H. SCHMIDT and E. MANN: Anisotropic Electron-Phonon Intcraction in Cu 561

phys. stat. sol. (b) 93, 661 (1979)

Subject classification: 6; 13.1; 14.1; 21.1

InstiCut fiir Physik, Max- Planck-Institut fiir Metallforachung, 8tuttgart’) (a ) and Iltstitut fiir theoretische und angewandte Physik der Universitcit 8tuttgart (b)

Theory of Anisotropic Electron-Phonon Interaction in Copper7

BY

I. Electron-Phonon Matrix Elements

H. SCHMIDT (a, b) and E. MA” (a)

The electron-phonon interaction in copper is calculated on the basis of realistic wave function? and taking into account all anisotropies of the system. The electron wave functions are derived from a combined interpolation scheme, in which the wave functions are assumed to be composed of both, plane waves and Bloch sums of atomic d-functions. Because of the “pseudo” character of the wave functions thus obtained, these have to be combined with a pseudopotential in order t~ render the correct electron-phonon matrix elements; this contrasts with an earlier paper in the literatiire. Taking a slightly modified Cohen-Heine pseudopotential and the plane-wave contribu- tions only, leads to a satisfactory agreement between calculated and observed anisotropic quan- tities. The other contributions to the matrix elements are critically discussed. Applications will be considered in a subsequent paper.

Die Elektron-Phonon-Wechselwirkung in Kupfer wird oon Grund auf und unter Berucksichtigung aller wesentlichen Anisotropien berechnet. Die als Kombinationen von ebenen Wellen und Bloch- siimnien atomarer d-Funktionen angesetzten Wellenfunktionen werden mit Hilfe eines Inter- polationsschemas bestimmt. Wegen ihres Charakters als Pseudowellenfunktionen mussen sie bei der Berechnung der Elektron-Phonon-Matrixelemente mit cinem Pseudopotential kombiniert werden; dies steht einer anderen in der Literatur geiiul3erten Meinung entgegen. Eine leicht modi- fizierte Form des von Cohen und Heine angegebenen Pseudopotentials ergibt bereits mit den ebenen-Wellen-Anteilen der Wellenfunktionen allein eine gute Peschreibung verschiedener aniso- troper MeBgrol3en. Der EinfluB der ubrigen Beitriige wird dislrutiert. Uber Anwendungen wird in einer nachfolgenden Arbeit berichtet.

1. Introduction

Refined experiments in the past few years have revealed an extremely high anisotropy in the electron-phonon scattering of the noble metals. For copper the measurements by Koch and Doezema [l] and by Gantmakher and Gasparov [2] have shown that a t low tenipcratures the electrons on the necks of t.he Fermi surface are scattered about thirty times stronger than the (110) belly electrons. Even on the belly, the (100) electrons are scattered about ten times stronger than the (110) electrons.

With increasing temperature this high anisotropy decreases, but even in the limit of very high temperatures there remains an anisotropy characterized by a factor of a t least two. This may be seen from a semi-empirical analysis by Lee [3] for t.he elec- tron-phonon mass enhancement factor (at T = 0 K) which is proportional to the quasiparticle scattering rate at high temperatures.

Evidently there is a large variation of the scattering anisotropy with temperature, which inust reveal itself in appropriate experiments. Saeger and Luck [4] and Hurd and Alderson [5] have measured the temperature dependence of the Hall coefficient of

1) Heisenbergstr. 1, D-7000 Stuttgart SO, BRD. *) Based in parts on the dissertation of H. Schmidt, Universitiit Stuttgart, 1977.

36 pliyaica (b) 93p

562 H. SCHMIDT and E. NANN

copper and have found a characteristic local maximum below 50 K for certain orienta- tions of the magnetic field. According to Hurd and Alderson [5] this effect is due to the variation of the scattering anisotropy in connection with the transition from the low-field to the high-field region with decreasing temperature. This interpretation has been confirmed in an extensive study by the present authors [6]. The calculations performed with a variety of parameters characterizing the external conditions demonstrated both for pure copper and dilute copper alloys that anisotropic electron- phonon scattering may, a t least qualitatively, account for all the observed effects.

As a matter of course, for calculating the anisotropy phenomena all relevant aniso- tropies inherent in the problem have to be included; these are the anisotropies of the electron system, of the phonon system, and of the electron-phonon matrix elements.

I n the past we had developed and applied a theory in which the electron-phonon matrix elements were parametrized in terms of Wannier matrix elements of the dis- placement potential [6 to 91. This formulation was exact in principle. However, the necessary restrictions - for sake of simplicity - to a one-band formalism and to only a few adjustable parameters (two to four) permitted a quantitatively satisfactory description of the averaged quantities only but not of the anisotropy [6].

I n order to account more quantitatively for the real anisotropies of the system, one has to look for a more fundamental theory, based on real wave functions and potentials. We adopted the “combined interpolation scheme” of Mueller [lo] for the description of the band structure and the wave functions of copper. This method had been already applied by Das [ll] to the calculation of the scattering rate and the mass enhancement of copper. I n spite of satisfactory results, however, we believe that Das’ paper contains some severe inconsistencies. Das uses a Chodorow potential, and one may simply estimate that the published data for the matrix elements lead to results which are too high by a factor of about 30.

An analysis of this problem shows that the wave functions derived from the coin- bined interpolation scheme have to be interpreted as pseudowave functions and hence have to be combined with a screened pseudopotential (instead of the true Chodorow potential) in order to give the correct electron-phonon matrix elements. Using the published pseudopotential form factors leads indeed to the right order of magnitude for the observable quantities. This method resembles, in some sense, that of Nowak [12] which is based on a phase shift pseudopotential formulation and which is the only other one that describes the anisotropic scattering rate and mass enhancement of copper in a satisfactory way.

Our results will be presented in two parts. I n this paper we regard first the definition of the electron-phonon interaction, give an outline of the interpolation scheme sup- plying us with the wave functions, and discuss the wave functions and potentials in the light of pseudopotential theory (Section 2) . In Section 3 we consider the various contributions to the electron-phonon matrix elements and present the results. Scat- tering and transport properties will be dealt with in a subsequent paper [29] (hereafter referred to as Part 11).

2. Theoretical Basis

2.1 Electron-phonon interaction

I n the usual Bloch model [13] the electron-phonon interaction is described by means of the coupling function (cf. also [S])

(1) where N is the number of atoms in the crystal in the case of a Bravais lattice, et the phonon polarization vector in the phonon branch 7, for the phonon wave vector q ,

&k, k’) = N e i . ~ ( k , k’) ,

Theory of Anisotropic Electron-Phonon Interaction in Copper

and

563

the matrix element of the displacement potential a V(r)/aR, between unperturbed Bloch states (R, is the position of the origin atom) ; V ( r ) is the periodic crystal poten- tial (or pseudopotential, in which case the Bloch states must be replaced by “pseudo” Bloch states [14, 151; see also Sections 2.3 and 3). A usual approximation is to write

V ( r ) = v ( r - R ) , (3) R

that is, the total potential is assumed to be composed of rigid atomic potentials, properly screened. Then (2) reads (with R, = 0 )

I (k , k’) = - ’$*(‘)“) D ~ v ( r ) l y k ( r ) d r . (4) I n terms of the coupling function J t ( k , k’) one may express the transition proba-

bility W ( k , k‘) for transitions in the electron system from state k to state k’, caused by the interaction with phonons of wave vector

q = k’ - k + K , ( 5 ) where K is a suitable reciprocal lattice vector to reduce q to the first Brillouin zone, cf. [6]. By means of the transition probability the transport properties may be cal- culated. Other scattering properties, depending even more directly on the coupling function J& are the scattering rate p(k, T) and the renormalization (mass enhance- ment) factor A(k, 0) . These k-dependent, anisotropic quantities are best suited for a comparison between theory and experiment. These applications will be considered in Part I1 [as].

Whereas in our former treatment [6] the matrix elements (4) were parametrized in terms of Wannier matrix elements, we now wish to calculate them explicitly in order to achieve a more fundamental description. If the true potential v ( r ) were known, one would have to use the true Bloch functions y ) k ( r ) , taken, e.g., from a corresponding “ab initio” band structure calculation such as the APW method [16] or the KKR method [17]. Such a calculation is far too laborious to be performed easily. Therefore we use Mueller’s combined interpolation scheme [lo] as a model band structure cal- culation, which yields - because of the “pseudo” character inherent in the ansatz of the scheme - “pseudo” Bloch functions y:(r) . These have to be coupled with an appropriate pseudopotential to give a good description of the electron-phonon interac- tion. I n the following we give an outline of the interpolation scheme as far as it is necessary for an understanding of the subsequent applications.

2.2 Interpolation scheme

The main idea of Mueller’s combined interpolation scheme [lO]3) is to take into account the hybridization effects in the band structure of, e.g., the noble metals through an ansatz for the wave function that combines s- and d-like parts

Iylk) = CkKlk, K ) + ck,ulk, p ) (6) K P

( K are reciprocal lattice vectors, of which all up to second neighbours are considered; p = 1, ... , 5 corresponds to five d-functions). The combined basis set { Ik, K ) , lk, p) } , which is assumed to be orthonormalized, includes “s-like” basis states lk, K ) chosen

3, 9 similar scheme has been proposed by Hodges et al. [18, 191.

361

564 H. SCHMIDT and E. MANN

I n ( 7 ) the plane waves p + K ) = 9 - l i 2 ei(k+K).r (9)

(Q = NQ,,; 9, is the volume of an elementary cell) are orthogonalized to the d basis states only. This last point is responsible for the “pseudo” character of the wave function (6) (see Section 2.3). The “d-like” basis states are formed by Bloch sums of atomic d-states

I n the actual calculation the radial part R( r ) of the atomic d-functions is taken as a linear combination of Slater-type orbitals, similarly as in the calculation of Das [ll] (see Appendix).

The symmetrical Hamiltonian matrix formed with these basis states is par- ametrized in terms of several parameters as described in the papers [lo, 111, which gives rise to a “model character” of the Hamiltonian. It takes the block form

where the diagonal blocks describe non-hybridized d- and s-bands, respectively, and the non-diagonal blocks are responsible for the hybridization of s- and d-bands. The band structure and the wave functions (for general k-vectors) are determined through the secular problem

P’ c [ H p p ’ ( k ) - &(k) 8 p p r ] c k p ’ 0 (14)

det { H P p l ( k ) - e ( k ) S,,,} = 0 (15) ( p , p’ = K or p) . The parameters of the Hamiltonian matrix are determined by fitting Burdick’s APW band structure [lo, 11, 161, calculated for special k-vectors. I n our calculation we used the parameters as determined by Das [ll] (after correcting some likely errors; see the following footnote).

Two form factors enter the interpolation scheme, an orthogonalization form factor f ( k ) defined by

n (b, p I 7 c + K ) = f(lk+Kl) yZPW + K ) (16)

(apart from a multiplicative constant this is identical with the radial part of the Fourier transform of the atomic d-state @Jr), see Appendix), and a hybridization

Theory of Anisotropic Electron-Phonon Interaction in Copper 565

form factor g ( k ) defined by

The representation (17) of the hybridization terms is not exact but is a reasonable one as shown by Mueller [lo] and Ehrenreich and Hodges [19]. I n the Appendix we give an analytical form of the form factor f ( k ) for the analytically given radial wave func- tion R(r) , which seems to have not yet been published in the literature.

In this context an inconsistency with Mueller’s and Das’ calculations arises. On the one hand, we re-derived the explicit form of the Hamiltonian matrix Hpp, (k ) carefully and obtained the same formal result as that given by Mneller [lo]. On the other hand, in our calculation the form factor f ( k ) turns out to be negative semi-definite ( f ( k ) 2 0). Regardless of the sign of f ( k ) itself, we found that the sign of the ratio of the two form factorsf(k)/g(k) ( k $. 0) should be negative for reproducing the correct band structure. This is in contrast to the published data of Mueller [10]andDas [ l l ] . 4 )

2.3 Pseudopotential concept

The interpolation scheme wave function yf is not explicitly orthogonalized to the 1s to 3p core states as should be the case for the true wave function yk. The expansion (6) must rather be regarded as an ansatz for a pseudowave function describing the conduction bands as well as the d-bands. The parametrization of the corresponding pseudo Hamiltonian matrix and the fitting of the parameters to the real band struc- ture ensures that this procedure has physical significance.

For the reason of comparing our results with those of Das [ll] we will also consider the construction of the true wave function. To obtain the true wave function in the OPW representation one will have to orthogonalize the interpolation scheme wave function to the “deep” core states

Ik, c) E @ k , $ ( r ) = N-112 c eak. @ c( Y - R) ( c = Is, ... , 3p) , (18) R

where @Jr) denotes the atomic core states. One should write

The normalization constants N k , like the NkK of (8), will be only slightly different from unity.

The Schrodinger equation for the true wave function with the real crystal potential, simulated for example by a “muffin-tin” Chodorow potential, will transform into a pseudo Schrodinger equation for yf with an additional repulsive orthogonalization pseudopotential (operator)

which is non-local, in principle, but may be approximated by a local potential. Then in analogy to (3) one may write

_.___

*) According to our opinion the parameter A as published by Nueller [lo] and Das [ll] should be negative. Furthermore, the numbers for A and B in Das’ Table 2 should be interchanged.

H. SCHMIDT and E. MANN

For calculating the electron-phonon interaction the matrix elements (4) are of interest. Following the lines of Sham and Ziman [14, 131 we may derive, with the definitions (18) and ( lg ) , (&(k) = &k)

566

( y k ‘ ( r ) I v ~ ~ ( ~ ) IY)k(r ) ) = - ___ {(P:,lv ivwi> + (vwi,l?J Iwi ) + N k N k ’

+ 5 (&k - &c) [ (wf ’ I @c) (@c I vwf> f (vd’ I @c> <@e I d)]} * (23) c=1s

The same result may be obtained in a more direct and general way (without having to assume non-overlapping between neighbouring atomic functions and/or atomic potentials [I 41) by defining an atomic repulsive-potential operator in the sense of (22). From (21) we might define

1 (24a) V:tp(r) = (Ek - & c ) __ l@c> (k, c /

e 1/N or

1 O P vTep(r) = C (Ek’ - e,) __ lk’, c> <GCi . e jilv

This gives in the conception of pseudopotentials (with &k’ = &k)

+ C (Ek - &c) [ (wk I @c> < v @ e 1 wi> f <&‘ I V @ c > (@c I Vr!)]} (25)

Here we used

(26) -1 1

( k , c I y ~ i ) = - C e--ik.R @z(r - R) &(r) dr = 1/N @z(r) y i (r)dr, 1/N s

which results from Bloch’s theorem yi(r - R) = e--ik.R yi(r). Equations (23) and (25) are identical; (23) follows from (25) by partial integration. From both expressions we see that the atomic pseudopotential operator may also be defined in the simple form

V:tp(r) = 2 (&k - &c) I@c> (@cl *

,yL(p. w.)(y) = C CkK(NQO) - 1/2 ei(k + K ) . T ,

(27) c

I n the case that yf would be approximated by plane waves only,

(28) K

we would obtain from (23)

Theory of Anisotropic Electron-Phonon Interaction in Copper

and

567

1 P

The second term in the brackets of (29) may in the local approximation ( 2 2 ) be inter- preted as the Fourier transform of the repulsive potential vrep(r)

;rep C (Fk - E c ) b,kKb&,K, Zrep(k‘ + K - k - K ) , (32) C

which defines the Fourier transform of the local pseudopotential

c p s ( ~ ) = v(q) + &ep(q) * (33) An estimate of the order of magnitude of the repulsive-potential terms (32), using

the data for the atomic core states @&r) and core eigenvalues F, (properly shifted) as published by Wahl et al. [20] for k, k’ on the Fermi surface (&k = F ~ ) , shows that Zre,(y)_is of the same order of magnitude (but with opposite sign) as the Fourier trans- forni oChod.(y) of Chodorow’s p ~ t e n t i a l . ~ ) This is in contrast to the statement of Das [ll] who regards the repulsive terms as totally negligible. We could show (see next section) that using Chodorow’s potential alone yields the wrong order of magnitude for the electron-phonon coupling.

Instead of using the pseudopotential (33) with EreP(y) in the form (32), which would have been uncomfortable, we used niodifications of given pseudopotential form factors taken from the literature [21 to 251. This procedure corresponds to an empirical fitting of a model pseudopotential to experimental results. All form factors used gave the correct order of magnitude for the electron-phonon coupling (apart from a factor 2 or 3 a t most). Both the anisotropy structure and the order of magnitude of the scattering rate and the mass enhancement could be best reproduced by slight modifica- tions of the Cohen-Heine form factor [21]. This will be discussed further in the fol- lowing section where we report the calculated electron-phonon matrix elements.

3. Electron-Phonon Matrix Elements

3.1 General formulation

As shown originally by Sham [14], the electron-phonon matrix elements between the true displacement potential vrw(r) and the true Bloch states may, within the framework of the OPW method, also be formed between the gradient of the OPW pseudopotential (locally approximated) and the OPW pseudo-functions, which are linear combinations of plane waves. The same theorem applies, as shown in the pre- ceding section, to our case with vf replacing the OPW pseudo-function, i.e. we have

The normalization constants Nk have to be introduced since, contrary to the original work [13, 141, we assume to be normalized. Because of Nk z 1, we shall drop these factors in the following. The pseudopotential w p s ( r ) is a sum of the true potential ~ ( r ) and the repulsive potential $tP (defined, e.g., through (27)), locally approximated,

ups ( r ) == ~ ( r ) + Urep(r1 * (35) ~~ ~~~

5, It does not matter whether the sum in (32) is extended from 1s to 3p or to 3d states, since the 3d states give a contribution of less than 1%.

568 H. SCHMIDT and E. ~ I A N N

Writing the interpolation scheme wave function as a linear combination of pure plane waves Ik + K ) and d basis states lk, p),

with coefficients

we obtain three types of matrix elements, such between pure plane waves, such between d-states, and mixed ones:

<vi,I Vrvps(r) Iw’,) = C aZ,h-,ak~(k’ + K ’ I Bops Ik + K ) + KK’

+ aZ‘ ,u‘akp(k’ , p’1 Vvps I k , p) a

Y&’

We consider the different types of matrix elements separately.

3.1.1 Plane-wave matrix eleme.lzts I n terms of the Fourier-transformed potential (or pseudopotential)

V(q) eiq.r dq ,

we get, similarly as in (29), for the plane-wave matrix elements i

N (k’ + K’I Vrwps Ik + K ) =- ( k ’ + K ’ - k - K ) Vps(k’+K’ - k - K ) . (40)

3.1.2 Plane-wave-d-state matrix elements One of the mixed terms is

1 (k‘, pl vvps Ik + K ) = ~ C e--ik‘.R (@p@ - WI W p s Ik + K ) f (41) 1/N n

With the Fourier transform of the atomic d-function

and with aid of the relation J ei(k-k’).r d r = (2n)3 S(k - k’) the term (41) may be brought into the form

where VB = ( .Z~Z)~/L?~ denotes the volume of the Brillouin zone. By means of the relation

(44) 2 ei(k--k’).fi = Jl R K ’

C S(k - k‘ - K’) , B

Theory of Anisotropic Electron-Phonon Interaction in Copper

where K‘ denotes the reciprocal lattice vectors, we obtain in a close form

569

(k’, PI P P S lk + K ) = i - -- - (k’ + K’ - k - K ) GpS(k’ + K’ - k - K ) &.,(kt + K’) . (45). N tQ0 K’

According to the Appendix &Jk) may be also written as the product of the form factor f ( k ) and the spherical harmonic FzP

By neglecting in (41) the overlap of the atomic potential with d-functions localized a t R $: 0, one would obtain

This is an integral approximation to the sum (45) and is the form that has been con- sidered by Das [ll].

3.1.3 d-d matrix elements

I n a similar manner we get for the d-d contributions

1 (k’, p’l v w p S Ik, p) = C ei(k.R-k’.R) ( @ J r - R’)I vvpS I@Jr - R ) ) =

RR’

K ) Gps(k’ + K‘ - k-

I n an approximation one would have to consider a t least the overlap with the nearest neighbours R =+ 0, because the term with R = R‘ = 0 vanishes for reasons of sym- metry. Such an approximation has been studied by Das [ll].

For the calculation of the coupling function

one needs besides the matrix elements just considered the expansion coefficients L%kK and L%kp. These are obtained by solving the eigenvalue problem of the Hamiltonian matrix (14), (15) for each electron wave vector k.

The phonon spectrum enters via the phonon polarization vectors e; which are the eigenvectors of the dynamical matrix

The dynamical matrix is given by

Dag(q) = C @&) eiq. (51) R

with the matrices of force constants @&g(R) taken from the experimental work of Sinha [261.

The shape of the Fermi surface and the values of the Fermi electron velocities were taken fro; the work of Halse [27].

570 H. SCHMIDT and E. MANN

3.2 Results and discussion

As a measure of the strength of the electron-phonon interaction we consider, like Das [ll], the “coupling constant”

For a comparison with Das’ work we re-calculated Ig(k, k’)I2 for k, k‘ on the Fermi surface in some symmetrical directions with the screened Chodorow potential. When retaining only the plane-wave contributions, we attained nearly perfect agreement. Das [ l l ] considered also the mixed terms in the form (47) and found them to give a correction of roughly 20% a t most. The d-d terms were estimated by Das to be negligible. I n strong contrast to this, we calculated that including the mixed terms increases 1gI2 by an order of magnitude. Apart from this large difference, Das’ result cannot be correct in that it does not display the necessary symmetry in interchanging k and k’.

I n any case, our calculations show that it is not permitted to use a Chodorow potential in connection with the interpolation scheme wave function. Even in the case when we retain the plane-wave contributions only, we obtain a value of 1912

which is too large by a factor of about 30. This may be seen by comparison with the corresponding quantity obtained in the Wannier representation [6] or by calculating the scattering rate or inass enhancement (see Part 11). It is hard to understand that Das [11] arrives a t the right order of magnitude for the scattering rate or mass enhance- ment by employing the coupling functions J i calculated with Chodorow’s potential.

As discussed in Section 2.3 one has to use an appropriate pseudopotential instead of the Chodorow potential. We have applied two modifications of the pseudopotential form factor given by Cohen and Heine [all, denoted by CH1 and CH2 in Fig. 1. We modified the original Cohen-Heine form factor in three ways: (i) I n the q + 0 limit of the screened pseudopotential, screened, for example, via Lindhard’s dielectric function,

we utilized the band structure density of states N ( c F ) rather than the free-electron value; (ii) we shifted the zero q,, ( l /qo being a measure for the extension of the pseudo- potential in r-space), and (iii) we varied the height of the maximum following the zero qo. The form factor CH1 represents only a slight modification, determined such that the scattering rate, calculated with the plane-wave part of the wave function only (confer below), yielded the correct value for the k-point (100) on the Fermi surface (see Part 11). The form factor CH2 was determined to give a good overall description of the scattering rate.

-ad-

Fig. 1. Pseudopotential form factors CH1 and CH2, as empiri- cally determined by modifying the original form factor of Cohen and Heine (CH) [21]

Theory of Anisotropic Electron-Phonon Interaction in Copper 571

Fig. 2. “Coupling constant” Ig(k, k’)l’ for k and k’ varying on the Fermi surface of Cu, calculated with the plane-wave part of the pseudo wave function in connection with the pseudo potentials a) CH1 and b) CH2. The points denoted by 11111 refer to the centre of the “neck” a t the Brillouin zone boundary and do not belong to the Fermi surface. k: o [ l l l ] , A [loo], [110]

The functions \g(k, k’)I2, calculated with the pseudopotentials CH1 and CH2 and retaining the plane-wave matrix elements only, are shown in Fig. 2a and b. Calcula- tions including the mixed as well as the d-d ternis have not been carried through in full generality. Besides for the reason that this program would be rather laborious and expensive, it is impeded by the fact that for these calculations also the pseudopotential for larger g-values than shown in Fig. 1 is needed, which, however, is quite uncertain in this regime. Nevertheless we have performed calculations with inclusion of the mixed terms in the integral approximation (47) and assuming ZPs(q) to be zero for q/2k, > 1.8. The result is that the mixed terms contribute to Ji nearly to the same extent as the plane-wave terms, so that Ig(k, k’)I2 becomes 3 to 4 times larger than with the plane- wave terms alone. Furthermore we have estimated the d-d terms and found them to be of the same order of magnitude as the mixed terms (contrary to the statement of Das [ll]). Since, on the other side, we get a rough agreement with experimental data by using the plane-wave terms alone, we have to conclude that mixed terms and d-d terms largely cancel. Thus we shall use in the following applications the plane- wave contributions only; little adjustments have been made by slightly modifying the original Cohen-Heine pseudopotential, as discussed above.

Although the two potentials CH1 and CH2 do not differ too much, the functions Ig(k, k’)12 in Fig. 2 a and b show a rather different picture. Nevertheless the calculated scattering rate and mass enhancement, to be reported in the following paper [29], show nearly the same anisotropy structure for the two potentials.

Appendix Orthogonadixationformfactor f(k); Fourier transform of the atomic d-functions

I n the Hamiltonian matrix (13) enters the overlap of a tight-binding d-function with a plane wave, which can be shown to be proportional to the Fourier transforni of the atomic d-function, i.e.

(k, ,u I k + K ) = sZ,1/26”,(k + K ) The Fourier transform is given by (cf. (11))

57 2 H. SCHMIDT and E. MANN

Table 1 Parameters of the radial d-functions after [20]

P GP Ep

1 0.03670 12.600 2 0.36232 6.681 3 0.52013 3.599 4 0.25074 1.896

where 6k,r denotes the angle between k and r , and dQn, is the solid angle element a t r . The radial part of the d-functions is given by a “Slater-type” expansion

R(r) = c C P R P W (A3)

Rp(r) = A,r2 e-tp* (A4f

P with

and with parameters c p and 5 , as given by Wahl et al. [20] (see Table 1). The normaliza- tion conditions

m a3

J r2R2(r) dr = J r2Rg(r) dr = 1

A , = (6!)-ll2 ( Z E p ) 7 / 2

0 0 yield

and the relation

which reduces the number of independent parameters by one. With the plane-wave expansion [28]

m M 7

involving the usual spherical harmonics Yzm(8’, q ) and the spherical Bessel functions jl(z), the angular integration in (A2) may be performed immediately using the con- nections between the angular parts Yz,(8, y ) and the spherical harmonics for 1 = 2 . One gets

,.m

0 6 p ( k ) = --4nF2,(k) J r2R(r) j,(kr) dr . (A9)

I n Mueller’s original scheme [lo] the radial integral in (A9) was not calculated but parametrized, whereas Das [ 111 seemingly has calculated it numerically. However, using the expansion (A3) and

3 2 2

j,(x) = ($ - :) sin x - - cos x,

the integral may be performed analytically yielding

6 p ( k ) = - k Z ~ z , ( ~ ) c CpApIp(k) P

with

Theory of Anisotropic Electron-Phonon Interaction in Copper 573

The orthogonalization form factor f ( k ) defined by n

(k, p I k + m = Ye,(lC + K ) f(lk + XI)

f ( k ) = --4nQn,1'* c C p A p l p ( k ) , is then given by

P

which is clearly negative definite for k =+ 0 ( c p , 5 , > 0) . The form factor (A14) differs from Das' one by the factor -4n/I /z = -1.415 (at. units) but is otherwise numeri- cally identical.

Acknowledgements

The authors wish to express their graditude to Prof. A. Seeger for his continued interest and support. They are indepted to Dr. R. Bauer and Dr. H. Teichler for valuable dis- cussions. They wish also to thank Prof. F. M. Mueller and Dr. S. G. Das for corres- pondence and for making available some of the ANL reports.

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(Received illarch 8, 1979)