The Electron-Phonon Interaction in Metals (Selected Topics in Solid State Physics XVI) [1 ed.] 044486105X, 9780444861054

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THE ELECTRON-PHONON:

INTERACTION

IN METALS

GORAN GRIMVALL

SERIES OF MONOGRAPHS ON SELECTED TOPICS

IN SOLID STATE PHYSICS

Royal Institute of Technology

Editor: E. P. WOHLFARTH

Stockholm, Sweden

1 Magnetostatic principles in ferromagnetism, W.F. Brown, Jr. 2 The physics of magnetic recording, C.D. Mee 3 Symmetry and magnetism, R.R. Birss

4 Ferromagnetism and ferromagnetic domains, D.J. R.S. Tebble 5 The growth of crystals from the melt, J.C. Brice

Craik

i

4 ‘A

‘

i 3

and

6 X-ray determination of electron distributions, R.J. Weiss

7 8 9 10 Il 12 13

Ferroelectricity, E. Fatuzzo and W.J. Merz Experimental magnetochemistry, M.M. Schieber Experimental methods in magnetism, H. Zijlstra Group theory and electronic energy bands in solids, J.F. Cornwell The electrodynamics of magneto-electric media, T.H. O’ Dell The growth of crystals from liquids, J.C. Brice Soft modes in ferroelectrics and antiferroelectrics, R. Blin c and

B. Zeks 14 Magnetic bubbles, 4.H. Bobeck and E. Della Torre 15 Magnetism and metallurgy of soft magnetic materials, C.W. Ch en 16 The electron-phonon interaction in metals G. Grimvall

50

ig)

3 ‘gy

muNe 1981

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM: NEW YORK -OXFORD

‘

© North-Holland Publishing Company, 1981 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the copyright owner. ISBN

PREFACE Between 1960 and 1980 more than 1000 articles have been published within the field covered by this book. About half of them are referred to here, It has been my aim to quote older papers which are historically or scientifically significant, as well as the most recent work. It is hoped that these references will make it possible to trace other work which has been omitted, inadvertently or because of lack of space. I am grateful to many colleagues who have read parts of the manuscript. My special thanks go to H. Citrynell who has read it all. I am also most grateful to my children Eva and Mats and my wife Siv who, without any serious complaints, have let me spend endless evenings and weekends typing and retyping the manuscript, checking formulas and references etc. It is the more admirable as Siv has been completing her thesis at the same time.

0 444 86105 X

PUBLISHERS:

NORTH-HOLLAND PUBLISHING CO. AMSTERDAM, NEW YORK, OXFORD SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA! ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017

Goran Grimvall

Stockholm, March 1980

Library of Congress Cataloging in Pubheation Data

Gramvall, Gdran, The electron-phonon interaction in metals, (Selected

topics

in

solid

state

Physics ; 16) Bubliography: p. Includes iniex. i. Blectron-phonon interactions. 2, Metals-~Electric properties. I, Title. II, Series: Serres of monographs on selected

topics

QC793.5 .R628G74

ISBN O-Whl-86105-X

in solid

530.412

state

physics

81-365

(Elsevier North-Holland)

PRINTED IN THE NETHERLANDS,

‘AACR

3 16.

List of most important symbols

electron velocity (12)

LIST OF MOST IMPORTANT SYMBOLS* phonon creation and annihilation operators (269) electron creation and annihilation operators (269) phonon Green’s function (193)

transport coupling function (212)

electron energy Fermi energy (7) phonon density of states (23) energy— wavenumber characteristic (53)

1/kgT

electronic heat capacity coefficient (1)

as above, ‘bare’ band contribution (125)

OOo

Fermi-Dirac distribution function (13) reciprocal lattice vector

phase shift (73) energy gap in a superconductor (153) ‘bare’ electron energy (95)

electron Green’s function (92)

dielectric function (272) McMillan— Hopfield parameter (50)

§ eS - aE

electron—phonon coupling (47) electron—phonon matrix element (47)

m,

my N

N(E) NE) n n

P q

R, 6 Tr T,

4;

Vv

Boltzmann’s constant

Fermi wavenumber (8)

mass enhancement parameter (95) phonon branch index

1on mass

electrical resistivity (210)

electron—electron interaction (158)

Thomas—Fermi wavenumber (8)

electrical conductivity (208) scattering time (217)

electron self-energy (92)

free electron mass electronic band mass (11) electronic thermal mass (125) total number of ions

electronic density of states, per spin electronic density of states, per spin Bose-Einstein distribution function number of electrons per unit volume

wave vector wave vector

Debye temperature (25)

(10) and atom (11) (24) (7)

position of the jth ion electron density parameter (7) Fermi temperature (9) transition temperature for superconductor (176) displacement of the th ion (19) crystal volume

*(page where the quantity is defied within parenthesis) vi

solid angle atomic volume

A)

ion plasma frequency (30) average phonon frequency (175)

Sa

2s

E Ey

form factor (52) ion valency renormalization function (153) renormalization constant (120) Eliashberg coupling function (49)

are

b*,b et,ec D

Fermi velocity (8)

phonon frequency for mode (q, A) (21) electron plasma frequency (8)

'

‘

Units Usually, there is no ambiguity in the units used in this book. However, we adopt the convention of writing e? instead of e?/(47ey) with

&) =8.85410~'? Fm™! which should be used if the unit charge e is

Chapter |. Some observable effects of the electron-phonon interaction...

1.1.4... 1

1. The heat capacity of metals.

2. 3.

The electrical resisitivity. The transition temperature of ‘superconductors.

Chapter 2. Electrons in metals Basic concepts 1.

2.

The free electron gas

The electronic density of states N(E). . . 2.1. From k space to energy integrals. 2.2,

23.

24.

ay

cipressed ini SI units. Unless otherwise stated, the quantity f is explicitly written out. Whenever a phonon frequency w appears in the combination hw, it is assumed that w is given in radians per second. When there is no risk of ambiguity, as for instance in the combination Bw=hw/ksT in eq. (1.10) and similar relations, we write w instead of hw.

5.

.

An angular decomposition of N, ME).

The effective mass.............

2.5, An integral transformation. . Thermodynamic functions. . . Transport properties

Probing electronic properties at the Fermi level.

Chapter 3.

1,

N(E) for free electron:

Lattice vibrations—A brief review...

66.

The conventional theory of lattice vibrations. . .

eee

.

11 The dynamical matrix. Exgenvalues and eigenvectors.. 1.2 Normal coordinates and second quantization operators. 2. Thermodynamic properties... 0.00... e eee eee ee eee 2.1. Phonons as a Bose-Einstein gas... ...... 2.2. Frequency moments and Debye temperatures 3.

2.3,

Anharmonic effects... ...

3.2.

Point-lke positive charges..........

Lattice vibrations in metals 3.1. The bulk modulus of a free electron | as.

ne

.

19

Chapter 4,

The electron-phonon interaction. .........

The adiabatic approximation...

ve

YN

1.1. 1.2.

A conventional approach. . A refined treatment.

6...

eee

beeen

eee

eee

Non-diagonal matrix elements and the electron phonon scattering.

eee

:

Evaluation of the scattering matrix element.

3.1, 3.2.

The rigid-ion approximation Plane-wave electrons The Mott—Jones model. 3.4, Electron-phonon coupling functions The diffraction model.

ee

3,

.

Some useful relations for M,, and A.

4.1.

Is there an excitation spectrum?.

4, The excitation spectrum and the interpretationof A(w, p). . . .

5.

4.2. The interpretation of (n,>. . 4.3. The interpretation of A(w, p)-

oe

212 113

wee

3

114

The Migdal theorem. ..........-..+.+-Diagrams leading to superconductivity. . .

ALS ~iI7

6.1.

Introduction.

-

.

The quasi-particle excitation spectrum. The screened electron-phonon matrix clement.

The thermodynamic potential and the entropy. The electronic heat capacity The Azbel’-Kaner cyclotron resonance.

The de Haas—van Alphen effect.

The tunnelling rate between normal ‘metals.

6.1 . 6.12. 6.13, 6.14,

oe

»

. “25 » 128

131

«134

+ 135

The high-frequency electrical conductivity . General results for DC transport propert es. The Paul spin susceptibility. The Landau Fermi-liquid theory..

+. 136 . 139 . 140 . 141 . 144

6.16. The positron effective mass

Chapter 5. Many-body interactions... . Single particle states in perturbation theory.. 1.1. The excitation spectrum in second order perturbation theory. « 1.2.

M,, for an Einstein model and fora real metal...

2.2.

Green’s functions for electrons and phonons.

......-

Green’s functions and simple concepts for many-body systems. . . 2.1. Brillouin— Wigner perturbation theory . . .

Quasi-particles........0-2--0000 eee

Chapter 6, Superconductivity. 1.

20.

146

occ cece eee eee ees 147

Hamiltonians for electrons in metals 1.1. Introduction. . 1.2. The FrohlichHamiltonian. eee 1.3. The Bardeen-Pines Hamiltonian.

14,

1.5.

gy

.

6.10, The DC electrical conductivity..

oe

117 118 119

The shape of the Fermi surface

6.15. The electron—electron renormalization effects.

23.

MO

5.2, 5.3.

6.2.

8.1. Companson between reudopotential and phase shift methods

8.3. Rigid and deformable potentials. . . . 8.4. The deformation potential. . . . . 8.5. The Bloch and Frohlich descriptions.

oe

- 110

5.1. Corrections already included in G, Dandg. . . .

Band structure effects... .. 2... . Further improvements of the Gaspari--Gyorffy method eee

:

Higher order corrections to the self-energy.

The Gaspari—Gyorffy formula for (17>

oe

.

5.4. Multr-phonon processes. Lee 5.5. The effect of static impurities. . . . 6. Renormalization of electronic properties.

O(a)e(q)exp(ig-R°

(3.10)

aj

The equation of motion is wan

orthogonal and normalized, and the eigenvalue can be written

o(q,)= ze. ala e°(4, Ae%(q, A).

UsAR,5 Rj) = Use Rj Ri) = Usp Ry — Rs 0)= Usg(R, —R,)

21

matnx is simplified to

aU,

Dap) = q >’ [1-exp(—ig-R,)] axtaxF J

(3.12)

2’ denotes a sum with the term R, =0 omitted. The derivatives of U, are taken at the equilibrium atomic positions. The form (3.12) is convenient for short range forces, when only nearest neighbours contribute. For long range forces, it is better to work with the Fourier transform of eq. (3.11). We let the choice of the argument in U, (i.e. R or q) distinguish between the pair potential in real space and its Fourier transform, and similarly for the dynamical matrix D,,. It follows that

Dsl

79

[(G+4)"(G+9)"U,1G+4) -6°G°U,(\G) J, (3.13)

where G is a reciprocal lattice vector. In certain symmetry directions, the lattice vibrations are strictly longitudinal or transverse (e(g, A) either parallel or perpendicular to g). In that case, the phonon frequencies can be obtained directly from eq.

(3.10) without the explicit diagonalization of the eigenvalue equation. As an example, one finds in the [1 0 0] direction of a lattice with cubic symmetry

(4 )e*(q).

(3.9)

There are three eigenvalues and eigenvectors, A=1,2,3, and therefore also three normal coordinates Q(, A) for each g. The eigenvectors are

(q,4)=D,x(q)-

(3.14)

In any direction g, the phonon frequencies obey the sum rule

Derr) = EPala),

.

(3.15)

2

Lattice vibrations— A brief review

2. Thermodynamic properties

which combined with eq. (3.13) gives

and

Tora d)= FE [(Grayry(Grah-GFU,le)]. 1

G16)

G

The generalization of the theory of lattice vibrations to lattices with more than one atom per unit cell or several kinds of atoms is straight forward. Whenever relevant, we let the index A in w(g,A) and eq, A) refer not only to the polarization of the acoustic modes but also to distinguish between acoustic and optical branches.

Although the theory, as it has been outlined above, gives the correct phonon frequencies w(q, A), it is not the best formalism for the evalua-

tion of electron-phonon scattering matrix elements. One reason is that

our normal coordinates Q(g, 1) must be complex, Q*(g,\)=Q(—q, A), to give real displacements uj in eq. (3.6). It is therefore better to

introduce new quantities,

bla, d)=[2ho(a, A)]-"| ala NOW+12 .AI O(-4,2)|

BG@,)=[2 "| ala ne(4 O(-4.d)-i ,)]L 00.2) ], (3.17b) where b(q,A) and 6*(qg,A) have the properties of annihilation and creation operators in the second quantization formalism of quantum mechanics (appendix D). The Hamiltonian in the new operators is

(3.18)

This is the sum of Hamiltonians for independent harmonic oscillators with frequency spectra hw(g, A)[} +n], n being 0, 1, 2 etc. In the nth quantum state of such an oscillator,

b*|n> =(n+1)'7|n4+1)

.

From eq. (3.17): h

o@.»)= [53255

1/2

[b(g,4)+6*(—g,A)].

(3.20)

The operator for the amplitude u, of the jth ion has by eqs. (3.6) and (3.20), components

H(0-0"| aaa) where e(g,A)

1/2

e(g, A)[e'*Fib(g, A)

te FRib+(g,r)],

is the polarization

(3.21)

vector (of unit length) of the mode

@, ) and we have used that w(g,)=«(—g, A) and e(g, \)=e(—g, A)

for a lattice with one atom per unit cell.

2. Thermodynamic properties

(3.17a)

aa

(3.19b)

h

1.2. Normal coordinates and second quantization operators

H= > hw(g,d)[}+6*(g,A)@,A)].

bln> =(n)'7|n-1).

23

(3.19a)

2.1. Phonons as a Bose-Einstein gas For many thermodynamic properties of lattice vibrations, the only important characterization of the vibrational eigenstates is the density of states F(w),

F(w)=

v

(ny

HD f¥48([email protected])-0)

Q,

ds

»

(any x J 1V,o(9,A)|’

(3.22)

where the integral over d’g is restricted to the first Brillouin zone and the surface integral dS is over that surface in the first Brillouin zone on which w(g,)=w. For a lattice with one atom per unit cell, F(w) is

normalized as

[O"Fo)do=3.

(3.23)

2. Thermodynamic properties

Lattice vibrations— A brief review

24

The lattice vibrations in a crystal can formally be considered as a gas of independent phonons obeying Bose-Einstein statistics, ie. with an occupation number

n(w(q,))=n(q.)= exp[

J :

64)

The total energy E is

E=> [}+n0@,.r)]hota, qa

aN [3

+n(o) ]hoF(w)de. (3.25)

The heat capacity C=dE/0T at high temperatures is

1{fe\l,

0

(3.26)

A useful expression for the entropy S is

qa

(3.27)

with the high temperature expansion

GEE)

ww Ke? M

4 (ts) +..]reree

2.2.

(3.28)

max 0

[A

2

fies ) +. [A

kyT B

@2

dw.

(3.30)

Frequency moments and Debye temperatures

We can summarize the information contained in F(w) further by the introduction of frequency moments 1,

[ror

w)dw

(3.31)

1, ae

F(w)de

For practical purposes, it is often better to define a Debye cut-off frequency w,(n), which in a Debye model gives a moment equal to that

obtained from eq. (3.31) for the true phonon spectrum. One finds that

on(”)=(

S=ky > {[1+n(q,d)]In[1+2(g, )]—n(q,A)Inn@q,d)},

sana f [ss

—

f

1 (hw \* + |Feoree atop) + ( 255) coy [ [1-5 =

In the high temperature limit,

3.24

hw(q, A)/kpT]

25

n+3

3 tH)

if n> —3 and nO.

\l/n

(3.32)

For n=0,

the definition is

1

[07 F(e)in(w) do

3

f ™" F(w)do

op(0)=exp| 5 +

|.

(3.33) Z

We also define Debye temperatures @p(7) as

In classical mechanics, the time averaged kinetic and potential energies of a harmonic oscillator are equal. A similar result holds in quantum mechanics, where the expectation values of the kinetic and the potential energies are equal for every eigenstate of the oscillator. It then follows immediately that the average of the (squared) vibrational amplitude in a lattice with cubic symmetry (which implies isotropy) is

sign

fwo(qsd)/2

0

(3.34)

This definition can be extended to include n= —3, with Op(—3) being

identical to the Debye temperature that gives the low temperature limit of the lattice heat capacity, cf. eqs. (1.4) and (1.5). It is common practice to interpret, e.g., heat capacity data in a Debye

model. The measured values at different temperatures are put equal to

wan p» [3+0@ 0) a@any/a = [sme] apg Flos.

k,@p(n)=hop(n).

the theoretical Debye result. Since the true phonon spectrum differs from a Debye model, we can not get a perfect fit with a single Debye

(3.29)

temperature ©) for all temperatures. It may

still be possible to repro-

duce the experimental data with a Debye model if we let the fitting parameter @,, be temperature dependent. In this way, we define a ‘heat

Lattice vibrations— A brief review

2. Thermodynamic properties

capacity Debye temperature’ OS(7). In an analogous way, we can use a Debye model in a fit to entropy data and define an ‘entropy Debye temperature’ @§(T) etc. It follows from eq. (3.26) that the high temper-

an additional temperature dependence of @,(n; T). We can express the

26

ature limit of @S(T) is @p(2) as defined in eq. (3.34) Also, @S(T) at

high temperatures is ©(0), while the vibrational amplitude, eq. (3.30), at high temperatures is related to @,(—2). Selected values of Debye temperatures are given in table 3.1. Figure 3.1 shows a typical variation of @(n) with n.

effect of anharmonicity, as it is manifested in the thermal expansion, by a Griineisen parameter y,(g, 4) for each phonon @, A);

— ¥o(9, A)= soled)

h

¥o(2)= sO)

@§(—3; 0 K) is based on the phonon part of the low temperature heat capacity

234K

Tepresentative sample of the entire phonon spectrum at these temperatures. However, real crystals always show anharmonicity. This implies

n

(3.36)

0) KI

—

160 + 150 + 140

+

+

3240123456

n

Fig. 3.1. @p() for sodium, based on data from Dolling (1965) and rescaled to @,(0)= 149 K as given in table 3.1. Figure 3.2 exemplifies how @,(7) may vary with temperature, mainly as a consequence of the thermal expansion. Obviously, the Griineisen

parameter y,(7) is not independent of n.

From thermodynamics, one can derive a relation for the ‘thermodynamic’ Griineisen parameter yg,

wo Ei) aE

2.3. Anharmonic effects For strictly harmonic lattice vibrations, the characteristic Debye temperatures @,(n) tend to constants when T2@)/3, i.e. we get a

(3.35)

|

where V is the crystal volume. An obvious generalization to moments over the whole frequency spectrum is

Table 3.1 Debye temperatures for some elements. @p(0) is calculated from entropy data at 298.15 K (Hultgren et al. 1973) corrected for the electronic part of the entropy. @5(—3; 298 K) is based on measured elastic constants (Gschneidner 1964) and Phillips 1971). The values for @p(—3), in particular @5(—3), are more uncertain than @ (0). Element (0; 298K) @5(—3; 298K) @§(-3; 0K) Li 370 350 335 153 164 149 Na K 86 1 a1 Rb 53 55 56 40 40 39 Cs Be 967 1367 1481 Mg 318 363 405 327 231 217 Zn 209 160 144 cd 2 sat Hg Al 383 403 430 322 89 226 Ga 110 85 114 In Tl 88 55 19 198 184 148 Sn Pb 84 81 105

2

em

©

where B is the bulk modulus of the crystal and £ is the coefficient of

volume expansion. For non-magnetic materials, y, from eq. (3.37) in the

high

and

low

temperature

limits should

Yo (0) and yg(—3) respectively.

provide

good

estimates

of

28

Lattice vibrations— A brief review

The inclusion of anharmonic effects via the Griineisen parameter is called the quasiharmonic approximation, since the lattice vibrations are

assumed to remain harmonic but with altered interatomic forces as the lattice expands. In a complete theory we must include not only the frequency shifts caused by the thermal expansion but also shifts which are explicitly due to the increased vibrational amplitude with increasing temperature. Thus, the latter effect is present also at constant crystal volume. We can write

@(g,A)=w°(g, A) +Aw(g, A; V)+Aw(g,d; T)—iy(g,A;T) (3.38) where w°(q, A) is the harmonic phonon frequency at T=0 K, Aw(g, A; V) is the quasiharmonic frequency shift which is caused by a change in the volume, and Aw(q, A; 7) is the frequency shift which explicitly refers to the vibrational amplitude. The imaginary term —iy(g, A; 7) leads to a damping of the phonons, since (g,) does not correspond to an exact eigenstate in an anharmonic crystal. The Debye temperatures in fig. 3.2 are based on neutron scattering data at 80 K and 300 K. The difference between the Debye temperatures should in principle not be represented by a Griineisen parameter alone, since this only takes into account the shift Aw(g, A; V). However, Aw(g, A; T) is usually small compared with Aw(q,A;V), and the quasi-harmonic approximation gives an adequate description.

One can show that the frequencies and widths of phonons as measured by neutron scattering are given by eq. (3.38). Often the damping term iy is small. Thus we can get experimentally determined phonon

ey tk]

3. Lattice vibrations in metals

29

frequencies which include both Aw(g,A;V) and Aw(g,A;T). An important question is whether these temperature and volume dependent frequencies can be used in the standard expressions for the thermodynamic properties of harmonic oscillators, in an attempt to account for the anharmonic corrections. Such a procedure is usually incorrect (Barron 1965, Cowley and Cowley 1966). One can show that only the entropy is correctly obtained in this way. In other quantities, such as the energy, there is a complicated double counting of the contribution from Aw(g,; T). When only the quasiharmonic shift is included, the corrections for anharmonicity are simple. As an example,

(3.39)

G=C(1+Br6(0)T) when TZ9p. 3. Lattice vibrations in metals

This section deals with some simple results for the elastic and the vibrational properties of metals, mainly for homogenous systems (jellium) and point like ions in a free electron gas. Many-body aspects on the lattice vibrations in metals are treated in ch. 7. 3.1.

The bulk modulus of a free electron gas

Consider a homogeneous electron gas in a uniform positive background, ie. a jellium model. Because of local charge neutrality, there is no Coulomb energy. If only the kinetic energy of the degenerate electron gas is taken into account, the bulk modulus B is

420-

a=

aP)-¥( aE\_10E 2 )= relation. This is not surprising since we have left out any influence of the crystal

structure and the nature of the ion core. A lattice with point-like charges in a uniform negative background can sustain transverse vibrations. Consider the dynamical matrix for lattice vibrations,

D=D, -D,.

(3.50)

D,, represents the interaction between the point-like ions in a rigid electron gas background and D,, represents the screening of the displaced positive charges due to the redistribution of the electron gas. In directions of high lattice symmetry, the vibrations are strictly longitudinal and transverse and we can write (cf. eq. (3.14))

@?(g,)=05(q, A) —wi(q, A).

G51)

32

Lattice vibrations A brief review 10

3. Lattice vibrations in metals

33

+

Al

1

B [10° Nm?)

Na

w

O5

-

0.3

0.2

o

15

Fig. 33. The bulk modulus B plotted versus 7, for simple metals. The dashed curve 1s the 7,5 law. The full line corresponds to Bacr,-*4, B is taken from Gschneidner (1964). The natural unit of w

is the ion plasma frequency 2.

In this unit,

wg, A) depends only on the type of the lattice, e.g, fec or bec, and not on the magnitude of the point charge or on the lattice parameter. Tables of w2(g, A) in the symmetry directions of fec and bcc lattices have been

published by Vosko et al. (1965). Figure 3.4 shows w3 and w? —w?, for aluminium (fcc) and sodium (bcc) in the [1 0 0] direction. The ‘bare’ (i.e. unscreened) longitudinal frequency is reduced by the screening

from the value 2, for small q to a value linear in |g|, which is in accordance

with

the

result,

eq.

(3.48),

obtained

from

the

dielectric

function. The bare transverse vibrations are linear in |g| for small q even in the absence of screening. For sodium, with a relatively weak screening (k, is small due to the low electron density), even the bare transverse

frequencies give a fair representation of the experimental results for all wave vectors q.

The bare frequencies obey the Kohn sum rule quoted by Bardeen and

Pines (1955):

Deiar)=G.

Uo

0.1

0.2

(3.52)

0

Lew

02

4

04

06

08

10

ql27a) Fig. 34a. 0°(q), in umts of the plasma frequency 92, for the [1 0 0} transverse mode in sodium (bec) and aluminrum (fcc). The full curves are w? and the dashed curves are experimental data (Vosko et al. 1965).

02

7

Za

Na 7

-7

~T

meencd

-—

-

04

06

08

1.0

4 (2"/0) Fig. 3.4b. As in fig. 3.4a but for the [1 0 0) longitudinal branch.

The quantity

Kei) = 705- ww & ebao(@ 0)

(3.53)

is therefore a measure of the average amount of screening . This is misleading. We have until now considered the ions as point charges. For Na, Al and

CHAPTER 4

THE ELECTRON-PHON

34

INTERACTION

Lattice vibrations — A brief review This chapter deals with the quantum mechanical basis of the electron-

Table 32

The average unscreened vibrational frequency and the average phonon frequency from experiments,

expressed

as the equivalent Debye

phonon interaction in metals. The main purpose is to give expressions for the electron—phonon matrix element. In the first half of the chapter

tem-

we derive some general results and in the last part we discuss the evaluation of the matrix element with emphasis on free-electron-like metals and transition metals.

peratures ©,(2) and ,,,(2) respectively. Experimental data from Dolling (1965) for Na, Gilat and Nicklow (1966) for Al, Nicklow et al. (1967) for Cu. For Pb, the value of ©§(0) from table 3.1 is used (room temperature).

Element

Na Al Pb cu

@,(2) (K)

®cxp(2) (K)

189 821 291 213

1. The adiabatic approximation

150 397 84 315

1.1. A conventional approach

Pb the size of the ion core (i.e. the electrons which are bound

to the

nucleus) is small compared with the size of the atomic cell. Therefore there is negligible overlap between core electrons on adjacent ions. We

In thermodynamics, an adiabatic process is one which takes place so slowly that the system under consideration remains in almost perfect equilibrium. There is no heat transfer, and the entropy is constant. On a microscopic level, this means that a system which is initially in some quantum state

remains in this state even though the wave function ¥,,

(3.54)

and the energy eigenvalue E, may change during the adiabatic process. A simple example is provided by the electron states in molecules. The total electronic wave function depends on the instantaneous relative positions of the vibrating nuclei in the molecule. The electrons move so rapidly that they adjust adiabatically to the much slower vibrations of

where the repulsive interaction due to the core overlap is represented by

developed by Born and Oppenheimer (1927). The adiabatic approxima-

call such

metals

simple

metals.

This

description

fails for

the

noble

metals, but still the core—core overlap is small enough to allow a simple generalization of eq. (3.51) (e.g., Toya 1961) to the form

w*(q, A) = 05 (q, A) + 02.(q, A) we(Q, A)

w,. For transition metals, the electronic structure is very complex, and no simple representation of the type of eq. (3.54) seems possible.

the

nuclei.

The

quantum

mechanical

theory

for

such

systems

was

tion, to be defined later, is therefore often called the Born—Oppenheimer approximation. We shall develop the theory for electrons and ions in solids. The main result is of course that we may treat the electrons and the phonons as (approximately) separate quantum mechanical systems. A strict proof 1s lengthy. It is therefore instructive first to assume that

the total wave function can be written as a product of an electronic part

and a phonon part, as implied by the adiabatic approximation. We then 35

s

36

1. The adiabatic approximation

The electron-phonon interaction

show that this wave function is an approximate solution to the total Hamiltonian with an error which involves the small parameter m/M, the ratio between the masses of the electron and the ion. Consider N like ions, with electron cores which can be regarded as tightly bound to the nuclei. The positions of the ions are R;=R} +4; (j=1,...,N) where u, is the deviation from the equilibrium position

Rj. Each

ion is assumed

to contribute Z conduction

coordinates r, (i=1,..., ZN).

and the ions is

a

Ayo.

Im 2a

The

total Hamiltonian

re

By

jal

ZN

electrons with

(4.1)

The first two terms are the kinetic energies of the electrons and the ions. The next term is the direct electron—electron Coulomb interaction between the conduction electrons. U,(R,) is short for Uson-ron(R»-++) R,>---» Ry), ie. the potential energy for direct ion—ion interaction. U,.(%3 Ry) = Uxeltys---5 tees zw Riv) Rye) Ry) is the from

energy

of the ZN

conduction

electrons

the nuclei and the ion core electrons, when

instantaneous

positions R, (j=1,...,N).

We

moving

would

(4.4)

Here a and f are quantum numbers which specify the quantum

completely. We shall now argue that the wave function (4.2) is a good approximation to the exact solution for the Hamiltonian H,,,. We first

consider that part of H,., which contains electron coordinates,

in the field

the ions take the like to solve

the

tion. Then it is assumed that the total wave function ¥,,, for conduction electrons and ions is the product of two wave functions, Y, for the conduction electrons and ¥,,, for the ions; (4.2)

Since the eigenfunctions ¥, form a complete set, we can always write Vio, aS an expansion in Y, with expansion coefficients depending on the coordinates of the ions. Equation (4.2) amounts to the approximation that only the dominating term is kept in such an expansion. ¥, depends

not only on the coordinates r, of the electrons but also parametnically on the positions R, of the ions;

Woe = Vea tireees ieee tans Rises Ry

Ry).

(4.5) (4.6)

HM, = EM, ‘ave,a?

where E, =E,(R))=E,(Rj,..-, Ry... Ry)- With this definition of Y,,

we insert the Ansatz (4.2) in the complete Schrédinger equation

Fro M or = Exo ror

(4.7)

When eq. (4.7) is multiplied from the left by ¥*, and integrated over all

the electron coordinates r,, (d°r,...d°x,...d°rzy =dr), the result is N

>| -a7 "+ Us(R,)+ELR)) Yon 9 +(OH en 9 Ean jar ne

j

complete Schrédinger equation with H,,, as in eq. (4.1). This is too difficult, and we shall have to be content with the adiabatic approxima-

‘tot =VMYon

states

and assume that we can find solutions ¥,, to the equation

hd

+U,(R,)+U(45 R,)-

potential

Vion, p= Vion, p(Riy-++5 Ryseees Rey t)-

+U,(15 Ry),

in—-l

nyet

Yo, is a function only of the coordinates of the ions and of the time #;

for the electrons

e?

DW

37

(4.3)

(4.8) where

how

(QH)%on.0 =~ Fag Ds (f WaVR 097 Yon. p j=l

25> he

N

(f%2.F—%.047) Ve Mone

(4.9)

y=

We shall argue below that (AH)¥,,,, , can be neglected in eq. (4.8). Then, eq. (4.8) is a Schrédinger equation for the ions, in which the energy eigenvalue E,(R,) of the conduction electrons acts as an effective potential for the movement of the ions. It might seem at this stage

38

1. The adiabatic approximation

The electron-phonon interaction

that we have achieved very little since the quantum state a of the conduction electrons enters the Schrédinger equation for the ions via E,(R,). In practice this is no serious objection because, on a relative scale, the energy E, is practically the same for the ground state of the electrons at T=0 K as it is for ‘normally’ encountered excited states, e.g, due to a finite temperature. We can therefore drop the label a on E,(R,) in eq. (4.8). Hence we have achieved the goal of separating the Schrédinger equation of the ions from that of the electrons. Of course, this does not mean that the conduction electrons do not affect the motions of the ions. On the contrary, they give rise to the very important screening of the longitudinal vibrations, and also transverse modes are partly screened. This effect is included in E,(R,) but it is the same for the ground state as for the excited states a of the conduction electrons. It remains for us to show that (AH)¥,,,, g is indeed negligible in eq. (4.8). In the absence of a magnetic field, we can choose ¥, , to be real and then the last integral in eq. (4.9) is zero since

J %aVeSeadt=3 Va f Yeoh adt= Ve (1)=0.

(4.10)

The first integral in eq. (4.9) is more difficult to treat strictly. We

first

note that ¥,, is independent of the ion coordinates R, in the free

electron model, and in this approximation vanishes. It is therefore plausible that the electrons are tightly bound to a particular ion. wave function contains terms of the form $(r,—

the integral in eq. (4.9) worst case is when the Then the total electronic R,). Since

-F7Vbon -R)=- ZF vie, -R,), 2

2

(nn)

the first term in eq. (4.9) is

(/M Eg aon, >

(4.12)

where E,\. is the kinetic energy of the conduction electrons. The parameter m/M is small which is one reason to neglect (AH)Vion, pin eq. (4.8). More important than the smallness of (4.12) 1s the fact that Ein depends very weakly on the positions of the ions and therefore only adds a constant to the equation for the lattice vibrations.

39

1.2. A refined treatment

The

previous

section

served

as an

introduction

to some

important

concepts in the adiabatic approximation but the arguments were not very stringent. We would like to have a theory with a small parameter «

so that the validity of the adiabatic approximation is expressed by the

power of « that appears in the correction terms. Such a theory has been developed by Chester and Houghton (1959) and further discussed by

Chester (1961). The most satisfactory formulation makes use of canonical transformations. In that way one can clearly see the connection with the Hamiltonians discussed in ch. 6, which form the basis for the theory of superconductivity. An alternative procedure, which we shall follow here, is to repeat the analysis in the previous section in a rigorous way. As in § 1.1 we assume

that we

know

the solutions ¥, ,(7;; R;)

of the

Schrodinger equation for the conduction electrons when the ions have the positions R;. We write R,=Rj +xi, where « is a dimensionless

expansion parameter and «a, is the deviation of an ion from its equilibrium position. We want « to be such that there is no displacement of an ion when its mass is infinite. A possible choice for « is (m/M)’. The exponent y is determined later. We now use the fact that the eigenfunctions ¥, , form a complete set in r space and expand the total wave function as

Yor = DX(Ry)¥e,0°

(4.13)

In § 1.1 we only kept one term in eq. (4.13). In analogy to the method

used in § 1.1 we consider

HM

ot = Evo

or

(4.14)

multiply both sides of eq. (4.14) from the left by ¥*,, and integrate over all r,. One immediately finds that the correction term corresponding to eq. (4.9) contains integrals

faa, a7

(4.15)

SMa Va%e,ad7.

(4.16)

and

40

The electron-phonon interaction

2. Non-diagonal matrix elements and the electron-phonon scattering

All terms in the Schrédinger equation are now expanded in the parameter «. For the lattice vibrations to be described by harmonic oscillators,

it is necessary that the kinetic energy of the ions is of the same order in « as are the quadratic terms a,. It turns out that this requires y=}. Further analysis shows that corrections to ¥,,, beyond the adiabatic approximation are of the order x? and hence the corrections to the energy are of the order x®. We note that « does not contain the strength of the electron-phonon interaction but only the mass ratio (m/M)'/4, which is less than 0.1 for all metals. The adiabatic approximation therefore seems to be adequate for both free-electron-like metals and transition metals. This result is of course expected since any textbook on solid state physics assumes that we can consider the electrons and the phonons as separate systems, e.g., when we calculate the total heat capacity. Still, there are cases when the adiabatic approximation is inadequate and that is in fact the main theme of this book. 2. Non-diagonal matrix elements and the electron—phonon scattering

second order perturbation theory (which involves non-diagonal terms). The electronic excitation spectrum turns out to be strongly modified in the vicinity of the Fermi level which affects, e.g., the heat capacity. Such

effects are considered in ch. 5. Suppose that ¥.(r; Rj +u,) can be expanded in a rapidly converging series in powers of u,. Since

Ve =V,>

contribution to (4.18a). In (4.18b) a term linear in u, in the expansion of Y, is sufficient to yield a contribution. Thus (4.18b) is the important

part of the non-diagonal electron-phonon matrix element. In the evaluation of (4.18b) we need an explicit expression for ¥,, when the

ions are displaced from their equilibrium positions. Ordinary second order perturbation theory gives

Vea = Kea

3 |E

aw DS Js bn. VE (V2%e,0) Vion, 247% I Tons 2 ff %n.0B

*[ U(r; RG +u,)— U(r; R2)]¥e dt

TEE

(4.17)

where H,,, is the total Hamiltonian (4.1) and the wave functions |a, 8) are the products of separate electronic and phonon parts in accordance with the adiabatic approximation. If we make the adiabatic approximation also in H,,,, and hence neglect the term (AH) defined by eq. (4.9), the matrix element (4.17) is zero unless a=a’ and B=£’. Thus the nonvanishing part of (4.17) with (AH) kept is

ae

(4.19)

we must include terms of the order of u? to get a non-vanishing

Consider the matrix element

-

3.1.

43

The rigid-ion approximation

We assume that the potential U, introduced in eq. (4.21) follows tigidly the motions of the ions so that, for small displacements uj,

U(r; Ry +u;)— U(r; RZ) =U,(r—R? —u,)— U(r-R§)

=u Va V(r—-R,)= —uy VV(r—R3).

(4.23)

4.

Hence Equation (4.23) is perhaps most easily verified if we first show (trivially) the commutator relation

[H,xJ=-—

(4.24)

for a Hamiltonian H of a particle with mass M moving in a potential of arbitrary form V(x). The momentum operator is p, = —ih(d/dx). Equation (4.24) and the fact that the lattice vibrations are described by

a Hamiltonian with the required property leads to eq. (4.23). Since energy is conserved in a

scattering process, E,, —E,, =E,. — Es,

and we finally get the non-diagonal matrix element

= & .

(4.25)

This decomposition into two parts referring to the displacements u, of the ions and the effective electron-ion interaction W(r; R}) is usually called Bloch’s relation.

(426)

W(r; R?)=—V,V(r—R?).

(4.27)

This is the rigid-ion approximation, first used by Nordheim (1931). In eq. (4.26) we introduced the notation V=U, for the rigid potential. Almost all calculations of the electron-phonon interaction in metals rely on the rigid-ion approximation, although it is not known precisely how good it is. Some comments on its validity are found in § 8.3. 3.2. Plane-wave electrons

We assume that the conduction electrons can be described by a product of plane waves. We consider the scattering of a single electron and let a

and a’ refer to the initial and

final wave vectors k and k’. The

index is suppressed since the scattering potential from the displaced ions does not change the spin. Let the electronic wave function be normalized to unity in the crystal volume V. We make the rigid-ion approximation and multiply by 1 =expfi(k—k’)-R} Jexpl ~ik-R§ expfik’- R? 1 After a simple rearrangement of terms one obtains

/2M=n/2M

(4.52)

Here wax is the maximum phonon frequency and , is taken, one obtains eq. (4.52). A second important moment

Com aR) 5

On

is

Ml Ex I?? May

structure factor which specifies the positions of the atoms. An analo-

gous approach is possible for the properties of nearly free electrons in a metal. This is the basis of the diffraction model (Sham and Ziman 1963, Harrison 1966).

We

use

second

(4.53)

where

(4.54) 1s assumed

to be perturbed by a weak potential

W(r),

which is the sum of potentials from different lattice sites R,,

(4.55)

W(r)=dw(r-R,). Jj

The perturbed wave functions, to a first order in the perturbation, are

Yy(r)=|k>+ DAG. WA+ED,

(4.56)

with the expansion coefficients

ck +q|W(r)|k>

(4.57)

40) am)42 =[kergP) The sum in eq. (4.56) goes over all plane wave states |k+q)

and the

prime on the summation sign means that the term qg=0 is to be omitted.

The energy of the state k, to second order in the perturbation, is

242 hk AO

=A,

51

ROA 9

CAI

RD ++>

Z

¢—((?/2m)(k? —|k+q|?)

.

(4.58) We now use eq. (4.55) and simplify the matrix element (k+q|W|k).

= T ferry

w(r—R,)e* "dr i

=F 1 > eB fe

ere) (RD (r—R, Je

OB) dr.

(4.59)

52

The electron-phonon interaction

5. The screening of electron-ion potentials

In the last integral, r—R, 1s a dummy variable. One can simplify that integral further if w(r) is a local potential (rather than an operator as a rigorous pseudopotential theory would require) and write (k+q|W|k>

The first term 1s the kinetic energy of free electrons. The second term is

im

geometrical

compact

notation

independent of k when W is

also independent

as

=S(q).

(4.60)

NG

and the form factor (k+q|w|k>

only

enters

in the last term,

(4.65)

When all the ions are at their equilibrium positions R?, the structure factor is

=5, 6,

(4.63).

25)

Alwlkta>

FO~ Dame —ikeal?) ketal A> gag. = 2%. f het

has been decomposed

may overlap.

J

with

(4.62)

into two parts; the form factor which is a property of the individual ion in the solid and the structure factor which depends on the positions of the ions but not on their nature. In most of our applications we shall assume that w(r) is a local spherically symmetric potential. Then the form factor is a function only of g=\|g| and we denote it w(q). In some basic expressions, however, we retain the notation (k+g|w|k> to remind us that w(r) should have the properties of an operator. The integration in eq. (4.62) goes over the entire crystal volume V=NQ,. Thus the potentials from different ions

De

of the atoms

The

q

Cetglwlky ag fem) ar.

Sg)=

configuration

Ey, = >’ S*(q) Sq) Fg),

defined by

In this manner, the matrix element

of the positions R, of the scattering centers.

It is

acteristic Fg). We have

(4.61)

Sen#®

the sum of local potentials w=w,(r).

which is called the band-structure energy of an electron in the state k. The total band-structure energy EF, (per ion) is usually expressed in terms of the structure factor Sq) and the energy-wavenumber char-

We have introduced the structure factor S(g) defined by

sat

33

ry J (1? /2m)(k? —|k+q|?)

+

4.66

469)

The sum (integral) in eq. (4.66) is over all occupied states k and the

prefactor 2 comes from spin degeneracy. F(q) is a function only of the

magnitude potential.

We

that:

g=|g|

when

w(r)

is a local

and

spherically

symmetric

close this section by quoting two relations. One easily verifies

A(q, k)=A*(—q, —k)-

(4.67)

When w(r) is a local potential it follows that:

w(q)=w*(—q).

(4.68)

where G=0 or a reciprocal lattice vector.

We now rewrite the total energy of the perturbed state |k> in terms of the form factor and the structure factor as

wR?

E(k) = 5% + +>’ q

S*(q)S(q)

(i2/2m)(K = lk+gl?)

5. The screening of electron-ion potentials In the prece .ing section we introduced a weak scattering potential w(r)

“eo

but with i» attempt to show how this potential is to be found in a self-consistent way. In this section we shall derive expressions for the form factor and the energy-wavenumber characteristic in terms of a bare (i.e. unscreened) electron—ion potential w°(r) and the Hartree

54

The electron-phonon interaction

5. The screening of electron-ion potentials

screening function e°(q). The new relations we shall prove are

= ch+q|w*|k> @(a)

electrons.

From

the first order

perturbation

expression

55

(4.56) for the

wave functions, we obtain the charge density caused by the electron in the state k (one spin direction, lowest order in the coefficients A):

4.69 (4.69)

eV), =— $ {1+ B’ [Aa ayerrr tara, kie']}.

and

Fg)=— with

the

2e iD!

bop

__ 2447

Seale tale leo

Hartree

dielectric

function

(also

approximation (RPA) dielectric function)

e(q)=1+

2

2ak,h?x?|

(:

x

_y2

2x

q

(4.70) called

the

4/41]

random

The total charge density is obtained by a summation over all occupied

states K and over the spin. The constant term —e/V in eq. (4.74), when summed over all k and spin, yields a uniform charge density — Ze/Q, which cancels the average positive charge density from the ions. It is the remaining term which is of interest here. The corresponding charge density p(r) is

phase

(4.71)

1-x

2e = , Spex LAG ve ler +A*(g, 4am k)e'#"] gr dk. Gap

e(r)=—

Here x=q/(2k,). The dimensionless prefactor on the right hand side

(4.75)

can be written alternatively as

me?

1

Qakph?x?

2aagkpx?

KE

We expand the charge density p(r) and the screening potential W*(r)

(4.72)

2g?”

(4.74)

in Fourier components as

e(r)= D>’ o(g)e'*"

In appendix F the dielectric function is generalized to be a function also of the energy (frequency) w. This version of e(q, w) is usually called the Lindhard dielectric function. Its static mit e(g,0) is identical to the Hartree dielectric function e°(q).

and

(4.76)

‘

(4.77)

W*(r)= >’ W(q)e't". q

5.1.

The screened form factor

:

We now derive eq. (4.69). Consider a total bare potential W°(r) which is the sum of bare potentials w°(r—R,) from each ion. This is the spacially varying potential an electron would see if the other electrons were uniformly distributed as in a free electron gas. However, the electrons redistribute under the influence of the bare potentials. Thus a non-uniform charge density is established, which by Poisson’s equation

can be transformed into a potential that affects the electron we are considering. Since all electrons interact mutually via this potential, it

has to be determined self-consistently. We assume

tive potential acting on an individual electron is

that the total effec-

(4.73)

W(r)=W°(r)+W*(r), where

W*

is

the

screening

potential

due

to

the

redistribution

of

As usual, the argument r or g distinguishes between a function in real space and its Fourier transform. In an undisturbed periodic lattice, g is either zero or a reciprocal lattice vector.

W*(r)

has the dimension

of

energy, i.e. the electronic charge is included in W**. Poisson’s equation

yields

VW*(r)=4mep(r).

(4.78)

We insert eqs. (4.76) and (4.77) in eq. (4.78) and equate the coefficients for each Fourier component exp(ig-r). Hence

—PW*(q)=4rep(q) = ~8ne7/(2n)° fi

|kl kt

S@)st005“ (/2m)—(K2 [ke gl?) *=(q)/?,

(4.88)

where w refers to a single ion site. The integral representation for e°(q) which follows from eq. (4.80), and the two relations

= +k+q|w*|k>

(4.89)

5. The screening of electron-ion potentials

58

The electron-phonon interaction

w*(r). We write for the Fourier components of the effective potentials:

and

ht g)w|a> = rab &°(q)

(4.90)

allow us to rewrite E,, in eq. (4.88) as

=’ S*(q@)SQ)F@)

(4.91)

q

Q,q?

°

Fea)=— 2258m? chet glwo|ay Pt (9)

(4.92)

which is the desired result (4.70). We note that this simple form requires that is independent of k.

It is useful to define a normalized energy—-wavenumber characteristic Fy) such that Fy(q)->1 when q—0. From the relation ¢k+q|w°|

ky) /e°(q)>—(2E,/3) when q—0, and the low-g limit k2/q? of e°(q), one obtains

Frq)=—-

g

meeZ a).

e)

jee og)

(4.94)

where W*(q) is obtained from eq. (4.78) and X(q) is —eW*(q)/p@q). There is no theory which gives X(q) explicitly for all wave vectors g, but much work has been done in the low-g and high-g limits.

The Slater approximation is:

charge density from the conduction electrons was smoothly modulated

by the presence of a weak perturbative potential from the discrete lattice, and that a particular electron experienced a screening potential calculated from this electron density. This is the Hartree model. It

neglects the fact that the electrons repel each other so that the instantaneous electron density in the immediate vicinity of a particular electron is lower than the average electron density used in the Hartree approximation. That is the correlation effect. The Hartree approximation also effects

of

exchange,

which

in

our

important than the correlation. Let us assume effects

can

be approximately

(4.95)

This expression only includes exchange effects, which dominate the correlation effects. It is derived for small g and in the limit of small % (high electron density). The Hubbard (1958) approximation:

X¥(q)=—2e?/(q? +k2)

(4.96)

correctly reduces the screening by a factor of

3 in the limit of large q.

There are several prescriptions for how to interpolate between the low-g and the high-g limits of X(q). A useful expression is (Geldart and Vosko

1966):

XN(q)= —2ne?/(q? +8).

(4.97)

Heine and Abarenkov (1964) and Sham (1965) used the form (4.97) with

this choice does not have Slater’s low-g limit. There are numerous other

In the derivation of the screened form factor (4.69) and the screened energy~wavenumber characteristic, eq. (4.70), it was assumed that the

correlation

+X(q)

&? =k? +k?, where k, is the Thomas~Fermi parameter. We note that

5.3. Exchange and correlation in the screening

the

W*(q)+W*(q)=—

[ss

X8(q)=—3me?/2k2.

with

neglects

59

case

are

even

that the exchange

incorporated

more

and

in an effective

central potential W**(r) which is to be added to the screening potential

prescriptions

for the handling

of exchange

and

correlation,

see for

instance Geldart et al. (1972) and works quoted by them. Repeating the derivation of the screened form factor, one finds (cf.

Harrison 1966, Wallace 1972) that it retains its analytical form (4.69), if the Hartree dielectric function e°(q) is replaced by a new dielectric

function e(q) which includes exchange and correlation as

e(q)

1+

mk, | 4me? ee |e2

-e(¢)+ FQ e(@)-1]. 2

TE

4ke-@? |In|2kp +g |

+X(q [eake q e

+

(498)

60

The electron-phonon interaction

In the limit gk,

6. Pseudopotentials

we get

theory.

(4.99)

7 =

eq)=

which is the same limiting value as for the Hartree dielectric constant

°(q), irrespective of the choice for X(q). The energy-wavenumber characteristic, on the other hand, is more complicated when exchange and correlation is included. The final result

°

Bre Kk+q|w°|k>|?

F(q)=

Te

eq)

tli

x(a)

Ane?/q?

one often

encounters

a definition

which includes exchange and correlation 1s such a way that there 1s no

extra multiplicative factor [1 +X(q)/(47e?/q7)] ~' in eq. (4.100). With a

conventional notation (e.g., Jones and March

1-G(4)Q0(4)”

G(q)=-

(477/q")

to Harrison

(1966),

rd

esr)

which

each

xo-aee.

that it is adequate

electron

can

be

to use a one-particle

considered

as

moving

in some

picture in

superposition of potentials v(r—R,) centered around each ion R,,

Vir)= & o(r—-R,).

(4.104)

yal

The Schrédinger equation (4.103) has solutions ¥y(r—R,)

values

With this definition, F(q) with exchange and correlation included has the same form as eq. (4.70) if e°(q) is replaced by e*(q).

6. Pseudopotentials In our treatment of the diffraction model (§ 4) it was assumed that there

are real metallic systems for which the electrons can be approximately described by plane waves |k which are scattered by a weak (screened) potential W(r). The success of the free-electron-model for non-transition

metals infers that such an assumption is plausible, but it remains to give a rigorous theoretical derivation. This is done in the pseudopotential

effective

potential V(r) which results from (1) the Coulomb potential of the ion nuclei and (2) some average interaction with all the other electrons, including the core-electrons which are bound to the ions. V(r) is a

(4.101)

(4.102)

Heine

(4.103)

N

where Qo(q)=e°(q)— 1. The relation between G(q) and X(q) is:

X(q)

[-

1973) we write:

et(q)=1+ ___Q09)

is referred

Let the Schrédinger equation for an electron in a metal be:

Thus we assume

of the dielectric function

the reader

6.1. Definitions and some basic results

. (4.100)

Thus we can not merely replace e°(q) by e(g) according to eq. (4.98), but there is also an extra correction factor which contains X(q).

However,

details,

(1970), Cohen and Heine (1970) and Heine and Weaire (1970). Here we shall only outline the main features and concentrate on those aspects which are of importance for the electron-phonon interaction.

1s

297

For

61

£,, which

correspond

lattice

G.

to core-states

particular ion jy. There are also through the entire crystal and have is free-electron-like, we expect that Fourier expansion in plane waves ciprocal

vector

Such

and

with eigen-

are localized

at a

states ¥,(r) which can propagate energy eigenvalues E,. If the metal ¥,(r) can be well represented by a exp[i(k+gq)-r], with g=0 or a re-

a series

expansion,

however,

would

converge very slowly. Each propagating state Y, has to be orthogonal to

each bound core-state ¥¢, which requires that ¥,(r) oscillates rapidly in the region where V{(r) is localized. It would therefore be necessary to include terms with very high G-values. A better approach is to expand ¥,(r) in a set of basis functions x, (orthogonalized plane waves, OPW’s) which are already orthogonal to the core states. We define x,

as:

ther’

xa) = a - Sun f WIT er

(4.105)

6. Pseudopotentials

The electron-phonon interaction

62

It is easily verified that the orthogonality condition

(4.106)

PAV Pr=0 is satisfied. In conventional notation, eq. (4.105) is written

(4.107)

Xn=lk>— Dla>. a

It is convenient to introduce the projection operator

(4.108)

P=Z |a>. q

contributions to the effective potential acting on a conduction electron

We note that the eigenvalues E, of the pseudo-Hamiltonian are identi-

W(r)=V(r)+ Dd (E, —E,)|@> suffices to describe the pseudo wave function, it follows that:

Ck|W(r)|k> =k

PY V(r)

PRD.

the pseudopotential is a local potential. It is an important result, and its validity is taken for granted in the subsequent treatments of the electron -phonon interaction for free-electron-like metals.

6.3.

Scattering near zone boundaries

When

of a Brillouin zone

(ie. k=~G/2 where G is a reciprocal lattice vector) we must use two plane waves in the expansion of the pseudo wave function. Thus we make the Ansatz:

$4 = B(0, k)|k> +B(—G, k)|k-G). Let w(q)

be the Fourier

transform

(4.116)

of a local pseudopotential,

with

w(—q)=w*(q), eq. (4.68). Standard perturbation theory for two almost

degenerate states gives the energy eigenvalue:

—kY+k2

2

_ py

pe?

z,-~ 22m UG-w'+ k)2 e] gy leaf ed +|W(G)?} el _{(2m wa 4 ]

1/2

measurements

of the

the Fourier components of the pseudopotential. These components can for Pb and Ashcroft and Lawrence (1968) for In). The ratio of the expansion coefficients in eq. (4.116) is

BO,k)

___w(G)

B-G.4) 7 aR am

(4.119)

At the zone boundary,

Ro vee

V/y2.

=+1,

(4.120)

on the sign of w(G). Hence,

Consider the scattering between

the wave vector k lies close to the boundary

Hence,

band gaps for free-electron-like metals provide direct information about

(4.115)

This relation is not trivial. It was proven by Sham (1961) in a lengthy but algebraically straight forward calculation under the assumption that

is 2|w(G)|.

65

surface when

g=k—k’

| B(O, G/2)|=|B(—G, G/2)|=

two states k and k’ on

the Fermi

is close to a reciprocal lattice vector G. Figure

4.2 shows schematically the geometry in k space. Since both k and k’ lie

close to zone boundaries we must use at least a two-plane-wave approximation for the wave functions. With

$4 = B(0, k)|k> +B(—G, k)|k-G)

(4.121a)

on = B(0, kk’) + B(G, kk’ +G),

(4.121b)

and

it is easy to show that J(k, k’)=0 (J is defined in eq. (4.43)), and hence g(k, k’;q,4)=0,

(4.122)

in the limit when g—G. This result is expected since there can be no real

(4.117) with E,_, identical to E, except for a plus sign in front of the square root in eq. (4.117). At the zone boundary, k=G/2 and 2

(4.118a)

2

(4.118b)

#,-£(£)-imey, Fo-s=2-(£)' +116).

Fig. 4.2. Scattering between states k and k’ near the Brillouin zone boundaries.

66

The electron-phonon interaction

6. Pseudopotentials

scattering between two electronic states (in the same band) which differ in their quantum number k by a reciprocal lattice vector. Such states are

a form of x, which gives a rapidly converging series for the plane-wave

to be considered as identical. 6.4.

The screened form factor in the limit q=0

We assume a local isotropic pseudopotential. The limiting value w(g=0) can be written in a very simple form. The bare pseudopotential w? is

part ¢,. It can be hoped that the best, or ‘optimized’, choice is obtained when ¢,(7r) is in some sense smoothest. Cohen and Heine (1961) defined the smoothest wave function to be the one which minimizes

fi¥ex)a%r

of the orthogonalization condition that leads to a weak pseudopotential.

In appendix C it is shown that the Fourier transform of w°(r) for small q is only determined by the long range behaviour of w°(r). Thus,

we (q)= —42Ze?/q?Q,.

(4.123)

In eq. (4.99) we obtained the low-q limit of the dielectric function,

eq) =4me*k p/h. Hence, the screened form factor in the limit g=0 is (w, =w

ion) is:

w(q)= —7?Zh?/mk,.Q, = —FEp.

(4.124) refers to one

(4.125)

6.5. Non-uniqueness of the pseudopotential

@

(4.126)

were used in the rewriting of the Schrédinger equation in a form which

contains a pseudopotential. This rewriting is not unique, since one can add any combination of core wave functions to x, in eq. (4.126) and

repeat the transformation to a pseudo-Hamiltonian. It can be shown that all such choices of x, lead to the same energy eigenvalues E, for the band structure. Different choices of x, yield different pseudopoten-

tials and different plane-wave expansions of the pseudo wave function

64= DRG Kk+q>. q

(4.128)

Corresponding optimized pseudopotentials have been calculated and tabulated by Shaw (1968) for a number of simple metals and also in an improved version by Appapillai and Williams (1973) for 33 elements. We note that the non-uniqueness of the pseudopotential does not mean that the electron-phonon matrix element is arbitrary. The expansion coefficients B(g, k) in the plane-wave representation, eq. (4.127), of ¢, depend on the pseudopotential, and it is only when this plane-wave series is truncated that a non-uniqueness in the electron-phonon matrix

element arises. Mathematically it is due to the fact that we expand x, in an overcomplete set of basis functions with both plane waves and core functions. 6.6. Non-locality and the ‘on-Fermi-sphere’ approximation

The concepts of locality and non-locality have the following meaning: let V(r) be a local potential; when V(r) operates on say a wave function

In § 6.1 orthogonalized plane waves

Xe=lk>— Dla>

.

floeer ,

Coulombic, w? = — Ze?/ r, at large distances from an ion j with valency

Z. In the core region, the bare potential 1s less singular than 1/r because

67

(4.127)

In applications of the pseudopotential theory, it is of interest to choose

,(r), the effect is simply to multiply $,(7) by V(r). In the definition of

a pseudopotential, however, there are also terms When |a> is a function only of |q|, or |qg|/2kp. This defines the so called OPW form factor. For |g|>2k,, we take |k|=kp and g The non-locality enters both in the bare potential and in the screening. The total non-local screened form factor can be written (Heine and

Weaire

1970):

=

+a x

2_|

Ae? 2 +x(o|

Fourier transform of this potential is

Ss —2etalwik

[ejck, (27/2m)(k? —|k-+q|?)

(4.130)

.

_a w°(q) kyo wee

eq)

(4.131)

eq)

Consider the sum over k in eq. (4.130). If |g|>2k,,

nominator

has

its minimum

value when

The Ashcroft or empty core model: Ashcroft (1966, 1968a) has introduced a model potential which has gained much popularity because of its very simple analytical form. The bare potential is assumed to be Coulombic,

—Ze?/r, for r larger than some distance R,. Inside R,, usually called the empty-core radius, the potential is assumed to be zero (fig. 4.4). The

For a local pseudopotential, is a function of g only, and can be taken outside the summation in eq. (4.130). We then recover the result

k

Icke

(F?/2m)(k? —|k-+q|?)

(4.132)

Fig. 4.4. The unscreened Ashcroft and Heine-Abarenkov pseudopotentials.

70

7. Strong electron scattering and phase shift methods

The electron-phonon interaction

There is also a non-local

form

of the Heine-Abarenkov

which wi4(")= —Ze?/r for r>R, and

What) = — 42 — (Ag ~A2) Po — (A, — Aa) Pi

n

potential in

(4.135)

for r

xp f Vr )PL iil) jc)

This is a poor approximation when the scattering potential V(r) is strong. It is better to let ¥,(r) be the solution of the Schrédinger equation in a muffin tin potential, ie. a potential which is spherically symmetric

where the last integral is, by eq. (B.7),

Be [Ve (kr) “fara kr)] ar.

and equal

to

V(r) inside the muffin

tin radius r, and is a

constant for r>r,. For convenience, we take the constant potential to be zero and count the Fermi energy relative to this level. In an angular representation, the wave function ¥,(r) is

4n

&

DD

V2, (=0 m=-1

i expli(Z,))

X RATS Eg) Yin (Ons Pi) Vim Os $e)

(4.143)

(4.144)

an

In the preceding section we calculated the Bloch matrix element for a single scatterer when the electronic wave functions ¥, are plane waves.

alo

tira (kr )jrg2(kr) +... dr

with

in the single-scatterer model for a strong potential

Yr) =

j=0

(4.146)

and then perform the integration in spherical coordinates angular decomposition (eq. B.14) of (singr)/(qr).

momentum

Cka-V,V(0)|k) = 9-447 S (21+ 1)P,(c08 Oyu) F

B

(4.147)

Outside the muffin tin radius, the radial part of the wave function is

Ri(r; Ex) =cos(3,(E,)) ier) — sin(5,( E,))n,(«r), where

x= [2mE, /h?]'”,

(4.148)

(4.149)

14

The electron-phonon interaction

7. Strong electron scattering and phase shift methods

n,(«r) is a spherical Neumann function and 6,(E,) is a phase shift which depends on the energy eigenvalue E,. We note that in the limit of small phase shifts, the wave function (4.147) reduces to a plane wave,

Evans et al. 1973a, corrected in sign by Butler et al. 1976). Provided that

eq. (4.141).

I(k, k’) can now be calculated as was done for the free electron case in § 7.1. One finds that (factors exp(+id,) suppressed in eq. (4.151))

Kk, k)= ~4ni (kk): A"

D& (21+ 1)P,(cos Oy IR,

i=0

(4.150)

with

75

only contributions with /=0, 1 and 2 enter, re

Tint = Fpagy Sin 8 Ex) — 84 Fe) )-

(4.154)

An elementary but rather tedious proof is given by Evans et al. (1973). Briefly,

they

differentiated

the

differential

equation

for

rR,(r),

i.e.

essentially the radial Schrédinger equation, and multiplied the result by rR,,,(r). Likewise, the differential equation for rR,, ,(r) was differentiated and multiplied by rR,(r). On adding these two results, a contribu-

Bao [VOR Rist RierRiga iT

Q,

lo

TESTED

f+1S 42

t - rtdr wee

(4.151)

.

*

Since the potential is constant, V’(r)=0, outside the muffin tin radius it suffices in eq. (4.151) to integrate to r=r,. We now seek Se QE, 2 S68 21+ 1)(2U' +1 () NO(Ep) 320) 22! thers)

x fda, f

M0

P,(c08 8g4-) P:(COS By y-)(1 — COS Og.)

X sin yy 27 dO TPTP.

(4.152)

With the recurrence relation (B.9) for Legendre polynomials, eq. (4.152) simplifies to

amy pya{2my ©

(2) x

BE, Fe we

element

can

be written as an integral

where

the

integrand contains rR,, rR,,,, V(r) and E,. Elimination of V(r) and E, by the use of the radial Schrédinger equation, and some relations for spherical Bessel and Neumann functions, finally leads to the desired result (eq. 4.154). In the definition (4.44) of 0 and —1 if x

(5.17)

The sum rule (5.15) is trivially fulfilled in this case. It is not difficult to show the relation converse to eq. (5.14), i.e. the Green’s function G expressed in terms of the spectral function A. The

relation is

G(, (@,

°

p) f.

where again 6

p)=

Alo, p)

o

, dwt | (° Al’, P) dw’, f

is a positive infinitesimal.

(5.18)

94

Many-body interactions

2. Green's functions and simple concepts for many-body systems

As an example, consider a spectral function of the form

I,

Alw, p= $2.

Awop)

(5.19)

{o-[E(p)— Ee] + Tp

This is a Lorentzian shaped curve with half-width 2F, (fig. 5.2). If its

wo

width 21, is small, the spectral function represents an approximate eigenstate with quantum number p, excitation energy E,=E(p)-E¢

Fig. 5.3. A quasi-particle peak on an irregular background.

and lifetime t=h/2T,. The time dependent wave function of such an

the quantum number p. We now look for haviour when the electron self-energy M(w, erties: M(w, p) is isotropic and depends M(w, p)=M(w); Re M(w) is zero for w=0

approximate eigenstate can be written:

¥(r,1)=¥(r) exp(—i£,t/h) exp(—T,t/h). It

is meaningful

to

say

that

eq.

(5.20)

95

represents

(5.20) an

for small w; Im M(w)

approximate

(5.21)

A(w,

that the spectral function (5.19) exhausts the sum rule (5.15). A basic idea in many-body theory is that we can start from a system of non-interacting particles in quantum states p with energies E( p).

1

.

This

[Im M(w)|

p)= + ———Pn se

(op)

part

ate

-gR [eT

(5.23)

shape

of a

7 [ o—e( p)+Aw]?+ [Im M(w) ]*

These different states are characterized by spectral functions A(w, p) which are delta-functions in energy. As the many-body interactions are

faa

(5.22)

For small w we can write A(, p) as:

It is trivial to show, by direct integration or by the calculus of residues,

which may be quite irregular. However, it may be so that the final shape of A(w, p) 1s essentially that of a Lorentzian peak superimposed on some background (fig. 5.3). This peak constitutes a quasi-particle with

(this A is in

A= _ aM(w) 8 lono

can quantify this by requiring that

slowly turned on, the spectral function for a particular state p broadens, first into a more or less Lorentzian form and then into a final shape

1s small. We define a parameter A as

principle identical to the A in the mass enhancement factor (eq. 1.7)):

eigenstate only if the phase E,t/h goes through many multiples of 27 before the wave function has decayed by an appreciable amount. We

E,t/h=E, /20, >27.

such a quasi-particle bep) has the following proponly weakly on |p|, i.e. and varies linearly with

of the

spectral

function

has

the

approximate

quasi-particle state with the energy e(p)/(1+A), lifetime A(1+ A)/{2|lm M[e, /(1+A)]]} and strength 1/(1+A). The function e( p) is the excitation energy E( p)— Er in the absence of many-body self-energy corrections. In practice we can consider e( p) to be the excitation energy that results from an ordinary band-structure calculation. 2.4.

The spectral function in second quantization

Before we discuss the spectral function for real systems, it is instructive

Fig. 5.2. A Lorentzian shaped peak.

to give an alternative definition of A(w, p) in terms of creation and annihilation operators (appendix D). Let |0; NV) be the total ground state at zero temperature for an interacting electron~phonon system

96

2. Green's functions and simple concepts for many-body systems

Many-body interactions

with N

electrons, and let |n; N+1)

be the nth excited state of N+1

electrons fully interacting with the phonons. If there were no many-body interactions, these eigenstates would be properly symmetrized products of single-particle wave functions. The operator c;* creates a ‘bare’ electron of momentum gp, i.e. an electron in the single-particle (non-

interacting) state p. The operator c, annihilates a bare electron in the state p. We now let of operate on |0; N). The result is a state with N-+1 particles, but it is not an eigenstate of the interacting system since

the added electron was introduced in a bare eigenstate. We shall use the spectral function to describe this new state. To do so, we first expand cp |0; N> in |n; N+1). The identity relation

Dla; N+1> |?8(w—w*!)

+ 2 [[78(w+ w¥—!).

J Ale, p)do=

(5.24)

Similarly,

(5.25)

.

¢p|0; N> =D |n; N—1>. n

0; Megas N+ 165 N+ lez 0s ND + D0; Nlof |n; N-1>,

EN+)

EN =( EN!

EN*1)4EN) (E=aN* Nt !4+E !N

(5.27)

and

EN! Efl=(ES(Ell Em BF-1. (5.28) o!") !—B-

’*! and wy"! are positive. We assume that the Fermi The quantities w*! energy (strictly: the chemical potential) varies slowly with N,

EN =EN*!-EN~EN= EN EN,

(5.29)

The time evolution of the state (eq. 5.25) is

which by the anti-commutation relation for c eq. (5.24) leads to

(5.33) 2.5.

The momentum distribution function

In the interpretation of the spectral function, it is helpful to refer to a momentum distribution function

+ 6.42)

[Im M, , pT ReMa(wo, piT)= 2 [~ co PE Malm PT) 4, wry

00

(5.45)

102

3.3.

3. Evaluation of the electron self-energy

Many-body interactions

“Mop

Me for Einstein and Debye models

We now make some drastic approximations to get an analytic expres-

0.75

energies we "and an electron— phonon coupling a7 F(w)=(a?F)°5(w—

0.50

sion

for

M,.,-

Consider

an

Einstein

phonon

spectrum

with

phonon

w,). The integrations in eq. (5.40) are easily performed. One obtains

Re M,,()= —(a2F)In os Im M,,(0)=—m(a?F)°sgn(w)

if

|w|>we,

(5.47a)

Im M,,(w)=0

if

|wl|w,.

(5.50)

The corresponding spectral function is

d + Ato, p)=a{ 0one p)+*5# in| S*2e]) W-

WE

0

05

10

15

20

25

w

Fig. 5.5. The self-energy M,, in an Einstein phonon model, with we as the energy unit andA=0.5.

A typical value of A for many metals is A=0.5. With this choice, M,,(w) and an example of A(w, p) for the Einstein model are plotted in figs. 5.5 and 5.6. The average momentum occupation number has to be calculated numerically from A(@, p) by eqs. (5.35a) and (5.35b).

The result is shown in fig. 5.7. Engelsberg and Schrieffer (1963) were the first to calculate M,, along

the lines of this section. They also obtained analogous expressions for a Debye phonon spectrum. The qualitative changes in M,,(w, p), A(w, p)

and when the Einstein model is replaced by a Debye model are essentially the following: The weak logarithmic singularity in Re M,, is

removed but Re M,, is peaked when the electron energy is of the order of the Debye phonon energy. Im M,, is finite for all energies and increases as w’ up to the Debye energy. The sharp delta-function in Alan) 0.06

when

|w|w,.

|otos|]?

|e

| +|

nee [

:

-1

(5.52)

“S43

2-40

4

2

39

4

wy,

Fig. 5.6. The spectral function A(w, p) in an Einstein model for ¢, =0.lwp. The peak at w=, /(1+A)= 0.067, has the weight (area) #.

104

3. Evaluation of the electron self-energy

Many-body interactions

1 04

24

0

1

2

3

105 4

~ReM(w)T=0 K

unit’ 10'3 rad/sec

elpl/or,

Fig. 5.7. The momentum occupation number 1,

(5.58)

ify«l.

(5.59)

zeta function with {(3)=1.202.

given in fig. 5.10. The peak value is 1.20 for y=0.26.

A plot of G(y)

is

3. Evaluation of the electron self-energy

Many-body interactions

108

We now consider Im M.,. A general expression is

Gy) 12

{Im M,,(#;T)| meh f

4

109)

nan

dw’ a?F(w’)[1—f(w—w") +2n(o') t+f(w+w')].

0.8 0.6

positive function

T.,(; T)=|Im M,,(; T)].

020

02

04

06

08

10

12

(5.64)

At zero temperature:

y

Fig. 5.10. The function G(y) which gives the temperature dependence of A(T) for an Enmstein phonon model.

I,,(@; T=0)=ah J 0? F(w’) da’.

(5.65)

At the Fermi level (w=0),

For an Einstein model,

hop)

|

T,,(0; 7) =27h J “a2 F(w’)[ f(w’) +n(w') do’.

eo

are ae)"

ment measures the electronic properties in some Fermi-Dirac-smeared ‘window’ at the Fermi level (ch. 2 § 5). Allen (1971b, 1975) has considered the case when A(T) is weighted over electron states ‘en-

closed’ by (—Af(w)/dw). If a2F(w’)ow”, the ratio (5.60) for small T is

enhanced by a factor of 2 (Allen 1975). Further comments on ‘Fermi smearing’ are found in Allen (1975) and Goy and Castaing (1975). In terms of the coupling function g( p—k; A) we have:

dk

|e(p—k3A)|?

EM] tw(p-ery Sweet

aan

seamen [ite fieceaceae dQ,

k-

: a

0? F(0')=3Aw"*/wi, Then, |

for

w’- Likewise, since

w(k) and is ~2/w(k) for

|ko|D In[ -D~(Q)]. (5.100) P

discontinuity. We obtained an analogous discontinuity in caused by the electron-phonon interactions. It is this step m which defines the Fermi surface. 6.4.

shown

that the screened

factor w(q) has the long-wavelength limit (q¢—0)

OS

pseudopotential

=a:

z = fayVg

form

where p is the wave

2-2 Efay

Heine et al. (1966) showed that the

(5.98) (5.99)

is not observable since it is cancelled

The sum over ¢, =(2+ l)ivkg7 can be transformed into an integral in the complex w-plane,

The thermodynamic potential and the entropy

Using a method developed by Luttinger and Ward (1960) and| Eliashberg (1962) for the thermodynamic properties of a many-body

system, it can be shown (Grimvall 1969a) that the total thermodynamic 2

(or

free

energy)

for

interacting

(5.103)

S. = —(82,/8T),.

the electrical resistivity.

potential

(5.102)

,

et al. (1963, ch. III, § 19) for details. In addition to , and Q,, in eq. (5.100) there are terms which, to the order of \/(m/M), are either small or cancel (Grimvall 1969a). The separation of 2,,, in two parts referring to electrons and phonons is not unique since the ‘interaction energy’ can be said to belong to both the electrons and the phonons. The separation we have chosen is a natural one because of its similarity with the single-particle results. We obtain the entropy of the electrons by:

against the renormalization of the density of levels in properties such as

6.5.

¢, is an imaginary

where (q, A) 1s the usual labelling of a phonon and w,,=2mirk,T, m being an integer. The logarithms in eq. (5.100) have imaginary parts which depend on the phase of their complex arguments; see Abrikosov

Note that this result holds for all g and then, as a special case,

this renormalization

of an electron and

Bosons. Then,

correct inclusion of electron-phonon renormalization in eq. (5.97) is as

However,

vector

(5.101)

+1)ivkgT and n is an integer. An analogous summation rule holds for

corrections

[ (0) eeormataes= Tex [ (0) Janenomatze

functions

quantity which replaces the energy w in G(w, p). For Fermions, e, =(27

when one includes electron—electron renormalization effects, but this is

[ (4) Tesomaines™ [5 [9(4) Junresormaied

Green’s

summation over P means

It can be proven (Heine et al. 1966) that eq. (5.97) remains valid even

which only holds in the limit 0.

are electron and phonon

with the self energy due to electron—phonon interactions included. The

(5.97)

due to a rather fortuitous cancellation of the many-body

2

G(w, p) and D(w, p;)

The screened electron-phonon matrix element

In ch. 4 § 6.4. it was

123,

electrons

and

phonons

'

Ss. ~ oot | asbr rf of -

al

x [InGa(w, p;T)—InG,(o, p;T)] do.

(5.104)

124

Many-body interactions

6. Renormalization of electronic properties

Gp and Gy are so called ‘retarded’ and ‘advanced’ electron Green’s

6.6.

functions. They are defined as: G

and

.

ale PiT)

Ga(-, pT)

=

—w-e

w—«( p)— Re M,,(«, ps T)—ilm M,,(@, p;T) (5.105a)

first treatments starting from a quasi-particle picture are those of Eliashberg (1962) and Nakajima and Watabe (1963a). Later work by Prange and Kadanoff (1964) and by Grimvall (1969a) extends the analysis to all temperatures, i.e. also intermediate temperatures T~Op for which the quasi-particle description breaks down. We obtain the electronic heat capacity C, from

1 _____—______.,_ p)+ReM,,(o, p;T)—ilm M,,(@, p;T)” (5.105)

C.=T7(0S, /8T),.

with G, as the complex conjugate of Gp, G,=GR. The factor of two

multiplying the right hand side of eq. (5.104) comes from a summation

We first consider very low temperatures (T,

:

135

Let us consider low temperatures, so that only excitations close to the Fermi level take part in the charge transport and scattering processes. Then,

a quasi-particle description

is correct. The

relaxation

time, 7, in

eq. (5.137) is not quite the same as the lifetime, 7, for quasi-particles given in eq. (5.91). However, the only essential difference is the Fermi surface average of a geometrical factor, 1—cos@, which gives more weight to back-scattering over a large angle 8. One can argue that 1—cos@ is not affected by many-body interactions. Thus, 7 in eq. (5.137) 1s expected to be renormalized by the same factor of 1+A as the quasi-particle lifetime (eq. 5.91). If we also renormalize the electron mass in eq. (5.137), we see that the factors of 1+) in + and m cancel. Hence, o is unaffected by the renormalization. The same conclusion is reached if we calculate the resistivity p=1/o from the well-known Ziman formula (8.18). In a nearly-free-electron model, with a screened pseudopotential form factor, w(q), the resistivity can be written: 3hQ,

=

On Be’ MkivekgT = Sa x

2

42,

9

[a-4(g.A) P1w(@)I?

{exp[ ho(q, A)/kgT] —1} {1—exp[ -ho(g, d)/kgT]}

.

(5.138) The integral is taken for g vectors connecting two points on the Fermi

136

6. Renormalization of electrome properties

Many-body interactions

surface S;. The Fermi velocity vp and the form factor w(q) are both renormalized (§ 6.2 and 6.4), and it follows immediately that this renormalization cancels in the expression for p. The rather crude arguments presented here have been pursued in more detail by Nakajima and Watabe (1963a) Holstein (1964), Prange and Kadanoff (1964) and Heine et al. (1966). The same conclusion can be reached if one starts from the Kubo formula for the electrical transport (Mahan 1980). The

absence of renormalization effects persists even at temperatures T~O for which the quasi-particle picture is not valid.Finally, we should note that there is no renormalization in 7 or in e in eq. (5.137). The number

of quasi-particle states is the same as the number of bare-particle states,

and each quasi-particle carries the bare electron charge (Prange and Kadanoff 1964). 6.11

For small t, g(t)~—4 and we recover eq. (5.139). In eq. (5.141) we expect the plasma frequency, and also v, and 7 in the argument of g(r), to be renormalized.

electron gas Gp Ku, is

with

an

electronic

relaxation

time

7, and

in the

limit

q)=3—— m

o(o.g)=

ake

1

OT

ve = 1-i(w—g-v_)t

(5.140)

(0.9)

2

qe

Ter ioe 78 (sh):

where w, is the electronic plasma frequency, wo =4ne?/m, the function

a(t) = Shee = 1m

-21|.

in a non-trivial

First, m, has to be replaced by the ‘optical’ band mass and second the mass enhancement can not be expressed simply by the derivative

—9M,,(v)/dv

taken

at the Fermi

level »=0.

It is customary

to de-

termine an optical electron mass from experiment by the high frequency relation

(5.143)

A standard expression for the optical mass in a metal with a nearly spherical Fermi surface is

Mop =

3nh?

as

—.

[2/2m)'] f. asiv,£(4)|

(5.144)

We now include the electron-phonon many-body interactions. They are probed at energies differing by hw (cf. the conductivity formula (5.123) for the Azbel’—Kaner cyclotron resonance). Joyce and Richards (1970), cf. also Farnworth and Timusk (1976), performed an important experi-

ment in which the volume absorption of the radiation was measured for normal and superconducting lead. In the theoretical analysis we follow Allen (1971a) and define two parameters, \,, and 1,,, as

where E is a unit vector in the direction of the applied field and the integral is over the Fermi surface. The integration in eq. (5.140) yields

3

enter

Boltzmann equation to electronic transport phenomena at high frequencies. There are no simple renormalization factors 1+A of the type encountered in previous sections. In particular, we can not replace the electronic mass by the quantity m,(1+A) and this is for two reasons.

(5.139)

We expect that the renormalization effects cancel in ner/m, i.e. in the DC limit of o(w), but there should be a renormalization present also in the term iwr. A generalization of eq. (5.139) to include the g-dependence is (cf. Pippard 1966):

dS (vp-£)

these corrections

o(w) =ine?/womor-

o(w, q)=0(w) =ne*r/m(1—iwr).

ne

However,

way, as has been clarified by Holstein (1964) in his generalization of the

The high frequency electrical conductivity

A standard textbook result for the conductivity o(w,q) of an isotropic

137

Ay=—

[Re M(e+e)—ReMi(e)] / hw

(5.145)

and

6.141) and g(t) is

(5.142)

PF argleto) +150).

(5.146)

The label ir denotes that we are interested in conductivities at frequen-

cies in the infrared. This corresponds to hw~hwp. Mg and Ty, differ from the usual M,, and T,, by a factor which, under some simplifying conditions, reduces to the geometrical factor 1—cos@ that appears in

138

Many-body interactions

6. Renormalization of electronic properties

the standard expressions for electrical transport. We define generalized

electron-phonon

and

coupling functions a2 F(w) which become a2 F(w)

when the simple factor 1—cos@ is adequate. Neglecting anisotropy, we

Re Mg(e)=h{™ do' a Fo!)

(5.147) °

and

Te(e)=ah f lead F(w’).

(5.148)

In terms of A,,(e, w) and 1,,(e, @), the formal solution of the Boltzmann— Holstein equation (Holstein 1964) leads to (Allen 1971a):

o(w,q)= BEMEs)

(on-8)

xf # he

hw

q-¥_ +o[1+4A,,(e, 0) ]+i/7,(e, 0)

\,

(5.149) where the brackets denote a Fermi surface average. All effects of the ;

electron—phonon interaction are included in A;, and A,,, i.e. there is no

additional renormalization of N(E,) or vf in eq. (5.149). We now consider some approximations to eq. (5.149). When g=0 and

either w>wp or w0).

The real and imaginary parts of the gap function A(w) at T=0 are shown in fig. 6.1 for lead. The data are obtained from tunnelling experiments. At zero temperature, A(w) is a real quantity up to the gap edge Ay

given by

A(4o; T=0)=Ao. therefore

often

(6.19) | sees

the

gap

equations

written

with

the

lower

integration limit Ag instead of 0. It is easy to check that the Eliashberg equation (6.17) gives the normal state properties when A(w)=0. Then, Z,=Z, and eq. (6.17) is just M,,=[1—Z(w)]w which is given by eq.

(5.40). For weak coupling superconductors, Z,(w)=Z,(w). This is easy

to see by the following important energy range », Then, Z,(w) is the same as a?F(w’+Aq). Now, Ay is

3

42%@

Usually a good

dence on the magnitude of the wave vector in A(w) and in Zw),

One

2

Fig. 6.1. The energy gap in a BCS model (hatched area) and the real and imaginary parts of the gap function for lead, plotted versus the phonon Debye energy. Data from expenments by McMillan and Rowell (1969).

phonon interaction but still small enough to avoid the divergence of the

integral due to the contribution containing N(E,)U,.

1

argument: Since A(v)Ag. Z,(w) if the latter function is calculated from much less than typical phonon frequencies,

:

and we obtain the desired result. Bardeen and Stephen (1964) have estimated the difference between Z,(w) and Z,(w). Using an Einstein

phonon model (frequency w,) and at T=0, they obtain:

Re{2(«)-Z,(«)]~( 22)

n( 28),

(6.20)

when wT,. Thus we can linearize the gap equation (6.22) in terms of A(w; 7) near T=T, by the replacement

1966),

—AGT ) Me

tone

157

(6.22)

and

The repulsive Coulomb interaction

The repulsive Coulomb interaction U,, often called the ‘Coulomb pseudopotential’, has not yet been specified. A term of this type was introduced by Bogoliubov et al. (1958) and Morel and Anderson (1962). The instantaneous bare electron—electron interaction 1s 47e?/q?V. We expect the screening from the conduction electrons to reduce this interaction to an effective Coulomb potential U,(q, ); 4ne? ULaie)= Ta oy"

[1-Z(o ( ;7))]]o=

[” nl [o

»

x{[m(w)+A-»)][

dw’ 02 a2F(w’)go?

oto’ +v+id

+

+[n(w)+/0)][

Oman

ol

ow —y tid

1 ow +410 | J

TO

4ne”

(6.26)

nay

We now define U, as an average over all q vectors connecting two points

on the Fermi surface and N(E,)U,. One obtains:

wto’—r+id

+

where e(g,) is the wave vector and frequency dependent dielectric function. Morel and Anderson approximated e(q,w) by the static Thomas-Fermi expression cf. eq. (4.72), which gives UG

1

(6.25)

B= N(Ep)U,=

(623)

introduce

MEs) 2

V2Kz Jog?

a dimensionless

4ne” qdq=

+k?

KR I

8k2

parameter

ake +k3

KR

y=

.

(6.27)

158

Superconductivity

2. The Ehashberg gap equations

This treatment is not quite correct since we have kept the frequency

dependence (i.e. the retarded character) in the electron-phonon interac-

tion but have taken neous. The

the electron-phonon

characteristic

frequency

interaction to be instanta-

in e(q,w)

frequency w,. It is reasonable to assume

anisotropy parameter

is the electron plasma

&

wey +pln(, /o,)” Often, one encounters

w* defined

. with w, replaced

model.

The

electron-phonon

coupling

Von 1s written in an anisotropic form as

temperature one finds, to lowest order in a(k),

(6.29) Fermi

surface. At zero

A(k)=A(k)[1+a(k)]

(6.30)

where

B(k)=}14+| 1 - 3 |ca?(k)> fg

in eq.

(6.28), and sometimes w, is replaced by wp. For transition metals, it is reasonable to replace w, by half the width of the d-band. Thus there is some ambiguity in «*, both because different authors use different cut-off frequencies w, and w,, and because the detailed nature of the effective Coulomb repulsion is not very well known (Allen and Dynes 1975a). Fortunately, the various definitions of u* have only a weak influence on the magnitude of »* and on the solution of the gap equations. McMillan (1968a) assumed that u*=0.13 for all transition

(631)

NM Ep) Von

and denotes an average over the Fermi surface. relation for the transition temperature can be written: T, ==1.130, 1.130, on

metals and »* =0.10 for all simple metals. Allen and Dynes (1975a) took p»*=0.10 for all metals.

-

1

——— awant} MEDE

The

BCS

( 6.32 )

In a more precise theory (Kus 1978b) Ag. The effects of anisotropy are then washed out in A(w). Bennett (1965) has written down the Eliashberg

equations in their anisotropic form (see also Garland

BCS

where k and k’ specify points on a spherical

(6.28) by E;/h

the

Valk, k’)=V,[1+a(k)][1+a(k’)],

that U, is a constant up to

=, and zero for higher frequencies. Thus we should use w, as a cut- -off frequency in the Eliashberg equations instead of the lower. value @, ~10wp. With the use of w, the integral is cut off before the Coulomb term has been fully accounted for. One can approximately correct for this by changing p to a new effective interaction »* obtained as +=

within

159

1977). For a pure

metal, the anisotropy in A(w) is usually very small, although not always

negligible in its influence on measurable properties. Following Markowitz and Kadanoff (1963) and Clem (1966) we first investigate the

,

temperature. These results were obtained by pseudopotential calculations. Similar conclusions were reached by Truant and Carbotte (1974) for zinc. Experimental results for the gap anisotropy, reviewed by

Bostock and MacVicar (1977), show a disturbingly large disagreement between values obtained by different techniques. Theoretical calculations for strong coupling superconductors are also inconclusive re-

garding

the importance

and Chandrasekhar

of anisotropy

(Butler and Allen

1977, Gurvitch et al. 1977).

1976, Farrell

160 2.5.

3. The electron density of states and tunnelling in superconductors

Superconductivity

vast} Nw/NEp)

Strong energy dependence in the electron band structure

\

In our formulation of the gap equations, it has been assumed that there is no strong energy dependence in the electron density of states N(E)

near the energy in the Fermi Carbotte dence of structure E, hwy.

1.40

Fermi level, when compared with the rapid variation with the denominators of K ,(v, ). In this case, it suffices to take level value N(E,). Horsch and Rietschel (1977) and Lie and (1978) have made attempts to incorporate the energy depenM(E). It has been thought that T, can be enhanced if a in N(E) suitably coincides with energies of the order of However,

such an argument

may

1.05 1.00 095)

be misleading (Ho et al.

1978). The reason is that a strong electron—phonon interaction also means a short lifetime 7 for excited electrons. Since the single-particle energy bands are blurred by an amount f/r, a sharp structure in N(E) may be washed out in a superconductor with a high T..

3. The electron density of states and tunnelling in superconductors

w-

Fig. 6.2. The density of states N(w) in superconducting lead and the Eliashberg coupling function «?F(w); based on experimental data from McMillan and Rowell (1969). The smooth N(«w) curve is the BCS result. Schrieffer et al. (1963). They get (cf. also Schrieffer (1964)):

Before we consider solutions of the Eliashberg equations, it is necessary

to discuss some basic results for the density of states and the tunnelling

in superconducting junctions. Further details about tunnelling can be found, e.g., in the reviews by McMillan and Rowell (1969), Duke (1969) and Wolf (1978).

N(o)=

NAer) J ae(a)}im6(, 4)

==

l" |. Ee) Rel —_ [a ~le so)

where for brevity o=£—E,.

3.1. The density of states

According to the BCS theory, the density of states N,(£) in a superconductor with the gap Ay is

m(eyen(eeyRe| — EBs

[(EpE-4]

161

(633)

Subscripts s and n refer to the superconducting and the normal state. A generalization of this result, valid also for strong coupling superconduc-

tors where the quasi-particle picture breaks down, has been derived by

(6.34)

When A(w) is real and energy indepen-

dent, we recover the BCS result. To see the general structure of N,(w),

we make the approximate expansion

Nw)=M(ED{ 1+ [ReA(w)}*=[Im A(w)]}? |.

(6.35)

20?

At a typical phonon

frequency,

there is a strong increase in Im A(w)

related to the emission of phonons, and also a sharp decrease in Re A(w) (fig. 6.1). Both these effects cooperate to give a structure in N,(w) which

is superimposed on the smooth coupling

superconductor

such

BCS result N2S(w).

as lead,

this

structure

For a strong

changes

N,(w)

typically by 5%. In the immediate vicinity of the energy gap Ay = A(A,)

3. The electron density of states and tunnelling in superconductors

Superconductivity

162

‘normal’),

we get:

The

densities

of states N,(w)

and

N,°(w)

163

are supposed

to

vary slowly with w on the energy scale eV, so that we can take the Fermi

|o| 1 (a) 1/4 4 —lel = +5 aa )ons, gy Nw) “En 0 1/ 0a._, (2 =N: NBCS, w[+3(¥)

level values N,{(Ep) and N°(E,) at all energies. We get (dI/dV),,,, Nj(Ep)N2(Ep) independent of the temperature. If now one of the

(6.36)

metals (say b) is in a superconducting state and the other is in the normal state, we obtain the normalized conductance ratio o(V) at T=0 as

= (AI/V on _ No Er t+ eV)

= aT]ex

For lead, the enhancement factor over N,BS in eq. (6.36) is 1.025. 3.2. Conductivities of tunnelling junctions

(6.41)

NECEp)

The ratio (d//dV),,, /(dI/dV),,, can be obtained experimentally from

We consider the tunnelling current /(V) between

two metals, a and b,

with a bias potential V. The standard theory has been reviewed, e.g., by Schrieffer (1964) and Meservey and Schwartz (1969) and the reader is referred to them for details and references to the original work. Refinements of the theory (for instance Caroli et al. 1975, Feuchtwang 1976 * and Arnold 1978a) are of little importance for the applications discussed here. We describe the tunnelling system by the Hamiltonian (Cohen et

tunnelling

between

two

different

metals

and

at

two

temperatures. At the higher temperature, both metals are normal, and at the lower temperature one of them is superconducting. Since one then works at finite temperatures, there is a thermal smearing in o(V) due to the Fermi-Dirac factor in expression (6.39). Another method, with better resolution, is to tunnel between two superconducting metals. At T=0 one obtains

IgV ENS Ep)NO(Ep) f

al. 1962)

H=H*+H°+H',

measurements

(6.37)

eV Ay

‘Ay

where H* and H° refer to the metals a and b and H' is a coupling or tunnelling term. We let H' contain matrix elements 7(p, p’) for the tunnelling of an electron from a state p in the metal a to a state p’ in the

ne

et

[(w-evy—a3]'?

| (6.42)

ae. xe 2} [wt-a] 1/72

metal b. The Golden Rule gives a tunnelling current

— Ey). KV)= Ze = \T(p. p’)?[ (Ep) “SE, +eV)]8(E,

3.3.

(6.38)

We assume that |7(p, p’)| is a constant T and obtain:

HV) [~ N%w)N%(w+eV)[ fw) —f(wteV)]do. 0

Consider

ev 0

first

the superconducting metals are described by the strong coupling theory.

(6.39)

Hr) Be fda, [7 doy{ Leen, meR,

xe

N*( Ey +o—eV)N%( Ep to) dw. tunnelling

Nam (1967) derived an expression for the DC Josephson current in a symmetrical superconductor—insulator—superconductor junction when His result for the Josephson current J,(T) can be written:

At zero temperature we get:

uv ye f

The DC Josephson current

between

two

normal

(6.40) metals

(n denotes:

@,—@

A(o3T)

fx

fla =fe»)| @, +H

A(w,;T)

[wars 7)}'2) [fad -ar(w,s 79]? (6.43)

Superconductivity

164

R

4. Calculations of M(w;T) from the Eliashberg equations

is the

normal

state junction

resistance.

begaokar and Baratoff 1963), JPS(T) = mao(T) tn 2eR, Fulton and McCumber

SolT)

2kaT

The

BCS

limit is, (Am-

electron.

(1968) used data from

McMillan

The Rowell experiment (Rowell and McMillan 1966) is analogous to the Tomasch experiment but tunnelling is into a normal metal of thickness d backed by a superconductor. The density of states is

and Rowell

M(w)=N(E-){ 4 Re

(1969) for A(w) in eq. (6.43) and found that the Josephson current J,(0)

at T=0 was reduced from the BCS weak coupling value by ~20% for lead and by ~10% for tin. Lim et al. (1970) extended these calculations to finite temperatures for lead and obtained reasonable agreement with their own experiments for J,(7T). Ginsberg et al. (1976) have also considered strong coupling corrections.

3.4.

nounced

Then

he

+

4.1.

first for

aluminium

and

later

more

pro-

*

‘

interaction

in superconducting

metals

(McMillan

1969). The analysis will be discussed in some sections.

and

Rowell

1965,

detail in the following

The Tomasch effect

The BCS limit

We shall first check that the BCS limit is contained in our formulation of the Eliashberg gap equation (6.22). The BCS relation for the energy gap A(T) is

A(T) =N(Ep)FyA(T) f°? SE[1-24(2)],

for lead (Giaever et al. 1962), that J(V) has a fine-structure

when eV is of the order of a typical phonon frequency. These experiments were continued by Rowell et al. (1963) who turned them into the most important method for the investigation of the electron-phonon

3.5.

limit of a thin film we essentially recover the standard result (6.35).

the first to utilize tunnelling experiments to probe

observed,

(46

4, Calculations of A(«; 7) from the Eliashberg equations

the density of states N,(w) near the gap edge in a superconductor. He found a sharp increase in the tunnelling current as eV=Ag, and thena gradual decrease in /(V) with increasing V, as expected from the BCS theory.

so ls))}

where F is an oscillating function with period ~27. F(0)=1, i.e. in the

Tunnel junction experiments

Giaever (1960) was

there is a slight non-

linearity in a plot of w? versus n because the renormalization of vp is energy dependent.

!

(6.44)

I

For strong coupling superconductors,

165

(647)

where E=[e?+°(T)]'/7. We get eq. (6.47) from eq. (6.22) when the _

following assumptions are made: U, =0, Z,(w)=1 (the extreme weak coupling limit), A(w; 7) real and energy independent (w=0), the electron—phonon coupling as given by A(v) in eq. (6.76) approximated by X@) and identified with N( Ex Vy, kT much less than typical phonon energies and finally w, =wp. If we set M(Ep)V,, =A, we get from eq. (6.47) at zero temperature:

Ay = 2hwpexp(—1/d).

Tomasch (1965, 1966) noted that there were oscillations in the effective

,

(6.48)

tunnelling density of states, which were a function of the film thickness,

when the tunnelling was into a superconductor of thickness d backed by a normal metal. McMillan and Anderson (1966) osciilations should occur at energies w, given by

o, =[ 84 (nahe,g/d)?]'”,

proved

that

such

4.2.

Direct solutions of the gap equations

The gap equations provide values for, e.g., the transition temperature T,

(6.45)

where 7 is an integer and vf is the renormalized Fermi velocity of an

and the energy gap A, if we know a?F(w) and U, in the normal state. Morel and Anderson (1962) approximated a?F(w) by an Einstein peak

with the frequency w, and used a constant electron—electron interaction

166

4. Calculations of M(w;T) from the Ehashberg equations

Superconductivity

of the form (6.28). In the weak coupling limit (A, 4*.

_

1

(6.97)

:

Leavens (1977a) set out to clarify which moments are best to use in: approximate formulas for T,. The first moment of a2F(w)/w, ie. Aw) in our notation, is found to be the most important parameter for strong coupling superconductors. In the limit of extremely strong coupling, the: second moment dominates (i.e. Aw?) in our notation, in with eq. (6.90)). For weak coupling superconductors, the best parameter to describe the electron—phonon interaction is A. Leavens (1975, 1977a) calculated upper bounds to 7, expressed as moments of a?F(w)/w. We exemplify these relations by a strict result for the case »*=0. It reads:

kT. .

but the interac-

tion parameter A in eq. (6.99) is independent of M (an explicit factor of

M in A=n/(M{a*)>) is cancelled against M~' from w(g, A)). When u*=0, we therefore get a=} from eqs. (6.99) and (6.101). This is the normal isotope effect. However, a< 4} when the dependence on M in p* (enters via wp) is included (Bogoliubov et al. 1958, Morel and Anderson 1962). Using McMillan’s formula for T, and eq. (6.100) one easily obtains:

ve ay? awit p= EMCEE HOEY . 2 [A—p*(1+0.62A)]

(6.102)

In fig. 6.7 we plot a as a function of A for several u*. We expect to see a

(6.98)

0

The equality sign holds for an Einstein spectrum with a?F(w)=4))

and A=1.14, i.e. a physically very reasonable value

for A. Similar results were given by Leavens for other moments, also in the case u* >0. Although work such as that of McMillan, Bergmann and Rainer,

Leavens has helped to clarify the dependence of T, on the strength and shape of a?F(w), we are still quite far from a deeper understanding of

what gives a high transition temperature. For instance, what is the

fundamental relation (if any) between lattice instabilities (cf. Allen 1980b).

7, and

phonon

softening

or

Fig. 6.7. The exponent

0

0.5

100

a in the isotope effect, according

A to McMillan’s

7, relation.

Experimental data for « in Ga, Cd and Zn (Fassnacht and Dillinger 1970) fall within the

square box.

186

12. The thermal conductivity of superconductors

Superconductivity

trend towards lower a-values as the electron-phonon coupling gets. weaker. This is verified experimentally, although our account of the variation in a is too simplified for a detailed quantitative analysis (cf. Gladstone et al. (1969)). Leavens (1974) obtained « in a theoretical’ solution of the Eliashberg equations for some strong coupling metals. Since a—} is small in this case, it is not meaningful to make comparison with the rather uncertain experimental data. 10. The nuclear spin-lattice relaxation rate

In ch. 5 § 6.12 we noted that the nuclear spin-lattice relaxation rate R,

in a normal metal is correctly described by the weak coupling theory, i.e. there are no explicit mass enhancement factors 1+) and no correc« tion due to the failure of the quasi-particle picture. Often, one is

interested in the ratio R,/R,, for the spin-lattice relaxation rates in superconducting and in the normal state of a metal. Hebel and (1957) derived a BCS type result for R,/R,. Their theory had a logarithmic singularity. Fibich (1965) showed that the singularity could 1.0

be relieved by the quasi-particle damping phonons. Fibich obtained the relation

Rk, Rw 2A)

A 14 ppl

due

to thermally

4at/3

JA) In ———_].

(t/v2)""

187 excited

(6.103)

where A=A(T), f is the Fermi—Dirac distribution function and T=I(T) depends on the electron-phonon coupling and on the phonon spectrum. An approximate expression for (7) is

r(T)=

mA 15 1+A

(kaT)(hep),

(6.104)

where wp is the Debye frequency. Since I(T) depends on A and wp, not even a BCS model leads to a universal curve for R,/R,, as a function of T/T,. A typical form of R,/R,, is shown in fig. 6.8. Gap anisotropy, paramagnon effects etc. may also wash out the logarithmic singularity in the Hebel~Slichter theory, see Hasegawa (1977) and references therein. 11, The proximity effect

s/n

The proximity effect refers to the situation when a normal metal is backed up by a superconducting metal. Cooper pairs can leak into the normal metal from the superconductor. The theory of the proximity effect is complicated and relies on a number of approximations which are often of poorly known accuracy. The reader is referred, e.g., to McMillan (1968b) who gives a BCS-type theory for tunnelling through an oxide layer separating the normal and the superconducting metals, and to treatments of the ultrasonic attenuation by Kratzig (1971) and the diamagnetic response by Deutscher (1971). Deutscher and de Gennes

0.8 +

0.6 F 04

(1969) have reviewed many aspects of proximity effects. More recently, Arnold (1978b) has given a detailed account of the theory. See also

0.2

some experimental work referred to in ch. 10 § 6 and references in these

0

0.2

04

We

06

os

10

Fig. 6.8. The universal BCS curve for the ratio between the thermal conductivities x, and’

x, in the superconducting and the normal states. The corresponding ratio for the lattice relaxation rate is no universal function of 7/7,, but the curve shown is typical many metals, ¢.g., aluminium.

papers.

12, The thermal conductivity of superconductors It is probably superfluous to say that although a superconductor has an infinite electrical conductivity, it does not have an infinite thermal

CHAPTER 7

188

Superconductwwity

PHONONS IN METALS

conductivity. Nevertheless, the major part of the heat current just below T, is carried by the electrons (in not too impure systems). As the temperature is further lowered, the energy gap widens and fewer electrons are excited above the gap to be available for thermal transport. Then, the phonons dominate the thermal conductivity, but electrons still play a role because phonon-electron scattering yields a contribution to the damping of phonons. At very low temperatures, this mechanism is frozen out and the phonon mean free path is limited, e.g., by the finite size of the sample.

In experiments near T,, it is convenient to consider the ratio K¢/KS, between the electronic contributions to the thermal conductivities in the superconducting and in the normal state. There are two cases, depending on whether electron-impurity or electron—phonon scattering dominates. The first one is the easiest to treat theoretically, and it is also the usual condition for experiments on weak coupling superconductors.

In this case, the difference between x and «® arises because of the difference in the electronic excitation spectrum. The theory for x has been worked out by Geilikman (1958) and by Bardeen et al. (1959) within the framework of the BCS model. They get:

ve

K i

0

"eB (af/0E) de

f° Paf/aeyae 0

_

f° P°@f/2E)AE 7a

[OEM@s/aE ae

(6.105)

o

where as usual E=(e? + A*)'/?. This expression, plotted in fig, 6.8, is in

very good agreement with experiments for weak coupling superconduc-

tors. For metals like lead and mercury, with a high 7, the electron— phonon interaction is the dominating scattering mechanism near T, for pure samples. Experiments show that x{/«§ falls off much more rapidly than predicted by eq. (6.105) for decreasing temperature. The theoretical description is complicated, and a strong coupling theory is required. It has been worked out by Ambegaokar and Tewordt (1964) and applied to lead by Ambegaokar and Woo (1965) and Vashishta and Carbotte

(1972).

The quasi-particle character of the conduction electrons in states near

the Fermi level is strongly influenced by the electron-phonon interaction. There is no corresponding strong influence by the phonon-electron interaction on the quasi-particle character of the phonons. The most important feature is that the conduction electrons screen out the ‘bare’ longitudinal vibrations, thereby changing ionic plasma oscillations to

longitudinal sound waves, but the description of this effect does not necessarily invoke many-body theory. In § | of this chapter, we describe how phonon frequencies are calculated in real metals and how measured frequencies can be used to derive an empirical electron-phonon interaction. In § 2 we relate the phonon Green’s function to the Feynman diagrams that describe the screening effects. The damping of phonons is considered in § 3. In § 4 we review some properties of the dynamical structure factor, a quantity of particular interest for the subsequent chapter on electronic transport properties. 1. Calculations of phonon frequencies

1.1. Free-electron-like metals In ch. 3 it was shown that the phonon frequencies w(g, \) are obtained

as eigenvalues of a dynamical matrix with the elements D,,(q);

Dgla)= gE [(G+4)"(G+4)UsG+4)-G°6"U,16+4))]. (7.1) U, is the Fourier transform of a

local isotropic pair potential between

into

referring

the ions. For free-electron-like systems it is convenient

three

terms

(ch.

3 § 3.2)

189

to the

to divide D,,

long-range

ion—ion

CHAPTER 7

188

Superconductivity

PHONONS IN METALS

conductivity. Nevertheless, the major part of the heat current just below T, is carried by the electrons (in not too impure systems). As the temperature is further lowered, the energy gap widens and fewer electrons are excited above the gap to be available for thermal transport. Then, the phonons dominate the thermal conductivity, but electrons still

play a role because phonon-electron scattering yields a contribution to the damping of phonons. At very low temperatures, this mechanism is frozen out and the phonon mean free path is limited, e.g., by the finite

size of the sample.

In experiments near T,, it is convenient to consider the ratio K°/K< between the electronic contributions to the thermal conductivities in the superconducting and in the normal state. There are two cases, depending on whether electron—impurity or electron-phonon scattering

dominates. The first one is the easiest to treat theoretically, and it is also

the usual condition for experiments on weak coupling superconductors.

Fol,

In this case, the difference between « and x arises because of the difference in the electronic excitation spectrum. The theory for «° has been worked out by Geilikman (1958) and by Bardeen et al. (1959) within the framework of the BCS model. They get:

[PEOs/E)de 0

eS

f e?(0f/de)de 0

_

important feature is that the conduction electrons screen out the ‘bare’ longitudinal vibrations, thereby changing ionic plasma oscillations to longitudinal sound waves, but the description of this effect does not necessarily invoke many-body theory. In § 1 of this chapter, we describe how phonon frequencies are calculated in real metals and how measured frequencies can be used to derive an empirical electron-phonon interaction. In § 2 we relate the phonon Green’s function to the Feynman diagrams that describe the screening effects. The damping of phonons is considered in § 3. In § 4 we review some properties of the

dynamical

structure factor, a quantity of particular interest for the

subsequent chapter on electronic transport properties.

[PEMAs/aE)AE

7a

f E*(0f/8E)dE 0

6.105

(6.105)

where as usual E=(e? + A*)'/?. This expression, plotted in fig. 6.8, is in

very good agreement with experiments for weak coupling superconduc-

tors. For metals like lead and mercury, with a high 7,, the electron— phonon interaction is the dominating scattering mechanism near T, for

pure samples. Experiments show that «{/x¢ falls off much more rapidly than predicted by eq. (6.105) for decreasing temperature. The theoretical

description is complicated, and a strong coupling theory is required. It has been worked out by Ambegaokar and Tewordt (1964) and applied to lead by Ambegaokar and Woo (1965) and Vashishta and Carbotte

(1972).

The quasi-particle character of the conduction electrons in states near the Fermi level is strongly influenced by the electron-phonon interaction. There is no corresponding strong influence by the phonon-electron interaction on the quasi-particle character of the phonons. The most

1. Calculations of phonon frequencies 1.1. Free-electron-like metals

In ch. 3 it was shown that the phonon frequencies w(g, A) are obtained as eigenvalues of a dynamical matrix with the elements D,,(q); 1

Dap a= 3g

G

1G +4) —G°6°U,(|6+4))}[(G+4)"(G+9)U, (7.1)

U, is the Fourier transform of a local isotropic pair potential between the ions. For free-electron-like systems it is convenient to divide D,, into three terms (ch. 3 § 3.2) referring to the long-range ion-ion 189

Phonons in metals

Coulomb interaction, the intermediate-range 1on—electron term and the short-range ion—ion interaction due to overlapping core wave-functions.

For simple metals, one often neglects the core overlap. The direct ion—ion interaction is taken between point-like charges Ze immersed in a uniform negative background. It can be calculated by the standard Ewald method. The remaining ion—electron term is the major cause of the uncertainty in a theoretical calculation of the phonon frequencies. With a local and isotropic pseudopotential we can replace the ion— electron part of U, by 2F(\g+G|) (Wallace 1972), where F is the energy—wave number characteristic, eqs. (4.92) or (4.100). The problem 1s now reduced essentially to the finding of a representative F(|q+G)). This in turn contains two parts; a bare pseudopotential and its screening. Several such bare potentials and screening functions were presented in ch. 4. However, modifications of this approach may be necessary. We assumed a local potential, but non-local effects can be significant (see, e.g., Sham (1965), Vosko et al. (1965), Bortolani and Pizzichini (1969)). We also stopped at the second order perturbation term in the electron— phonon interaction, eq. (4.64), although higher-order terms are not always negligible (see, e.g., Brovman and Kagan (1967, 1974), Bertoni et al. (1974), Finnis (1974)). Usually, the stmple local and second-order theory reproduces the phonon frequencies to within 10-20% of the experimental values when it is based on some ‘first principles’ potential and to better than 5~10% when the theory contains adjustable parameters. Among early calculations, one notes those of Toya (1958) for Na, Sham (1965) for Na and Vosko et al. (1965) for Na, Al and Pb. The field has been reviewed, e.g., by Harrison (1966), Joshi and Rajagopal (1968), Wallace (1972) and Brovman and Kagan (1974). 1.2.

191

e.g., the shape of the Fermi surface or the temperature dependence of the electrical resistivity. Schneider and Stoll (1966a, b) were among the first to carry out such a program for the alkali metals. Cochran (1963) and Cowley et al. (1966) in an analogous way derived an empirical screened form factor for potassium. We shall exemplify the fitting of a model potential to a phonon spectrum by reference to a calculation by Wallace (1969) for aluminium. Wallace tried the bare pseudopotentials of Harrison, Heine and Abarenkov, and Ashcroft, (ch. 4 § 6.7) and used a Hubbard—Sham correction to the Hartree dielectric function. The short range core—core interaction was allowed for by a repulsive Born— Meyer type potential. The free parameters in the bare potentials and in the core-core interaction were varied to get the best fit between the calculated and the measured phonon frequencies. There was also a fit to the average squared frequency. It was found that the inclusion of the core repulsion never improved the fit, so this term was discarded. The Heine- Abarenkov potential gave the best fit when its well depth is zero, ie. when it comcides with the Ashcroft potential. An equally good fit was obtained with a particular choice of parameters in the Harrison potential. Figure 7.1 shows the theoretical dispersion curves for the best fit and also the experimental frequencies. Figure 7.2 shows the screened form factor for the best fit. It is interesting to note that the two form factors equally well reproduce the dispersion curves, although they differ significantly for q>2k,. However, it is not possible to cut off the

The fitting of a potential to the frequency spectrum

Many physical quantities, such as the high temperature electrical resistivity or the transition temperature of a superconductor, express in a

single number the result of a complicated averaging of the electron— phonon interaction. Thus they are less good tests, if one wants to ascertain the adequacy of a particular form for the electron-phonon interaction. The phonon frequencies are much

w(l0'8/sec)

190

1. Calculations of phonon frequencies

richer in physical infor-

mation. Many authors have therefore chosen a screened interaction with

some adjustable parameters which are varied until the calculated frequencies agree satisfactorily with the experimental data. The resulting electron-phonon interaction then 1s used to calculate other properties,

Fig. 7.1. Comparison of calculated and measured phonon frequencies of aluminum. (From Wallace, 1969, Phys. Rev. 187, 991; by permission.)

192

Phonons in metals

2. The phonon Green's function

papers

by, e.g., Sinha

and

Harmon

Varma and Weber (1977, 1979). xx FERMI SURFACE — ASHCROFT =~ HARRISON

+02 +03)

cs

+05 ay

1 4

-04

-

4

@ 1

2

ashe

10 3

1

14 4

|

Fig. 7.2. Screened form factor curves for two pseudopotentials whose parameters have been determined to give the best over-all fit to the measured phonon frequencies of aluminium. (From Wallace, 1969, Phys. Rev. 187, 991; by permission.)

form factor at q=2k, since that gives rise to large discontinuities in w(g,) (Wallace 1969). It is necessary to choose a form factor which falls off smoothly to zero for q>2k, (oscillations are permitted) but the detailed shape is less important in this range of q values. We note that w(q) was uniquely obtained for gT, in the latter case). A number of useful results can be derived from eqs. (7.22) and (7.23),

such as the relation

yq,e)\_=v 1 (x22)

(7.23)

201

x(q.) 772 >AN(Er)/3 aay

(7.24)

for the average relative line-width y(q,0)/w(q, 0). The average phonon

202

4. The dynamical structure factor

Phonons in metals

frequency

a

S(q,0)=(1/20N) [™ dtet"D Ce“ METRO,

1972, Allen and Dynes

1975b, Pickett and Allen 1977) to search for some general relation between A and y(g,o). One can not expect an exact formula since a knowledge of y(g,w) only specifies ImII(g,o) at the frequency w= @(qg,0) and the full frequency dependence II(w,g, 0) is required in the Kramers—Kronig relation. Because of the importance of the parameter A, it is of value to see if there is some approximate relation. Allen and Dynes (1975b) conjectured that \ can be written:

27(q,0)—07(4,0)

thermal average. For simplicity, we assume that there is only one kind

.

equilibrium positions in a solid and write:

S(g, 0) =S(g, w) + SG, 0) +---,

strength,

N,(£;).

We

can

also

compare

with the relation w* = 7, —w?, for simple metals, eq. (3.51), and with the analogous eq. (7.6).

Butler and Williams (1978) used the relations for y(g,o) discussed above to derive an approximate expression for that part W2 of the lattice thermal resistivity W? which is limited by the interaction between

phonons

and

electrons.

semi-empirical formula

For low temperatures,

WE =K(Op/TY(2,)'PAN,( Ex), where k is a constant and Q, is the atomic volume.

they obtained

a

(7.26)

(7.29)

where

/(MKa?)), if 27g, 0)— «wg, 0) is taken as some measure of the eleccoupling

of ions, with mass M. S(g,w) can be obtained directly from neutron scattering experiments, since the scattering cross section is proportional to S(q, w). This is of practical importance for liquids, but for solids one usually calculates S(g,w) in the ‘one-phonon’ approximation from the measured phonon dispersion curves.

We expand R,(t)=R} +u,(t) in the (small) displacements from the

where 2(g,c) is some kind of a ‘bare’ phonon frequency and f is supposed to be a constant for a given class of materials. Equation (7.25) is related to the McMillan—Hopfield formula, \=N,(E,) tron—phonon

(7.28)

Here, R,(¢) is the position of the jth atom at time #, and ¢ >; denotes a

(725)

«(g,0)

(7.27)

SQQ)=(1/N) Dee

telative damping due to phonon-electron interaction can dominate over

the real and

203

Sg, 0) =(1/N Je-?"'8(w)& exp[ —ig-(R? —R5)] *

as

=e ?™$(w) DeinFi

2 ?™'8(w) NB, g-

(7.30)

The lowest order contribution (in ,) gives SY, w):

Sq, 0) =(€-7"/2aN) > exp[ —ig-(R} —R>)]

x [> dre" q-u,)] [aru (t)])

(7.31)

The prefactor exp(— 2M’) is the Debye-Waller factor (see, e.g., Ashcroft

204

Phonons in metals

4. The dynamical structure factor

exp(—2M’)=exp{ —([q-u]°),}-

(7.32)

We note that S® in eq. (7.30) differs by a factor of N from the static

(7.33)

S'(q, 0) =S(q, w)—-S(q, 0).

S’@,) can be divided into two parts; one which is peaked at the phonon frequencies w= + w(g, d) and is referred to as the ‘one-phonon’ term, and another part which is more or less smoothly varying with w and is referred to as the ‘multi-phonon’ term. Corrections due to interference between one-phonon and multi-phonon parts have been considered by Ambegaokar et al. (1965). When the phonons have sharp frequencies w(qg, ), the one-phonon part can be written:

-

h

A(q,@)=

Le@,A)-a7

« D epheykgTya14 (9.0),

(7.34)

vq.)

+—__1@ 0)

[w+w(g.d)P+y7(g, A)

|.

(737)

In many scattering problems, it is convenient to introduce yet another

structure factor S(q) defined as

1 {asEgT Trexp[ 1mexp[ ea)

.

The structure factors obey a number of important relations, such as the

2

Placzek sum rule (Placzek 1952):

f- S(q, 0)wodw=(hq?/2M).

+n(q,A)8(w—o(g, r))}.

(7.36)

In this form, with Bose~Eninstein factors n(g, \) explicitly written out, we clearly see how the two parts of A(g,w) are related to the (stimu-

emission

and

the absorption

of phonons.

(7.40)

~0

x {[1+2(g,)]8(w+(q,d))

lated)

eq.

he

with

A(q, 0) =(h/2Mw)|[ 8(@+0(g, A)) +8(@— (9, d))].

|

2Mom \ [o—o(g,r)P+¥7g.A)

practice to make

these two somewhat inconsistent definitions. The reason is that, in different applications, we want both Sq) and S“(q, w) to be independent of N, the number of atoms in the crystal. We are primarily interested in the quantity

SO, w)=e

205

picture, with a finite phonon width y(g, w), we get

and Mermin (1976) for how this term is split off),

structure factor S defined in eq. (7.27). It is common

*

In a quasi-particle

The principle of detailed balancing (Baym

1964) implies that

S'(—q, —w)=exp(—hw/kgT)S'(q,w).

(7.41)

In the literature one finds expressions for S(q, ) which differ in the sign

convention for w. The two alternatives are related by eq. (7.41). Baym

(1964)

has

shown

how

S’g,w)

can

be

used

in the calculation

of

CHAPTER 8 206

.

electronic properties

Phonons in metals

such as the electrical resistivity and

THE ELECTRICAL CONDUCTIVITY OF METALS the electron

self-energy. The relation for the resistivity is given in eq. (8.19). S’q, ) contains the effects of anharmonicity, cf. eq. (7.37). In actual calculations, however, one usually takes the harmonic version (7.34) and neglects the Debye-Waller factor. Since multi-phonon corrections are of the same order of magnitude but opposite in sign to the Debye-Waller

correction (cf. ch. 8 § 5.5), it would not be correct to keep only one of

these terms. Shukla and Muller (1980) have analyzed in detail how various contributions to S(q, w), i.e. anharmonicity, Debye-Waller factors, multi-phonon scattering and interference effects, affect the electrical resistivity. For further general aspects on S(q, w), sum rules etc, see for instance Maradudin and Flinn (1963) and Ambegaokar et al. (1965). Electronic transport properties formulated in terms of Sg, w) are con-

sidered by Baym (1964), Greene and Kohn (1965) and Dynes and Carbotte (1968). Computer simulations of S(g, w), with full inclusion of anharmonicity, have been published by Jacucci and Klein (1977) for aluminium and by Glyde et al. (1977) for potassium. These calculations give results which are in good agreement with high temperature neutron scattering data, but they give broader peaks than predicted by the anharmonic one-phonon description, eq. (7.37).

We noted in ch. 5 that the electronic DC transport properties of a metal are described by the weak coupling formalism, i.e., without any manybody renormalization effects. The electron—phonon interaction only provides a scattering mechanism for the conduction electrons. In this chapter we shall deal with the electrical conductivity in some detail but, for lack of space, leave out other transport coefficients. They can be described in an analogous manner. The emphasis is on the steps leading to the actual computation of the conductivity, rather than on the formal transport theory. We first briefly review

the Boltzmann

equation and

the variational

method for its solution. In § 2 we present several versions of the well-known Ziman formula for the resistivity of simple metals. Other expressions for the resistivity are given in § 3 and § 4. In § 5 we discuss

the accuracy of a calculation based on Ziman’s formula, and in § 6 the

progress in numerical calculations of the resistivity during the latest decades is surveyed. In § 7 we comment on the unexplained phenomenon of resistivity ‘saturation’ at high temperatures. Two aspects on the resistivity of dilute alloys, the deviation from Matthiessen’s rule and the

generation of Joule heat, are treated in § 8. In § 9 we briefly refer to transport coefficients other than the electrical conductivity. The chapter closes with a short section on the technique of point-contact spectroscopy. 1. Some basic results for transport phenomena 1.1. The Boltzmann equation We shall review some standard results for the Boltzmann equation. It is assumed that an applied electric field E changes the Fermi-Dirac distribution function f, of an electron with wave vector k by an amount 207

208

Sy —fg.. Here fg is the Fermi-Dirac function in the absence of a The change in f, takes place only near the Fermi convenient to introduce a new function ®, by

level. Hence,

field.

it is

(8.1)

Ju Se = — PC Ofe /8 4).

(8.2)

3 on) Safe)» b= —(2e/V k

where the factor of two comes from the summation over spin directions. We shall assume that ®, is linear in the applied field,

(8.3)

=—e(v,-E)r(k).

Equation (8.3) defines yet another function 7(k) (=1(k, €)) which is a form of relaxation time to be discussed later. ®, is obtained by solving the linearized Boltzmann equation:

(ey ENA /8e4)=1/kaT)

(®, —®,)P(k, k’),

(8.4)

where (kK, kK) +P (kk)

(8.5)

refers to the scattering of electrons by phonons

(P‘?), by impurities

(P"™?) and by other mechanisms (P*) such as electron—electron scattering. In this section we shall be concerned with P*?. In a system having cubic symmetry, the conductivity tensor is a scalar

o, with

j=oE. One

(8.7) can be rewritten as

system. eae

rie =f wkT

o=

de EH a(k, F(QLI-F()]ofl

|

(88)

The factor n/m in front of the integral is just another way of writing

The electric current density is

P(k, kK’) =P?(k, kh’) +P

209

1. Some basic results for transport phenomena

The electrical conductivuy of metals

(8.6)

can therefore average over all directions of E and get, from eqs.

(8.2) and (8.3),

om — (262/3V) Deg eur kV OE /de,).

(8.7)

We shall frequently refer to free-electron-like systems for which the Fermi surface is approximately spherical in shape, but still with allowance for anisotropy. Let kf, vf and S° be the Fermi wave vector, the Fermi velocity and the area of the Fermi surface for a free-electron

S°vg/(127h).

When

71(k, e) is a constant 7, and for a free-electron

system, we recover the familiar formula

(8.9)

o=ne?r/m.

We shall now write the Boltzmann equation in the form of a Fredholm integral equation. Let E be a unit vector along the z-axes so that vf{k)=y" -E. Equation (8.4) can be expressed as

1

(af). 1 sf, 2k) oe)

apa)

0(k) HH) Iu,

k 5D

,

6.10)

For a material with cubic symmetry, o is independent of the direction E. Hence we can take £ and k to be parallel. If r(k’)/1(k) does not depend strongly on & and k’, and if (k’) and k’ are parallel, we get from eq. (8.10) 1

(afe\__

abl r)-

ar"

_

COS O,4)P(k, k’).

(8.11)

Without a term cos, €q. (8.11) yields the relaxation time of a quasi-particle (cf. (4.48)). Mahan (1966) has shown how the part con-

taining cos ,,- arises from an infinite sum of vertex corrections to the Feynman diagram expression for the self-energy of a quasi-particle.

1.2. The variational method Until very recently, the Boltzmann equation has been solved almost exclusively by the lowest order variational expression. The variational method was outlined by Kohler (1948, 1949) and Sondheimer (1950) and exploited in detail by Ziman (1960). Briefly, the idea is as follows: Suppose that an integral equation can be written in the form

X=Po,

(8.12)

where X is independent of © and P is an integral operator satisfying

210

The electrical conductivity of metals

2. Ziman’s resistivity formula

conditions which are not crucial here. Consider all functions which obey the relation

@p.

(8.38)

In the limit of high and low temperatures, the scattering is effectively elastic and we can obtain the resistivity from the scattering rate as given by the Golden Rule, without reference to the variational method. This

Fig. 8.2. The coupling function a2 F(w)=Cw* which yields the Bloch-Griineisen law, compared with the calculated coupling function (Hayman and Carbotte 1972) for two crystallographic directions in sodium.

The electrical conductivity of metals

5. How accurate 1s Ziman’s formula?

coupling function a2 F(w) we have approximated by Cw‘, not only the scattering accounted for by Bloch’s original assumptions. In this way, many details left out by Bloch can in fact be incorporated in his formula without changing the temperature dependence of pgg(T). It is easy to see that the power law spectrum a2 F(w)=Cw" gives the low

temperatures (T>©p/10). Experience shows that eq. (8.42) is a reasonable approximation, and one can even take (1—cos@> = 1 for many

20

temperature limit pgg(T)«7"*! and the high temperature limit pgg(T) «T. For some metals, one gets a better fit to experimental data with

formula p§2(T), with n as an additional fitting parameter. Such expres-

sions may serve the purpose of convenient interpolation formulas for that they do not add much

metals. At low temperatures, eq. (8.42) fails because «”F(w)®,.

However,

it has been

theories, we

observed

by Mooij

(1973), Filippow (1973), Fisk and Lawson (1973) and others that this temperature dependence is not at all obeyed by many highly resistive metals and alloys. It is almost a rule that dp(T)/d7 14 /dInV)~(dIn¢w? > /dInV)=~ 26, we obtain from eq.

din®,

~({0&)-2 dinV

dinV

= (FRE) #22000). din K,

(3)

(9.9):

din 7,

Tiny 716 +10. u*){ S22 +276}.

Here, yg(A) and y¢(p) are generalized Griineisen parameters relevant

for the average phonon frequencies as they are weighted in ) and in p.

A general

expansion is

expression

for the coefficient

6)

1

a, )

“=a e sey

2.1.

r

(0.7) :

n=( anh) * rex( ain’)

this appears

(Munn

to be

1969). Often,

less reliable

than

the

thermal

the volume dependence

expansion

of K, and

method

K, is much

weaker than that of ©, and @,. If we also neglect the difference in the

weighting

Yo(P)~Ye

246:

of the phonons,

we

can

make

the approximation

Yo(A)=

and replace the right-hand sides of eqs. (9.4) and (9.5) by

If the transition temperature 7, of a superconductor is derived from a

McMillan-type relation, we get

(a (“ani e )a 0-0) 9 Gir)

2.2.

;

under

pressure

provide

Theoretical calculations for simple metals

There are three different reasons for a volume dependence in A or p. One is the volume dependence of the phonon frequencies. The other two, which appear together in (dIn K,/dInV) or (dIn K,/dInV), refer to changes in the screened form factor and to changes in the Fermi surface properties (the shape of the Fermi surface, v,(k) and N(E,)). The phonon frequencies were discussed in § 1. We should only add, that the parameter y,(0) obtained from the entropy (ch. 3 § 2.1) 1s usually a better choice than the ordinary thermodynamic Griineisen parameter yg, as an approximation also

(9.9)

superconductors

(Svistunov et al. 1979). Galkin et al. (1970) published tunnelling data for Pb and TI without the evaluation of a?F(w). Svistunov and Tarenkov (1977) determined dA/d P for Pb from the Tomasch effect.

The term y,, is just the ordinary Griineisen parameter yg, eq. (3.35). At very low temperatures, C, >C,,, and the thermal expansion is dominated dependence of d (cf. Collins and White 1964). For superconductors, one can also obtain y, from the pressure dependence of the critical field, but

on

Pb—In (Hansen et al. 1972, 1973, Wright and Franck 1977) and Pb-Bi

08)

by the electronic part in eq. (9.6). Thus, low temperature thermal expansion data can be used to get some information about the volume

experiments

direct information about the volume dependence of a?F(w). Such experiments have been performed on Pb (Franck et al. 1969, Zavaritskii et al. 1971, Svistunov et al. 1977b), Sn and In (Zavaritskii et al. 1971),

yo(1+4A)T= S(T), eq. (5.110), we get A_(dind

Tunnelling data

Tunnelling

with an analogous expression for the phonon part (‘ph’). From C(T)=

diny,

(9.10)

2. Applications to real metals

where C, is the electronic heat capacity and

_1(

din

For many transition metals, (dIny/dInV) is larger than 2yg and negative, which results in d7,/dV and 1/N,(E,). Butler (1977) calcu-

lated and compared (J?) and N,(E,) for 11 4d transition metals. , Nv Ee) l> Mao >p

Myo >y

11.3

>

ar)

Ay =0.78 gives \=0.82 for the alloy in question. As another example,

where the parameters refer to the sites for R and Y atoms.

obtained A=1.3 for Nbo75Zro2s (where T, has its maximum), \=0.93 for Nb and A~0.4 for Zr. In this case, the increase in A could be traced to a lower , while N,(E,)1 and r,B