TRANSITION AND NOBLE METAL COMPOUNDS EMMANUEL JOSEPH DANIYANG GARBA

Transcription

1 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

2 DEDICATED TO MY FAMILY

3 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 I am indepted to Z. Vucic for stimulating conversations and for making available some of his unpublished experimental results.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.

4 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

5 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.

6 VI CONTENTS Page Dedication Acknowledgements Abstract List of Figures List of Tables ii iii iv ix xi CHAPTER 1 CONCEPTS IN ENERGY BAND THEORY AND PHASE TRANSITIONS Introduction The one-electron approximation Bloch's Theorem and the concept of energy band Pauli Exclusion Principle and elementary consequences of Band Theory The Muffin Tin Potential Concept of phase transitions Discussion The tight-binding method and th e Slater-Koster interpolation Scheme. 23 CHAPTER 2 THE EXPERIMENTAL PROPERTIES OF Cu Se RELATED TO THE 27 ENERGY BANDS 2.1 Introduction Phase transitions Structural Properties Electrical properties 37

7 2.5 Magnetic properties X-ray Emission and Adsorption Diffusion of Copper in copper selenide 48 CHAPTER 3 THE ELECTRONIC STRUCTURE OF Cu Se Introduction Methods of calculation and Approximations Results of the Calculations Comparison with Experiment 79 CHAPTER 4 PHASE TRANSITIONS IN Cu Se Introduction Experimental phase diagram Landau Theory Phase transitions Conclusions 94 CHAPTER 5 ELECTRONICS STRUCTURE AND SITE PREFERENCE OF TRANSITION 96 METAL IMPURITIES IN Fe^Si 5.1 Introduction Outline of calculations Discussion and Comparison of Results Summary and Conclusion 133

8 CHAPTER 6 BAND STRUCTURES OF SOME TRANSITION METAL-METALLOID 134 COMPOUNDS 6.1 Introduction Details of Calculation Results Discussion and comparison of the results 151 REFERENCES 155

9 IX LIST OF FIGURES PAGE Equilibrium phase diagram of the Cu^ ^Se system The planes stacking sequence along the b 32 supperlattice axis [111]_ direction fee Crystal structure of acu^se a) Se M x-ray spectra and x-ray photo o emission spectra of valence electrons in the solid solution Cu^ Se 2-x b) Energy spacing A between the peaks in the 45 Selenium M spectrum 4 f D Forbidden-zone width at room temperature v 46 composition of alloy Cu ^Se Temperature dependence of the width of the 47 forbidden band of Copper Selenide Cu Se (x = 0.001) 2_X Electronic conductivity (CT ) versus 51 temperature (T) of Cu^ Se for various values of x Valence and conduction bands for the ideal 70 Cu^Se crystal structure along certain symmetry directions The Brillouin zone for the simple monoclinic 73 lattice a) Conduction band for the case of one inter- 74 stitial atom and one vacancy b) Conduction band for the case of two inter- 74 stitial atoms and two vacancies The Se p partial density of states The Cu d partial density of states Total density of states Total density of states of ideal Cu^Se convoluted 78 with a Gaussian of width Ryd A graph of CF^ extrapolated to T = 0 against x. 84

10 X A graph of discontinuity at the phase transition versus x The crystal structure of Fe^Si Energy bands for paramagnetic Fe^Si The total density of states for pure Fe^Si for spin up The total density of states for pure Fe^Si for spin down The local density of states of up-spin electrons on an Fe atom or a Co or Mn impurity situated on a T site The local density of states of down-spin electrons on an Fe atom or a Co or Mn impurity situated on a T site The local density of states of up-spin electrons on an Fe atom or a Co or Mn impurity situated on a C site The local density of states of down-spin electrons on an Fe atom or a Co or Mn impurity situated on a C site a) The structure of cementite 138 b) The (100) projection of the structure of the cementite (Fe^C) a) Regular trigonal prismatic coordination polyhedron b) Edge-sharing of polyhedra observed in the 139 Fe^C, cementite structure Electronic structure of Fe^C along certain high 146 symmetry lines and planes Electronic structure of Ni^B al n<3 certain high 149 symmetry lines and planes X-ray photo-electron and x-ray spectrum of iron carbide 154

11 XI LIST OF TABLES Page The tight-binding matrix elements in terms of overlap integrals a) Slater-Koster parameters used in the computation (Ryd) b) Additional Slater Koster parameters for nearest neighbour displacement (Hi) and (3/2H i) used in the computations (Ryd) The one interstitial one vacancy case Vacancy is shifted relative to interstitial atom The two interstitials and two vacancies case Nearest neighbour (n.n) configuration for both Fe and Si sites in Fe^Si Matrix elements in terms of overlap integrals for the body-centred cubic lattice Fe3Si Slater-Koster parameters used in the computations (Ryd) 98 1 m \j Continued fraction coefficients for orbitals Calculated magnetic moments on different atoms on the two sites T and C The values of U. and E.(R) iq i The band parameters, Lattice constants and 142 positions of the sixteen atoms per unit cell in Fe^C The band parameters, Lattice constant and 143 positions of the 16 atoms per unit cell in Ni^B Slater-Koster parameters used in the computations (Ryd). 144

12 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 a (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 ) (1.1.2 )

13 2 formed by the three translation vectors which when displaced by a vector r^ 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 the atoms in this volume is called the basis of the lattice. A useful unit cell 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 translation vector r^ from a given point in the lattice to the nearest equivalent sites. This is definitely the smallest volume around a given site contained within the planes which perpendicularly bisect the set of vectors {r }. If there is only one atom per unit cell, the atomic sites n can 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 thesis. This is the type of lattice we shall be concerned with in 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.

14 3 1.2 The One-electron Approximation There 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

15 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-Fock equations 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 manybody treatment. This leads to the solution of the Schrodinger equation for the electrons in a perfect crystal constituting a very complicated many-body problem.

16 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 functions ijj, (_r) is subjected to the periodic boundary ic conditions'

17 6 \b, (r + r ) = \p_ (r) Yk s k (1.3.5) such that r = s a + s a (1.3.6) S 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. r e = 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.

18 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 n e 1 ( )

19 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 (r_+r_ Uk+K ---m (r) (1.3.16) \ +K ---m and (r) is defined by m -ik, r_ k+k =e (1-3 17) ---m -ik >_r The 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

20 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) 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

21 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 depend upon the band structure.

22 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 Fermi energy, E, and all the energy levels above E are unoccupied. The F F surface 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 ~ 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

23 12 per atom or per spin in a small interval of energy (E,E + AE) and is given by N(E)AE =,.e +Ae _ f(e - E (k))de d3k 3 n B. Z.. E n 87T (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 by N = C(E ) F E 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 ds N(E) = z f n 3 n J grad E (k) 8tt j ' 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 on the 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

24 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.

25 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. neighbour distance. It is usual to take the radius as half the nearest 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 potential V (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. is approximated as a local potential, the Slater approximation The exchange form (1.5.1) where the crystal electronic density p(r) is obtained as a superposition 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)

26 15 pat The muffin-tin sphere is S. V(r) = VT (r) - VAVQ (1.5.3) where VAVG 3 Jr V r) r2dr/(r03 - ri3) (U5-4) 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 Va vg 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

27 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).

28 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

29 18 space group Gq. The remaining of the rules determine whether the transition will be of first or second order. phase transition in superionic conductors. and minor changes of the lattice symmetry. O'Keefe (1976) also d iscussed He distinguished between major 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.

30 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.

31 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

32 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 various model 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

33 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.

34 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 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 $ n m (_r)d' = 0 if f K_ (1.8.3) where H is the crystal Hamiltonian defined as H = H (r) + E V(r-r.) (1.8.4) o 3

35 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 by Z a $ (r) m k m m (1.8.7) 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 n

36 \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 by 1 (_3z2-r2 ) sf x, y, z, xy, yz, zx, x2-y2 and y 3. The function kn do not form an ortho-normal set, but Lowdin (1950) has shown that 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

37 26 (n/m) ($. H $ ) iin L exp(lk.r.)j X*(r ) i ~ ' 3 n ~ (1.8.10) (p,q,r) Z exp(ik.r.) E 1 nm 3 ' 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 nm of 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. n As 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 (1.8.10) have to be evaluated. However, Slater and Koster have proposed using it as an interpolation scheme, with the integrals in (1.8.10) 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 transition and noble metal compounds.

38 27 CHAPTER 2 THE EXPERIMENTAL PROPERTIES OF Cii Se RELATED TO THE ENERGY BANDS x 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 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 measurement of Takahashi et al (1976), (2) the electronic conductivity measurement of 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

39 Fig Equilibrium phase diagram of the Cu^ TrSe System: Second order 28 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)

40 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 reaches a 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 ) 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 phase transition at ~ 140 C (Line 3) is characterized by a discontinuity -elects c in 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

41 30 Vucic et al finding 140 C at stoichiometry. the material has a tetragonal structure. Below this temperature The temperature of the transition was found to depend on departure from stoichiometry. Thermal capacity (C ) measurements on Cu.Se and Cu Se by Kubaschewski p 2 1.ybo (1973) suggest a weak temperature dependence of C far below the P phase-transition temperature T^ and a saturation of above T^. There was an extra term in C observed within several tens of a P degree Kelvin below T^ ascribed to Frenkel pair formation. Vucic and Ogorelec (1980) obtained an activation energy of 50 k J mol ^ and 4 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---3 nicy U K i C L U i e 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 orderdisorder transformation in the cation subsystem (Takahashi et al (1976)). Above these second order phase transitions two effects

42 31 are observed simultaneously. One is an exponential temperature dependence of ionic conductivity characterized by an activation -2 energy 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 ev (100 k J mol ^). We shall discuss this phase diagram in more detail in Chapter 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 until Chapter 3.

43 The planes stacking sequence along the b superlattice axis ([111]f direction). Se cage planes-a; Cu cage plana planes of mobile Cu-ions: tetrahedrally co-ordinated-d and octahedrally co-ordinated-b. the open and full symbols are used to visualize the variation of occupancy between the (ar c, b, d) sandwiches of planes. The arrows indicate pairing between tetrahedral vacancies and octahedral ions along [111] directions fee d a c b d a T o V o V O T o V T o V o o d0-t c b d a c b d a c b d o V V o V o V V <0 o o V <0 v T o V <0 o V o o o dsuper=2d{f C

44 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 nldh9s of this suhisttic v ^ ^ J uiu 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 <111> direction. The plane stacking sequence along the b superlattice axis <111> fee direction are shown schematically in figure from data of Vucic et al (1981). At the structural phase transition temperature T = 140 C c (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