ELECTRONIC STRUCTURE OF COPPER

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ELECTRONIC STRUCTURE OF COPPER

A. Boring, E. Snow

To cite this version:

A. Boring, E. Snow. ELECTRONIC STRUCTURE OF COPPER. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-89-C3-93. �10.1051/jphyscol:1972312�. �jpa-00215046�

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JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-89

ELECTRONIC STRUCTURE OF COPPER (*) A. M. BORING and E. C. SNOW

University of California, Los Alamos Scientific Laboratory Los Alamos, New Mexico 87544

Wksume. ^-On presente une discussion des rksultats de quelques rkents calculs self-consistants de la structure de bandes du cuivre basks sur diffkrentes approximations de l'echange. Des approximations dependantes de I'knergie (type Hedin-Lundqvist, Liberman et Bohm-Pines) et des approximations n'en dkpendant pas (Xor et X@) ont CtB utilisees. On s'est intkress6 au spectre des valeurs propres electroniques. Une tentative a kt6 faite pour relier l'utilisation de ces operateurs d2change aux theories courantes concernant les ophateurs monoelectroniques de self-energie deductibles de la thdorie a plusieurs particules (Hedin et Lundqvist et Kohn et Sham).

L'energie totale statistique, dans le formalisme de Kohn et Sham, est utilisk pour calculer la constante du rkseau, I'knergie de cohesion et I'isotherme T = 0. On discute le r61e dans ces calculs du thtor6me du viriel et du principe variationnel et on commente les differents modes d'introduction des corrections de corrClation.

Abstract. - A discussion of the results of some recent self-consistent field calculations of the energy band structure of Cu based on various exchange approximations is presented. Energy dependent (Hedin-Lundqvist type operator, Liberman exchange operator, and Bohni-Pines type operator) and non-energy dependent (Xu, Xab) exchange approximations were used in making the calculations. In these calculations the interest was on the one-electron eigenvalue spectrum generated. An attempt is made to relate the use of these operators to the current theories in single- particle self-energy operators derivable from many particle theory (Hedin and Lundqvist, and Kohn and Sham).

Via the Kohn and Sham formalism, the statistical total energy expression is used to calculate the lattice constant, the cohesive energy, and the T = 0 isotherm for the local exchange operator (Xor). The role of the virial theorem and the variational principle in these calculations is discussed, and some comments are made on the various ways in which correlation corrections may be included.

I. Introduction. - A study is being made of the electronic structure of the transition metals. Copper has been chosen for the calculations because its ground state electronic structure is not complicated by magne- tic or superconducting effects ; relativistic effects and the effect of non-spherical components of the one- electron potential are small in this crystal. The initial assumption is that conventional band theory, aided by modern many-particle theory, will yield a calculated electronic structure in good agreement with experimental data. Specifically, we think the time is ripe for calculation of the excitation spectra (comparable to photoemission data) and the ground state properties (those properties that are derived from the total ground state energy). Such a calculation, however, is not presented in this paper.

Given a single-particle potential, many energy band calculational methods (such as APW, KKR, and OPW) can now routinely generate a self-consistent band structure. If there were only one many-particle theory that would give the single-particle potential to be used in band calculations, along with the prescrip-

(*) Work performed under the auspices of the U. S. Atomic Energy Commission.

tion for including higher order effects in a simple way, the electronic structure of a metal like Cu would have been determined 5 or 6 years ago. The complication of nonuniform charge densities in real metals - as opposed t o the uniform electron gas problem where explicit exchange-correlation functions have been obtained - has led t o the use of several single-particle exchange-correlation functions. Before discussing the use of five such functions consideration is given t o the self-consistency problem.

11. Self-Consistency. - To show that the self- consistent field method works in practice as well as in principle, the potentials generated from two atomic configurations for Cu (3 d l 0 4 s1 and 3 d9 4 s2) were taken as starting potentials in a self-consistent field test. All the occupied electronic states were calculated at each SCF iteration. The core states (1 s, 2 s, 2 p) were determined in the crystal potential by a Herman- Skillman [I] type program using boundary conditions appropriate to the Wigner Seitz cell. The 3 s, 3 p Bloch states were calculated at 32 points in the Brillouin zone (BZ) via the APW [2] method, and the band states (3 d, 4 s) were similarily calculated at 2 048 points. We judged the calculations self-consistent

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972312

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C3-90 A. M. BORING AND E. C. SNOW when all eigenvalues ^Edetermined in iteration n satis-

fied I ^q(n)^-^q(n- 1) I < 0.002 Ry. The condition that the two starting potentials give the same final results was considered met when the results differed by less than the self-consistency criteria. In these calculations the Xa (see below) operator was used. The final charge densities were also compared and good agreement was obtained. In figure 1 the E(k) curves are shown along the x-direction for the two starting potentials and for the final potential. Now that we have shown that the results of a self-consistent field calculation do not depend upon the starting potential, we will demonstrate that the final results depend very much on the functional form of the potential.

111. Single-particle potentiah. - Basic to our consi- derations is the formal development by Sham and Kohn [3] of an approximate solution to Dyson's equation for a system of interacting electrons. Sham and Kohn [3] have shown that the self-energy operator Z(rl, r ; E) can be written in such a form as to provide the following Schrodinger type one-particle equation :

+ I dr' M(rt, r ; E) t,b(r', E) = 0 , where V(r) = Vne(r) + Ve,(r) is the ion-electron arid electron-electron coulomb interaction. All the non- local exchange and correlation effects are contained in the self-energy operator M(rt, r ; E). For E = p (chemical potential) they obtained :

where V,,(n) = dldn (ne,,(n)), n is the uniform density, and

exchange and correlation effects, and ~,,(n) is given by electron gas theories. Although some of the functions used for the Vx,(n) operator can be identified in the limit of uniform charge density as the exchange part only, all of the single-particle operators used here will be called exchange-correlation operators simply because they are not the exact Hartree-Fock operators.

The first approximation considered is the well known cc p+ ⁾⁾ or Xa potential (Slater [5]), i. e.,

where k, is the Fermi wave vector. In this paper three calculations with this operator are considered (a = 516, 0.721, 213).

The next approximation is the so-called X@ potential of Herman [6]. This is given by

in which

The third approximation considered is a Hedin- Lundqvist [7] type exchange-correlation operator,

where y = klk,, and k is the wave vector of the electron under consideration. In the present work A(y, r) has been approximated by

where Cis a constant determined by the density and C' is just the slope of the Hedin-Lundqvist function [8]

for y ,< 2. y(r) is determined from k,(r) = (3 n2 n(r))*, and k(r) = JE - V(r), where

The function v,":(r, E), but not necessarily our approximation to it, contains both long range correlation effects (dynamic screened exchange) and short range correlations effects (coulomb hole).

The next approximation is the energy dependent exchange operator proposed by Liberman [9],

where

Again y = klk,, and the operator will be labeled VL

when kE is given by k, = (3 n2 n(r))+, and v:= ^when

kE is given by, E,,[n] is the energy functional (Kohn and Sham [4])

of the ground state charge density that represents kE = dE ^-VE(r), V,(r) = Vne(r) + ^Ke(r)^-^{4 Vxs(r)}.

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ELECTRONIC STRUCTURE OF COPPER C3-91 The last approximation is a screened version of this

operator,

where F B P ( ~ , r) = FL(q, r) - p k < kF - kc and

k > kF - ^{kc and}

The free electron gas from of this expression is obtainable from the theory of Bohm and Pines [lo], [ll], where k, is the screening length.

In figure 2, the E(k) curves are shown (in the x-direc- tion) for the V:c, v&, v . ~ and :v: operators to illus- trate the differences obtained. In Table I are shown the bandwidths (s-p and d) as given by the solutions to the above differential equation using these five different approximations. Also shown are the photoemission data [12], [13] giving the d bandwidth.

Bandwiths ["I Experimental

d bandwith

Spicer [b] 0.250

Eastman ["I 0.220 s-p bandwith Self-consistent APW Calculations

S. P. Operator

r] Energies in Ry.

[b] Ref. [12].

['I Ref. [13].

IV. Comparison of the Eigenvalue spectra. -

From Table I it is seen that the five exchange-correlation approximations give quite different bandwidths.

The interesting features to note in Table I are the following. For the ViC operator the d bandwidth varies with a variation of the a parameter, while the s-p bandwidth remains constant. The XaP scheme (for one value of p) gives bandwidths that are similar to the Xa (a = 213) bandwidths. The Hedin-Lundqvist operator gives a d bandwidth in agreement with the Xa(a = 516) calculation, which also agrees with photoemission data, but a narrower s-p bandwidth.

The Liberman operator (either FL or F') gives both s-p and d bandwidths that are wider than those obtained by use of the other single-particle operators.

It is our belief that these operators do a good job of simulating the Hartree-Fock operator. Finally it is seen that the s-p bandwidth varies with kc in the use of the FBP operator, while the d bandwidth remains constant. This result seems to be in codict with the observation made by Phillips [14] that ( ( p 3 >> operator is similar in its behavior to a screened exchange operator.

V. Excitation spectrum. - Now that the eigenvalues of the differential equation have been compared, the corrections to obtain the theoretically correct excitation spectra must be considered. These corrections are of two types : many-particle corrections to Bloch states and corrections to account for the fact that the Bloch representation may not be the best description of the d band electrons. First is must be pointed out that the Hedin-Lundqvist results need no such corrections, as the v,: operator is already theoretically correct for the excitation spectra.

For the V,", and C8, operators the correction given by Sham and Kohn [3] should be applied, i. e.,

where m* is the density of states mass, Gk(p) is the eigenvalue, p is the chemical potential, and EK is the 0.258 0.797 excitation energy. It is not clear what Bloch state corrections should be applied to the V$ and v':

0.225 0.774 eigenvalues.

The non-Bloch state corrections come from obser- vations of the use of these operators in isolated atoms.

0.395 1.580 No criteria are given in this paper to show when these 0.285 1.221 corrections are applicable, but it seems reasonable that they should be considered in the case of narrow bands.

he first correction is to obtain the equivalent of 1.207 Koopman's energy, i. e.,

1.041

where E(N) and E(N - 1) are the total energies of the N particle and N - 1 particle system calculated with the same eigenfunctions, and in the Hartree-

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A. M. BORING AND E. C. SNOW

Fock theory ^E is the eigenvalue of the differential equation. This correction has been applied to the Xa eigenvalues of atoms by Slater 1151 and to the eigenvalues of an Fe energy band calculation by Klein- man [16]. The second type of correction is the result of an attempt to obtain the difference between the total energies of the N particle system and the N - 1 particle system when each is calculated with the correct eigenfunctions. This is a correction for relaxation, and Slater [15] has shown that the so-called transition state >> calculation gives this energy very satisfac- torily in atoms. At present no such calculation has been attempted in crystals. Relaxation in the isolated Cu atom for d states amounts to several electron volts.

Next, consideration will be given to the ground state properties.

VI. Ground state. - Since at present there is no prescription for calculating the ground state energy for the energy dependent one-particle potentials (v:

:, V& vZc, v,":), this discussion will deal only with the ViC and v:! calculations. The three properties of the ground state to be calculated are the T = 0 isotherm, the T = 0 lattice constant, and the cohesive energy. These are obtainable from the total ground state energy as a function of lattice spacing, i. e.

and

and TK is the kinetic energy operator and

If the zero-point motion is neglected, then the pressure, which has only an electronic contribution for T = 0, is obtained from the virial theorem (Ross [IS]), i. e.

where EK = S TK q(r) dr, V is the volume, and U is the total potential energy. To date we have only prelimi- nary results for these calculations.

For the Vc: operator the pressure has been calculated at the observed lattice constant for several values of a.

These calculations seem to indicate that we will obtain P E 0 at the observed lattice constant (normal density) for a cz 0.7. The a = 516 calculation, which gave a very reasonable band structure, gives very poor results for the pressure.

For the ! v: operator we have obtained P r - 100 kbars E(a') , A' = (1 + ^{6) a,},

for fl = 0.004 0 at normal density, but we have some and a, is the experimentally observed lattice constant. reservations about the ca~culation and will repeat it.

The variational principle gives for the v:c and v,"! The calculated value of cohesive energy is obtained potentials the following total energy expressions [17]. from the energy diff~~ence between the total energy of the isolated atoms and the total electronic energy Ea = 1 ^[TK⁺^V(r)⁺314 a~.~(r)] ~ ( r ) dr of the Wigner Seitz cell. Since there are many ways of obtaining these total energies (isolated atom and

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ELECTRONIC STRUCTURE OF COPPER C3-93 crystal) we will simply describe the simple scheme used

and the results obtained. The total energy (Ea) was computed in the atom and in the crystal for the same value of a and the difference taken. The first calculation was done for a = 516, and good results were obtained. One difficulty with this calculation was that the atomic energy Ea (and hopefully the crystal Em) was some 15 Ry below the experimentally determined energy. It was discovered that this 15 Ry was in the core energy, but the calculation has not been repeated for a lower value of a.

VII. Concluding remarks. - For future work, in order to obtain a more reasonable electronic structure for Cu than has yet been presented, we suggest that the following calculations be performed. First, a free- electron gas correlation function should be added to the EC (a = 213) operator, which makes the calcula- tion no more difficult, to obtain the ground state properties. Then the Bloch corrections should be applied to obtain the excitation spectrum. Similar calculations should also be carried out for the v:!

operator.

[I] HERMAN (F.) and SKILLMAN (S.), Atomic Structure Calculations (Prentice-Hall, 1963).

[2] SLATER (J. C.), Phys. Rev., 1937,51,846.

[3] SHAM (L. J.) and KOHN (W.), Phys. Rev., 1966, 145, 561.

[4] KOHN (W.) and SHAM (L. J.), Phys. Rev., 1965, 140, A 1133.

[5] SLATER (J. C.), Phys. Rev., 1951, 81, 385; Semi- Annual Progress Report No 71, SSMTG, MIT

(USA).

[6] HERMAN (F.), VAN DYKE (J. P.) and ORTENBUR-

GER (I. B.), P h y ~ . Rev. Letters, 1969, 22, 807.

[7] HEDIN (L.) and LUNDQVIST (S.), Solid State Phys., 1970,23, 1.

[8] HEDIN (L.) and LUNDQVIST (S.), reference 171, p. 148.

[9] LIBERMAN (D.), Phys. Rev., 1968,171, 1.

[lo] PINES (D.), Phys. Rev., 1953,92,626.

[ l l ] PINES (D.), Solid State Phys., 1955, 1, 394.

[12] BERGLUND (C. W.) and SPICER (W. E.), Phys. Rev., 1964, 136, A 1044.

[I31 EASTMAN (D. E.) and CASHION (J. K.), Phys. Rev.

Letters, 1970,24,310.

[14] PHILLIPS (J. C.), Phys. Rev., 1961,123,420.

[15] SLATER (J. C.), Computationa2 Methods in Band Theory (Eds. P. M . Marcus, J. F. Janck and A. R. Wil- liams, Plenum Press, 1971), p. 447.

[16] KLEINMAN (L.) and SHURTLEFF (R.), Phys. Rev., 1971, B 3 , 2418;