copper, silver and gold with higher valency metals, certain alloy phases occur at particular epa values. The rx boundary (primary FCC solution) occurs at epa ~ 1·4; the f3 phase (BCC) at 1·5; the y phase (complex cubic) at about 1·6; and so on. Jones pointed out that when the epa is raised by such alloying, so that the expanding Fermi surface begins to touch the inside of a zone boundary, the filling of the cosine-like standing wave functions should save energy. But, when this epa is raised still further, the then

Introduction

5

necessary filling of sine-like functions, on the outer side of the zone boundary, should lead to a steep rise in energy. Thus, a given crystal structure should be particularly favoured at alloy compositions where the Brillouin zone boundary is just touched. Simple calculations then suggested that this could explain the compositions of the Hume-Rothery phases. TheJones theory is conveniently expressed in terms of the density of states, i.e. the number of quantum states, per unit volume of metal, in a given narrow range of energy. The argument of the theory is then that the density of states climbs high, in the energy range approaching a Brillouin zone boundary from below; and then drops sharply to zero at higher energies as EF surpasses the energy limits of the zone. Time has not dealt kindly with either of these theories. The first one implied, absurdly, that an imaginary sodium crystal, the atoms of which have a one-mile lattice spacing, would still be a metallic conductor. The second was blighted by Pippard's experimental demonstration, in 1957, that the Fermi surface already touches the zone boundary in pure copper. 1 Something of the Jones theory has survived however. It probably does explain some complex alloy structures, such as that ofy-brass, in terms of a Brillouin zone which resembles a facetted sphere, which can be fitted well by a quasi-spherical Fermi surface. More recently, Heine and Weaire" have successfully accounted for several metallic crystal structures by using Brillouin zone concepts.

1.4 THE FREEDOM

OF ELECTRONS

Despite the Bloch theory, the freedom of metallic electrons still provided a challenge. Experiments similar to that of Pip pard proved that in many metals, e.g. aluminium and lead, the Fermi surface was very little different from that of a free electron sphere. Why do the enormously deep potential energy wells, at the centres of the atoms, play so small a part? The answer was provided by pseudopotential theory which, in its application to metals, began in about 1959. Later lectures will deal fully with this. Briefly the answer is that, as a conduction electron passes through the central region of an ion, in the metal, it not only experiences the strong electrostatic attraction to the nucleus. It also experiences a strong repulsion from the core electrons. This repulsion is primarily quantummechanical, due to the Pauli principle which forbids this electron to have an atomic-like u(r) function similar to those already occupied by the core electrons. It can pass through the core only with a u(r) that is orthogonal to these atomic orbitals, in the sense that it must have more nodes. The sharp bending, to and fro, of its wave function, so as to fit all these nodes into the core region, implies that the electron has additional kinetic energy due to this Pauli repulsion. In 'good' metals such as Na, Mg and AI, these two opposite energy contributions almost cancel, so that the ions in these metals appear to be nearly transparent to the conduction electrons.

6

Electron Theory in Alloy Design

It thus became possible, in Schrodinger's equation, to replace the true potential by a pseudopotential in which the Pauli repulsion is introduced to cancel out most of the Coulomb potential of the nucleus. An added advantage of this is that, in 'good' metals, the pseudopotential is so weak that it approximates almost to the simple Vo = constant of the free electron theory, so that u(r) ~ A and the Bloch wave function differs only slightly from a plane wave. This opens the way to a high-quality treatment of such metals by a simple perturbation treatment.

1.5 SPATIAL DISTRIBUTION

OF ELECTRONS

A weakness of the k-space theory is that it virtually ignores the distribution of electrons in the real space of the metal; its absurd prediction of metallic properties among widely spaced atoms stems from this. The problem was solved by Mott in 1949 and the subsequent Mott-Hubbard theory has given a quite different criterion for the difference between metals and insulators.v" Start with a lattice of widely spaced, neutral, atoms. Take an electron off one and give it to another. This costs a large ionisation energy and delivers, in return, only a small electron-affinity energy. However, we can give this electron to anyone of all the other atoms. In other words, we can put it into a Bloch state at the bottom of the 'conduction' band. Because of the bandwidth of this, there is an energy gain for an electron at the bottom of the band. lfthe lattice spacing is large this gain is too small to compensate for the ionisation-affinity loss. The neutral state then has lower energy and the material is an insulator. But if the spacing is small, the gain from the large bandwidth can then more than compensate the ionisation-affinity loss. The state in which such electrons leave their parent atoms, to roam through the material in Bloch states, is then preferred. This is the Mott transition. While very different from the Brillouin zone theory of insulators and metals, it does not supersede the latter. Even when the electrons are free to move about in Bloch states, the original condition at the Fermi surface has still to be satisfied before the material is a metal. The final criterion for a metal remains that there should be a finite density of states at the Fermi energy.

1.6 THE INDEPENDENCE

OF ELECTRONS

Another audacity of the free electron theory was its one-electron nature. The potential in the true Schrodinger equation should contain terms representing all the mul titudes of electron-electron interactions. Since each such term depends on where each electron is, relative to the others, it depends on tfr2, i.e. on the solution of this equation. The true equation is thus an impossible many-electron one. In all one-electron (i.e. Hartree) theories this problem is

Introduction

7

by-passed by assuming that an electron moves in the time-averaged potential of the cores and other valence electrons. We then recover a one-electron Schrodinger equation with a fixed V distribution which can be found by repeated solution until a self-consistent one is attained. In this, it is necessary to take proper account of the fermion symmetry of the electrons, which leads to exchange interaction between them and to the Hartree-Fock refinement. The effect is that each electron is surrounded by a small exchange hole in which no other electron of the same spin is likely to be found. This theory is still unsatisfactory, however, because it does not take proper account of the electrostatic interactions between electrons. This has proved to be one of the most difficult parts of the subject. An important advance was made in 1951 by Bohm and Pines5 who showed that these interactions, at long range, produce a collective motion of the electron gas, rather as ifit were a kind of elastic medium. These are the plasma oscillations, with frequency of about 1016 s -1. When the totality of electron-electron electrostatic interactions is partitioned into plasma oscillations and the rest, the rest turns out to be short-ranged and representable as a correlation hole, round each electron, inside which no other electron is likely to be found, and of size approximately given by the Thomas-Fermi screening radius, which is =

rTF

E (~ 6nne

)1/2

( 1.8)

~

(EF = Fermi energy, n = number of electrons per unit volume). The recognition that the electron-electron interactions could be largely dealt with by representing each electron as a quasi-particle, in which it is surrounded by a neutralising exchange-correlation hole, encouraged the view that a oneelectron theory which somehow included these interactions might be possible after all. This hope was realised in the importan t density-functional theory, introduced by Hohenberg and Kohn6 in 1964. By making some reasonable approximations, including representing the exchange-correlation terms as an effective electrostatic potential

~c

=

e2 _ - [3n2 n(r)] 3

1/3

(1.9)

where n(r) is the local value at r of the (generally variable) electron density, they were able to return to a solvable one-electron Schrodinger equation with V:c added to the core potentials. This theory has proved to be extremely powerful and successful; and is the starting point of most modern advances in the electron theory of metals. One useful simplified version of it is the effective medium version of the embedded atom theory. Suppose that we wish to know the cohesive energy of a foreign atom, embedded in a metal. Then, in this theory, we simply first evaluate the local

8

Electron Theory in Alloy Design

electron density, no, in the metal, at the embedding site, before the atom is embedded, and then, from standard results which have now been published for most of the lighter elements, we deduce the energy of interaction of this atom with a uniform electron gas of the same density, no' In the absence of strong covalent bonding this simple method generally gives over 80% of the cohesive energy; and can often then be improved to give much more accurate values. 1.7 COVALENT

BONDING

The other main line of development which has greatly increased the utility of the modern theory has been to exploit the analogy with the covalent bond theory of molecular chemistry. We recall how the hydrogen molecule, H2, can be represented as a linear combination of atomic orbitals (LCAO), i.e. by a superposition of the two Is orbitals of the combining atoms. When these orbitals begin to overlap, spatially, they may do so as wave functions of the same sign, giving a bonding state between the atoms; or opposite signs, giving an anti-bonding state. The degenerate Is energy level of the two atoms is thus split, in the molecule, into two levels, bonding and anti-bonding, respectively below and above it. Exactly the same principles can apply when we bring a large number of atoms together, to form a solid, except that there is multiple splitting into a quasi-continuous band of levels. This approach is best justified when the atomic orbital overlaps are not too extensive; and so is particularly suited for dealing with the bonding in transition metals, between the partly-filled d-shells of these atoms. The theory in fact goes back to the earliest days of the Bloch theory, where it became known as the tight-binding approximation, but it has since been made much more useful by the subsequent work of Slater and Koster," Friedel,8 Pettifor," and others. Two examples of this: (i) As follows directly from the chemical covalency theory, the bonding is strongest when the bonding orbitals are all filled and the anti-bonding ones are all empty. Thus the cohesion of the transition metals is greatest in the middle of the transition rows, in the periodic table. (ii) A good first approximation to the theory shows that if an embedded atom bonds 'covalently (and equally) with 3 neighbours, then its total bonding energy is times that with a single neighbour. This allows one to estimate cohesive energies, e.g. at free surfaces, by simple bond counting and then taking the square root of the number counted. In this form it has become the basis of a second version of the embedded atom theory, also very successful. 10

.J3

1.8 CONCLUSION The aim of this introduction has been to put these various developments of the electron theory into perspective. They will now be developed systematically in the following lectures.

Introduction

9

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

A.B. Pippard, Phil. Trans. R. Soc. A, 1957, 250, 325. V. Heine and D. Weaire, Solid State Phys., 1970, 24, 249. N.F. Mott, Proc. Phys. Soc. A, 1949, 62, 416. J. Hubbard, Proc. Roy. Soc. A, 1963, 276, 238. D. Bohm and D. Pines, Phys. Rev., 1951,82, 625. P. Hohenberg and W. Kohn, Phys. Rev. B, 1964, 136, 864. J.C. Slater and G.F. Koster, Phys. Rev., 1954, 94, 1498. J. Friedel, in The Physics of Metals I-Electrons, (ed. J.M. Ziman), Cambridge Press, 1969. 9. D.G. Pettifor, ]. Phys. (Paris) C, 1970,3, 367. 10. M.W. Finnis and J.E. Sinclair, Phil. Mag. A, 1985,50, 45.

University

2

Modern Electron Theory M.W. FINNIS Max-Planck-Institut fur Metallforschung, Institut fur Werkstoffwissenschaft, Seestrasse 92, D-7000, Stuttgart 1, Germany

2.1 FIRST PRINCIPLES THEORY: PSI IN THE SKY? The kind of electron theory which is relevant to alloy design has to do with the calculation of electron densities and total energies. We are talking about the energies of atoms bonded together into structures which range from perfectly crystalline, through heavily deformed to completely amorphous. The discussion in this chapter is mainly about the energy of a system when the electrons are in their ground state, which means the equilibrium state of electrons at zero temperature for a given frozen configuration of the atomic nuclei. Fortunately the ground state theory is of wider applicability than might be expected, and the effects of temperature can often be taken into account, as we shall see later. We shall focus on how calculations of the total energy can be done fromfirst principles, that is without any parameters which have to be adjusted to fit experimental data. Simplifications can be made along the way, and these will be discussed more later. They can be used to develop semi-empirical models of the energy, which in their simplest form, for example balls and springs models, do not involve any explicit description of the electrons. An important role of the most accurate first principles calculations is to provide benchmarks against which more empirical models of the energy can be tested. The total energy of a system together with the electron density are the two fundamental quantities calculated by theoreticians. However, many related quantities of more practical interest can be coaxed out of the same theory, such as magnetic moments, elastic properties and the energies of formation of point and planar defects. Furthermore, with the addition of some statistical mechanics, calculations from first principles are beginning to shed light on the origin of phase transitions and the structure of phase diagrams in alloys.

10

Modern Electron Theory

11

2.2 THE IDEA OF A DENSITY FUNCTIONAL In the fifties and sixties the theory of metals was of necessity domina ted by a study of electronic states or band structures. Reliable methods of calculating total energy first came on the scene in the late 1960s and 1970s. The branch of electron theory which has grown rapidly over the last twenty-five years to meet the challenges of metallurgy is based on density functional theory. Most quantum mechanical calculations of metallurgical relevance employ density functional theory, although the technical ways in which it is applied are many and varied. The purpose of this chapter is to acquaint readers with its main concepts and jargon, and to introduce some of the ways in which it is implemented in practice. For more complete accounts of the theory introduced here the reader is referred to very good review articles.v' The concept of a density functional is not rooted in quantum mechanics; in fact a very similar formalism has been developed to describe classical liquids. While a minimal knowledge of quantum mechanics is necessary to understand the density functional theory of electrons, it is easier to grasp than the conventional quantum mechanics of many-body systems, of which it is one aspect. Since the early days of quantum mechanics in the 1920s and 1930s it was understood that the ground state of the N electrons in a piece of matter is described in principle by a wave function H' (rl , r2, ... r N) which is a function of all their coordinates. The position of the nth electron is denoted by r n : The spin of each electron must also be specified, which introduces an additional set of variables in an explicit treatment. The exact wave function could never be calculated, but let us take it for granted that it exists. The Pauli exclusion principle now imposes the condition of anti symmetry under the exchange of the coordinates of any pair of electrons with the same spin: \f (. . .

r i, . . . rj'

. . .)

==

-

\f (. . .

rj'

. . . ri,

(2. 1)

. • .)

which has the effect of keeping electrons of like spin out of each other's way. To see how this works, we recall that 1\f12 is proportional to the probability that the electrons will have any particular set of positions. According to the above equation \f must tend to zero as any pair of r., rj approach each other, which we interpret to mean that two electrons with like spin tend to keep apart. Electrons are also of course also directed out of each other's way irrespective of their spin by their mutual Coulomb repulsion. We want now to take a step away from the wave function description of electrons and discuss instead the electronic charge density n(r), where r denotes a point (x,y, z). Unlike the wave function, n(r) is a measurable quantity which depends uniquely on the wave function according to the prescription: n(r)

=

N

f dr2 . · . f drN'P*

(r, r2,

• ••

rN)'P (r, r2,

•••

rN)

(2.2)

The converse is not true: the wave function is by no means a unique function

12

Electron Theory in Alloy Design

of the charge density. This fact seems at first sight unfortunate but it is actually exploited in the development of density functional theory. As we shall see later, it is postulated that there are simple wave functions corresponding to hypothetical non-interacting particles which give the same charge density as more complicated wave functions of real interacting electrons. Since we are interested in the total energy of systems, let us now ask the question: what is the electrostatic energy of this charge density? The classical answer, called the Hartree energy, is given by E

=

~

2

H

II drdr' n(r)n(r') Ir - r'l

(2.3)

Although EH is part of the answer to our question, it does not take account of the fact that the charge density n(r) is an average result of the motions of electrons whose instantaneous positions are correlated. The actual electrostatic energy for a given n(r) will be lower than EH• Firstly there is a contribution due to the Pauli exclusion principle which by virtue of the antisymmetry of the wavefunction keeps the electrons automatically out of each other's way. This contribution to the lowering of the electrostatic energy is called the exchange energy. Secondly the mutual repulsion of electrons lowers the energy a step further than would be required just from the antisymmetry of the wavefunction. This further lowering of the energy is called the correlation energy. The sum of these two corrections is what electron theorists mean by exchange and correlation energy, usually written Exc. It should be noted that the exchange and correlation energy arises from nothing other than the antisymmetry of the wavefunction and the Coulomb repulsion between electrons; it is an additional term to EH by virtue of the fact that their positions in space are correlated. Thus the total electron-electron interaction energy is given by (2.4)

The total energy, denoted by F, associated with the ground state charge densi ty n (r) is the sum of electron-electron interactions and the kinetic energy of the electrons T:

F

=

T + Eee

(2.5)

The quantities entering F are assumed to be given uniquely by the particular density n(r). This is one of the basic assumptions of density functional theory, which it is beyond the scope of this book to discuss in more detail. We can see that it is trivially true for the Hartree energy, since EH is an explicit double integral over the density. Since n(r) is itselfaJunction ofr, the unique dependency of a given quantity on n is described by saying that the quantity is afunctional of n, which is usually expressed by putting the argument n in square brackets. Thus we write EH[n], F[n] and so on. A functional is a function ofa function.

Modern Electron

Theory

13

Having in trod uced the energy of the inhomogeneous electron ,gas F[ n], we now ask: what is the energy of a piece of matter, in which a number of fixed atomic nuclei-ranging from 1 in a single atom to 1022 or more in a baked bean -are embedded in the inhomogeneous electron gas? We ignore-for the time being the energy of motion of the nuclei. Besides F[ n] the energy of a collection of atoms contains two further terms, namely the interaction of the electrons with the nuclei, denoted Eep, and the Coulomb interaction between the nuclei, denoted Epp. As far as the electrons are concerned, the nuclei present a potential just like any other external potential, and it is therefore usually denoted V;xt (r). The electron-nuclei energy is therefore

e; The nuclei-nuclei

=

(or internuclear

f drn(r)

proton-proton)

-.!. L

E

pp -

Vext(r)

2··

1,)

(2.6)

energy is given by

ZjZj

IR. - R·I J

(2.7)

1

where Zi is the charge on the ith nucleus and R, is its position. We use atomic units, in which the charge on an electron is unity and the unit of energy is the Hartree = 27·2 eV. If the nuclei are the only source of external potential to the electrons we have (2.8)

The final result for the total ground state energy of a system of electrons and fixed nuclei is therefore given by E[ n]

=

F[ n]

+ Eep [ n] + Epp

(2.9)

We now present the most fascinating and important property of this density functional which was proved by Hohenberg and Kohn," and in the more general form stated here by Levy." It comprises two statements: (1) The ground state energy EGS is given by equation (2.9) when n(r) is the ground state charge density nGS (r). (2) For a very large class of functions n (r) corresponding to a fixed total number of electrons, namely all those functions which are obtainable from some antisymmetric wave function by the standard equation (equation (2.2)), E[n] ~ EGs• In other words, the ground state energy is got by minimising E[n] with respect to the density. This variational principle is the central result of the HohenbergKahn paper. As an aside we note that the effect of temperature can be included in a rigorous way in nand E[n], although its most important effect appears

14

Electron Theory in Alloy Design

as the kinetic energy of the nuclei. The nuclei in a solid are constantly dancing under the influence of electrostatic forces from the electrons and other vibrating nuclei. Because the electrons are so light they follow the nuclei so closely that within the rather good Born-Oppenheimer approximation the instantaneous electron density is the same as if all the nuclear positions were frozen. This happy result makes the solution of the equations of motion of the nuclei a tractable problem in many cases, because the electronic and ionic motions are effectively decoupled. Powerful though the Hohenberg-Kohn result is, there were some major questions to answer before it led to practical tools. These concerned how best to describe T and Exc without solving the full many-body problem. Conversely if reasonable approximations to T and Exc can be developed as functionals of the density, we have reduced the relevant part of the many-body problem to a classical problem in functional minimisation. One direct approach to solving problems within this framework is the Thomas-Fermi (TF) method and its extensions, as described in the next section. We discuss the TF method first because it helps to explain some useful ideas about functionals, although it is not nowadays regarded as the most reliable quantitative way for doing total energy calculations. After the discussion of the TF functional, we shall derive some further physics of density functionals, which is not dependent on any explicit functionals or on wave-functions. Included here is a discussion of the forces on nuclei and the calculation of energy changes induced by a change in the external potential, that is the idea of perturbation theory. The section after that will introduce the theory of Kohn and Sham," which provides the most widely used functional for modern calculations of ground state properties. Wave functions rear their heads again at this point, but fortunately these are single electron wave functions. The many-body problem has now been reduced to an effective one-body problem. This sets the stage for a summary of the way theoreticians have put the Kohn-Sham theory into practice for calculating such things as electron densities, magnetic moments and total energies of perfect and defective solids.

2.3 THE THOMAS-FERMI

MODEL AND ITS EXTENSIONS

Two approximations to the electronic energy F[ n] can be introduced which lead to the model of'Thomas'' and Fermi. 7 The first is to treat the kinetic energy T[ n] as a local quan tity. In other words the total kinetic energy is assumed to consist of a sum of little bits from all regions of space, whereby the contribution of each little bit depends only on the local electron density*. Formally, this is *The meaning of local and non-local can be appreciated by considering the form of EH, which is a simple non-local functional; in other words it is not the sum of little bits which depend only on the density of electrons at their local positions.

Modern Electron Theory

15

expressed as T[ n]

=

f

(2.10)

drt(n(r))

The local kinetic energy density t(n) is taken to be that of a hypothetical uniform gas of non-interacting electrons with density n. The energy of such a uniform free electron gas is all kinetic, and even at zero temperature it is non-zero, because of the wave nature of electrons and the Pauli exclusion principle. The elementary treatments of a free electron gas describe how the electrons can be thought of as occupying plane waves exp (ik • r), each having energy (2.11) The plane wave states are doubly occupied with electrons of mass m having up and down spin, and the states are uniformly distributed in k-space (the space of wave numbers) up to the Fermi energy EF• These considerations lead to the result

t[ nJ

(2.12)

where (2.13) The local kinetic energy per unit volume in this model is proportional to n(r)5/3 which per electron is n(r)2/3. We can easily see the reason for this scaling behaviour. If we compress the volume of ft box containing electrons, their wavelengths 2n/k scale as the linear dimension of the box L. Their kinetic energy, proportional to k2, therefore scales as L - 2. Since the density n is proportional to L - 3, the kinetic energy must be proportional to n2/3• The second approximation of the TF theory IS to ignore exchange and correlation. Thus the complete TF functional is ETF[n]

=

f drt(n(r))

+t

If drdr'

~~r:(::i +f

drn(r)

v.xt(r)

+

Epp (2.14 )

Notice that the nucleus-nucleus term Epp is independent of the electron density and has no part to play in the minimisation of the functional with respect to n[ r] which defines the ground state. Without it however the incomplete functional would not scale linearly with the size of the system, because there is a large degree of cancellation between Epp and the previous two electrostatic terms. The Hartree energy is of the same order of magnitude as Epp, and these are each about half the size of and opposite in sign to Eep.

16

Electron Theory in Alloy Design

Following the Hohenberg-Kohn theory, to obtain the ground state energy we now minimise ETF[n] with respect to n(r) subject to the condition that the total number of electrons is conserved, that is

f

drn(r)

=

N

(2.15)

The constrained minimisation can be done explicitly by the method of Lagrange multipliers and leads to the TF equation ..5..C 3

knGS

()2/3 r

+

f dr ' \r _(r')r'\ nGS

+

TT ( Yext

r

)

-

Jl

=

0

(2.16)

The parameter Jl is the Lagrange multiplier, and it corresponds physically to the chemical potential or Fermi energy of the electrons. The solution of this equation has been found to give a rough description of the electron density, which becomes exact in the high density limit, but it is unsatisfactory for quantitative applications at normal densities. For example the energy versus separation curve of atoms does not have a minimum, so no binding is predicted." In order to make a more realistic total energy functional at least two kinds of extensions to the TF functional have been tried. The first is to improve the kinetic energy functional by adding a term to t[n] which incorporates the inhomogeneity of the electron gas. The simplest conceivable term which will do this is one proportional to the squared gradient of the electron density. The kinetic energy functional is then written in the form T[] n

=

c

k

f drn ()5/3r

+

h2A,

16m

f dr IVn(r)12 n(r)

(2.17)

The parameter A is not unambiguously defined. In the limit of long wavelength density variations it can be shown that A = 1/9.9 For short wavelength density fluctuations on the other hand A = 1.10 It has sometimes therefore been treated as an adjustable parameter within this range. The second improvement which can easily be made to the TF functional is to add the exchange and correlation energy, using a local approximation by analogy to the kinetic energy (2.18)

(n) is the exchange and correlation energy per electron in a homogeneous electron gas, and it is a function for which satisfactory approximate forms are known from many-body theory. This local density approximation or LDA as it is called is a central approximation of all the practical density functional schemes we shall be considering.

8xc

Modern Electron Theory

17

With these improvements a much more realistic energy is obtained than the simplest TF equation, and properties such as the cohesive energy, lattice constant and bulk moduli can be calculated with moderate success.'! In practice for closed shell systems such as rare gases and ionic materials the gradien t correction to the kinetic energy does not seem to be a significant improvement. An important application to these systems was made by Gordon and Kim.12,13 They obtained the binding energy and internuclear separation of rare gas dimers and ionic crystals in good agreement with experiment. Instead of trying to minimise the extended TF functional for this purpose they used an approximate charge density obtained by overlapping the calculated HartreeFock charge densities of free atoms. Their approach with small refinements is still used to obtain short ranged repulsive potentials suitable for modelling the behaviour of ceramic materials, the cohesive component of which comes from the long ranged Coulomb potential between the ions.

2.4 FORCES ON ATOMS 2.4.1

THE HELLMANN-FEYNMAN

THEOREM

The theory of total energies is more useful if the energy can be minimised with respect to the nuclear positions, leading to a prediction of the equilibrium structure either of a perfect crystal or a defect. In general this is best done by calculating the forces on the nuclei. The nuclei can then be moved around in a systematic way to minimise the energy, at which point the forces on the nuclei vanish. With a theory of forces it is also possible-especially if one makes approximations to the full density functional theory-to calculate the dynamics of moving atoms ('molecular dynamics') and study such things as radiation damage processes, point defect migration and dislocation mobility. The way to calculate forces from electron theory is in principle very simple, no different in fact from the way they would be calculated using classical electrostatics, and this is the remarkable result known as the Hcllmann-Feynman'Y'" theorem, which was discovered independently by several others, perhaps first by Born and Fock.16 ' The idea is to consider the effect on the energy of a small change in the external potential, which goes from ~xt(r) to ~xt(r) + b~xt(r). It can be thought of as due to a small displacement bx of one of the atomic nuclei. In this case we get directly the force on that nucleus if we divide the change in energy bE by the displacement bx. The change in energy can be split into its components from equation (2.9): (2.19) We can further split the second term into the parts due to variation in charge

18

Electron Theory in Alloy Design

density and variation in the potential (2.20) Now suppose that the undisturbed solid was in its ground state. Hohenberg and Kohn's variational principle tells us that the energy is at a minimum with respect to variations in the charge density, so from equations (2.6) and (2.9) we deduce bF[ n] Equation

+

f drbn(r)

v,xt (r)

=

0

(2.21 )

(2.20) together with (2.21) reduces (2.19) to (2.22)

This result is exactly what we are seeking. The right hand side is nothing other than the change in the electrostatic energy due to a change in the external (that is, nuclear) potential calculated as if the electrons were just a fixed classical charge distribution n(r). All the subtleties of quantum mechanics, wavefunctions and exchange and correlation, have apparently disappeared! That this formula must be true on a macroscopic scale is necessary for consistency with classical electrostatics, but it may come as a surprise to realise that it is precisely true also on the scale of individual atoms and molecules. We can immediately write down the force on a nucleus due to the electrons by dividing the first term of equation (2.22) by the displacement of the nucleus in question (2.23) to which must be added the electrostatic force due to the other nuclei (2.24 ) Two things should be noted about the Hellman-Feynman theorem, which is equation (2.22). Firstly, although we have described b r:;xt as arising from the displacement ofa nucleus, the derivation of equation (2.22) holds for the energy change due to any potential applied externally to the electrons. In the most general case Epp should include all electrostatic energy which does not involve electrons, which means the interaction of the nuclei with each other and with any external charges or applied fields. Secondly the result is only true when the electrons are in their ground state; that is when the density functional has been minimised with respect to variations bn(r). This has important consequences in

Modern Electron Theory

19

Fig. 2.1 Schematic diagram of the electron density profile (solid curves) as a block ofjellium is cleaved.

practice because calculated forces are often completely wrong unless the variational condition is accurately satisfied by the electron density. 2.4.2

EXAMPLE:

METALLIC

COHESION AND JELLIUM

As a simple example of the concepts introduced so far, let us consider the adhesion between two pieces of metal. Let us further suppose that the pieces of metal are identical and perfectly joined, so that the work required to pull them slowly and reversibly apart is just twice the surface energy. This quantity has a controlling influence on the strength of a metal, although of course irreversible processes accompany the decohesion in practice. The surface energy can nowadays be calculated with an accuracy which is probably better than any measurements. Our purpose here is to illustrate how it is related to the electrostatic forces holding the material together. The simplest model of a metal is the uniform electron gas, which we have already called upon in order to obtain the kinetic energy and exchange and correlation energies for use in the local density approximation. We must imagine that to balance the charge of the electrons there are also atomic nuclei present, but these are smeared out into a uniform background of positive charge which terminates abruptly at surfaces. The electronic charge density does not of course terminate abruptly but tails out into the vacuum. This is the jellium model. In fact the electrons in the core of the atom can be lumped in with the nucleus for this purpose, so that only the outer electrons are treated explicitly; this is the business of pseudopotential theory, which we shall be discussing more later. We can now imagine creating two surfaces in the jellium model by cleaving the positive background and pulling the two halves apart (Fig. 2.1). The energy needed to do this per unit area of surface created will be the surface energy in thejellium model. The energy may be calculated in principle by calculating the force on unit area of either of the blocks of positive charge as it is pulled away

20

Electron Theory in Alloy Design

Fig. 2.2 The distribution of electron density across the surface of jellium, illustrating the effect of the kinetic energy of the electrons, which in a wave mechanical treatment leads to oscillations in the charge density (Friedel oscillations).

from the other, and integrating the force over the distance the block has moved, from 0 to infinity. Now as Fig. 2.1 illustrates, applying the Hellmann-Feynman theorem (that is, classical electrostatics) to one of the blocks (block B) tells us that at any instant the force on block B is the electrostatic force due to the electron density and to the other positive block (block A). The force on block B due to electrons on block A up to midway between the blocks is exactly cancelled by the force due to the positive charge of block A, so we need only consider the force from electrons in the half space containing block B itself If this electron density is rather smooth, as in Fig. 2.1, we can easily predict the sign of the force on block B. The electrons which are already spilled out on the outer, preexisting surface of the jellium (marked SI) are pulling the blocks apart, while the electrons at the freshly cut interface (marked S2) are holding them together. There are more SI electrons, so the net force is such as to cleave the jellium. In other words the surface energy of this hypothetical material is negative. This is the situation for the jellium model of a metal with relatively high electron density such as AI, with three conduction electrons per atom. For AI, the jellium model would predict spontaneous disintegration of the metal. The jellium model does however have a positive surface energy at lower densities, as first calculated by Lang and Kohn.V Our force argument shows that this is only possible if there is an overshoot of electron density inside the jellium surface. Such an overshoot indeed occurs; there are Friedel oscillations in the charge density, an interference effect due to the wave-like nature of electrons and the finite width of their energy distribution (Fig. 2.2). The oscillations are most pronounced in a low density metal such as Na. The direction of the net force of the electrons on the positive background is no longer obvious, but it is in fact reversed by the oscillation, so that the jellium model of Na has a positive surface energy. The above discussion highlights the inadequacy of the jellium model for describing cohesion. Ifhowever the uniform positive background is replaced by discrete ions, the resultant electrostatic forces tend to bind the two surfaces together and the catastrophe in Al is avoided.l" We conclude that both wave

Modern Electron Theory mechanical effects (beyond Thomas-Fermi) are important in the cohesion of metals. 2.4.3

PERTURBATION

THEORY:

ANALOGIES

21

and the discrete lattice structure

AND EXAMPLES

A very useful result can be obtained easily and without introducing wave functions by means of the Hellmann-Feynman theorem, namely the total energy in second order perturbation theory. We can regard equation (2.22) as the result of first order perturbation theory, which becomes exact in the limit of small b ~xt (r). Now we ask what is the term in the energy of order b Jt:xt (r)2? A 'bootstrapping' trick to get it is to let the external potential be changed by Ab ~xt (r) where A is a parameter we are going to vary from 0 to 1. The perturbation is now switched on gradually by means of this variation in A, so that the electrons at each instant are in their ground state. For each infinitesimal change dA the energy change is exactly given by applying equation (2.22): dE[ n]

=

f drn(r)b

v.xt (r)dA +

dEpp

(2.25 )

In this equation the charge density is now also a function of A. We next make the approximation of linear response, according to which we can write n(r)

=

no (r) + Abn(r)

(2.26)

where no (r) is the unperturbed charge density corresponding to A = 0 and Abn (r) is the charge density ind uced by the curren t perturbation Ab Jt:xt (r). Now by substituting (2.26) into (2.25) and integrating with respect to A from 0 to 1 we obtain (2.27) The new second term is the result of second order perturbation theory. It is exactly analogous, also in its derivation, to the classical result for the energy stored in a capacitor containing a charge Q(the induced charge) at a potential V (the external potential), namely tQ.V. A second interesting classical analogy is provided by a particle of charge q (the source of external potential) at a distance z outside a flatmetal surface. As we know from elementary electrostatics, the particle induces on the metal surface a charge - q which appears to the particle as ifit were at the image position, a distance z below the metal surface. The interaction energy between the external charge and its image is therefore - q2/2z. By analogy with the second order term in equation (2.27), the total energy is got by dividing by 2, which gives - q2J4z. This can be understood as the result of adding to the direct interaction of the external charge with its image, - q2j2z, the self-energy of the induced charge. The latter turns out to be exactly + q2j4z.

22

Electron Theory in Alloy Design

These two classical cases are more thanjust analogies to equation (2.27); they are actually examples of equation (2.27) operating on a macroscopic scale. It may be asked what has happened to the first and third terms of equation (2.27) in these examples? The answer is that they cancel each other out. The third term is (2.28) where np represents all the positive charges of the nuclei or ions. This contribution cancels the first term of equation (2.27) on the macroscopic scale because of the local charge neutrality of the unperturbed capacitor or metal surface. To prove this we have to average the charge densities over a region of space large enough to contain a few atoms, but small enough for c5li;xt(r) to be effectively constant, so that we obtain (np (r) = (n(r) where the brackets represent the average value of the density (c.f. equation A.27). Equation (2.27) is the basis for a description of the energy of a bulk metal in terms of pairwise interactions between the ions. For this purpose it has to be assumed that the difference between a real metal and jellium can be obtained by linear response, during a hypothetical process of switching on the real ions in the jellium, while the positive background is switched off. This assumption can be justified for simple metals (those without d-electrons) by the pseudopotential theory, as will be described in the next chapter. The simple part of this idea is that the core electrons are counted together with the nucleus, so forming an ion with an effective positive charge equal in magnitude to the valency of the metal, leaving only the conduction electrons to be treated explicitly by quantum mechanics. Less obvious, but essential to the pseudopotential idea, is the fact that this net positive charge of the ion can be distributed within a sphere in such a way as to present a weak perturbation to the nearly free conduction electrons. The true potential of the atom core is thus replaced by the pseudopotential of an ion, which must be constructed so as to induce the same electron density outside the atomic core as the true atomic potential. The search for an optimal way of doing this has been intensive. The solutions-a large number have been proposedinvolve a careful mathematical trade-offbetween the accuracy with which the real electron density can be reproduced outside the atomic cores and the goal of making the pseudopotential a weak perturbation to jellium. When it is achieved, we have a model of a metal in which the electron density is a superposition of the spherical charge densities induced by each ion around itself. If the ions are shifted around in this model, the energy change is given by the Hellmanri-Feynman theorem. It is just the attractive electrostatic interaction of these spherical induced charge densities, sometimes called pseudoatom charge densities, with the other ions, plus the direct Coulomb repulsion between the ions themselves. These are the second and third terms of equation (2.27). The first term of equation (2.27) is simply a constant in the bulk of a

>

>

23

Modern Electron Theory

metal, independent of the structure although dependent on the density. It is only in the bulk of a metal that the model is valid at all, because near a surface, or a large perturbation such as a cavity, the charge induced when an ion is switched on is no longer spherically symmetric. In bulk non-transition metals the pairwise force model has been very successfully applied to calculate phonon frequencies, for which the atoms are not seriously perturbed from perfect lattice sites. The metals Li, Na, K, Mg and Al are perhaps the best candidates for it to work well. The simplicity of the model has also enabled the simulation of complicated situations, for example at grain boundaries, which display a variety of metastable structures. 2.5 THE KaHN-SHAM 2.5.1

FUNCTIONAL

THE LOCAL DENSITY APPROXIMATION

We have already indicated that the Thomas-Fermi approach to calculating total energies is not considered accurate enough for predictive work. The functional which has proved more successful was introduced by Kohn and Sham.? As we shall see, it requires us to solve the Schrodinger equation explicitly, making the method more complicated than one which works only with the electron density. However, the Schrodinger equation to be solved is not the one for the many-electron wave function, which remains a formidable problem, but rather a special Schrodinger equation for just one electron moving in a fixed effective potential. For solving such a one electron Schrodinger equation, several standard techniques have evolved. They will be introduced later. Let us first look at the Kolin-Sham functional. The improvement actually consists in choosing a better kinetic energy functional. The kinetic energy T[ n] is now approximated by To [n], which is that ofa system offictitious non-interacting electrons having the density n(r) of the real system. If non-interacting electrons are to have the same density as a system of real electrons, they must be moving in a fictitious, effective external potential, which is denoted ~ff (r). This parallel reference system of noninteracting electrons, a purely theoretical concept, was an essential stepping stone for several further developments. Of course, we are no better off unless we can determine ~ff(r), but this is just what Kohn and Sham showed how to do. First let us write down the new density functional, using the notation already introduced E[n]

=

To[n]

+

f drn(r)

V:xt(r)

+t

f f drdr'

~;r~(::i+

Exc[n]

+

Epp (2.29)

We recall for comparison that in the most extended form of Thomas-Fermi theory, the energy functional is the same as equation (2.29) but with To [n]

24

Electron Theory in Alloy Design

described by equation (2.17). We can also write down the density functional for the system of non-interacting electrons, which is (2.30) The ground states of these functionals must by definition correspond to the same charge density, although they are not required to have the same energy. At their ground states equations (2.29) and (2.30) must satisfy the condition that for any arbitrary small change bn(r) which conserves the total number of electrons, the variation in the energy is zero

c5E[n]

=

0

=

c5To[n] +

f

drc5n(r)T1.xt(r)

+

If drdr'

bl:(r2n~?

+ c5Exc[n] (2.31)

and

c5Eo[n]

=

0

=

c5To[n] + f drc5n(r) V~(r)

(2.32)

These mathematical problems are equivalent and therefore have the same solution for n(r) if we define our effective potential as follows (2.33) There are some important points to make about this equation for the effective potential. Firstly the notation for the last term is standard but misleading, because in general it is not simply the quotient of bExc [n] and = (E - Ee)ll/Ie) the Austin-Heine-Sham Hamiltonian reduces to the pseudo Hamiltonian originally derived by Phillips and Kleinman" via an expansion of the valence electron states into orthogonalised plane waves. Cohen and Heine25 proposed to exploit the arbitrariness in the choice of a pseudopotentialby constructing an 'optimised' pseudopotential giving the smoothest possible pseudo orbital. The smoothest pseudo orbital minimises the kinetic energy

< .

0::

-4.36

I

55

Vo(exp.) I

UJ

-4.50 -4.44

100

100

120 Volume

120

140

160

(c.u.)

Fig. 3.3 Total energy of a-Ga, Ga-II and p-Ga as a function of volume. The common tangent construction for the pressure induced z-Ga ---+ Ga-II transition is indicated. The results given in (a) are from a self-consistent total energy calculation (Ref. 68), those given in (b) have been obtained using perturbation theory (Ref. 69).

structure. The general rule is that under pressure the lighter elements undergo a transition to the structures of the heavier homologous elements. The most striking examples are the transitions of C from graphite to diamond, of Si and Ge from diamond to fJ-tin and further (through a series of intermediate structures) to close packed metallic lattices, of Ga from the orthorhombic et-Ga to the tetragonal In structure, and of P from a base centered orthorhombic structure to the rhombohedral As and further to a simple cubic structure.l' The early history of the pseudopotential theory of crystal structures has been summarized by Heine and Weairc." recent reviews of the calculation ofstructural energy differences using perturbation theory and self-consistent total energy calculations have been given by Hafner+'" and Ihm.8 Tight-binding approaches to the structural trends have been described by Allan and Lannoo'" and Cressoni and Pettifor.f Quite generally, the self-consistent total energy calculations confirm and extend the arguments brought forward on the basis of the much simpler perturbation calculations. As an example, Fig. 3.3 compares the total energies for orthorhombic et-Ga, the tetragonal high pressure Ga-II phase and the metastable {3-Gaphase (monoclinic, produced by crystallizsation from a supercooled melt) as obtained very recently using self-consistent total energy calculations using a norm conserving pseudopotential'" and as calculated years earlier'" using perturbation theory and orthogonalised plane wave pseudopotentials and model potentials. The phase transition from the et- to the Ga-II phase is predicted correctly by all calculations, with the structural energy difference derived from the total energy calculations being in slightly better agreement with thermochemical estimates. The self-consistent calculation of the electron

56

Electron Theory in Alloy Design

density distribution also leads to a very interesting interpretation of the bonding in a-Ga in terms ofa mixture of metallic and molecular (in the shortest bonds in Ga pairs) bonding, which is of course not accessible to a perturbation study. The perturbation calculations produce a remarkable argument for the existence of the metastable {3-Ga: this monoclinic phase is predicted to be the lowest energy phase at densities intermediate between the zero pressure It phase and the high pressure Ga-II phase. However, the energy is higher than that of a macroscopic mixture of a-Ga and Ga-II. The density at which {3-Gais lowest in energy is very close to that of liquid Ga. Hence this result appears to supply a convincing explanation for the formation of metastable {3-Gafrom the melt. I t is not completely clear why the energy of the {3phase is much higher in the self-consistent calculation. One possibility is that the supercell approach chosen to circumvent the Brillouin zone integration problem has not produced a completely converged answer. This isjust one example of the state of the art in predictions of phase stability, for a more extensive list of references, see Ref. 10. Here we concentrate on the simple physical mechanism behind the structural trends. The argument may be formulated either in direct or in wave number space. For example, the occurrence of a distorted structure for Ga may be explained in terms of the position of the first reciprocal lattice vector relative to the wave vector qo where the pseudopotential matrix element passes through zero. If the reciprocal lattice vector falls close to zero, its contribution to the ground-state energy to second order in the pseudopotential is small-hence it will be energetically favourable to distort the structure in such a way that no vector of the reciprocal lattice falls close to qo. The analogy between this argument and the discussion in terms of an opening of a gap at the Fermi surface a la Mott, Jones,14 and Peierls" is evident. Here we shall discuss a closely related interpretation of the structural trends in terms of systematic variations of the interatomic forces with valence, atomic number and density. 3.4.2

FORCES AND STRUCTURES

The features of the interatomic pair potential which are most important in determining the structural energy differences are the diameter of the repulsive core, the occurrence and position of a minimum around the nearest neighbour distance, and the amplitude and phase of the Friedel oscillations. Recently, Hafner and Heine28,59 have shown that the variation of(R) across the Periodic Table is determined by two parameters only, the electron density and the effective radius of the ionic core. For a local model potential the interatomic potential (R) (see equation (3.11)) may be written as

(R)

;c- {2

32 f IWn (qW [X(q) je(q) R ~--------------,--------------- red (R)

Jl sin qR)

dq}

(3.21)

Nearly-free-electron Approximation

57

where ui; (q) is a normalised pseudopotential matrix element. For an empty core model potential'? we have (3.22) where R, is the empty core radius (the radius of the ion to a good approximation). The terms in the curly brackets define a reduced pair interaction with the Coulomb part factored out. In Fig. 3.4 we show the variation of.

5

-

~

~

0

~

e

U I

~

5 0

~

C U

~ 0::: ~ 0:::

cr .s

1

~

-10 '6i

o

2345678

R CA)

Fig. 3.8 Continued.

alkali atoms with group IV elements (e.g. NaPb) or in AB2alloys of an alkaline earth metal with a group IV element (e.g. CaSi2), the generalised valence rule requires the formation of a tricoordinated anion sublattice. This may be realised in the form of Pb4 tetrahedra (as in white P), or by the formation of an As type layer structure in the case of CaSi2. Correspondingly, in the equiatomic alkali-pnictide compounds, the anions form spiral chains like the chalcogen-elements. For a more extensive tabulation of polyanionic compounds (see e.g. Pearson.l'" p. 216). The I-III and I-II compounds have been discussed repeatedly in the literaturelOB-1ll and the original Zintl-Hiickel conjecture has been fully confirmed: the band structure stabilisation of the N aTI phases arises from the formation of more or less complete Sp3hybrids. On the other hand, the electrostatic effects tend to stabilise the CsCI structure, and this explains the switch from NaTI to esCI stability under compression. The cluster compounds formed by I-IV alloys have been discussed by Springelkamp et al.112 and Tegze and Hafner.1l3 It is found that the valence states split into a narrow low-lying IV s

Nearly-free-electron Approximation

71

band and a IV p band at the Fermi level. The p band may be indexed in terms of the electronic eigenstates of an isolated icosahedron, demonstrating the strong covalent character of the intra-cluster bonds. The I-V compounds are characterised by a (SSG bonded) S band and a p band split into bonding, non-bonding and anti-bonding parts (Fig. 3.9), the Fermi level falling into the narrow non-bonding-anti-bonding gap.114This shows again the similarity with the bonding in the group VI elements. Due to the strong covalent character of the bonding, second order NFE perturbation theory and pair interactions describe the bonding in these compounds only qualitatively. At least the leading third order terms associated with the filling of the Jones zone must be included to explain for example the stability of the NaTI vs. the CsCI structure.lO,110

3.7 APERIODIC

ALLOYS

The investigation of the structural and electronic properties of aperiodic (liquid, amorphous, quasicrystalline) alloys is perhaps one of the most important applications of NFE theories. 3.7.1

LIQUID ALLOYS

As for the elements, many of the important trends in the structures of crystalline compounds are reflected in the local order of the liquid alloys. Hence the computer simulation of the liquid alloys allows for interesting investigations of electronically driven structural effects that are not biased by a choice between a finite number of structural alternatives. A very basic example of electronic influence is given by the structure and heat of formation of homovalent liquid alloys. i1H is positive throughout, except for alloys of the divalent metals. The reason is that for a random close packed structure of divalent metals we have = 2kF, where is the position of the first peak in the static structure factor. This means that the liquid structure satisfies a generalised Hume-Rothery criterion confering the alloy extra stability. Many liquid alloys show strong chemical short range order.l1S In Li-Pb and Li-Bi alloys for example the maximum ordering occurs at the 'stoichiometric' composition Li4Pb, and Li3Bi respectively. The corresponding crystalline compounds are characterised by salt-like ordering. The ordering tendency is generally well described by the pseudopotential-derived interatomic potenrials!" (c£ Section 3.5.2). As an example we show in Fig. 3.10 the neutron scattering intensity for liquid LixPb1_x alloys. The maximum ordering occurs for x = 0·8, which is also close to the composition of the 'zero alloy' (zero average neutron scattering length) where only the concentration fluctuations contribute to the scattering intensity.

C4

C4

72

Electron Theory in Alloy Design 30~~~~~--~~~~--~~~~--~~~~~ 20 10 Or---~------~~------~---------+~------~ 20 10 Or-----~--~--~~------~--~---+~------~ ,....-....

~

~

Q)

20

10

0~----~--~~~--------~~~~~--~---4

~

30

~

20

o:

10

+->

o ~

0~----~--~~~--------~~~~~~-----4 50 40 30

20 10

O~----~--~--~~------~~~---+~------~ -10

-5

o

E (eV) Fig. 3.9 Electronic density of states of the alkali-mono-antimonides, with the LiAs and NaP structures showing the separate s-band and the p-band split into bonding, non-bonding, and antibonding (pP(J) complexes (after Ref 114).

Nearly-free-electron

Approximation

73

S(q)

o o

Lio.aPbO.2

o o

3

o

2

05

1.5

2.0

Fig. 3.10 Neutron diffraction interference function S(q) in liquid LixPbl_x' Full linetheory; open circles-experiment. The intensity of the first peak is a measure for the short range chemical order (after Pasturel et al., Ref. 116).

The tendency to the formation of poly anionic clusters or sub-lattices subsists in the liquid state.Il7 Tetrahedral clusters are found e.g. in liquid (K, Rb, Cs)(Pb, Sn) alloys. Their formation is also predicted by the strong bonding forces between anions reflected in the interatomic potentials.'!" In Li-rich liquid Li-Ga alloys computer simulations'" predict the existence of short, broken Ga chains similar to those found in liquid Se and Te. Their existence is suggested by the form of the interatomic potentials (see Fig. 3.7), and the investigation of the electronic as well as the structural properties confirms the existence of a generalised Zintl principle for liquid alloys as wel1.85

74

Electron Theory in Alloy Design o-CoZn, T=286 K ---Co-Co, _._.- Co-Zn -Zn-Zn

Or-~~--~~--~~--~~--~~--~~ 2

o

2

6

8

Fig. 3.11 Partial Ashcroft-Langreth (a) and Bhatia-Thornton (b) structure factors Sij (q) for amorphous CaZn alloys. The prepeaks in SZnZn (q) and SNN (q) near q ~ 1 A-I indicate a topological short range order (after Re£ 123).

3.7.2

GLASSES

A possible electronic origin of the glass forming ability of metallic alloys has been a much debated topiC.119 Simple metal glasses like Mg-Zn or Ca-Mg and glasses formed by the noble metals and polyvalent elements indeed satisfy a generalised Hume-Rothery rule Qp = 2kF. For the atomic structure this leads to a 'constructive interference' between the pair potentials CIlv· (R) and the pair correlation functions gij (R) (whose maxima fit into the minima of the potential), conferring an extra stabilisation to the random close packed structure.V'' In the electronic structure the peak in the structure factor induces a weak minimum in the electronic DOS at the Fermi level-both effects describe the same physical effect. The existence of the DOS minimum has been confirmed by electronic structure calculations'" and by photoelectron spectroscopy.121,122 In weakly interacting systems such as Mg-Zn, the geometrical requirements of sphere packing lead to a random poly tetrahedral structure. In systems with a more pronounced difference in the electron densities, chemical effects in the pair interactions (mainly the reduced screening of the electronegative component) lead to chemical ordering, large size differences may in addition induce topological short range ordering. An interesting example is found in Ca-Zn melts and glasses: the reduced electron density leads to a strong Zn-Zn interaction and the formation of strong Zn-Zn bonds dominating the structure. 123This is reflected in a pre-peak in the partial Zn-Zn structure factor of the liquid and of the glass (Fig. 3.11). The theoretical prediction is well confirmed by diffraction studies.V" The local order in the glass seems to consist in a trigonal prismatic arrangement of the larger Ca atoms around the Zn atoms, the

Nearly-free-electron Approximation

75

combination of the trigonal prisms being dominated by strong Zn-Zn bonds. Again the situation is completely analogous to the crystalline Ca-Zn compounds: for compositions in the glass forming region they are all of trigonal prismatic type. Again the comparative study of the electronic structure of the crystalline and amorphous alloys confirms the conclusions drawn on the basis of the pair-potential studies.l'" 3.7.3

QUASICRYSTALS

The recent discovery of quasicrystals, 126i.e. of materials with long range bond orientational order, but only quasiperiodic translational order has given solidstate theory a new dimension. It has been shown that a structure based on a quasiperiodic tiling of 3d space with two fundamental units (a 'Penrose tiling') reproduces the essential features of the quasicrystalline diffraction patterns.127 However, many essential questions remain open. Why should nature prefer quasiperiodic to periodic order? What is the character of electronic and vibrational eigenstates? It was realised quite soon that the electron-per-atom ratio of many quasicrystalline alloys falls into the range eja ~ 2·1-2·4, and "this suggests a Hume-Rothery-like mechanism for the stabilisation of quasicrystalline structures." The most straightforward approach to the problem came from Smithr'" based on the ideas of Hafner and Heine,28 he constructed three-dimen-

Fig. 3.12 Phase diagram showing the stable structures as a function of valence Z and core radius R, (expressed in units of Rs). For simplicity only the coordination numbers of the stable structures are given. The hatched area marks the region of quasicrystalline stability (after Smith, ref. 34).

76

Electron Theory in Alloy Design

sional structure maps defining the stability range of many crystalline and of idealised mono-atomic quasicrystalline structures. A section of such a structure map is shown in Fig. 3.12 for Jt: = 120 a.u.: it is found that quasicrystals are the stable ground-state in a narrow region separating close packed and open crystalline structures. The point is that although no element lies in this region, the virtual crystal parameters of the known quasicrystalline alloys fall into the proper parameter range. The argument has been confirmed by detailed molecular-dynamics studies of AI-Zn-Mg, AI-Cu-Mg, and AI-Cu-Li quasicrystals using realistic pair interactions.I28,I29 It was shown that the quasicrystalline state is stabilised by a Hume-Rothery mechanism similar to that discussed for the metallic glasses. The argument may even be carried further: the electronic structure may be calculated for a hierarchy of complex crystal structures approaching the quasicrystalline structure as closely as possible. In practice, such 'rational approximants' can be treated for systems up to about 12,000 atoms per cell." The results (Fig. 3.13) show that the dispersion relation of AI-Mg-Zn quasicrystals follows very well the pattern of a quasiperiodic free electron model where every point of the quasiperiodic reciprocal lattice is at the origin of a free electron parabola.P" Very close to the Fermi energy one finds a quasiperiodic sequence of highly degenerate free electron states (marked by arrows in Fig. 3.13). The lifting of the degeneracy leads to the formation of a pseudo gap at the Fermi level which contributes to the lowering of the energy 5

>OJ c

0

-5

k..

w I w -10 -15

o

5

10

15 ~ Ik

20

I

25

30

(i n un its

Fig. 3.13 Quasiperiodic dispersion relations for valence Mg phase along a 2-fold symmetry axis. The full and free electron model, the dots the positions of the peaks The arrows mark highly degenerate free electron states Krajci, Ref. 130).

35 (2

TT /

40 d))

electrons in the icosahedral AI-Znbroken lines show the quasiperiodic in the calculated spectral functions. at the Fermi level (after Hafner and

Nearly-free-electron Approximation

77

of the quasicrystalline phase. This is a beautiful case where an old quantum mechanical concept has found a new and unexpected application. 3.8 CONCLUSION In this chapter I have reviewed the nearly free electron theory of cohesion and structure of metals and alloys. I have tried to show that even today, when total energy calculations for fairly complex systems can be done from first principles, with no other assumption beyond the local density approximation, NFE theory remains a valuable tool for research. It permits the achievement of a decomposition of the total energy into a sum ofvolume-, pair-, and many-body forces. This double tracked approach of self-consistent total energy and interatomic potential calculations allows the reconciliation of the physicists' conception of crystal structure, which has its origin in the geometrical packing of spheres and is now expressed in terms of interatomic quantum forces, with the chemists' view based on bonds and bands. It also allows the inclusion of the aperiodic phases of matter (solid solutions, melts, glasses, quasicrystals) in our consideration. For the elements, a few well-defined structural principles, covering the solid and liquid state have emerged. I am convinced that NFE theory will be very helpful in achieving the same goal in the far more complex field of alloy phase stability. ACKNOWLEDGEMENT I would like to acknowledge the long-time support of my research work by the Fonds zur Forderung der wissenschaftlichen Forschung in Austria and by the Jubilaumsfond der Osterreichischen N ationalbank.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

M. Born and J.R. Oppenheimer, Ann. Phys., 1927,84, 457. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, B864. W. Kohn and L.J. Sham, Phys. Rev., 1965,140, Al133. R.O. Jones and O. Gunnarson, Rev. Mod. Phys., 1989, 61, 689. H. Hellmann, Einfuhrung in die Quantenchemie, Leipzig: Deuticke, 1937. R. Car and M. Parrinello, Phys. Rev. Lett., 1985,55, 2471. J. Hafner, From Hamiltonians to Phase Diagrams, Berlin, Springer, 1987. J. Ihm, Rep. Prog. Phys., 1988, 51, 105. D.A. Young, Phase Diagrams oj the Elements, Berkeley, University of California Press, 1991. J. Hafner, in Cohesion and Structure, vol. 2, (ed. D.G. Pettifor and F.R. de Boer), Amsterdam, North Holland, 1989. 11. J.A. Majewski and P. Vogl, in Cohesion and Structure, vol. 2, (ed. D.G. Pettifor and F.R. de Boer), Amsterdam, North Holland, 1989.

78

Electron Theory in Alloy Design

12. P. Drude, Ann. Phys., 1902, 7, 687. 13. H.A. Lorentz, Elektronentheorie, Leipzig, Deutike, 1909. 14. N.F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, Oxford, Oxford University Press, 1936. 15. E. Fermi, Nuouo Cimento, 1934, 11, 157. 16. H. Hellmann, Acta Physicochim., 1935, (USSR) 4, 913. 17. H. Hellmann, Acta Physicochim., 1938, (USSR) 4, 225. 18. E. Antoncik, ]. Phys. Chem. Solids, 1959, 10, 314. 19. J.C. Phillips and L. Kleinman, Phys. Rev., 1959, 116, 257; ibid., p. 880. 20. W.A. Harrison, Pseudopotentials in the Theory of Metals, New York, Benjamin, 1966. 21. V. Heine and D. Weaire, Solid State Phys., 1971,24, 247. 22. A. Blandin, in Phase Stability in Metals and Alloys, (ed. P.S. Rudman, J. Stringer, and R.I. Jaffee), New York, McGraw-Hill, 1965, p. 115. 23. J. Hafner, ]. Phys., 1976, F6, 1243. 24. M.H. Cohen, in Metallic Solid Solutions, (ed. J. Friedel and A. Guinier), New York, Benjamin, 1962, p. IX-I. 25. M.H. Cohen and V. Heine, Phys. Rev., 1961, 122, 1821. 26. M.S. Daw and M.l. Baskes, Phys. Rev., 1984, B29, 6443. 27. K.W. Jacobson, J.K. Norskov and M.J. Puska, Phys. Rev., 1987, B35, 7432. 28. J. Hafner and V. Heine, ]. Phys., 1983, F13, 2479. 29. J. Hafner and G. Kah1, ]. Phys., 1984, F14, 2259. 30. J. Hafner, Phys. Rev. Lett., 1989, 62, 784. 31. J. Hafner and L. von Heimendahl, Phys. Rev. Lett., 1979, 42, 385. 32. P. Haussler, F. Baumann, J. Krieg, G. Indlekofer, P. Oelhafen and H.J. Guntherodt, ]. Non-cryst. Solids, 1984, 61 + 62, 1249. 33. J. Friedel and F. Denoyer, C.R. Acad. Sci. (Paris), 1987,305, 17l. 34. A.P. Smith, Phys. Rev., 1991, B43, 11635. 35. J. Hafner and M. Krajci, Europhys. Lett., 1992, 17, 145. 36. G.B. Bachelet, D.R. Hamann and M. Schluter, Phys. Rev., 1982, B26, 4199. 37. A.M. Rappe, K.M. Rabe, E. Kaxiras and J. Joannopoulos, Phys. Rev., 1990, B41, 1227. 38. N. Troullier and J.L. Martins, Phys. Rev., 1991, B43, 1993. 39. D. Vanderbilt, Phys. Rev., 1990, B41, 7892. 40. G. Kresse, J. Hafner and R.J. Needs, ]. Phys: CondenseMatter, 1992 (submitted). 41. D. Vanderbilt, Phys. Rev., 1985, B32, 8412. 42. M.L. Cohen and V. Heine, Solid State Phys., 1970,24, 37. 43. M.L. Cohen andJ.R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, Berlin, Springer, 1988. 44. N.W. Ashcroft, Phys. Lett., 1966, 23, 48. 45. B.J. Austin, V. Heine and L.J. Sham, Phys. Rev., 1962, 127, 276. 46. R.W. Shaw and W.A. Harrison, Phys. Rev., 1967, 163, 604. 47. W.C. Topp and J.J. Hopfield, Phys. Rev., 1974, B7, 1295. 48. H. Ehrenreich and M.H. Cohen, Phys. Rev., 1959, 115, 786. 49. E.G. Brovman and Yu Kagan, Soviet Phys. ]ETP, 1970,30, 721. 50. J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Med., 1954, 28, 28. 51. P.W. Anderson, Elementary Excitations in Solids, New York, Benjamin, 1964. 52. L. Hedin and Lundqvist, Solid State Phys., 1969, 23, 1. 53. R. Taylor, ]. Phys., 1978, F8, 1699. 54. J. Callaway and N.H. March, Solid State Phys., 1984, 38, 136. 55. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett., 1980, 45, 566. 56. S.M. Vosko, L. Wilk and Nussair, Can. ]. Phys., 1980, 58, 1200.

Nearly-free-electron 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95.

96.

Approximation

79

S. Ichimaru, Rev. Mod. Phys., 1986, 16, 5779. S. Ichimaru and K. Utsumi, Phys. Rev., 1981, 24, 7385. J. Hafner and V. Heine, ]. Phys., 1986, FI6, 1429. G. Jacucci and R. Taylor, ]. Phys., 1981, FII, 787. W.A. Harrison, Phys. Rev., 1973, B7, 2403. R. Evans, in Electrons in Disordered Metals and at Metallic Surfaces, (ed. P. Phariseau, B.L. Gyorffy and L. Scheire), New York, Plenum, 1979, p. 417. D.G. Pettifor, ]. Phys., 1986, CI9, 285. O.K. Andersen, in Electrons at the Fermi Surface, (ed. M. Springford), Cambridge, Cambridge University Press, 1980. J.C. Cressoni and D.G. Pettifor, J. Phys: Condens. Matter, 1991, 3, 495. S. Lee, Acc. Chern. Res., 1991, 24, 249. G. Allan and M. Lannoo, ]. Physique (Paris), 1983, 44, 1355. X.G. Gong, G.L. Chiarotti, M. Parrinello and E. Tosatti, Phys. Rev., 1991, B43, 14277. J. Hafner and O. Hittmair, Anz. Osterr. Akad. Wiss. Math. Naturw. Kl., 1975, 4, 1; Ch. Regnault, J.P. Badiali and M. Dupont,]. Physique (Paris), 1980,41, C8-603. R.E. Peierls, Q,uantum Theory of Solids, Oxford, Oxford University Press, 1955, P.B. Littlewood, Crit. Rev. Sol. Stat. Mater. Sci., 1983, 11, 229. Y. Waseda, The Structure of Non-crystalline Materials, New York, Mcflraw-Hill, 1980. R. Bellisent, C. Bergman, R. Ceolin and J.P. Gaspard, Phys. Rev. Lett., 1987, 59, 66l. M.e. Bellisent-Funel, R. Bellisent and G. Tourand, ]. Phys., 1981, FI1, 139. W. Jank and J. Hafner, Phys. Rev., 1990, B41, 1947; ibid., B42, 6926. J. Hafner and W. Jank, Phys. Rev., 1990, B42, 11530. J. Hafner and W. Jank, Phys. Rev., 1992, B45, 2739. J. Hafner, Phys. Rev. Lett., 1989, 62, 784. J. Hafner, ]. Phys. Condens. Matter, 1990, 2, 1271. L. Vegard, Z. Phys., 1921,5, 17. E-an Zen, Am. Mineral., 1956, 41, 523. J. Hafner, ]. Phys., 1985, F15, L43. D.G. Pettifor and C.D. Gelatt, in Atomistics of Fracture, (ed. R. Latanision), New York: Plenum, 1983, p. 296. D.G. Pettifor, in Physical Metallurgy, (ed. R.W. Cahn and P. Haasen), Amsterdam, North Holland, 1983, vol. 1, p. 137. L. Pauling, The Nature of the Chemical Bond, 2nd Ed., Ithaca, Cornell University Press, 1952, ch. 12.5. J. Hafner and W. Jank, Phys. Rev., 1991, B44, 11662. W. Biltz and F. Weibke, Z. AUg. Anorgan. Chern., 1935, 223, 321. E. Zintl and G. Brauer, Z. Phys. Chern., 1933, B20, 245. W. Klemm, Proc. Chern. Soc. (London), 1958, 329, 328. V. Heine, in The Physics of Metals, vol. 1, (ed. J.M. Ziman), Cambridge, Cambridge University Press, 1969, p. 1. R. Evans, P. Lloyd and S.M.M. Rahman, ]. Phys., 1979, F9, 1939. S.M.M. Rahman, ]. Phys., 1981, FII, 1191; ibid., 1011. M.M. Rahman and S.M.M. Rahman,]. Phys., 1985, FI5, 477. M. Kogachi, ]. Phys. Chern. Solids, 1973, 34, 67. J.S. Faulkner, Prog. Mat. Sci., 1982, 27, 1. D.D. Johnson, D.M. Nicholson, F.J. Pinski, B.L. Gyorffy and G.M. Stocks, Phys. Rev. Lett., 1986, 56, 2088; P.E.A. Turchi, M. Sluiter, F.J. Pinski, D.D. Johnson, D.M. Nicholson, G.M. Stocks and J.B. Staunton, Phys. Rev. Lett., 1991, 67, 1779. G.M. Stock, M. Boring, D.M. Nicholson, F.J. Pinski, D.D. Johnson, J.S. Faulkner and

80

Electron Theory in Alloy Design

B.L. Gyorffy, in Noble Metal Alloys, (ed. T.B. Massalski), Warrendale, PA, The Metallurgical Society of AIME, 1986. 97. C.S.G. Cousins, ]. Phys., 1974, F4, 1. 98. C.W. Krause and J.W. Morris, Acta Metall., 1974, 22, 767. 99. F.C. Frank andJ.W. Kasper, Acta Crystallogr., 1958, 11, 184; Acta Crystallogr., 1959, 12,483. 100. W.B. Pearson, Crystal Chemistry and Physics of Metals and Alloys, New York, Wiley, 1972. 101. L. Pauling, Acta. Crystallogr., 1957, 10, 374. 102. A. Simon, Angew. Chem., 1983, 95, 94. 103. J. Hafner, Phys. Rev., 1980, H2I, 406. 104. J. Hafner, ]. Phys., 1985, FI5, 1879. 105. T. Ohba, Y. Kitano and Y. Komura, Acta Crystallogr., 1984, C40, 1. 106. J. Hafner, Phys. Rev., 1977, HI5, 617. 107. W. Hiickel, Structural Chemistry of Inorganic Compounds, Amsterdam, Elsevier, 1952, p. 829. 108. A. Zunger, Phys. Rev., 1978, HI7, 2582. 109. N.E. Christensen, Phys. Rev., 1985, B32, 207; ibid., p. 6490. 110. J. Hafner and W. Weber, Phys. Rev., 1986, B33, 747. Ill. Z. Paw1owska, N.E. Christensen, S. Sapathy and O. Jepsen, Phys. Rev., 1986, B43, 7080. 112. F. Springelkamp, R.A. de Groot, W. Geertsma, W. van der Lugt and F.M. Mueller, Phys. Rev., 1985, B32, 2319. 113. M. Tegze and J. Hafner, Phys. Rev., 1989, B39, 8263; ibid., B40, 9841. 114. M. Tegze and J. Hafner, ]. Phys. Condense Matter, 1992,4, 2449. 115. P. Chieux and H. Ruppersberg, ]. Physique (Paris), 1980, 41, C8-32l. 116. A. Pasturel, J. Hafner and P. Hicter, Phys. Rev., 1985, H32, 5009. 117. M.L. Saboungi, W. Geertsma and D.L. Price, Ann. Rev. Phys. Chem., 1990, 41, 207. 118. J. Hafner, ]. Phys. Condense Matter, 1989, 1, 1033. 119. H.J. Giintherodt and H. Beck, eds., Glassy Metals I, Berlin, Springer, 1981. 120. J. Hafner, ]. Phys., 1988, FI8, 153. 121. J. Hafner, S.S. Jaswal, M. Tegze, A. Pflugi, J. Krieg, P. Oelhafen and H.J. Giintherodt, ]. Phys., 1988, FIB, 2583. 122. P. Haussler, F. Baumann,J. Krieg, G. Indlekofer, P. Oelhafen and H.]. Giintherodt, Phys. Rev. Lett., 1983, 51, 714. 123. J. Hafner and M. Tegze, J. Phys. Condense Matter, 1989, 1, 8277. 124. U. Mizutani, T. Shimizu, T. Fukunaga, T. Koyano, K. Tanaka, M. Yamada and T. Matsuda, ]. Phys. Condense Matter, 1990, 2, 7825. 125. M. Tegze and J. Hafner, J. Phys. Condense Matter, 1989, 1, 8293. 126. D.S. Shechtman, 1. Blech, D. Gratias and J. Cahn, Phys. Rev. Lett., 1984,53, 1951. 127. D. Levine and P. Steinhardt, Phys. Rev., 1986, B34, 596. 128. J. Hafner and M. Krajci, Europhys. Lett., 1990, 13, 335. 129. M. Krajci and]. Hafner, Phys. Rev. B, 1992 (in print). 130. J. Hafner and M. Krajci, Phys. Rev. Lett., 1992, 68, 2321. 131. T. Fujiwara and T. Yokokawa, Phys. Rev. Lett., 1991, 66, 333.

4

The Tight Binding Approximation: Concepts and Predictions D.G. PETTI FOR Department of Mathematics, Imperial College of Science, Technology and Medicine London SW7 2BZ

4.1 INTRODUCTION The growing dialogue between electron theorists and alloy designers is due to the new found ability of electron theory not only to make quantitative predictions but also to provide reliable concepts. We have seen in Chapter 2 that the solution of the local density functional (LDF) Schrodinger equation allows the cohesive and structural properties of simple metals, transition metals and intermetallics to be predicted accurately from first principles. These calculations have helped make sense of some unexpected results in alloy development. For example, cubic pseudo binaries based on Ti Al, remain brittle even though they have the same close packed crystal structure as ductile single crystals of Cuj Au or Ni3Al. Recently their elastic constants have been calculated.' It was found that whereas the ratio J.lIK of the shear modulus on the slip plane to the bulk modulus is only 0·33 for Ni3AI, it takes the much larger value of 0·63 for TiA13. This is sufficient to put it amongst the inherently brittle cubic metals which are found'' to have il/K > 0·5. This large value of il/K for cubic TiAl3 reflects the strong directional character of the bonding which is observed in the LDF charge density.' In general, however, these accurate LDF calculations do not make direct contact with our physical or chemical intuition regarding the importance of such factors as the electronegativity difference or atomic size mismatch in controlling the stability of metallic phases. In order to understand the trends in cohesive and structural properties approximations must be made. In the preceding chapter simplifications to the solution of the LDF equations were achieved by regarding the valence electrons as a gas of free electrons that is only weakly perturbed by the underlying ionic lattice-the so...called nearly free electron (NFE) approximation. In this chapter we consider the other limit in which the valence states are formed by the weak overlap of atomic orbitals -the so-called tight binding (TB) approximation. Whereas the NFE approxi81

82

Electron Theory in Alloy Design

mation is a valid description of the sp-bonded simple metals, we will see that the TB approximation is a reasonable description of the d-bonded transition metals and most binary intermetallics. The TB approximation starts from the overlap of the valence orbitals of the constituent atoms. In Section 4.2, therefore, the three important concepts of atomic size, angular character of the valence orbitals, and electronegativity are introduced with respect to isolated free atoms. In Section 4.3 the characteristics of bond formation are illustrated with respect to the heteronuclear diatomic molecule, thereby introducing the concepts of bond charge density and bond order. In Section 4.4 the cohesion of elemen tal transition metals and their heats offormation are understood within a simple TB d-band model. In Section 4.5 the moments of the local density of states are shown to provide a powerful link between trends in structural stability and the underlying topology of the lattice, thereby allowing a direct explanation for the observed structural trends across the periodic table for the elements and across the stoichiometric structure maps for the binary compounds. In Section 4.6 an angularly dependent many-atom potential is derived within the TB approximation that will be very important for the atomistic simulation of defects within the transition metals and intermetallics. In Section 4.7 we conclude.

4.2 THE CONSTITUENT

ATOMS

The free atomic valence orbitals .p(r) may be obtained by' solving the LDF Schrodinger equation (with h2/2m = 1 in atomic units), namely

[ - V2 + V(r) ]tfJ(r)

=

E.p(r)

(4.1 )

(see, for example, Herman and Skillman"). As is well known, because the potential V(r) is spherically symmetric, the solutions take the separable form (4.2) where r, f} and c/> are spherical polar coordinates and n, t and m are the principal, orbital and magnetic quantum numbers respectively. The angular character is, therefore, determined solely by the spherical harmonics Jim (f), c/» whose probability clouds are shown in Fig. 4.1 for t = 0 (s orbitals), t = 1 (p orbitals) and t = 2 (d orbitals) respectively. The size of the free atom can be measured by the extent of the valence orbitals through the radial function Rnl (r). Figure 4.2 shows the radial functions Rnl (r) for the Is, 2s and 2p states of hydrogen. We have also plotted the probability of finding the electron at some distance r from the nucleus (in any direction) which is given by the radial probability density Pnl (r) = r2 R~I (r). We see that there is maximum probability of locating the electron at the first Bohr

The Tight Binding Approximation

83

z

s

x l=0 p

l=1 d

m=O z

z

z

Pz

Px

Py

m =±1

m=O z

z

z y x

x

x

dzy

dxz

d3Z2 - r2 m=O

m =±1

z

z

x

x

l =2

dX2

_y2

m=±2

dxy

Fig. 4.1 The probability clouds corresponding to s, p and d orbitals.

radius a, for the Is state and at the second Bohr radius a2 for the 2p state. The average or expectation value of the radial distance r is given by (4.3) so that TIs = 1·5al, r2s = 1·5a2 and r2p = 1·25a2• Therefore, the 2s orbital is more extended than the corresponding 2p orbital as is evident from Fig. 4.2. This is due to the fact that all solutions of the Schrodinger equations must be orthogonal to one another, i.e. if t/I nlm and t/I n'l'm'

84

Electron Theory in Alloy Design Bohr radius an

r (a.u.)

5

10

Fig. 4.2 The radial function Rnl (dashed lines) and the probability a function of r for the Is, 2s and 2p states of hydrogen.

density Pnl (solid lines) as

are any two solutions and t/I* is the complex conjugate of l/I, then

f

t{I~lm

(r)t{ln'l'm' (r)dr

=

0

(4.4)

If the states have different angular momentum character then the angular integration over the spherical harmonics guarantees orthogonality. But if the states have the same angular momentum character then the orthogonality constraint implies that (4.5) Thus, for the 2s radial function to be orthogonal to the Is radial function it must

The Tight Binding Approximation

85

0 ~.:~

.....~=:::.~.~....~~.~. - s:s

~

...........•.•................ ·····-4s ...•.•.....•. 3s

~oc:(

~~~..•...•.

-,

·=::::·:::~~:.~~~-:::::::s~ ........ , •.........•.• ···-4p ' ........ 3p

QJ

N

'Ci)

-R-1 S

QJ l...

a

2s

u

QJ

Vl

l... QJ

> c ......

-10

IA IIA

ns

Li Be No Mg K (0 ..... Zn Rb Sr--Cd (s Bo-'-Hg

na B

NS"SlBmBWB (

Si Go Ge In Sn Tl Pb

Al

F Ne 0 S (I A Se Br Kr Sb Te I Xe Bi Po At Rn

N P

As

-R

-1

P

2p

IA IIA Fig. 4.3 The negative of the inverse sand p pseudopotential

re mSNBWVlBWB radii (after Zunger").

change sign, thereby accounting for the node at r = 2 a.u. in Fig. 4.2. Similarly, the 3s radial function must be orthogonal to the 2s and, therefore, has two nodes, the 4s has three nodes, etc. Just as the energetically lowest Is state has no nodes, so the 2p, 3d and 4f states are nodeless, since they correspond to the states of lowest energy for a given t. The position of the outer node of the valence electron's radial function may be used as a measure of an t-depend en t core size, since we have seen that the node arises from the constraint that the valence state must be orthogonal to the more tightly bound core states. This reflects Pauli's exclusion principle which states that no two electrons may occupy the same quantum state. A not unrelated measure of size to that of the outer node has been adopted by Zunger4 who determined l-dependent radii R, directly from first principles pseudopotentials (cf Chapter 3). Figure 4.3 shows his values of -Rs-1 and "_R;l for the sp-bonded elements. We see that the inverse core radii of free atoms vary linearly across the periodic table, unlike the nearest neighbour distance or equilibrium atomic volume in bulk systems which varies parabolically (see Section 4.4). As expected, the sand p radii contract across a period as the nuclear charge increases and they expand down a column as additional full orbital shells are pulled into the core region. Figure 4.3 clearly demonstrates that the sizes of the second row elements B, C, Nand 0 are a lot smaller than those of the other elements in their respective groups which accounts for their propensity to alloy interstitially.

86

Electron Theory in Alloy Design

o~~~----~~~~~~~~~~~~--~~~~~~~~~~ > QJ

uj -10

> ~ L1J

r

Ep

l!:J

a: L1J

z

UJ

~ ~ a ~(R) oc [h(R)]2 is suggested by the Wolfsberg-Hclmholtz/" approximation SiJ OC HiJ so that the overlap repulsive contribution Hv·~·i (cf~ equation (4.37)) varies as Hij. In practice, the repulsive pair potential Cl>(R) in transition metals varies more strongly than [h(R)]2 due to the strong ion core repulsion experienced by the valence sp electrons which arises from orthogonality constraints (see Fig. 3 (b) of Pettifor26 ). The prefactor N in h (R), and hence N2 in Cl> (R), is suggested by the linear dependence of the bond integral on the atomic potential. The binding energy U is obtained by substituting equations (4.53) and (4.54) into equations (4.52), (4.50), (4.37) and (4.36). The following are the resultant equilibrium expressions for the band width, nearest neighbour distance, cohesive energy and bulk modulus:

W

(3b2j5a)N(10

Ro

K-

u:

1

- N)

In{IOa J3/[~b(lO

(3b2/200a)[N(lO

- N)]2

(4.55 ) N)]}

(4.56) (4.57)

and B

= (2J2K2/9Ro)

c:

(4.58)

Figure 4.9 compares the theoretical predictions with the experimental values

98

Electron Theory in Alloy Design

across the 4d series assuming one valence electron per atom and taking 3 12 corresponding to close packed lattices. The 'experimental' values of the band width are taken from first principles LDF calculations." The ratio b2ja is obtained by fitting a band width of 10 eV for Mo with N = 5 so that from equation (4.55) b2ja = ieV. The skewed parabolic behaviour of the observed equilibrium nearest neighbour distance is found to be predicted by values of the inverse decay length K that vary linearly across the series, namely K

=

0·435

+

0'075N

(4.59)

The ratio ajb in equation (4.46) is obtained by fitting the observed Mo Wigner-Seitz radius of2·92 a.u., giving alb = 18·0. It follows that a = 216 eV and b = 12 eV for the 4d series. Thus, we see that the simple rectangular d band model of Friedel is able to account qualitatively for the observed trends in cohesive energy, equilibrium nearest neighbour distance and bulk modulus across the non-magnetic 4d (and Sd) transition metal series. In particular, the parabolic behaviour of the cohesive energy reflects the initial filling of the bonding d states, followed by the anti-bonding states in Fig. 4.8. The skewed parabolic behaviour of the equilibrium nearest neighbour distance, on the other hand, reflects the competition between the attractive bonding term, which varies parabolically with band filling, and the repulsive overlap term, which at a fixed internuclear separation decreases monotonically across the series as the size of the free atom contracts (cf Fig. 4.3). We must note, however, that the d-bond model is a poor description at the noble metal end of the series where the neglect of an explicit sd hybridisation contribution leads to sizeable errors in the predicted cohesive energy and bulk modulus (see Fig. 4.9). Nevertheless, since the sd hybridisation term is approximately constant across the 4d series at about 2 eV/atom,27 it may be neglected to a good approximation when predicting the trends in transition metal heats of formation in the next sub-section.

4.4.4

RECTANGULAR

D BAND MODEL

FOR AB ALLOYS

The heats of formation of equiatomic AB transition metal alloys may be predicted by generalising Friedel's rectangular d-band model for the elements to the case of disordered binary systems.16,24We saw in the lower panel of Fig. 4.S that the AB heteronuclear diatomic molecule has bonding and anti-bonding levels separated by an energy WAB such that from equation (4.18) 2 WAB = (1 + b )1/2Wwhere b = dElw and w = 21hl. Within the spirit of the Friedel rectangular d-band model the disordered AB alloy may be characterized by ad-band of width WAB as illustrated in the bottom right hand corner of Fig. 4.10.16,24 Following the relation between the second moment of the

The Tight Binding Approximation

99

E

o -...~ > ~3

s: o

-:5

Fig. 4.9 The theoretical (.) and experimental ( x ) values of the equilibrium Wigner-Seitz radius, cohesive energy and bulk modulus of the 4d transition

band width, metals.

density of states and the hopping integrals in equation (4.47), we have (4.60) where the average energy E == t(EA + EB) has been taken as the reference energy zero. The first term represents the hopping to the 3 nearest neighbours and back again, the second represents hopping twice on the same site which leads to the contribution (- tl.E/2) 2 for A atoms or (+ tl.E/2)2 for B atoms. Substituting in equation (4.51) for the elemental band width W, the alloy band width is given by (4.61 ) where b == llE/W. This common band model of the binary alloy is very different from the ionic model which is illustrated in the upper part of Fig. 4.10. In the ionic picture the binary alloy density of states is assumed to be a rigid superposition of the elemental ones so that charge flows from the B site to the A site in order to equilibrate the Fermi energy, thereby setting up an ionic bond through the electrostatic Madeiung interaction. In the common band picture, on the other

100

Electron Theory in Alloy Design

Ionic Bond E

E

E

E

EO B

EO B +

EO A

EO A

nB B

ns

nA

A

+

AB Metallic Bond

E

E

E

E

E~

Es EA

E~ ns

nA

Fig. 4.10 Schematic representation of ionic and metallic bond formation in binary AB systems. EX and E~ give the free-atom energy levels, whereas the positions of EA and EB in the metallic bond reflect the small shift which takes place on alloy formation in order to main tain local charge neu trali ty.

hand, the binary alloy density of states comprises skewed partial densities of states located on the A and B sites as illustrated in the lower panel of Fig. 4.10. The amount of skewing is such as to guarantee local charge neutrality (LeN) as required for a metal with its perfect screening. Generalising equation (4.24) for the heteronuclear diatomic molecule to the bulk case as in Section 7 of Pettifor," the resultant normalized atomic energy

The Tight Binding Approximation

101

level mismatch is found to be given by (4.62)

Thus, provided the A and B transition metal contituents are not too far apart within the transition metal series so that IlN = NB - NA is not too large (4.63) Hence, in the vicinity of a half-full common band where energy mismatch for LeN will be given by

.N ~ 5, the atomic (4.64)

choosing a value of 10 eV for the bandwidth. This is to be compared with the difference in the d energy levels ofJree transition metal atoms of about - leV per valence difference across the 4d series." The heat of formation per atom IlH is obtained by comparing the binding energy of the disordered AB alloy with that of the elemental A and B transition metals, namely (4.65)

We will assume for the moment that the size factor difference between the constituent atoms is negligible, so that with respect to a fixed lattice there will be no change in the repulsive energy contribution on alloying. Moreover, with this assumption the elemental band widths WA and WB take the same value W as illustrated in the bottom left hand corner of Fig. 4.10. Then the heat of formation will be determined by the change in the bond energy alone, that is (4.66) where (4.67) and from equation (4.50) u,.A(B) bond

- ioWNA(B)

(10 -

NA(B))

(4.68)

The bond energy of the AB alloy may be expressed directly in terms of the average total density of states per atom nAB (E) rather than the partial densities of states nA (E) and nB (E) since it follows from equation (4.67) that (4.69)

102

Electron Theory in Alloy Design

Assuming a rectangular total density of states nAB (E) with a band width ~B corresponding to an energy level mismatch llELcN as illustrated in the lower panel of Fig. 4.10, the bond energy becomes (4.70) Substituting in equation (4.61) for WAB and using equation (4.63) to write J1j\/ in terms of bLCN, the bond energy is given to second order in bLCN by AB Ubond

=

(4.71)

where from equation (4.50) i{ond

(N)

--LWN(10 20

(4.72)

This expression has a simple interpretation. ~ond (N) is just the bond energy in the AB alloy written within the virtual crystal approximation (VeA), in which all atoms are assumed to be identical and described by the average properties of the A and B constituents. The prefactor 1/(1 + 3b~CN) 1/2 represents the loss of bonding with respect to this average VeA state due to the actual mismatch in the atomic energy levels J1ELCN on the A and B sites. It is equivalent to the reduction in bond order which we have already found for heteronuclear molecules in equation (4.30). The heat offormation follows directly from equation (4.66) after substituting in equations (4.71), (4.68) and (4.63). It is given to second order in llN by (4.73) where the explicit parabolic behaviour of the bandwidth (equation (4.55)) is inferred by W(N). We have labelled it the bond order contribution to the heat of formation since it reflects the change in bond energy (equation (4.39)) arising from the change in bond order rather than the bond integrals which we have assumed fixed within our neglect of size mismatch. The first contribution inside the curly brackets represents the change in the bond energy within the VeA in which each atom is characterized by N electrons. I1HvcA is attractive since the bond energy is parabolically concave upwards as a function of N so that

The second contribution inside the curly brackets is repulsive as it represents the loss of bonding due to the atomic energy level mismatch in the alloy. Figure 4.11 shows the different contributions to the normalized heat of

The Tight Binding Approximation

,

103

z ••.....

Fig. 4.18 The titanium-aluminium

At_xBx

neighbourhood

t--+--+--P"""'--

maps.

~

~oG--+--I--I

Cd Hg

116

Electron Theory in Alloy Design ~ ~ ~

>.

CsCl-

NaCl··· ....·..·..· 2 FeSi _ ..NiAs _._._._.

0'1

~

.

OJ

13 L.

::J •.•.... u

MnP-·FeB ------CrB --

2

l/) -2

o Fig. 4.19 The structural

8

N

12

16

energy of pd bonded AB compounds as a function of band filling = Ep - Ed = O. (From Pettifor and Pod-

N for seven different crystal lattices with Epd loucky.32)

accounts for their stability above N ~ 5·5. The NaC} and NiAs structural energy curves are relatively similar as might be expected from the similarity in their Jensen notation, namely NaC16/6 and NiAs6/6, respectively. This indicates that the symmetry of the co-ordination polyhedron about the Na, CI, or Ni sites is that of an octahedron (denoted by 6), whereas that about the As site is that of a trigonal prism (denoted by 6') .47The minimum in the two curves at N ~ 6 corresponds to a minimum in their density of states at the Fermi energy when all the pd-bonded orbitals are occupied.f" The structural energy depends not only on the electron per atom ratio or N, but also on the atomic energy level mismatch Epd = Ep - Ed' Curves similar to Fig. 4.19 have, therefore, been calculated for values of the atomic energy level difference in the range from -10 to +5eV (in steps of 2·5eV). However, rather than plotting the most stable predicted structure on a structure map of Epd versus N, the rotated frame of axes Np versus Xd is used in order to make direct comparison with the experimental results. Np and Nd are th~ number of p and d electrons computed on the CsCI lattice for a given choice of Epd andN. The theoretical predictions are compared with experiment in Fig. 4.20 where we see that broad agreement is obtained between the topological features of the two-structure maps. In particular, NaCI in the top left-hand corner adjoins NiAs running across to the right and boride stability down to the bottom. MnP stability is found in the middle of the NiAs domain and towards the bottom right-hand corner where it adjoins CsCI towards the bottom. The main failure of this simple pd TB model is its inability to predict the FeSi stability of the transition metal silicides which is probably due to the total neglect of the valence s electrons within the bonding. The TB model has successfully accounted for the structural trends not only within the pd bonded AB compounds above, but also within the AB2 and AB3 structure maps. 18,34,35,50 As expected, the structural stability of the binary phases

The Tight Binding Approximation

2·5

b

~B

x,p NB 1-5

Xd

0·7 1mB VIB ~

2.0

11 7

mB~

~~

~

I

~

~t. ~

~

m*

C

00

~~i 0

0

I

mA NA VA

C

o

C 0

C

c

• FeSi NaCl v NiAs ~ CrB o esCl II MnP ~ FeB

c

2

4

Nd

6

8

10

Fig. 4.20 The upper panel shows the structure map (Xp' Xd) for 169 pd bonded AB compounds, where Xp and Xd are values for the A and B constituents of a certain chemical scale, X, which orders the elements in a similar way to the Mendeleev number M. The lower panel shows the theoretical structure map (Np, ~) where Np and ~ are the number of p and d valence electrons respectively on the CsCllattice. (From Pettifor and Podloucky.Y)

is found to be controlled by four factors, namely the average number of valence electrons per atom (or band filling), the atomic energy level mismatch (or Mulliken electronegativity difference), the atomic size mismatch, and the angular character of the valence orbitals. Classic ionic Madelung terms appear to play little role in determining the structures of in termetallic phases. 4.6 ANGULARLY-DEPENDENT

MANY-ATOM

POTENTIALS

The atomistic simulation of defects such as dislocation cores and grain boundaries is essential for understanding the mechanical properties of metals and alloys (see, for example, Vitek'"}, Electron theory can help guide the choice of the interatomic potentials which are used in the simulations. In Chapter 3 we saw that the nearly free electron (NFE) approximation provides density dependent, oscillatory pair potentials which accurately describe the structural properties of bulk crystalline, liquid and glassy simple metals and their alloys. In this section, following the pioneering work of Moriarty,52 we will see that the

118

Electron Theory in Alloy Design

tight binding (TB) approximation provides a prescription for generating angularly dependent many-atom potentials for modelling transition metals and their alloys. The second moment approximation provides the simplest TB description of the cohesive properties of transition metals.19,S3Within this approximation it follows from equations (4.36), (4.37), (4.50) and (4.51) that the binding energy can be written in the Embedded Atom form,54 namely (4.83) (R)is the usual repulsive pair potential and F(p) is an attractive square root embedding function, representing the resonant nature of the metallic bond (cf. equation (4.52)) namely F(p)

=

_ pl/2

(4.84)

The argument of the embedding function in equation (4.83) is given by Pi

=

L Ch

2

i:;6

j

(Rij)

(4.85)

where C is a constant which takes the value (J3/10) N (10 N) within the Friedel rectangular band model for transition metal elements. Daw and Baskes54had derived the embedding term on the right-angle side of equation (4.83) by assuming that it represented the energy of embedding an atom at site i in the local charge density Pi which comes from the tails of the free atomic charge clouds on neighbouring sites, i.e. Pi = ~ii=j Patorn (Rij). F(p) could, therefore, in principle be calculated by embedding an atom in a free electron gas of density p. Such calculations have been performed by Manninen et al.55 within the related effective-medium theory. On the other hand, equation (4.84) and (4.85) show that within a TB description the embedding function results from the bond formation which takes place between the embedded atom i and its surrounding neighbours j. The explicit square root dependence of equation (4.84) reflects the fact that the bond energy associated with atom i does not vary linearly with the number of bonds or neighbours due to the resonant character of the metallic bond.23 The embedded atom method has been very successful in modelling fcc noble metals, aluminium and their alloys (see, for example, Chen et al.56 and references therein). In particular, it accounts naturally for metals with positive Cauchy pressures C12-C44which would be identically zero within a pair potential descriptionr" However, there are many important systems with negative Cauchy pressures, such as the brittle cubic intermetallics based on TiAl3 which we mentioned in the Introduction. In the case of the titanium tri-aluminides the angular character of the pd bonding is of paramount importance as indicated

The Tight Binding Approximation

119

both by the LDF charge density and the large values of the shear to bulk modulus /1/ K. Explicit angular character is ignored within the Embedded Atom method because either spherical atoms are embedded in an electron gas of density p or the second moment approximation is made to the local TB densities of states. Angularly dependent many-atom potentials have recently been derived which give, in principle, an exact representation of the bond order within the TB approximation.P'f" They have the very important feature that the lowest term in the many-atom expansion is already angularly dependent since it is the (J, ti or b bond which is being embedded in the local environment (c£ Fig. 4.7) rather than a spherically symmetric atom. For sp valent systems with zero sp splitting, the first term in the expansion for the (J bond order between atoms i and j is given by (4.86) where Nand

X2 is a reduced susceptibility which depends primarily on the band filling

with a, band c constants which are defined in terms of the ratios PP(J/ss(J and ppn/ss« (see equations (32)-(34) of Ref. 60). The predicted angular character of the embedding function G for the (J bond follows very closely that obtained by Tersoffi1 in hisfitting of an empirical bond order potential for the semi-conductor silicon. It has a shape similar to that shown in Fig. 4.16 for the three-atom contribution to the sp valent fourth moment. This explicit angular character of the bond order potential is probably responsible for the binary inter-metallic phase LiAl taking the Zintl structure type NaTl which consists of two interpenetrating diamond sublattices. As we have seen in the previous chapter this ordering of the lithium and aluminium atoms with respect to an underlying bee lattice, so that we have four like and four unlike nearest neighbours, cannot be modelled by central pair wise potentials. Currently these TB bond order potentials are being developed for the Li-AI system, and also transition metals and their alloys. 4.7 CONCLUSION The TB approximation has allowed the quantum mechanical first principles calculations to be linked directly with our physical and chemical intuition. In the process we have seen that some well-accepted concepts such as atomic size can be placed on a more fundamental footing, whereas others such as elec-

120

Electron Theory in Alloy Design

tronegativity and ionicity must be treated with great caution when dealing with metals and alloys. In addition, important new concepts have arisen, such as those linking the electron per atom structural trends with the local topology through the moments of the density of states. The development of simple, yet reliable concepts is essential for understanding the complex world of alloy design.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 2l. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

C.L. Fu, ]. Mater. Res., 1990, 5, 971. A.H. Cottrell, Mater. Sci. Techno I, 1991,7, 981. F. Herman and S. Skillman, Atomic Structure Calculations, New Jersey, Prentice Hall, 1963. A. Zunger, Phys. Rev. B, 1980, 22, 5839. R.S. Mulliken, ]. Chern. Phys., 1934, 2, 782. R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, New York, Oxford University Press, 1989. L. Pauling, The Nature of the Chemical Bond, 3rd. edn., Ithaca, Cornell, 1960. C.A. Coulson, L.R. Redei and D. Stocker, Proc. Roy. Soc. A, 1962, 270, 357. J.C. Phillips, Rev. Mod. Phys., 1970,42, 317. W.A. Harrison, Electronic Structure and the Properties of Solids, San Francisco, Freeman, 1980. C.A. Coulson, Proc. R. Soc. A, 1939, 169, 413. A.J. Skinner and D.G. Pettifor, J. Phys.: Condense Matter, 1991, 3, 2029. D.G. Pettifor, ]. Phys. F: Met. Phys., 1977, 7, 61.3. O.K. Andersen, W. Klose and M. Nohl, Phys. Rev. B, 1978, 17, 1209. J. Friedel, in The Physics of Metals, (ed. ].M. Ziman), New York, Cambridge University Press, 1969, p. 494. D.G. Pettifor, Phys. Rev. Lett., 1979, 42, 846. A.H. Cottrell, Introduction to the Modern Theory of Metals, London. The Institute of Metals, 1988. F. Ducastelle, Order and Phase Stability in Alloys, Amsterdam, North Holland, 1991. F. Ducastelle, J. Physique, 1970,31, 1055. See, for example, A.P. Sutton, M.W. Finnis, D.G. Pettifor and Y. Ohta,]. Phys. C: Solid State Phys., 1988, 21, 35. F. Cyrot-Lackmann, Ado. Phys., 1967, 16, 393. F. Ducastelle and F. Cyrot-Lackmann, J. Phys. and Chern. Solids, 1970, 31, 1295. V. Heine, I.J. Robertson and M.C. Payne, Phil. Trans. R. Soc. Lond. A, 1991, 334, 393. D.G. Pettifor, Solid State Phys., 1987, 40, 43. M. Wolfsberg and L. Helmholtz, ]. Chern. Phys., 1952,20, 837. D.G. Pettifor, J. Phys. F: Met. Phys., 1978, 8, 219. C.D. Gelatt, H. Ehrenreich and R.E. Watson, Pkys. Rev. B, 1977, 15, 1610. F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema and A.K. Niessen, Cohesion in Metals: Transition Metal Alloys, Amsterdam, North-Holland, 1988. L. Brewer, Science, 1968, 161, 115. A.R. Williams, C.D. Gelatt and V.L. Moruzzi, Phys. Rev. Lett., 1980, 44, 429. D.G. Pettifor, ]. Phys. C: Solid State Phys., 1986, 19, 285. D.G. Pettifor and R. Podloucky, Phys. Rev. Lett., 1984, 53, 1080. J.K. Burdett and S. Lee, J. Am. Chern. Soc., 1985, 107, 3063. S. Lee, ]. Am. Chern. Soc., 1991, 113, 101.

The Tight Binding Approximation

121

35. S. Lee, Acc. Chern. Res., 1991, 24, 249. 36. j.c. Cressoni and D.G. Pettifor, ]. Phys.: Condense ~atter, 1991, 3, 495. 37. F. Ducastelle and F. Cyrot-Lackmann, ]. Phys. and Chern. Solids, 1971, 32, 285. 38. P. Turchi and F. Ducastelle, in The Recursion Method and its Applications, (eds. D.G. Pettifor and D.L. Weaire), Berlin, Springer, 1985, p. 104. 39. j.F. Gaspard and P. Lambin, in The Recursion Method and its Applications, (eds. D.G. Pettifor and D.L. Weaire), Berlin, Springer, 1985, p. 75. 40. D.G. Pettifor, in Physical Metallurgy, (eds. R.W. Cahn and P. Haasen), Amsterdam, Elsevier, 1983, ch. 3. 41. H. Hasegawa and D.G. Pettifor, Phys. Rev. Lett., 1983, 50, 130. 42. R.J. Weiss, Proc. Phys. Soc., 1963, 82, 281. 43. C. Zener, Phys. tu«, 1947, 71, 846. 44. L.M. Hoistad and S. Lee, ]. Ann. Chern. Soc., 1991, 113, 8216. 45. D.G. Pettifor, Mater. Sci. Technol., 1988, 4, 2480. 46. D.G. Pettifor, Mater. Sci. Technol., 1992, 8, 345. 47. W.B.Jensen, in The Structures of Binary Compounds, Amsterdam, North Holland, 1989, p. 105. 48. P. Villars, K. Mathis and F. Hulliger, in The Structures of Binary Compounds, Amsterdam, North Holland, 1989, p. 1. 49. D.G. Pettifor and R. Podloucky,]. Phys. C: Solid State Phys., 1986, 19, 315. 50. Y. Ohta and D.G. Pettifor, ]. Phys.: Condense Matter, 1989, 2, 8189. 51. V. Vitek, in Dislocations and Properties of Real Materials, (ed. M.H. Loretto), London, The Institute of Metals, 1985, p. 30. 52. J.A. Moriarty, Phys. Rev. B, 1990, 42, 1609. 53. M.W. Finnis and j.E. Sinclair, Phil. Mag. A, 1984,50, 45. 54. M.S. Daw and M.l. Baskes, Phys. Rev. B., 1984, 29, 6443. 55. M. Manninen,j.K. Nerskov and C. Umrigar,]. Phys. F: Met. Phys., 1982,12, L.1. 56. S.P. Chen, A.E. Voter, R.C. Albers, A.M. Boring and PJ. Hay,]. Mater. Res., 1990,5,955. 57. R.A. Johnson, Phys. Rev. B, 1988,37, 3924. 58. D.G. Pettifor, Phys. Rev. Lett., 1989, 63, 2480. 59. D.G. Pettifor and M. Aoki, Phil. Trans. R. Soc. Lond. A, 1991, 334, 439. 60. D.G. Pettifor and M. Aoki, in Structural and Phase Stability of Alloys, (eds. j.L. Moran-Lopez andJ.M. Sanchez), New York, Plenum, 1992, p. 119. 61. J. Tersoff, Phys. Rev. B, 1988, 38, 9802.

5

Order and Phase Stability in Alloys FRANQOIS DUCASTELLE Office National d'Etudes et de Recherches Aerospatiales (ONERA) BP 72 92322 Chatillon Cedex, France

5.1 INTRODUCTION Due to the presence of interatomic interactions, any alloy should eventually order or phase separate at low temperature. This is observed in many systems. When the ordering interactions are strong, the alloy may remain ordered up to the melting point (Ni3Al for instance). When they are very weak, the solid solution can be the only observable phase since at low temperature atomic diffusion is no longer efficient (Cu-Ni). In the intermediate regime we have a more interesting situation with one or several ordered phases at low temperature which disorder at a critical temperature before melting (eu-Au, Pd-V, Ni-Fe, etc.). In the simplest case all phases are built on a fixed underlying lattice such as the fcc or bee lattice, but more frequently, several structures are involved and many compounds display structures which are not observed in elemental metals (Laves phases, (J phases, A15 phases, etc.). Ifis of great interest, both from a fundamental and a practical point of view, to understand what stabilizes a particular structure at a given concentration and temperature and then to predict the phase diagrams of alloys. Until recently, most theories in this field were based on phenomenological models. The stability of several crystalline structures has been interpreted using geometrical arguments by looking for the most effective way of filling space with hard spheres of different radii. Size effects are also known to account for the limits of stability of solid solutions. Chemical or electronic parameters have also been used. One of the celebrated Hume-Rothery rules for instance relates the stability of several structures to definite values of the electronic ratio (number of valence electrons per atom). Finally various electronegativity scales have been introduced. As far as phase diagrams are concerned, most approaches have been based on phenomenological thermodynamic models using regular solution models or various improvements on them. Recent research indicates that it is now possible to tackle these problems from

122

Order and Phase Stability in Alloys

123

microscopic theories based on first principles. This confidence is based on the success of first principles electronic structure calculations concerning elemental metals as well as intermetallic compounds, and on the development of efficient thermodynamic tools. In fact 'ideal' methods starting from the Schrodinger equation and from general thermodynamics are generally not so useful when one is interested in a physical understanding of fairly complex systems, and it is still quite valuable to define intermediate models containing a few relevant parameters. Many modern studies use an approach similar to that presented in this short review which can be summarised as follows. We start from electronic structure models based on various well defined approximation: the one-electron approximation based on the so-called local density approximation, the tight-binding approximation when necessary, etc. These approximations are now well controlled in the case of elemental metals or simple ordered compounds and provide total energies with a very good accuracy. The next step is to calculate the energy of any atomic configuration which obviously cannot be done exactly. At this stage it is shown that this energy can be obtained through a generalised perturbation expansion starting from the disordered state. The latter is described within the coherent potential approximation which is the simplest method that allows us to treat the scattering of electrons in strongly disordered systems, i.e. in systems where the atomic potentials strongly differ from one atom to the other. At the end of this process, the electronic degrees of freedom are eliminated and we are left with the desired expression for the energy of any configuration as a function of pair and multiple interactions. In the case of a binary alloy we recover the celebrated Ising model or direct generalisations of it, were it not for the concentration dependence of the interactions which is completely unavoidable in the theory. The implications of this dependence can be considerable, in particular when studying phase diagrams (principally because of the common tangent rule), but they are still far from being well characterised, both experimentally and theoretically, and it is hoped that at least locally on the concentration axis the previous theory offers a good justification for the use of Ising models. The next step is therefore a thermodynamic one and consists in exploring the properties of the Ising model, i.e. studying the nature and the stability of ordered structures as a function of concentration, temperature and of the interactions. Exact results can be obtained at zero temperature but approximations are required at finite temperature. In the case of short-range interactions the modern developments of generalised mean field methods such as the cluster variation method now provides very accurate results, as tested by Monte Carlo simulations. We are thus able in principle to go from electronic structure calculations to phase diagrams through a set of well controlled approximations. At least this is so when ordering effects can be assumed to take place on a fixed, rigid lattice.

124

Electron Theory in Alloy Design

These restrictions are rather important, since they are necessary in order to speak of Ising models with discrete numbers of degrees of freedom per site. However even if this has still not been worked out in detail there is no difficulty in principle in extending the whole chain of arguments to include elastic and relaxation effects; it is also possible to compare ordered structures on different types of lattices, i.e. to treat structural effects. The fact that such an approach can now be implemented is certainly an important result of recent years. It should be stressed however that the number of approximations introduced to derive in the end models with a few physical parameters, necessarily implies some loss of accuracy when studying concrete situations. Phenomenological models remain necessary to account in detail for experimental data, but on the other hand they cannot be reliable globally if they are not consistent with what is learned from more fundamental theories. The present review is somewhat schematic. Details can be found in a recent book.' Several other important topics not discussed here can be found in the review by de Fontaine, in the books by Khachaturyan and Hafner and in those of the series Cohesion and Structure. 2,3,4,5 Sections 5.2 to 5.5 are concerned with the electronic structure of alloys and with the derivation of effective multi-atom interactions. The thermodynamics of ordering effects is discussed in sections 5.6 and 5.7. Finally a few applications are presented in section 5.9. 5.2 TIGHT-BINDING

MODEL FOR ALLOYS

The tight-binding formalism described in Chapter 4 is particularly well adapted to a semi-quantitative discussion of strong alloying effects. It also applies quite naturally to transition alloys. Furthermore there is no difficulty in principle, if not in practice, to derive the appropriate extensions involving full band structure calculations. We consider here binary alloys Al_cBcon fixed underlying lattices. For convenience we assume crystalline structure with a single atom per unit cell (e.g. fcc or bcc). Each atomic configuration is characterised by the value of the so-called occupation numbers p~ equal to unity or zero depending on the presence or not of an atom or type i at site n; obviously p~ = 1 The simplest tight-binding hamiltonian H can then be written in the Dirac notation (see Appendix) as

r:

H

=

L

In)

en

}). Let us use alloy variables and define the concentration fluctuations bCn = Cn - c. We now compare the corresponding canonical free energy with that of the disordered

"

n

T Fig. 5.14 Variations of the order parameter as a function of temperature, order transitions; (b) for first order transitions.

(a) for second

Order and Phase Stability in Alloys

147

state (5.61 ) with

+ k BTL

{Cn

log c;

+ (1 - Cn) log (1 -

Cn)}

n

We recall that ~m

=

4Jnm.

Expanding

st; c, {bcn})

in powers of bcn, we obtain

(5.62)

Using then Fourier transforms of bCn and of Vnm

s;

=

L r-s.; k

V(k)

=

L

=

V(m - n)

eik,R

V(R)

(5.63)

R

we obtain

(5.64)

where the prime in the sum means that k, + k2 + k3 should be equal to a reciprocal vector of the underlying lattice. The coefficient of the quadratic part of bFis positive at high temperature and then the disordered phase characterised by bCk = 0 is stable. When the temperature decreases, it becomes unstable at a critical temperature I: (ko) for the concentration fluctuations of wave vector ko such that (5.65) Consider

J

=

again the BeC

V/4 and when

C =

lattice with positive first neighbour interactions 1/2. The most unstable concentration wave is then

148

Electron Theory in Alloy Design 1\

of

T>

r,

Fig. 5.15 Free energy as a function of the order parameter 11 for a second order transition. 110 and -110 are the equilibrium values of 11 below I:.

[100] in units of 2n/a, where a is the lattice parameter and we recover the critical temperature kB = 8J. Notice here that the wave [010] and [001] are strictly equivalent to [100] in the sense that they differ from a vector of the reciprocal lattice of the bee lattice. Up to a normalisation factor we see that the order parameter 1] is nothing but the amplitude of the concentration waves which allows us to describe the B2 ordered structure.1,18,23In the present case there is a single wave to consider and the order parameter is scalar. Keeping only this wave in the free energy we obtain an expansion of the form

I:

(5.66) where odd powers of 1] are absent since changing 1] into -1] amounts to replacing one variant of the B2 structure by the other. Hence the familiar behaviour of the free energy as a function of the temperature (Fig. 5.15). In the fcc lattice and at stoichiometry A3B, positive first neighbour interactions favour ordering according to the L12 (Cu3Au) structure (Fig. 5.18), (in fact other interactions are necessary to lift the degeneracy of this structure with other ones);' To describe this structure we need the three waves [100], [010] and [001] with a common amplitude 11; actually in the fcc lattice they are not equivalent modulo a vector of the reciprocal lattice. On the other hand the sum of these wave vectors is equal to such a vector, which means that cubic terms in the Landau expansion no longer vanish. Using standard arguments this proves that the L 12order-disorder transition should be of first order and occurs

Order and Phase Stability in Alloys

149

1\

of

o Fig. 5.16 Free energy for a first order transition.

above the temperature of the free energy

~(ko).

Figure 5.16 shows the corresponding behaviour

(5.67) Notice that the presence of three waves indicates that the order parameter in this case should be considered as a vector with three components. This is important when studying for example interfaces between different variants of

LI2•24,25

Landau theory therefore allows us to make predictions concerning the nature of the ordered phases and of the order of the transitions. This is described in detail elsewhere. 1,18,23 We now consider short-range order. In principle the simplest mean field theory neglects it but some information can be recovered by using equation (5.44) which relates ((In(Jm)c to the derivative o«(Jn)/ohm. Using the MFA equations in the presence of inhomogeneous external fields hn we can obtain a closed formula for the Fourier transform lJ..(k) of the SRO parameters lJ..nmwhich only depend on m-n in the disordered phase. This is the so-called KrivoglazClapp-Moss (KCM) formula lJ..(k)

=

[1 + c(1 - c) V(k)/kB

T]-1

(5.68)

This is obviously not a consistent treatment ofSRO since, for instance, lJ..(ko) diverges at the critical temperature (ko) whereas the initial MFA approximation corresponds to set lJ..(k) = 1. However the KCM formula is exact to order

I:

150

Electron Theory in Alloy Design

Fig. 5.17 Schematic

structure

map for AsB compounds.

1/ T at high temperature and provides us with a reasonable first approximation. The fact that a(k) -1 is proportional to the coefficient of the quadratic part of the Landau free energy functional is not fortuitous at all since both quantities measure the response of the disordered state to concentration fluctuations in a harmonic analysis. Similar although more complicated formulae can be derived within the CVM. 5.9 APPLICATIONS All the ingredients to understand the behaviour of specific systems are now available. We discuss some applications which will hopefully give a flavour of the present state of the art. 5.9.1

ORDERED

STRUCTURES

OF TRANSITION

ALLOYS

Transition alloys are well described qualitatively within the tight-binding scheme presented in the first part of this chapter. We recall that the important alloy parameters are the alloy parameter b / Wand the filling of the band N.. Since b is itself approximately proportional to 11~ = ~B - ~A, it is quite natural to classify the structures by using the coordinates 11~ and N. .26 Such maps are very successful in that the different ordered structures generally belong to different non-overlapping domains. We comment here on the map at c = 1/4 (Fig. 5.17); other maps are discussed in Refs. 1, 26 and 27. Lists of

Order and Phase Stability in Alloys

151

0022

Fig. 5.18 L12 and D022 structures.

structures are available in Refs. 26 and 27. We shall consider here just the simplest ones. At c = 1/4 we keep the familiar L12 and D022 structures built on the fcc lattice (Fig. 5.18) as well as the equivalent structures on the hcp (D019 and DOa) or doublehcp lattice (D024), and the A15 structure. The ordering tendencies are quite clear. A15 appears when AN: > 0 and all other non-magnetic structures when AN: < O. Note that our sign convention is such that for instance AN: is positive for PtV3 but negative for Pt3 V (we always assume a compound A3B). It is also striking that all D022 structures, plus the 'equivalent' DOa structure occur at the same point, N. = 7·75, AN: = - 5 and therefore correspond to elements of the column of vanadium and of that of nickel. The non-magnetic Ll2 structures are all in the range 6·5 ~ N. ~ 7·5 whereas most equivalent hexagonal structures occur for N. = 7·25. From the theoretical side, consider first the fcc lattice. We know that in general Il'tl ~ I~, Vs, ~I ~ I~, ~, .. ·1, so that ordered structures only appear when l't > O. Calculating l't from the electronic structure defines a domain in which non-magnetic systems may order. Now, on the fcc lattice and for c = 1/4, the only possible ground states in the presence of interactions up to fourth neighbours are L12 and D022. The two structures have identical numbers of AB first neighbour pairs, which means that the difference in energy between them is much smaller than l't. It is given by the combination = ~ - 4 Vs + 4 ~, and moment arguments show that should have at least four zeros as a function of band filling. Tight-binding calculations then clearly show a stability exchange between L12 and D022 for N. about seven, in good agreement with the observed structures (Fig. 5.19). Large values of N. are irrelevant (since there is no order) as well as small ones (since the lattice is no longer fcc). These results have been confirmed by more sophisticated electronic structure calculations. The stability ofD022 for large electron numbers is therefore explained. Ifwe now compare different crystalline structures, the GPM is of no direct use and full calculations should be undertaken although there is some hope that structural energies can also be analysed in terms of pair and multi-atom interactions.

e

e

152

Electron Theory in Alloy Design ~H ( L 12 20

-

DO

22 )

(meV)

10

o

Fig. 5.19 Calculated energy difference between L12 and D022 as a function of the number of d electrons," Continuous line: full calculation; dotted line: calculation with pair interactions up to ~.

Simple, arguments tell us on the other hand that BCC or more complex structures should be favoured when.N. ~ 5-6 and it is therefore not surprising that A 15 is the most stable structure here. Such simple arguments inherited from the study of elemental metals do not apply in the case of the cubic-hexagonal competition when ~~ < O. Actually both the fcc-hcp and the L12-D022 energy differences are very small and the interplay between chemical and structural effects has still to be investigated in detail. There are numerous other examples showing that simple theoretical arguments predict the correct structural trends. For example the hierarchy between the interactions on a fcc lattice shows that it is generally necessary to include third and fourth neighbour interactions as soon as second neighbour interactions are introduced. This is frequently verified, in the case of transition alloys at least. Actually many observed ordered structures require at least fourth neighbour interactions to be stabilised. Conversely a structure like that of CuPt(L 11) requires second neighbour interactions comparable with the first neighbour ones and is in fact never observed except precisely in the case of CuPt! Other hierarchies are expected in different types of compounds. The case of substoichiometric NaCI transition carbides and nitrides is very instructive in this respect. Here ordering of vacancies takes place in the fcc sublattice occupied by interstitial (carbon or nitrogen) atoms but these atoms principally interact with the metallic atoms of the other sublattice. A simple moment analysis shows that we should then have the hierarchy If; I > I~ I ~ ... , which favours structures very different from those met in transition alloys. In particular the CuPt-type structure is found to be very stable, which is actually observed.29,so Pseudobinary semiconductor alloys form another category of compounds which has been studied in detail." As another example illustrating how we have elucidated trends, consider the nickel based transition alloys, in particular their phase diagrams as compiled

Order and Phase Stability in Alloys

153

5

by de Boer et al. When 11N; is small, ordering is absent except in Pt-Ni. Small values of 11~ imply small almost unobservable transition temperatures, and anyway large values of .N. induce a tendency to phase separation. For the case of 4d and 5d alloying elements, except for Pt-Ni, there is indeed no ordering tendency up to Tc and Re. Ordering effects in Ni-Fe and Ni-Mn are clearly correlated with the appearance of magnetism (an extensive discussion of the interplay between magnetic and chemical ordering has been given recently by Bieber and Gautier'"). Weak ordering effects appear when 11N: increases., Ni-Mo and Ni-W present ordered phases whereas SRO effects have clearly been detected in Ni-Cr. This is consistent with the fact that the order of magnitude of ordering energies should increase with the bandwidth, i.e. when going from the 3d to the 5d series. The same phenomenon exists for the vanadium column: on the Ni-rich side of the diagrams there are ordered structures on the fcc lattice which disorder at high temperature in the case of Ni-V but not in the case of Ni-Nb and Ni-Ta. Finally many very stable ordered compounds appear when 11~ is equal to 6 and 7. Thus, except for the ordering in Pt-Ni which is still a subject of controversy, almost all is understood qualitatively for the case of Ni based alloys. This is certainly a tsomewhat favourabie situation, but many other successful analyses have been 'made. For example, the fact that off-diagonal disorder induces phase separation is well verified in the case of Cr-Mo and Cr-W alloys. If one is interested in a particular system rather than in trends, then first principles calculations of total energies are obviously preferable if they are accurate enough, which now begins to be the case. By performing severa-lsuch calculations it is also possible to determine through an inverse procedure the interactions of an effective Ising model. 33 An advantage of the first principles methods is that they can handle non-transition elements as well. Aluminium based alloys, in particular , have been considered in some detail. 5.9.2

DIFFUSE SCATTERING

AND SHORT RANGE

ORDER

Diffuse X-ray or neutron scattering is a very efficient tool to measure shortrange order effects. In fact, provided that other contributions due to atomic displacements or to magnetic effects are separated out, the diffuse intensity is proportional to Ci(k), the Fourier transform of the SRO parameter. If the Krivoglaz-Clapp-Moss formula (equation (5.6.8)) is valid, this yields the Fourier transform of the pair interactions V(k). For a given set of interactions it is then IPossible to relate Ci(k), measured in the high temperature disordered phase, to: the observed ground states at low temperature. For example, assuming interactions up to the fourth neighbours in the fcc lattice, it is known that the stable ordered structures at the A3B stoichiometry are L12 or D022 depending on the sign (negative or positive, respectively) of the quantity = V; - 4Vs + 4~, provided that V. is positive and sufficiently large. The KCM formula then predicts that Ci(k) is maximum for wave vectors

e

154 020

000

Electron Theory in Alloy Design Ni3V

(a)

022

002

Fig. 5.20 Diffuse intensity in the (110) reciprocal (Laue units): (a) for Ni3V; (b) for Pd3V.38

(b) plane obtained

from neutron

scattering

of type (100) for L12 and (1¥J) for D022• This is observed in many systems, for example in Ni3V which is D022 at low temperature (Fig. 5.20a). Surprisingly enough measurements on Pd3 V which also orders according to D022 show maxima at positions (100) (Fig. 5.20b). This breakdown of the usual mean field theory has been successfully explained using more accurate CVM approximations. It turns out that if ~I Vi is positive but small enough, the short range order above D022 should actually be of the L12 type." Experiment thus tells us that ~ ~ 0 in Pd3 V. Qualitative electron microscopy studies indicate that the situation is the same for Pt3V. Using the structure maps, one can then predict that alloys with a slightly lower number of d electrons should order according to L12• This has been recently verified by considering pseudobinary alloys (PtRh)3 V and (PtRh)3 V alloys. " Diffuse scattering experiments are now fairly accurate. Combined with appropriate CVM or Monte Carlo analyses they now provide us with reliable estimates of the pair interactions. Recent results concerning Cu-Zn, Au-Ag and Ni based alloys are given and discussed in Refs. 36 to 38. 5.9.3

PHASE DIAGRAMS

I t is more difficult to calculate phase diagrams than to characterise the single phases, one reason being that it is not sufficient to know the interatomic chemical interactions. The free energy of the disordered state is also required. Although it can be calculated in principle, using the CPA for example, additional contributions related to elastic, vibrational and other effects have to be included. Another serious difficulty is that real phase diagrams frequently

Order and Phase Stability in Alloys

T(K)

Experiment

Ti

T(K)

155

Ni

Theory

2000

1000

Ti

Ni 43

Fig. 5.21 Experimental and calculated phase diagram of Ti-Ni.

involve different crystalline structures, and at the moment there is no elaborate theory to describe the interplay between structural and chemical (ordering) effects. In practice, ordering effects are treated on fixed lattices and the free energies for different structures are compared. This is a correct procedure but the uncertainties concerning the structural energies or entropies are difficult to estimate, and generally some parameters have to be fitted to experimental data in order to produce reasonable phase diagrams. I t is clear also that it is difficult to treat the liquid state on the same footing as the solid phases. Despite all these problems several phase diagrams have now been calculated.

156

Electron Theory in Alloy Design

There is no room here to undertake a comprehensive review of these results. We just mention some calculations concerning the Ti-Rh,39 Ni-Cr,40 Pd-Rh, and AI-Li41 phase diagrams and the recent detailed discussions concerning the Ni- Ti and Ni-Al phase diagrams.42,43 The experimental and calculated diagrams of Ti-Ni are shown in Fig. 5.21. REFERENCES 1. F. Ducastelle, Order and Phase Stability in Alloys, Cohesion and Structure, Vol. 3, (ed. F .R. de Boer and D.G. Pettifor), North-Holland, Amsterdam, 1991. 2. D. de Fontaine, Solid State Phys., 1979, 34, 73. 3. J. Hafner, From Hamiltonians to Phase Diagrams, Springer, Berlin, 1987. 4. A.G. Khachaturyan, The Theory of Structural Transformations in Solids, Wiley & Sons, New York, 1983. 5. F.R. de Boer, R. Boom, W.e.M. Matens, A.R. Miedema and A.K. Niessen, Cohesion in metals: Transition Metals Alloys, Cohesion and Structure, Vol. 1, (ed. F.R. de Boer and D.G. Pettifor), North-Holland, Amsterdam, 1989; J. Hafner, F. Hulliger, W.B. Jensen, J.A. Majewski, K. Mathis, P. Villars and P. Vogl, Cohesion in metals: Transition Metals Alloys, Cohesion and Structure, Vol. 2 (ed. F.R. de Boer and D.G. Pettifor), North-Holland, Amsterdam, 1989. 6. F. Ducastelle, Electronic Structure, Effective Pair Interactions and Order in Alloys, in Alloy Phase Stability, (ed. G.M. Stocks and A. Gonis), NATO-AS I Series E, Vol. 163, Kluwer, Dordrecht, 1989. 7. A. Bieber, F. Ducastelle, F. Gautier, G. Treglia and P. Turchi, Solid State Communications, 1983, 45, 585. 8. M. Sluiter and P.E.A. Turchi, Phys. Rev., 1989, B40, 11 215. 9. L.G. Ferreira, S.H. Wei and A. Zunger, Phys. Rev., 1989, B40, 3197. 10. S.H. Wei, L.G. Ferreira and A. Zunger, Phys. Rev., 1990, B41, 8240. II. A. Genis, X.C. Zhang, A.J. Freeman, P. Turchi, G.M. Stocks and D.M. Nicholson, Phys. Rev., 1987, B36, 4630. 12. H. Dreysse, A. Berera, L.T. Wille and D. de Fontaine, Phys. Rev., 1989, B39, 2242. 13. M. Asta, C. Wolverton, D. de Fontaine and H. Dreysse, Phys. Rev., 1991, B44, 4907; C. Wolverton, M. Asta, H. Dreysse and D. de Fontaine, Phys. Rev., 1991, B44, 4914. 14. P.E.A. Turchi, G.M. Stocks, W.H. Butler, D.M. Nicholson and A. Gonis, Phys. Rev., 1988, B37, 5982; P.E.A. Turchi, M. Sluiter, F.J. Pinski, D.Johnson, D.M. Nicholson, G.M. Stocks and J.B. Staunton, Phys. Rev. Lett., 1991, 67, 1779. 15. A. Bieber, These de Doctorat d'Etat, Unioersite Louis Pasteur, Strasbourg, 1987. 16. A. Bieber and F. Gautier, J. Phys. Soc. Japan, 1984, 53, 2061 and Zeischrift fur Physik, 1984, B57, 335. 17. See e.g. P. Haasen, Phys. Metall, Cambridge University Press, Cambridge, 1978. 18. D. de Fontaine, Solid State Phys., 1979, 34, 73. 19. D. Gratias, 'Introduction aux Methodes de Champ Moyen: 1a Methode Variationnelle des Amas, in L'Ordre et Le Desordre dans les Materiaux, Les Editions de Physique, Les Ulis, 1984, p. 119. 20. J.M. Sanchez, F. Ducastelle and D. Gratias, Physica, 1984, 128A, 334. 21. D. de Fontaine, 'The Cluster Variation Method and the Calculation of Alloy Phase Diagrams, in Alloy Phase Stability, (ed. G.M. Stocks and A. Gonis), NATO-ASI Series E, Vol. 163, Kluwer, Dordrecht, 1989.

Order and Phase Stability in Alloys

157

22. A. Finel, These de Doctorat d'Etat, Uniuersite Paris VI, 1987. 23. A.G. Khachaturyan, Progress in Mater. Sci., 1978,22, 1, and The Theory of Structural Treansformations in Solids, Wiley, New York, 1983. 24. F. Ducastelle, Thermodynamics of surfaces and interfaces, in Structural and Phase Stability of Alloys (ed. J.L. Moran-Lopez, F. Mejia-Lira and J.N. Sanchez), Plenum Press, New York and London, 1992. 25. A. Finel, V. Mazauric and F. Ducastelle, Phys. Rev. Lett., 1990, 65, 1016. 26. A: Bieber and F. Gautier, Acta Metall., 1986, 34, 229l. 27. M. Sluiter, P. Turchi and D. de Fontaine, J. Physics F: Met. Phys., 1987, 17, 2163. 28. S. Pei, T.B. Massalski, W.M. Temmerman, P.A. Sterne and G.M. Stocks, Phys. Rev., 1989, B39, 5767. 29. J.P. Landesman, G. Treglia, P. Turchi and F. Ducastelle, Journal de Physique, 1983,46, 1001; D.H. Le, C. Colinet and A. Pasturel, Physica B, 1991, 168, 285. 30. C.H. De Novion andJ.P. Landesman, Pure and Applied Chem., 1985,57, 1391; Nonstoichiometric Compounds; Advances in Ceramics, Vol. 23, The American Ceramic Society, 1987. 3l. S.H. Wei, L.G. Ferreira and A. Zunger: Phys. Rev., 1990, B41, 8240. 32. A. Bieber and F. Gautier, J. Magnetism and Magnetic Mater., 1991, 99, 293. 33. Z.W. Lu, S.H. Wei, A. Zunger, S. Frota-Pessoa and L.G. Ferreira, Phys. Rev., 1991, B44, 512, and references therein. 34. F. Solal, R. Caudron, F. Ducastelle, A. Finel and A. Loiseau, Phys. Rev. lett., 1987,58,2245. 35. E. Cabet and A. Loiseau, private communications and to be published. 36. L. Reinhard, B. Schonfeld, G. Kostorz and W. Biihrer, Phys. Rev., 1990, B41, 1727; P.E.A. Turchi, M. Sluiter, F.J. Pinski, D.D. Johnson, D.M. Nicholson, G.M. Stocks and J.B. Staunton, Phys. Rev. Lett., 1991, 67, 1779. 37. B. Schonfeld, J. Traube and G. Kostorz, Phys. Rev., 1992, B45, 613. 38. R. Caudron, M. Sarfati, M. Barrachin, A. Finel, F. Ducastelle and F. Solal, to be published in Journal de Physique, 1992. 39. M. Sluiter, P. Turchi, F. Zezhong and D. de Fontaine, Phys. Rev. Lett., 1988, 60, 716. 40. N.C. Tso, M. Kosugi and J.M. Sanchez, Acta Metall., 1989, 37, 12l. 4l. D.D. Johnson, P.E.A. Turchi, M. Sluiter, D.M. Nicholson, F.J. Pinski and G.M. Stocks, Mater. Res. Soc. Symp. Proc., 1991, 186, 21; A. Gonis, P.E.A. Turchi, M. Sluiter, F.J. Pinski and D.D. Johnson, Mater. Res. Soc. Symp. Proc., 1991, 186, 89. 42. D.H. Le, C. Colinet, P. Hicter and A. Pasturel, J. Phys: Condensed Matter, 1991, 3, 7895 and 9965. 43. A. Pasturel, C. Colinet, A.T. Paxton and M. Van Schilfgaarde, J. Phys: Condensed Matter, 1992,4,945; C. Colinet, P. Hicter and A. Pasturel, Phys. Rev., B45, 1571; see also M. Sluiter, P.E.A. Turchi, F.J. Pinski and G.M. Stocks, to be published in J. Phase Equilibria.

6

Point, Line and Planar Defects A.T. PAXTON SRI International, 333 Ravenswood Avenue, Menlo Park, CA 94025, USA

6.1 INTRODUCTION This chapter attempts to bring together the theory of alloys, as seen from the density functional viewpoint, and practical problems in physical metallurgy. These two are separated by so wide a gap that many metallurgists take the view that quantum mechanics can play no part in the design of alloys. It is becoming clear though that while many properties of metals are amenable to a classical description (see chapter 7), when it comes to intermetallic compounds (in which there is intense interest at present) some of their most devastating properties such as intrinsic brittleness are a consequence of their electronic structure.1,2,3,4 Therefore, like it or not, alloy designers may be forced to take a quantum mechanical viewpoint in order to understand the behaviour of their materials on a fundamental level. The purpose of the present volume is to make this viewpoint more accessible. In the past two decades it has become possible to calculate from first principles, and with striking accuracy, the structural properties of metals. This is because we have learned how to find numerical solutions to Schrodinger's equation, the quantum mechanical equation of motion, in a way in which all the errors can be controlled, while the only approximation remaining is the local ansate; to the electron-electron exchange and correlation (see Chapter 2 and section 6.2.2.1 below). This is still so expensive in terms of computer resources that we are reduced to solving problems involving at most a hundred or so atoms with imposed periodic boundary conditions. It is not surprising, therefore, that these advances are greeted with some scepticism by practising alloy designers. It is however clear from the literature that theoretical work is being taken very seriously, although there is considerable confusion surrounding the plethora of methods employed. These range from the wholly empirical and essentially non quantum mechanical approaches such as the embedded atom method, to those density functional schemes that involve rather un158

Point, Line and Planar Defects

159

controllable shape approximations, and finally to the essentially exact solution to the local density functional equations. The title of this chapter evidently encompasses the entire field of physical metallurgy, even when restricted to aspects involving electron theory. Rather arbitrarily, I have decided to discuss two subjects in detail, which are presented in two separate sections which can be read quite independently of each other. Some overall conclusions will follow. The first topic is the structure and energy of planar faults in intermetallic compounds having the Ll2 crystal structure. These arise from the dissociation of superdislocations of the ordered lattice, and it is the nature of these dissociations-the extended core structure of dislocations-which determines the deformation behaviour. Fault energies appear as parameters in most theories of the yield behaviour of intermetallics, and while a few have been measured these are largely unknown quantities. The zero temperature structure and energy of planar faults are accessible within current implementations of density functional theory and I will describe some new calculations designed to demonstrate this. The second part of this chapter examines point defect dislocation interactions in the context of solution hardening of metals and alloys. There are few, ifany, accurate quantum mechanical calculations of the forces between dislocations and point defects while there is some evidence that not all experimental data can be reconciled within classical models, particularly the hardening of intermetallics. The purpose of this second section is to examine this in some detail with the aim of motivating future calculations in this area.

6.2 PLANAR FAULTS

IN L12 INTERMETALLICS

Some early high temperature superalloys were single crystal Ni3AI or twophase alloys containing a high volume fraction of the famous 'gamma-prime' Ni3Al phase. The yield stress in Ni3AI increases with temperature in a range roughly between 80 K and 500-1000 K. This anomalousjlow has now been found in a number of metals, particularly intermetallic compounds." Many intermetallics are now the subject of active research as they show promise of even better high temperature properties than the original nickel and cobalt based superalloys." Very few intermetallics, however, have the same good room temperature ductility as single crystalline Ni3AI, and it has become important to understand the mechanisms of deformation and fracture at a fundamental, microscopic level. The idea is that when this has been achieved it will be possible to design new, lighter alloys with both room temperature ductility and increasing strength with temperature

extending as far as possible to the melting

point. The question we are addressing in this book is to what extent electron theory has a role to play in alloy design. We intend to give a very practical example of this in what follows. The local

160

Electron Theory in Alloy Design

density approximation will be applied to the problem of calculating planar fault energies in three L12 intermetallic compounds: Ni3AI, CU3Au and Pt3Al. Special attention will be paid to giving details of the calculation and estimating errors. This will assist the metallurgist in assessing the validity of reported calculations, who should be aware, for example, of the errors involved in the shape approximations inherent in methods such as 'LMTO-ASA' and 'KKR'. 7 It is essential, also, to ask questions about details such as convergence of the energy with respect to Brillouin zone integration, basis-set size and size of supercells, if these are not given. We will not attempt to review the very active current debate surrounding the mechanisms of deformation in intermetallics, and in particular anomalous yield. Instead the reader is referred to some recent reviews. Vessyiere" has described the field from an electron microscopy point of view and reviewed the theory up to 1989. Dimiduk9 has given the most recent critique of theories of anomalous yield, while Hirsch'" makes a more quantitative evaluation and advances a new theory." A very readable account by Hazzledine and Sun has also recently appeared." Let us now recall some important facts that will motivate our calculations. The L12 crystal structure is adopted by a number ofintermetallics with A3B composition. It is an ordered structure based on an fcc lattice in which the B atoms occupy one simple cubic sublattice, and the A atoms the other three. In other words, the B atoms occupy the corners and the A atoms the face centres. In this arrangement, there are no B-B first neighbour pairs. Some L12 intermetallics, for example Ni3AI and CU3Au, show anomalous yield;12,13others, like Pt3AI, do not.12,14The flow stress T in the latter metals is 'normal' as a function of temperature T, in that it decreases monotonically. In the anomalous alloys, there are three regions in the T( T) curve that have been identified.Y' At low temperatures, 'region I,' the behaviour is normal and dT/dT is negative. At intermediate temperatures up to a peak temperature Yp, in 'region II,' dT/d T is positive and the behaviour is anomalous. Above Yp, in 'region III,' the flow stress decreases once again with temperature. The extended core structures of dislocations (i.e., their dissociation reactions) are central to all theories of the yield anomaly. It is observed.' that in regions I and II the active slip system is (110){Ill}. In region I, the superlattice dislocations are observed dissociated into superlattice Shockley partials separated by a superlattice intrinsic stacking fault (SISF). In region II, the core structures are different: the superdislocations dissociate into perfect dislocations of the fcc lattice separated by an antiphase boundary (APB). These in turn may dissociate into Shockley partials bounding a complex stacking fault (CSF). Details of these reactions are given below in equations 6.1-6.3. In region III the slip system changes to cube slip on {001} planes. It is not yet known what causes the yield anomaly, but all theories agree that the extended structure of dislocations is important. These depend, of course, on the energies of the bounding planar fault: the change from SISF to APB

Point, Line and Planar Defects

161

coupled partials may reflect a decrease in the APB energy with temperature as will be discussed later. It has been suggested that an unstable CSF may be associated with normal yield in Pt3AI,15 and we will have something to say about this too. 6.2.1

STRUCTURE

COMPARISONS AND SYMMETRY RELATED

PROPERTIES

We have indicated the importance of planar faults in the Ll j lattice in theories of the deformation of high temperature alloys. Of central importance are four defects: APB on {Ill} and {OOI} planes, CSF and SISF on {Ill} planes. The calculations to be presented here concern defects on {Ill} planes. To begin with, we will describe the atomic structure of these defects. A great deal can be learned before doing any computer calculations simply from examining this and the accompanying symmetry, and a very thorough treatment has been given by Yamaguchi et al.16 The close-packed fcc and hcp structures are characterised by ... ABCABC ... and ... ABABAB. . . stacking of {Ill} planes. In As B alloys, the L12 and D019 structures are the cubic and hexagonal derivatives of these when every fourth atom in the planes is of a different type, and these are arranged in a triangular net in order to stay as far apart as possible (see Fig. 6.1 a). All the defects we are concerned with here are 'translation defects' which can be constructed notionally by making an infinite, flat cut parallel to a crystallographic plane and displacing the semi-infinite crystal on one side of the cut with respect to the other by a vector p, the 'fault vector.' After making the displacement, we allow the two halves to stick back together again, and under the action of interatomic forces a further displacement may occur which we will refer to as the 'relaxation vector' T. 6.2.1.1 The antiphase boundary Antiphase boundaries occur in all ordered crystals that allow the formation of crystallographically equivalent domains. Imagine a disordered fcc crystal of A3B composition beginning to order in different regions which then grow together as the whole crystal slowly orders into the L 12 structure. The fcc structure comprises four interpenetrating simple cubic lattices and the B atoms must decide which of these to occupy so that the A atoms can occupy the other three. All four are equivalent, so four possible domains can form and when these grow together their boundaries will form a network in the crystal. Three points are noteworthy for us. (i) The APB can exist in any orientation. (ii) The fault vector p is of t(110) type, which is an fcc lattice vector, so that if all the atoms were the same, or the crystal disordered, the APB would not be a fault. (iii) If the APB lies parallel to {Ill} , there will be B- B neighbours across the boundary; ifit lies in {OOI}the first neighbour arrangements will be unaffected. This has led to the proposal" that the APB energy will be highly anisotropic in crystals in which B-B pairs are unfavoured (i.e., in strongly ordering systems) although recent measurements in Ni3AI contradict this supposition."

162

Electron Theory in Alloy Design

(b)

Fig. 6.1 Close-packed planes in A3B alloys. (a) Triangular arrangement of the minority atoms showing A, Band C stacking positions .... ABC ... stacking leads to the cubic L12 structure; ... AB ... and ... ABAC ... to the hexagonal D019 and D024 structures. Fault vectors a, C and S of the APB, CSF and SISF in L12 are shown; as well as two unit cell vectors in the fault plane: c, = [011] and C2 = [110]. (b) Rectangular arrangement of minority atoms found in the close-packed (112) planes of the tetragonal D022 structure, which is obtained from the stacking sequence ... AbCaBc ... as shown.

The (001) antiphase boundary has a t[IIO] fault vector which corresponds locally to the arrangement found in the D022 structure which is a tetragonal lattice made out of two L12 cubes on top of each other displaced by t[IIO] (see Fig. 6.1b). By symmetry, 16 the only possible relaxation vector is t[OO I], in other words an expansion at the boundary. No in-plane relaxation can occur. The scalar t measures the amount of relaxation. The (111) antiphase boundary has a t[OII] fault vector which brings two B atoms into neighbouring positions. Apart from an expansion normal to (Ill) the only allowed relaxation vector is16 it[211], which is perpendicular to the fault vector. The APB also arises from the splitting of a superlattice dislocation into 'superpartials,' which are perfect fcc dislocations but partial dislocations of the L 12 structure, in the reaction [011]

--* t[OI 1] + APB + t[OI 1]

(6.1 )

This can in principle occur on any plane although usually only {Ill} and {OOI} APBs are observed. In contrast to stacking faults, however, the APB is also a grown-in defect and need not be associated with dislocation dissociation.

6.2.1.2 'Superlattice intrinsic' and' complex' stacking faults If all the atoms were the same then the SISF and CSF would become the familiar fcc intrinsic stacking fault on {Ill} in which locally the stacking becomes hcp. The same effect can be produced with a fault vector of type 112). Only three of the six variants in the fault plane produce stacking faults. These are called the 'positive intrinsic fault vectors'; the other three result in

i
O. J2 can have either sign, but polytypes are only observed if J2 < 0, in fact the condition JI/J2 = - 2 leads to a state of 'multiphase degeneracy' whereby an infinite number of stacking sequences have the same energy. This is what leads to polytypism. LDA calculations" show that this condition is very nearly met in SiC which indeed displays a wide variety of poly types. In Si, Ge and AI, J2 is negative but more than five times smaller than Jl.20,23 In these cases polytypes could only arise, if at all, at finite temperatures. In the case of AsB intermetallics, the condition J2 < 0 is the same as b24 < b19' D024 is closer to L12 in energy than DOl9 in Pt3AI, CUsAu and the transition metal trialuminides," but it is not usually so close for J2 to be negative. Surprisingly, in Ni3AI J2 is greater than JI as a consequence of the very low value of b19• In this regard Ni3AI is anomalous compared to other L12

t

Point, Line and Planar Defects

177

intermetallics. Because 12 > 0 we do not expect (Ill) polytypes in intermetallies even at finite temperatures. We can still say nothing about (001) stacking, although this is important, since cross-slip of screw dislocations on to cube planes results in the formation of (001) APBs. We note from Table 6.5 that the normal alloy PtsAI has both b22 and b19 much larger than the anomalous alloys, which is in confirmation of a very early speculation.V Application of the ANNNI model on (001) will prove very interesting. It will be possible to make the connection between (001) APB energy and energy differences between D022 and D02s, and L12. We also expect 12 < 0 at least in Als Ti and CUsAu which show a rich variety of (001) polytypes in their phase diagrams. In these cases the study would become quite involved: we would expect 12 ~ - tll in which case higher 1n would also be needed to converge the total energy expansion (equation (6.4)). 6.3 POINT

DEFECT

DISLOCATION

INTERACTIONS

Density functional theory has not yet intruded into the physical metallurgy of point and line defects except for a few calculations of their atomic structure. Both these defects have long ranged strain fields so that convergence with respect to supercell size becomes a particular problem. There have been a few local density functional calculations of the structure and formation energies of vacancies in A1.2s,5sThe object of the following sections will be to review the point defect dislocation interaction in its role in the hardening of metals with a view to stimulating future quantum mechanical calculations. We will begin with the classical theory of solution hardening. 6.3.1

CLASSICAL THEORY

OF SOLUTION

HARDENING

6.3.1.1 The form oj theflow stress Let us consider the hardening of a metal due to the presence of dissolved impurity atoms. In the classical theories.i" these are considered to be distributed randomly through the crystal and to provide obstacles to the glide of dislocations. These theories are known as 'fixed discrete obstacle models.' Under an applied shear stress L the force on a dislocation whose Burgers vector has a magnitude b will be tb per unit length, but extra forces will arise due to the presence of obstacles in the glide plane having a concentration c per unit area. The force arising from the interaction between the dislocation and obstacle will have a range wand a maximum valuefm. Iffm is taken as positive, then the dislocation encounters c attractive potential wells per unit area, of width 2w and depth approximately wfm. The line tension (energy per unit length) of the dislocation will be denoted T. The critical, extra shear stress (flow stress) Lc' over and above the friction stress of the lattice, to move the dislocation through

178

Electron Theory in Alloy Design

the field of obstacles in the glide plane can only depend on these quantities, and a dimensional analysisf shows that (6.8) where ff is a function of its two dimensionless arguments. Two major problems arise in developing a theory: a determination of the concentration dependence of LO and a measure oflm that can be related to experimental quantities. In this section we will consider the first problem assuming that 1m is given. We will see how different models can give rise to different concentration dependencies. In the following section, we will return to the physical interpretation oflm. A very simple model for the hardening can be worked out as follows.i'' If T ~ 1m so that the dislocation remains almost straight, then it will interact with obstacles an average distance Lo apart. Lo is easily found by noting that an area ofwidth 2w and length Lo will contain one obstacle so that 2wLoc = 1. To drive the dislocation through the field of obstacles requires an applied force per unit length sufficient to overcome forces I; from obstacles distance Lo apart. The balance of forces requires LcbLo = I; so that (6.9) This 'almost straight dislocation model' can also be seen to result directly from equation (6.8). Since 1m / T is very small it can be neglected. If c is also small enough that IF, which now depends only on this, can be linearised, then we obtain equation (6.9) apart from the prcfactor 2. This model is important firstly as an easily worked example, and secondly because in common with early models.i" it predicts a linear increase of flow stress with concentration. This, and the linear dependence on I; is also often in agreement with experiment as we will see below (Table 6.6 and Figs. 6.5 and 6.6). We can obtain another limit from equation (6.8), namely the 'point obstacle model' where we let w ~ O. In thise case :F depends only on Im/T. This model has been developed by Friedel,58,56and is illustrated in Fig. 6.4a. The length of dislocation considered to move independently is just that segment pivoted between two obstacles and breaking away from another in between these two pinning points. In steady state, the dislocation moves through the field of obstacles bulging between them and breaking away before encountering further obstacles. Under these circumstances, Friedel finds

wJ

(6.10)

PointJ Line and Planar Defects

•

•

•

•

•

0

•

•

•

•

•

•

•

D

179

•

• •

•

(a)

0

0

0

o~--------------2L------------~ (b) Fig. 6.4 The Friedel and Labusch limits in solid solution hardening theory. (a) The dislocation interacts with point obstacles separated by an average distance L along the dislocation line. In one event, the dislocation pinned at A and C, unpins from obstacle Band is pinned again at D. Only the segment of line between points A and C move in this event. (b) In the Labusch limit, a segment of dislocation of length 2L moves as a unit through a large number of diffuse obstacles. After Nabarro."

This displays the well-known square root dependence on concentration." which was also found in computer simulations'" in which the obstacles were treated as mathematical points. If the range of point defect dislocation interaction or the concentration are relatively large, whereas the line tension is not large compared to L, then the dislocation will take on a wavy shape as it lies in the long ranged field of many obstacles. This is illustrated in Fig. 6.4b where, in contrast to the point obstacle model, a segment moving approximately independently of the rest of the line interacts with a quite large number of obstacles. This 'diffuse obstacle model,' in which IF depends strongly on both its arguments, bears a close analogy with the related problems of pinning of domain walls in ferromagnets, and flux-line pinning in type II superconductors which was solved by Labusch and applied to the question of solution hardening." In the solution given by Nabarro'" the

180

Electron Theory in Alloy Design

result is 7: c

(Wi: C2)1/3 2

b = ~

T

(6.11)

While the theory of solution hardening is complex, it can be seen to reduce essentially to the knowledge of two quantities: w.Jc andimlT. In fact these are sometimes combined into the single dimensionless parametcrf'r"

and Nabarro'" shows that the Friedel limit (equation (6.10)) is valid only if f3 ~ 1 while the Labusch limit (equation (6.11)) applies only if f3 ~ 1. So far the theory is purely classical whereas it is clear that w andfm are microscopic quantities, intimately connected with the interatomic forces in the solid, and therefore ultimately accessible only through the application of electron theory. There is however an ingenious classical interpreation of I; which we will describe now. 6.3.1.2 The point defect dislocation interaction Before looking at some experimental data, we will turn to the question of interpreting im' the maximum force exerted between a point defect and a dislocation. This would seem to be an exemplary area for the application of electron theory and atomistic simulation. However, I know of only two applications of quantum mechanics to this problem.62,63,64 In both these, numerous difficulties in theory meant that no unequivocal conclusions were reached. There is however a well developed classical theory which is very clearly explained by Fleischer59 so we need only quote the salient details. Three mechanisms of interactions have been identified: elastic, electronic and chemical. Chemical interactions arises through the migration ofsolute atoms to stacking faults which lowers the stacking fault energy locally. An extra stress must then be applied to move the dissociated dislocation into a region of crystal where the stacking fault energy is higher. This is known as 'Suzuki locking'. In the present discussion, treating dislocations as undissociated, we will not consider Suzuki locking quantitatively. The electronic interaction is important to us since it is quantum mechanical in origin and hence related to electron theory. Here an interaction arises between a charged point defect and the electric dipole of an edge dislocation. The charge on a point defect arises as the host electron gas responds to screen the charge of the solvent nucleus if its valence is different from the host metal. An edge dislocation has a region of compression next to one of tension; because of the volume dependence of the kinetic energy of an electron gas, electrons in the compressed region will have

Point, Line and Planar Defects

181

their energy levels raised with respect to those in the region of tension and will flow into these regions. One can say that electrons are squeezed from the region of compression to that of tension, setting up a charge dipole. The electronic interaction between solute atom and edge dislocation has been estimated'" but found to be rather smaller than the classical elastic interaction. But it was able to account for the 'valence' effect in which solutes with similar atomic radius to the host still act as potent hardeners. This effect is also accounted for, though, by including the modulus interaction (see next paragraph) so that the electronic effect is not thought to be important nowadays." The purely elastic interaction between solute atoms and dislocations divides into two contributions, the size misfit and the modulus interactions. If the solute atom has a different size from the host atoms its dilatational elastic field will interact with the elastic field of a dislocation so that the energy needed to put the solute into the host crystal will be affected by the presence of a dislocation.* There will also be an interaction from the difference in elastic moduli of the impurity and host. This is modelled by treating. the impurity atom as an inclusion having differing elastic constants from the surrounding matrix. Interactions of both sorts are worked out in detail by Fleischer'" in linear isotropic elasticity for both edge and screw dislocations. The only non vanishing interaction to first order in b is the size misfit interaction with edge dislocations. All others have their leading terms in second order. In all four cases, the interaction energy depends linearly on size or modulus misfit parameters, which need to be carefully defined. The size misfit C,b' the fractional change in atomic radius between solute and solvent atoms, is conventionally related to the bulk property

(6.12)

in which the lattice parameter change is measured as a function of concentration x. (Note that x ~ cb2.) The modulus misfit parameter c,~ is rather more equivocal since it is anyway hard to envisage the elastic properties of a single impurity atom in a metal. However, in analogy with the size misfit, one may relate e~ to the measured change in shear modulus G with concentration

1

dGI

G dx x=o

(6.13)

"If the point defect has also a deviatoric component of strain the hardening will be much greater, the interaction being to first order with both screws and edges." Here we will only be concerned with 'gradual hardeners' 59 whose strain fields are purely dilatational.

182

Electron Theory in Alloy Design

Fleischer'" obtains (6.14 ) where the absolute value neatly takes care of cases where the solute is both 'harder' or 'softer' than the matrix. In summary, the maximum forcefm between a point defect and a dislocation in classical linear elastic theory has two contributions (from size and modulus misfit) each being different for edge or screw dislocations. The strongest of these is the (first order) size misfit interaction with edge dislocations. The forces depend linearly on the misfit parameters and combining the size and modulus contributions leads to a generalised misfit parameter e for which (6.15) where a ~ 16 for edges and a ~ 3 for screws.i" The forces decay with the inverse cube of the distance between dislocation and point defect. The factor 16 arises from the stronger size misfit interaction for edges, and the factor 3 was found by Fleischer66 by plotting d'tcldxlx=o against lee - ae, I and varying a until the data fell on a straight line (see Fig. 6.5). These data were for the hardening of Cu by a number of solutes, the conclusion being that since the observed IX was less than 16, hardening was due to interaction with screw dislocations the size and modulus effects being roughly equally important. 6.3.1.3 Assessment of the classical theory As we have already mentioned in the last paragraph, one may relate the classical theory to experiment in two ways. Firstly, for a given impurity in a given host, one may measure the concentration dependence of the flow stress. Secondly, amassing data from a number of impurities in the same host, one can plot the rate of hardening against a combined size and modulus parameter e in order to assess further the mechanism of hardening. Some data from the literature are shown in Table 6.6. Although the high temperature data indicate adherence to the Labusch limit, we will see in a moment that this is less clear than it seems. The low temperature data favour both c and c dependence: in fact systems as similar as Al in Cu and Al in Ag differ in this respect. While it may be that the data could fit equally well to either relationship, it seems the Friedel limit applies in some instances, while in others the range of the point defect dislocation interaction is larger. This is not surprising since the range of interaction has to be less than b for the Friedel statistics to apply. The Friedel limit implies, in other words, that the point defect dislocation interaction is significant only when the solute atom lies within the dislocation core and vanishes even if the atom lies one lattice spacing away. .J

Point, Line and Planar Defects

183

160

80

~P'0 ~

40

!

20

0

Ni

10 0.02

Zn

0.04

0.08 Eb

0.16

160

80

0.8 Es = (3Eb-EO')

1.6

3.2

80

~Pro ~ ~ ~ ~

40

~ ~ 20

10

3.2

160

~P~

1.6

(b)

(a)

'0

0.8 Ea'

0.4

0.32

40

20

0.4

0.8

1.6

EE = 116Eb-EO'1 (c)

3.2

6.4

10

0.2

0.4

(d)

Fig. 6.5 The rate of hardening by solutes in Cu plotted against various misfit parameters. 't is half the tensile flow stress or lower yield stress at room temperature, x is the atom fraction of solute. (a) A plot against size misfit alone shows no correlation of the data. (b) A slightly better correlation is shown in a plot against the modulus misfit parameter (equation (6.14)). (c) and (d) show plots against the combined size and modulus misfit parameters (equation (6.15)) for edges and screws. (c) shows the best fit obtainable for a value of e appropriate to edges. (d) shows the best fit available for ~ ~ 16 which show the hardening to be due to interactions with screws and that ~ = 3 is appropriate for screw dislocations in equation (6.15). After Fleischer.66

There are some serious objections to the classical theory which we will indicate briefly now. The classical approach to the interaction forcefm has the usual weaknesses associated with linear elastic theory applied near the cores of dislocations. Clearly the theory is invalid when the defect approaches within

184

Electron Theory in Alloy Design

Point, Line and Planar Defects

185

the core radius, but this is precisely the situation in the Friedel limit and ~ hardening. This will remain a difficulty of Fleischer's theory until the point defect dislocation interaction can be properly calculated in atomistic simulations. It is paradoxical that at distances at which the linear theory may begin to apply, the interaction energy will become vanishingly small due to its rapid decay with distance. Some serious objections have also been raised by Kocks.71 Firstly, he points out that experiments tend to show either the critical resolved shear stress or hardening increment in single crystals, neither of which are as relevant as the lower yield stress to the theory of solution hardening. He also questions the validity of the fixed discrete obstacle model itself, on which all the above theories are based. Kocks points out that the characteristic shape of yield stress against temperature in all crystals shows a rapid drop followed by a plateau of flow stress Lp (possibly with a hump in it). Presumably the rapid drop with temperature in the fixed discrete obstacle model is associated with thermally assisted breakaway from the obstacles. There must come a temperature at which fixed impurities no longer act as obstacles to dislocations. Writing the total interaction energy as wfm with w ~ 3b, in the Friedel limit Kocks finds the critical temperature above which the obstacles can no longer contribute to hardening since dislocations can be thermally assisted through the obstacle field. This temperature is much lower than the temperature at which the plateau begins and becomes yet lower as to ~ 0 as it should in the Friedel limit. Kocks enumerates a number of other objections to the fixed discrete obstacle model, advocating instead a 'trough model' in which a dislocation is pinned along its length in a trough of low line energy T and escapes from the trough under applied stress by nucleating a bulge which then expands along the dislocation line. Thus the dislocation moves jerkily between troughs, explaining a number of experimental phenomena (the yield plateau and Portevin-Le Chetalier effect), although the nature of the troughs is somewhat undefined. While the high-temperature data in Table 6.6 seems to confirm the Labusch statistics at first sight, in light of the above remarks, it seems more likely that the fixed discrete obstacle model cannot account for the hardening, instead continual locking and unlocking associated with rapid solute diffusion is occurring. This is confirmed by the observation that the concentration dependence of hardening by Al in Cu is different in the two temperature regimes indicating that two mechanisms are at play. 6.3.2

THE NEED TO GO BEYOND

THE CLASSICAL APPROACH

Notable deviations from classical behaviour have now been reported." especially in the hardening of ordered intermetallic alloys which are the primary subject of this chapter. Maybe it is not surprising that a theory concerning undissociated dislocations should fail in alloys in which the core structure of dislocations is extended due to complex dissociation reactions. Furthermore, it has been

186

Electron Theory in Alloy Design

2

4

4

5

EE

2 EE

(a)

(b)

3

4

Zr: 8.5 Hf: 7.5

Fig. 6.6 The rate of hardening by B subgroup and transition metal solutes in Ni and Ni3Al. 1:' is half the tensile 0·2% flow stress at 77 K and x the atom fraction of solute. (a) and (b) After Mishima et al.,68 show the hardening rate for Ni and Ni3Al plotted against the generalised misfit parameter for screw dislocations (equation (6.16)). (c) After Shinoda et al.,63 shows the data for Ni from (a) plotted against the size misfit parameter (equation (6.l2)). Two interpretations of these data are possible: either the transition metal solutes provide an 'extra hardening, ,63 or Sb, Sn and In are the anomalous solutes.

suggested" that point defects in, say, Ni3Al may produce a tetragonal distortion of the lattice so as to produce a first order interaction with screw as well as edge dislocations: these are the 'rapid hardeners' of Fleischer's theory." This is an area in which quantum mechanical calculations would be useful in determining the structure of point defects. It has also been suggested recently that solutes interfere directly with the mechanism of anomalous yield in Ni3A1.74 Recently, experiments were done to determine the rate of hardening by substitutional solutes in Ni3AI and Ni.67,68 In both cases the 0·20/0flow stress at 77 K proved to depend linearly on concentration (exceptions were Ti, Nb and Ta in Ni-see Table 6.6). d'rcldxlx=o was then plotted against two generalised misfit parameters appropriate for edge and screw dislocations. These are not

Point, Line and Planar Defects

187

the parameters of Fleischer (Equation (6.15)) but those resulting from a more general discussion." and which are made to depend on the concentration and misfit dependence of the flow stress. When 7:c depends linearly on both, then the appropriate parameters are

for screws, and (6.16) for edges. The factors 3 and 16 transfer from Fleischer's theory, but the modulus misfit parameter e~ differs in detail " from eo (Equation (6.14)). The data were very scattered when plotted against the misfit parameter for screws, but when plotted against BE the data gathered nicely as shown in Figs. 6.6a and b. The data for Ni are shown again in Fig. 6.6c plotted against the size misfit parameter only. Firstly, note that a generalised misfit parameter can be found for which hardening is consistent with the almost straight dislocation model (Equation (6.9)) showing a linear dependence on both c and 1m. Secondly, transition metal hardeners exhibit enhanced hardening rates which the authors call the 'extra contribution to hardening.' Even with such an impressive gathering of experimental data, the interpretation is still by no means easy. Shinoda et al.63 pursued the notion of 'extra hardening' by transition metals by proposing a contribution from the effects of d-electron bonding. Although in a very simple quantum mechanical calculation they were able to reproduce the experimental effect in Ni very convincingly, once they refined their calculations somewhat'" they found to the contrary that there was no extra effect from transition metals that could explain the experimental results. In fact Fig. 6.6 admits an alternative interpretation that has not been considered. Rather than dividing the solutes into transition metal or B-subgroup hardeners, one may regard all the solutes as obeying one law except for Sb, Sn and In which, for some unknown reason, harden less effectively than the others. The data shown in Table 6.6, Figs. 6.5 and 6.6 and the preceding discussion illustrate the difficulties and uncertainties associated with the theory of hardening. Although until now attempts to make quantum mechanical calculations of the point defect dislocation interaction have failed, they have highlighted the obvious need for such an approach to be pursued. We have therefore made a quite detailed excursus into classical theory here in order to identify precisely which quantities of interest in the theory of solution hardening need to be calculated by means of electron theory. We intend that this should stimulate future work in this area.

188

Electron Theory in Alloy Design

6.4 CONCLUDING

REMARKS

I have presented two examples in which electron theory can make a contribution in physical metallurgy. The theory of solid solution hardening is a classical one, and still controversial. 71 I t is not disputed though that a central parameter is the energy of interaction between point defects and dislocations as a function of their separation. This quantity is a prime candidate for investigation by atomistic simulation, since although an ingenious classical theory exists it relies on concepts such as the elastic constants of a single impurity atom and the structure of the dislocation core which are not well defined in continuum theory. Also, there is some evidence of deviation from classical behaviour as we have seen. Whether this can be directly attributed to quantum mechanical effects is not yet clear, as the calculations so far done are too simplified. However, they represent pioneering work and show the way that improved calculations can be done. The structure of point defects and dislocations and their interaction forces can now be calculated using methods described in this volume. Completely ab initio LDA calculations are at the moment too expensive and periodic boundary conditions are less appropriate in these applications. However, more approximate schemes such as empirical tight binding can be used; and indeed construction of these schemes can be aided by using appropriate LDA results.i'' The calculation of planar defect energies is of great importance in the theory of the deformation of intermetallics. It is also possible to do these calculations from first principles as we have seen. We have also seen that energy differences of interest in physical metallurgy are very small and that errors must be carefully controlled. In particular, we must avoid shape approximations to the total energy functional and take great care in testing supercell and k convergence. Once these are in hand, it is possible to make a sufficient number of calculations that trends in energies and translation states across a series of intermetallic compounds can be obtained. We expect that the calculations of planar fault energies that we have presented will provide useful benchmarks in the future and stimulate further work in due course. We have also seen that within the context of the ANNNI model, even such abstract quantities as the energies of metastable phases can be usefully and quantitatively worked into theories of yield. It will be very interesting to do calculations similar to the ones presented here for (Ill) stacking to study (001) polytypism in intermetallics. Not only would such work demonstrate the origin and size of the (001) APB energy in L12 alloys, it would also lead to understanding of those polytypes that are actually observed in intermetallics, namely (001) stacking polytypes. Another example of interesting calculations would be fault energies on basal and close-packed planes in D022 transition metal trialuminides which could be used to understand the brittleness of many of these phases."

Point) Line and Planar Defects

189

As much as anything it is hoped that this chapter has provided stimulus for further density functional calculations of use in the design of alloys. ACKNOWLEDGMENTS The results presented here are the consequence of a long collaboration with Michael Methfessel to whom I am grateful for generously sharing his thoughts and computer programs. Mark van Schilfgaarde provided his version of the full-potential programs suited to large supercells, and has given innumerable suggestions and advice. I am grateful to Adrian Sutton and Peter Gumbsch for telling me about homogeneous shear boundary conditions, and to Mike Finnis for a number of valuable suggestions during the preparation of this chapter. Also I would like to thank Arden Sher for his hospitality, encouragement and provision of facilities. Financial support has been provided under ONR contract NOOOl4-92-C-0006 and AFOSR contract F49620-88-K0009. REFERENCES 1. C.L. Fu, J. Mater. Res., 1990, 5, 971. 2. D. G. Pettifor, in Intermetallic compounds: Structure and mechanical properties, (ed. O. Izumi), Jpn Inst. Met., Sendai, 1991, p149. 3. D.G. Pettifor, in NATO ASlon Ordered intermetallics-physical metallurgy and mechanical behaviour, (ed. C.T. Liu and R.W. Cahn), Kluwer Academic Publishers, 1992. 4. A. T. Paxton and D.G. Pettifor, Scripta Metall., 1992, 26, 529. 5. P.M. Hazzledine and Y.Q Sun, Mater. Res. Soc. Symp. Proc., 1991, 213, 209. 6. R.L. Fleischer, D.M. Dimiduk and H.A. Lipsitt, Ann. Rev. Mater. Sci., 1989, 19, 23l. 7. M. van Schilfgaarde, A.T. Paxton, A. Pasturel and M. Methfessel, Mater. Res. Soc. Symp. Proc., 1990, 186, 107. 8. P. Veyssiere, Mater. Res. Soc. Symp. Proc., 1989, 133, 175. 9. D.M. Dimiduk, J. de Physique Ill, 1991, 1, 1025. 10. P.B. Hirsch, J. de Physique III, 1991, 1, 989. 11. P.B. Hirsch, Philosophical Magazine A, 1992, 65, 569. 12. D.-M. Wee and T. Suzuki, Trans. Japan Institute of Metals, 1979, 20, 634. 13. D.P. Pope, Philos. Mag., 1972, 25, 917. 14. F.E. Heredia, G. Tichy, D.P. Pope and V. Vitek, Acta Metall., 1989, 37, 2755. 15. V. Vitek, Y. Sodani and J. Cserti, Mater. Res. Soc. Symp. Proc., 1991, 213, 195. 16. M. Yamaguchi, V. Vitek and D.P. Pope, Philos. Mag. A, 1981,43, 1027. 17. P.A. Flinn, Trans. Metallurgical Society AIME, 1960, 218, 145. 18. J.W. Christian, The theory of transformations in metals and alloys, Pergamon, Oxford, 2nd. edition, 1975, chapter 6. 19. C. Cheng, RJ. Needs and V. Heine, J. Phys. C, 1988, 21, 1049. 20. P.].H. Denteneer and W. van Haeringen, J. Phys. C, 1987, 20, L883. 21. J .M. Ziman, Principles of the theory of solids, Cambridge University Press, Cambridge, 1964, p30 1ff. 22. J. Yeomans, Solid State Phys., 1988,41, 151. 23. P.J.H. Denteneer and ].M. Soler, Solid State Communications, 1991, 78, 857. 24. A.T. Paxton, in Atomistic simulation ofmaterials, V. Vitek and D.]. Srolovitz ed., Plenum, New York, 1989, p327. 25. A. Loiseau, G. van Tenderloo, R. Portier and F. Ducastelle, J. de Physique, 1985, 46, 595. 26. M. Methfessel, Phys. Rev. B, 1988, 38, 1537.

190 27~ 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74.

Electron Theory in Alloy Design M. Methfessel and M. van Schilfgaarde, unpublished. M. Methfessel, D. Hennig and M. Scheffler, Phys. Rev. B, to be published. R.J. Needs, R.M. Martin and O.H. Nielsen, Phys. Rev. B, 1986,33, 3778. A.T. Paxton, M. Methfessel and H.M. Polatoglou, Phys. Rev. B, 1990, 41, 8127. M. Methfessel and A.T. Paxton, Phys. Rev. B, 1989,40, 3616. M.P. Allen and D.J. Tildesley, Computer simulation ofliquids, Clarendon Press, Oxford, 1978, p246. C.L. Fu and M.H. Yoo, Mater. Res. Soc. Symp. Proc., 1989, 133, 81. T. Hong and A.J. Freeman, Phys. Rev. B, 1991,43, 6446. A.J. Freeman, T. Hong, W. Lin and J. XU, Mater. Res. Soc. Symp. Proc., 1991,213, 3. O.K. Andersen, Solid State Communications, 1973, 13, 133. O.K. Andersen, Phys. Rev. B, 1975, 12, 3060. A.R. Williams, J. Kubler and C.D. Gelatt Jr., Phys. Rev. B, 1979, 19, 6094. M.S. Hybertsen and S.G. Louie, Phys. Rev. Lett., 1985, 55, 1418. P. Bagno, O. Jepsen and O. Gunnarsson, Phys. Rev. B, 1989, 40, 1997. XJ. Kong, C.T. Chan, K.M. Ho and Y.Y. Ye, Phys. Rev. B, 1990,42, 9357. M. van Schilfgaarde, unpublished results. R. Jones and O. Gunnarsson, Rev. Mod. Phys., 1989,61, 689. A.T. Paxton, M. Methfessel and M. van Schi1fgaarde, to be published. V.L. Moruzzi,J.F.Janak and A.R. Williams, Calculated Electronic Properties of Metals, Pergammon, New York, 1978, p14. P. Veyssiere, J. Douin and P. Beauchamp, Philos. Mag. A, 1985, 51, 469. D. Dimiduk, to be published. N. Ba1uc, R. Schaublin and KJ. Hemker, Philos. Mag. Lett., 1991, 64, 327. M.A. Crimp and P.M. Hazzledine, Materials Research Society Symposium Proceedings, 1989, 133, 131. G. Tichy, V. Vitek and D.P. Pope, Philosophical Magazine A, 1986, 53, 467. D. Caillard, N. Clement, A. Couret, P. Lours and A. Coujou, Philosophical Magazine Letters, 1988, 58, 263. D.M. Dimiduk, Ph.D. Dissertation, Carnegie Mellon University, 1989. MJ. Gillan, J. Physics: Condensed Matter, 1989, 1, 689. A.J. Ardell, Me tall. Trans., 1985, 16A, 2131. R. Labusch, Acta Metall., 1972, 20, 917. F.R.N. Nabarro, Philos. Mag., 1977, 35, 613. A.H. Cottrell, Dislocations and plastic flow in crystals, Clarendon Press, Oxford, 1953, p 125ff. F.R.N. Nabarro, J. of the Less Common Metals, 1972, 28, 257. R.L. Fleischer, The strengthening of metals, ed. D. Peckner, Reinhold, London, 1964, p.93. AJ.E. Foreman and MJ. Makin, Philos. Mag., 1966, 14, 911. R. Labusch, Physica Status Solidi, 1970, 41, 659. D.M. Esterling, D.K. Som and A.K. Chatterjee, J. Physics F, 1978, 17, 109. T. Shinoda, K. Masuda-Jindo, Y. Mishima and T. Suzuki, Phys. Rev. B, 1987, 35, 2155. T. Shinoda, K. Masuda-Jindo and T. Suzuki, Mater. Res. Soc. Symp. Proc., 1991, 186, 283. A.H. Cottrell, S.C. Hunter and F.R.N. Nabarro, Philos. Mag., 1954,44, 1064. R.L. Fleischer, Acta Metall., 1963, 11, 203. Y. Mishima, S. Ochiai, N. Hamao, M. Yodogawa and T. Suzuki, Trans. Japan Institute of Metals, 1986, 27, 656. Y. Mishima, S. Ochiai, N. Hamao, M. Yodogawa and T. Suzuki, Trans. Japan Institute of Metals, 1986, 27, 648. D.L. Wood and J.H. Westbrook, Trans. Metall. Soc. A/ME, 1962,224, 1024. L.A. Gypen and A. Deruyttere, Scripta Metallurgica, 1981, 15, 815. V.F. Kocks, Metall. Trans., 1985, 16A, 2109. D.M. Dimiduk and S. Rao, Mater. Res. Soc. Symp. Proc., 1991, 213, 499. R.L. Fleischer, Acta Metall., 1962, 10, 835. F.E. Hereida and D.P. Pope, Journal de Physique III, 1991, 1, 1055.

7

Mechanical Properties of Metals at the Nanometre Scale A.P. SUTTON, J.B. PETHICA, H. RAFII-TABAR and J .A. NIEMINEN* Department

of Materials, Oxford University, OXl 3PH, UK *Department of Physics, Tampere University of Technology, P.O. Box 692, SF-33l0l Tampere, Finland

7.1 INTRODUCTION Surfaces are rarely atomically flat. When they are brought into contact so as to form a joint, or slid against each other as in friction, the area of real contact is generally a small fraction of the area of apparent contact. The mechanical properties of the asperities on the original surfaces affect the area of real contact and hence the adhesion and friction. This is one example where the mechanical properties of very small volumes of matter may affect macroscopic material behaviour. Another example is the response of an atomically sharp crack to an applied load. The energy balance controlling the movement of the crack includes the long range elastic field. But the mechanism by which the crack may relieve stress at the tip through blunting is concerned with the mechanical response of the very small, highly strained region right at the tip. For example the energy required to generate dislocations at the crack tip may be prohibitively high and the material fails in a brittle manner. In both of these examples a monolayer of impurities can dramatically influence the adhesion of the contact or the propensity for brittle failure. This is one of the most important unsolved problems of materials science. It has remained unsolved largely because no understanding exists of the mechanisms by which a layer of atomic thickness can influence the mechanical response of the much larger adjoining regrons. The motivation for understanding the mechanical response of very small, and sometimes highly strained, volumes of matter has, therefore, existed for a long time. But it is only in the last ten years or so that experimental and theoretical techniques have been developed to address such fundamental 191

192

Electron Theory in Alloy Design

questions of mechanical behaviour. One of the most significant developments has been the use of single tips of micron to nanometre dimensions interacting with flat surfaces to probe adhesion, sintering, friction, indentation and fracture. There is now an explosion of activity in the area of 'tip-surface' interactions which of course includes scanning tunnelling and atomic force microscopies. On the theoretical side most of the effort has concentrated on atomistic simulations using interatomic potentials to describe atomic interactions. While the length scales of the experiments have been decreasing those of the simulations have been increasing, thanks to enhanced computing power, and they are now directly comparable. This is one of the main reasons why this field is such a good testing ground for descriptions of interatomic forces and for various techniques of computer modelling. At the same time the usefulness of continuum descriptions of deformation is not at all obvious at such small length scales. By the same token the mechanisms of irreversible deformation and the limits on reversible deformation are by no means obvious in these ultra small volumes. For example, the role of dislocations is not clear since there may not be any in such small volumes and because their nucleation energy may be extremely high. We shall see that there are circumstances under which dislocations can be nucleated in these small volumes but most of the irreversible deformation is diffusional creep of a rather exotic kind. This conclusion could not have been reached without atomistic simulations and it is one of the main results of this work. Another key result of this work is the behaviour of a metallic tip under tensile loading. After the tip has been brought into contact with a substrate it is pulled off to determine the force of adhesion. It is found that the tip effectively melts under the tensile loading in the sense that rapid diffusion is seen and the profile of the tip is well described as that of a liquid drop under tension. This has enabled us to develop a simple model of the effect of a layer of impurities between the tip and the substrate on the force of adhesion. Contrary to the usual analysis of this problem based on the Yeung-Dupre equation, but in agreement with the simulations, we find that increased wetting of the tip on the substrate, due to a layer of impurities, decreases the force of adhesion. In comparing this result with experimental data we are immediately struck by the sparsity of single point contact measurements of the force of adhesion carried out under URV conditions. Although a variety of continuum based models has been developed.vP inspection of the experimental data shows remarkably little definitive material. The data of ref. [1] is the only clear cut description of the influence of monolayers in controlled, URV conditions, and it does seem to agree with our analysis. All the work to date on atomic force microscopy on tip-surface contact has been in non-URV conditions, and the forces and responses concerned are widely recognized as due to ill-defined contaminant films. Our simulations offrictional sliding of a tip on a substrate have shed light on the fundamental mechanisms of wear and lubrication by a solid film. We see

Mechanical Properties of Metals at the Nanometre Scale

193

that wear is caused by the tip adhering to the susbtrate so that sliding takes place within the tip rather than at the interface between the tip and the substrate. A film on the surface acts as a lubricant provided the shear strength of the interface between it and the tip is greater than the shear strength of the tip, and provided the compressive strength of the film is sufficient to prevent penetration of the film by the tip. Before we describe these simulations and relate them to relevant experiments we describe, in section 7.2, the interatomic forces that we have developed for these simulations and, in section 7.3, the application of molecular dynamics to studying interactions between tips and substrates.

7.2 INTERATOMIC SIMULATIONS

FORCES IN METALS FOR ATOMISTIC OF TIP-SURFACE INTERACTIONS

The models we shall use to describe atomic interactions in metals are dictated by our wish to consider thousands, even millions, of atoms in molecular dynamics. With current computing power it is possible to do this only by using interatomic potentials. N-body Finnis -Sinclair potentials/ and embedded atom poten tials" represent the current state of the art in the description of cohesion in metals for computer simulations involving up to a million atoms. In both approaches the total internal energy is represented by a cohesive functional of pair interactions and a predominantly repulsive pair potential. Using arguments based on the second moment approximation in tight binding theory, Finnis and Sinclair/ argued that the functional should be the square root (cf. Equation (4.84)). The functional gives rise to N-body interactions in the sense that the force exerted by one atom on another depends on the disposition of all neighbours to both atoms concerned. By contrast, a pair potential gives rise to a force that depends only on the separation of the two atoms concerned. The N-body nature of the potential leads to an increase in the bond strength when the coordination number is reduced, which is a fundamental feature of metallic bonding. This leads to a much more physically correct description of relaxation at free surfaces of metals than can be achieved with a pair potential. For example, it generally leads to an inward relaxation of the surface layer, in agreement with experimental observations of many transition metal surfaces, and in contrast to non-oscillatory pair potentials." Modelling the approach of an atomically sharp tip to a flat substrate, and subsequent adhesion and fracturing of the tip from the surface, demands a potential that can describe atomic environments ranging from 1 to 12 or more nearest neighbours. The potential should be capable of describing the long range van der Waals interactions when the tip is separated from the substrate. For these reasons we have developed Finnis-Sinclair potentials that have a long range of interaction.v'' The intention is to combine the superior description of short range interactions afforded by an N-body potential, to obtain a good

194

Electron Theory in Alloy Design

description of surface relaxation, with a van der Waals tail to give a correct description of long range interactions. Van der Waals interactions are normally represented by an attractive 1/r6 pair potential. The obvious method of introducing them into a Finnis-Sinclair description is therefore to add a negative l/r6 tail to the pair potential component. However, we prefer to keep the attractive contributions to the total energy separated from the repulsive contributions and therefore we write the total energy in the following Finnis-Sinclair form: (7.1 ) where V(r)

(7.2)

and p, l

=

L [~

J#i

rv"

J

m

(7.3)

Here, rv" is the separation between atoms i and j, c is a positive dimensionless parameter, e is a parameter with the dimensions of energy, a is a parameter with the dimensions of length, and m and n are positive integers, such that n > m. The pair potential, V, is purely repulsive and the N-body term is purely cohesive. Let p' denote the value of p, given by the sum in equation (7.3), for an atom belonging to a free surface. The cohesive contribution this atom makes to the total energy of the surface is then l = - efl. If an additional atom is now placed above the surface, and at a separation R from our surface atom, the value of l changes to p' + (a/R)m. For large vaues of R, compared with a, we may expand the square root:

ecJ

(7.4) Equation (7.4) shows that when an atom approaches the surface it interacts at large separations in a pairwise fashion, although the magnitude of this pair potential is affected by the number of neighbours to each surface atom. As the separation, R, decreases the expansion in equation (7.4) becomes increasingly invalid and the interaction changes smoothly to an N-body form. Thus, by choosing m = 6 in equation (7.3), we can achieve our objective of the long range interactions between two clusters of atoms being described by an attract-

Mechanical Properties of Metals at the Nanometre Scale

195

ive l/r6 pair potential, whereas at small separations the interaction is N-body in nature. Furthermore the transition between these two limits is smooth and continuous. We note that the physical origin of the van der Waals interaction is quite distinct from the unsaturated covalent bonding that gives rise to the N-body form of the energy at short range. In our potentials these two interactions have been combined into one term. We regard this as a convenient mathematical property of the potential, which should not be interpreted physically as meaning that van der Waals interactions become covalent bond interactions at short range. There are further advantages in choosing power laws in equations (7.2) and (7.3), which are directly analogous to the scaling properties of Lennard-Jones potentials." We may define e and a in equation (7.1) as our units of energy and length. For a given crystal structure the Finnis-Sinclair potential of equation (7.1) is defined completely by the exponents m and n. That is because the equilibrium condition for a particular crystal structure fixes the parameter c. If the potentials of two metals, with the same crystal structure, share the same values of the exponents, m and n, then the results obtained for one metal may be directly converted into results for the other simply by rescaling the units of energy and length. For example, we shall see that Cu and Ni are both described by the (m, n) = (6, 9) potential, while Pt and Au are both described by the (m, n) = (8, 10) potential. Therefore, a simulation performed with the Cu potential maybe translated directly into a simulation for the Ni potential simply by rescaling the units of length and energy. Moreover, the relative stabilities that our potential predicts for the fcc, bee and hcp crystal structures are also determined entirely by m and n. Let S[ denote the following lattice sum in a perfect fcc crystal

sf

=

~

[:;J

(7.5)

where the sum is taken over all atomic separations, rj' from an arbitrary atom. Setting af equal to the fcc lattice parameter then fixes our unit of length. The equilibrium condition is that the total energy of the crystal does not change to. first order when the lattice parameter is varied. This leads to c

(7.6)

The cohesive energy per atom is then given by

est 2m

[2n

- m

]

(7.7)

196

Electron Theory in Alloy Design

Table 7.1. Long-range Finnis-Sinclair potential parameters for fcc metals." The cohesive energy per atom, E!, in eV and the lattice parameter, ai, in Angstroms, are fitted exactly. Bi is the calculated bulk modulus in eVjA3, and the experimental value is in brackets. The calculated elastic constants Cll, Cl2 and CH are in eVjA3, and the experimental values are in brackets.

Ni eu Rh Pd Ag Ir Pt Au Ph Al

m

n

s (eV)

6 6 6 7 6 6 8 8 7 6

9 9 12 12 12 14 10 10 10 7

1·5707E-02 1·2382E-02 4·9371E-03 4· 1790E-03 2·5415E-03 2·4489E-03 1·9833E-02 1·2793E-02 5·5765E-03 3·3147E-02

ai

Ei (

39·432 39·432 144·41 108·27 144·41 334·94 34·408 34·408 45·778 16·399

and the bulk modulus,

Bi

Cll

4·44 3·52 1'22( 1·17) 1·44( 1·63) 3·50 3·61 0·89(0·89) 1·06( 1·10) 5·75 3·80 1·68( 1·68) 2·12(2·63) 3·94 3·89 1·25(1·22) 1·55(1·46) 2·96 4·09 0·69(0·68) 0·88(0·82) 6·93 3·84 2'28(2·31 ) 2·97(3·74) 5·86 3·92 1·73(1·80) 1·96(2·23) 3·78 4·08 0·99(1·03) 1·12(1·17) 2·04 4·95 0·26(0·26) 0·31 (0,29) 3·34 4·05 0·47(0·48) 0·51 (0·67)

s', is given Bf

CJ2

Cf4

1·11 (0·94) 0·81 (0,78) 1·45( 1·20) 1·10(1·10) 0·60(0·61) 1·94( 1·60) 1·61(1'59) 0·92(0·97) 0·24(0·24) 0'45(0·38)

0·50(0·82) 0·36(0·51 ) 0·89(1·21) 0·58(0,44) 0·37 (0·32) 1·32(1·68) 0'46(0·48) 0'26(0·28) 0·10(0·09) 0·10(0·18)

by (2n - m)n8S!

=

(7.8)

36Qf

where Qf is the atomic volume, (af)3/4. Using equations (7.7) and (7.8) we obtain the following relation between the bulk modulus and the cohesive energy

The pressure-volume equations (7.1-3,5,6

QfBf

nm

Eic

18

(7.9)

relation for the fcc crystal is also readily obtained from and 8)

p

6Bf

=

(2n -

m)

[[ v

]m1

Vo

6

[

/

v ]n 3

Vo

]

(7.10)

where v is the volume of a sample of the crystal, the volume of which at equilibrium is Vo' Bf is the bulk modulus at the equilibrium volume and is given by equation (7.8). Simple expressions for the elastic constants CII, Cl2 and C44 may also be derived. Elastic stability of the crystal requires C44 > 0, Cll > 0 and Cll > C12• It may be shown that these conditions are satisfied for a cubic crystal provided n > m. Table 7.1 contains the parameters 8, C, m and n that have been fitted to ten fcc metals. For each metal the experimental cohesive energy and crystal lattice parameter were fitted exactly. The indices m and n were restricted to integer values, such that mn is the nearest integer to I8Qf BfIE{ (see equation (7.9)).

Mechanical Properties of Metals at the Nanometre Scale

197

This meant that the bulk modulus was not fitted exactly, but by restricting m and n to integer values we are more likely to find the same values of m and n for different metals. In that case we will be able to take advantage of the scaling properties of the potential. We further restricted m to be greater than 5 because otherwise the potentials would have too large a range for computer simulations. The fitting procedure was as follows. For each value of m greater than 5, n was set equal to the nearest integer to 18n!BfIE:. Provided n was greater than m, e was then fitted to the cohesive energy, equation (7.7), and c was obtained from the equilibrium condition, equation (7.6). The bulk modulus, equation (7.8), was always found to be within a few percent of the experimental value. Several potentials with different values of m and/or n were sometimes found for each metal, although no potentials were found with m > 10. The selections appearing in the table were made on the grounds that they gave the closest agreement with the experimental elastic constants Cll, C12 and C44• Better agreement with the elastic constants could be achieved for Ni, Cu, Rh and Al by allowing m to be less than 6. For those metals the best agreement was obtained with (m, n) = (4, 13), (5, 11), (4, 18) and (5,9) respectively. We have also considered the relative stabilities of the fcc, bee and hcp crystal structures. It is well known" that the relative stabilities of these crystal structures are determined by higher moments of the local density of states than the second. Therefore, if our potential predicts that the stable crystal structure for a known fcc metal is indeed fcc then this may only be considered as fortuitous. Nevertheless, for the purposes of computer modelling using these potentials it is important to ascertain the relative stabilities that the potentials predict. For all potentials with 4 ~ m ~ 10 and m + 1 ~ n ~ 20 it is found" that the nearest neighbour spacing is always predicted to be smaller in the bee structure than in the fcc structure. It is also found that the cohesive energy of the bee structure is always predicted to be less than the fcc structure. This is why we have fitted only fcc metals in the table. For an hcp crystal it is necessary to minimize the cohesive energy, with respect to the atomic spacing, ah, in the basal plane and the cia ratio. It was found" for all potentials with 6 ~ m ~ 10 and m + 1 ~ n ~ 20, that Ech jE: differs from unity by less than 0·001, that a'[a! differs from 2-1/2 by less than 0·001, and that the cia ratio deviates from the ideal value, (8j3)1/2, by less than 0·001. Not surprisingly, therefore, the (Ill) intrinsic stacking fault energies in fcc are practically zero. As shown by Finnis and Sinclair," effective pair potentials may be defined from equation (7.1) by expanding the N-body term about a reference density. Our expansion in equation (7.4) stems from the same observation. Taking the reference density to be an ideal fcc crystal, and substituting equation (7.6) for the parameter c, we obtain the following effective pair potential

E;,

(7.11)

198

Electron Theory in Alloy Design

The effective pair potential is therefore a Lennard-Jones m-n potential. We believe this is the reason why the stacking fault energy is practically zero. As noted by Ackland et al.9 it is necessary to make the effective pair potential positive in the vicinity of the third neighbours in order to get a positive stacking fault energy. Ackland et ale achieved this by making suitable adjustments to the pair potential component of their fcc Finnis-Sinclair potentials. However, we note that although, for our potentials, the stacking fault energy is virtually zero, and the energy of the hcp phase is virtually the same as the energy of the fcc phase, there is still a very significant energy barrier to shearing one crystal structure into the other on the close packed planes. For many simulations, though not all, this is the only requirement for meaningful results. In reality, the stacking fault energy is determined by higher moments of the local density of states than the second, just like the fcc-hcp energy difference, and we should not be surprised that it is virtually zero in a second moment model. Rafii- Tabar and Sutton" showed how potentials for binary alloys of the ten fcc metals listed in the table may be constructed without any further parameter fitting. A consistent set of four rules was used to derive the parameters for interactions between unlike atoms from those given in the table for the elemental metals. Assuming an unrelaxed, random alloy model, the lattice parameters, elastic constants and mixing enthalpies throughout a range of compositions for all 45 binary combinations of the ten fcc metals were calculated to test the potentials.P Reasonable agreement with available experimental data was obtained. Equations (7.1-7.3) are readily generalized to describe binary A-B alloys by expressing the Hamiltonian in the following form .'Yl'

=

~ [ ~"'~ p;jJj

+

VAA (rij)

[ft; (1 - Pj)

+ Pj(l

dAA ~ ft; L~/jepAA

dBB

~

+

(1 - ft;)

(1 - Pi) (1 - Pj) vBB (rij) - Pi)] vAB (rij) ]

(rij)

+

(1 - Pj )epAB (rij)

L~i

(1 - h)epBB(rij)

+

J2

pjepAB(rij)J2 (7.12)

The site occupancy operators

_ {I

ft; -

Pi

are defined as follows

if site i is occupied by an A atom

0 if site i is occupied by a B atom

(7.13)

Mechanical Properties of Metals at the N anometre Scale The funcjtions VAA, VBB, VAB, VAA (r)

=

cpAA(r)

e

[

-

[

(r)

=

r

VBB

'

7 ' AAJmAA

=

q>BB

AAJrzAA

a

AA

q>AA,

and e

are defined as follows

q>AB

aBBJnBB

BB [

-

r

=

'

7 ' BBJmBB

cpBB(r)

=

[

199

({JAB

e

AB

[

aABJnAB r (7.14 )

-

(r) (7.15)

The constants dAA and dBB are defined as follows (7.16)

eAA, eAA, aAA, mAA and nAA are equated with the parameters e,

C, a, m and n of the pure A metal, given in the table. Similarly, eBB, eBB, aBB, mBB and nBB are equated with the parameters e, c, a, m and n of the pure B metal. Thus, only the four parameters eAB, aAB, mAB and nAB remain to be determined. These parameters are determined by assuming that the functions VAB and ({JAB may be expressed as follows

(7.17) These relations lead to the following expressions for the remaining parameters AB

m

nAB

AB

a

AB

e

AA

+

AA

+

!(m

!(n

BB

m

)

BB

n

)

~ ~

(7.18)

Thus, all parameters in the Hamiltonian, equation (7.12), may be obtained from the parameters for the pure metals given in the table. Our assumption that VAB and ({JAB are given by equation (7.17) was made on purely empirical grounds. For example, we found that setting eAB = (eAA + eBB) /2 led to very unsatisfactory predictions for the lattice parameters of the alloys. 7.3 THE APPLICATION OF MOLECULAR DYNAMICS TO SIMULATING INTERACTIONS BETWEEN TIPS AND SUBSTRATES The reader who is unfamiliar with molecular dynamics techniques may wish to study Ref 7 for an excellent introduction. In this section we shall focus on two

200

Electron Theory in Alloy Design

(a)

(b)

Fig. 7.1 (a) To illustrate the use of periodic boundary conditions for adhesion and fracture simulations. The emboldened cell is repeated in three orthogonal directions. (b) Schematic illustration of displaced periodic boundary conditions showing 9 simulation cells, for simulations of frictional sliding.

issues of central importance in our simulations, namely (a) how one controls the position of the tip relative to the substrate and (b) how one controls the temperature in the simulation. 7.3.1

CONTROLLING

THE POSITION

OF THE TIP

In our simulations we either bring a tip into contact with a substrate and then pull it off, or we slide the tip along the surface. An obvious, but rather crude, way of controlling the relative positions of the tip and substrate is to 'clamp' the positions of certain atoms in the tip and the substrate as far as possible from their region of interaction. By fixing the positions of the clamped atoms in the substrate and moving the clamped atoms in the tip rigidly the tip will be forced to move relative to the substrate. The disadvantage with this method is that there are artificial interfaces in both the tip and substrate between atoms that are clamped and atoms that are free to follow Newtonian dynamics. These interfaces reflect all thermal vibrations and mobile defects such as dislocations. Our solution to this problem is shown in Figs. 7.1a and b. We use periodic boundary conditions in all three directions and no atoms are clamped. In both Figs. 7.1a and 7.1b the tip is attached to the underside of a slab. By reducing the length of the computational cell normal to the slab in Fig. 7.la the tip in one cell is brought closer to the top surface of the slab in the image cell below. In this way the slab acts as both a tip holder and the substrate with which the tip interacts. In Fig. 7.1b we show how displaced periodic boundary conditions may be used to simulate the frictional sliding of a tip over a surface. In this case a rigid

Mechanical Properties of Metals at the N anometre Scale

201

displacement of each computational cell is introduced, with respect to the cell beneath, in the direction of the frictional sliding. Again, the slab acts as both the tip holder and the surface over which the tip slides. The disadvantage with using periodic boundary conditions is that events in one cell may influence events in neighbouring cells in a way that one may not wish. For example, during indentation of the tip into the substrate the tip may start to interact with its periodic image on the other side of the slab if the slab is not of sufficient thickness. This problem is not so much a consequence of periodic boundary conditions but the finite size of the computational cell. Analogous difficulties arise with other methods of controlling the tip, such as clamping some atoms, stemming from the finite size of the model. 7.3.2

CONTROLLING

THE TEMPERATURE

In classical molecular dynamics the temperature of the system is proportional to the total kinetic energy of the atoms. When the tip makes contact with the substrate and adhesion takes place there is a very significant release of potential energy as two surface energies are converted into an interfacial energy. During indentation or sliding, work is done on the system by an external loading mechanism and this work is eventually converted into heat. Unless some form of tempera ture control is in trod uced in to the simula tions the temperature will rise and may even result in melting of the system. On the other hand we are familiar with heat being generated by friction and therefore it is not obvious to what extent we should restrict rises in temperature. In fact the question of energy dissipation in adhesion, fracture and friction has quite profound implications and how it should be modelled properly is still not known. For example, in a metal the electronic contribution dominates the thermal conductivity. But in classical molecular dynamics heat is transported only by thermal vibrations and the electronic contribution is ignored. There is still no satisfactory solution to this problem, but several ingenious methods have been introduced to control the temperature through immersing the system effectively in a heat reservoir. Finnis et al.1O have considered the interplay of the electronic and ionic contributions to the thermal conductivity in metals. They model the exchange of heat between the ion and electron subsystems in molecular dynamics by introducing a frictional force}; = - lliVi on ion i. The coefficient u, varies with the 'local' temperature of ion i, as measured by its kinetic energy, and it may be positive or negative. One difficulty with this approach is that the region in which one attempts to define a 'local' temperature is arbitrary. Finnis et ale take the ion itself, but one could also take the ion and its first neighbour shell for example. Nevertheless, it is physically obvious that some definition of a local temperature is desirable in order to model a process that is spatially localised. Otherwise we scale the particle velocities throughout the entire system in response to a surge in kinetic energy in a small region of the system. This is in fact what the Nose-Hoover thermostat does, and which we have implemented

202

Electron Theory in Alloy Design

in our simulations of contact formation and fracture. However, for the simulations of frictional sliding we have used a different strategy. In those simulations we rescale the particle velocities in the bottom three layers of the substrate to keep them at the required temperature. Heat that is generated elsewhere in the computational cell is conducted to the bottom three layers by atomic vibrations. In this way we hope to simulate the production of heat at the sliding interface and its dissipation in the adjoining regions. We do not regard either of these solutions as very satisfactory, but at present they are as good as one can do. In the Nose-Hoover thermostat!' we imagine that the entire system is immersed in a heat bath which accelerates or decelerates each atom, so that on average the temperature of the whole system is equal to that of the heat bath. In a metal we may think of the heat bath as being the electrons, which are able to conduct heat more efficiently than the ions. The position ri and momentum Pi of the i'th particle are given by the following equations of motion .

(7.19) where Fi is the force acting on the i'th particle. These are the usual Newtonian equations of motion except for the 'friction' term -1JPi' The parameter 1] also satisfies an equation of motion

LP~/mi Q

gkT

(7.20)

where (7.21 ) Here g is the number of degrees of freedom, which is 3 (N - 1) where N is the number of particles in the system. T is the thermostat temperature and 1: is a 'relaxation time' which is specified by the user and controls the rate at which the thermostat damps temperature fluctuations in the system. Note that the frictional damping coefficient 1] is the same for all atoms in the system. This is the feature that we feel should be improved in the future. We have implemented the Nose-Hoover thermostat in the velocity Verlet algorithm' for solving the equations of motion. Since we are not aware of this implementation in the literature we have given a summary in our Appendix.

Mechanical Properties oj Metals at the Nanometre Scale

slab

203

t

!

h

slab

1

t

Fig. 7.2 Schematic illustration of two parallel slabs of thickness t, separated by a gap of h. The shaded regions indicate that the upper surface of the upper slab and the lower surface of the lower slab are clamped in position. The two inner surfaces are allowed to relax towards each other.

7.4 BRINGING TWO SURFACES OF THE SAME MATERIAL TOGETHER AND PULLING THEM APART 7.4.1

A CONTINUUM

MODEL

Consider two flat parallel slabs of continuous matter and thickness t, separated by a distance h, as shown in Fig. 7.2. For t ~ h it is easy to show that for van der Waals forces acting between the slabs their energy of interaction is proportional to - l/h2 per unit area. If the slabs are clamped as shown in Fig. 7.2 they will undergo a vertical strain of 8 = bit under the action of their mutual attraction. At large separations, h, the attraction between the slabs is compensated by the elastic strain energy, which is proportional to 82• However, at sufficiently small separations, h, the van der Waals attraction overcomes the elastic restraint and the two slabs jump into contact. At that point the slabs become one continuous, uniaxially strained slab. The instability is characterized by the second derivative of the total energy of the system reaching zero. At that point the effective spring constant of the whole system is zero. This is the 'jump-to-contact' phenomenon which has been known for a long time.12,13 It is a ubiquitous feature of adhesion, and it was first modelled atomistic ally by Pethica and Sutton.l" Writing the energy of attraction per unit area between the slabs as - AI(h - 2b)2, where A is a constant, and the linear elastic strain ~nergy of the two slabs, per unit area, as 2 x (C82tI2) = Cb2/t, where Cis the appropriate·

204

Electron Theory in Alloy Design

Young's modulus, the total energy is given by (7.22) There are additional terms in the attractive energy which are negligible provided t ~ h. Mechanical equilibrium is attained when (oE/Ob)h is zero, i.e. 2A/(h

-

2b)3

=

Co]!

(7.23)

provided (82E/Ob2)h is positive. Equation (7.23) has two solutions between b = 0 and h/2, but (o2EjOb2)h is positive only for the solution nearer to b = 0, as one would expect physically. The second derivative is given by (7.24 ) When b is such that (a2E/Ob2) h is zero the system is mechanically unstable and the jump to contact occurs. Thus, the instability is given by the condition h -

2b

=

14 /

12AtJ [- C

(7.25)

If the right hand side is greater than h then the system is immediately unstable, otherwise h must be reduced by moving the slabs closer together until this equation is satisfied. It follows that the thicker the slabs or the smaller the elastic stiffness, C, the further apart the two slabs will be when the instability occurs. This is physically obvious since both conditions lead to a reduction in the elastic strain energy for a given displacement b. Once the slabs have combined to form a single slab of thickness 2t at equilibrium, we may ask what happens when we try to pull them apart by moving the clamped surfaces apart by 2b. Another instability occurs at a critical value of b whereupon the two slabs jump apart. It is again characterized by the second derivative of the total energy of the system reaching zero. In this case, however, while the elastic strain energy is again increasing with b the van der Waals attraction is decreasing with b only because the densities of the slabs are decreasing. Moreover, the initial separation, h, of the two slabs does not appear in the total energy anymore: all memory of h is lost once the jump to contact has occurred. This means that there is a hysteresis in the instability points characterizing the jump to contact and the jump to separation. In a cycle of bringing the slabs together and pulling them apart again the slabs would heat up no matter how slowly we tried to complete the cycle. This hysteretic behaviour in adhesion and fracture is not confined to continuum models or slab geometries. It is seen in all the simulations below and in point contact experiments. I The continuum model above is clearly not able to describe the atomic

Mechanical Properties of Metals at the N anometre Scale

205

processes that take place on contact formation and fracture and the atomic mechanisms giving rise to the irreversible nature of the process. For example, without such insight it is not easy to see how a monolayer of impurities can have a dramatic influence on the force of adhesion, which is the force required to pull the tip off the substrate. It is also not clear how a continuum model could distinguish whether wear or lubrication occurs in frictional sliding. These are processes involving the formation and breaking of atomic bonds and we have no option but to consider the problem atomistically. 7.4.2

SIMULATION

USING A LENNARD-JONES 15

POTENTIAL

Sutton and Pethica carried out a simulation of a tip interacting with a slab using the three dimensional periodic boundary conditions shown in Fig. 7.1a and a Lennard-Jones 6-12 pair potential. The slab had dimensions llaf by lla! by 3a! and contained 1694 atoms in an fcc structure. The tip was constructed by selecting atoms in an fcc crystal falling within a paraboloid with a radius of curvature of It! and a height of 5a!. The tip contained 273 atoms, giving 1967 atoms altogether. The normal to the slab was (00 I) and the tip had the same orientation and fcc crystal structure as the slab. The potential was cutoff at 2af, leading to 140 interacting neighbours in the perfect fcc crystal. In Lennard-Jones reduced units the temperature of the simulation was 0·2 and the time step was 0·01. The simulation temperature was estimated." to correspond to 0·3 of the bulk melting point of fcc 'Lennard-Jonesium'. In this simulation the temperature was maintained constant in every time step by rescaling the particle velocities. An equilibration run was performed for 2000 time steps. During this run the computational cell in the z direction (normal to the plane of the slab) was sufficiently large that no interactions between adjacent cells occurred in this direction. The area of the computational cell in the x and y directions was allowed to change during the equilibration to relax the pressure arising from thermal expansion. After this equilibration no further changes in the cell x and y lengths were allowed. The tip was then brought within range of the slab in the adjacent image cell at a constant strain rate. The strain rate was such that the cell length decreased by laf over 5000 time steps. The strain was applied homogeneously to all atoms in the cell every tenth time step. This was continued until the force of attraction between the tip and the substrate became almost zero. The tip was then pulled off the substrate by reversing the sense of the strain but keeping its rate constant. This was continued until the tip broke free of the substrate. Changes in the structure of the tip were recorded as snap shots taken in projection along [100]. As the tip came within range of the substrate atoms at its base strained increasingly towards the substrate. At the same time those atoms underwent larger amplitudes of vibration, with correspondingly lower frequencies. Eventually, when the lowermost tip atom was approximately laf from the

206

Electron Theory in Alloy Design

potential

energy

end of tip

t

1

substrate

Fig. 7.3 Schematic illustration of the potential double well for the atom on the end of the tip as the tip approaches the substrate. When the potential barrier in the middle of the well becomes of order kT the atom can pass freely between the end of the tip and the substrate.

substrate on average, it jumped across the gap onto the substrate. It then underwent low frequency oscillations between the tip and the substrate, with rest times on both the tip and the substrate. This is the jump to contact instability, but in this case it involves, at least initially, just one atom. It arises when the potential barrier separating the two potential wells for the atom at the base of the tip and on the susbtrate is comparable to kT, as shown schematically in Fig. 7.3. At that separation the atom is then in a double well with a small curvature associated with the barrier. Another way of describing the instability is to say that the vibratory mode, normal to the substrate, of the atom at the tip base becomes soft. Continuing the run with a fixed cell length for a further 10,000 time steps, more atoms at the base of the tip became quasi-detached and underwent these large amplitude, low frequency oscillations between the tip and the substrate. Figure 7.4a shows a snapshot in which 5 atoms are undergoing these oscillations. We are seeing here the initial stages of sintering of the tip to the substrate. The sintering is being effected not so much by surface diffusion as by soft mode vibrations of the least coordinated atoms at the tip base. As the tip was brought closer to the substrate, at the constant homogeneous strain rate, the area of contact increased and a (001) stacking fault was produced because the total number of (002) planes in the tip and substrate was 15 and therefore odd. The fault produced a bending and rotation of all planes in the tip. A defect was nucleated at one end of the fault and after a further 2000 time steps both the defect and the fault had vanished and the total number of (002) planes had been reduced to 14. The tip was then in perfect epitaxial contact with the substrate and the force of attraction between the tip and the substrate had reduced to zero. A Burgers circuit analysis revealed that the dislocation was a 1/2[010] dislocation. The dislocation moved by glide and

Mechanical Properties oj Metals at the Nanometre Scale

••••

,

~

'.

,

,

,.

06

•

~

,.,

~

•

II'

~

.. . " . •

"

207

••

.;.

III

•

Ie

-

..

..

,

•..

" ~ '.-"-

. . .•..

•

•

v

. .• •. •..

(a)

(b)

Fig. 7.4 (a) Snapshot of the tip and substrate seen in projection along [100]. The encircled atoms are undergoing large amplitude, low frequency vibrations between the tip and substrate. (b) 1/2 [0 10] edge dislocation moving by glide and climb along the path shown. On the left of the dislocation there is a (001) stacking fault between the tip and the substrate. The dislocation centre is at C and the extra half plane can be seen by viewing along the vertical planes of the figure.

climb diagonally through the tip and emerged from the top left corner of the tip in Fig. 7.4b. The dislocation is seen in Fig. 7.4h approximately midway along its path. Although the strain to pull the tip off the substrate was applied homogeneously it became localised at the base of the tip very rapidly. The lowermost 3-4 planes of the tip became increasingly diffuse and widely spaced, as shown in Fig. 7.5a. Atoms in these layers performed low frequency, large amplitude vibrations normal to the surface of the substrate. On further straining, these layers merged into each other and reformed to produce an extra layer at the tip base. A snapshot taken soon after the formation of this new layer is shown in Fig. 7.5b. It is seen that the layers are now less diffuse because their spacing has been reduced. On further straining a neck started to form above the second layer of the tip base. Eventually the tip broke free leaving a considerable number of tip atoms on the surface of the substrate, as shown in Fig. 7.5c.

208

Electron Theory in Alloy Design

. .•. .. .

.".

..

r v

.

..

. I

,

:

• •

•

• • •

A

#

••

•

.•

•

., ... ,/

&

.:

,

;

•.

..•. +

,

.~

.. \-

'-~. -'"

-~

" ...

-

.

'.t:

,.

Mechanical Properties of Metals at the Nanometre Scale 7.4.3

SIMULATIONS

209

USING N BODY POTENTIALS

We have repeated the simulation described in the previous section using N body potentials described in section 7.2. In this section we summarise the results of two simulations in which the tip and substrate are again of the same material. In the first case the material is Pb and in the second it is Ir. We chose these potentials because the elastic constants they predict differ by more than a factor of9. In later sections we shall describe simulations of indentation and the effect of a monolayer on the force of adhesion using these potentials. In this section we explore the significance of the N body nature of the interactions compared with the pair potential of the previous section. We also explore the significance of the elastic constants. The tip and substrate are of the same size and geometry as in the previous section. The temperature of the simulation is 300 K and is controlled by a Nose-Hoover thermostat. The time step is 10-14S• For both Pb and Ir, during the equilibration stage before adjacent cells start to interact, it is seen that the tip contracts rather than expands as it did in the Lennard-Jones case. This is to be expected because of the predominance of under-coordinated atoms and the bond contraction that this leads to with an N body potential. In the case ofPb this contraction almost cancels the thermal expansion of the slab, whereas in Ir the thermal expansion is much smaller and the slab does contract overall. The second, but related, difference compared with the pair potential simulation, concerns the instability during the tip approach. Whereas with the Lennard-Jones potential a single atom jumped from the tip base to the substrate we find with the Pb potential that a small cluster of 2 or 3 atoms detaches from the tip base. In the case of the Ir potential the instability is very weak at 300 K-- and occurs only slightly above the substrate. This is because of the greater depth and curvature of the potential well confining the Ir atom on the tip base or on the substrate. The barrier between the potential wells, shown in Fig. 7.3, is therefore a larger multiple of k'T, Apart from these differences the Pb and Lennard-Jones simulations were qualitatively very similar, and the sequence of events described in the previous section was repeated with the Pb potential. In Fig. 7.6a we show the Pb tip being pulled off the Pb substrate and the beginning of the formation of a neck in the highly disordered lower half of the tip. But the Ir simulation was quite different. Most of the tip retained an fcc crystal structure throughout the cycle of contact and pull-off, whereas much greater disorder was seen with the other two potentials. We believe that this is a consequence of the greater elastic constants in the Ir case, or, to put it another way, the simulation temperature of 300 K is about 11% of T M (the bulk experimental melting point) for Ir compared with 500/0 of T M for Pb. The homologous temperature of the simulation is the important parameter. In consequence the mechanism by which the tip establishes a contact area with the substrate in the Ir case is different, in that it more obviously involves surface diffusion. In contrast to the Pb and Lennard-Jones tips the Ir tips retains the

210

Electron Theory in Alloy Design

Fig. 7.6 (a) Snapshot of the Pb tip necking down on the Pb substrate. Note the structural disorder in the lower halfofthe tip. (b) Snapshot of the Ir tip taken soon after initial contact has been made with the Ir substate. Note the crystalline order and the surface ledges in the tip.

ledge structure it had when it was constructed geometrically. When the initial contact is made atoms cascade down these ledges, as shown in Fig. 7.6b. Thus the contact area is established quickly by surface diffusion along the ledges of the tip. During the pull-off the part of the Ir tip in contact with the substrate became highly disordered as a relatively small neck was formed, which again resulted in some tip atoms being left behind on the substrate. But because less of the Ir tip was involved in the formation of the neck, fewer atoms were left behind. This was also reflected in a smaller tail in the force displacement relation shown in Fig ..7.7a for the Ir case, compared with the Pb case shown in Fig. 7.7b. We summarize the main points as follows: 1. The dominant mechanism of irreversible deformation is diffusional creep driven by the stresses acting in the tip which are at the limit of the theoretical strength.

Mechanical Properties of Metals at the N anometre Scale Ti

30

-Substrate

Distance

vs

Force

z

IR

211

IR

20

10

-10 -20 -30 -4 -40

-0.6

-0 . 4

-0.2

0.2 Distance

0.4 0.5 (NanD-meter)

(a)

0.8

1.0

1.2

-0.5

0.5 Distance

1.0 1.5 (NanD-meter)

2.0

2.5

(b)

Fig. 7.7 (a) The force vs. distance curve for the Ir tip and Ir substrate. The 'distance' is defined as follows: let the separation of the tip base and the substrate when they are well separated be So and let the corresponding computational cell length be bo• For a computational cell length of b, the 'distance' between the tip and the substrate is defined as So + b, - bo• The arrow to the left shows the approach and the arrow to the right shows the pull-off. All force-distance curves have been smoothed by an averaging procedure. (b) The force vs. distance curve for the Pb tip and Pb substrate.

2. As in creep of bulk specimens the rate and extent of creep deformation is controlled by the homologous temperature. 3. At low homologous temperatures the diffusional creep involves surface diffusion. At higher temperatures diffusion within the tip is involved as well. In both cases the 'diffusion' is closely related to soft-mode behaviour. 4. Dislocations are nucleated only when the shear strain is maintained at the theoretical limit in a sufficiently large volume. 5. Much greater diffusional creep is seen under the tensile stresses operating during the pull-off. The free volume arising from the tensile strains allows more extensive atomic rearrangements within the tip leading to necking and eventual separation. 6. During pull-off a fraction of the tip in contact with the substrate effectively melts. The larger the homologous temperature the larger the fraction. The profile of the part of the tip that melts closely resembles that of a liquid drop under tension. 7. The overall irreversibility of the cycle is very evident in the simulations, especially in the transfer of tip material to the substrate during pull-off. 8. No indentation of the substrate by the tip was seen. This is in accord with standard plasticity theory where indentation occurs only if the stress in the substrate reaches about 3 times the flow stress. This does not happen because the tip deforms and relieves the applied stress. Consequently the substrate remains almost unaffected throughout the entire cycle. 9. Some of the results reported here are expected to be dependent on the size and curvature of the tip. For example, we expect blunt tips to give rise to the

212

Electron Theory in Alloy Design

instability on approach described in section 7.4.1. By 'blunt' we mean the radius of curvature is of the order of a few nanometres or more. We also expect the diffusional flow to become more confined to the surface of the tip as its radius of curvature increases. 7.5 THE INTERACTION SUBSTRATE 7.5.1 Ir

TIP AND

Pb

BETWEEN A HARD TIP AND A SOFT

SUBSTRATE

In this section we report two simulations of an Ir tip approaching and indenting a Pb substrate followed by retraction and separation of the tip. The first simulation was carried out at 300 K and the second at 20 K, and a NoseHoover thermostat was used. The time step was 10-14s. The tip and substrate. geometry is exactly the same as in section 7.4.2. We concentrate on two aspects: (a) the effect of temperature and (b) the effect of the very different elastic constants in the tip and the substrate. During the initial equilibration of 4000 time steps at 300 K (in which adjacent cells are not interacting) the Ir tip embedded itself within the underside of the Pb substrate by excavating a hole, as shown in Fig. 7.8a. The Pb atoms that were removed diffused down the sides of the tip and wet it. We have repeated this simulation with a slab of twice the thickness. After 5000 time steps the tip became lenticular in shape and half buried in the substrate as shown schematically in Fig. 7.9. The driving force for this process is presumably the establishment of the equilibrium contact angles ex and /3, which appears to be more inhibited in the smaller slab. It is remarkable that this process takes place in 50 ps! I t is another demonstration of the astonishing atomic mobility in these systems in the presence of driving forces which arise only from interfacial stresses. This is particularly surprising in view of the fact that the simulation temperature is only 11% of the experimental bulk melting point oflr. By using computer graphics techniques we have sectioned through the tip in the thicker substrate after the tip has buried itself. We have found that there are Pb atoms in the Ir tip and that the interface is thus a few atoms wide. The Ir atoms are about 22% smaller than the Pb atoms, and we think that the mixing of some Pb into the Ir tip reduces the size mismatch between. them. We were unable to detect any strain in the thicker slab around the Ir tip after the embedding. However, .considerable compressive strain is visible in the thinner slab, as seen in Fig. 7.8a, where the embedding is not so complete. Our analysis is that in the thinner slab the misifit is accommodated elastically, but the thicker slab is beyond the critical thickness at which the misfit must be accommodated plastically. The freedom from this elastic constraint enables the tip to embed itself further in the thicker slab. Figures 7.8 and 7.10 show a sequence of snapshots taken from the 300 K and 20 K simulations respectively. At both temperatures wetting of the tip by the

Mechanical Properties oj Metals at the N anometre Scale

213

Fig. 7.8 Six snapshots of an Ir tip (dark) interacting with a Pb substrate (light) at 300 K. (a) a section taken through the tip and slab to reveal the embedding of the tip in the substrate above before it starts to interact with the substrate below. Note the elastic (coherency) strain in the slab and the wetting of the tip by the Pb atoms. (b) A projection showing the Pb slab rising to meet the Ir tip. (c) A section showing the tip indenting the substrate below; note the diffusional disorder introduced into the slab. (d-f) show retraction of the tip and the formation of a neck of Pb atoms and extensive wetting of the Ir tip by substrate atoms.

substrate atoms takes place. Also, in both cases the instability on approach is marked by the substrate rising to meet the tip, Fig. 7.Bb and Fig. 7.1Ob. This simply reflects the softness of the substrate compared with the tip. Indentation proceeds by diffusional creep of the soft substrate around the hard tip. This diffusion is more extensive at the higher temperature, but it is remarkable how extensive it is even at 20 K. On retraction of the tip there is a larger neck formed from the substrate material at the higher temperature. This results in a larger hysteresis in the force displacement curve for the higher temperature

214

Electron Theory in Alloy Design slab /

vacuum

Fig. 7.9 Schematic illustration of the embedding of the tip in a thick substrate establishing equilibrium contact angles ct and p.

simulation. It was found by sectioning through the model that although many substrate atoms were transferred to the tip the damage within the substrate healed very quickly and the surface of the substrate recovered almost immediately, concealing the damage beneath it. Throughout the cycle at both temperatures the tip retained its structural integrity and virtually all the deformation was confined to the soft substrate. It is clear in Figs. 7.8 and 7.10 that the slab is not sufficiently thick to study indentation in detail because the deformation in the slab is being constrained by the next tip. We expect that diffusional creep will still occur in thicker slabs, but the tip will also be accommodated at least in part by dislocation emission in the substrate. 7.5.2 Ni

TIP AND

Ag

SUBSTRATE

In this simulation the Ag substrate has a [111] normal. The Ni tip has the same crystal orientation as the substrate. The simulation was performed at 300 K with a Nose-Hoover thermostat and a time step of lO-l~. During the equilibration phase the originally atomically sharp tip faceted on the (111) plane, parallel to the substrate. On approaching the tip to the substrate the instability was again marked by the substrate rising to meet the tip. But in this case a whole tetrahedron of the substrate slid out of the substrate to meet the tip, as sketched in Fig. 7.11. The tetrahedron was bounded by (Ill) on the top and (11I), (1II) and (Ill) on the sides. The side of the tetrahedron was 5 atoms long. The distance the tetrahedron slid was approximately half of a (Ill) interplanar spacing, and it was therefore a 'stacking fault tetrahedron'. The tetrahedron's lifetime was less than 1ps, and it slid back into the substrate once contact with the tip was achieved. On decreasing the cell length further the tip did not indent the substrate but increased its contact area. As the tip was pulled off the substrate another tetrahedron of greater size was produced, which resulted in the susbtrate becoming one (Ill) plane thicker in the area of the contact. In other words the initial tensile strain was accommodated in the substrate whose elastic constants are approximately half those of the tip. The tetrahedron is in fact a prismatic dislocation loop, with a Burgers

Mechanical Properties oj Metals at the N anometre Scale

215

Fig. 7.10 Six snapshots of an Ir tip (dark) interacting with a Pb substrate (light) at 20 K. Compare the sequence shown in Fig. 7.8. Note the tip still embeds itselfin the substrate even at 20 K (a), and the substrate rises to meet the tip in (b), the disorder produced in the substrate by diffusional flow at 20 K during indentation is still substantial though less than at 300 K, and during retraction (d-f) only a very small Pb neck is formed, though extensive wetting of the tip by substrate atoms still occurs.

vector of 1/3[Ill]. On further straining the tetrahedron did not vanish, and the tip necked down leaving the contact area almost the same. On separation Ni atoms of the tip were left behind on the Ag substrate. This simulation shows the importance of the crystal orientation at the contact. Planes of easy slip can be exploited in nanostructures just as in bulk specimens.

216

Electron Theory in Alloy Design

W Ag slab

Fig. 7.11 Schematic illustration of a tetrahedron

7.6 THE REDUCTION

IN ADHESION

of Ag slab rising to meet the Ni tip.

DUE TO A SOFT LAYER

In section 7.4.3 we discussed the interaction between an Ir tip and an Ir substrate at 300 K. In this section we discuss another simulation at 300 K in which a layer ofPb is introduced on the surface of the Ir substrate. The purpose is to investigate the effect that this layer has on the adhesion of an Ir tip to the substrate. The Pb was introduced by 'transmuting' the top layer of Ir atoms of the substrate and equilibrating the system. Because of the greater size of the Pb atoms many of them popped out of the original Ir layer producing about 3/2 monolayers of Pb. During the approach to contact the instability was marked by some of the Pb atoms jumping up to meet the Ir tip. On bringing the tip closer to the substrate the tip established a contact area with the substrate as in the Ir tip-Ir substrate simulation. Despite compressing the tip much more than in earlier simulations we found that the tip did not penetrate the film of Pb except at one atomic site. Consequently, almost no direct Ir-Ir bonds were formed between the tip and the underlying Ir substrate. This is the key to understanding what happens during the pull-off. Because the Ir-Pb bonds are considerably weaker than the Ir-Ir bonds the tip does not adhere to the Pb layer since that would involve breaking strong Ir-Ir bonds within the tip. It is therefore no surprise that on pulling the tip off no Ir atoms of the tip were left behind on the substrate. A relatively small neck of Pb atoms was formed and several Pb atoms were transferred to the tip. We conclude that the main effect of the Pb layer was to prevent direct Ir-Ir bonds between the tip and substrate from forming. This reduced the force required to pull the tip off and the hysteresis in the force displacement curve (see Fig. 7.12a) compared with the case of no intervening Pb layer between the Ir tip and Ir substrate, shown in Fig. 7.7a. Since the contact area that the tip establishes with the film on the substrate is not maintained during the pull-off, and the tip leaves the substract intact, we may say that the tip does not wet the substrate. This interpretation is developed in section 7.8.

Mechanical Properties of Metals at the N anometre Scale Ti

50

-Substrate

Distance

vs

Force

z

IR PB IR

Ti

-Substrate

Distance

vs

Force

z PB IR

1.0 1.'5 (Nano-meter)

2.0

217 PB

40

]

!

2 1

~ o·r---~~----~--~*-~~~~ B-1 OJ

~ -2 -3 -4

-30 -0. B

-0.6

-0.4

-0.2 Distance

0.2 (Nano-meter)

0 .4

0.6

O. B

(a)

-0. '5

0.'5 Distance

2.5

(b)

Fig. 7.12 (a) Force vs. distance curve for the Ir tip and Ir slab with an intervening Pb layer. Note the reduction in the force of adhesion compared with Fig. 7.7(a). (b) Force vs. distance curve for the Pb tip and Pb slab with an intervening Ir layer. Note the reduction in the force of adhesion compared with Fig. 7.7 (b).

7.7 THE REDUCTION

IN ADHESION

DUE TO A HARD LAYER

The result of the last section may have been anticipated. In this section we consider a Pb tip and a Pb substrate as in section 7.4.3, but we introduce an Ir layer on the surface of the substrate. Although there is now strong wetting between the Pb tip and the Ir layer we find that there is a reduction in the force of adhesion and in the hysteresis of the load displacement curve compared with the Pb tip-Pb substrate case, discussed in section 7.4.3. The simulation was again performed at 300 K with a Nose-Hoover thermostat. At first we tried introducing the Ir layer by transmuting the top layer of the Pb substrate. But during the equilibration the Ir layer diffused into the substrate and disappeared from the top surface! Therefore, we increased the thickness of the slab by one layer and transmuted the top two layers to Ir. Although the Ir still diffused into the Pb, sufficient Ir remained on the surface to prevent direct Pb-Pb bonds from forming with the tip. During equilibration the area of the slab contracted significantly due to the much greater strength of the smaller Ir-Ir bond. The underlying Pb slab was amorphised in the process. The instability on approaching the tip to the substrate was marked by atoms at the base of the Pb tip jumping into contact with the Ir layer. The Pb tip very rapidly established a large area of contact with the layer. No crystal structure within the tip was discernible: it appeared to have the structure and properties of a liquid drop wetting a surface. Wetting was energetically favourable because of the greater energy of Pb-Ir bonds than Pb-Pb bonds. Virtually nothing happened in the substrate under the Ir layer during contact and pull-off of the tip. During the pull-off the tip necked down and separated from the substrate

218

Electron Theory in Alloy Design

Fig. 7.13 Snapshot of the Pb tip being pulled off the Pb substrate with an Ir layer (dark). Note the reduction in the neck radius compared with Fig. 7.6, and the larger area of contact with the substrate. Note also the liquid like appearance of the tip both in its profile and in its internal structure.

leaving more Pb atoms on the Ir layer than were left in the absence of the Ir layer. This was because the larger area of contact was kept almost the same throughout the pull-off thereby maintaining constancy of the angle of contact between the Pb tip and the Ir layer, as shown in Fig . 7.13. The constancy of the contact angle imposed a constraint on the necking down of the tip. For the same cell length in the z direction the radius of the neck in the presence of the Ir layer was less than in the absence of the layer, because in the former case the contact area was greater and the contact angle was smaller. This may be seen by

Mechanical Properties of Metals at the N anometre Scale

219

comparing Fig. 7.13 with Fig. 7.6a. Thus, the force required to pull the tip off the surface in the presence of the layer was smaller than in the absence of the layer. This is seen in the force displacement relation shown in Fig. 7.12b, which should be compared with Fig. 7.7b. I t is also seen that the hysteresis in the force displacement curve was reduced in the presence of the Ir layer. We conclude that even when wetting is favourable there is a reduction in the force and work of adhesion compared with the absence of the layer. Taking into account the result of the previous section we conclude that the maximum force of adhesion between a tip and substrate of the same material occurs when there is no intervening layer between them. This is a central result of the chapter. In the next section we describe an analytic model which reproduces the result.

7.8 LIQUID

DROP MODEL FOR THE REDUCTION

IN ADHESION

Let us first recapitulate the results of the simulations concerning the force of .adhesion. During the pull-off the Pb tip effectively melts at 300 K. By this we mean that (a) no crystalline structure is discernible within the tip, and (b) the atomic mobility within the tip is very high so that the creation of new surface area during necking is accomplished readily by diffusion. This implies that there is no distinction between the surface energy and surface tension of the tip, as in a liquid drop. However, the substrate remains crystalline and in the absence of an intervening layer the interface between the substrate and the tip resembles a solid-liquid interface. The contact angle between the Pb tip and Pb substrate during pull-off remains approximately 90°. In the presence of an Ir layer on the surface of the Pb substrate extensive wetting of the layer by the Pb tip takes place and a small contact angle is established. During pull-off the liquid like Pb tip necks down while maintaining the contact angle and area approximately constant. The force of adhesion, i.e. the force required to pull the tip off the substrate, is reduced by the presence of the Ir layer. The force of adhesion between the Ir tip and Ir substrate is reduced by an intervening Pb layer. The Ir tip retains an ordered structure in the presence of the Pb layer, and a small neck is formed during pull-off involving only Pb atoms. However, rapid diffusion is seen in the Ir tip to enable its change of shape. Wetting of the Pb layer is not favourable because it would involve breaking stronger bonds within the tip: the Ir tip is pulled off the Pb layer intact. In the absence of the intervening Pb layer the lowermost 3-4 layers of the Ir tip become highly disordered and effectively melt. Some of the Ir tip is transferred to the Ir substrate and the force of adhesion is greater than in the presence of the Pb layer. The conclusion is that irrespective of whether wetting between the tip and the layer of impurities occurs, theforce of adhesion is always reduced relative to the case where there is no intervening layer. The experiments of Pashley et al.1 seem to be in agreement with this con-

220

Electron Theory in Alloy Design y

y=y(x)

9

Fig. 7.14 Liquid drop between two plates at x rotating the curve y = y(x) about the x-axis.

=

0 and x

=

x

g. The drop is obtained

by

elusion. In those experiments a tungsten tip was brought into contact with aNi substrate under UHV conditions. The UHV conditions are essential to eliminate spurious capillarity forces between the tip and the surface due to water vapour etc. They compared the force of adhesion in two cases: (a) when the tip and sample were atomically clean and (b) when oxygen was released into the chamber forming approximately 2 monolayers of oxide on the surface of the substrate. They found that the force of adhesion was roughly halved by the presence of the oxide layer. We are not aware of any other point contact experiments on adhesion carried out under URV conditions. Our simulations suggest the following model for the force of adhesion. The tip may be modelled as a liquid drop that establishes a certain contact angle with the solid substrate. During the pull-off the drop necks down at constant volume while maintaining the contact angle with the substrate. The force of adhesion is the maximum tensile force the drop can sustain throughout the necking process. The work of adhesion is the energy associated with the increase in area of the free surface of the drop during necking, and any change in the area of contact between the drop and the substrate, until the point of mechanical instability when the drop breaks. The assumption of constant volume of the drop is an approximation, but we believe it is a better approximation than to assume that there is no constraint on the change of volume of the drop. For example, we have tried modelling the tip as a soap film being stretched between two open rings and found no success in reproducing the results of the simulations. Figure 7.14 shows a liquid drop of volume Va between two solid plates at x = 0 and x = g. The drop is obtained by rotating the curvey = y(x) about the x-axis. The boundary conditions we assume at the left plate are that the radius of contact is fixed at R} and that the contact angle is 90°. They are consistent with the assumption that in a real tip only the bottom part of it melts

Mechanical Properties of Metals at the Nanometre Scale

221

during the pull-off. They are also consistent with the simulations. boundary condition we impose at the right plate is that t/I t/lo where

The

(7.26) where "Is is the surface energy of the substrate, 'Ysd is the energy of the interface between the substrate and the drop and d is the surface energy of the drop. t/I is the angle between the x-axis and the tangent to the curve y = y(x), i.e. tan t/I = dy/dx. The contact angle is 90 - t/lo. Equation (7.26) is obtained from the condition that the total surface and interfacial energy of the system is a variational minimum. If t/lo > 0 wetting is favourable, if t/lo < 0 wetting is not favourable. We now imagine that the interfacial energy, "lsd' may be varied at will, while the volume of the drop is kept constant. This changes the profile of the drop, and in particular it changes the radius of the neck. We shall show that the maximum tensile force the drop can sustain is determined by the radius of the neck. For a given t/lo the neck radius decreases monotonically as the separation between the plates, g, is increased. But the crucial point is that there is a maximum possible neck radius for given contact radius R, at the left plate, contact angle 90 - t/lo at the right plate and volume Va. It is this maximum possible neck radius that determines the force of adhesion. We shall show that the neck radius, and hence the force of adhesion, is a maximum when t/lo = O. In the simulations we see that this corresponds to the absence of a layer of any kind between a tip and substrate of the same material. It may be shown by a variational analysis that the equation governing the curvey = y(x) is the following "I

2ny T cos t/I

-

ny2p

=

F

(7.27)

Here p is the pressure inside the drop and F is the tensile force applied to the drop. This equation expresses the condition that the net force acting on a circular cross-section of the drop is zero. Equation (7.27) is a differential equation whose solution was given by Lamb," and it was the subject of much attention in the last century. 17 However, we shall see that it is not necessary to solve the equation to obtain the condition we seek. Setting a

equation

+ /3

2T/p

and

rx/3

Flnp

(7.28)

(7.27) becomes

y2

=

(rx + f3)y cos t/I

-

rxf3

(7.29)

It follows thaty2 satisfies f32 ~ y2 ~ rx2. Thus, to describe necking we may take

222

Electron Theory in Alloy Design

p

to be the radius of the neck and rJ, to be the maximum radius where t/J = O. Therefore fY., = R1, the radius of the drop at the left plate. We may set fY., = 1 to define our unit of length. Hence /32 ~ 1. It follows from equation (7.28) that

F J = 2n T =

f3

I+7i ~

1

2

(7.30)

We see that the maximum value of the applied force, F, is given by the maximum value of the neck radius {3,and the upper bound on Fis tt'T (i.e. n'T« if a is not taken as the unit length). For example, for T = 1J 1m2 and o: = 1 nm we obtain F ~ tt nN. Another useful inequality follows from equation (7.27). In order for rx and f3 to be real numbers we must have (7.31 ) For example, for F = 1 nN and T = 1J/m2 we obtain p ~ nGPa. These values of the applied force and the pressure in the drop are of the same order of magnitude as we find in the simulations of the tip, which gives some confidence in the physical sense of the model. Let the radius of the drop at the right hand plate be R2 (again in units of rx = R1). Since t/J = t/Joat the right hand plate R2 must satisfy equation (7.29) with l/I = t/lo: (7.32) where we have set rx = 1. Real roots for R2 are not obtained for values of f3 above a certain value. The condition for real roots is that f3

:::;;

1 - [sin t/lol 1 + [sin t/I 0 I

(7.33)

Thus the greater the absolute value oft/lo the smaller the maximum possible neck radius p. Finally, using equation (7.30) forfwe deduce

f ~

(1 - [sint/loI) 12

(7.34 )

It follows that the maximum value ofJis reached when t/lo = 0, and that any positive or negative deviation oft/lo reduces] This is precisely what was found in the simulations. What is most unusual about the liquid drop model is the role played by the Young'-Dupre equation, equation (7.26). Usually fracture of the contact is assumed to take place at the interface between the tip and the substrate. In our model this occurs only when wetting is not favourable. The ideal work of fracture is usually expressed as A (Y d + ,y s - Y sd) = Ay d (1 + sin t/J0) where A

Mechanical Properties of Metals at the N anometre Scale

223

is the area of the contact. For a given contact area the ideal work of fracture is a maximum when there is complete wetting (t/lo = 90°) and zero when there is complete non-wetting (t/lo = - 90°). The reason why the usual analysis does not agree with the simulations, when wetting is favourable, is that fracture does not take place at the interface between the tip and the substrate but within the tip, leaving tip material on the substrate after fracture. This crucial feature is reproduced by our liquid drop model. In our model the role of the YoungDupre equation is to introduce a constraint on the necking down process which imposes a maximum neck radius. By analogy with our liquid drop model we speculate that in a macroscopic system, when wetting is favourable, the role of the Yeung-Dupre equation is to introduce a constraint at the interface, which (a) ensures that the interface remains intact (b) restricts the amount of plastic deformation in the adjoining media. This may be relevant to the mechanism of embrittlement by those solute atoms that lead to strong bonds at interfaces.

7.9 FRICTIONAL GROWTH

SLIDING AND WEAR: STATIC JUNCTION

In this section we discuss two molecular dynamics simulations we have carried out of a tip sliding on a slab." We have used the three dimensional displaced periodic boundary conditions shown schematically in Fig. 7.1b. The vertical direction is along [001] and successive unit cells in the vertical direction are relatively displaced along [110]. Each slab consists of six (002) layers of an fcc crystal, and in each unit cell each layer comprises a 16 x 16 square array of atoms along

ca

E

20

c a .~

~

1ti

C/)

10~~~~--~~~~~~~~--~~~~ La Ce Pr Nd PmSm Eu Gd Tb Dy Ho Er Tm Yb Lu

Fig. 8.7 Experimental saturation magnetisation for R2Fel4B compounds' results from a simplified theoretical estimate (drawn line, see text).

(closed dots) and

R atoms in antiferromagnetic, because then the total 41 moment and its spin component are coupled parallel. As an example, Fig. 8.7 shows the experimentally observed variation of the total magnetic moment per formula unit of R2Fe14B compounds and the magnetic moment which would follow from equation (8.4) (full curve) by assuming a constant TM contribution (dashed curve, corresponding to the magnetic moment ofY2Fe14B). It follows that the trend is predicted very well with tetravalent Ce as an exception. In reality, the TM contribution depends slightly on the R atom, as discussed in section 8.5. The highest magnetisation is obtained for Pr and Nd compounds.

8.4 BAND STRUCTURE MAGNETISM

THEORY

OF TRANSITION

METAL

Within the itinerant electron picture electronic states in periodic systems are characterised by wave functions t/I(k, s, i), where k, s (1 or l) and i are the wave vector, spin quantum number and the band index, respectively. The set of energy levels E, (k, s), with a certain band index i, is called an energy band. The band structure formalism has already been developed in earlier chapters for non-magnetic systems (wave functions and energy levels independent of s). Here we will concentrate on those aspects of band structure theory which are related to the formation of magnetic moments. In band structure calculations the ground state energy Eo, the charge density

Permanent Magnets n(r) and the spin density m(r) Schrodinger equation

243

are obtained by solving self-consistently the

(8.5) for a single electron with quantum numbers and band index (k, s, i) a local periodic potential which is given by VS (r)

=

~xt (r)

e2

+ 4

n80

fdr'

I

n(r'),

r - r

I

+ ~~(n(r),

m(r))

==

a in

(8.6)

The external potential V:xt contains the attractive potential due to the nuclei. The second term in equation (8.6) represents the repulsive interaction of an electron at r with the average electrostatic field due to all electrons. The exchange correlation potential V:~, which depends on the spin s, is a function of the local electron and spin density. This is an approxima tion, the local spin density approximation (LSDA), because in an exact treatment of the problem V:~(r) is a (universal) function of the charge and spin density of the entire crystal. An explicit expression for the exact potential is not known. The exchange correlation term takes in to account that the actual electron density around a certain electron at a certain point is lower than the average electron density at that point, due to the direct Coulomb repulsion between electrons and the exchange interaction. The exchange interaction results from the Pauli exclusion principle, which states that the total wave function of in teracting electrons (fermions) is an tisymmetric for the exchange. of two electrons. This leads to a reduction of the average Coulomb repulsion between two electrons with parallel spins, compared to two electrons in similar orbitals, but with antiparallel spins. If m(r) i= 0 the exchange correlation term is different for spin up and spin down electrons, leading to spin-dependent solutions of the Schrodinger equation. The solutions of equation (8.5) should be self-consistent, i.e. the total charge and spin densities which follow from the wavefunctions of the occupied electron states should lead to the same potential V (r) as the potential from which these wavefunctions were calculated. This situation is reached by solving equation (8.5) iteratively, for a dense grid ofkpoints in the first Brillouin zone and for both spin directions. Note that band structure calculations using the LSDA are ab initio, in the sense that the only input parameters are the lattice parameters, the atomic positions and the atomic numbers. The most commonly used bandstructure methods for crystallographically complex systems, such as R2FeI4B compounds with 68 atoms in the unit cell, employ the atomic spheres approximation (ASA) (see Appendix). Examples are the LMTO (linearized muffin tin orbitals) 10 method and the ASW (augmented spherical wave)ll method. Within these methods the crystal is divided into S

244

Electron Theory in Alloy Design 350 -----~--....,

E o ~

300

CD

g ~ ~

50

CD

c::

W

OL..--_.J.....-..;::::...-~_-----a

0.85

0.90

0.95

1.00

V/Vexpt Fig. 8.8 Total energy of ferromagnetic and hypothetical the volume, from ASW band structure calculations.

nonmagnetic

Fe, as a function of

overlapping 'Wigner-Seitz spheres', centered around the atomic positions, with a total volume that equals the volume of the solid. The potential inside the spheres is taken to be spherically symmetric. This is usually a very good approximation in densely packed metals. Although the potential inside each sphere is spherically symmetric, the charge density which is calculated after each iteration step is in general aspherical, and could, for example, be used to study crystal field effects (section 8.6). As an example, we will discuss briefly some results of LSDA based band structure calculations for Fe, Co and Ni. In Fig. 8.8 the calculated total energy offerromagnetic and (hypothetical) nonmagnetic bee Fe is shown, as a function of the volume. In agreement with experiment, the ground state is ferromagnetic. At the calculated equilibrium volume the calculated magnetic moment is 2·15 flB' whereas the experimental spin contribution to the magnetic moment is 2·12 flB. For Co and Ni the calculated (experimental) spin magnetic moments are 1·54flB (1·57flB) and O·611lB (0·56 JlB), respectively. The agreement between theory and experiment is quite good. However, as can be seen from Fig. 8.8, the calculated unit cell volume is slightly too small. Similar small discrepancies (4-6%) have been found for Co and Ni, and for a large number of binary compounds (e.g. for Y-Fe compounds"). Related to this error, the magnetic moments, if calculated at the experimental lattice parameters, tend to be slightly too high. The origin of these small errors lies in deficiencies of the LSDA in the treatment of interatomic exchange and correlation effects, and have been discussed extensively in a recent review paper by Fulde et al.12 We already anticipated this point in section 8.2, where we saw that at the end of the 3d series the condition Ueff ~ W for itinerant electron systems is not quite fulfilled.

Permanent Magnets 8.5 MAGNETIZATION

245

OF R2Fe14B COMPOUNDS

Band structure calculations of the magnetization of rare earth transition metal intermetallic compounds require the use of an approximate method for handling the 4] electrons. As we have shown in section 8.2 the strong interatomic correlation and exchange effects in the 4] shell are not treated adequately within the local spin density approximation. It is possible to circumvent this problem by studying the magnetism of the transition metal sublattice by performing calculations for the yttrium or gadolinium compounds. Y is chemically very similar to the real R metals (valence, electronegativity, atomic radius), but it does not contain a 4]shell. Gd has a completely filled majority spin 4jshell, and an entirely empty minority spin 4fshell, resulting in a 4jspin moment of 7 IlB' in agreement with Hund's rules. A more generally applicable practical method for handling the 4] states has been proposed recently by Brooks et al.I3 They treated 4f states as open core states, i.e. states which are not allowed to hybridise. The number of 4] states is a fixed integer, and the spin polarisation of the states is determined by the requirement that the spin contribution to the magnetic moment is equal to the value which follows from atomic theory, using the Russel-Saunders scheme to couple Sand L. For Nd (4f3), e.g. the total magnetic moment is 3·27 IlB jatom (see Table 8.2), with a spin contribution of 2 (gL - 1)] = - 2·44 IlB jatom. This implies that the occupation numbers are 2·72 for majority spin and 0·28 for minority spin electrons. With these occupation numbers as a constraint the 4J majority spin and minority spin densities are calculated self consistently in the band structure calculations. What is the effect on the 3d-sublattice magnetisation of interchanging R atoms in R2Fe14B compounds? This was investigated using self-consistent band structure calculations by Jaswal (R = Nd and Y),I4 Coehoorn (R = Gd and y)15 and Hiimmler and Fahnle (R = Nd, Gd, Tb, Dy, Ho, Er).16 All groups found that the effects on the magnetic moments on Fe atoms are very small. Hiimmler and Fahnle carried out self consistent LMTO band structure calculations and treated the 4f magnetism according to the method which we discussed above. Coehoorn carried out self consistent ASW calculations, and treated 4fstates in Gd2Fe14B as normal band states. Figure 8.9 shows the results of both groups. The Fe moments were obtained by integrating the spin density within the atomic sphere around each atom, i.e. within a radius of about 1·40 A. The spin magnetic moments appear to depend strongly on the Fe site,; varying from about 2·01lB for the 4(e) sites to about 2·61lB for the 8(j2) sites. Fe atoms on the 16(k1) and 4(e) sites, for which the lowest magnetic moments are found, have 1 and 2 B nearest neighbours, respectively, whereas the other sites have no B nearest neighbour. Fe atoms at the 8(j2) site, with the highest magnetic moment, have their nearest neighbours at relatively large distances. This implies that for this site the partial density of states (DOS), i.e. the DOS which is obtained after separating the total DOS into contributions from the

246

Electron Theory in Alloy Design R2Fe14B

3.0 r----------.-, ASW

LMTO

• 2.5 •.. ~j1

• 2.0

•

• • • •

• ....• lz

.-.

• •.• k2

.--.

• •..• k1

•

.-.

---.c

--'j2 C

• •

•e

---. ---.

---.e

-.k1

1

Nd

Gd Tb Oy Ho Er,

Fig. 8.9 Magnetic moments on Fe sites in R2FeI4B, band structure calculations.

k2 j1

J

Gd Y

as obtained from LMTOl6

and ASW7,15

atomic like states from which the charge density within each of the atomic spheres is constructed, shows a relatively narrow band width. Partial densities of states for Y2Fel4B, calculated using the ASW method, illustrate this point (Fig. 8.10). The figure also shows that the Fe-B hybridisation leads to the mixing of B(2s) states, 2 sharp peaks around - 8 eV, with Fe(kl) and Fe(e) states (mainly 4s). Figure 8.9 shows that the LMTO and ASW results are in good agreement with respect to the size of the magnetic moments. A comparison between calculated Fe moments in Y2Fe14B, obtained using ASW band structure calculations, and experimental moments in Y2Fel4B and Nd2Fe14B, obtained from spin polarized neutron diffraction experiments by Givord et al.l7 is shown in Fig. 8.11. The large differences between moments on the corresponding Fe sites in the Y and Nd compounds seems quite unrealistic, in view of the calculational results shown in Fig. 8.9, and indicate in our opinion difficul ties in the analysis of the experimental data for these very com plica ted systems. Taking the experimental uncertainties into account, it can be seen that there is a fair correlation between theory and experiment, the largest discrepancy being found for the moment at theJ2 site. The experimental moments tend to be 0·1 and 0-2 J.lB per atom higher than the calculated ones, because the negative charge density in the interstitial areas is (see e.g. Fig. 8.4) not probed sensitively by the experiment. The site and rare earth dependence of Fe moments in R2Fel4B compounds which was found theoretically is also supported by the variation of Fe hyperfine fields, as obtained from 57Fe Mossbauer spectroscopy. The discussion of these

Permanent Magnets

c .0..

247

0

en

E

0 +-'

ro

>

--

~

3

en

Q)

rn +-'

s: en CD

rn (j)

0

'0 ~

'w c: Q)

"C

3

(ij

'ero

a...

0

3 ~~~~~~~~~~~~~~~~~~~~~~

-5

o

5

-5

o

5

-5

0

5

Energy (eV) Fig. 8.10 Partial densities of states of non-equivalent

sites in Y2Fe14B6.

results is beyond the scope of the present chapter, and the reader is referred to Refs 7 and 18. The calculated total magnetic moments for R2Fe14B compounds agree within 1·0 ,uB /formula unit for the LMTO and ASW calculations shown in Fig. 8.9. For Y2Fe14B, e.g., the total calculated moment is 30·6,uB/£U., and the experimental value is 31·0 ± 0·5,uB/f.u. It should be noted that the Y and B sites carry magnetic moments, too, which are induced by hybridisation with Fe-3d states. The calculated values are - 0-43 ,uB, - 0-34,uB and - 0·15 f.lB for Y(f), Y(g) and B sites. The negative polarisation on the Y sites (mainly of Y-4d

248

Electron Theory in Alloy Design

~ 2.5 3 o

(ij OC])

u,

E

2.0 2.0 mFe(~B)

2.5 (neutron diffraction)

Fig. B.l1 Calculated Fe moments in Y2Fe14B6, as a function of the moments in Y2Fel4B (filled circles) and Nd2Fe14B (open squares), as obtained from neutron diffraction."

character) leads (as we have seen in section 8.3) to an antiparallel exchange coupling of the R and Fe spin magnetic moments. This is a very general phenomenon, which is also found, for example, in binary Y-FeI9 and V-Co compounds. The negative sign of the polarisation can be understood from a consideration of the Y(4d)-Fe(3d) hybridisation. In the absence of this hybridisation the Y (4d) band lies essentially above the Fermi level. The hybridisation with the Fe(3d) minority spin band, which is partly unoccupied, is stronger than the hybridisation with the Fe(3d) majority spin band, which is almost completely occupied, and lies at lower energies. This results in a larger mixing in ofY(5d) character into the occupied part of the minority spin Fe(3d) band, than into the occupied part of the majority spin Fe(3d) band, and thereby to a negative polarisation on the Y atoms. The V-partial DOS for Y2FeI4B in Fig. 8.10 illustrates this point. A quantitative method for calculating the R-Fe exchange interaction has recently been presented by Brooks et al.I3 8.6 CRYSTAL FIELD INTERACTIONS AND RARE EARTH MAGNETOCRYST ALLINE ANISOTROPY Within the method for handling open 4f shells in band structure calculations which we described in the previous section the interatomic exchange interactions, leading to Hund's first rule, are taken into account properly. However, orbital correlations (leading to Hund's second rule) and spin orbit coupling (Hund's third rule) were neglected. As a result of the latter two interactions the charge density of rare earth ions in a magnetic field (or an exchange field) IS aspherical. The angular dependence of the 4j charge density can be calculated within atomic theory and is shown in Fig. 8.12. The large magnetocrystalline

Permanent Magnets

Ce

Pr

Nd

Pm

8m

Tb

Dy

Ho

Er

Tm

249

Gd

Vb

Lu

Fig. 8.12 Angular distribution of the 4f charge density of rare earth atoms, for MJ (effective moment parallel to the vertical axis), normalized per electron."

=

J

anisotropy (MeA) of Nd2Fe14B, SmCos and Sm2Co17 permanent magnets originates mainly from the anisotropy of the interaction energy between the aspherical4J charge clouds and the aspherical electrostatic potential at the rare earth sites. This contribution to the MCA, which results from the crystal field interaction, is called the rare earth single-ion anisotropy. For a long time it had been thought that the main contribution to the crystal field comes from the charges of the neighbour atoms ('point charge model'). However, recently it has become clear that the crystal field is predominantly due to the aspherical charge density of the 6s, 6p and 5d valence electrons of R atom itself.6,7,10-22 Of course, this sphericity is determined by interactions with the neighbouring atoms. The contribution to the crystal field due to the charge density outside the R atomic radius ('lattice' contribution) is in metals relatively small because charges are screened on atomic scale by the conduction electrons. First principles calculations of the rare earth single ion MCA in Nd2Fe14B and SmCos have been carried out recently. We will discuss here briefly the methods and results. To a good approximation the problem can be treated in a perturbative way: (i) the spherical potential can be calculated for systems with a spherical 4J charge densi ty and (ii) it may be assumed tha t the R- TM exchange interaction is much stronger than the crystal field interaction. Point (i) assumes that the crystal field is not affected by the asphericity of the R atoms. This assumption is supported by the experimental observation that in the series R2Fe14B (R = Nd, Tb, Dy, Ho, Er) the most relevant crystal field parameters do not depend significantly on the type of R atom. Therefore one can carry out the calculations of the crystal field using the methods for handling the 4f electrons which were discussed in section 8.5. Point (ii) implies that, to a good approximation, the 4J charge densities (Fig. 8.14) are not perturbed by the crystal field. Under this condition, which is often, but not always, fulfilled, the first order

250

Electron Theory in Alloy Design

anisotropy constant (at T KI

=

=

0) is given by

-tNCiJ4fA~(2J2

-

J) + higher order terms

(8.7)

where Nis the density of rare earth atoms. The lowest order term is due to the interaction between the axial quadrupole moment of the R atom and the term in the spherical harmonics expansion of the electrostatic potential at the R site. This contribution to the crystal field is indicated by the crystal field parameter A~. Note that A~ is zero for sites with cubic point symmetry. The sign of the second order Stevens factor r/.,J indicates whether, to second order, the 4f charge cloud is 'discus-like' (r/.,J < 0, e.g. Nd), or 'rugby-baIl-like' (CiJ > 0, e.g. Sm). Values for CiJ for all rare earth atoms in their ground state have been tabulated by Hutchings." Thermal fluctuations lead to a decrease of the time-averaged asphericity of the 4f shell. A generalisation of equation (8.7) to finite temperatures reveals that the higher order terms, go to zero much more rapidly than the lowest (second) order term." Therefore A~ is often the most relevant crystal field parameter at room temperature. The exact expression for 4JA~ is

r~

which looks a bit complicated because of the overlap between the 4f charge densities, P4J(r)~ and the charge density due to all other electrons, p(R) with R = (R, fJ, QJ). Here r < (r , are the smaller (larger) of rand R, respectively, and the integrations extend over the entire crystal. Band structure calculations for systems which arc as complex as, for example, R2Fe14B compounds are presently only feasible, within a reasonable amount of computer time, if the atomic spheres approximation (ASA) is used. In discussions on the origin of the crystal field, it is therefore natural to separate A~ into two parts: A~ (val), which is the valence electron contribution to A~ (obtained after integration up to R = rws(Gd), the Wigner-Seitz radius ofGd atoms, about 1·9A) and A~ (lat), which is the remaining ('lattice') term. An overview of results from ASW band structure calculations of A~ (val) for a series of systems is given in Table 8.3. We have used the units for A~ which are conventionally used in the field. Furthermore, we have also followed the convention to discuss A~, instead of the more relevant product 4fA~. In practice the 41 radial expectation value does not depend strongly on the compound .studied (about 1·0 a~ for Gd), which makes a separate discussion of A~ meaningful. Positive values of A~ (val) imply that the valence electron densi ty in the z-direction (c-axis) is larger than in the x and y directions (see equation (8.8)). In this situation R atoms with CiJ < 0, such as Nd, Tb or Dy, have their moments parallel to the c axis, in order to minimise the electrostatic 4f-valence electron repulsion energy. Indeed, equation (8.7) shows that KI is

Permanent Magnets Table

8.3

251

Comparison

of calculated values of structure theory, and values of A2• Units: Kao2. details, and references to the papers, have been given in Refs. 6, 15,

A~ (val), from ASW band experimental Calculational experimental 24.

Gd2Fe14Ba Gd2 Fe13SiBa

cace,

Gd2Fe17 Gd2Fe17N3

A6 (val)

A2 (expt)

376 500 -691 -302 -475

300-330 (400) - 200 ± 50 - 60 ± 40 (- 200)h

a Values of A2 are averaged over the f and g sites. hRough estimate (M.W. Dirken, R.C. Thiel, R. Coehoorn, T.H. Jacobs and K.H.J. Buschow, ]. Magn. and Magn. Materials, 1991, 94, LI5).

positive in such cases. Table 8.3 shows that calculations of A~ (val) can be used to predict successfully which type of R atom should be used, e.g., Nd and Sm for R2Fel4-B and RCo5 permanent magnets, respectively. Calculations of A~ (lat) using LMTO and ASW bandstructure theory are computationally much less straightforward than those for A~ (val), and have not yet been performed. Recently complete calculations of A~ were performed by Daalderop for the crystallographically relatively simple system GdCo5, using the self-consistent full potential augmented plane wave (FLAPW) method.21,24- A~ (lat) was shown to be indeed much smaller than A~ (val). The total calculated value for A~ had the right sign, but its magnitude was higher than the experimental value by a factor 4. Whereas the ab initio theory for A~ needs to be developed further, the new insight that A~ is mainly due to the asphericity of the rare earth valence electron charge density has already led to a simple modelfor trends in Ag, which can be quite useful in the search for materials with higher magnetocrystalline anisotropy.6,7 Within the spirit of Miedema's 'macroscopic atom' model of cohesion in metals'" one can view the formation of a compound as a process in which atomic building blocks (Wigner-Seitz cells) with different charge densities at the cell boundaries are brought together. Within the Miedema model these charge densities are indicated by model parameters nws, which depend only on the type of atom, and are in fact strongly correlated with the actual charge densities at the edge of the Wigner-Seitz cell in elemental metals. Upon compound formation the charge density differences at the cell boundaries between neighbouring atoms must be eliminated. If a certain atom is surrounded aspherically by neighbours with different values of nws, the process of eliminating charge density differences is expected to involve a charge distribution between orbitals with the same t, but with different azimuthal (m)

252

Electron Theory in Alloy Design

.' :. . '. '::::::~:!:;:'::" '. '. '.' ·:·neutr~I·. :::

z

.':: . ~neutrar.

: . ' .. : .. ::.f(::~~t;.:.;;· : ... '.. : . :~::::: :::::...:.:.~~

t tr~~;,~::.~~}: LxlY Fig. 8.13 Schematic view of the charge distribution on a rare earth atom, leading to a positive value of the crystal field parameter A~. The squares denote Wigner-Seitz cell boundaries. The cells have been taken as cubic for simplicity. The central cell contains a rare earth atom. The asphericity of the electron density on the central atom results from differences in the electron densities at the cell boundaries of the neighbour atoms (indicated with different densities of points).

quantum numbers. In the schematic picture of Fig. 8.13, e.g.,pz or dz2 orbitals are expected to be more populated than Px and Py or dx2 _y2 and dX), orbitals. In the example of Fig. 8.13, with the neighbour atoms with the highest value of nws on or close the the z axis through the R atom, one has A~ > O. Figure 8.3 shows that the latter situation applies to Nd2Fe14B, where all Nd neighbours (nws = 1·73) are situated in the xy plane, whereas most Fe neighbours (nws = 5·55) are situated in directions which are closer to the z axis through the R atoms. If one replaces the Fe (4c) atoms in the xy plane around R atoms by atoms with a lower value of nws, such as Si (n,vs = 3·38), one expects that A~ increases further. It is possible to realize such a substitution experimentally, and Table 8.3 shows that A~ increases indeed, theoretically as well as experimentally. The opposite situation is found for the nitrogenisation ofR2Fe17 compounds. N atoms fill three interstitial holes close to the R atoms in the x/y plane. The expected decrease of A~ which results from the increased valence electron charge density in the x/y plane has indeed been observed (see Table 8.3). I t leads to a huge anisotropy field for R = Sm (see Table 8.1). Carbon interstitials (nws = 5·55) could yield even larger anisotropies than nitrogen (nws = 4·49). Theoretically, this effect has been found.f" but the experimental verification is hampered so far by the difficulty to fill all available interstitial sites by C. Further theoretical and experimental support for this model has been given in refs. 7 and 26. Another important result of the ab initio calculations of crystal field effects was the clarification of the relationship between A~ and the electric field gradient (EFG) at the rare earth nucleus. Measurements of the EFG, by means of 155Gd-Mossbauer spectroscopy, play an important role in the systematic quan-

Permanent Magnets

t

C

«

I 0 ::J..

-0

253

50 40 30 20

Q)

'+>. = 0-

e

10

0 en

·c

-c

100

200

Temperature

300

400

(K)

500

~

Fig. 8.14 Temperature dependence of the anisotropy field, HA, of Nd2 Fel4B and Y 2 Fe14B (Yang Fu-Ming et al., The Magnetic Anisotropy of Nd-Fe-B and their Y-Fe-B magnets, in Proc. 8th Int. Workshop Rare-Earth Magnets and their application, Dayton, 1985).

titative study of crystal field effects. A discussion of this subject goes beyond the scope of this chapter, and the reader is referred to Refs. 6, 7 and 15.

8.7 MAGNETO CRYSTALLINE ANISOTROPY TRANSITION METAL SUBLATTICE

DUE TO THE

The rare earth and transition metal contributions to the magnetocrystalline anisotropy (MCA) in R-TM compounds can most easily be enraveled by making a comparison with the corresponding Y- TM compounds. Figure 8.14 shows that at room temperature the contribution from the Fe sublattice in Nd2Fe14B to the anisotropy field is smaller than the Nd sublattice contribution, but still significant. The anisotropy field of SmC05 (25-44 T at room temperature) contains a very large contribution from the TM sublattice: the anisotropy field for YC05 is already 17 T. 30 Just as in the case of the rare earth single ion anisotropy, spin-orbit interaction is essential for the TM sublattice contribution to the MeA. However, the detailed mechanisms are entirely different. In section 8.6 we saw that in the case of rare-earth atoms spin-orbit interaction leads to a coupling between the orientation of the 4Jmagnetic moment and the orientation of the aspherical4J charge density. The magnetocrystalline anisotropy energy (MAE) could be derived from a perturbative treatment of the interaction with the crystal field. It was the task of the theory to calculate the strength of the perturbation (the crystal field parameters). On the other hand, in 3d- TM systems the spin-orbit splitting is only a weak perturbation on the bandstructure. The spin orbit Hamiltonian is ~o = ~L· S, where the spin orbit parameter ~ is about 50 and

254

Electron Theory in Alloy Design

v ce, OY e •

Co 2c Co3g

Fig. 8.l5 Unit cell ofYCo5•

70 meV for 3d states on Fe and Co atoms, respectively. The resulting energy shifts are much smaller than the 3d band width and the exchange splitting between majority and minority spin bands. In this case the strength of the perturbation can be calculated easily; ~ is determined by the gradient of the electron potential and by the radial wave function of the 3d orbitals. The most difficult task of the theory is to calculate the effect of spin orbit splitting on the band structure, and to derive from these calculations the MAE, i.e. the energy difference I1E

=

Eo(Mjjc-axis) - EoCM.jja-axis)

(8.9)

in the case of systems with uniaxial symmetry. Band structure calculations of the TM contribution of the MAE should be carried out to a formidable precision, because I1E is very small. For Co, e.g., I1E is only - 0·065 meV jatom. The exceptionally large anisotropy field of YCo, results from a MAE of only - O: 76 meV jCo-atom. Successful calculations of the MAE of elemental Co and a number of binary compounds containing Fe and Co including YCos have recently been reported by Daalderop et al.21,27,28 Nordstom et al.29 carried out calculations for YCos, too, and reported similar results. As an example, we will briefly discuss the case of YCos' YCos is a structurally simple hexagonal compound. As shown in Fig. 8.15 the Co atoms reside on two inequivalent sites: the 2(c) and 3(g) sites. It is a saturated (strong) ferromagnet, i.e. the majority spin 3d band is completely occupied. This can be seen from the total DOS in Fig. 8.16, which was obtained from an ASW band structure calculation.f The calculations were scalar relativistic: the relativistic mass velocity and Darwin terms were included, but spinorbit interaction was neglected. The majority spin Co-3d band extends from about - 4·5 to - 0·5 eV, and the minority spin Co-3d band extends from about - 4 to + 1eV. The structure in the DOS above the Co-3d bands is mainly related to the Y-5d states. The total spin magnetic moments which follow from these ASW calculations, selfconsistent LMTO calculationsi'r" and full poten-

Permanent Magnets

t

255

10

(/)r-"'I

2~ cae.

5

o~ >.0

0

.•...• en en.

:t= :t=

enc

C::J (])'

-0>

5

(ij~ .•...• en O(])

I-(ti

~

10

-10

-5

0

Energy (eV) Fig. 8.16 Total density of states ofYCo5,

5

~

as obtained from ASW band structure calculations.

tial APW (augmented plane wave) calculations" range from 6·7 to 7·0J1BI formula unit. However the experimental total magnetic moment is much higher, viz. 8·33 IlBlformula unit.30 The discrepancy points to a large orbital contribution to the magnetisation, about 0·30 IlBICo-atom on the average, which has indeed been observed from neutron diffraction experiments." Daalderop et al." and Nordstrom et al.29 included spin-orbit interaction in selfconsistent LMTO band structure calculations, and found orbital moments of 0·12(0·12) J1B and 0·14(0·10) J1B, respectively, on the 2c(3g) sites. These values are significantly lower than the estimate which we made above. Similar calculations for elemental Fe and Co also yield moments which are too small by a factor of two;" Strongly improved results are obtained if a semi-empirical term of the form Hop = - BmorhLz is included in the Hamiltonian.V This so called orbital polarisation (OP) term accounts, to some extent, for orbital correlation effects (Hund's second rule in atomic theory), which are neglected within the LSDA.33 The proportionality factor B is derived from the selfconsistently determined radial parts of the 3d-wave functions, and morh is the selfconsistently determined orbital moment. The inclusion of the OP term enhances the orbital moments, to 0·23 (0,23) J1B21 and 0·26(0·17) J1B29 for the 2c(3g) sites in YCos, resulting in a good agreement between the theoretical and experimental total magnetisations. Before we proceed with giving the results of calculations of the MAE, we will first examine in some detail the energy shifts and orbital moments of individual states, caused by the inclusion of the spin orbit (SO) and orbital polarisation (OP) terms. For this purpose it is convenient to write the SO Hamiltonian in the form (8.10)

256

Electron Theory in Alloy Design

where L± and S± are raising and lowering orbital and spin angular momentum operators. So SO interaction mixes majority and minority spin states. Within a perturbative treatment of the energy shifts of eigenstates f/J a (a = (k, r, i)) due to SO interaction we have to distinguish two cases. First, if the eigenstate is degenerate one can apply first order perturbation theory, Two degenerate states with dxy and dx2 _y2 symmetry, e.g., whose wave functions form linear combinations with m, = + 2 and m, = - 2 atomic-like states, will split into these m, = ± 2 states if the quantisation axis (z axis) is along the e axis, whereas they will remain degenerate if the magnetisation is along the a axis. The energy shifts are if Mile-axis. This splitting will only affect the total energy, and will only lead to a net orbital magnetic moment, if the unperturbed energy of the degenerate states is within a distance, from the Fermi level. For Yeo5, which is a strong ferromagnet, this means that only minority spin states will contribute by this mechanism to the MAE and the orbital magnetisation. OP then effectively increases the SO parameter to ~ = ~ + 2Bmorb' The total change of the energy due to the splitting of degenerate states is expected to vary roughly as ~2, which is the product of the individual shifts and the width of the region around EF which is effective. Similarly, also the anisotropy of the total energy, ~E, is expected to vary roughly as ~2. The orbital moments, and their anisotropy, are approximately proportional to ~. The second case involves the shifts of non-degenerate states. Second order perturbation theory reveals that SO interaction leads to shifts of the individual levels which are proportional to and that the orbital moments of the individual levels are proportional to ~.34 Only interactions between states below and above the Fermi level have a net effect on the total energy and the orbital momen t. The shift due to the interaction between two states, f/J 1 and l/12' is largest for states close to the Fermi level, because the interaction varies like (EI - E2) -1. In the limit ofa large exchange splitting (top of the majority spin 3d band far below EF), interactions between states of different spins become very ineffective. Just as in the degenerate case, only interactions between minority spin states will then contribute to the MAE and the orbital moments. In that case one may replace ~ by ~ in calculations which include OP, and it can be proven that the anisotropy of the total energy, ~E, is proportional to the anisotropy of the orbital magnetic moment

±,

,2,

~E

=

~ ~morb

---4

(8.11 )

JlB

with ~morb = morb(Mlle-axis) - morb(Mlla-axis), provided changes in the shape of the Fermi surface upon the inclusion of SO and OP are neglected." These considerations of the effect of SO and OP interactions provide a good basis for the analysis of the results of the LMTO band structure calculations of the MAE. Daalderop et al.21 and Nordstrom et al.29 obtained ~E = 0·5 and 0·35 me V jCo-atom, respectively, whereas the experimental zero temperature

Permanent Magnets

t

E 0

257

a)

0

co

"C

(V)

> .s

-1

Q)

UJ

-

0 CJ

....-.-_.-

..•....'

-0.5

-1.0 0

10

20

40

30

CHROMIUM

THICKNESS

50

(A)

Fig. 9.3 Oscillations of the exchange coupling between two iron layers in a FeCrFe sandwich at room temperature as a function of the thickness of the chromium spacer layer (after Grunberg et ai.). 7 The inset is an enlargement of the tail of the oscillations.

with a period of 2ML for Cr thickness between 15 and 45A (10-30ML), except for a one-layer phase slip after 21 to 24 ML which indicates that the period is in fact not exactly 2 ML. However in a poorer quality wedge Unguris et ale find a long period oscillation with the usual period of 10-12 ML. It seems clear that in poorer quality structures irregular local variations in spacer thickness wash out the 2 ML period component of the oscillation. This does not 0.40 0.20

,

"""""""""""""""""""",

~

!

r

t-

.,-.. 0.00 N

S

"-0.20 to-:)

8

•.........••. N