3- APPROXIMATIONS USED IN THE THEORY OF LATTICE DYNAMICS

Transcription

1 45 3- APPROXIMATIONS USED IN THE THEORY OF LATTICE DYNAMICS 3.1 INTRODUCTION:- The theoretical and experimental study of lattice dynamical problem has been of interest to physicists for understanding the thermodynamic, elastic, optical, electrical and numerous other physical properties of solids. 1,2 The dynamical theory of metallic crystal is still one of the most interesting area in the theory of lattice-dynamics, despite the effort which has been devoted to its study. The metallic crystal is supposed to consist an array of positive bare ions forming a periodic lattice which is embedded in a uniform sea of negative charge of conduction electrons. It is basically a many body problem involving the interaction of ions in the field of many moving electrons. The exact solution of equation of motion generated for this system becomes formidable and therefore, it is necessary to make some approximations in order to solve the problem. 3.2 THE ADIABATIC APPROXIMATION:- When a thermal wave disturbs the lattice, the atoms execute small oscillation about their mean position. It is assumed that the core-electrons belonging to the ions move rigidly with the nuclei and can not be excited at the energies available. The conduction electrons respond easily to screen out the local charge fluctuations generated by the vibrations of positive ions. A rigorous approach to predict the possible frequencies for the normal modes of vibration of lattice is to

2 46 express the basic Hamiltonian of the lattice and then to determine its eigen values. The basic Hamiltonian operator 3,4 H, for the whole system consisting of ions and electrons is written as H = Hii + Hee + Hie.. (3.1) Where ( ) and ( ) Where m is the mass of the electron and M is the mass of the ion, ri represents the position of i th electron and denotes the displaced position of the th ion. The indices and run over all the ions where as the indices i and j extend over all the valence electrons. The first term Hii includes the (i) the kinetic energy operator of ions and the (ii) potential energy UI (Rl - Rl ) of direct interaction between ions which includes the coulomb repulsion between the ions and the core exchange interaction. The second term Hee corresponds to the valence electrons which interact through coulomb potential. This also includes exchange interaction between conduction and core electrons. The

3 47 ionic cores and valence electrons interaction is represented by Hie. This interaction takes place through a potential VI ( ri - Rl). The singularities originating on account of the coulomb interaction present in equation (3.1) are avoided by taking the spatial average of the potential to be zero and by excluding the interaction of the particle with itself 5. The problem of determining the eigen-values of Hamiltonian (3.1) is a complicated task. The adiabatic approximation 6 introduce tremendous simplifications in the problem by separating the dynamic aspect of electron and ionic motion. The total Hamiltonian (3.1) may be expressed as H= HI + He.. (3.2) Where ( ).. (3.3) and ( ).. (3.4) Let the wave function for Hamiltonian H be ( ) ( ).. (3.5)

4 48 The crystal wave function ψq satisfies the Schrödinger equation for the electrons in a static lattice as given below. ( ) ( ) ( ) ( ).. (3.6) Now applying the operator H to the crystal wave function ψq and using (3.6) we get. ( ) [ ( ) ( )] ( ) [ ( )].. (3.7) Where G(R) ( ).. (3.8) If the second line of (3.7) is ignored, we can solve our complete eigen- value problem, H ψq = Eq ψq, by making ( ) to satisfy a schrodinger type equation [ ( ) ( )] ( )= ( ).. (3.9)

5 49 This is an equation for a wave function of the ions alone, which shows that the motion of ions is governed by the effective potential, G( r ) + Ee( R ), i.e. electrons contribute adiabatically to the lattice energy. Justification for dropping the terms is (3-7) can be shown in the following way. The second live of (3.7) contains two terms which contribute almost nothing to the expection value of the energy of the system in the state ψq. The first term vanishes because it produces integrals like... (3.10) Where ne is the total number of electrons. The second term is small because, at worst, the electrons would be tightly bound to their ions. Ψ( r, R ) = ψ ( ri - Rl ) Which would give a contribution like.. (3.11)

6 50 This is just m/m times the kinetic energy of the electrons. Since m/m is of the order of 10-4 or 10-5, which is negligible small in comparison to the ordinary thermal energies. In this way, adiabatic principle allows us to separate the ionic motion from the electronic motion, leaving only a residual interaction between the electrons and the phonons. So we can treat the electrons and lattice waves as nearly independent entities and allot the electrons the same coordinate as that of ions. 3.3 LATTICE VIBRATION IN HARMONIC APPROXIMATION AND SECULAR EQUATION :- In addition to the adiabatic approximation already discussed the theory of lattice vibration is based on the so called harmonic approximation viz; when the displacement of the nuclei are small this assumption postulates that the potential energy of the crystal is dependent only on second order terms in the displacement of atoms from their equilibrium position. As a result of thermal fluctuations, each ion in the lattice is displaced from its equilibrium position. The total potential energy of the crystal is assumed to be some function of the instantaneous positions of all the atoms. However the potential energy of a crystal is expanded in power of the amplitude of the atomic vibrations and all the terms higher than those which are quadratic in amplitude are

7 51 neglected, the remaining expansion is called the harmonic approximation. Let us confine ourselves to the crystal lattice with a basis and having no impurities, vacancies, interstitials or dislocations. The equilibrium position of the k th ion in the l th cell is given by R 0 (l,k) = R 0 (l) + R 0 (K).. (3.12) The constituent ions in a crystalline solid execute small oscillations about their equilibrium positions as a result of thermal fluctuation at finite temperature. Let the displacement of the (l,k) th ion from the mean position be u (l,k), the displaced position of ion is given by R (l,k)= R 0 (l, k) + u (l, k).. (3.13) Distance between the k th ion in the l th cell and k th ion in the l th cell at any instant is given by R (l,k) - R (l, k )= R * +.. (3.14) As the two ions vibrate about their equilibrium positions, the potential energy V of the whole lattice will be a function of R * +. The total kinetic energy of the lattice is expressed by

8 52 T = ( ).. (3.15) Where Mk is the mass of the k th atom and ux (l,k), uy(l,k) and uz (l,k) are the components of u (l,k) along x, y, z axes respectively. This approximation gives the expression for potential energy as follows. V( R ) = V0+ V1+ V2.. (3.16) Where V0 is the equilibrium potential energy of the crystal and V1= Vα ( l, k ) Uα ( l, k) = ( ) uα ( l, k ).. (3.17) and V2= ( ) ( ), ( ).. (3.18 a) where ( )= (( ) ( ).. (3.18 b) The subscript zero means that the derivative are to be evaluated for the equilibrium configuration. The coefficient Vα(l,k)

9 53 represents the negative of the force acting on (l,k) th ion in the α (=x,y,z) direction in the equilibrium configuration. The coefficient Vαβ(l k,l k ) is the force constant acting on the ion at R (l, k) in the direction when the ion at R (l,k ) is displaced a unit distance along β direction. As in equilibrium the force on any particle must be zero as Vα( l, k )=0 The equation of motion for (l,k) atom, using above argument is, Mk α (l,k) = ( ) ( ) = { ( ) ( )} ( ).. (3.19) This forms an infinite set of simultaneous linear differential equitation. Their solution is simplified by the periodicity of the lattice. Let us consider a running wave solution of (3.19) in the form uα (l,k) = ( ) * ( ) ( )+.. (3.20) Where uα(k) is independent of l and q is wave vector of the disturbance. Substitution of (3.20) in (3. 19) gives (for monatomic crystal) [ ( ) ( ) ( ( ) ( )].. (3.21)

10 54 Where δαβ and δkk are Kroneeker delta s and dynamical matrix elements Dαβ (q, kk ) are, ( ) ( ) * ( ) ( )+.. (3.22) Here Vαβ (lk, l k ) do not depend on l or l due to the translational symmetry, but only on their difference (l - l ), Because of general properties of invariance under arbitrary translation, the force constants must satisfy the relation 7. ( ).. (3.23) Equation (3.22) includes the terms for which l - I =0 and k=k. such arise due to the interactions of atoms with themselves and obtained their meaning from (3.23) and are, as a matter of fact the reaction of an atom to the force exerted upon it by the crystal as a whole. Because of mathematical difficulties, it would be desirable to avoid such terms, and this can be obtained by subtracting (3.23) from (3.22). we also set l =-0 for the sake by simplicity, to obtain ( ) ( )* ( ) +

11 55 ( ).. (3.24) While for k k, we retain the form of (3.22) with l=o. The equation (3.24) is more convenient through less simple than (3.22), since it avoids the term Vαβ (Ok, ok). Therefore the secular determinant for a monatomic hexagonal crystal can be written as det ( ) ( ).. (3.25) Dαβ (q, kk ) is a element of the dynamical matrix of order 6 6. The solution of the secular determinant along the symmetry direction of a crystal becomes easy as the secular determinantial equation in these directions are factorisable in three 2 2 equations. If Q is a general vector in reciprocal space, we can take a Fourier expansion of the interaction potential U(R) between two ions separated by distance R. U(R)= ( ) ( ).. (3.26) The lattice potential energy V to be the sum of two body potential U { R(l,k)-R(l,k )} given as Vαβ(lk, l k ) = - ( ) R = R (l, k) R (l, K ).. (3.27)

12 56 Substituting Vαβ (lk, l k ) with l =0, in (3.24) with the help of (3.27) and using the standard relation. N -1 ( ( ) ) We have ( ) ( ) ( ) ( ) ( ) ( ) ( ).. (3.28) Here N is the total number of unit cells in the crystal and summations extend overall the reciprocal lattice vector. In general a three dimension crystal with n atoms per unit cell has 3n frequencies for each wave vector. The acoustic modes are 3 in number and for them ω tends to zero as q goes to zero. The remaining [(n-1).3] frequencies are called as optical modes and they are different from zero when q = 0. For a quantum mechanical description, the harmonic Hamiltonian for the crystal lattice can be quantized in the fashion of second quantization in field theory 8. This leads to a description of lattice vibration in terms of phonons, which can be thought of as travelling quanta of vibrational energy.

13 THE SELF-CONSISTENT FIELD APPROXIMATION:- In the present theory, the most decisive approximation is the self consistent field approximation. The electrons in a metal form, a kind of gas inside the metal, which is different from a perfect gas on account of strong interaction of electrons among themselves and also with positive ions. Because of this large interaction, a gas of this kind must be too complicated for simple mathematical treatment. However, in quantum-mechanics, the effect of one electron on all other to a large extent can be averaged; one can treat each electron as moving in the field of the other electrons. The average potential depends on the distribution of electrons and upon the states which are occupied by them. These states in turn depend upon the potential, thus we must compute the potential self- consistently. Ultimately the only important interaction between electrons is the coulomb repulsion, but this is conveniently divided into three distinct contributions. First is the Hartee potential 9, obtained by computing the time average of the electrons distribution and then using Poisson s equation to determine the corresponding potential. Second is the correction for the potential seen by an electron due to the Pauli principle. If an electron with given spin is at a position r, no other electron of the same spin can lie at that point, simply because of the antisymmetry of the wave functions. This effectively gives a hole in the electron distribution and gives rise to the exchange interaction. The third contribution arises from the correlated motion of the electron, which is known as correlation energy. These two corrections fall under the well known Hartree-Fock approximation. 10 In the Hartree-Fock approximation the effect of correlation has not been properly considered.

14 58 However, on account of the use of determinantal wave function the correlation of the parallel spins only is taken into account while the antiparallel spin is still lacking. Such type of correlation has been considered at the first time by Seitz 11. Bohm and Pines 12 based on collective vibrational approach, have shown that the long rang part of coulomb interaction leads to a coherent motion of the electrons which can be described in terms of plasma oscillations. Toya 13 has incorporated the correlation energy of both the spins in an approximate way using the reduction of density of states near the Fermi-surface to meet the defects of the H-F approximations arising from the neglect of proper correlation on the screening. 3.5 PERTURBATION THEORY:- The displacement of the ions in the crystal caused by lattice vibrations provide perturbation in the electronic Hamiltonian (3.2). This theory can be used to calculate the conduction band state. We can then write the perturbed Hamiltonian, total electronic wave function, and the total energy occurring in equation (3.9) as... (3.29 a) ( ).. (3.29 b) ( ).. (3.29 c) Where the superscript refers to the order to which u occurs in the expansion terms. The contribution to the lattice vibrational potential energy made by ion-electron-ion interaction will be.

15 59 The total electronic wave function ψ(r, R) in the Hartree approximation is written as a product of one-electron wave function (r, R). The satisfies the self-consistent Schrödinger equation. Where H1 ψk ( r, R ) = E ( K ) ψk (r, R ).. (3.30) H1= ( ) ( ) ( ).. (3.31) The bare ion potential VI is the energy of interaction with the ions, and the self consistent Hartree potential Ve( r ) is due to interaction with all the other electrons. They can be written as VI (r, R ) = ( ( )).. (3.32 a) and Ve(r, R )= ( ) r.. (3.32 b) Here K includes the spin quantum number of the electron when it is needed. The displacement of the nuclei are so small that the core electrons are not distorted and core electrons plus nucleus can still be considered as a single entity producing a one particle potential UI for conduction electrons. The total energy Ee is given in terms of one electron Hartree energies E( k ) by the following expression,

16 60 Ee= ( ) * ( ) +.. (3.33) Where nf (k) is the probability that the state k is occupied at T= O 0 K, we know that nf(k)( has a step function behaviour with nf(k)=1 for k < kf and nf( k)=0 for k > kf, here kf is Fermi wave vector for electrons. If we make the perturbation expansion corresponding to (3.29) for the one-electron Hamiltonian and for the wave function as ( ) ( ( )).. (3.34 a) The first order term is the sum of two interaction potentials, due to the motion of ions and originating from the response of conduction electrons = = ( ) ( ( )) + = ( ) ( ( )).. (3.34 b) 0 The second order terms are = = ( ) ( ( )) ( ) ( ) We also write.. (3.34 c)

17 61 ψk=.. (3.34 d) We have used the following expressions for, and as ( ) ( ) { ( ) ( ) }.. (3.35 a) ( ) ( ) {( ( ) ( ) ( ) ( )}.. (3.35b) and ( ) ( ) *( ( ) ( ) ( ) ( ) ( ) ( )+.. (3.35 c) Where represents a set of one particle Block functions for electrons which apply to a crystal for electrons with ions fixed in equilibrium positions. We can establish the following relation between the secondorder matrix elements of Ve, =.. (3.36) The second-order perturbation theory together with (3.33) can be used to obtain the lattice vibrational ion-electron ion potential

18 62 energy for the system in terms of one electron wave function, Hartree potential, and their perturbations due to thermal motion as. ( )( ( )) ( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ).. (3.37) If we use the relation (3.36) together with the following expression for the first order perturbed wave function, as ( ).. (3.38) ( ( ) ( )) Then the expression (3.37) may be written as ( )( ( )) ( ( ) ( )) ( ).. (3.39) Here the first two terms on the R.H.S. are caused by repeated one-phonon process and the last term is caused by intrinsic two phonon processes. Instead several approximation methods have been introduced to evaluate the electronic energy bands in solids based on

19 63 Hartree-Fock method. Complete discussion of these method may be found else where 14. The Hartree-Fock approximation was used by Toya to calculate the frequencies of normal modes in metals, in which the treatment is analogous to the one given above for Hartree approximation, so it will not be repeated again. The form of equation (3.39) remains unchanged inspite of introducing exchange term in the Hartree-potential energy. 3.6 THE SMALL ION CORE APPROXIMATION:- The small core approximation plays an important role in expressions of the dynamical matrix element in the frame-work of the Toya s Theory. In this approximation we make three assumptions as (i) (ii) (iii) According to this assumption, the variation of the potential due to the conduction electrons over the core-region is neglected. We assume that this potential shifts the energy of the core states but does not change their wave functions. In the second assumption the radial variation of the potential in evaluating the matrix element of smooth function over the core region is neglected. This is the most important assumption which enters in the calculation of dynamical matrix elements of electrons-phonon interaction is that the overlap repulsive interaction between ion cores is neglected due to small core-radii i.e. the spatialvariation of neighbouring core-potentials over the core region is ignored.

20 64 However, the effective potential in the present theory is expressible as a sum of pair potentials, which will not be the case if core-repulsion interactions remain present or three body forces are included in the theory. Estimates of ionic radii 15,16, in the metals to be investigated, are considerably smaller than half of the interionic spacing, therefore it is believed that the assumption is justified.

21 65 REFERENCES 1- Debye P.; Ann. Phys., 4,39, 789 (1912). 2- Born, M. and Karman, V.Z., Physik, 13,297 (1912) 3- Pines, D.; Elementary Excitations in Solids, Ch. I (W.A. Benzamin, Inc., N.Y.), (1963). 4- Ziman, J.M. Electrons and Phonon, Ch. II(Oxford clarandon press), (1962). 5- Bardeen. J and pines, D., Phys, Rev., 99, 140(1955). 6- Born, M. and oppenheimer, R, Ann. Physik, 84, 457 (1927). 7- Maradudin, A.A., Montroll, E.W. and Weiss, G.H.; In solid state physics, edited by seitz, F. and Turnbull. D. (Acad. Press, Inc. New York), (1963), (Suppl.3). 8- Kettel, C., Quantum theory of solids (wiley, New York), (1963) 9- Hartree, P., Phys. Rev., 36, 57 (1930). 10- Hartree, Fock, Hand buch der, Physic, 24/1, 349 (1933). 11- Seitz, F. phys. Rev. 47, 400 (1935). 12- Bohm, D. and Pines D, Phys. Rev, 82, 625 (1951). 13- Toya, T,J. Res. Inst. Cata. Hokkaido, 6, 161(1958).

22 Callaway, J., Energy Band Theory. (Academic press, New York), (1964). 15- Pauling, L., J. Am. Chem. Soc. 49, 765 (1927). 16- Hand book of chemistry and physics, (the chemical Rubber Published co., Cleveland), 48 th ed. Page F-143, 1967/68. **********