THE ELECTRONIC STRUCTURE AND PROPERTIES OF SOME

TRANSITION AND NOBLE METAL COMPOUNDS

BY

EMMANUEL JOSEPH DANIYANG GARBA

Thesis submitted for the degree of Doctor of Philosophy of the

University of London and for the Diploma of Membership of the Imperial College of Science and Technology

September 1984

DEDICATED

TO

MY FAMILY

ACKNOWLEDGEMENTS

I do express my deep gratitude to my research supervisor,

Dr. R.L. Jacobs for suggesting the subject of the thesis. I am greatly indepted to him for his guidance, encouragement and above

all, for the keen interest he maintained throughout the duration of

this work.My sincere gratitude also goes to the Head of Mathematical

Physics Section, Professor E.P. Wohlfarth for his invaluable advice and encouragement.

I am indepted to Z. Vucic for stimulating conversations and for making available some of his unpublished experimental results.I wish also to thank the entire members of staff and students of

the department for their moral support throughout the period this work was being done. I also offer my sincere thanks to

Mrs P.A. Easton-Orr and Miss Annie Macpherson who typed this work.

A scholarship and a study fellowship awarded to me by the Plateau State Government and Federal University of Technology Owerri are gratefully acknowledged.

Finally, I would like to thank my wife, Everista, and my children for the great price they paid to see me through.

IV

ABSTRACT

This thesis contains electronic structure calculations of the noble metal compound Cu^Se and the transition metal compounds

Fe^Si, Fe^C and Ni^B. The properties of these materials are

discussed in terms of the calculated electronic structures.

The electronic structures are calculated in the Linear Combination of Atomic Orbitals (LCAO) approximation with hopping

integrals deduced by means of the method of Andersen from first

principles calculations due to other authors.The compound Cu Se is a superionic conductor (i.e. a solid

state electrolyte) at high temperatures and undergoes various phase

transitions depending on the degree of departure x of the material

from stoichiometry. The electronic conductivity displays interesting behaviour as a function of both temperature and x. This thesis discusses the behaviour of the electronic conductivity in the

various phases of the compound in terms of electronic structure calculations for simple models of the atomic structure of the various

phases. The phase transitions are also discussed.The compound Fe^Si is of interest because there are two

inequivalent Fe sites in it and substitutional impurities from the

first transition series show a systematic tendency to occupy one type of site or the other depending on their positions in the series.

This site preference tendency is discussed in terms of the calculated electronic structure and a simple physical argument for the observed site preferences is presented in terms of total energies calculated in the one-electron approximation from local densities

V

of states for various impurities at various sites.The compounds Fe^C and Ni^B are of great practical importance

as they represent the major tonnage of the world's steel production

and amorphous allotropes can easily be formed from them. Their crystal structures are much more complicated (containing at least

16 atoms per unit cell) than that of the Do^-type such as Fe^Si.

These two compounds are interesting from a theoretical point of

view, both, as regards their crystallochemistry because three types of bond T-T, T-M and M-M (where T = transition metal and M =

metalloid, may be realised in them simultaneously and as regards their magnetic transformations. The results of the calculation

may lend itself to the discussion of the possibility of an electronic contribution to the stability of these compounds. The

study of the electronic structure also provide us with some

information about their mechanical properties. We have discussed

most of the above properties in terms of the calculated electronic structures.

VI

CONTENTS PageDedication iiAcknowledgements iiiAbstract ivList of Figures ixList of Tables xi

CHAPTER 1

CONCEPTS IN ENERGY BAND THEORY AND PHASE TRANSITIONS 1

1 . 1 Introduction 1

1 . 2 The one-electron approximation 31 .3 Bloch's Theorem and the concept of energy band 5

1.4 Pauli Exclusion Principle and elementary consequences of Band Theory

11

1.5 The Muffin Tin Potential 141 . 6 Concept of phase transitions 171.7 Discussion 20

1 . 8 The tight-binding method and th‘e Slater-Koster interpolation Scheme.

23

CHAPTER 2

THE EXPERIMENTAL PROPERTIES OF Cu„ Se RELATED TO THE 27ENERGY BANDS

2 . 1 Introduction 272 . 2 Phase transitions 272.3 Structural Properties 312.4 Electrical properties 37

2.5 Magnetic properties 402 . 6 X-ray Emission and Adsorption 422.7 Diffusion of Copper in copper selenide 48

CHAPTER 3

THE ELECTRONIC STRUCTURE OF Cu„ Se 50

3.1 Introduction 503.2 Methods of calculation and Approximations 553.3 Results of the Calculations 68

3.4 Comparison with Experiment 79

CHAPTER 4

PHASE TRANSITIONS IN Cu„ Se 86

4.1 Introduction 86

4.2 Experimental phase diagram 86

4.3 Landau Theory 874.4 Phase transitions 91

4.5 Conclusions 94

CHAPTER 5ELECTRONICS STRUCTURE AND SITE PREFERENCE OF TRANSITION 96

METAL IMPURITIES IN Fe^Si

5.1 Introduction 965.2 Outline of calculations 100

5.3 Discussion and Comparison of Results 1275.4 Summary and Conclusion 133

CHAPTER 6

BAND STRUCTURES OF SOME TRANSITION METAL-METALLOID 134COMPOUNDS

6 . 1 Introduction 134

6 . 2 Details of Calculation 140

6.3 Results 145

6.4 Discussion and comparison of the results 151

REFERENCES 155

IX

LIST OF FIGURES PAGE

2.2.1 Equilibrium phase diagram of the Cu^ ^Se system 28

2.3.1 The planes stacking sequence along the b 32supperlattice axis [1 1 1 ]_ directionfee

2.3.2 Crystal structure of aCu^Se 35

2.6.1 a) Se M x-ray spectra and x-ray photo- 454 § oemission spectra of valence electrons inthe solid solution Cu^ Se2-xb) Energy spacing A between the peaks in the 45Selenium M spectrum4 f D

2.6.2 Forbidden-zone width at room temperature v 46composition of alloy Cu ^Se

2.6.3 Temperature dependence of the width of the 47forbidden band of Copper Selenide Cu„ Se(x = 0 .0 0 1 ) 2_X

3.1.1 Electronic conductivity (CT ) versus 51temperature (T) of Cu^ Se for variousvalues of x

3.3.1 Valence and conduction bands for the ideal 70Cu^Se crystal structure along certain symmetry directions

3.3.2 The Brillouin zone for the simple monoclinic 73lattice

3.3.3 a) Conduction band for the case of one inter- 74stitial atom and one vacancyb) Conduction band for the case of two inter- 74 stitial atoms and two vacancies.

3.3.4 The Se p partial density of states 75

3.3.5 The Cu d partial density of states 76

3.3.6 Total density of states 773.3.7 Total density of states of ideal Cu^Se convoluted 78

with a Gaussian of width 0.025 RydA graph of CF extrapolated to T = 0 against x.3.4.1 84

X

3.4.2 A graph of discontinuity at the phase transi­tion versus x.

85

5.1.1 The crystal structure of Fe^Si 975.2.1 Energy bands for paramagnetic Fe^Si 1095.2.2 The total density of states for pure Fe^Si

for spin up121

5.2.3 The total density of states for pure Fe^Si for spin down

122

5.2.4 The local density of states of up-spin electrons on an Fe atom or a Co or Mn impurity situated on a T site

123

5.2.5 The local density of states of down-spin electrons on an Fe atom or a Co or Mn impurity situated on a T site

124

5.2.6 The local density of states of up-spin electrons on an Fe atom or a Co or Mn impurity situated on a C site

125

5.2.7 The local density of states of down-spin electrons on an Fe atom or a Co or Mn impurity situated on a C site

126

6 .1 . 1 a) The structure of cementite 138b) The (100) projection of the structure of the cementite (Fe^C)

138

6 .1 . 2 a) Regular trigonal prismatic coordination polyhedron

139

b) Edge-sharing of polyhedra observed in the 139Fe^C, cementite structure

6.3.1 Electronic structure of Fe^C along certain high 146symmetry lines and planes

6.3.2 Electronic structure of Ni^B al°n<3 certain high 149symmetry lines and planes

6.4.1 X-ray photo-electron and x-ray spectrum ofiron carbide

154

XI

LIST OF TABLES Page

3.2.1 The tight-binding matrix elements in terms of overlap integrals

57

3.2.2 a) Slater-Koster parameters used in the computation (Ryd)

61

b) Additional Slater Koster parameters for nearest neighbour displacement (Hi) and (3/2H i) used in the computations (Ryd)

62

3.2.3 The one interstitial one vacancy case 653.2.4 Vacancy is shifted relative to interstitial

atom66

3.2.5 The two interstitials and two vacancies case 675.1.1 Nearest neighbour (n.n) configuration for

both Fe and Si sites in Fe^Si98

5.2.1 Matrix elements in terms of overlap integrals for the body-centred cubic lattice Fe3Si

1 m• \j

5.2.2 Slater-Koster parameters used in the computations (Ryd)

108

5.2.3 Continued fraction coefficients for orbitals 1165.3.1 Calculated magnetic moments on different

atoms on the two sites T and C131

5.3.2 The values of U. and E.(R) iQ i 1326 .2 . 1 The band parameters, Lattice constants and 142

positions of the sixteen atoms per unit cell in Fe^C

6.2.2 The band parameters, Lattice constant and 143positions of the 16 atoms per unit cell in Ni^B

6.2.3 Slater-Koster parameters used in the computations (Ryd).

144

1

CHAPTER 1

CONCEPTS IN ENERGY BAND THEORY AND PHASE TRANSITIONS

1.1 Introduction

In this chapter of the thesis a brief introduction is given to the

concepts and notations used in the following chapters. This introductory chapter has been included not only to provide a brief account of the

background to this work, but also to enable the necessary terms and

notationsto be consistently and precisely defined, for it appears that in the literature different meanings have sometimes been attached to some of these terms.

The basic properties discussed shall be restricted to regular

crystalline solids, which are assumed to extend indefinitely in all directions, and to those with lattice defects which can be treated

quasi-periodically. From the regularity of the arrangements of the atoms, there exist non-coplanar basis vectors a^, a_ , a^ such that

the atomic structure remains invariant under translation through any vector r given by

r—n n,a, + + n a1— 1 2—2 3— 3 (1.1 .1 )

where n^, n^, n^ are integers. The translation vector defined above is

known as a primitive translation. The set of vectors fr^} form a space lattice, called the Bravais lattice. Since around each lattice site of the Bravais lattice there is an identical arrangement of atoms, the structure of the crystal is determined if we define a volume

£2 = a. . (a A a ) — 1 —2 —3 (1.1.2 )

2

formed by the three translation vectors which when displaced by a vectorr^ will cover uniquely each point of the crystal. This volume is called

the unit cell and the set of vectors which specify the positions of theatoms in this volume is called the basis of the lattice. A useful unitcell is the one where each point in the cell is nearer to one site of the

Bravais lattice than the others. This cell, the Wigner-Seitz cell, is

obtained by drawing the perpendicular bisector planes of the translationvector r^ from a given point in the lattice to the nearest equivalent

sites. This is definitely the smallest volume around a given sitecontained within the planes which perpendicularly bisect the set of

vectors {r }. If there is only one atom per unit cell, the atomic sites —ncan be taken to coincide with the lattic sites of the Bravais lattice.

A lattice with a basis is one in which there are several atoms in the unit cell. This is the type of lattice we shall be concerned with in this thesis. There are indeed many types of crystal structures and they are classified according to their symmetry properties, such as invariance

under rotation about an axis or reflection in a plane and translation.

The fact that metals have crystal lattices with these properties allows the problem of the calculation of the one electron energy levels to be

reduced to manageable proportions and will often help to simplify our theoretical computations of chapters 3, 5 and 6 .

The remainder of this chapter will be devoted to an introductory

discussion of the basic ideas and assumptions of the one-electron band

approximations.

3

1.2 The One-electron ApproximationThere are in general two approaches to the study of the physical

properties of metals and compounds : the one-electron approximation, which assumes that each electron moves in a potential field independent of the motion of the other electrons, and the many-body theory. Ideally, a fundamental understanding of these properties can only be achieved within the frame-work of the many-body theory. Surprisingly good agreement with experiments has been achieved solely within the frame­work of the one-electron approximation. The Landau theory of quasi­particles has been used to give a partial justification for this.

In this thesis we shall only be concerned with one-electron aspects of the behaviour of some transition and noble metal compounds. We shall not attempt to derive the one-electron approximations from fundamental considerations.

We shall assume a crystal which extends indefinitely in all directions and a one-electron potential field which can be considered to be some average field experienced by the electron through its interaction with the other electrons in the system. The total wave function of the system of electrons is a combination of functions that each involve the coordinates of only one electron. If V,(r) is the potential experienced by each electron with an additional assumption that each one electron wave function is a simple product of an orbital function then the SchrOdinger equation for the orbital wave function for each electron i is of the form

-V 2¥. (r) + V. (r) 'i'.(r) = E. (r) (1.2.1)l — i — i — i i _

where is the energy of each electron measured in rydbergs, (r_) is its wave function, other quantities are in atomic units and V (r_) is the

4

Hartree or Hartree-Fock potential depending on whether the total wave function is taken as simple product or as a determinantal function of the one-electron wave functions. The Hartree and Hartree-Fock approximations are now scarcely ever used in calculations of the electronic structure of solids. Most often used nowadays is the Local-Density Functional approach in which (r ) in equation (1.2.1) is a function of the electron density nir) evaluated at the point r_. This density n(r ) depends, of course, on the (_r)'s which are solutions to equation (1.2.1) so that a self-consistent solution is necessary. However, this solution is easier to obtain than the solution to the ’ .. j Hartree-Fockequations which depend in a non-local fashion on the ^ A r ) %s . The solutions are now recognised to be more accurate and useful for the ground state properties of solids than the Hartree or Hartree-Fock solutions (Slater 1951, 1953, Kohn and Sham 1965).

The nature of the solutions to (1.2.1) can be deduced from symmetry properties of the crystal lattice. In particular, these solutions can be classified according to the rotational symmetries of the lattice.

The central problem in the one-electron approximation is thus the construction of a suitable approximate potential V ( r ) for an electron.The correlations between the electrons give rise to the many-body effects and can in principle be superimposed onto this potential in the many- body treatment. This leads to the solution of the Schrodinger equation for the electrons in a perfect crystal constituting a very complicatedmany-body problem.

1.3 Bloch's Theorem and the Concept of Energy Bands

The band approximation is based on the assumption that the average potential felt by each of the itinerant electron in a perfect single crystal is a periodic crystal potential V(r) such that

V(r + r ) = V(r) (1.3.1)— —n

where _r is as defined by equation (1.1.1).This periodicity of the potential defined above imposes

the Bloch condition

ik.r\|j(r_ + r ) = e “ \J>(r_) (1.3.2)

which requires that

ilc.r\p(_r) = e U(r_) (1.3.3)

where

U(r + r ) = U(r) (1 .3.4)— —n —

and 1c is a vector for each eigenfunction of the equation (1.2.1).The eigenstates can hence be characterized byavector Jc i.e.we can label the eigenfunctions of equation (1.2.1) by ip (r_)and the corresponding eigenvalues E(k_) . The wave functionsiJj, (_r) is subjected to the periodic boundary conditions' ic

6

\b, (r + r ) = \p_ (r) Yk — —s k — (1.3.5)

such that

r = s a + s a (1.3.6)—S 1— 2 3— 3

where s , s ^ , are very large integers.It follows from (1.3.1) that

k.r = 2nn (1.3.7)

where n is an integer or equivalently

ik. re = 1 (1.3.8)

This requires that

K n,b , * n2b2 ♦ n3b3(1.3.9)

and

a . . b . = 2tt5 . . (1.3.10)- i “ 3 13

where , b_ , b are the basic primitive lattice vectors of the reciprocal lattice and a , a > are t*"ie ^asic primitive translation vectors of the crystal lattice defined in §1.1.

7

The _b_. are given explicitly by

b, = 2H a2 a a3

[ r a_2 '.^.3 1

b = 277 a3 a a (1.3.11)[ar a2,a3]

b3 = 2ir ^ a a2

[a.-] t 'iLj ^

where [a_ ' —2 3 ^s defined as ^ * (a^ A .3 ) and since the vectors a , a , a_ are non-co- planar these vectors are also non-co'planar

and defined a three dimensional space called the reciprocal space of the crystal.

The vectors

I'm = n, b + nu b„ 2 —2 m3 — 3 (1.3.12)

where , m^r m^ are integers generate a lattice called the reciprocal lattice of the original direct lattice. From (1.3.7) the allowed _k vectors are given by

k_ = (2TTtn1 /s 1 + (2TTm2/s2 )_b2 + (27Tm3/s3 )b_ (1.3.13)

It is clear, since

iK < r —m —ne 1 ( 1 .3 .1 4 )

8

that equations (1.3.2) and (1.3.3) are equivalent to

i (k+K )» r_(_r) = e “m U. v (r) (1.3.15)

K K*rK— ---m

■where

Uk+K ---m(r_+r_

\ +K---m(r) (1.3.16)

and (r) is defined bym

-iK , r_°k+K =e (1-3‘17)---m —

-iK > _rThe function e has the periodicity of the lattice and may be

absorbed into the function U, (r).k —The wave vector _k does not label a state uniquely, the vectors

1c and k + are considered equivalent. It follows that eigenstates can only be labelled by k_ up to the addition of a reciprocal lattice vector. We can therefore take the eigenstates to be multi­valued function of k in a unit cell of reciprocal space. A convenient choice of the unit cell is the Wigner-Seitz cell of

the reciprocal space called the Brillouin zone. It is known that

the Brillouin zone of a face centered cubic lattice is the Wigner- Seitz cell for the body centered cubic lattice and vice versa.

We will now assume that s^, s a n d s^ are so large that the

finite set of vector k_ defined by equation (1.3.13) tends to a

9

continuum. The eigenfunction and eigenvalues E(Jc) are then

functions of the continuous variable k , having full periodicity

of the reciprocal lattice. Bouckaert et al (1936) showed that

E(JO has the full rotational symmetry of the crystal lattice.

As a consequence of this symmetry, E(_k) is uniquely specified

if it is calculated in an irreducible part of the Brillouin zone.

The function E(k_) can be obtained anywhere else in the Brillouin

zone simply by applying rotation operations on ]c-vectors in the irreducible part.

In general, there could be an infinite set of eigenstates and hence an infinite set of eigenvalues for each k vector in the Brillouin zone. This set of eigenvalues can be put into

one-to-one correspondence with integers by arranging them in

ascending order. We then obtain a set of functions (e^K)}, each continuous through the Brillouin zone and satisfying the

inequalities

E (k) < E (k) < E (k) < -- < E (k) --1 — 2 — 3 — n —

for all k . This set of functions {E^CJO} is called the energy

bands-Jones (1960) proved that within the first Brillouin zone the

En()c) are analytic functions ' of k_ for each band n, except at the points where two or more bands touch; at these points the functions E (k) for the bands that touch are continuous but have

discontinuous gradients. This scheme for the labelling of energy

bands is known as the reduced zone scheme. This scheme may

be repeated in all, cells of the reciprocal space, giving functions

En(k_) that have the periodicity of the reciprocal lattice called

the periodically extended zone scheme. Alternatively, it is

sometimes convenient to consider E(k) as a single-valued function

of k, Jc then running over all the allowed values (1.3.7) of the reciprocal space. This is the extended zone scheme.

The main task of this thesis is to discuss the problems involved in calculating E^CJO for some transition and noble metal

compounds and to investigate the physical properties which dependupon the band structure.

1.4 Pauli Exclusion Principle and Elementary Consequences of Band Theory

In the ground state of the metal, the states of lowest E (]c) are

occupied according to Pauli Exclusion Principle, which states that no two electrons may occupy the same one-electron state; that is, no two

electrons of the same spin occupy the same Bloch state. Not more than two electrons may have the same orbital wave function and when an orbital

state is doubly occupied, the two electrons concerned must have opposite

spins.At absolute zero of temperature, the electronic energy levels are

completely filled up to a certain energy, which is called the Fermienergy, E , and all the energy levels above E are unoccupied. The F Fsurface in Jc-space enclosing the occupied region of energy states is

called the Fermi surface. The number of states contained by the Fermi

surface is exactly equal to the number of valence electrons. At a finite temperature electrons with energies ~ KT below the Fermi energy can be excited to states with energies ~KT above, where K is Boltzmann's constant, and the Fermi surface becomes blurred, i.e. at a finite temperature there are electrons outside the Fermi surface and some holes

within it. At room temperature KT ~ 0.002 Ryd which is small, for

example, in comparison with the width of d-band in transition metals

(~ 0.1 Ryd). Thus, the energy bands in the neighbourhood of the Fermi energy will contribute nearly all the excitations and are physically the

most important in determining the electronic properties of solids.

We will therefore be concerned mostly with those energy bands which straddle the Fermi surface when dealing with transitional metal compounds.

One of the most important quantities in the description of a solid is the density of states per atom. This is the number of electronicstates

12

per atom or per spin in a small interval of energy (E,E + AE) and is given by

N(E)AE =8 7T3 n B. Z..

,.e+Ae_ f(E - E (k))dE d3k E n — (1.4.1)

where f(n) «s the Dirac delta function. The function N(E) is called the

density of states per spin/per atom and is of considerable physical

importance. The cummulative density of states per atom C(E) is defined as the number of states per atom with energy less than E. Thus

N(E) = dC(E)/dE (1.4.2)

The number of valence electrons per atom N is given byE

N = C(E ) F N(E) dE (1.4.3)

If n is the number of atoms per unit cell, then the number of atoms per unit volume is n/ft. Note that (1.4.1) could also be written as

(n/ft) N(E) = 1

8tt'ds

|V E(k) 1 k —(1.4.4)

or

N(E) = z f|n 3 n J 8tt j '

dSgrad E (k) n —

where the integration is over the surface of constant energy E(lc) = E.

In this form we can see that singularities in N(E) occur at those

values of E where for one or more bands grad E^tlO = 0 at some point onthe surface E (k) = E. The singularities which arise if E (k) is n — n —analytic in the neighbourhood of these points are called Van Hove singularities and are described by Ziman (1964). The point k^ in k_ space at which V^E(k^) = 0 is called a critical point. At a degeneracy

13

the E^flO °f the degenerate bands have at least one discontinuous

component of gradient. Phillips (1956) has shown that the point at which such a degeneracy occurs must be considered as a critical point. It is agreed that critical points more frequently occur at symmetry axes and planes than at general points in the Brillouin zone. The

alkali and noble metals have topologically simple energy bands, and

the only critical points that occur in the energy range of interest

being an ordinary minimum at the centre of the Brillouin zone,Cornwell (1961 ).

Later we will use the density of states/atom aswell as the density

of states/spin/atom to explain the various observable physical

properties of some transition and noble metal compounds.

14

1.5 The Muffin Tin Potential

One of the central problems in energy band calculations is the

construction of a suitable potential which represents the best possible approximation to the actual crystal potential V(r). The problem of

solving the Schrodinger equation for a given periodic potential can be considerably simplified for a certain class of potentials the best of which is the muffin-tin potential. A muffin-tin potential is a potential

that is spherically symmetric within the spheres inscribed in each Wigner-

Seitz cell and flat in the interstices between these spheres. The radii

of these muffin-tin spheres are arbitrary, except that the spheres should not overlap. It is usual to take the radius as half the nearest neighbour distance. The constant value of the potential in the interstices is called the muffin-tin constant and it usually proves convenient to

choose the zero of energy so_that..the muffin-tin constant is zero.

The method frequently used for setting up the muffin-tin potential is

that suggested by Matheiss (1964). In this method the exchange potentialV (r) and the coulombic contributions V (r) are treated separately,x — c —V (r) is obtained as a superposition of spherical (Soulomb potential c —derived from the free atom Hartree-Fock Calculation. The exchange form

is approximated as a local potential, the Slater approximation

where the crystal electronic density p(r) is obtained as a superposition

(1.5.1)

of the Hartree-Fock atomic density p (r)- The total potential is then

given by

V (r) = V (r) + V (r)T c x (1.5.2)

15

patThe muffin-tin sphere is

S .

V(r) = VT(r) - VAVQ (1.5.3)

where

VAVG ■ 3 Jr° V r) r2dr/(r03 - ri3) ( U 5 -4)1

where r^ is the muffin-tin radius and r^ is the Wigner-Seitz sphere

radius. As earlier stated, the zero of energy is usually chosen so that VavG is zero. For further details and references on the construction of

V(r) see Loucks (1967).

Segall (1957) advanced some arguments supporting the choice of

muffin-tin potential as a reasonable approximation to the actual crystal potential. This will be briefly reviewed here. In the neighbourhood of

the atomic sites, the potential is atomic in character and is therefore spherically symmetric. At the outer regions of the muffin-tin sphere,

the overlap of atomic potentials about neighbouring sites will destroy

this spherical symmetry. But because of the symmetrical arrangement of

the neighbour about any lattice point, these deviations are largely cancelled. The deviations from spherical symmetry within the muffin- tin spheres will therefore be small, in the interstitial region between the muffin-tin spheres, the potential is flat and can be closely approximated by a constant VA . these points the actual crystal

potential is small. By shifting the zero of the energy scale so that

VAVG = an^ deviations from an arbitrarily chosen approximating potential can be accounted for by perturbation theory.

Ham and Segall (1961) tested the appropriateness of the approximating potential by comparing the exact eigen values calculated from a Mathieu

16

potential for~a simple cubic lattice with the approximate eigenvalues calculated from the corresponding muffin-tin potential. They found that

errors were small, and can be further reduced by a perturbation calculation.

There are, however, two problems connected with use of the muffin- tin approximation. The first is that of constructing the muffin-tin potential from a given periodic potential . The second is that of

estimating the errors due to the approximation and correcting for them.

As is seen above, the muffin-tin potential corresponding to a given

periodic potential is easily constructed by carrying out appropriate averaging procedures in the two regions inside and outside the inscribed

spheres. The second problem is reduced by a perturbation calculation.In our thesis the use of this potential is an indirect one which

came from the tight binding parameters derived from the APW method band

calculation of Williams et al (1982), Switendick (1976) and the KKR

calculation of Jacobs (1968).

17

1. 6 Concept of Phase Transitions

A characteristic property of crystalline solids is that at

sufficiently low temperatures the atoms in them execute only small

vibrations about the crystal lattice sites. There also exists in nature amorphous solids in which atoms vibrate about randomly situated points. These bodies are thermodynamically unstable and must ultimately become

crystalline. To simplify the discussion we shall assume the atoms to be

at rest when speaking of configuration of atoms or its symmetry.If the number of crystal lattice sites at which atoms of a given

kind can be situated is equal to the number of such atoms, the probability of finding an atom in the neighbourhood of each site is unity and the crystal is said to be completely ordered.

In a case where the number of sites that may be occupied by an

atom of a given kind is greater than the number of such atoms, the

probability of finding atoms of this kind at either the old or new site

is not unity. The crystal becomes disordered. This normally occurs at sufficiently high temperatures.

We shall concentrate our attention on the general features of the

order-disorder transition. The effects of order-disorder transition on

electric resistivity, magnetism and other physical properties have been adequately investigated and are well documented by Muto and Takagi (1955), Salamon (1979), Landau and LIfschitz (1980) and many others.

Landau (1937) established some rules for classifying phasetransitions. These were again reviewed in Landau and Lifschitz (1980) and are based on the symmetry changes which occur at the transition. The most fundamental of these rules requires that the space group G of the low temperature phase of the crystal be a subgroup of the high temperature

18

space group Gq . The remaining of the rules determine whether the

transition will be of first or second order. O'Keefe (1976) also d’iscussed

phase transition in superionic conductors. He distinguished between major and minor changes of the lattice symmetry. His subclass lib transitions

satisfy the first Landau rule while the subclass Ila transitions do not.

Our model of chapter 4 belong to the class Ila transitions. O'Keefe's

class III transitions pose more of a challenge, since no symmetry

breaking occurs. We will not be concerned with this class in our discussion. We shall briefly review the main features of these phases as discussed by Landau and Lifschitz (1980) below.

The main features of the second order phase transition in contrast

to first order phase transitions are (i) the sudden rearrangement of the

crystal lattice between crystal modifications, (ii) a continuous change

of state of the body, (iii) a possible discontinuous change of symmetry at transition point, (iv) a continuous change of the configuration of the atoms in the crystal, (v) the change of symmetry as a result of an arbitrary small displacement of the atoms from the original symmetrical

positions, (vi) the "own" and "other" lattice sites are geometrically identical (Landau and Lifschitz (1980)) and differ only in that they have different probabilities of containing atoms of the kind in question.

When these probabilities become equal (they will not be unity, of course) the sites become equivalent and the symmetry of the lattice is increased. The symmetries of the two phases in the second order phase

transition must be related, but in first order phase transition, this need not be the case. We will not discuss the second order phase transitions which bring about a transformation between two phases differing in some of the properties of symmetry.

19

To describe quantitatively the change in the structure of the body when it passes through the phase transition point, we can define a

quantity r|/ called the order parameter, in such a way that it takes non-zero (positive or negative) values in the unsymmetrical phase and

is zero in the symmetrical phase. The symmetry of the body is changed

(increased) only when r| becomes exactly zero. Any non-zero value of the

order parameter, however small, brings about a lowering of the symmetry.

A passage through a phase transition point of the second kind has a continuous change of a non-zero value r) to zero. The continuity of the

change of state in a phase transition of the second kind is expressed

mathematically by the fact that the quantity r\ takes arbitrarily small

values near the transition point.In chapter 4 applications are made of the concept discussed above

with some modifications. A critical discussion of the phase transitions

in Cuprous Selenide is presented.

20

1.7 Discussion

We now review briefly the methods available for the calculation of

the electronic band structures and summarize the remainder of this thesis.

The methods of calculating energy band structures of solids fall into two categories; (i) Those based on first principle calculations which

directly solve a given one-electron crystal wave equation. In this category

are the Green's function method also known as the Korringa-Kohn-Rostaker

(KKR) Method, the Augumented Plane Wave (APW) Method, the Orthogonalized

Plane Wave (OPW) Method, the Tight-Binding Method and the Cellular Method.We shall describe in Section 1.8 the Tight-Binding Method. Common to

these methods are the expansions of unknownfunctions in the sets of known functions like plane waves and products of radial functions and spherical

harmonics. The expansion coefficients which are chosen so that the wave

functions satisfy the boundary conditions imposed by the cell are used to evaluate eigenvalues. Group theoretical considerations are often used to simplify the calculations. The choice of functions and boundary conditions, however, vary from method to method. Similar band structures are almost always obtained by the different methods. (ii) Those based on interpolation

schemes which describe the bands in terms of minimal basis set and the

corresponding disposable parameters. Among these methods are the atomic- ■ orbital scheme of Slater and Koster (1954) and the combined interpolation schemes or Model Hamiltonian of Hodges et al (1966) and of Mueller (1967).

First principle calculations are very complicated algebraically and computationally. They are also tedious to apply and require a large

amount of computing time to calculate the energy eigenvalues. Hence,

normally the energy levels are calculated over a few _k-points of high symmetry. Suitable interpolation schemes are then used to obtain the

21

energy levels at more Jc-vector points.

These energy levels can then be used for calculating the density of states per atom per unit energy range, N(E), which is directly related to the physical properties of the solid.

Because of the difficulties discussed above, we will not adopt the first principle approach to calculating the energy bands of the transition

and noble -m etal compounds of chapters 3, 5 and 6 . Our approach will be

to use an interpolation scheme with parameters deduced from application

of similar schemes to similar materials for which first principles calculations are available . More specifically the Slater-Koster

interpolation scheme based on the tight-binding method for both s, p and d bands will be used.

In this work we discuss, in Chapter 2 very briefly those experimental properties of xSe which are of interest to us and which relate to the

electronic energy bands pointing out where future calculations and experiments may be of value.

In chapter 3 we apply the Slater-Koster interpolation scheme of chapter 1 to calculate the energy bands of Se, x 0, for variousmodel crystal structures. The results are compared with experimental

results' and certain elementary physical consequences are discussed. In an attempt to understand the various phase diagram (see chapter 2 ) and

especially the degeneracy in the phase diagram observed- by some authors we have, in chapter 4, used the Landau Theory of Phase Transitions.

In chapter 5 we have applied the Slater-Koster interpolation scheme again to calculate the energy bands of Fe^Si. We also calculate the

local densities of states of pure transition compound, Fe^Si for various sites and spin orientations from a tight-binding model using Haydock's

22

recursion method. Also, calculated are the local densities of states of

the systems with various impurities in the various sites. These are used to calculate one-electron band energies for the various situations and

the site preferences of the impurities are discussed in terms of the differences between the band energies. A simple physical interpretation

of the results is proposed.

Finally, in chapter 6 we calculate energy bands of Fe^C and Ni^B whose crystal structure is completely different from those of chapter 5.

23

1. 8 The Tight-Binding Method and the Slater-Koster

Interpolation Scheme

In the tight-binding method, proposed by Bloch (1928) the wave

function for an electron in a crystal is taken to be a linear

combination of atomic orbitals. Suppose \jMr_) is an atomic wave function, then the Bloch sum is given by

$k (r>-n

N 2 £exp( ik.>.r .) \Jj (r-r.) j --------3 n -------- j

(1 .8.1 )

where N is the number of atoms in the crystal, the sum is over all

atomic sites r__. m the crystal, _k is the ree±pic7C<Jl vector and _r the position vector.

The function $k (£.) satisfies the equations (1.3.3) and ' n

(1.3.4). The Bloch sums (t.8 .1 ) have the properties

J$* (r) $. , (r)dT = 6. , ,5 (1 .8 .2 )k — k — kJi' nm—n - m —

and

(r_) H $ (_r )d'—n — m

= 0 if f K_ (1.8.3)

where H is the crystal Hamiltonian defined as

= H (r) o — + E V(r-r.)3

H (1.8.4)

24

in which

H (r) = - V2 + v(r) (1.8.5)o

with the assumption that

H t (r) = E \Jj (r_)o m — m m — (1.8 .6 )

where m refers to one of the s, p, d-type atomic wave function

and the crystal potential is represented as a superposition of spherically symmetric atom potential.

The wave function for an electron in the crystal is given

The symmetry properties of the Bloch sums are of considerable importance. It can be shown that the Bloch sums have the symmetry properties of the atomic wave function from which they are formed

(Cornwell (1961)).

An atomic s function has the formssf(r) while atomic p-wave functions may be taken to bs xf(r), yf(r) and zf(r) and the atomic d-wave functions to be xyf(r), yzf(r), zxf(r), (x2-y2 )f(r) and (3z2-r2 )f(r) where f(r) is a normalized spherically symmetric function which is unaltered by the operations of a point group.A Bloch sum thus, transforms as a spherical harmonic Y , (because

by

mZ a $ (r)m k — m —m

(1.8.7)

n

\JM_r) = f(r)Yn(r_)) by which it may be conveniently labelled (see,

Cornwell (1961), Egorov et al (1968), Lendi (1974) and Nussbaum

(1966)). The normalized spherical harmonics are denoted by1___ (_3z2-r2 )

sf x, y, z, xy, yz, zx, x2-y2 and y 3 . The function

do not form an ortho-normal set, but Lowdin (1950) has shown knthat it is possible, by taking linear combinations of atomic

orbitals to construct orthogonalized atomic orbitals. The Bloch

sums may then be formed from these, having coefficients which

are the same as in ordinary Bloch sums and having the properties

(1 .8 .2 ) and (1.8.3) above.

Slater and Rosier (1954) have shown that the Bloch sums formed using Lowdin’s functions have the same properties as

Bloch sums (1.8.1) and so they may be transformed as and be labelled by spherical harmonics.

Using (1.8.2) and (1.8.3) it follows that

I [ ($. H $ ) - E 6 ]a = 0 (1.8.8)m k„ kjm nm -n

The eigenvalues E, are given by solving the secular equation

det{(n/m). - E6 } = 0 (1.8.9)k nm

where

26

(n/m) ($. H $ )iin

L exp(lk.r.)J X*(r)i ~ ' 3 n ~

Z exp(ik.r.) E— 1 nm3 '

(p,q,r)

(1 .8.10)

where p,q, and r are integers r^ is a translation vector and

E are the energy integrals. i.e. (n/m) are the matrix elements nmof the Hamiltonian H between Lowdin Bloch sums X (r) and X (r).n — m —The forms of the matrix elements of the Hamiltonian and of the identity operator are given by Wong (1969). We can factorize

(1.8.9) by using the symmetry properties of the spherical harmonics

Y .nAs a method for calculating the energy levels from scratch,

the tight binding scheme is almost prohibitively difficult, for a large number of extremely difficult energy integrals (1 .8 .1 0 ) have to be evaluated. However, Slater and Koster have proposed using it as an interpolation scheme, with the integrals in (1 .8 .1 0 )

being taken as disposable parameters to be fitted by comparison

with accurate calculations carried out by other methods. In most applications, three centre and overlap integrals are neglected. This approximation is found to be good only if the atomic wave functions are highly localized around each atomic core, with small overlap onto adjacent atoms (Fletcher (1952)).

We shall in Chapters 3, 4 and 5 apply Slater Koster interpolation

scheme to the calculation of energy bands of some transitionand noble metal compounds.

27

CHAPTER 2

THE EXPERIMENTAL PROPERTIES OF Cii Se RELATED TO THE ENERGY BANDS. --------------------------------- 2-x------------------------------

2.1 Introduction

It is the purpose of this chapter to discuss briefly some of

the physical properties of copper selenide which depend on .the electronic

energy bands, and to point out where present and future calculations may be of value. Most of the properties reviewed here are supported by ample experimental evidence.

2.2 Phase Transitions

The experimental evidence suggests that there are at least two phase transitions in Cu^ ^Se at each value of x. Figure 2.2.1 contains a phase diagram due to Vucic et al (1981). The nature

of the phase transition on each line in this diagram is still contro­versial. Some of the available investigations on the nature of

the phase transitions are: (1 ) the ionic conductivity measurementof Takahashi et al (1976), (2) the electronic conductivity measurementof Ogerelec and Celustka (1969), (3) the thermal expansion, electron

diffraction and electronic conductivity measurements of Vucic et al (1981), (4) the x-ray measurement of Tonejc et al (1978),(5) the differential thermal analysis (DTA) of Vucic and Ogorelec (1980) and many more to be referred to in the discussion. All

report the results of their investigations over certain ranges of temeprature and departure from stoichiometry.

For a stoichiometric sample (x = 0), the main feature accompanying the transition to a superionic state is a change of the type of

Fig. 2.2.128

Equilibrium phase diagram of the Cu^ TrSe System: Second order

phase transition-Lines (1) and (2); structural phase transition-Line

(3); electronic conductivity data (Vucic et al (1981)) - (#), ionic

conductivity data (Takahashi et al (1976)) - (4)/ electronic conductivity data Qshikawa and Miyatani (1977JJ— (q)

29

crystal structure from a low temperature 3-phase of tetragonal structure

to a high temperature a-phase with a face centred cubic lattice.

Detailed descriptions of these structures will be given later.There is also a decrease in the activation energy (E ) for the ionic conductivity. According to some authors (Vucic and Ogorelec (1980))

the structural change and the decrease in activation energy occur

simultaneously. According to others (Tonejc et al (1978)) the structural change takes place at slightly high temperature.

Measurements on the nonstoichiometric samples (x=j*0), however,

show that the activation energy of the ionic conductivity reachesa value characteristic of all good superionic conductors of fee

type, a few tens of degrees before the structural change occurs.

This sequence of phase transitions is unusual. (1) The phase

diagram (fig. 2 .2 .1 ) for electronic conductivity measurement of

Vucic et al (1981) suggests that the phase transition for stoichiometric

Cu^Se is degenerate, but after small composition changes a splitting

of the phase transition is observed. (2) The structural phasetransition at ~ 140°C (Line 3) is characterized by a discontinuity -elects cin ion-irC conductivity. Thediscontinuity is usually large. Above

the temperature of this phase transition the material is in its superionic state and displays both cation and electronic conductivity. There is a rearrangement of the immobile-ion sublattice at the superionic phase transition temperature in addition to the disordering of the mobile ion sublattice. (3) All authors (Takahashi et al (1976),Vucic et al (1981)) agree that the structural phase transition involve a change of lattice symmetry. They do not, however, agree on the temperature of the transition, Takahashi et al finding 110°C and

Vucic et al finding 140°C at stoichiometry. Below this temperature

the material has a tetragonal structure. The temperature of the

transition was found to depend on departure from stoichiometry.

30

Thermal capacity (C ) measurements on Cu.Se and Cu Se by Kubaschewskip 2 1 .ybo(1973) suggest a weak temperature dependence of C far below thePphase-transition temperature T^ and a saturation of above T^.

There was an extra term in C observed within several tens of aPdegree Kelvin below T^ ascribed to Frenkel pair formation. Vucic

and Ogorelec (1980) obtained an activation energy of 50 k J mol and4

an enthalpy of transition of 6.83 k J mol ** . They observed that the activation energy was almost concentration independent and also

that the concentration of the conductivity ions remained unchanged

thruUghi/ut the structural yhabe transition. m l ______ ________ C _______ ____________j - l . - J ---3nicy UKiCLUie uuuCiuucu

that the process characterized by energy could not contribute to ionic transport in the specimen.

The nature of the structural phase transition (on line 3) is also

in some doubt. Originally Vucic et al (1981) concluded that this phase transition is of first order. These authors now feel that the evidence is inconclusive and that the phase transition is as likely to be of second order (Vucic, private communication).

The second-order phase transitions at lower temperatures (lines 1 and 2 ) are characterized by (a) a power law divergence in the

specific heat; (b) continuity of the ionic conductivity at the phase transition, although the activation energy of the ionic conduction appears to change; (c) the fact that they relate only to order- disorder transformation in the cation subsystem (Takahashi et al (1976)). Above these second order phase transitions two effects

31

are observed simultaneously. One is an exponential temperaturedependence of ionic conductivity characterized by an activation

-2energy of the order of a phonon energy (10 ev). Another observed

by DTA is an extra term in exhibiting an exponential temperature

dependence with an activation energy of about 1.6605 ev (100 k J mol ).

We shall discuss this phase diagram in more detail in Chapter 3.

2.3 Structural Properties

Since many of the fundamental questions about the superionic

conductors of which cuprous selenide is a member are structural in nature, structural probes are essential in understanding this

material.One of the main structural properties of this material is

the disordering of the mobile Cu ions, normally at high temperatures.

It has long been known that above a certain transition temperature Cuprous Selenide (Cu^Se) exhibits superionic properties (Rahlfs

(1936)) (values of obtained by various authors range from 110°c

to 140°c). Thus, its electrical properties as well as structural properties have been widely investigated (Borchert (1945), Stevels

(1969), Bueger (1965), Vucic et al (1981), Okamoto et al (1969),Murray and Heyaing (1975), Boyce et al (1981) etc). Most of these

investigations tried to determine the structures of the stoichiometric samples at low and high temperatures. A few were devoted to non- stoichiometric samples (Vucic et al (1981) and Okamoto et al (1969)).

In our discussion of structures we shall defer the detailed description of the structures of nonstoichiometric samples untilChapter 3.

The planes stacking sequence along the b superlattice axis ([111]f direction). Se cage planes-a; Cu cage planaplanes of mobile Cu-ions: tetrahedrally co-ordinated-d and octahedrally co-ordinated-b. the open and full symbols areused to visualize the variation of occupancy between the (ar c, b, d) sandwiches of planes. The arrows indicate pairingbetween tetrahedral vacancies and octahedral ions along [1 1 1 ] directionsfee

d a c b d aT o ▼V— ♦ ▼▼ o ▼▼ o ▼V — —♦ ▼▼ O T▼ o ▼V — —♦ ▼T o ▼▼ o ▼V — — ♦ ▼▼ o ▼▼ o ▼

d0-T

c b d a c b d a c b do ▼ o ▼ o ▼o V ▼ o Vo ▼ o ▼ o ▼o ▼ o ▼ o ▼o V —♦ ▼ o Vo V o ▼ <0 ▼o ▼ o ▼ o V<0 v— T o Vo ▼ o ▼ <0 ▼o ▼ o ▼ o ▼o ▼ o Vo ▼ o ▼ o ▼o ▼ o ▼ o ▼

dSUPER= 2d{f|C

33

The low temperature phase (Bcu^SeJis generally described as

having a tetragonal structure (Borchert (1945), Boyce et al (1981),

Vucic et al (1981) and Stevels (1969)). Vucic et al (1981) have found that the room temperature 3 -phase of stoichiometric cuprous

selenide is a superstructure of the rhombohedrally deformed fee

which could be described as an ordered cation subsystem (containing

half the Cu ions) within the "Zinc-blende frame-work" of immobile Cu and Se ions - the cage. The cage contains the remaining half

of the Cu ions. Most of the Cu atoms in the ordered cation subsystem

reside on a distorted fee sublattice, this sublattice being only partially occupied. One in two of the planes of this fee sublattice

normal to the <111 > direction are fully occupied. The remaining

nldH9 S of this suhisttic ■ v _ __ ^ Juiu LUJ.1UO UUUU^ICU. The vaCdiiuie^

in these planes are hexagonally ordered. The remaining Cu atoms are also hexagonally ordered in sites which are octahedrally co­ordinated with Se atoms (i.e. they are not on the fee sublattice).

These sites are related to the empty sites of the fee sublattice (vacancies) by a displacement parallel to the <1 1 1> direction. The plane stacking sequence along the b superlattice axis <111>

fee direction are shown schematically in figure 2.3.1 from data of Vucic et al (1981).

At the structural phase transition temperature T = 140°Cc(Vucic et al (1981)) a transition from the low temperature 3-phase to the high temperature a-phase is observed. The evidence for this single phase transition are the symmetrical peak in the linear

thermal expansion coefficient (due to 1.4% volume contraction) measurement and the sharp increase of electronic conductivity

34

(g ) at T . The behaviour of the g in the a-phase was reported e c eto be essentially metallic.

According to various authors (Rahlfs (1936), Borchert (1945)

and Boyce et al (1981)) in the high temperature a-phase the Se

atoms and the Cu atoms of the cage lie on an undistorted zinc-

blende sublattice. The remaining Cu atoms are distributed randomly

over the four tetrahedral (c), four octahedral (b) and thirty-

two trigonal (e) vacant sites of this sublattice (see figure. 2.3.2).

According to recent E X A F 3 data of Boyce et al (1981)

the mobile Cu ions are preferentially found in regions of space

centred at the tetrahedral sites and the probability density for finding them extends in the directions of the neighbouring octahedral

sites, but even at elevated temperatures the concentration of Cu ions has no substantial octahedral occupation. Boyce therefore

concluded that Cu ions are confined predominantly to the tetrahedral

regions. A Cu ion at the centre of a tetrahedral site is at a distance -/3/4a from the four neighbouring anions. If this ion moves to the centre of an octahedron, it is then at a distance of a/ 2 from the six neighbouring anions, but a closer distance

of -/3/4a from other Cu ions occupying tetrahedral sites. Boyce's findings thus suggest a large Cu-Cu repulsion for Cu ions in the octahedral sites and make it energetically unfavourable for a Cu ion to occupy these positions. However occupation fraction

calculated for each type of site as parameters in fitting the experimental results of Vucic et al (1981) reveals a nonzero

occupation of the octahedral sites (b). Vucic takes this to be

35

Fig. 2.3.2.

Crystal structure of aCu .Se

^7 o

36

one of the main characteristics of superionic states. It is

hard to see how the material can conduct ionically without pathways in the lattice for the mobile Cu atoms passing through octahedral

sites.The possible diffusion path in the fee (close packed) structure

have been discussed by Boyce and Hayes (1979). In the fee unit

cell, there are 8 tetrahedral and 4 octahedral voids. Each tetrahedron

shares faces with 4 octahedra and each octahedron with 8 tetrahedra.

The diffusion paths available for the mobile Cu ions consists of alternating unlike voids:

tetrahedral -*■ octahedral -*■ tetrahedral etc.

The distribution of the cations inaCu^Se was carefully studied by Ralhfs as early as 1935 and was reviewed by Funke (1976)-

This was also confirmed by Borchert (1945) and recently by Vucic et al (1981). One half of the cations (inaCu^Se) occupy the sites plus fee) which correspond to a zinc blendelattice.

The remaining 4 cations within the fee unit cell are more or less randomly distributed. Besides the octahedral and tetrahedral

positions, Ralhfs also considered 32 sites called 16 a (1/3) and 16 a (2/3). Each of these is situated on a < 111 > passageway between a tetrahedral and a neighbouring octahedral site, at the

intersection with the face common to both anion polyhedra. The

16 a (1/3) positions form sets of 4 subsidiary sites around each

of the 4 occupied zinc blende sites, while the 16 a (2/3) positions are correspondingly associated with the 4 tetrahedral sites (3/4, 3/4, 3/4; plus fee).

37

The application to the present work where we calculate electronic

band structures is obvious.

2.4 Electrical Propertiesse_rAccording to the concept of the semiconducting bond of -Pearson

Pearsonand M ossajc (I960), the solid is a semiconductor if all the possible

valencies lead to filled subshells, whereas the metallic state is characterized by partially filled valence orbitals of the component

atoms. This valence bond treatment is successful for many inter-

metallic compounds in which the bonds are formed by s or p electrons on each atom, (Hulliger and Mooser (1965) and Suchet (1959)) but it is not applicable for compounds which contain transition elements

or rare earths. Okamoto et al (1969) extended bond treatment of intermetallic compounds to the situation where d orbitals

as well as s or p orbitals contribute to the valencies of the

compound. Hulliger and Mooser (1963) have discussed a number of pyrite and marcasite phases and concluded that semiconductivity occurs if cation d electrons are assumed to be localized, while

if the d subshells are less than half-filled, metallic conduction

results.Electrical conduction of noble metals such as Cu which have

one s electron per atom and for which the d band is full, is considered to be effected mainly by the nearly free s electrons travelling through the lattice, the resistance being caused by scattering of these electrons by the atoms, when displaced from their mean positions by thermal vibrations.

38

In this work the material which concerns us is cuprous selenide (Q^^Se) which has been reported to be stable over a wide range

of departure from stoichiometry (x= 0 to x- 0.15) and over a

certain range of temperature (Ishikawa and Miyatani (1977), Takahashi

et al (1976), Vucic et al (1981) and Vucic and Ogorelec (1980)). Various authors (Celustka and Ogorelec (1969) and Sobolev et al

(1969)) have found that at low temperature Cu^Se is a semiconductor

with a band gap greater than 1 ev.

Several interesting results were reported which relate the electrical properties of this material to temperature and crystal

structure.

We shall in this section review these properties.The conductivity of this material is mixed cationic and

electronic, with the electronic component dominating even inthe superionic a-phase. At 700°C for example, the ionic conductivityof Cu^Se has the high value of 4 (ftcm) ^, but this is only 5per cent of the total conductivity (Boyce and Hayes (1979).

It achieves this conductivity while still in solid phase.Okamoto et al (1969) found that Cu^Se is less conductive

than other Cu-Se compounds e.g. CuSe^ which exhibits mostly metallicconduction. This tendency has also appeared on a preliminarymeasurement of the thermoelectric power: + 50y V/°c for Cu^Seand 20P V/°c for Cu. _Se at room temperature, whereas only a1 . 0

few microvolts/°C for others (Okamoto et al (1969)). The differences in the resistivities according to Okamoto are related to the

binding energy of the Cu-Se bond. Similar experimental results were reported by Vucic and Ogorelec (1980) and Voskanyan et al

39

(1980). These authors also found for nonstoichiometric materials

that below the phase transition temperature (line 3, fig. 2.2.1)

resistivity and thermoelectric power both increase with temperature. For these nonstoichiometric materials there is substantial vacancy

transport of copper which becomes more significant at higher

tempeatures. This increases the electron-Cu scattering and

leads to an increase in the resistivity with temperature. The

substance becomes more electronically conducting with low cation content (i.e. increasing departure (x) from stoichiometry).

4The consequence of the copper deficit is an unfilled valence

band and thus the high conductivity can be ascribed to the introduction

of nonstoichiometry which moves the Fermi level into the valenceband and destroys the semiconducting properties.

Further when the temperature is raised above Tc (i.e. the

material is above line 3 in the phase diagram) there is a sudden increase in the electronic conductivity (Vucic et al (1981)).

This increase can be attributed to an increased concentration of Cu atoms in the octahedral sites (i.e. to the breaking of many Cu-Se bonds), and thus to the introduction of more holes into the valence band.

The ionic conductivity, however, has no discontinuity or

kink at Tc (Vucic et al (1981), Takahashi et al (1976)). Later in Section 2.7 we shall see that this can be attributed to a

grain boundary diffusion as the major mechanism for ionic conductivity at these lowish temperatures.

The band calculation presented later should enable some of these views to be investigated qualitatively.

40

2.5 Magnetic Properties

Experimental work on the gyromagnetic effect has indicated

that the magnetic properties exhibited by solids are largely

due to electron spin. There is also a small effect due to

their orbital motion. Associated with each half spin electron

is a magnetic moment and this must be quantized, either parallel

or antiparallel to the direction of a magnetic field (Debye,Sommerfeld (1916)). In non-magnetic material, where there is

no suspicion that atoms or ions carry localized magnetic moments,

if we ignore electron spin or assume that all spins are tightly

paired there still remains a contribution to the magnetic susceptibilityfrom the orbital motion of the electrons. The simplest case

is where each atom or ion consists of closed shells of electrons,and where the energy of excitation to a higher state is large.It is well known that this gives rise to a diamagnetism (negative

susceptibility). It has been pointed out that the necessary conditions for electronic paramagnetism are, the atoms and lattice

defects must possess odd number of electrons with non-zero total spin and an incomplete inner shell of the electrons, capable therefore of taking on a magnetic moment by proper orientation

of their spins. In a non-magnetic state the spins must just balance each other by setting themselves in antiparallel pairs.

When a magnetic field is imposed on the system, the energy

of parallel spins is lowered and that of anti-parallel spins is raised. This results in a positive contribution to the susceptibility i.e. paramagnetism. In a metal where there are free unpaired electrons, this contributions is called the Pauli paramagnetism.

41

The susceptibility due to this effect is only weakly temperature

(T) dependent and at low T measures directly the density of states

at the Fermi level N(E ). In principle, then, it ought to correlateFdirectly with the electronic specific heat coefficient, although when one makes correction for many electron interactions the exact equivalence is destroyed. The difficulty about making

comparison of this sort, in practice, is that the diamagnetic

correction is not negligible, and is not known very well. But

one can measure the spin susceptibility directly, in some cases,

by observing the electron-spin resonance of the conduction electrons.

For some other types of solids the paramagnetic susceptibility is temperature dependent and obeys the Curie law. This holds

especially where the magnetic atoms or ions are well separated

from one another in the crystal, so that the assumption of independence which lies behind the Curie law is valid.

Measurement of the susceptability (X) of Se for 0 < x ^ 0.5

and T < 4 00 K have been carried out by Vucic and Ogorelec (1980) and less extensively by Okamoto et al ( 1969). These measurements show that Cu^ ^Se is diagmagnetic with a temperature independent susceptibility (in the above T range) for x< 0.05. This indicates that the Cu d band is full for this range of concentrations.

Further a Curie-Weiss contribution to X is observed for x > 0.05. this probably arises from holes in the Cu d band.

42

2 . 6 X-ray Emission and Adsorption

When an electron is removed from a core level in an atom

of the solid, electronic transitions from higher occupied levels

into the core hole result in soft X-ray emission. Here we are

concerned with transitions from the valence bands of a metal into

core holes. The discrete energy levels of the atom are broadened

into bands and hence the emission spectrum consists of a continuum.

The band of frequencies of the emitted radiation reflects the band states occupied by the conduction or valence electrons.

The actual shape of the spectrum depends on the product of the

density of states in the bands with the square of the appropriate

matrix element for the transition (Ziman (1964) p. 276). This

matrix element may vary considerably through the band according to whether the core level is an s, p or d - state and according

to whether the valence band wave function ty (r) is more or less, s, p or d - like in the interior of the atom. There are also effects due to electron-electron interaction.

Ziman (1964) has explained that the exact shape of the emission

spectrum is not necessarily a direct measure of the density of states, although it should reflect some of the features of that

function, especially the sharp cut-off at the Fermi level.Soft X-ray emission has two major advantages in probing

the density of states of the conduction band. The first advantage is that by choosing the symmetry of the core-state we can select the symmetry of the valence band state involved in the transition and thus obtain information on partial densities of states.

43

For example, an s-core state provides information on the p-component of the valence band density of states, whereas a p-core state

provides information on the s and d-components of the valence

band state, the d-band density of states usually predominating

in materials with full or partially full d-valence bands.

The second advantage is that in an alloy or compound it

is possible to select core-states on one constituent of the material or another, thus providing information on the local density of

states in the material.Recent soft X-ray emission measurements on Cu,, Se have beenX

made by Terekhov et al (1983). They used a Se M core state4 f bi.e. on s-like core state which provides information on the p-component

of the valence band density of states on Se atoms. They found that this component is split into two roughly symmetrical peaks

(see fig. 2.6. la}. They also carried out X-ray photo emission measurements which provide information about the total density of states of the material which is, of course, dominated by the d-band density of states of the Cu atoms. They found that this density of states had a large peak precisely in the minimum between

the two peaks of the soft X-ray spectra. This peak must be attributed to the Cu d-band.

Further measurements at room temperature of the principal

optical absorption band edge in Cu^^Se by Sorokin and Idrichan (1975) give the band gap in this material as a function of x (fig. 2.6.2). This band gap increases with x.

Finally (and perhaps slightly out of the context in this section) thermal conductivity measurements of Cu^^Se by Zhukov

et al (1982) for very low values of x (i.e. an approximately stoichiometric material) and high values of temperature (T) give

the band-gap as a function of T. This is shown in fig. 2.6.3.

The band gap decreases as a function of T.

All the above measurements will be discussed in the context

of our calculation.

45

Fig. 2.6.1.

I ' Se (solid curves) x-ray spectra and (dashed curve) x-ray photoemission spectra of valence

f b) „ . _Energy spicing A be­tween the peak* B and Bi in the seleni* um M4i5 spectrum < i). and width of the band gap (2). as functions of the composition of the solid solutionCu,-xSe.

46

Fig. 2.6.2

Forbidden-zone width at room temperature Vs composition of alloy Cu^ xSe

j f , ev

47

Fig. 2.6.3.

Temperature dependence of the width of the forbiddenband of Copper Selenide Cu„ Se (X = 0.001)2-x

woor,n'JOO

900

48

2.7 Diffusion of Copper in Copper SelenideThe motion of the cation in cuprous selenide solid electrolyte

must be a highly correlated process. Each single diffusion step of an ion will result in a change of the local potential

seen by the ion itself and by its neighbours. These changes,

due to the electrostatic and the hard core interactions, are

comparable to the average energy needed for a diffusion step.Thus each step will not only influence the next step of the same

ion but also those of its neighbours.

Diffusion of copper in copper selenide samples have been investigated by Chatov et al (1980) and Celustka and Ogorelec

(1966). Fitting the experimental data for the diffusion constant

of copper vacancies at various temperature (ranging from 180°C to 600°C) with equation

D = Dq exp(-E/KT) (2.7.1)

where the pre-exponential factor Dq and the energy of activationfor diffusion E are the fitting parameters, Celustka and Ogorelec

_2(1966) found that the value of D is very high (0.5 x 10 cm2/sec at 450°C). This can be understood on the basis of the crystal

structure of cuprous selenide in its high temperature phase.

The number of the lattice sites that may be occupied by the Cu ions is much greater than the number of the Cu ions present,Rahlfs (1935). Such a crystal structure seems to make possible fairly free motion of the copper ions. The paths along which these mobile ions move have been described in Section 2.3.

Temperature (T) dependencies of the diffusion coefficient

D have also been investigated by Chatov et al (1980). He i d e n t i f i e d

49

two diffusion mechanisms: (i) diffusion along vancancies which

predominates at high temperatures and (ii) diffusion along

grain boundaries which is not very sensitive to temperature and

predominates below 450°C . The predominance of the vacancy diffusion

mechanism at high temperatures is supported by the fact that

in agreement with theoretical predictions, the diffusion coefficient of copper in Se increases on increase in the degree of non­stoichiometry. There is no discontinuity or kink at the transition

temperature in the temperature dependence of D for single-crystal

samples suggesting structure-insensitive diffusion along grain

boundaries. It was further confirmed that an increase in the

nonstoichiometry resulted in predominance of the vacancy diffusion mechanism.

The predominance of the grain boundary diffusion mechanism

at the transition temperature should also explain the absence of a kink in the ionic conductivity at the transition temperature (see Section 2.4).

50

CHAPTER 3

The Electronic Structure of Cu^ ^Se

3.1 Introduction

Superionic conductors have been studied intensively in recent

years from an experimental and theoretical point of view (Takahashi

et al (1976), Vucic et al (1981), Sobolev et al (1969), Ishikawaand

Miyatani (1977) and Celustka and Ogorelec (1969)). This study has been motivated by the potential practical application

of these materials to electrical storage batteries. The material

which concerns us here is cuprous selenide (Cu2 _xSe) which is stable

over a wide range of departure from stoichiometry (x = 0 to x - 0.15) (Takahashi et al (1976) and Vucic et al (1981)). This material has a phase diagram of a particularly interesting form and displays simultaneous ionic conductivity and electronic conductivity

a over certain ranges of temperature (T) and x (Takahashi et

al (1976), Vucic et al (1981), Sobolev et al (1969) and Chartov et al ( 1980) ) .

Our aim is to understand the behaviour of a in terms ofetheoretical models of the band structure. Presented in fig.2.1.1. is the phase diagrams of Cu2 _xSe in the x-T plane dueto Vucic et al (1981). Also presented in fig. 3.1.1. is thea versus T curve for various values of x. e

A nearly stoichiometric material (x - 0) has a structuralphase transition at T = 140°C (Vucic et al (1981)) from an orderedclow temperature phase ftC^Se in which the ionic conductivity

Electronic conductivity (a ) versus temperature (T) of.a 51

Fig. 3.1.1.

52

and the electronic conductivity G^ are both close to zero to

a disordered high temperature phase aCu Se in which CL and 0

both take on much larger finite values. When x differs from

zero the phase transition splits into two, a second order transition

on line 1 from the ordered phase (3Cu^ ^Se) to a partially ordered

phase and a structural transition on line 3 from partially ordered

phase to a disordered phase (aC^ Se). In the ordered phase

Q. is zero and G is finite and closely related to x. In the partially ordered phase CL is still very small and a is reduced

and in the disordered phase CL is finite and G^ increases again to a large value which also depends on x. Various authors (Sobolev

et al (1969), Celustka and Ogorelec (1969) and Okamoto et al (1969)) have found that BCu^Se is a semiconductor with a band

gap greater than 1 ev.The crystal structure of the various phases of Cu^ ^Se can

perhaps be best understood in terms of successive deviations from the following ideal model structure of stoichiometric Cu^Se.(1) In this ideal model (which we shall call structure (1))the Se atoms lie on a face centre cubic lattice and the Cu atoms lie on a simple cubic lattice in which alternate cube centres are occupied by Se atoms on their fee lattice. This simple

cubic sublattice is made up of two interpenetrating feesublattices. The successive deviations that produce the structures of the phases that are actually observed are discussed successively below.(2) The structure of the ordered low temperature stoichiometric

53

(ftO^Se) (structure (2 )) is obtained from this ideal structure

in the following way:

The lattice of Se atoms is unaltered and half of the Cu

atoms remain in this ideal positions on one fee sublattice, these

two sublattices together forming a wartzite (ZnS) structure

which we shall call the cage. The remaining fee sublattice

is only partially occupied. One in two of the planes of this

fee sublattice normal to the <1 1 1> direction are fully occupied. The remaining planes of this sublattice are two-thirds occupied.

The vacancies in these planes are hexagonally ordered. The remaining Cu atoms are also hexagonally ordered in sites which

are octahedrally co-ordinated with Se atoms. These sites are related to the empty sites of the fee sublattice (vacancies)by a displacement parallel to the <1 1 1> direction.

The exact details of this structure are reported elsewhere

(Vucic et al (1981)). It is sufficient for our purposes to say that there are 12 Cu and 6 Se atoms per unit cell and there

is a rhombohedral elongation of the whole structure along the

<1 1 1> direction.(3) The structure of the high temperature stoichiometric phase (aCu Se) (structure (3)) differs from that of ftC^Se (structure (2))

only in that the Cu atoms of the second fee Cu sublattice are distributed randomly over the tetrahedral and octahedral holes in the Wurtzite structure (i.e. the cage) comprising of the first Cu fee sublattice and the Se sublattice. The resulting structure

has cubic symmetry (on average) and there are two Cu atoms and one Se atom in the average unit cell (one of the Cu atoms being

of course mobile).

54

(4) The structure of the ordered low temperature non-stoichiometic phase (BCu^^Se) (discussed by Okamoto et al (1969)) appears

to be similar to that of the stoichiometric phase (structure

(2 )) with a random distribution of vacancies over some (unknown)

subset of Cu sites not in the cage.

(5) The structure of the partially ordered nonstoichiometric

phase differs from the above in that the hexagonal ordering of vacancies in the second fee sublattice and Cu atoms in octahedral

sites is lifted and these vacancies and atoms appear to be free

to move in their planes normal to the <1 1 1> direction.

(6 ) the high temperature nonstoichiometric phase (ctCu ^Se) is similar to that of stoichiometric material except that some

of the mobile Cu atoms are absent. The exact details of this structure are reported by Okamoto et al (1969).

It is worth noting that in all the nonstoichiometric phases the cage remains undisturbed.

In this chapter we shall present tight-binding (LCAO) cal­

culations of the electronic structure of Cu„ Se for several2-xmodels (to be described later) of the crystal structure of the various phases.

In the ideal structure (structure (1)) where there are only 2 Cu atoms and one Se atom in the unit cell the basis set will

consist of the s, p and d states on the Cu atoms and s and p states on the Se atom.

In the models for the less ordered phases where there are

at least 12 Cu and 6 Se atoms per unit cell the basis set consists only of s and p states on the Cu atoms. Nevertheless, by comparison

55

with results for the ideal structure we can draw certain conclusions

about the electronic conductivity.

The parameters of the tight-binding models are obtained

by extrapolation from the parameters for the same elements found •

by other authors in different systems.

3.2 Methods of Calculation and ApproximationsIn this section we discuss the method used to obtain the

band structure for the various models of the crystal structures discussed above.

A thorough study of the electronic structure should involve

a realistic calculation of the band structure to facilitate comparison

with experiment and aid in the interpretation of the main features

in a simple physical way. Unfortunately, there is only a limitedamount of experimental information on the band structure of Cu„ Se2-xin all its phases. Also the number of atoms per unit cell and the complexity of the compound prevent a first principles calculation of the band structure and even a complete LCAO calculation.Thus we do not carry out the full LCAO program for the actual

crystal structure, but at this stage our aim is only to find

seme fairly realistic, but computationally more tractable approximations to it.

As a first approximation we carry out the band calculation for structure (1) which has only 3 atoms per unit cell.

For the full band scheme we followed the LCAO approach of

Slater and Koster (1954), reviewed in Chapter 1. We use a basisset of twenty-two wave functions. One 4s and three 4p for

56

Se — and five 3d, one 4s and three 4p for each Cu+ ion, each

unit cell containing two Cu sites.

The 4s, 4p functions for Cu+ give rise to the conduction

band and the remaining basis functions give rise to the valence-band

as is obvious from simple chemical considerations.We take the matrix elements of the Hamiltonian H or (n|m)^,

between the functions of the basis set above and construct our

secular equation at any point in the Brillouin zone. The values

of E which satisfy (1.8.10) at any _k are then the energy eigenvalues. The secular equation we obtain contains a matrix which can be

divided into nrne submatrices.A B CTB D E

cT TE FI J

These are three 9 x 9 Cu-Cu tight binding, matrices A, C and F; two 9 x 4 hybridization matrices B and E between the tight-binding

Cu-Se functions and one 4 x 4 Se-Se tight binding matrix D.

Other matrices with subscript T are the transpose complex conjugates

respectively. The matrix elements of the Hamiltonian worked out for the s-, p- and d states are presented in Table 3.2.1 and will be used to obtain the various energy integrals. We have restricted our treatment to the first and second neighbours only for the Cu-Cu energy integrals and the first neightbours only for Se-Se, Cu-Se and Se-Cu energy integrals.

57

TABLE 3.2.1

The tight-binding matrix elements in terms of overlap integrals

Submatrix A

T = s + 4(ssa)^(cosXocsY+cosYcosZ+cosZcosX)s,s o 2

T = 2y/2i (spa)„(sinXcosY+sinXcosZ)s,x 2

T = -2>/3 (sdci) _ sinXsinYs,xy 2

T 2 2 = /3 (sda)_(cosXcosZ-cosYcosZ)s,x -y 2

T _ 2 2 = (sda)„(-2cosXcosY+cosXcosZ+cosYcosZ)s,3z -r 2T = p + 2(ppa)_(cosXcosY+cosXcosZ)+2(ppTT)_(cosXcosY+cosXcosZ+2cosYcosZ x,x o 2 2

T = -2[ (ppa) -(PPTT) ] sinXsinYx $ y £ £

T = / 6 i(pda) cosXsinY+2>/2i (pdir) sinYcosZx / xy zT = 0x, yzT 2 2 = / 3/2 (pda)0 isinXcosZ+2v/2 i(pd7T)_ [sinXcosY+jsinXcosZ]x 9 x A

T _ 2 2 = -/2 (pda)0 i[sinXcosY-|sinXcosZ]-/6 (pd7T) i sinXcosZX/jZ £ £

T 2 2 = -/3/2(pda)_i sinYcosX-2\/2i (pdir) _ [sinYcosX-^sinYcosZ ]y ,x -y 2 2

T _ 2 2 = -/2i(pda)_(sinYcosX-|sinYcosZ)-/6 i(pdiT).sinYcosZy, 3z -r 2 2

Tz,x2-y2 = [>/3/2 (pda)2 -/2 (pdTT)2 ]i sinZ(cosX-cosY)T 2 2 = [ 1/>/2(pda)^-h/6(pd7T)^]i(cosXsinZ+cosYsinZ)z, 3z -r 2 2

T = d +3(dda)^cosXcosY+2(ddTT)^(cosXcosZ+cosYcosZ)xy,xy o 2 2

+ (dd5)2 (cosXcosY+2cosXcosZ+2cosYcosZ)T = 2[-(ddir) +(dd5) - ] sinYsinZxy,zx 2 2

T 2 2 = 0xy,x -y**xy,3z2-r2 ~ f (cider)2“ (dd^)2 sinXsinY Txz,x2 -y2 ~ (dda)2~(dd5 )2 ] sinXsinZTxz,3 z2 -y2 = V 3 /2 [(ada )2-(ad6)2J sinXsinZ

)

58

T 2 2 2 2 = d +4 (ddIT) cosXcosY+ [ 3/4 (dd(7 L+ (ddTT) _x -y ,x -y o 2 2 2

+9/4(dd6 ) 2 (cosXcosZ+cosYcosZ)T 2 2 2 2 = /3/4[(dda) -4(ddTT)_+3(dd6 )„](cosXcosZ-cosYcosZ)x -y , 3z -r 2 2 2T 2 2 2 2 = d +1/4[(ddCT)+3(dd5)_](4cosXcosY+cosXcosZ+cosYcosZ)3z -r ,3z -r o 2 2

+3 (ddTT) [cosXcosZ+cosYcosZ)

Submatrix B

T = 4(ssQ)(cos^XcosiYcos^Z+isinlXsiniYsinlZ)s,s 2 2T = 4//3(spa)(icos^YcosfZsin^X+cosIXsiniYsinlz)

S f X

T = 4/3[ppa+2ppiT](cos|Xcos^YcosiZ+isiniXsinjYsin^Z)X § X

T = -4/3[ppa-ppil](icosIXcoslYsiniz+sinlXsinlYcosiZ)x,yT =-4/3/3(sda)(icosiXcosTYsinvZ+sinTXsiniYcosiz)xy,s 2T = -4/3/3 [/3pda+pd7T](icosjXsinjYcosjZ+siniXcosjYsinjZ)xy t xT = 4/3/3 (/3pda-2pd7T) (cosjXcosjYcosjZ+isinjXsinjYsinYZ)xy 1 zT 2 2 =0x -y ,sT 2 2 = -4//3pd7T(isiriYXcosiYcosjZ+cosjXsinlYsinYZ)x —y t xT 2 2 = 0x -y ,zT, 2 2 = 03z -r ,sT 2_ 2 = +4/3(pdTT)(isinjXcosjYcosjZ+cosIXsinjYsinjZ)

j 2 * IT f X

T 2 2 = -8/3pdiT(icosYXcos^YsinjZ+sinlXsin^YcosjZ)jz *r $ 2

X = k a, Y = k a, Z = k a x y zAll other matrix elements can be obtained from those given here

by appropriate permutations on X, Y and Z.

The matrix E is the transpose of B and submatrix F is equalto A.

59

Submatrix C

T = 2(ssa).(cosX+cosY+cosZ)s, s 1

T = 2i(spa!,sinXs ,x 1

T = 0s,xyT 2 2 = */3 (sda), (cosX-cosY)s,x -y 1

T _ 2 2 = (sda),(-cosX-cosY+2cosZ)s,3z -r 1T - 2(ppa),cosX + 2(ppt),(cosY+cosZ)

X , X 1 1T = 0x,yT = 2i (pdTT), sinYx, xy 1

T = 0x,yzT 2 2 = t/3 (pda), isinXx.x -y 1

T _ 2 2 = -(pda),isinXxf3z -r 1T _ 2 2 = 2i(pda),sinZz,3z -r 1T = 2(ddlT), (cosX+cosY)+2(dd6) ,cosZxy,xy 1 1

T = 0xy,yzT 2 2 = 0xy ,xTxy, 3zT 2 2x -yT 2 2x -y

-y2 2 = 0 -y2 2 = 3/2(dda),(cosX+cosY)+j(dd6),(cosX+cosY+4cosZ)#x -y 1 1- 2 2 = i/3/2(-(dda),+(dd6),)(cosX-cosY)#3z -y 1 1

2 2 2 = j(dda),(cosX+cosY+4cosZ)+3/2(dd6),(cosX+cosY),3z -r 1 1

Ts, s Ts ,x Tx,xTx, y

Submatrix D

= s + 4(ssa)^(cosXcosY+cosXcosZ+cosYcosZ)= 2>/2i(spa)2(si-nXcosY+sinXcosZ)= P1+2(ppa)2(cosXcosY+cosXcosZ)+2(ppir) (cosXcosY+cosXcosZ+2cosYcosZ)

= -2 [ (ppa) 2“ (PPTT) 21 sinXainY

60

The number of parameters used in the calculation is 23 and

these are listed in table 3.2.2 (a) in the usual notation.

The parameters are obtained by extrapolation from the parameters

for pure copper and pure selenium found by other authors (Jacobs— (5/ + & 1 + 1)(1968) and Joannopoulos et al (1975)) using R power

laws to describe the dependence of the integral on atomic separation.

We employ the method of Andersen et al (1978) to obtain the Cu-Se

interaction energy integrals from the above parameters. The

parameters determining the centres of the bands Es, Ep and Ed

are chosen to give a realistic band structure and a non-zero

band gap.For the less ordered phases where there are at least 12 Cu and

6 Se atoms per unit cell, the order of the full LCAO matrix is 132 x 132 so that a full band structure calculation over much of the Brillouin zone is not feasible. We thus make the simplification of using only 4s and 4p basis function on the Cu ions while ignoring

the states on Se ions. This reduces the size of the matrix to 48 x 48. This simplification is an attempt to describe the

conduction band accurately and relies on the fact that the

Cu d-band and the Se 4s and 4p bands are entirely full and the positions of the Se atoms are undisturbed relative to the ideal structure (provided we ignore the rhombohedral distortion of

the cage which we shall do henceforth and provided we ignore effects due to the fact that in some of the phases the number

of Cu ions is reduced). We can thus draw conclusions about the existence or non-existence of a band gap for any particular model crystal structure by comparing our results for the conduction

61

TABLE 3.2.2(a). Slater-Koster parameters used in the computations (Ryd.)

Se-Se Cu-Cu Se-Cu Cu-Se1 2

SSQ -0.046 -0.066 -0.0467 -0.0757 -0.0757

s p a 0.0341 0.0668 0.0334 0.1248 0.0916

ppa 0.0506 0.1351 0.0478 0.2135 0.2135

PP7T -0.0254 -0.0675 -0.0239 -0.1067 -0.1067

pda -0.0396 -0.0099 -0.0723 0pdTT 0.0324 0.0081 0.059 0

s d a -0.0131 -0.0046 -0.0199 0

dda -0.0155 -0.0027ddfT 0.0077 0.0014

ddS -0.0013 -0 . 0 0 0 2

ESe = sSe-1 .0 EP = 0.0

ECUs = 1.35 ECU = 1.0 P 4 “ ■ -0 . 1

Lattice constant a = 10.96 a^

62

TABLE 3.2.2(b). Additional Slater Koster parameters for nearestneighbour displacements (| j |) and (3/2 7 7 ) used in the computations (Ryd)

Cu-Cu

3 4

ssCT - 0 . 0 7 6 2 - 0 . 0 3 9 8

s p a 0 .0 7 7 1 0 .0 4 0 3

ppa 0 . 1 5 6 0 .0 8 1 5

PP7T - 0 . 0 7 7 9 - 0 . 0 4 0 7

63

band with those for the yalence band of the ideal structure which

are assumed to remain unaltered on going to our more complicated

model structures.Our model for the actual structure of gCu^Se is structure

(2) described in section 3.1. In this structure (Cu atoms fallon a simple cubic lattice with 1/3 vacancies in every 4th layer

normal to a (111) direction) one Cu atom out of 12 is placed

in an interstitial position at the centre of a cube with 7

Cu atoms at the corners (the eighth corner is a vacancy site

which was formerly occupied by the Cu atom which is now in an

interstitial position. For this structure the basis vectors

for the vacancy lattice are a (2 1 1 ), a (1 1 2 ) and a (1 2 1 ) andthe general form of the co-ordinates of the interstitial atoms

aare ^ + "J (1 1 1 ) where y^ are the vacancy position vectors in the lattice. Based on the above picture the co-ordinates of the vacancies in the unit cell are found to be (212), (320),

(241) (131). The positions of the normal atoms in the unit

cell are presented in table 3.2.3.For this structure we need some new parameters which describe

electronic hopping from interstitial Cu atom to the remaining Cu atoms. The tight-binding Hamiltonian in this case consist of a 48 x 48 matrix which describes hoppings among atoms of the same unit cell plus those from one unit cell to the other.We restrict our treatment to the first nearest neighbour hopping

integrals in which the displacements considered are ±(1 ,0 0 ),

(± Y ± 2 ± i) and their cyclic permutations. Also considered

64

are the second nearest neighbour hopping in which the displacements3are (± 1 , ± 1 , 0 ), ( ± ~ ± j ± j) and their cyclic permutations.

The parameters which are listed in table 3.2.2(b) are obtained

from those in table 3.2.2(a) with the help of the R ~ law.

All other parameters remain the same as before.

Our model of the partially ordered nonstoichiometric phase

(structure (5)) differs from the model of structure (2) above only in that the vacancy and the interstitial atoms are separated

by a greater distance, the size and the shape of the unit cell

remaining the same (i.e. we shift the position of the vacancy in the actual structure (2 ) relative to the interstitial atom)

see table 3.2.4. This greater separation of the vacancy and

interstitial is an attempt to mimic the fact that in this phase the interstitial atoms and the vacancies appear free to move

relative to each other. The additional parameters are those listed in table 3.2.2(b).

Finally our model of the fully disordered phase aCu^Se

(structure (3)) is a lattice with two interstitials and two vacancies

and a unit cell of the same size and shape as before. The positions of atoms in this case is presented in table 3.2.5. This model

should display the effects of interaction between electron states which are effectively localized on interstitial atoms. Bonding states based on close pairs of interstitials should have lower

energies than states on nearly isolated interstitials and thus tend to close the band gap. The additional parameters are thoselisted in table 3 .2 .2 (b).

65

TABLE 3.2.3. The one interstial one vacancy case

Position of atoms inAtomicindices

Central unit cell Cell 1 Cell. 2 Cell 3 Cell. 4 Cell 5 Cell

0 2 - U 2 - 4i 2 2 3i 12 12 11 31 2 - 11 11 12 41 1 1 3± 11 3! i~ 21 2 2 1 4 3 2 0 1 0 3 3 T 1 1 3 1 4 0 3 0

2 2 3 0 4 4 1 1 2 T 3 4 2 1 2 2 1 5 T 3 1

3 3 2 1 5 3 2 1 1 0 4 3 T 2 1 3 2 4 0 4 0

4 3 3 0 5 4 1 1 2 1 4 4 2 2 2 2 2 5 T 4 1

5 2 3 1 4 4 2 0 2 0 3 4 T 1 2 3 1 5 0 3 1

6 3 2 2 5 3 3 1 1 1 4 3 0 2 1 4 2 4 1 4 0

7 3 3 1 5 4 2 1 2 0 4 4 T 2 2 3 2 5 0 4 1

8 3 4 0 5 5 1 1 3 T 4 5 2 2 3 2 2 6 T 4 2

9 3 3 2 5 4 3 1 2 1 4 4 0 2 2 4 2 5 1 4 1

10 3 4 1 5 5 2 1 3 0 4 5 T 2 3 3 2 6 0 4 2

i i 4 3 1 6 4 2 0c. 2 n\J 5 A 1i 3 £ 3 *■% 5 0 5 1

Cell 1-6 are unit cells touching the central cell_and displaced with respect to by the vectors ( 2 1 1 ) a , ( 21 1 ) a, (1 1 2 1a, (112 ) a, (1 2 1 )a and (1 2 1 )a respectively.

6

312

1

2

1

2

32

1

32

2

it

66

TABLE 3.2.4. Vacancy is shifted relative to interstitial atom

Position of atoms inAtomicindices

Central unit cell Cell 1 Cell. 2 Cell 3 Cell. 4 Cell 5 Cell 6

0 2 - It 2 - 4t 2 t 3t X2 X2 U 3i 2 t X2 X2 4t 3£ U 3 2 X2 3?1 2 1 2 4 2 3 0 0 1 3 2 0 1 0 4 1 3 1 3 T 32 2 3 0 4 4 1 0 2 T 3 4 2 1 2 2 1 5 T 3 1 1

3 3 2 1 5 3 2 1 1 0 4 3 T 2 1 3 2 4 0 4 0 2

4 3 3 0 5 4 1 1 2 T 4 4 2 2 2 2 2 5 T 4 1 1

5 2 3 1 4 4 2 0 2 0 3 4 T 1 2 3 1 5 0 3 1 2

6 3 2 2 5 3 3 1 1 1 4 3 0 2 1 4 2 4 1 4 0 37 3 3 1 5 4 2 1 2 0 4 4 T 2 2 3 2 5 0 4 1 2

8 3 4 0 5 5 1 1 3 T 4 5 2 2 3 2 2 6 T 4 2 1

9 3 3 2 5 4 3 1 2 1 4 4 0 2 2 4 2 5 1 4 1 310 3 4 1 5 5 2 1 3 0 4 5 T 2 3 3 2 6 0 4 2 2

1 1 4 3 1 6 4 2 2 2 0 5 4 T 3 2 3 3 5 0 5 1 2

Cells 1-6 are unit cells touching the central unit cell and displaced with respect to it by the vectors ( 2 1 1)a, (2T T ) a , ( 1 1 2 )a , ( T T 2 )a, ( 12 1 )a and (T 2T )a respectively.

67

TABLE 3.2.5. The two interstials and two vacancies case

Atomicindices

Central unit cell Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Ce-11 6

0 2 £ i£ 2 - 4i 2 — 21 2 3£ 12 12 1£ 3i 2 - 12 12 12 4i U 3£ l£ 3i i~ 2 3i1 2 2 1 4 3 2 0 1 0 3 3 7 1 1 3 1 4 0 3 0 2

2 2 3 0 4 4 1 0 2 T 3 4 2 1 2 2 1 5 7 3 1 1

3 3 2 1 5 3 2 1 1 0 4 3 7 2 1 3 2 4 0 4 0 2

4 3 3 0 5 4 1 1 2 T 4 4 2 2 2 2 2 5 7 4 4 1

5 2 3 1 4 4 2 0 2 0 3 4 7 1 2 3 1 5 0 3 1 2

6 3 2 2 5 3 3 1 1 1 4 3 0 2 1 4 2 4 i 4 0 37 3 3 1 5 4 2 1 2 0 4 4 7 2 2 3 2 5 0 4 1 2

8 3 4 0 5 5 1 1 3 7 4 5 2 2 3 2 2 6 7 4 2 1

9 2 - 2 i U 4j 3 2 2 - c 2 12 1£ 12 3£ 3* i“ 2 1£ 1£ 3£ 1£ 4i 12 3£ 12 2 i10 3 4 1 5 5 2 1 3 0 4 5 7 2 3 3 2 6 0 4 2 2

1 1 4 3 1 6 4 2 2 2 0 5 4 7 3 2 3 3 5 0 5 1 2

Atoms 0 and 9 are in the interstitial positions. Cells 1-6 are unit cells which touch the central unit cell and are displaced with respect to it by the vectors a ( 2 1 1 ), a(277), a(1 1 2 ), a(1T2 ), a(T2?) and a(721) respectively.

68

3.3 Results of the calculations

(a) Ideal structure

The energy bands for the ideal crystal structure in various

directions of the Brillouin zone are shown in Fig. 3.3.1. We shall

examine only one of the directions, (T-X) say.

The lowest lying band from -1.642 to -0.816 Ryd has its origin

mostly in the Selenium 4S wave functions. The width of this Se

derived S band is .826 Ryd.

Above this is a complex of bands in the range -0.3054 to 0.2293 Ryd which is made up of Se 4p orbitals and Cu+ 3d orbitals. In

the lowest part of this range from -0.3054 to -0.147 Ryd these bands

are heavily hybridized; in the central part of the range from —0.147

Ryd to -0.0461 Ryd the bands are very flat and are predominantly of Cu+ 3d character; finally in the upper part of the range from -0.0461 Ryd to 0.2293 Ryd the nature of the bands varies from strongly hybridized bands near -0.0461 Ryd to pure Se 4p bands at 0.2293 Ryd which is the top of the valence band.

There is a direct band gap between occupied and unoccupied states at the centre of the Brillouin zone T of magnitude 0.2544 Ryd and one at X of magnitude 0.215 Ryd. however, the indirect band gap

from the top of the valence band at T to the bottom of the conduction band at X is much smaller being of magnitude 0.0382 Ryd.

The magnitude of the calculated gap is sensitive to various approximations and should not be taken too seriously in our non-self- consistent approach, but there is little doubt that the material in this hypothetical structure is a semiconductor with a finite band gap.

69

The eight conduction bands arise respectively from Cu+ 4S and

Cu+ 4p orbitals and display a considerable degree of degeneracy

at particular points (such as T) and in particular directions (such

as A).

CR

ud

s-}

Fig 3.3.1. Valence and conduction bands for the ideal Cu^Se crystal structure along certain symmetry directions.

71

(b) One interstitial atom and one vacancy

Figure 3.3.3(a) represents the conduction band for the case of

one interstitial atom and one vacancy (our model structure (2 )) plotted along a direction T-A of the Brillouin zone shown in Figure

TT3.3.3.2 where A is — (2,1,1).

There is a fairly strong Cu s-like character at the lowest energy band at T. The value of the lowest energy at T is 0.2523 Ryd. The density of states in the valence band is assumed to be the same

as that of Figure 3.3.1 (apart from the Cu d band) for the reason described above.

Results for our model of the partially ordered non-stoichiometric

phase (structure (5)) (where the vacancy and interstitial are more widely seperated) show no appreciable difference from the above case.

The lowest energy at V in this case being 0.2516 Ryd.

(c) Two interstitial atoms and two vacancies

The results of the conduction band calculation in this (model

structure (2)) case is presented in Figure 3.3.3(b). The lowest energy shows a strong Cu s-like character. The minimum value of the energy at Tbeing 0.1863 Ryd, much lower than in (b) and also lower than the value at V of the top of the valence band in (a) which is 0.2293 Rd.

The low lying bands in case (c) are generally lower in energy than in case (b) but the shift is not rigid. We may therefore

conclude that an increase in the number of copper vacancies reduces

the energy of the conduction band states due to the bonding effects

72

of states on neighbouring interstitial Cu ions. This results in a closing of the band gap and our calculation if accurate would indicate

that the resulting material is metallic with a fairly low electronic

conductivity. This will be discussed in the context of experiment later.

We have also calculated some partial densities of states (PDOS)

for comparison with experiment later. The method used was the

tetrahedron method (Lehmann and Tau (1972), Rath and Freeman (1976)). This gives numerically stable and highly accurate results for a

sufficiently fine mesh of points in the k-space. Our mesh contains

191 points which should be sufficient for our purposes.

In fig. 3.3.4 we present the Se p PDOS over a fairly wide range of energies, in fig. 3.3.5 we present the Cu d PDOS over the same

range and in fig. 3.3.6 we present the sum of the two which is an accurate approximation to the total DOS in the energy range from

-0.5 Ryd to +0.25 Ryd. The Se p PDOS shows two distinct peaks in agreement with the qualitative remarks made above. The Cu d PDOS has one large central peak with subsidiary structures on either side due to hybridization with the Se p states. This is also in agreement

with our qualitative remarks. Later, we shall compare these resultswith experimental information.

73

Fig. 3 .3 . 2 The Brillouin zone for the simple monoclinic lattice

Ener

gy C

&jck.

l

74

(a)

Fig. 3.3.3 (a) Conduction band for the case of one interstitial atomand one vacancy (model crystal structure (2 )) along a symmetry direction,

(b) Conduction band for the case of two interstitial atoms and two vacancies (Model structure (3)) along a symmetry direction.

1>es)

scZq

of

75

Fig. 3.3.4 The Se p Partial density of states.

76

Fig. 3.3.5 The Cu d Partial density of states.

77

Fig. 3.3.6 Total density of states

Figure 3.3.7 : The total density of states of ideal Cu2Se convoluted with a Gaussian of width 0.025 Ryd to represent instrumental broadening. This should be compared with the experimental photoemission spectrum in

Figure 2.6.1a.

79

3.4 Comparison with Experiment

In this section we shall compare the results obtained with the

available experimental results.

(a) Soft X-ray spectra

presented in fig 2.6.1a. This spectrum depends slightly on matrixelement effects, on many-body effects and on instrumental broadening.

However, these are small effects and the spectrum provides fairly«

direct information on the Se p partial density of states. The

experimental results for Se in fig. 2.6.1a should be compared with our calculation in fig. 3.3.4. The general shape of the two spectra are in excellent agreement;with two prominent peaks. The experimental

separation A of the two peaks is 4.1 eV for x = 0 which is to be compared with the theoretical value of 5.2 eV read off from the curve. This is an excellent agreement given the crude fashion in which our parameters (and in particular pda and pdfT for Se-Cu) were derived. Further Terekhov et al (1983) have presented a curve (fig.2.6 .1b) giving the peak separation A for Cu^ Se as a function of x. The following simple argument can be used to understand the curve. The two peaks

arise from bonding and anti-bonding states made up of Se p states and Cu d states. The hybridisation involved in this effect depends linearly on the concentration of Cu atoms. i.e. the peak positions are given by the roots of the following simple equation

The Se soft X-ray spectra of Terekhov et al (1983) are9

E -E P

80

for small x

A = 5.0 - 1.15x eV

which is to be compared with the following fit to experiment

A = 4.1 - 2x eV

The agreement when x = 0 is good but the slope which is of course

harder to obtain accurately is only given to within a factor of ~ 2 .

(b) Photoemission spectra

Terekhov et al (1983) have also measured the photoemission spectra for Cu^ ^Se at x = 0.2. Their curve is presented in fig. 2.6.1a.

This curve reflects not partial densities of states but the total

density of states (modified as before by many-body effects, matrix element effects and instrumental broadening). This curve should be compared with our total density of states curve presented in fig. 3.3.6. It is seen that the agreement is once again good. The large structure

in the centre is, of course, the Cu d-band. The left hand shoulder arises from the Cu d - Se p bonding states, the right hand shoulder from the corresponding anti-bonding states. Even the relative heights of these shoulders appear to come out correctly from our calculation once broadening effects are taken into consideration.

(c) The dependence of the energy gap Ae on x

In figure 2.6.2 we presented the experimental results for the energy gap Ae as a function of x due to Sorokin and Idrichan (1975).This curve can be understood in terms of our band structure as follows.

81

Firstly the top of the valence band is a pure Se p state in ideal

Cu^Se and will therefore not be affected much by the absence of Cu

atoms in Cu^ ^Se. The conduction band is made up mostly of Cu 4S and

4p states and this will be narrowed by the absence of Cu atoms and

its width will depend on the square root of the concentration of

Cu. Thus the bottom of the conduction band will be given by an expression of the form

E,( x) = E - w/l-±x b c

where E is the centre of the conduction band and W is its half width, c

With W = 0.72 Ryd taken from the band structure we get for the slope

of the curve

- f - (Ae ) = ±Wdx=0.18 Ryd = 2.5 eV

which is to be compared with an experimental value of 3.5 eV taken from

Sorokin and Idrichan.

(d) The dependence of the energy gap Ae on T

The temperature dependence of the energy gap Ae has been obtained

by Zhukov et al (1982) and is presented in fig. 2.6.3. This shows

that Ae (called Eg in the figure) decreases with T. This decrease can be understood in terms of our models although the magnitude of the slope cannot be obtained. As T increases so more Cu atoms move from their ideal positions to interstitial positions. As we have shown the lowest state in the conduction band moves down when the number of interstitial Cu atoms increases. If there are no interstitial Cu

atoms ( the ideal structure) then Ae = 0.0382 Ryd = 0.52 eV. If

there is one interstitial Cu atom and one vacancy per 12 Cu atoms then Ae = 0.0223 Ryd = 0.30 eV, a significant narrowing. If there

82

are two interstitial Cu atoms and two vacancies then the band gap

disappears. Our calculation cannot be relied on to give the slope

of the curve because there is no unique way of relating the number

of Cu interstitials and vacancies to T. Further it is based on the

assumption that the top of the valence band is undisturbed when Cu atoms move from ideal positions to interstitial positions. It is

thus very crude but it nevertheless gives the correct sign for the effect.

(e) The dependence of the electronic conductivity at low T on x

In fig. 3.4.1 we present a graph of extrapolated to T=0

against x. This is taken from the work of Vucic et al (1982, private communication). The electronic conductivity rises linearly with x from very nearly 0 at , x=0. This can be understood in terms of the population of holes in the valence band. In the ideal case CugSe the material is a semi-conductor with a finite band gap. With each Cu

atom that is removed, 10 d states are removed from the valence band

and 11 electrons are removed from the valence band. The net result is that with each Cu atom that is removed one hole is put in the valence band. It is this hole that is responsible for the

conductivity which clearly rises linearly with the number of holes i.e. x. The small deviation of from 0 at x=0 could be explained in terms of imperfection in the samples or impurities or other sucheffects.

83

(f) The discontinuity in fp at the phase transition and its dependence on x

In fig. 3.1.1 we present G^ versus T for various values of x

from Vucic et al (1981). At about 410 K there is a phase transition

which gives rise to a discontinuity in G^. This discontinuity is

small when x is small (except for x= 0 which will be discussed later)

and rises linearly with x^see fig. 3.4.2. This can be understood

as follows. At all temperatures the electronic conductivity G^ isdetermined essentially by the number of holes in the valence band.

Near x=0 this conductivity is close to zero as discussed in theprevious section. Thus at the phase transition the change in theconductivity AG^ will also be determined by the number of holesand thus by x. This explains the linear rise of Ag with x fromenear 0 at x=0. The only deviation from the rule is that experimentallyAgq appears to be large at x=0. According to Vudic (1984, private

communication) close to this concentration x=0 there is a two phaseregion above 410 K. This could easily lead to anomalous results forG above 410 K. e

84

Fig. 3.4.1 A graph of C7 extrapolated to T=0 against x.

85

^ 3-4~- fl graph °f discontinuity at the phase transition versus x

86

CHAPTER 4

PHASE TRANSITIONS IN Cu^ ^Se

4.1 Introduction

In this short chapter we will discuss briefly the experimental

phase diagram of Se. We attempt a theoretical discussion

in terms cf the Landau theory of phase transitions and will

discuss critically some experimental speculations relating to the phase diagram in the light of the Landau theory.

4.2 Experimental phase diagram

In fig. 2.2.1 the experimental phase diagram according to

Vucic et al (1981) is presented. The upper line labelled 3 is according to these authors a line of first order phase transitions. This statement is based on differential calorimetry measurements.

In other words the position of this line is determined by observation of the release of latent heat in an experiment in which the sample is cooled at a more or less constant rate. The other lines in the diagram are lines of second-order phase transitions and it is worth noting that according to these authors, at x= 0 the line of first-order transitions and the line of second-order transitions meet i.e. the transitions become degenerate. In the next section we will try to develop a simplified Landau theory with several order parameters and examine whether any symmetry of the crystal at x= 0 is likely to produce this type of degeneracy.

87

4.3 Landau Theory

The parameters which determine the state of the crystal are the thermal average populations n^ of mobile Cu atoms in the various types

of sites i that are available. The order parameters m_. are linear

combinations of the populations n^. The Landau theory assumes that

the free energy F may be expanded to an adequate approximation as a

low-order Taylor expansion in terms of the order parameters. The

behaviour of the coefficients in the expansion as functions of T and x determines the nature and the position on the phase diagram of the phase transitions.

We shall assume that there are six types of sites in the lattice

which are equivalent by symmetry in the basic Hamiltonian i.e. inthe high temperature phase. These six sites will be in planes

labelled d in the stacking sequence diagram of fig. 2.3.1. Three

sites labelled 1, 2 and 3 will be in one d plane and the remaining

three sites labelled 4, 5 and 6 will be in the adjacent d-plane. Theother d planes in the crystal remain at all temperatures and valuesof x considered equivalent to one or other of the above d-planes. Thesymmetries between adjacent d-planes and between some d sites inthe same plane are broken at low temperatures as explained in detail

in Chester 2. For the purposes of this extremely simple exercise we

shall ignore the occupation of sites in b planes by Cu atoms. Theresults we hope will, nevertheless, be illuminating. Finally it isworth noting that the parameters n. and the thermodynamic variable

x are not unrelated. If we assume that the population of Cu atoms onb sites remains fixed always at one Cu atom per six sites and thepopulation of Cu atoms in the cage also remains fixed always then we

6 Z

i=1can write n .l 5 - 6 x s C f > 0 (4 , 3- / J

88

The Landau expansion of the free energy is then

F = F + aZ(n.) + a Z (n.n.) + B£(n.) o . 1 . . . 1 3 , 11 i<3 i

+ 3' 2 (n2n.) + 3' ' Z (n.n.n ) + vZ(n.‘) itj i<j<k 1

+ y' Z (n3n.) + y'' Z (n?n2) + y''' Z (n2n.n_ ) i4=j i<j i4=j

i*kj<k

• v+ y Z (n.n.n, n ) ♦ higher order terms . , . 1 3 k 1i < 3j<kk<l

(4.3.2)

This is the most general expansion up to fourth order terms which

preserves the symmetry of the free energy to interchange of any

pair of the variables and omits the first order terms. This can be simplified with the help of the following relations which may

easily be derived:

Z(n.n .) = ic2 - ±Z(n2) 1 3 2 1

(4.3.3)

Z(n2n.) = cZ(n2) - Z(n3) 1 3 1 1

(4.3.4)

Z(n.n.n_ ) = 1/3cZ(n.n.) - 1/3Z(n2n.) 1 3 k 1 3 1 3

(4.3.5)

Z(n3n.) = cZ(n3) - Z(n4) 1 3 1 1

(4.3.6)

y , 2 2 j — 2 2 1 v t 4 .Z(n.n.) = a(Z(n.)) - 2^(n.) 1 j 1 1

(4.3.7)

Z(n2n .n ) = i[cZ(n2n.) - Z(n3n.) - 2 Z(n2n2)] 1 3 k 1 3 1 3 1 3

(4.3.8)

2(n. n.n.n ) = £[c2(n-n -ni ) - Z(n2n.n )] 1 3 k ! 1 3 k 1 3 k (4.3.9)

The restrictions on the summations in the above expression (4.3.3)

89

to (4.3.9) are omitted but may be understood to be the same as

in equation (4.3.2). On substituting (4.3.3) - (4.3.9) in equation (4.3.2) we get

F = (F + ia'c2 + 1/3! (3"c2 + 1/4! y,vc4)

+ (a - ia' + 3 'c - !B"c + ^ ' " c 2 - 1/6y 'vc2) Z(n2)l+ ($ - B* + 1/3B" + Y'c - y ' " c - 1/24y ,v c ) Z(n^)

+ (y - Y' - iY" + Y " ’ + tY,v) ^(n4)i+ (£y " - 5Y"' ” 1/4y ,v) (Z(n2 ) ) 2 (4.3.10)

ThusF = f (c) + f (c) Z(n2) + f_(c) Z(n^) + f.(c) E(n?) o 2 i 3 i 4 i

+ f_(c) (Z(n2 ) ) 2 (4.3.11)5 l

where the f's are appropriately defined.

The order parameters may be defined in terms of the linear combinations of the occupation numbers by a relation of the following form

x. = U..n. (4.3.12)i 13 3

where summation over the repeated index is understood and U is an orthogonal matrix i.e. we also have

n. = UT .x. (4.3.13)i 13 3

We now make plausible assumptions about the form of U whichare in line with the suggestions of Vucic et al (1981) concerningthe nature of the order parameters. One of the order parameters(labelled x below) must reflect the change in population of thedifferent d layers whereas others (labelled x„, x . x, and x^) must2 3 4 5

90

reflect changes in the relative populations of different sites in

a single layer i.e. we choose

X1 = 7e <ni + n2 + n3 ' n4 - n5 - V x2 = 7s <2ni - n2 - V x3 = 7 l <2n4 - n5 ■ V X4 = 7 I (n2 ' n3)

x5 = 7 I (n5 • V

x6 = 76 (ni + n2 + n3 + "4 + "5 + n61 (4.3.14)

x^ is not an order parameter since = ~7T c which is a constant o d yofor fixed values of x, the deviation from stoichiometry. The free

energy is of course symmetric with respect to change of sign of x , x^ and x_ so that these will normally be associated with second

order phase transitions and we shall assume that this is the case here.

The Free energy can easily be written in terms of these new

variables and with a little work we find2 . „(f o + 1/6 c f^ + f + 1/6 c f + c f,_)1 3 . . 4_ 4

_ ...... . 4 " C "55

+ (f2 + f3 + °2f4 +

+ f3 + 2/3cf4>(x2 + x2>. , 1 „ .W 3

'4[X1

2 2 2 2X, - X V - x^x.2 1 3 1 1 4 2 44 4 4. 5 4+ x0 + x„ + x^ ] + f_ Ex.3 4 5 5. , li=1

X3X5

, . 2 2 2 2 2 2 2 2 2 2 2 2 + f 4 (x lXz + Xlx 3 + x , x 4 + Xlx5 + x2x4 + x3x5

/91

2 2 2 2+ x_x_ + x0x. 2 3 2 42 2 2 2

+ X3X4 + X3X52 2, „ . 3 3 ,+ x x_) + 2/3f (x x - xx.) 4 5 4 2 1 3 1 (4.3.15)

This is the final expansion of the free energy on which our arguments are based.

4.4 Phase transitions

If in accordance with Vucic we assume that one of the important

phase transitions is associated with repopulation of the atoms between the layers and another with repopulation within the

layers and that one is a first order phase transition then we need only contemplate the variables x^, x^ and x^ changing from zero to

non-zero values as we lower the temperature. (In the high temperature phase of course all the x.'s are zero apart form x^).l 6

Thus if we put x^ = x_ = 0 in (4.3.15) we get the following

simplified expression for F

+ (f 2 2 2 2 2 cf3 + c f4 + 2c f5 )(x^ + x 2 + X3 }

(4.4.1)

In a simpler notation we can write

92

( 4 . 4 . 2 )

The c o e f f i c i e n t s g , g _ , g_ e t c . a r e a l l f u n c t io n s o f c ando 2 3

h ence o f x and may be r e la t e d b a c k t o th e c o e f f i c i e n t s in o u r

o r i g i n a l e q u a t io n ( 4 . 3 . 2 ) v i a ( 4 . 4 . 2 ) , ( 4 . 4 . 1 ) , ( 4 . 3 . 1 1 ) and

( 4 . 3 . 1 0 ) . T h e y a ls o depend on T .

We can now d e te rm in e th e c o n d it io n u n d e r w h ic h th e f i r s t o r d e r

and seco n d o r d e r p h a se t r a n s i t i o n s become d e g e n e ra te by e x a m in in g

f i r s t l y th e f i r s t o r d e r ch an ge a r i s i n g from d is c o n t in u o u s ch a n g e s

in x ? and x^ from z e ro and th e n s u b s e q u e n t ly th e ch an g e o f x 1

c o n t in u o u s ly from z e r o . F u r t h e r th e sym m etry o f th e f r e e e n e rg y

F ( x ^ , x ^ ) r e q u ir e s t h a t x ^ x ^ when x ^ = 0 . Th us we s e e k th e p o in t

w here s im u lt a n e o u s ly

( 4 . 4 . 3 )

and

( 4 . 4 . 4 )

from ( 4 . 4 . 2 ) we h ave

and th u s ( 4 . 4 . 3 ) g iv e s

(1 ) (3) 22g2 + 2g3x2 + U g ' 1’ + g' ) = 0 ( 4 . 4 . 6 )

and ( 4 . 4 . 4 ) g iv e s

93

(i ) (3 ) 94g2 + 6g3X2 + 4 (2 g 4 + g4 } x 2 = ° ( 4 . 4 . 7 )

S o l v i n g ( 4 . 4 . 6 ) and ( 4 . 4 . 7 ) s im u lt a n e o u s ly we f in d

9oX 2 X 3 n ( 1 ) . ( 3 ) .<294 + g4 1

and

( 4 . 4 . 8 )

9 2 o/o (1> A <3)i2 ( 2 g 4 + g 4 )( 4 . 4 . 9 )

E q u a t io n ( 4 . 4 . 9 ) d e te rm in e s T a s a f u n c t io n o f c o r o f x . E q u a t io n

( 4 . 4 . 8 ) g iv e s th e d is c o n t in u i t y o f x^ o r x^ a t th e p h a se t r a n s i t i o n .

To e xam in e th e p o s s i b i l i t y o f d e g e n e ra te f i r s t and seco n d o r d e r

p h a se t r a n s i t i o n s we ta k e x^ and x^ from ( 4 . 4 . 8 ) and s u b s t it u t e

b a ck in t o ( 4 . 4 . 2 ) t o g e t

Fo

F + o

[ 2g

, 2 ( 2 )

2 g 3g 4 (1 A (3 ) ,2

+ 9, 1

2 4X 1 + g 4X 1

( 4 . 4 . 1 0 )

H ere we h ave assum ed t h a t th e p h a se t r a n s i t io n s a r e d e g e n e ra te and

t h u s s u b s t it u t e d th e v a lu e s o f x^ and x^ a t th e p h a se t r a n s i t i o n

in t o ( 4 . 4 . 2 ) . The seco n d o r d e r p h a se t r a n s i t i o n o c c u r s when th e

c o e f f i c i e n t s o f th e q u a d r a t ic te rm d is a p p e a r s (Lan d au and L i f s c h i t z ,

1 980, p . 4 5 2 ) and o u r a s s u m p tio n s th e n a r e c o n s is t e n t o f th e f o l lo w in g

c o n d it io n i s s a t i s f i e d s im u lt a n e o u s ly w it h ( 4 . 4 . 9 )

0 2 ( 2 )2 g 39 4g0 + --- = 0’2 ro (1) A (3)i2f2^ + 94 1

( 4 . 4 . 1 1 )

The s im u lt a n e o u s s a t i s f a c t i o n o f ( 4 . 4 . 9 ) and ( 4 . 4 . 1 1 ) i s th e

c o n d it io n f o r th e d e g e n e ra c y o f th e p h a se t r a n s i t i o n s . E q u a t io n s

( 4 . 4 . 9 ) and ( 4 . 4 . 1 1 ) by e l im in a t io n o f g^ may be re d u c e d to

0 (4.4.12)2g(1 ) 4 + 8g ( 2 )

4 + g (3)4

This condition may be written in terms of the original expansion

parameters via equations (4.4.2) and (4.4.1) and (4.3.11) and (4.3.10) as

9y - 9y' + 11/2 y" - y* ' ' - £y,v = 0 (4.4.13)

where the y's are functions of T only. Thus the condition for

degeneracy is either never satisfied or is satisfied for all values of x.

4.5 Conclusions

In the previous section we found that the condition (4.4.13)

for the degeneracy of the first and second order transitions

can be written in terms of the fourth order expansion coefficients of equation (4.3.2). This condition as derived is independent of x. If higher order terms are taken into account in (4.3.2) then the condition would have some x dependence, but the condition can only be satisfied accidentally and there is no symmetry based reason to suppose that it can only be satisfied at x = 0 as claimed by Vucic et al (1981).

It is thus our conclusion that the results claimed by Vucic is extremely unlikely to be true in the form stated. Either the upper phase transition (line 3 in fig (2.2.1)) is not a line of first order phase transitions or the degeneracy does not occur at x = 0 .

The claim that line 3 is a line of first order transitions depends on differential calorimetry and it is possible that a sluggish

95

second order transition could resemble a first order transition in

this experiment. There also appears to be a two phase region in the phase diagram at high T near x = 0 (Vucic, 1984, private

communication) and this could invalidate conclusions about the phase

transition in this region.

It is clear that much experimental work remains to be done.

96

CHAPTER 5

ELECTRONIC STRUCTURE AND SITE PREFERENCE OF TRANSITION METAL

IMPURITIES IN Fe^Si.

5.1 Introduction

There have been several experimental and theoretical investigations

of the properties of pure and impure Fe^Si (Burch et al ( 1974),

Pickart et al (1975), Shabanova and"Frapeznikov (1975), Switendick

(1976), Ishida et al (1976), Niculescu et al (1977), Stearns (1978),

Himsel et al I, II (1980), Haydock and You (1980), William et al (1982) and Muir et al (1982)).

The crystal structure of this compound was established hy Burch

et al (1974) and can most easily be understood in terms of an under­lying body-centred cubic lattice. The crystal structure is shown in fig. 5.1.1. This lattice is made up of two interlocking simple cubic lattices. All of the sites on one of the simple cubic lattices are occupied by Fe atoms. The sites on the other simple cubic lattice

are alternately occupied by Fe and Si atoms. The Fe atoms on the first simple cubic lattice have a local environment of tetrahedron symmetry having four Fe and four Si nearest neighbours which lie

on the second simple cubic lattice. These sites of tetrahedral symmetry will be denoted by T. The Fe sites (and indeed the Si sites) on

the second simple cubic lattice have a local environment of cubic symmetry having eight Fe nearest neighbours on the first simple

cubic lattice. These sites of cubic symmetry will be denoted by C.A description of Fe and Si atoms local environments in Fe^Si crystal

97

• □ O - AFell Fel Felll Si

Fig. 5.1.1 The crystal structure of Fe^Si. The sites Fell and Felll are identical.

98

are summarised in table 5.1.1 below, based on fig. 5.1.1 (Niculescu

et al (1977)).

Table 5.1.1 Nearest neighbour (n.n) configuration for both Fe and Si sites in Fe^Si.

Sites First n.n Second n.n Third n

Si 4FeII, 4FeIII 6FeI 12Si

Fel 4FeII, 4FeIII 6Si 12FeIFell 4FeI, 4Si 6FeIII 12FeIIFelll 4FeI, 4Si 6FeII 12FeIII

Neighbour distance in A°

2.43 2.83 3.99

Burch et al (1974) also established that the moments of the two

types of Fe atoms were different, the moment of the C Fe atoms being2.4y and the moment of the T Fe atoms being 1.2y .B B

In the same paper and Pickart et al (1975) it was shown that substitutional impurities from the first transition row in thermal

equilibrium show a marked preference for one type of Fe site or the other, depending on their position in the transition row relative to

Fe. Transition metal impurities from the left of Fe such as Mn reside preferentially on the C sites and impurities from the right of Fe such as Co reside preferentially on the T sites.

There have been several calculations of the electronic band structure of this material aimed at explaining the experimental datadiscussed above. Switendick (1976) discussed the electronic structure

99

of the pure material. He also attempted a qualitative discussion of

the site preferences in terms of band structure. Williams et al

(1982) calculated the density of states of the pure material and

obtained reasonable values for the Fe moments on the two sites.

Haydock and You (1980) calculated the density of states of the similar material Fe^Al using the recursion method (Haydock et al

(1975)) again with an eye to calculating magnetic moments on various

sites. Ishida et al (1976) also calculated the band structure of Fe^Al in both paramagnetic and feromagnetic states.

In this work we calculate the band structure for pure Fe^Si and

local densities of states on various sites and for various spin orientations, from a tight binding model again using the recursion method. We also calculate local densities of states for Fe.Si with

various substitutional impurities on various sites. These are used to calculate one-electron band energies for various situations and the site preferences are discussed in terms of the differences between these band energies. A simple interpretation of the results is then

presented.

100

5.2 Outline of Calculations

The purpose of this section is to discuss the method for computing

the band structure and local density of states n. (E,R) for an atomiQof type i and spin a situated on site R. Here i could be an Fe, Co or Mn atom situated at a T or C site or a Si atom situated on the Si

sublattice. We also calculate the one-electron band energies

u. (R) = I E n. (E,R)dE (5.2.1)1Q IQ

The total energies of various situations will be discussed in terms

of U^a (R). We shall use the recursion method for calculating the n^(E,R) (Haydock et al (1975), Haydock (1980)).

(a) Application of the Slater-Koster interpolation scheme to the

energy bands of Fe^Si.We shall be concerned in this section with applying the Slater-

Koster interpolation scheme to the calculation of the band structure of paramagnetic Fe^Si. The mathematical methods involved in this scheme ha$£already been discussed in Chapter 1 and will not be

reviewed here.The basis set used consists of five 3d orbitals on each Fe atom

and one 3S orbital and three 3p orbitals on each Si atom; each unit cell contains three Fe sites. We take the matrix elements of the Hamiltonian H, between the functions of the basis set above and construct our secular equation (19x19) at any point in the Brillouin zone. The values of E which satisfy (1.8.10) at any k are then the eigenvalues.

The resulting matrix can be divided into sixteen submatrices

101

' A B c' DTB E F G

cT TF H Idt g t iT JV

where A is a 4x4 Si-Si tight binding matrix 7 B,C,D are 4x5 Si-Fe

tight binding matrices; F, G and I are 5x5 Fe-Fe tight bindingmatrices and E, H and J are 5x5 Fe-Fe tight binding matrices. E, H

and J are in our approximation diagonal matrices whose elements are

the Fe self energies because we consider only first and second

nearest neighbour hopping for Fe pair and an Fe atom of type I for

example has no Fel atoms in its first and second nearest neighbour

shells. The other matrices with subscripts T are the conjugate

transpose of the corresponding matrices.

The matrix elements of the Hamiltonian H are presented intable 5.2.1. This table gives the matrix elements in terms of energyintegrals such as SSQ, SPQ etc in the usual notation. In our case asin the bcc case it is necessary to include both nearest neighbour andsecond nearest neighbour hopping integrals because of the slow decayof the SSa, SPa, ppa and ppa integral as a function of pairseparation R. The number of parameters used in the calculation is

19 and these are listed in table 5.2.2 in the usual notation. Thehopping parameters are obtained by extrapolation from the parametersfor pure Fe and pure Si found by other authors (Dempsey et al (1975))

_(£«+£+ 1 )using R power laws to describe the dependence of the integral

on atomic separation. We employ the method of Andersen et al (1978)

= (-)S'+M+1(jl+il' )!2 ' ( 2 W 1 ) ( 2 & + 1 ) A t ’ & ' A t &(&'+M) ! (JUM) ! (H+M) ! (&-M) fj X(R}

k .S &' +£+1

(5.2.2)

102

where p, d stand for Z , Z ' = 1,2 and CT,TT, 6 for M=0,1,2 to obtain the Fe-Si hopping energy integrals from the above parameters. The self

energy parameters are obtained by requiring that our density of

states resemble that of the modern self-consistent calculation of

Williams et al (1982). The resulting energy bands are present in Fig. 5.2.1.

103

TABLE 5.2.1. Matrix elements in terms of overlap integrals for the body-centred cubic lattice Fe3Si

(Si/Si) Submatrix A Si Si

(s/s) = so + 4sscr (cos£cosr)+cos£cos£+cos£cosr|)(s/x) = 2-/2 i spa3 (sin£cosr|+sin£cos£)

(x/x) = + 2ppa3 (cos£cosn+cos£cos£) +2ppTT3 (cos£cosr)+cos£cosC+2cosr|cos£)(x/y) = - 2 [ppa3 ~ppa3 ] sin^sinri

£ = k a, r| = k a, £ = k a x y z

All other matrix elements can be obtained from those given here by appropriate permutations on x, y and z.

(Fel/Si) Submatrix B Fel Si

( x y / s ) = ( y z / s ) = ( z x / s ) = 0

2 2(x - y / s ) = s d C L .(co sk a -c o s k a)2 x y

2 2(3 z - r / s ) = s d a „ ( 2 c o s k a -c o s k a -c o s k a )2 z x y

( x y /x ) = 2 p d u ^ s in k a 2 y

( y z / x ) = 0

( z x /x ) = 2pdTT^ s in k a 2 z2 2

(x - y / x ) = -/3pdCF2 s in k ^ a

2 2(3 z - r / x ) = -p d a 2 s in k^a

( x y / y ) = 2pd7T2 s in k ^ a

( y z / y ) = 2pd7T2 s in k ^ a

( z x / y ) = 0

2 2(x - y / y ) = - / 3 p d a 2 s in k ^ a

104

2 2(3z -r /y) = -pda2 smk^a

(xy/z) = 0

(yz/z) = 2pd7T sink a4 y(zx/z) = 2pdiT sink a

jL X, 2 2 , ,(x -y /z) = 0

2 2(3z -r /z) = 2pdao sink a 2 Z

Submatrix C Si Fell

Here d = a/2(111), d = a/2(1lT), $3 = a/2(11 1 ),ik.d„ ik.d„ ik.d_, ik.dw = e - -1,w = e - -2 , = e - -3, w4 = e - -

(Si/Fell)

(s/xy) =/3/3 sda, (w^w^-w^+wjI I £. J H

(s/yz) = /S/Ssda^(w i+w 2“W3“W 4 )(s/zx) = i/3/3sda (v^-w +w -w )

2 2 2 2 (s/x -y ) = (s/3z -r ) = 0(x/xy) = 1/3 (/3pdai+pdiT1 ) (w1-w2+w3-w4 )(x/yz) = 1/3 (/3pdai+pd7T1 ) (w^+w2+w3+w4 )

(x/zx) = 1/3 (/3pda.)+pd7T1 ) (w1-w2 ~w3+w4 )2 2(x/x -y ) = 1/i/3pd7T (w^+w2+w^+w4)2 2(x/3z -r ) = -1/3pdir, (w -w +w -w )

(y/xy) = l/3(pdal+1//3pdTTl ) (w +w -w -w )(y/yz) = 1/3(pdal + 1//3pdTTl ) (w1-w2~w3+w4)

(y/zx) = 1/3 (pdal“2//3pdTT1 ) (wl+w2+w3+w4 )2 2(y/x -y ) = -1//3pd7Tl (w -w +w -w )2 2(y/3z -r ) = -1/3pdiTl (w1-w2+w3~w4)

(z/xy) = 1/3 (pdai-2/>/3pdTT1 ) (w1+w2+w3+w4 )

= a/2(Til)

4

105

(z/yz) = 1/3(pd(Jl+1//3pdTTl ) (w.j-w +w -w )(z/zx) = 1/3 (pda^ + 1/>/3pd7T ) (w^+w^-w^-w^ )

2 2(z/x -y ) = 0 2 2(z/3z -r ) = 2/3pdTTl (w1-w2~w3+w4 )

Submatrix D Si - Felllc 1 = a / 2 ( T T T ), c 2 = a / 2 ( T 1 1 ) , c 3 = a / 2 ( H I ),0 ik.c, 0 ik.c„ 0 ik.c Q ik. 1 2 3 4(Si/Felll)

(s/xy) = 1//3sdC7l (8 1 —^2~8 3+p4 )(s/yz) = i//3sda1 (.8 1+8 2”8 3-8 4)

(s/zx) = i//3sda1 (8 1-8 2+8 3~8 4)(s/x2-y2) = s/3z2-r2) = 0

(x/xy) = -1/3 (pdai+1//3pd7Tl ) (8 ^ 8 2 +B3"8 4)

(x/yz) = -1/3(pdai~2//3pdTT1 ) (8 1+8 2+8 3+8 4)(x/zx) = -1/3(pdal+i//3pdTT1) (8 1"8 2“8 3+8 4) (x/x2-y2) = -1//3pd7T1 (8 1+8 2"8 3'8 4)

(x/3z2-r2) = 1/3pd7Tl (8 1+8 2 "8 3-8 4 )

(y/xy) = -1/3 (pdai+1//3pdTTl ) (8 1+8 2"8 3"8 4 )

(y/yz) = -1/3(pdai+1//3pdTTl ) (8 1-8 2~8 3+8 4)(y/zx) = -1/3(pdai~2//3pdTTl ) (8 1+8 2+8 3+8 4) (y/x2-y2) = 1//3pdTTl(8l-82+83-84)(y/3z2-r2) = 1/3pd7T1 (8 1"8 2+8 3-8 4)(z/xy) = -1/3(pdal-2//3pdTTl 1 (8 ^ 8 2 +8 3 +8 4 )(z/yz) = -l/Stpda^ 1//3pdiT1 ) (8 ^ 8 2 +8 3 -8 4 )(z/zx) = -1/3 (pda1+i//3pd7Tl) (81+82-83-84)

a/2 (1 1T)

4

106

(z/x2 -y2) = 0

(z/3z2-r2) = -2/3pdir1 (B^-32-B^+B4)

Submatrix F Fel - Fell

This has the same geometrical relations as Si Felll.

= a/2 (1 1 1 ), a2 = 3 /2 (1 1 1 ),ik.a„ ik.a^= e - "1 , e2 = e - -2 , e3

a3 = a/2 (1 1 1 ),ik.a„ e - -3, £ = e

a. = a/2 (1 1 1 ) ~4ik.a3

(Pel/Fell)

(xy/xy) = 1/9 ( 3ddcr+2ddTT.+4ddS )(E +£ +£ +£ )1 1 1 1 2 3 4(xy/yz ) = 1/9 ( 3ddCT -ddTT -2dd6^ ) (£^-£2+£3~£^ ) (xy/zx) = 1/9 ( 3ddQ -dd7T -2dd6 ) (e^+e^-£^-e^ )

2 2(xy/x -y ) = 0

(xy/3z2-r2) = 2-/3/9 (-ddTT tddS^ ) ( -e 2-E3+E4)(yz/xy) = 1/9(3dda^-dd7T^-2dd6^ ) (£^-£2+£3-£^)

(yz/yz) = 1/9 ( Sddcr^+2ddTT^+4dd5^ ) ( +e2+e3+e^ )

(yz/zx) = 1/9 (3dda^ -ddTT^-2dd5^) (£ ~ £ 2 “ £ 3 + £ 4 )(yz/x2-y2) = 1/3(-ddTT +ddS )(£+£-£-£ )1 1 1 2 3 4(yz/3z2-r2) = /3/9(ddir -dd6 1 )(el+ e 2 ~ £ 3 ~ £ A}

(zx/xy) = 1/9 ( 3ddCJ1-dd7T.|-2dd61 ) (e +e2 - e 3 ” £ 4 ) (zx/yz) = 1/9(3ddai-ddTT1-2dd6l ) (ei-eo-e3+e4)

(zx/zx) = 1/9(3ddC +2ddTTl+4dd61 ) (£1+£2+£3+£4)

(zx/x2 -y2) = 1/3 (dd'n‘1-dd6 1) (e1~e2+e3_e4)2 2 2 2 2 2 2 2 2 2 (x -yVxy) = (x -y /3z -r ) = (3z -r /x -y ) = 0

(x2 -y2/yz) = 1/3(-dd7T.]+dd61 ) (e1+e2"e3"e4)(x2-y2/zx) = l/3 (ddTTl-dd6 1) (ei-e2+e3-e4)(x2 -y2/x2-y2) = 1/3 (2dd7T1+dd6 l ) (ei+£2+e3+e4)

107

(3z2-r2/xy) = -2/3/9 (ddT^ -dd<51 )(e - £ 2 - ^ 3 + e 4 )

(3z2-r2/yz) = /3/9 (ddTT -dd5^ ) ( +G2_£3_£4 )(3z2-r2/zx) = /3/9(dd7T1-dd6l) (e1-e2+e3-e4)

( 3z2-r2/3z2-r2 ) = 1/3 (2dd7T1+ddS1 ) ( +£2+G3+£4 )

Submatrix G____Fel - Felll

This submatrix is the same as the complex conjugate of the submatrix F (i.e. e -► e*, e2 -> e*, •+ e* and e4 -+ e*)

Submatrix I____ Fell - Felll

a . = a ( 1 0 0 ) , a _ = a ( T 0 0 ) , a , = ( 0 1 0 ) , a , = ( o T o ) , a c = ( 0 0 1 ) , a = ( 0 0 ? )— 1 ~2 ~3 4 ~5 ~ 6

(II/III)

(xy/xy) = ddir (2 cosk a+2cosk a) + 2dd6 „cosk a J 2 x y 2 z

2 2 2 2(xy/yz) = (xy/zx) = (xy/x -y ) = (xy/3z -r ) = 02 2 2 2(yz/xy) = (yz/zx) = (yz/x -y ) = (yz/3z -r ) = 0

(yz/yz) = 2dd7T^(cosk a+cosk a) + 2dd6 ^cosk a2 y z 2 x2 2 2 2(zx/xy) = (zx/yz) = (zx/x -y ) = (zx/3z -r ) = 0

(zx/zx) = 2dd7Tdcosk a+cosk a) + 2dd6 cosk a)2 z x 2 y2 2 2 2 2 2(x -y /xy) = (x -y /yz) = (x -y /zx) = 0

2 2 2 2(x -y /x -y ) = 3/2ddCfdcosk a+cosk a) + ?dd<5„(cosk a+cosk a+4cosk a)2 x y 2 x y z2 2 2 2(x -y /3z -r )=- /3/2ddcr„(cosk a-cosk a) + /3/2dd6dcosk a-cosk a)2 x y 2 x y

(3z2 -r2/xy) = (3z2-r2/yz) = (3z2-r2/zx) = 0,, 2 2. 2 2 2 2 ., 2 2 ,(3z -r /x -y ) = (x -y /3z -r )

2 2 2 2(3z -r /3z -r ) = |dda~(cosk a+cosk a+4cosk a)+3/2dd6_(cosk a+cosk a)

108

Table 5.2.2

Slater-Koster parameters used in the computations (Ryd)

Neighbours Considered1 2 3

sso 0 0 -0 .0 1 0 8spa 0 0 0.015sda -0 .0 1 2 9 -0 .0 0 8 4 0ppa 0 0 0.0628PP7T 0 0 -0 .0 2 9pda -0 .0 8 7 8 -0 .0 4 9 4 0pdfT 0 .0507 0.0285 0dda -0 .0 5 6 3 -0 .0 2 3 2 0ddTT 0.0257 0.0083 0ddS -0 .0 0 3 -0 .0 0 0 8 0

4 = 0.051, EP . = Si 0.882, 4 e = 0.735

A = 0.047 Ed = Mn 0.765, 4 = 0.705

Fig. 5.2.1

Energy bands for paramagnetic Fe3Si 109

110

(b) Local density of states and one-electron energy bands

We shall use the recursion method for calculating the n (E,R)

and thus U (R) (Haydock et al (1975), Haydock (1980)). This method

depends on developing particular diagonal matrix elements of the

Green function

G(E) = (E-H) " 1 (5.2.3)

as infinite continued fractions and does not depend on the use of the Bloch theorem or knowledge of the band structure energies E(k). It

can thus be used in a situation where translation symmetry has been

broken (such as the neighbourhood of an impurity) as well as in a

pure material where translational symmetry still holds. This method depends on having the Hamiltonian II in a tight-binding representation

where for perfect ordered crystal

H Z t c + c1

(5.2.4)

is given as a spin dependent matrix whose elements are self-energies

or near neighbour hopping integrals. Cubiotti et al (1973, 1975) and Jacobs (1973, 1974) has extended (5.2.4) to the study of disordered system where

H z e .cTc. +. i l l Z t . . c T c ., l] l J 17*3+ H2 (5.2.5)

where i and j are site indices£is an annihilation operator for electrons, E^ are self energies and t are near neighbour hopping intergrals. The

equation (5.2.5) has been reviewed by the above authors and will not bereviewed in detail here.

In the case of an ordered crystal the basis set used to obtain

the continued fraction is defined as follows:

|0) = |o >

| 1) = N [ H | 0) - | 0 ) ( 0 | H | 0 ) ]

| 2 ) = N [H|U - | 1 ) (1 | H | 1 ) - | 0 ) ( 0 | H | 0 ) ]

|3) = N [H|2) - |2)(2|H|2) - 11 )(1 |H|2)]

Since |0)(0|H|2) = 0

n- 1I n) = N [H|n-1) - Z Ii ) (iIHIn-1)] (5.2.6)i n 1 1 1 1

1=0

The main properties of the set (5.2.6) are

(i) (iIj) = 5..

(ii) (i+n|H|i) = 0 1

f if n)2 (5.2.7)(i|H|i+n) = 0

The relevant part of the matrix is the tri-diagonal. The relevant part of the resulting secular equation is of the form

"(0 IHI 0)-E ( 0 | H | 1 )(0 | H | 1 ) ( 1 |H|1 )—E ('I

0 ( 2 | H | 1 ) (2 |0 0 <3|

with (i j H|j) = 0 unless |i—j| < 1

We can understand why the matrix

by examining one matrix element

0 0 0

H | 2 ) 0 0H|2)-E ( 2|H|3) 0H | 2) (3|H|3)-E (3|H|4)

(5.2.8)

(5.2.9)of equation (5.2.9) is tri-diagonal

;uch as (2|H|0). From (5.2.6) we

112

H I 0 ) = N 1 I 1 ) + | 0 ) ( 0 I H I 0 )

( 2 | H | 0) = N~1 ( 2 | 1) + ( 2 | 0 ) ( 0 | h | 0 ) = 0 (5.2.10)

because the basis set is orthonormal.

The electronic density of states of any system can be written as

p(E) = — Im Z < n G(E-ie) n > ttn 1 1n(5.2.11a)

where N is the number of atoms in the system and |n> is the Wannier

function situated on atom n. Taylor's theorem can be used to develop

an expansion for the matrix elements of (5.2.11a). They are given by expression of the form

.2< 0 | G | 0> = < 0 l' + "^' + '^J + ••• l0>

EZ EJ(5.2.11b)

which shows that a basis set derived from the ket |0 > by repeated operation af h i is sufficient to determine the diagonal matrix

element <o |h |o> . P(E) can also be determined if we use distinct atom in the crystal. Using the theorem which says that the inverse matrix can be written as

A B

C D

-1 -1 -1 (A-BD C)

-1 -1 -1 -D C (A-BD C)

- (D-CA 1B) ^ A - 1 "1

(D-CA 1B) 1 (5.2.12)

We can write the following expression for the matrix elements that enter the formula for p(E) (5.2.11a) as

r ,<0I GI 0> Ie- ( o| h| o) - (o |h h ) (11 h| o)

J E- (1 IHI 1 ) - (1 lH|2 ) ( 2 IH|1 )L E - ( 2 |HI 2) - ( 2 ! h [ 3 ) ( 3 [ h | 2)

1

E-J

(5.2.13)

113

which can be written more compactly as

<0 |G)0> 1

E-ao

(5.2.14)

Thus the diagonal matrix elements of the Green function arecalculable as a continued fraction. The successive coefficient inthe continued fraction (a.,b.) hereafter called Haydock coefficientsi idepend on the atomic configuration in successively larger shells of

atoms surrounding the central atom.

In our calculation we use a cluster of 931 atoms which issufficient to ensure that the coefficients in 14 levels of the

continued fraction are obtained exactly. The calculation of thecontinued fraction (a.,b.) was performed on the Amdahl 470/V8i icomputer of the University of London.

In order to obtain the total densities of states, it is

necessary to add local densities of states for each type of orbitalon each type of atom. Due to the symmetry of the crystal there isa considerable degeneracy of these local densities of states

e.g. the xy, yz and zx local densities of states on atoms of type IIand III are all degenerate. It is thus necessary to calculate onlyone continued fraction to obtain these local densities of stateswhich is then given a weighting factor of six in the calculation of

2 2 2 2the total density of states. Similarly orbitals x -y and 3z -r on atoms of the type II and III are degenerate giving a weighting factor of four for the single continued fraction. When all the degeneraciesare taken into account we find on six independent continued fractions.

114

The continued fraction coefficients and their associated weighting

factors are listed in table 5.2.3.

The continued fraction is terminated by putting deeper

coefficients equal to estimated asymptotic values which are chosen

to give the correct band limits which can be derived simply from the band energies at symmetry points in the Brillouin zone (see section 5.2a). The terminator is then the solution of a quadratic

equation. Similar terminators (square root terminators) have been discussed by Haydock et al (1975). The terminators a^ and b for

each spin orientation are given in table 5.2.3. The formalism

discussed above needs as input the form of the tight-binding matrix

and the values of the self energies and hopping integrals in it. We

obtain the form of the tight—binding matrix for each spin

polarization from the Slater and Koster (1954) atomic orbital scheme using a basis set consisting of five 3d orbitals for each spin on each Fe (or Co or Mn impurity) atom and one 3s orbital and three

3p orbitals on each Si atom. This results in a 19x19 secular equation for each spin in the pure material where the Bloch theorem

applies.The hopping integrals are given in terms of ssQ, spa, ppa, ppTT,

pda, pdir, sda, dda, ddir, dd5 parameters (see table 5.2.2). The self­energies that enter the matrix are E^. , e J3. and E^a = E^ + aA,

a = ±1 for up and down spin electron respectively and A is the Stoner exchange splitting. The energy interval used is dE = 0.002 Ryd.

The exchange splitting parameter A is chosen to give the correct

observed total moment. That is A is chosen such that N4- - Nt = 4.8yBwhere

115

fT, w. p. (E ) = Nt1 1 F 1

and

Z w.pt (E ) = N+ (5.2.15)l i Fl

and are the orbital weightings, p(E) is as defined in (5.2.11a), i is the band index and E M.81 Ryd) is the Fermi level chosen such

thatNt + m = 26 (5.2.16)

In the presence of an impurity we also need an additional self energy d dparameters E_ or Ew .Co Mn

The impurity parameters are chosen so that the overall change in the total number of electrons in the system is +1 in the case of

Co and -1 in the case of Mn.

The local density of states on an impurity atom on a given site

is obtained from very nearly the same continued fraction as that usedfor an Fe atom m the perfect crystal on the site. The onlycoefficient that changes is the highest diagonal coefficient (a in5.2.14 becomes a ± 0.03) because only that coefficient contains oinformation refering co the self-energy on the impurity site.

The parameters used in the calculation are presented in table 5.2.2. The exchange splitting parameters A(0.09 Ryd) is closer to the value (0.07 Ryd) obtained for pure Fe by Gunnarson (1976).

1 1 6

CONTINUED FRACTION COEFFICIENTS FOR ORBITALS:Spin up

T a b l e 5 . 2 . 3 .

xy on Fe atom II. Associated v/eightinc factor 5.

ai7.820000E-01 8 ♦704624E-017♦360981E-01 3♦753387E-01 7♦139556E-01 7 ♦228663E-01 3•880728E-016«709560E-013•797533E-01 6 ♦757361E-01 6♦958383E-01 3 ♦742489E-01 7♦143599E-01

bi1 ♦OOOOOOE+OO 1•789140E-02 5♦632492E-02 1 ♦088262E-01 9♦721428E-02 5 ♦ 694335E-02 1♦244875E-01 1•226876E-01A L O O i 0 " 7 C _ A ?*T « U / ^ ii ' ' L. V k .

1 .293958E-01 1 ♦ 116187E-01 5♦047202E-02 1♦407728E-01 1 .012717E-01

(The weighting factor of 6 arises iron the fact that the local density of states associated with the above orbital is degenerate with 5 other local densities of states associated v/ith orbitals yZ and Z:: on Fe atoms II and III and also with

orbital xy on Fe atom^n)’

117

9 2— v on Fe atom j.— * Associated weighting factor

7.820000E-01 7♦98931IE-01 8♦867554E-01 7♦445563E-01 4♦053241E-01 6♦592794E-01 7♦571278E-01 4♦271538E-01 6♦011735E-01 7♦366378E-01 4♦230530E-01 5♦928286E-01 7♦585667E-01 4♦493504E-01

1 ♦OOOOOOE+OO 1 ♦024458E-02 4♦574284E-02 4♦275577E-02 8♦352125E-02 1»475716E-01 5♦028030E-02 8♦619428E-02 1 ♦588420E-01 5♦134264E-02 8♦183140E-02 1 .607015E-01 5♦129702E-02 9♦510046E-02

jcv on Fe atom Associated weighting f a c t o r 3 .

7♦820000E-01 8♦355485E-01 9♦174517E-01 6 *704733E-01 3♦910952E-01 7♦422923E-01 7*435395E-01 4♦036300E-01 6♦302770E-01 7♦357905E-01 4♦076377E-01 6♦075433E-01 7♦630554E-01 3♦943794E-01

1♦000000E+00 1 .290769E-02 2♦515233E-02 5♦275817E-02 8♦302754E-02 1 .432428E-01 4♦754016E-02 9♦368867E-02 1 ♦477495E-01 4♦815549E-02 9•315675E-02 1♦479813E-01 5«093884E-02 9•652001E-01

118Spin down

xy on Fe atom II. Associated weighting factor 6ai

6 ♦ 880000E""01 8 ♦ 436350E-01, 7♦236323E-01 3♦699974E-01 6 *876990E-01 7.143705E-01 4♦010825E-01 5♦977346E-01 7♦113395E-01 3♦955876E-01 6 ♦098486E-01 6♦806871E-01 4♦468345E-01 5♦643882E-01

bi

1 *OOOOOOE+Off 1 .789140E-02' 6♦384665E-02 8♦695012E-02 1 .049355E-01 6♦419843E-02 9♦772819E-02 1♦309096E-01 5♦895770E-02 8♦422637E-02 1♦419919E-01 5♦964854E-02 8♦535576E-02 1»545866E-01

x -y on Fe atom II. Associated weighting factor 4ai

6♦880000E-01 7♦678264E-01 8♦618289E-01 7♦183912E-01 4♦682086E-01 5•580956E-01 7 ♦ 534463E-01 4♦950871E-01 4♦863725E-01 7♦292434E-01 4♦995302E-014 ♦ 662047E-01 7 ♦ 422750E-015 ♦ 444835E-01

bi1 ♦000000E+00 1 ♦024458E-02 5♦323709E-02 3♦970724E-02 7♦845628E-02 1 *480656E~01 5♦936005E-02 7♦006991E-02 1 ♦499866E-01 6♦990784E-02 6♦009392E-02 1 ♦401144E-01 9♦215832E-02 5♦087049E-02

xy on Fe atom I. Associated weighting 3ai

6♦880000E-01 7♦652092E-01 9♦291508E-01 6♦384970E-01 4♦513698E-01 6 *570508E-01 7♦555753E-01 4♦134089E-01 5♦686672E-01 7♦473246E-01 4♦094834E-01 5♦703206E-01 7♦287693E-01 4♦574997E-01

bi1 *000000E+00 1 ♦290769E-02 2♦933604E-02 3♦847975E-02 9♦087867E-02 1♦464903E-01 4♦983333E-02 7•932138E-02 1«441042E-01 5 *676928E-02 7♦186073E-02 1♦484424E-01 6.306899E-02 7»228351E-02

y on Fe atom I associated weighting factor 21 1 9

ai7♦820000E-01 7♦445088E-01 8 * 339902E-01 3♦553503E-01 6 ♦693155E-01 8♦203842E-01 3 ♦ 455862E-01 6♦523463E-01 7♦502595E-01 3♦657685E-01 6 ♦596S07E-01 7♦110487E-01 4♦008546E-01 + ♦ 727210E-01

bi1♦OOOOOOE+OO 1♦107937E-02 6.152055E-02 1♦317307E-01 5♦687562E-02 5♦941840E-02 1.094557E-01 1♦070542E-01 5♦195322E-02 1♦119864E-01 1♦193020E-01 4♦596852E-02 1.316639E-01 1♦208921E-01

S on Si atom associated weighting factor 1ai

5 ♦ 100000E-026 ♦311086E-014 ♦ 548352E-01 7«016140E—0 15 ♦738251E-014 ♦ 471372E-01 7♦269462E-01 6*236683E~01 3♦920830E-01 7♦289098E-015 ♦ 950330E-014 ♦ 274509E-01 7 ♦ 508807E-015 ♦ 733833E-01

bi1♦000000E+00 5♦85431&E-03 1 »705766E-01 1 ♦285971E-01 3 ♦ 771862E-02 1♦549870E-01 1 *083410E-0i 5♦313179E-02 1 ♦579468E-01 7 ♦ 922i&iE""$2 5.663287E-02 1♦719212E-01 7♦455987E-02 5♦803474E-02

X on Si atom associated freighting factor 3ai

8 * 820000E-01 8 ♦ 098796E-01 6♦626730E-01 4♦126277E-01 7♦321695E-01 6♦805465E-01 3♦659523E-01 7♦235097E-01 6 ♦545775E-01 3♦817279E-01 7♦307034E-01 6 ♦ 472747E-01 3♦789695E-01 7♦458102E-01

a°o = .788

bi1 ♦000000E+00 6♦612796E-02 3♦107584E-02 1♦577663E-01 9♦527045E-02 5♦478501E-02 1♦342177E-01 1♦010512E-01 4♦903472E-02 1♦477381E-01 9♦366459E-02 5♦877162E-02 1 ♦385802E-01 9♦916645E-02

b» = 0.02806

x -y on Fe atom I. Associated weighting factor 2 120

ai

6♦880000E-01 7♦144183E-01 8♦40901IE—01 3♦023217E-0i 6♦970494E-01 7♦676339E-01 3♦559983E-01 6♦382226E-01 6♦999865E-01 4♦236694E-01 5♦668318E-01 7♦131205E-01 4♦624994E-01 5-S06153E-01

bi

1 ♦ OOOOOOE+OO 1 ♦ 107937E-02 7 ♦ 624656E-02 9 ♦ 191918E-02 5♦734673E-02 8♦174157E-02 7♦045162E-02 1♦299939E-01 6♦087658E-01 7♦639533E-02 1♦439788E-01 6♦224691E-02 7♦409000E-02 1♦630089E-01

S on Si atom. Associated weighting factor 1ai

5«100000E-02 6♦029352E-01 4♦890702E-01 6*148197E-01 6 ♦147743E-01 4♦393701E-01 6♦683208E-01 6♦484208E-01 3 ♦ 539045E-01 6♦912296E-01 6♦522620E-01 3.709449E-01 6♦902442E-01 6♦396323E-01

bi1♦000000E+00 5 ♦ 854316E-03 1♦660480E-01 1 ♦ 299282E-01 3♦583179E-02 1 »497920E-01 1♦103914E-01 5♦949609E-02 1 ♦ 146117E-01 1 ♦014021E-01 6♦053172E-02 1♦177772E-01 1.090538E-01 5♦646050E-02

x on Si atom. Associated weighting factor 3ai bi

; 8 ♦ 8 2 0 0 0 0 l .~017♦496144E-01 6♦963433E-01 3♦822076E-01 7.057221E-01 6♦743863E-01 3»719663E“01 6♦526889E-01 6♦783645E-01 3.913276E-01 6♦432142E-01 6♦679214E-01 4*I66105E-01 6 * 141895E-01

1♦000000E+00 6♦612796E-02 3* 425330E-02 11 307960E-01 1 »044968E-01 5♦885446E-02 1♦021600E-01 1.203684E-01 5♦807033E-02 9♦30212IE-02 1♦409279E-01

560793E-02 8 ♦ 979559E-*o^1♦466667E-01

a°° = 0 . 8 8 2 = 0.02806

140

120

1 0 0 -

1 60-<-u.D

0)

d 2 0 -

/Energy 0 lyc] )

The total density of states for pure

Feyjj for spin up.

Fig. 5.2.2

Fig. 5.2.3.

The total density of states for pureFe^Si for spin down

mCM

1 4 0 —

120 -

100 -

Energy C&jd)

Fig. 5.2.4.

The local density of states of up-spin electrons on an Fe atom or a Co or Mn impurity situated on a T site

FeCoM n

1 0 0

<N

80

60 -

C d y d - )

Fig. 5.2.5.

The local density of states of electrons on an Fe atom or a

impurity situated on a T

dow-spin

Co or Mn

site

T-1-j-1-1-1-1^1—1 - 1 1 - 2

Fig. 5.2.6.

The local density of states of up-spin electrons on an Fe atom or a Co or Mn impurity

situated on a C site

1 00 —

fcnQxgy C f y d - )

• Fig. 5.2.7.

The local density of states of down-spin electrons on an Fe atom or a Co or Mn

impurity situated on a C site

t— i— |— rn— i— r1-1 1-2

127

5.3 Discussion and Comparison of Results

We have calculated the following quantities:

(a) The electronic band structure for pure Fe^Si presented in

figure (5.2.1 ).

(b) The total densities of states for pure Fe^Si for each spin

polarisation. These are presented in figures (5.2.2) and (5.2.3).

(c) The local densities of states of up-spin and down spin electrons

on an Fe atom or a Co or Mn impurity situated on a T site.

These are presented in figures (5.2.4) and (5.2.5).(d) The local densities of states of up-spin and down-spin electronson an Fe atom or a Co or Mn impurity situated on a C site. Theseare presented in figures (5.2.6) and (5.2.7).

We have also calculated the quantities (R) defined inequation (5.2.1) where i = Fe, Co or Mn, O = ± 1 and R = C or T.From these latter quantities the site preference in thermal equilibrium will be deduced.

The main features we observe in the figures are (i) the general similarity of our band structure to that of Switendick (1976) and the similarity of our local densities of states curves of those

for Fe^Si of Williams et al (1982) and for Fe^AL of Haydock and

You (1980) and Ishida et al (1976). The total density of states for pure Fe^Si for spin up has three distinct peaks while spin down has five. The peak in the down spin falls just below the Fermi energy (0.81 Ryd) while for the up spin the Fermi energy lies well into the peak.

The energy bands along T can be classified as follows: the lowest lying band from-0.08 to 0.1 Ryd has its origin in

128

the silicon 2s state. The next lowest band from 0.49 to 0.61

Ryd is derived largely from the silicon 3p states. The third

band 0.6 to 0.73 Ryd is largely associated with the d states

as are the next two. Finally above the Fermi energy there are

states corresponding to the antibonding combination of the Si (3p) and Fe (3d) orbitals,

(ii) the local density of state for each spin on T sites in

pure Fe^Si narrow (due to the small number of nearest neighbours) and has a large central peak,

(iii) the local density of states of each spin on C sites in pure Fe^Si is broader and has two (or more) widely separated

peaks resembling in this the density of states in pure Fe and(iv) the main change in the density of states when an Fe atom

on the central site is replaced by a Co or Mn impurity is a shift

of the peaks to higher energies for a Co impurity and lower energies for a Mn impurity.

The calculated magnetic moments on different atoms on thetwo sites are presented in table 3.3.1. The moment on eachsite increases as we move from right to left across the periodictable. The moments on Fe atoms on T and C sites should be comparedwith experimental values of p(T) = 1.2p , ] i {C) = 2.4y obtainedB Bby Burchet et al (1974).

To investigate the site preferences of impurity atoms, we need to calculate the change in the one-electron band energy when the impurity is moved from one site to another. For an impurity of type i this is given by the change in

129

E.(R) = U. (R) + LJ. (R) - U (R) - U (R) (3.3.1)1 it it Fet Fet

where R is changed from T to C. Thus the site for which E^(R)

is smaller is the preferred site. The values of U. (R) andlGE^(R) are presented in figure 5.3.2. The result for the Mn

impurities is that E^(R) is smallest on the T site so that this

site is correctly predicted to be preferentially occupied by Mn atoms. For Co impurities E A R ) is smallest on the C site

so that this site is correctly predicted to be preferentially occupied by Co atoms.

The results in the case of the Co impurity are simple to analyse.

The changes in U. (T) and u. (C) when an Fe atom is replaced it itby a Co atom are small because the up-spin band is nearly full

with the main peaks well below E particularly on the T site.FThe main effect is then just a downward displacement of the d-band

d dby an amount of approximately E^n“Epe = 0.03 Rydberg which resultsin a decrease in U. (R). This effect is shown in our calculationitof U. (T) and U. (C). The changes in U. (T) and U. (C) are it it 14- 14-much larger. This is due to the fact that lies well into

the band so that when the band is shifted downwards on replacingFe by Co many states cross the Fermi level resulting in an increasein U. (R). The effect is particularly large on the C site because i 4-of the presence of a peak in the local density of states justabove E which moves below E on making the substitution. The F Fnet site preference energy E n (T) - E n (C) is large enough tobe well outside the range of error of the calculation.

130

The results in the case of Mn impurities are slightly less

clear. The changes in both U.,(T) and U.,(C) are large for

the same reason as in the case of Co. These changes are both

negative because the number of down spin electrons is reduced.

The greater effect takes place on the T site owing to the closer

proximity of the nearest occupied peak to the Fermi level.

The changes in both IL^(T) and IL^(C) are also significant owing

to the upward movement of the peaks below the Fermi level.This results in a positive effect due to the upward movement of occupied states and a negative effect due to the fact that

some hitherto occupied states become unoccupied. The negative

effect is larger on the C sites owing to the close proximity

of the peaks below the Fermi level. The net result for up spin electron is a positive effect on T sites and a negative effect

on C sites.This is large enough to outweigh the differences for the

down spin electrons and results in a net preference of Mn atomsfor the C sites. The energy difference which governs this preferenceE (T) - E , (C) is much smaller in magnitude than the difference Mn Mnobtained for the Co atom and is perhaps just outside the range of error of our calculation. It is pleasing to note that itis however in the right direction.

131

Calculated

y(C)

y(T)

Table 5.3.1

Magnetic Moments on Different

on the Two Sites T and C

Mn Fe

2.16 1.9

1.69 1.2

Atoms

Co

1.15

0.93

132

Table 5.3.2

The values of U. and E. (R) ---------------------------------------iG --------------- 1 -------

i U. (T) U. (C)it it

Fe 2.94 2.96

Co 2.895 2.945

Mn 3.04 2.895

U . (T ) U.(C) E .(T)i4 i4- l

2.337 1.806

2.568 2.442 0.186

1.925 1.453 -0.312

E. (C)

0.621

-0.418

133

5.4 Summary and Conclusion

In this chapter we have calculated the band structure of Fe^Si,

the local densities of states of the pure system for various

sites and spin orientations from Haydock's method and similar

densities of states of the system with various impurities in

the two nonequivalent Fe sites, one (C) with cubic symmetry and another (T) with tetrahedral symmetry. We have also calculated

the one-electron total energies and compared them. We found

that our calculation gives the correct experimental site preferences

in terms of the energies. In this intermetallic compound Fe^Si we found that impurities from the first transition row show a

marked tendency at thermal equilibrium to occupy one or other of the two types of sites preferentially; impurities from the left of Fe such as Mn preferring the C site and impurities from the right of Fe such as Co preferring the T site. Our calculations

also agree with experiment (Pickart et al 1975) in regard to

the Fe atoms in Fe^Si compound carrying two distinct moments(2.19 and 1.2u ) as compared to experimental values of 2.4 andB1.2y^. The calculated magnetic moments on different atoms on

-D

the two sites increase as we move from right to left across the periodic table. We have given a simple physical interpretation of our results which explain the experimental results successfully.

134

CHAPTER 6

BAND STRUCTURES OF SOME TRANSITION METAL-METALLOID COMPOUNDS

6 .1 Introduction

A great deal of effort has been made to elucidate the electronic

structures of transition metal-metalloid compounds. Many physical

properties of these compounds have been interpreted on the basis of

accurate knowledge supplied by band calculations. Most of these

calculations are, however, limited to alloys of the Cs Cl - type structure and their DO^ - type superstructure (Ishida et al (1976),

(Fe^Al), Switendick (1976), (Fe^Si), Williams et al (1982), (Fe^Si),

Haydock and You (1980) (Fe^Al), Bylander et al (1982) (Ni^Si), Himsel et al (1980) and Blau et al (1980) (Mn^Si), (Fe^Si) and (Fe^Al).

There are, however, compounds of great practical importance such as cementite (Fe^C) and nickel boride (Ni^B) where the crystal structure

is much more complicated (containing at least 16 atoms per unit cell) and where amorphous allotropes can easily be formed. Little attention

has been paid to the study of their electronic structures however. These calculations have been hindered by the structural complexity of the compounds.

However, Riley et al ( 1979) have calculated a partial density of states for a Si atom embedded in a Pd matrix in an attempt to model amorphous compounds of the approximate formula Pdg^Si^. Kelly and Bullett (1979) have calculated densities of states and partial

densities of states for crystalline and amorphous alloys containingthe same two elements.

135

Our aim is to calculate the electronic structures of Fe^C and

Ni^B using an established procedure (the LCAO method). There is also

the hope that it will provide some guidance into the possibility of

evaluating the elctronic structure of the more complex compounds (e.g. Ni^C). These two compounds are interesting from a theoretical

point of view, both, as regards their crystallochemistry because three

types of bond T-T, T-M and M-M (where T= transition metal and M =

metalloid) may be realized in them simultaneously and as regards their

magnetic transformations (Kostetskiy and L'vove (1972) and Shabanova and Trapeznikov (1973). The results of the calculation may lend

itself to the discussion of the possibility of an electronic

contribution to the stability of these compounds. The study of

electronic structure also provides us with some information about their mechanical properties. Cementite (Fe^C) is one of the primary phases,

in addition to ferrite, in plain carbon steels and mildly alloyed steels, which represent the major tonnage of the world's steel production. However, little is known about its properties except that it is hard and brittle. The primary reason underlying the lack of

knowledge of the properties of this compound is that, in the binary

system, iron-carbon, this compound is metastable at all temperatures in respect to its products of decomposition, graphite and iron saturated with graphite (Ozturk et al (1982). Nickel boride (Ni^B)

is known to be paramagnetic (Kostetskiy and L'vove (1972)) while the alloy Fe^C is known to be ferromagnetic with an average magnetic

moment of 1.78 MB ( Shabanova and Trapeznikov (1973)). Published data on the physical properties (e.g. electronic heat) of these compounds is extremely scarce and so some reasonable empirical evidence

136

regarding the electronic structure of these compounds can only be

drawn from the results of studies of their magnetic characteristics

by Kostetskiy and L'vove (1972) and from x-ray photo electroscopy measurements of Shabanova and Trapeznikov (1973). For example these

authors attributed the magnetic properties of these compounds to the presence of uncompensated 3d electrons in them (i.e. partially full

d-bands). These compounds also possess metallic conductivities. Experimental and theoretical studies of the electronic structure of

the valence band of these materials could provide an additional tool to study the topological structure of amorphous semiconductors.

The crystal structure for Fe^C (Ni^B) is relatively complex with

four formula units per unit cell: eight Fe sites, four Fe . sites

and four C sites each with different local coordination (see Fig 6.1.1). The Bravais lattice of this compound is orthorhombic. The iron atom of type I has twelve iron nearest neighbours at an average

distance of 2.62A° while the iron atom of type II has eleven iron

nearest neighbours at an average distance of 2.60A° (Lecaer et al

(1976) and Shabanova and Trapeznikov (1975). One way of understanding the principal on which its cell is put together is in terms of a trihedral prism (a nearly regular trigonal prism (see Fig. 6.1.2a)}

containing a carbon atom in its centre, surrounded by six iron atoms

at an average distance of 2.0A° (according to Wyckoff (1964) pp.112, the iron-carbon distance lies between 1.85 and 2.15A° and distance

between iron atoms lies between 2.49 and 2.68A°). The prisms are linked together to make layers of prisms by sharing their corners and edges (see Fig. 6.1.2b). In this structure two kinds of prism layer A and B (see Fig. 6.1.1b) which are related by a twofold

137

screw axis in parallel with the C axis are stacked with a gap between

adjacent layers (Nagakura and Nakamura (1983)).

In this chapter we present the results of calculation of the

electronic structure of these transition metal-metalloid system using the LCAO method. In §6.2 we outline the methods used for calculating

the electronic band structure. In §6.3 we present the electronic

structure for the alloys and discuss salient features of the results and in the final section we give a critical comparison of our results

with experimental data, other calculations and draw our conclusions.

138

a) The structure of cementite

b) The (100) projection of the structure of the cementite(Pe^C). Large circles: iron atoms. Small circles: carbon atoms.

FIG. 6.1.1

(b)

139

a) Regular trigonal prismatic coordination polyhedron.b) Edge-sharing of polyhedra observed in the Fe^C, cementite

structure. Note that the Fe atom G for the upper prism occupies an Fe site in relation to the lower.

FIG. 6.1.2.

140

6 .2 Details of calculation

The electronic structures of Fe^C and Ni^B have been calculated

using the linear combination of atomic orbitals (LCAO) method.Detailed description and some discussion of the accuracy of this

method are presented in Chapter One.We use a basis set of seventy six wave functions. One 2s and

three 2p for each carbon (boron) ion and five 3d for each iron (nickel)

ion, each unit cell containing four carbon (boron) sites and twelve

iron (nickel) atoms. With this choice of basis function, the

paramagnetic Hamiltonian H obtained consists of a 76 x 76 secular

determinant at a general point of the Brillouin zone. The iron-iron(nickel-nickel) d-d blocks are parameterized in terms of three two-

center integrals dda, ddTT and dd6 . Also included in the diagnoal d-dblock is E , the position of the d-bands relative to the conductionbands. H also contains the iron-carbon (nickel-boron) hybridization

blocks and the hybridization integrals in terms of sd(7, pda and pdTT.

Also included in H are the carbon-carbon (boron-boron) blocks whichare parameterized in terms of ssQ, spa, ppa and ppn. Included in

s pthese blocks are E and E , the positions of the s and p bands relative to the conduction bands. In this calculation we consider only interactions between the first nearest neighbours of the same

kind as well as of different kind and neglect all interactions between carbon sites separated by more than 3.10A°.

The number of parameters used in the calculation for each compound is 13 and these are listed in table 6.2.3 in the usual notation. In columns 1 and 2 of this table are parameters for Fe^C

142

Table 6.2.1

gAtomic configuration Fe(3d ), C(2sz,2p2)

Lattice constantsa = 4.523A0

b = 5.089A°

c = 6.7428A°

lattice vectorsin the unit cell, in units of the real space

si S 2 S3ci 0.43 0.87 0.25

C 2 0.57 0.13 0.75

C3 0.07 0.37 0.25

C4 0.93 0.63 0.75Fe' 0.833 0.040 0.25

Fe' 2 0.167 0.960 0.75

Fe* 3 0.667 0.540 0.25Fe' .4 0.333 0.460 0.75Fe, 0.333 0.175 0.065

F e 2 0.667 0.825 0.935

Fe3 0.167 0.675 0.435Fe . 4 0.833 0.325 0.565

Fe5 0.667 0.825 0.565

F e 6 0.333 0.175 0.435Fe7 0.833 0.325 0.935

Fes 0.167 0.675 0.065

143

gAtomic configuration Ni(3d ), B(2sz,2p)

Lattice constants

a = 4.389AC

b = 5.211 Ac

c = 6.6T9A'

Table 6.2.2

lattice vectors

E - s ^ + a ^ + s ^

in the unit cell, in units of the real space

Bi 0.433 0.889 0.25

B 2 0.567 0 . 1 1 1 0.75

B3 0.067 0.389 0.25

B4 0.933 0.611 0.75Ni' 0.864 0.028 0.25

Ni ' 2 0.136 0.972 0.75Ni 1 3 0.636 0.528 0.25Ni' , 4 0.364 0.472 0.75Ni, 0.347 0.178 0.061

N i 2 0.653 0.822 0.939Ni3 0.153 0.678 0.439

Ni4 0.847 0.322 0.561Ni5 0.653 0.822 0.561

N i 6 0.347 0.176 0.439Ni? 0.847 0.322 0.939

N 1 8 0.153 0.678 0.061

144

Table 6.2.3

Slater-Koster parameters used in the computations (Ryd)

Compounds for which energy bands are computedSK parameters T e C 3^ Ni3B

ssG -0.0144 -0.0144spa 0.0267 0.0267ppa 0.1492 0.1492ppn -0.0706 -0.0706sda -0.02053 -0.01677pda -0.1631 -0.1332pdTT 0.094? 0.0769dda -0.04002 -0.0267ddTT 0.0183 0.0133dd6 -0 . 0 0 2 1 -0.0018

ESc = 0.051, EP c = 0.822, 4eb = 0.051, EP = 1.172, 4

0.735

0.676

145

6 .3 Results(A) Energy band for Fe C

Figure 6.3.1 shows the energy bands for the paramagnetic Fe^C calculated for a grid of 21 k points in the irreducible

Brillouin zone. The symmetry is given at T , X, Y, Z, Z, Z', Z''

and A where these are the directions [ 000 ] , [100] , [010 ] , [001] , [ 110 ] , [101 ] , [011] and [111] in the reciprocal space respectively. The bands are too dense to list the symmetries along symmetry lines.

The four lowest lying bands form 0.02866 to 0.05918 Ryd have

their origins mostly in the four carbon 2s wave functions. The

width of this C derived S band is 0.03052 Ryd.Above this are twelve bands in the range 0.2512 to 0.5356 Ryd

which is made up of carbon 2p orbitals hybridized with iron 3d

orbitals.The third set of bands between 0.6192 and 0.8729 Ryd are very

flat and are predominantly of the iron 3d character; finally abovethis in the range 1.016 to 1.342 Ryd are further twelve hybridizedpd bands. The two sets of twelve pd bands have clearly thecharacter of bonding and anti-bonding hybrids.

By accomodating 56 electrons per unit cell we have determinedthe Fermi energy E = 0.82 Ryd which intersects the Fe 3d band.F

146

FIG. 6.3.1

Electronic structure of Fe^C along certain high symmetry lines and planes

rcUJ

r x r y r z

147

FIG. 6,3.1 (continued)

o . o***•••••#•

r z r i

148

(B) Energy band for Ni^B

The band structure of Ni^B is shown in Fig. 6.3.2 along certain

high symmetry lines and planes having the same labelling as in (A).

The lowest set of bands from .03145 to .06065 Ryd are associated with the boron s wave functions. The width of the B derived s

band is 0.0292 Ryd. The next twelve bands found in the energy range

0.4325 to .569 Ryd are made up of boron p bands hybridized with nickel 3d orbitals. The top of this band touches the bottom of the predominantly 3d bands which range from 0.595 to 0.769 Ryd

along the direction X'. These predominantly 3d bands are very flat. Finally in the range 1.135 to 1.475 Ryd are twelve bands associated with boron p wave-functions. These bands are also hybridized with nickel 3d orbitals.

neyu O<tyd)

149

FIG. 6.3.2

Electronic structure of Ni^B along certain high symmetry lines and planes.

S . U~

r x r y r z

150 FIG. 6.3.2 (continued)

r Z r z

151

6 .4 Discussion and comparison of the results

We have performed tight-binding calculations using basis set of seventy six wave functions. One 2S and three 2p for carbon and

boron ions and five 3d for iron and nickel ions, each unit cell

containing four carbon or boron sites and twelve iron or nickel

atoms. These calculations give results which are in good agreement with x-ray photo-electron and x-ray spectra of iron carbide of

Shabanova and Trapeznikov (1975) (see Fig 6.4.1). Although we have

not calculated the density of states it is clear that our calculations

would give peaks at the following positions relative to E , an s peak

at -0.78 Ryd compared to the experimental value of -0.92 Ryd (e), p-peaks between -0.58 and -0.28 Ryd compared to the experimental

values -0.59 and -0.48 Ry (c,d) and d-peaks at between -0.22 and 0.0 Ryd compared to the experimental values of -0.22 and -0.11 (a,b) obtained by these authors.

Our results for Fe^C should also be compared with those of Kellyand Bullett (1979) for crystalline Pd^Si and those of Riley et al(1979) for Pd Si . In the first of these papers the transition o 1 19metal d-band is broader and lower relative to the metalloid p complex than ours and the bonding-anti-bonding splitting of the p complex is of smaller magnitude. The broader d-band arises from the fact that the d-resonances in second row transition metals such as Pd are broader than in first row transition metals such as Fe. The lower d-band of Kelly and Bullett can be understood in terms of the fact that Pd lies to the right of Fe in the periodic table and thus must

have more d electrons and consequently a lower d band. The smaller splitting of the pd complex arises from the greater size of the Si

152

atom. This results in a larger Si-Pd interatomic distance

(r . , = 2.50A° whereas r „ = 2.0A) and consequently in smallerSi-Pd Fe-CpdQ and pdTT hopping integrals. An estimate of this effect can be made

-4as follows. From Andersen et al (1978) we know that pda ~ R . Thus(pda)KB(pda) (2/2.5) - 0.41

GJ

The splitting between the bonding-anti-bonding peaks in our

calculation is 0.8 Ryd. Taking account of the distance dependence only we find that the bonding-anti-bonding splitting in Pd^Si is 0.328 Ryd or 4.46 eV which can be compared with Kelly and Bullett's splitting of 5.9 eV. The remaining discrepancy is presumably due to

the greater width set of the d-resonance in Pd than Fe and the p-resonance in Si than C.

The main difference between our Ni^B band structure and our Fe^C

band structure is that the d-band in Ni^B is lower and consequentlythe gap in the spectrum between the pd and d bands below the Fermilevel is absent. The gap between the pd and d-band above the Fermilevel is also larger. The reason for this is that Ni is further tothe right of the transition series than Fe and trivalent B is further

to the left of the periodic table than tetravelent C. These twofacts give a d-band. in Ni^B lower with respect to the p complex.

\nFinally it is well known that the coesive energies and heat ofA

solution of the materials under discussion are very high (Watson and

Bennett ( 1979) and Gaskell ( 1 979 );. The reason for this is clearly apparent in our one-electron calculation. The strong p-d hybridization depresses the energy of the bonding pd-band and elevates that of the anti-bonding pd-band. The anti-bonding band is above the Fermi level so the net results is a large saving in energy relative to pure Fe

153

and C or pure Ni and B.

This effect should also give rise to a strongly attractive force

between unlike atoms in these materials relative to the force between like atoms. It could thus account for the high degree of short range order observed in the amorphous materials (Vincze et al (1979)) and

possibly justify the speculations concerning the circumstances under

which it is easy to form amorphous materials discussed in the paper of Giintherodt et al ( 1980).

154

FTG. 6.4.1X-ray photo-electron and x-ray spectrum of iron carbide.

r

30 2 0 15 10Binding energy(ev)

25 5 0

155

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