Sound propagation in an anharmonic metal

ANNALS

OF

PHYSICS:

71, 25-92

Sound

(1972)

Propagation

in an Anharmonic

II. The Collision-Dominated

Metal.

Regime*

ROLF SANDSTR~M Department of Theoretical Physics, Royal Institute of Technology, S-100 44 Stockholm 70, Sweden Received April 26, 1971

The generalized transport equations derived in Z for the coupled electron-phonon system are solved in the collision-dominated regime. The solution is obtained without making any relaxation-time-approximation. The sound velocity and the ultrasonic attenuation are calculated with the help of the equation of motion for the lattice deformation. In particular, we derive the lowest order anharmonic contribution to the sound velocity. For the attenuation, the contributions due to the viscous properties of the phonon gas and the electron gas are considered. We also take into account the thermoelastic damping.

CONTENTS 1. 2. 3. 3.a 3.b 3.c 3.d 3.e 3.f 4. 4.a 4.b 4.c 4.d 4.e

Introduction ............................ Solution of the generalized transport equations .............. Transport coefficients ........................ Heat conductivity .......................... Longitudinal electrical conductivity .................. Longitudinal dielectric function .................... Transverse electric conductivity .................... Transverse dielectric function ..................... The local temperature deviation .................... Evaluation of the relaxation times ................... TherelaxationtimesforSE = Q = 0. ................. R-dependent relaxation times ..................... Q-dependent relaxation times ..................... Matrix elements of R-W x ...................... Range of validity of the assumption to express R-l in terms of a few distribution functions ............................ 5. Sound waves in the collision-dominated regime ............. 5.a Longitudinal dielectric function .................... 5.b The local temperature deviation ....................

* Research sponsored by the Swedish Natural Science Research Council.

25 0 1972 by Academic Press, Inc.

26 28 39 42 43 44 47 47 48 50 52 56 57 58 60 61 62 63

26

SANDSTRijM

5.c Longitudinalsoundvelocities..................... 5.d Attenuationof longitudinalsoundwaves................ 5.e Attenuationof transversesoundwaves................. 6. Conclusions ............................ Appendix.Equationof motionfor the latticedeformation...........

64 69 74 79 80

1. INTRODUCTION In this paper we will study the propagation of sound waves in metals by solving the generalized transport equations derived in Ref. [l] (henceforth referred to as I). The background of the problem is described in detail in I and will only be briefly commented on here. The transport equations for the electron-phonon system were derived from first principles using the Dyson equations for the phonon and the electron propagators. Many-body corrections are thus fully included. These transport equations are in general quite complicated. However, when the wavevector of the sound wave Q is much smaller than the Debye wavevector qD , the transport equations take a simplified form which is quite close to the semiclassical transport equations. In this paper we will solve these equations in the collision-dominated regime, i.e., when the wavelength of the sound wave is much longer than the mean free paths of both phonons and electrons. The casefor shorter wavelengths will be investigated in a forthcoming paper. The velocity and the attenuation of a sound wave can be obtained from the real and imaginary parts of the phonon self-energy. Rather than calculate the phonon self-energy by summinginfinite setsof diagrams, we will, according to the discussion in Section 1.2, calculate it by using the transport equations. How the phonon self-energy can be obtained from the transport equations is indicated diagrammatically in the Fig. I.lb, 1.3, 1.16, and 1.17. Equivalently it is possible to use the equation of motion for the lattice deformation, which we will do. The equation of motion is derived in an appendix. In this expression for the phonon self-energy, there appear the electron charge density and the local temperature variation (T(U)) induced by the sound wave. These quantities must be calculated by solving the transport equations. The generalized transport equations will be solved by a method due to Guyer and Krumhansl [2] which has also been used by Niklasson and SjSlander [3-51. In this method, the collision operator is inverted in a spacewhere its eigenvectors with zero eigenvalues have been projected away. This inverse is called R-l. In Section 2 the solution of the transport equations is expressedin terms of matrix elementsof R-l. The frequency and wavevector dependenceof the transport coefficients is needed when calculating the sound attenuation. Thus in Section 3 generalized formulas

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Thus JE = J,” + JEW= +

(2.32)

* (fi(Q~)>,

where V is the matrix QJW) 4l”d w 0

v=

() (2.33) E’(k) i ’

We notice that according to Eqs. (2.13) and (2.19), G can be written (2.34)

G=Q-V.Q. The local temperature deviations from equilibrium systems TD and Te have the following form

of the phonon and electron

= (w 1 a> T/(w ( w),

Tp = pEP/Cv”

(2.35)

p = p$lC”e = (w I r> T/(w I w).

The specific heat C, , the energy density pE and the local temperature for the whole system are now easily obtained as

deviation T (2.36)

C” = C”P + eve = + (% I %), PE

=

PEP

+

fEe

=

(%

(2.37)

/ y>>

(2.38) Here Eqs. (2.20) and (2.24) have been used. (1 j 1) is the electron density of states N(EF) at the Fermi surface a%)

=

(1

I 1)

=


I 732)

=

2

j

&

(-

(2.39)

*,.

The electron charge density and current can be written (1.10.53):

pp -e = (1 I r> =

(~2

of

I v> = 2 j &

( - aE,

1 y(k

(2.40)

Q@,

.

Je = (E’(k) I Y) = --e

(932

I V I T> = 2 j &

(- p)

E’(k) r& Q.n).

(2.41)

We denote the local change in the Fermi energy by ,Z F;(Qsr)

=

P,/(-~N~F>>

=

(v2

I v>K-e

<

(~2

I ~32)).

(2.42)

28

SANDSTR&i

However, estimating these relaxation times one finds that they have a similar temperature dependence. Further a recent detailed numerical calculation for potassium has verified this fact [9]. In this paper, the results of Steinberg [7] and Bhatia and Moore [8] will be generalized to an anisotropic system. For such a system, the longitudinal and transverse electron attenuations will no longer be proportional to the same relaxation time. The neglect of anisotropy may be one reason why Rice and Sham [9] got a slight disagreement with the experimental results of Natale and Rudnick [lo]. We will also discuss how the sound velocities and the attenuation depend on the correlation effects in the electron polarization.

2. SOLUTION OF THE GENERALIZED

TRANSPORT

EQUATIONS

To find the charge density p,(QsZ) and the local temperature variation T(QsZ) induced by a sound wave (u(Rt)) = (u(QS2)) eiQ*Re-iJ)t we are going to solve the generalized transport Eqs. (1.8.1-2) for the electron-phonon system. In I, it was found that these equations could be simplified to the form below if IR < wn and Q < qD , k, . For larger frequencies the solution has already been obtained in I with the aid of perturbations methods.

= -jQ Aj(qw) ( - ?$$-) W

zzz- &igkd) (

4qj)

Q * y”(q) * WQs2)h

(2.1)

(1.10.49)

E’(k) * eE&QJ$(-i).

(2.2)

(1.10.50)

In (2.1) and (2.2) the index 1 on&(k, Qsz> and rr(k, QSZ) has been dropped because f (k, QL& and y(k, QSZ) will not be used in this paper. Otherwise the notation is unchanged. The harmonic approximation will be used for the phonon spectral density function A,(qw) which thus consists of two delta functions [see Eq. (1.2.33)]. Thus the o-dependence of (2.1) can be removed but we will keep it for formal reasons. n(w) andf(w) are the equilibrium number densities n(w) = l/(&J - 1);

f(w) = l/(eE” + 1).

(2.2)

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29

The nonequilibrium number densities nj(qw, QL?) andf(k, QQ) are related to the phonon and electron distribution functions oLj(qw,QD) and y(k, Qs2) in the following way (1.6.13; 1.10.22; 1.10.48; 1.10.51):

w(qj) is the harmonic energy (1.10.14):

phonon frequency and E(k) is the electron quasiparticle E(k) = 44 + ~$8 2% E(k)),

(2.4)

where E(k) = k2/2m - t.~ is the one-particle energy and L&w) is electron selfenergy.
+ iQA(Q@.

(2.5)

A(QSZ) is the vector potential, and U,n(QG) is an effective potential (1.7.1 l), which includes, besides the Maxwellian potential U(Qs2), also a correlation potential

Uc,,,(QQ) [cf. (I.74 (2.6)

p,(QSZ) is the nonequilibrium electron charge density and G(Q) is a function which is related to the pair correlation function in the electron gas (1.7.5). In the collision operator the electron-phonon, the phonon-phonon and the electron-electron interactions are explicitly taken into account. The corresponding collision terms are given by the following expressions [(1.6.18), (I.10.25-27), (1.10.31-34), (1.10.48)]:

30

SANDSTRGM

dk3 dk4 z(k3) z(kg) d(k, . 19 13.(k3 , k,)12 ?. -f(EkJ(l Adqw) A4 w

QQ)

* /%4w

-

y(k,

,

- k4 - q) S(&, -

-fb%J)(l

Q-f-4 +

w)

+ n(w))

QQ> +

y&4,

Ekz -

(k,

-

k4)

.

(ti(QQ))),

L?j"" . cljh, 452)

. Aj3;;w3) Aj4z:w4) +p

= (2s + 1)z(k)j & . I(& . (1 * BMk

-

(23)

(1 + n(w))(l + n(w3))(1 + n(w4))

&

% 44 z&J z@s)

/ 0,’ 1W4>124k, - k, + k, - k> &%, - Ek, + Ek, - Ed f(E,Ml

QQ)

+

f@k,)>fcEk,).f(Ek,) y&3

9

QQ)

-

r&,

3

QS2) -

rOr4

3

QQ):.

(2.10)

The many-body corrections that appear in (2.1) and (2.2) have been summarized in I in the following way: (a) The one-particle energies are replaced by the quasiparticle energies; (b) Explicit z(k) factors [Eq. (I.10.15)] appear in the collision integrals; (c) Vertex corrections are included in the scattering matrix elements; (d) The Maxwellian electric field E(QSZ) is replaced by an effective field Eerr(QSZ) which contains a correction due to the correlation effects between the electrons. Bandstructure effects are not taken into account in (2.1) and (2.2). Before we continue discussing the properties of the transport equations, a simplifying notation involving bras and kets will be introduced. For the electron variables, I fi> = .h , etc.,

(.AIf,) = (2s + 1)j +

2 f&w) (- $$$) .L*(kw)hQw>> (2.11)

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31

II

where (2s + 1) is the spin factor and &km)

= 27r S(w -

E(k)).

277Aj(qw) (-

*)

For the phonon variables,

(%

I G

=

F

I,,,

&F

g

+*(qjw)

n,(qjw).

(2.12)

In (2.11) and (2.12) the w integration can again be immediately performed. A compact notation for the transport Eqs. (2.1) and (2.2) will be used. For the flow terms, Gj” 1 a) = (fi

-

“(‘j)

“?I

* Q, 1 aj(qw,

QQ)),

(2.13) For the collision terms, Aj(qw) (- *,

iry

f / q) = ioEpj”“(Qs2)- oli(qw, QL?),

Ai

illj””

* I4

Ai

(- *) (-

*)

i&”

= =XQQl

+ I Y) = =f’“(QQ)

* (0, 4w,

Q.n>>,

- (y(k, Qsz>, (9,

(2.14) - ~W%> iAep . I o$ = icY”“(QQ) . (0, aj(qw, QD)), ( aEk 1 (

- -a! (EIJ iAee . 1y) = iPp(QS2) . (y(k, QL?), 0), ah 1

The thick dot indicates convolution. The phonon-phonon collision term and electron-electron collision term are written as I’fp and P. Notice that the electronphonon collision term (denoted by (1) in both the phonon and electron equation has been split up into two parts, each operating either on the phonon system or the electron system. The first p or e index indicates the type of transport equation and the second the type of function it is operating on.

32

SANDSTRijM

We write the force terms as

Ai(qw)(- +) = A,(qw)

I W * (-

w)

+

Q . yjj(q) . (ia( (2.15)

( - w)

=

I Fe) * Eeri(QJ-2) (-

af@A

-)

(4)

eE’Or)

* &dQQ),

i.e.,

I W = w”(nj) Q * VW/w, 1Fe) = (-i)

eE’(k).

(2.16)

Now, using (2.13), (2.14) and (2.16), we can rewrite (2.1) and (2.2) as G,” 1CX~)+ (iI’;’

+ i/l:“)

* 1aj) + iAf’” * 17 - k * (ti)) = I F”) . (ti),

(2.17)

G” 1r) + iAe’ * 1aj) + (iTee + iA”“) . 1y - k * (ii)) = I F”) * Eeff s In (2.17), iP I r) has been replaced by iP 1y - k * (ti)) which is possible since the electron-electron collision term (2.10) conserves momentum. The fact that it is y - k * (ii) and not y which appears in the collision operator is due to the collision drag effect. It makes the electrons scatter towards a local equilibrium following the sound wave. The transport Eqs. (2.17) may be written in terms of a matrix formalism in the following way:

where the matrices G and r are given by (2.19) G and r operate according to the ordinary rules for matrix multiplication. easily checked that the full collision operator r is hermitian. Thus

where

It is

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33

11

and

Due to the presence of the energy delta functions (2. lo), r satisfies

in the collision integrals (2.7)-

It is also directly seen that

whereI yz>= (,:,).

(2.21)

We will see later that the conservation of energy and charge, follows from (2.20) and (2.21). If the momentum too were conserved the following relation would have been satisfied (2.22) but due to the presence of the umklapp processes this is not the case. Next we turn to the problem of how to solve Eq. (2.18). On moving the collision drag term to the right hand side, Eq. (2.18) takes the following form: G I y,i + jr I y> = I @bxti,

(2.23)

where

and (2.25) Since r has eigenvectors with eigenvalues equal to zero, it has no inverse. Instead of inverting the operator G + ir directly which thus has the disadvantage of being singular for small Q and Q, we are going to solve Eq. (2.23) by a method due to Guyer and Krumhansl [2]. This method has also been used by Niklasson and Sjiilander [3-51. The idea of this procedure is to invert the collision operator in a space where the eigenvectors with zero eigenvalues have been projected away. More explicitly, we consider the space of all different phonon and electron distributions 1 y) [Eq. (2.24)]. From this space we construct a new space A, of all distribution functions (1 - P) / y) where P is the following projection operator (0, (1 I). 595/71/I-3

(2.26)

34

SANDSTR6M

In A, , the distribution functions q0 and y2 [Eq. (2.21)] are missing and there the collision operator rcan be inverted. When formulating (2.26) we have assumed that (2.27)

(0 I 1) = 0,

which is not strictly true. This fact may of course be taken into account by some orthogonalization procedure. The only property which is affected to some extent by this deficiency is the thermopower. But even for the thermopower, the effect is small at least for low temperatures and we will not take it into account (see also the discussion in Section 4). In addition to yg, and ‘pZ, other distribution functions could also have been projected away. Niklasson has thus also projected away ] q), since the phonon momentum is approximately conserved in insulators at low temperatures. In practice we have found it most convenient to choose this set as small as possible. The distribution function F can now be written I y,> = co I To> +

I y2>

c2

(2.28)

+ (1 - f? I q>,

where the last term is the part of q~in the space A, . The scalars co and c2 have a simple physical interpretation which can be seen in the following way. The specific heats C,p and CVe for the phonons and the electrons may be written as [see Eqs. (2.11, 12)]:

(2.29) Cve = f

(w I w) = f

s &

,&%)(1

- f&))

4c2

Mei) = +4s.i))). The energy densities pEf’, pEe and currents JEp, JEe have a simple form in the bra and ket notation [cf. (I.l04Ob,41b,52)]: PEP

=


1 a>

=

c

~2

= +J I Y) = 2 j &

JEp = cw , w(qj)

5

fiBz

w’W

d’

=cJ

j

1B.z09

J% (I4

d

Co

dw Aj(qw)

&

e)

+

Ai

(-

w (-

-?$$-)

aj(qu, Qsz>, (2.3Oa)

y(k

QQ),

(2.3Ob)

I v5W>

*



+f$)

. k4.0 w’W> Mw QS2)+ w”(si>v5W * <~(Qsz)>l, JEW= (w I E’(k) I r> = 2 1 $j$

JW’W (- w)

rOr, QG).

(2.31a) (2.31b)

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35

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Thus JE = J,” + JEW= +

(2.32)

* (fi(Q~)>,

where V is the matrix QJW) 4l”d w 0

v=

() (2.33) E’(k) i ’

We notice that according to Eqs. (2.13) and (2.19), G can be written (2.34)

G=Q-V.Q. The local temperature deviations from equilibrium systems TD and Te have the following form

of the phonon and electron

= (w 1 a> T/(w ( w),

Tp = pEP/Cv”

(2.35)

p = p$lC”e = (w I r> T/(w I w).

The specific heat C, , the energy density pE and the local temperature for the whole system are now easily obtained as

deviation T (2.36)

C” = C”P + eve = + (% I %), PE

=

PEP

+

fEe

=

(%

(2.37)

/ y>>

(2.38) Here Eqs. (2.20) and (2.24) have been used. (1 j 1) is the electron density of states N(EF) at the Fermi surface a%)

=

(1

I 1)

=


I 732)

=

2

j

&

(-

(2.39)

*,.

The electron charge density and current can be written (1.10.53):

pp -e = (1 I r> =

(~2

of

I v> = 2 j &

( - aE,

1 y(k

(2.40)

Q@,

.

Je = (E’(k) I Y) = --e

(932

I V I T> = 2 j &

(- p)

E’(k) r& Q.n).

(2.41)

We denote the local change in the Fermi energy by ,Z F;(Qsr)

=

P,/(-~N~F>>

=

(v2

I v>K-e

<

(~2

I ~32)).

(2.42)

36

SANDSTRGM

Now, if we multiply that

Eq. (2.28) by (v,, ] and (QJ~1,we find from Eqs. (2.38) and (2.42) co = T(QQP’,

~2

= F(Q52).

(2.43)

co and c2 thus give the local deviation of the temperature and the Fermi energy. Here we have made use of the assumption that ~~ and ~~ are orthogonal to each other and to the space A, . The conservation rules follow directly from (2.20) and (2.21). In fact, multiplying (2.23), respectively, by (v. 1and (y2 /, and using (2.13), (2.19) and (2.34), one finds the following two equations: ~
I QJ>- Q * (?. I V I v> = Q * . (ii(QsZ)>, WY, I T> Q * (972 I V I TJ>= 0.

(2.44) (2.45)

Using (2.32), (2.37), (2.40) and (2.42), we see that (2.44) and (2.45) are just the energy and charge conservation rules [see also Eqs. (1.10.46,47)]: (2.46) (2.47)

&E=Q*JE, QP, =Q*L.

G + iI’ [Eq. (2.23)] will be denoted by S. After adding -SPp, to both sides of Eq. (2.23) and multiplying by (1 - P), it takes the following form (1 - P)S(l

- P) yJ = -(l

- P)SPy

+ (1 - P)a%xt.

(2.48)

Each eigenvector of the collision operator r with eigenvalue zero corresponds to a conserved quantity. y. and y’z are connected to the conservation of energy and charge. Since there are no other conserved quantities, no other eigenvectors of I’ with eigenvalue zero can exist. Thus, if S has no such eigenvectors except possibly y. , y2 , the operator has an inverse R-l in the space A, . Thus (2.48) can be rewritten as (2.49) (1 - P)IJJ = -R-lSPp, + R-l@&. Here we have used that R-l(l

- P) = (1 - P) R-l = R-l.

(2.50)

If Eq. (2.49) is inserted into (2.28) we get q~ = c,(l - R-Y?) &, + c,(l - R-lS) v2 + R-l@;?xt .

(2.51)

To obtain the coefficients co and c2 we substitute (2.51) for y in Eq. (2.48) c,S(l - R-?S) y. + c,S(l - R-lS) y2 = (1 - SR-l) %t .

(2.52)

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If (2.41) is multiplied

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II

by yO and y2 , we get the following two equations: cog00 -r czgo2 = A’, cog20

+

c2g22

=

(2.53)

.fi',

where go0 = (To I w

- R-T

I To>,

g20

=

CT2

I w

-

R-1

I TJoh

go2

=


I w

-

R-W

I v2>,

g22

=

CT2

! w

-

R-W

I v2>,

Al'

=


I (1

-

SR-9

(2.54)

I @&d,

(2.55)

fi' = (~2 I (1 - SW) I @&t). By direct inspection it is seen that =

go2

g20

(2.56)

.

Solving Eq. (2.53) gives us co and c2 co =

Uo’g22

-

fi’go2Ykoog22

-

d2>,

c2

(f2'goo

-

folgo2Y(goog22

-

d2),

=

(2.57)

Unfortunately the Eqs. (2.53) have no simple physical interpretation. They only give a convenient way of obtaining co and c2 . If (2.57) is inserted into Eq. (2.51), the final solution of the transport equations is obtained. According to (2.43), Eq. (2.57) connects in a formal way the local deviations from equilibrium of temperature and Fermi energy to the lattice deformations (u(QsZ)) and the effective field Eerf(QQ).

We are next going to evaluate the expressions (2.54,55). We introduce following convenient notation [cf. Eqs. (2.22), (2.36) (2.39)]: TOO =

(90

I TO> =

~11 =

(q+

I 42

+

v22

(92

I 9)2)

=

=

the

TCv, (kr

I k,)

=

(~1

(2.58)

I 6

W&),

p -p--O _ (w I m> _ Too

@r)2= (kz I kc) ; %I

eve cv

r2 + r12 = 1, ’

(2.59)

s2+d2=

1.

38

SANDSTRGM

The factors r, r’, s and s’ are used for scaling between the phonon and the electron system. It is well-known that the phonon contribution dominates the heat capacity above liquid helium temperatures, thus making r2 m 1. At lower temperatures, the electron contribution dominates. Since 1 (42 I 4e) m 3 c,2

and

(kz I k,) SWy

(1 1 1)

one can easily estimate s2 and s12, and one finds that s2 > s’~ above about @/lo. At lower temperatures, s2 goes to zero as T4. The following two quantities, which have the dimension of velocity, will appear in many places: CII

=

(43e

I 44A

%‘@md<~

0, = W,‘(k) I Wv’(l

I wx40

I 4x),

I I>& I k,).

(2.60)

(2.61)

For an isotropic Debye model and a free electron gas, respectively, cII and z), are given by CIK= G/d%

(2.62)

v, = v,/2/3.

(2.63)

c, is the velocity of sound, and VFis the Fermi velocity. We will only consider cubic crystals and also assume that the sound wave is propagating along a high-symmetry direction. This enables us to consider either the pure longitudinal or the pure transverse modes. We choose the x-axis to be along the direction of propagation (parallel to the Q vector) which is a highsymmetry direction according to the above assumption. The y and z directions are chosen to be parallel to eigenvectors of the transverse modes along the Q-direction. By introducing certain relaxation times, (2.54) and (2.55) can be written as

fo’

go0= %o{Q + ~d,Q2~oo>, (2.64) go2= g2, = d9300~22 iw~~To2Q2ry a2 = 9322{Q + iv,2Q2T22>, Q - <02(qj) I VW> . (I> + (P)~I VS-‘ir I k,) Q * - Q . EdQQ) ~~~~~~ rwwo2e + Q (To / VJ-lI

f2’

-Q

+

Q . y’?(q))

. Eeefe722v,2v22 + Q * (ii(QG)>(E,‘(k)

+ C Q,Q,4zi,>(~,‘W I R-l I + -BY

(2.65)

. (ir(Qii’)),

&

I R-W

I k,),

SOUND IN AN ANHARMONIC

METAL.

39

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where (2.66)

01 and /3 are Cartesian coordinates. Since we are more often going to use the longitudinal components 7:: , T:: , and Ttt than the transverse, we just denote them as T$c?@@ = TOO@% (2.67)

%:z”(Q~n)= ‘-odQ52>, %“,“(Q@ = T,,@-%

The transverse component

is written in the following way:

Til

%(Qfi)

(2.68)

= +dQ-@

Due to our choice of having the x-axis along the propagation direction (parallel to Q), the nondiagonal elements of T$ , T;: , T$ vanish. This fact has considerably simplified the form of Eq. (2.65). If Q = 0 one has full cubic symmetry and thus T;;(o,

G) =

T;;((o,

Q),

(2.69)

etc.

Too9Toz7722 will be called relaxation times because of their physical interpretation for Q = D = 0. As we shall see in the next section, Too appears in the heat conductivity, 702 in the thermopower and 722 in the electric conductivity. So far Eqs. (2.66) are only formal definitions because we have not given any explicit procedure to calculate R-l. However, this will be done in Section 4. 3. TRANSPORT COEFFICIENTS

In this section we are going to derive generalized expressions for the heat conductivity and the electrical conductivity. To find these transport coefficients, the heat and the charge currents will be expressed in terms of the local temperature deviation, the effective field and the lattice deformation. The total heat current JH is given by J,,

=

J#

+

J$

=

<4-&4j)

I a>

+

WV4

I r>

=


IV

I v>.

(3.1)

The difference between the energy current JE and the heat current JH is equal to the energy transport by the lattice deformation. From (2.32) and (3.1) we get JE

=

JH

+


I VW>

. <3QW.

(3.2)

40

SANDSTRGM

The solution 1v> of the transport equations is given by Eq. (2.51). On inserting it into (3.1), we find that JaH =
- R-3)

I q~oo> co +
R-IS) I yz>

c2

+

The second equality follows on inserting Eq. (2.57) for c2 . For the third equality we have used Eqs. (2.64) and (2.65). We have also rewritten the matrix elements using the definitions (2.66) and the relation S = a - Q . V + ir. Before we discuss the results we make the same manipulations as above for the electric current j, . j, is given by Eq. (2.41). Inserting (2.51) into (2.41) gives us .e Jo1 -e = (932 I V,(l - R-W I TO> co + (932 I Vdl - R-W I ~2) ~2 + =

I

=

I Vm/,(l -

(~2

+

I KR-l

(~2

(~2

I

VRl

I @At) R-W

I vo>

I %xt) i

+

(E,‘(k)

+ (F2

~22~0;~vVe2-

I Vu/,(1 -

(+

R-W

I

i

El

~2)

co

2

ive2T22Q2 ) QacO

I2 + iv,2r22Q2

v,~T,,Q,QB 2 y22722ve2Eew I Q + iv,2T22Q

k) * (1 - Ri~2$$!T22)

I R-lirl

I VJ-l

(972

+ (932I V,(l - R-W I 932)2

- irvec11r02 T----p)ooy22 1 -

-4 0

-

Q

. uj’(q))

.

(ia(Q

.
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Equation (2.57) was used for the second equality and Eqs. (2.64-66) again for the third. Recalling that c,, = T/T, we can write the rather unwieldy Eqs. (3.3) and (3.4) compactly as JaH = c I---i&.,Q, B

f

-

eKl,&erf,s

+ K&(z&) ifim, - K&,(zi,) i $$f ve2mb!. .iee= C I- eEla, B

T

Q, T + e2Ko,&eff,B - z

(3.5) KL(4d

1.

(3.6)

The Greek indices represent components along the different coordinate axes. We have introduced an effective electron massmb by assuming that E’(k) = k/m, on the Fermi surface. From this definition it follows that

l/m = (h I Ez’(W(h I &J = NEd ~.~/n,

(3.7)

where also Eqs. (2.39) (2.61) and (3.38) have been used. The transport coefficients in (3.5) and (3.6) are given by (3.8a)

(3.8b)

-Klas = dyoovz2 rv,cII (7:: - To2L?ive2T22QarQo + iv,2r22Q21’

(3.8~)

kia = ha 2

(3.8d)

isZm,K& = (cpoI V,R-W

I k,) + c (y, ( V&l 6

I*

Q&(q)),

(3.8e)

(3.8f)

(3.W The notation for the transport coefficients is taken from Ziman [21] and somewhat extended. If it were the Maxwell field E instead of the effective field Eerr (their difference being the correlation field E,,,, (Eq. (2.6)) that appeared in (3.5) and (3.6),

42

SANDSTR6M

we would interpret the transport coefficients in the following way: KoDIBand KzaB are related to the electrical and the heat conductivities eUPand R,, through ud3 = e2KOaB, GE = Kz,,IT, Kl is a thermoelectric

(3.9) (3.10)

coefficient and it is related to the thermopower Stherm by Stherm

=

-

[KdO,

(3.11)

0)1(1/T)

W&6X

(K,(O, 0) means that we have put Q = D = 0, letting Q go to zero first). The transport coefficients become scalars for Q = Q = 0. We will see later that

Ko(O, 0) = Ko’C-4 O),

(3.12)

K,(O, 0) = K,‘(O, 0) = K;(O, 0),

which explains the choice of notation. It might be thought that there is a certain arbitrariness in the definition of the transport coefficients, because T according to Eq. (2.57) can be expressed in terms of Eerr and (ii(Q But this arbitrariness is removed with the aid of the Onsager relation (3.8d). Now it is Eerr which appears in (3.5,6) instead of the Maxwell field E, which is the field that is measured. So, in principle, the correlation field should be removed [with Eq. (2.6)] and then Eq. (2.57) should be used to make the new quantities satisfy the Onsager relations. However, to interpret the transport coefficient it is not necessary to make such modifications in all cases, because most properties are measured under the condition Q M 0 and then the correlation field vanishes anyhow. For the interpretation of the electrical conductivity it is however necessary to take it into account. a. Heat Conductivity According to Eqs. (3.8) and (3.10), the heat conductivity form for Q = 52 = 0 (again letting Q go to zero first): K= =

T

takes the following

&T 0 = cyc~~Too(o9 0) 9

(3.13)

’

where (2.58) has been used. The interpretation of TV as a relaxation time is clear from (3.13). 7oo is defined in Eq. (2.66). Since R-l is a 2 x 2 matrix, there are four different contributions to TV . However, the nondiagonal elements are negligible (see Section 4). The two diagonal elements give the following contributions to the heat conductivity K =

Kg

+

K,

=

C,%T&(O,

0)

+

C,“V,2T&(O,

01,

(3.14)

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11

where TC,“&&(Q!2)

= i(w(qj)

TC,eve2~&,(QL’)

= i(wE,‘(k)

w,‘(qj)

I R-l I 4-U)

/ R-l / E,‘(l)

w,‘(qj>>,

w).

(3.15) (3.16)

In (3.15,16) the explicit forms of q,, and V have been used [(Eqs. (2.20) and (2.33)]. The two terms in (3.14) give the heat conducted by the phonons and the electrons, respectively. b. Longitudinal

Electrical

Conductivity

We next consider the longitudinal electrical conductivity (a and fi along the Q-direction). To start with we neglect the local temperature deviation T. If there is no sound wave present, Eq. (3.6) takes the form

.h = e2Koo [&I + i $ QWQ)pe]. In (3.17) we have inserted Eq. (2.6) for the correlation that we are investigating longitudinal phenomena. current jeo are connected via the continuity equation QP, =

Qh

field. The Q indices indicate The charge density pe and

.

(3.18)

If we also use (3.8a) for K,,o we find the following relation between the charge current and the electric field i-0 =

(~a((&?)

EQ,

(3.19)

where the electrical conductivity oo(QL?) is given by (3.20) kTF is the Thomas-Fermi screening radius, which is defined by k& = 4ne2N(EF).

(3.21)

Comparing (3.9) and (3.20) one seesdirectly that

$$~o~QW~o(QJ31= Q2/[Q2- k%Q)I,

(3.22)

and

uo(QQ) = cdQQ>.

(3.23)

The correlation field thus affects the conductivity in the limit J2 -+ 0 but not when Q -+ 0.

44 C. Longitudinal

SANDSTRiiM

Dielectric

Function

When studying sound propagation in metals it is necessary to calculate the electron charge response to the lattice deformation (u(QsZ)) since the charge density appears in the equation of motion for (u(QSZ)). For later comparision, however, we shall first consider the charge response to an external electric field E ext. If we neglect the response of the ions, the total electric field E inside the crystal may be written EO = Eext,o - $ Using the following definition

QP, .

(3.24)

of the dielectric function

~cdQJ4= &xt.oIE, ,

(3.25)

we find directly from (3.24) that pdQs;Z) = -i $-

‘QtQzoT 0

’ Eext, O .

From Eqs. (3.19) and (3.26), and the continuity Eq. (3.18) and the definition of the dielectric function (3.25), one obtains the following standard relation between the longitudinal dielectric function and the conductivity in an electron gas 47ri

l ,(Qsz)= 1 + sz ao(QQ). Inserting (3.20) for the conductivity,

(3.27)

one finds that (3.28)

For Sz = 0, Eq. (3.28) becomes simplified

to

We notice that E&QO) does not depend on the form of the collision operator and hence ho has the same form both in the collision-free and the collision dominated regimes (when the temperature deviations are neglected). It is easily verified that (3.29) agrees with the ordinary dielectric function (1.7.5~) in the collision-free regime for Q < kF which is the limit of validity of Eqs. (2.1) and (2.2). If one puts G(Q) = 0 in Eq. (3.29) one recovers the RPA approximation for

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eo(QO). In a forthcoming paper [Eq. (III.5.5)] the following approximate expression for T&Q@ in the coliision-free regime (Qv,T,, > 1) is derived ~~~(Qi-2) w 6

QllFT

.

(3.30)

If (3.30) is inserted into (3.28) we find that also the .L’ dependence of cO(QQ) agrees with (1.7.5~) in the collision-free regime for a < EF, Q < kp . The form of 722 in the collision-dominated regime will be given later in this paper. We now proceed to find the charge response to a deformation wave. Such a wave generates, according to Eqs. (1.7.8,9) and (1.7.12), the following unscreened field:

E.2.Q zzzf%(Q) Ze

vi,(Q) is the ion-electron

potential. EQ

The definition

=

Q2(uQ(QQ))

(3.31)

The total field E. is now given by [cf. (3.24)] - $

%a

Qpe .

of the dielectric function is directly taken over from (3.25):

and hence (3.26) is formally unchanged

ps= -i 2

Ed

-

1 Ei

EQ(QQ)

In the presence of a sound wave, (3.17) is modified temperature deviation is still neglected] jeQ

=

e2&o

[EQ

+

i $

(3.34)

Qf '

[see (3.6) where the local

QGCQ) p.]- z

(3.35)

GQ(~Q>.

The last term in Eq. (3.35) comes from the collision drag effect. Combining (3.35) with Eqs. (3.18) (3.31-3.34) we find the following expression for the dielectric function: 47rie2 KOQ - i74rre2 & 1-t 7

l Q(QQ)=

oG@)

(3.36)

1- i y

K,,QG(Q) -

\TFi;z/ ze

t 2

Ki,

.

46 Using (38a,b)

SANDSTRiiM

for K,c and K& , we can write (3.36) as

When deriving (3.37), we used Eq. (3.21), as also the following can be obtained by integration by parts: n = (E,’ I k,) = 2 j & For the interpretation

E,‘(k)

k, (-

T).

equation

which

(3.38)

of (3.37), we notice that for small Q uie(Q) = -(4ne2Z/Q2)(l

where S is of the order of magnitude -(4nZe2/vie(Q)

+ SQ2) + e-e,

(3.39)

l/kr2 and thus

Q2) SW(1 - SQ”)

for

Q < kF

(3.40)

is quite close to 1. Comparing the dielectric functions (3.28) and (3.37), one finds that the difference between them is the term in the denominator of (3.37), which is proportional to [(E,‘(k) ( R-W I k,)]/[(E,‘(k) 1kz)]. This term comes from the collision drag effect. The importance of this term is very different in the collision-free and the collisiondominated regimes. It will be shown in a forthcoming article that the collision drag term is negligible in the collision-free regime. This means that the two dielectric functions (3.28) and (3.37) are identical, which agrees with the fact that it is the dielectric function of the ordinary electron gas which is used to calculate phonon frequences and line widths in the collision-free regime. On the other hand, it will be seen in Section 5 that in the collision-dominated regime the collision drag term is not negligible and that there is even a large cancellation between the two Q-dependent terms in the denominator of (3.37). What is the physical reason for the difference between the dielectric functions (3.28) and (3.37) ? For (3.37) the driving field comes from the moving ions. But the moving ions do not only produce a field, they also modify the scattering of the electrons due to collision drag. The modification of the scattering is, of course, more important in the collision-dominated regime than in the collision-free one, where the properties of the collision operator have little significance.

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Conductivity

The transverse conductivity is simpler to obtain than the longitudinal one, because for transverse modes both the correlation field E,,,, and the local temperature Tvanish. This follows from Eqs. (2.6) and (3.15). Thus from (3.6) and (3.8a) we directly get the following expression for the transverse conductivity: a,(QSZ) = e2Ko,(Q1;2) = e2N(E,) v,~~:~(QSZ).

The relaxation time &(QQ)

(3.41)

will be evaluated in Section 4.

e. Transverse Dielectric Function

The definition of the transverse dielectric function is not unique. Pines and Nozibres [ll] define it from a relation between the displacement field D and the electric field E D, = c C&B ’ 19

(3.42)

However, we will define l JQ!ZJ) in analogy with (3.33):

Q(QQ) = Ei.JEl-

(3.43)

The definition (3.42) agrees with our definition (3.33) for the longitudinal case but not with (3.43) for the transverse case. From (3.42), a relation of the type (3.27) follows immediately. Our definition (3.43), however, does not yield a relation analogous to (3.27) for the transverse modes. For the transverse case we consider directly what happens when the driving field is generated by a deformation wave. The total field is given by [see Eqs. (1.7.7) and (1.7.12)] (3.44)

where the driving unscreened field is [see Eqs. (1.7.7,8), (1.7.12)] EisI = $

47TQ2 (-l)

we(Q) Q2 @dQW. Z4Te

(3.45)

Q2 - c2 Equation (3.6) takes, in this case, the form (3.46)

48

SANDSTRGM

From the Eqs. (3.43-46),

we get at once the following l-

expression

iS24n-e2K0, c2Q2 - fin2

for E~(QSZ):

(3.47)

w&, 1 Inserting

11722

(3.8a,b) for K,, and K& and using (3.38) we find that

cdQ-9)

1 _ iQ4n-e2N(EF) ve2,i2 c2Q2 - Q2 = l _ (-4rre2Z) (E,‘(k) / R-W 1k,) ’ vie(Q) Q2 6% I kd

(3.48)

We recall that the x axis is along the Q direction. Due to the cubic symmetry, (Ez’ j k,) = (E,’ 1k,). In order to get the dielectric function for the case of an external field we just drop the second term in the denominator in analogy with the longitudinal situation. Using Eqs. (3.43-45), we can write the electron current je , due to the lattice deformation, in the following simple way: jel =

1 ( cJQJ-3

-

1) h,l

jion is given in Eq. (1.7.8). The analogous relation true. Hence

.

for longitudinal

Since e(Qm> is a large quantity for small Q, the electron current, following a deformation wave, are approximately opposite in sign. The electrons are dragged along with difference in sign of the currents is of course a result of the charge carriers.

modes is also

current and the ionic equal in magnitude but the moving ions. The difference in sign of the

f. The Local Temperature Deviation T In Section 3.d, when deriving the induced charge density (3.34) the local temperature deviation T(Qs2) was neglected. Corrections due to the presence of T(QA2) will be discussed in this section.

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T can be expressed in terms of Een and (ti) with the aid of Eq. (2.57), remembering that T = Tc, . Using the expressions (2.64), (2.65) and (3.8), we can, after some simple manipulations, write (2.57) in the following form:

f (yoos2+ iQ * K, * Q) = Q . (J-(qj) I y"(q))>. (ii

- Q * K, . he

+ Q - K,’ . (it) i!Zm, - Q * Kc . (ti) i -Q2 ce2mb . !2

(3.51) Equation (3.51) can also be obtained in another, perhaps more direct, way. If (3.2) is used, the energy conservation rule (2.46) can be written as

f vodz - Q *JA = Q - (w2GU)I yW> . .

(3.52)

Inserting (3.5) for JH into (3.25) gives us directly (3.51). It follows from (3.51) that T = 0 for a transverse mode. All the matrices K, , K1’, KI and (w”(qj) 1y”(q)) are diagonal. For K, , K1’, and K’; this follows directly from their definitions [see also the remark below Eq. (2.68)]. With the help of (1.10.36), it is possible to show that $(qz, qv, qz) = -y&(-q5 , qr , qz), etc., which also makes (w2(qj) I yjj(q)) diagonal. The dominating contribution to T(QS2) comes from the first term on the right hand side of (3.51). This term appearsbecausethe phonon frequencies are changed by the lattice deformation, and this change in the phonon frequencies modifies the energy density and thus the local temperature. We will seein Section 5 that the threelast term in (3.51) give thermoelectric corrections to the first one. It is now straightforward to extend the discussionin Section 3c and to take into account T, by modifying (3.35) according to Eq. (3.6). One then finds the following expression for the dielectric function which appears in the charge responseto a deformation wave [Eq. (3.34)]:

where

%,Q = K,, 595/7dI-4

%62

iQ”K,“* i- iQ2K20

’

(3.54)

SANJXTRijW

50 and R’ 00 -= 722

OK,, $+

Q2

9),j,# + iQ2&Q

’

= >.

(3.55) (3.56)

All collision drag effects enter via I?& . We can, of course, make (3.53) more explicit by inserting all the coefficients in (3.8). This would, however, lead to a very lengthy equation and we will see in Section 5 that it is enough to investigate some special cases, when calculating the ultrasonic attenuation.

4. EVALUATION OF THE RELAXATION TIMES To get explicit results for the different transport coefficients in Eq. (3.8), we have to evaluate the matrix elements of R-l, which appear in the definition 2.66 of the relaxation times. To calculate the electron contribution to ultrasonic attenuation it is necessary to evaluate

to second order in Q, as we will see. For the phonon contribution to the damping, on the other hand, the Q and Szdependence of these matrix elements is not needed. That is the reason why Niklasson [4, 51, in his investigation on insulators, did not consider this Q and Sz dependence. The evaluation of the matrix elements of R-l cannot of course be done exactly, and we shall use the following approximate procedure. R-l is defined through the relationship (1 - P)(G + ir> R-l = R-l(G

+ ir)(l

- P) = (1 - I’),

(4.1)

where P, G and r are defined in (2.19) and (2.26). We assume that R-l can be expanded in the following way:

G!Ji=l.N*is a set of a few distribution

functions with the following properties:

(1) The &‘s are orthogonal, i.e., (& / I&) = &(& ] &). Any nonorthogonal set may of course always be orthogonalized. (2) All $+ are orthogonal to v. and ?‘z , otherwise Eq. (2.50) is violated. The range of validity of the assumption (4.2) will be discussed in Subsection 4.e.

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According to Eqs. (2.51) and (4.2), the solution v of the transport equations can now be written in terms of v0 , y’z and the set {&>. I v> = co I vo) +

c2

I v2>

+ c a, I 50.

(4.3)

The coefficients co , c2 , (ai} are easily obtained by inserting (4.3) into the transport Eqs. (2.23) and then multiplying from the left by the different distribution functions and solving the resulting system of equations. To get the coefficients R;’ we insert (4.2) into (4.1) and multiply from the left by (I& 1and from the right by I I&). We find that

where

From (4.4) we see that the matrix R;l/#ii$ll is the inverse of the matrix Sij . The procedure here is directly related to a variational method. Let us say that we want to calculate

<#a I R-l I b>.

(4.5)

If we insert Eq. (4.2) for R-l into (4.5) we find that (4.6) The result could equally well have been obtained from the following by making its variation vanish

expression

I(?& I F>l”/(F I s I y>,

(4.7) when 1 y)) is varied. I v> is assumed to have the form (4.3) with co = c2 = 0. Without loss of generality we can assume that I #a> is orthogonal to v. and yz since it does not affect the value of (#, I R-l I #,). We can furthermore assume perfectly generally that

(F I s I TP>= (p’ I #a)

(4.8) because I y> appears quadratic in both the numerator and denominator in (4.7). Now varying the coefficients {ai} in (4.3) and using (4.8), we find that the variation of (4.7) vanishes when

C S&j =
(4-9)

52 According

SANDSTRiiM

to Eq. (4.4), the matrix Sij has the inverse R;l/($jj#ll)

and thus (4.10)

It is easy to see that on substituting from (4.10), (4.7) becomes identical to (4.6). We have thus verified that our approximation (4.2) for R-l gives the same results as a variational method would. The variational principle described here was first introduced by Kohler (see Ziman [12]). If -8 is hermitian (J2 = Q = 0), this principle gives an upper bound for i($, 1R-l 1 t,!~~). Other more complicated variational methods can give both an upper and a lower bound even when -iS is non-hermitian (see Ref. (13) and references therein). a. The Relaxation Times for Sz = Q = 0 For Sz = Q = 0, R-l is denoted by R;‘. The first relaxation time we are going to evaluate is 722, which appears in the electrical conductivity [Eqs. (3.8a) and (3.20)]. 722 is defined in Eq. (2.66). We want to apply Eq. (4.6). The question is what functions should we include in the set {&}. It is obvious from Eq. (4.6) that only #i with a large overlap with #a are important when evaluating (#, 1R-l 1 #,). According to Eq. (2.66), $a = E,‘(k) for 722 . The only distribution function & , which is usually considered, is k, . From (4.4) and (4.6) we then find that 7&

0) = 6% I kJl(k,

I r I k,).

(4.11)

A better choice than k, would probably be E,‘(k), which however is more difficult to handle. For free-electron-like metals, the difference between using k, and E,‘(k) is unimportant. Equation (4.11) gives the standard approximation for the resistivity (Ziman [12]). Several authors have taken into account more distribution functions (see, e.g., Kohler [14], Sondheimer [15], and Greene and Kohn [16]). They have found that (4.11) is, in general, quite satisfactory, and it involves an error which is less than 10% in simple metals. This suggests that a few {&} can be sufficient for calculating matrix elements of R-l. However, there is at least one situation where (4.11) is not a good approximation. For very low temperatures in the alkali metals, Umklapp processes are frozen out and thus also momentum is approximately conserved in the scattering process. This means that the phonons can no longer be described by an equilibrium distribution. They are drifting along with the electrons (this effect is usually called phonon-drag, not to be mixed up with collision-drag which is something entirely different). Thus, also including the phonon distribution function I qe) in the set {#i} gives us, according to Eq. (4.2), the following expression for R;l:

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where

With the definitions

(2.61) and (2.66) one now finds directly that (4.14)

Equation (4.14) leads to another standard expression for the electrical conductivity where the phonon-drag is fully taken into account; see Ref. [12, Eq. (9.13.6)]. Next we consider 700 which appears in the heat conductivity K according to Eq. (3.13). From the definition of T,,~, Eq. (2.66), one can easily see that it is possible to divide the contributions to K into the following two parts:

where TK~

=

TK,

=

TCypcf& = i(4-U) wz'(qj> I R-l I w(W) ~s'WD, TCvev,2& = i(wE,'(k) 1R-l1 E,‘(k) w).

(4.16) (4.17)

Strictly speaking, two other terms appear also. These cross-terms-they involve both electron and phonon distribution functions-are odd in the frequency W. Estimating them, one finds that they are quite small [cf. also Eq. (2.27)] and they are thus neglected. The two parts in (4.15) correspond to conduction by phonons and electrons. The electron part usually dominates [12]. Since w(qj) w’(qj) has the same symmetry as q in the Brillouin zone, Eq. (4.12) for R;l can also be used for evaluating 7:’ . From (4.16) we then find that T;o)o(o, 0) = (rkdAK42 I 42).

(4.18)

When deriving (4.18), Eqs. (2.60) and (2.29) have been used. Next we consider T,fo which appears in the electron part of the thermal conductivity K, [Eq. (4.17)]. The natural choice of distribution function I+&is 1wEZ’(k)) [12, Chap. 91. Equations (4.4) and (4.6) give, in this case, TK

e=

TC

l W-f&@)>? Yev e2?e 00 = lb&‘@) (wE,'JI+JEyj-'

(4.19)

This standard expression, where only one distribution function is taken into

54

SANDSTRijM

account, is known to give an error which is roughly smaller than 25% (Sondheimer [15], Klemens [17]). T,,~is defined in (2.66). This definition may be written in the following way, if we substitute for V from (2.33)

A term which is odd in o has been dropped in (4.20). Since E’(k) has the same symmetry as k, and q as w(qj) o’(qj), the obvious choice for R;l is (4.12). It gives the following expression for 702 : 702@

0)

=

4rhl/4

a!

I kJ<%!

I 4z:).

(4.21)

In (4.21), Eqs. (2.60) and (2.61) have been used. T~~(O,O) appears in the thermopower &herm [Eqs. (3.11) and (3.8)]. Also, using (4.14) and (4.21) we find that (4.22) Equation (4.22) is the so-called phonon-drag contribution to the thermopower [12, p. 4091. Other contributions [12, Section 9.121 involving matrix elements of r which are odd in o are usually much smaller, at least at low temperatures and have been neglected. If normal processes were to dominate the collisions instead of Umklapp processes, the momentum would also appear as a conserved quantity. This would strongly affect the transport properties. One can expect that normal scattering dominates in the alkali metals at very low temperatures (this is further discussed in Section 5). To distinguish between the contributions from the normal and Umklapp processes we write r,, , etc., as

r,,=r,“,+rK. Since in normal processes momentum

(4.23)

is strictly conserved, we get

r& = -rk”, = -r,$ = rg

(4.24)

(phonon-phonon and electron-electron scattering give no contribution to these elements). On the other hand, for Umklapp processes it is reasonable to expect that rk and r& are essentially larger in magnitude than r&. This may be understood by noting that all k and q vectors give positive contributions to I’g and rg, but of alternating signs to rg , which gives a cancellation effect (see [12, p. 4111). For the two special cases where Umklapp and normal processes dominate, we thus obtain for (4.14), (4.18) and (4.21) [using (4.13)]:

SOUND

(1) Umklapp

IN

AN

ANHARMONIC

METAL.

scattering: 1r& 1 > 1I$

II

55

1, etc.,

(TU < TN) 9

(4.25a)

Go@, 0) = (42 I 9d/r44 9

(4.25b)

T22(0, 0) = @, I bJ/rkk

~02(0~0)= -(rk,/rkkr*,) (2)

Normal

scattering: I r;

l/kc I MqLz I c&J

(4.25~)

/ < j r& 1, etc.,

(Tu > TN)

In (4.26) Eq. (2.59) has been used.

TV

and


TN

TN

have also been introduced: = kz I kJ + <4r I q2) . rNkk

(4 27)

Three different kinds of scattering have been taken into account: electronphonon, phonon-phonon and electron-electron. The electron-phonon scattering contributes to all four elements of the matrix r [Eq. (2.19)], but the phononphonon and the electron-electron scattering only to the upper left (rll) and the lower right (r,,) elements, respectively. From the calculations of phonon line widths in metals [18-201, one can get an approximate idea about which kind of scattering gives the dominating contribution to r,, . One has found that the electron-phonon interaction is most important for low temperatures and the phonon-phonon interaction at higher temperatures, they being equal at about e/10, where 19is the Debye temperature. Theoretical arguments [12] suggest that it is important to take into account the electron-electron scattering for r,, at liquid helium temperatures in simple metals. This has however been difficult to verify in resistivity measurements because it is masked by impurity scattering which can give the same temperature dependence for the resistivity. One can of course artifically assume that the electron-phonon interaction is very weak (which is certainly not true). Our model for a simple metal will then approximately become a model for two independent systems: The electron gas and phonons in an insulator. In this way it is possible to compare with the results for these systems. Thus, for example, it can easily be verified that our results agree in this special limit with those of Niklasson [4, 51 where a comparison is possible.

56

SANDSTRGM

b. O-dependent Relaxation

Times

To calculate the ultrasonic attenuation, it is necessary to evaluate the Q- and Q-dependence of certain relaxation times. To do that we rewrite Eq. (4.1) as R-l = R;’

- R,l(L’

- Q * V) R-l.

(4.28a)

Here the definition of G, Eq. (2.34), has been used. First we consider only the Q-dependence, thus putting Q = 0. R-l in this case will be denoted by R;l and from (4.28a) we get R;l = R;l - R,~L~R;~.

(4.28b)

We assume that &’ and R;l are expanded in the same set {&} according to (4.2). Equation (4.28b) then takes the following form: (4.29)

or

where T is the inverse of the matrix with the elements

To evaluate 7&O, .n>, T&(O, LZ), and ~,,~(0,1;2), the set of distribution functions (1 qz), 1k&} will again be used. If this set is inserted into Eq. (4.29), one gets a system of equations. Solving this system one finds the following generalizations of Eqs. (4.14), (4.18), and (4.21):

T22@,Q) =

T&O, 0) (1 - isz (q;! q”) ) QB [ 1 - zx-2((ICE 1k.J 2

+ (qz 1 q,& 9)

7&(0,0) (1 - il2 ($/

‘a) ) ,

[ 1 - is2 ((/& 1k,) J$ + (q2: 1q&J 2) Q) =

, (4.31) ’ k”]

kk

~,“,
702@

- 522(q= ’ qsp

~02(0~0)

- 322 (4% ’ qzy

(4.32)

’ kz)] . (4.33)

SOUND

IN

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ANHARMONIC

Using the single distribution function directly obtained [cf. Eq. (4.19)]:

METAL.

/ w&(k)),

57

II

the expression

for T&(O, 52) is

T&(0, 52) = T&(0, O)/[l - iL?T$(O, O)]. c. Q-dependent Relaxation

(4.34)

Times

Finally we also take the Q-dependence of R-l into account. An alternative of (4.1) is the following, which can be derived in the same way as (4.28): R-l = R;l $ R;lQ

. VR-I.

form (4.35)

Once again we use an expansion of the type (4.2) for R;l and R-l. Equation (4.35) then gives a convenient way to connect the two matrices (R& and R;‘. More precisely, we get the following two equations:

(R-l)ij = (R& + ; 9 (Q . VR-I),?

= c

(Q

m

’ V;‘Q

(Q - VR-% , ’ vhm CR-%nj

.

(4.37)

mm

In (4.36) and (4.37), we can eliminate (Q * VR-l),j and we obtain the equation CR-‘)ij = c ~idR;%

(4.38)

,

where U is the inverse of the matrix with the elements 6ij _ c (K% 1 bl

(Q * VKIQ hi

* Vu *

(4.39)

In the same way as (4.31-33) were derived, we are going to use Eq. (4.38) to calculate T&Q@, $,,(Q@, and T,,(QS~). These quantities will be needed in Section 5. The resulting expressions are quite lengthy and we shall neglect the coupling to the distribution function I q,(w(qj) w,‘(qj)/w)>, which can be expected to be quite small because the sound velocity is much smaller than the Fermi velocity. From (4.38) we then find the following equations: (4.40)

(4.42)

58

SANDSTRijM

In (4.40-42) the relaxation time ~1Ql(s2)has been introduced. In Section 5 we will find that the electron part of the longitudinal ultrasonic attenuation is proportional to ~fi(L’) in the collision-dominated regime. T$G’) has, except in special cases (see Section 5), about the same magnitude as ~~~(0, Sz). Tg(G) is defined as (4.43) We will also need the @dependence generalized to

of

T&,

. Using (4.38), Eq. (4.34) is easily (444)

where

d. Matrix

Elements of R-W

Due to collision-drag, the transport coefficients K,‘, K,‘, and K,” appeared in the equations for the currents (3.5) and (3.6). According to Eqs. (3.8b and f), these coefficients contain the expressions E’ck)

I R-W

I 44,

(4.46a)

I k,).

(4.46b)

and CR, I V3-r

Due to the assumed cubic symmetry and the fact that the x axis is along the Q direction, (4.46a and b) are only nonzero if 01= /3. To obtain (4.46a and b), we write Eq. (4.1) in the form R-lir(l

- P) = (1 - P) - R-l@

- Q . V)(l - P).

(4.47)

From (4.47) and (4.37) we can directly derive the following equations: (k, 1R-W

I k,)

= (k, Ikz)I1 (kg I R-W = 6%

I k,)

-

(k, I R-l I kc) . Q _ (k,E,’ I R;’ I k,E,‘) es] 1, (4.484 (k,Ik,) II (kz I kd

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METAL.

59

II

In (4.48) the same approximation has been used as in (4.4&42). The longitudinal component K& = K&, and the transverse component K& = K&, of K&, (Eq. 3.8b) can be written in the following way, if Eqs. (4.48) and (3.7) are used: (4.49a) K& = ~~~~~~~~~~ 1 + iT,l,(J? + iQ%,%:,)}.

(4.49b)

The second term in (3.8b) has been dropped. It is unsymmetrical in both k and w, and it therefore gives a very small contribution. ~19is defined in (4.43). T:, is the corresponding transverse quantity

When deriving (4.49a) we have also used the fact that T&Q)

= i[(kz I R-l I b)/&

may be written

722

(4.51)

I kz)],

which directly follows from its definition (2.66) and the fact that E’(k) = k/m, around the Fermi surface. In the same way, we find for T& [Eq. (2.68)] that (4.52)

dz(QG) = i[(k, 1R-l 1k,)/@, t b)l.

We will see in Section 5 that it is the Q-dependent terms in the curly brackets in (4.49) which give the electron contribution to the attenuation. This explains why it is necessary to take into account the LA and Q-dependent parts of R-l. For evaluating K;, [Eq. (3.8e)], we need an expression for (4.46b). Using(4.47) and (4.37), one finds after some simple manipulations that
[
I kd %%i)

/ %z)/d%-k%

m

On deriving (4.53) we have used the fact that

+ ~~'e2Q%).

i7,2@ T,,~

(4.53)

may be written as

I k&q, I cdl.

T&Q@ = ik& I R-l I kJ/d(k

(4.54)

(This follows from 2.60, 2.61, 2.66 and 4.12.) Neglecting the last term in (3.8e), which is small due to its nonsymmetry in q and which describes the coupling to more complicated distribution functions, we get for KiQ , .2 K;Q

=

dC,~N(E,)

cIIv,T~~

*

+

“z

20 Q

T1l

.

(4.55)

The equations that have been used to obtain (4.55) are: (3.7), (2.29) (2.39), (2.60),

60

SANDSTRijM

(2.61) and (3.38). In precisely the same way, we find the following expression for K,“a [Eq. (3.8f)], using, in addition to the above mentioned equations, also (4.48a).

K;', = v"C,"N&~

CIIV,~02

J2{1 + i722@ + iQ"ve"&>> Q + ir22v,2Q2 ’

(4.56)

For the sake of comparison, we also write down the expression for K,o [Eq. (3.8c)l: (4.57) Equation

(3.12) follows directly from (4.55,57).

e. Range of Validity of the Assumption to Express R-l in Terms of a Few Distribution Functions We want, in conclusion, to discuss the range of validity of the results in this section. We have nowhere explicitly assumed that Q * V or JJ should be small compared with some inverse relaxation time. All the same, the results will not be valid outside the collision-dominated regime. Let us look at an example. From (4.40) we get the following expressions for T~~(Q!S) in the collision-free regime Qv,T~~(O, 0) > 1 depending on whether SZT~ is bigger or smaller than unity and using the fact that &52) has the same type of Q-dependence as Tie(Q) in Eq. (4.34). ~zz(Q@ - [l/v,“Q”~~(o>l~

fidi(o)

< 1;

(4.58)

~zz(Q1(2)= (- Wv:Q”);

ITS

> 1.

(4.59)

If now, (4.58) and (4.59) are inserted into the expression for the dielectric function Eq. (3.28) one finds that both (4.58) and (4.59) give wrong results. In a forthcoming paper the correct formula for Tag in the collision-free regime will be derived, namely, Eq. (3.30). One can also verify that if (3.30) is inserted into (3.28) it agrees with the correct expression (1.7.5~) in the collision-free regime. As we see, neither (4.58) nor (4.59) is the same as (3.30). What is the reason for the breakdown of the methods of this section in the collision-free regime? We write the solution of the transport equations in the following schematic form: I v> = [.n -

Q * V + (i/T>]-’ I @&A

In (4.60) we have made a relaxation-time l/~ is the diagonal matrix

approximation

(4.60)

for the collision operator.

(4.61)

SOUND

IN

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METAL.

61

II

In our derivation we have assumed the same set of distribution expanding R-l [Eq. (4.2)] for all Q and 1;2.For

functions

{&} for

(4.62)

Q+,

this is no longer a good approximation, because of the singular behaviour of (4.60). A few distribution functions cannot describe the solution 1 y) for varying Q. To evaluate the relaxation times T&QL?), T&(QL?) in the collision-free regime, a more general method will be used in a forthcoming paper, which is correct also in the collision-dominated regime, where the results agree with those derived in this section. Throughout this section only one or two functions have been taken into account for the set (&}. For certain cases this is known to be quite sufficient. For other situations, the symmetry properties have simplified the choice of distribution functions. Of course, one has to investigate numerically as to how many distribution functions have to be taken into account in each separate case. It is quite straightforward to generalize the formulas by increasing the number of distribution functions. This would naturally lead to very lengthy expressions, but this presents no problem to a computer.

5. SOUND

WAVES

IN

THE

COLLISION-DOMINATED

REGIME

In this section we are going to calculate the attenuation of phonons in the collision-dominated regime. These phonons are usually referred to as first sound. For low frequencies the number of collisions during one period of the sound wave is so large that a local equilibrium is built up. Certain conditions must be satisfied for establishing a local equilibrium. The mean free paths of the thermal phonons (A,) and the electrons (de) must be much smaller than the wavelength of the sound wave l/Q:

Q4 < 1, Q4 < 1.

(5.1)

Equation (5.1) gives the condition for being in the collision-dominated regime. A, and A, can be written .A, M cI1’* and A, = v,r, , where r9 and 7, are typical relaxation times for phonons and electrons, respectively. For a sound wave, G w c,Q, and Eq. (5.1) implies thus that (5.2)

62

SANDSTReW

The quantities Qcrrr, , etc., in (5.2) can thus be used as expansion parameters in the collision-dominate regime. How is the sound wave damped ? The sound wave induces locally varying phonon and electron densities. These densities are not quite in phase with the sound wave. The larger the number of collisions, the better the phonons and the electrons follow the sound wave, and the less is its attenuation. The attenuation is thus inversley proportional to the number of collisions and proportional to the relaxation times in the collision-dominated regime. This description emphasizes the viscous nature of the attenuation. In the collision-free regime the situation is directly opposite. The attenuation of high frequency phonons is proportional to the number of collisions. Before we can calculate the attenuation of first sound, we must know the charge density and the local temperature deviation induced by the deformation wave. a. Longitudinal Dielectric Function According to Eq. (3.46) we can obtain the induced charge density p,(QL?) using the dielectric function E&Q&?) [and the continuity Eq. (3.191. ec(QL?) is given by (3.52). This equation becomes extremely lengthy if all K coefficients are written out explicitly. Instead of using the full expression, we are going to expand ec(QS2) in powers of Q and Sz, which is possible according to Eq. (5.2). We will see that it is necessary to expand E&Q@ to the two lowest nonvanishing orders in Q and L? when calculating the ultrasonic attenuation. To this order, Eq. (3.52) takes the following form if Eqs. (3.8), (3.53) and (3.54) are used: 1

1

co(QQ) =

.

(E,‘(k)

1 R-W’ 6% I L)

+KIV

29 -KOv 4rre2n (

. If I

1k,)

+ K,, 3

Q2s n

4vre2Z

vie(Q) Q2

i?S r 22 v B2k2T F !I

) Q2(-i)

K” - K;b ‘$ Qrpoo + iQ2K2v

v,$i’&
iQ2K$ ’

’

Kov
1

+

iQ2K2J ! *

I (5.3)

If Eq. (4.48a) is inserted into (5.3), we see that the lowest order terms that appear in (5.3) are of second order. It is not possible to expand the factor (GJJ,,, + iQ2K2v)

SOUND

IN

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METAL.

63

II

because Q2Kzp can be larger than L&D,,, even in the collision-dominated regime. All terms in (5.3) do not, of course, have the same magnitude, and certain terms may be neglected directly. Thus the factor (3.40) and the expression

can both be replaced by 1. The error involved is smaller by a factor ( ~/E~~T,,T’,‘> < 1 than the leading term, and hence completely negligible. Still, to third order in Q and Sz, Eq. (5.3) can now be considerably simplified. In fact, several terms cancel each other. Using (3.8), (4.48a), (4.50), (4.51), (4.55) (4.56) and (4.57), we find that 1

=

4QSZ)

+ g

-G(Q)

- -..&? TF

v e 2k2T F
+ &

2

- i -2%

I r>

v 8 2k2T F

Q”v:T,“,

Q

[A-&, + iQ2Kzo]

’ + K,&&,,,

iQ2Kto + iQ2Kzo) ’ (5.4)

1

Ifsz * csQ, the third term in (5.4) is smaller by a factor (c,/v,)~ M IO-6 than the second term and can therefore be neglected. According to (3.50) and the continuity equation, the electron charge density following the sound wave is given by

=

1 (eo(QQ) -

l)( -

v$Qig

) $ ne(zio(QJ2)).

(5.5)

b. The Local Temperature Deviation The local deviation from the equilibrium temperature Tin a sound wave will be calculated using Eq. (3.51). However, we must first evaluate Eerr . According to Eqs. (2.6) and (3.31-34), the effective field ~!&rr,~ is given by

QG(Q)PAQQ) Q[> - 1) + 11pron Q[Q/~dQQ>>+ G(Q)1Pion-

Eeff.0 = EQ + i(4n/Q2)

=

-i(4rlQ2)

M --i(4nlQ2)

Using Eq. (5.4) for E&Q& and Q, be written as E eff.0

=

-

54, [ Q2eve2-

-

&rr,o can, to the two lowest nonvanishing

(5.6)

orders in IR

G’rn, - i.f2Q2v,2&mb

Inserting (5.7) into (3.51) and again taking into account terms to the two lowest

64

SANDSTR6M

nonvanishing orders gives us the following deviation T(QsZ)

expression for the local temperature

(5.8a) For the time being we assume that the frequency JJ of the sound wave is so low that Szv,,, > Q2Kzo. According to Eqs. (3.13) and (3.14) this means that (5.8b)

ZQ u,r’2T~o < 1.

The condition (5.8b) can be essentially stronger than (5.2). Equation (5.8b) enables us to simplify (5.8a) to

(co(QQD 11 - $&

&

[Kzo - $11.

(5.8~)

There is an almost complete cancellation amongst the last three terms in (3.51). The only contribution that survives is the second term in the square brackets in (5.8~). The replacement K2o

-+

Kzo

- W~oIKoo)

W)

means that it is the heat conductivity which is measured at zero electric current instead of at zero electric field that appears in (5.4) and (5.8) [cf. Eqs. (3.5,6) and (3.10)]. This modification is usually of minor importance (see Ref. [21, p. 1971). More explicitly, (5.8~) can be written as (5.10) where we have introduced the ordinary Griineisen parameter yor

_ (w” I r>

YGC=-
Equation (5.10) has the same form in insulators (see, e.g., Ref. [22, Eq. (27.2)]) except for the factor Cyn/C, , which is close to unity above liquid helium temperatures. c. Longitudinal

Sound Velocities

We have already in Section 2 restricted ourselves to the case where the sound wave is propagating along a high-symmetry direction in a cubic crystal. With this

SOUND

IN

AN

ANHARMONIC

restriction, it is possible to separate transverse ones. We start by studying denote the longitudinal components motion (A.26) for (u(Qfi)) takes the

METAL.

65

II

the modes into pure longitudinal and pure the longitudinal modes, and, as before, we by an index Q. In this case, the equation of form

Here we have neglected both the last term in (A.26) and the terms in (A.26) that involve nonzero reciprocal lattice vectors. In the collision-dominated regime, a simple estimate shows that the ignored terms are quite negligible. h,(QJ2) is given by Eq. (A.27). If the phonon propagators in this equation are taken in the harmonic approximation and the w integration is performed, one finds, for small Q and Q after some straightforward but rather lengthy manipulations that

.

1~

a@,

+

-(q++)

-Q 9++ j

j1

2 w2(sjd - 44.L)

jI

q-+

Q

j2

.i

[ 2WI) + 1 _ 24w2) + 1 ] t1 - 6,&)/* 4qjd 4wTJ (5.13)

For small Q, the anharmonic coefficients can be expressed in the following way in terms of the effective ion-ion interaction V&R): Qi Q a

-tq+% B Cp (y

=dcQ.R(l R -pQ

;

,“)

= -&c

R

hj(Q.n> is directly related to the isothermal

-cosq.R)V$,V,fn(R), (Q * R)2 (1 - cos q . R) V,$,,,V,n(R). (5.14) sound velocity ci*“‘(Qj):

A,(QQ) = [c’i”(Qj)]2 Q2. (5.15) s &(Qln) has the same form as in an insulator (see Cowley [23] and Niklasson [5]), 595/71/I-5

66

SANDSTRaM

the only difference being that the effective ion-ion potential Verr is used in metals instead of the atom-atom potential that is used in insulators when evaluating the harmonic frequencies w(Qj) and the anharmonic coefficients (5.14). Using (A.28) and (1.10.36) the second term in (5.12) can be written as

4qj)

=-

where p is the mass density and $ 4qj) (

r%l) w

r%l)

o1 , I >

6J

P (

= ~2~ . Inserting (2.51) for 1 01), we find that

j o1) = cd4qj)

I Ah)>

+ cg ( w2hj)/%d

+ c2( +

Since we only need terms attenuation, it is possible the underlined terms then (2.25) that gives a nonzero be rewritten as

where we have introduced (see Niklasson [4, 5])l

w2(qj~&d

4qj~i%q)

/ &-IQ

. v , ,&

1 RilQ

. v

1 Ril

, QLxtj.

lowest orders in Q and Sz to calculate the Sz = 0 in R-l (= R;l). Due to symmetry, the same reason it is only the upper term in to the last term in (5.17a). Thus (5.17a) can

the phonon contribution w2hj>

y%(q) w

(

, v2)

(5.17a)

ol, = co(w2 I ye) - i&honQ

w

7$:9”,: = i

(

to the two to put Q = vanish. For contribution

w”(crj) rt%s) (

(5.16)

R;l

to the viscosity tensor $t$ w2(ti’)

I

&
I

(5.17b)

>

9

(5.18)

7jshon = r]g!y* . 1 The phonon contribution to the lattice viscosity $$a phonons in the phonon gas v$$~~~ gn8which is given by 4qjhYti)qp

T~hOllOllg&S = i &,v8

(

w

is not the same as the viscosity of the

R-l I

0

4ti)wy’W)q8 I

w

>.

From the structure of these quantities one can expect that they are of the same order of magnitude.

SOUND

IN

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METAL.

67

II

After inserting c,, (cO = T/T) from Eq. (5.&c), we find from (5.16) and (5.17b) that

1 _

Q2

i

Q%o

[

K (

--

2Q

K?Q

KOQ

_

irlPghon

11

Q I

. ctio,,

/-

(5.19)

Next we consider the last term in Eq. (5.12), pe is given by Eq. (5.5) and c,(QO) by Eq. (3.29). Expanding EJQO) to the same order in Q as pe gives us p(QO)+ e =

G(Q) ‘cf~f~)&

iNsz, 4rre ” ___- 1 ( EQ@@

+

Q2

KIQ ___~

K OQ

1

~

4QO)


1 Y)

1

’ Q(uQ>

pion

1

(5.20)

Too

where Eqs. (3.7) and (3.19) have been used. If we insert (5.19) and (5.20) into the equation of motion for (no(QJ2)), i.e., Eq. (5.12) it takes the following form Q2(~~(Qln>> = MQ’(QJW~Q(QQ)> = G’Q(Q) - irdQsz>)2 =

h,(QG)(u,(QS2))

(5.21) where M/(QS2) is the retarded phonon self-energy and a,(Q) = cXQj)Q is the frequency of the sound wave. Using Eqs. (2.58, 59), (5.11), (3.10), (3.11), (5.15), we get the following expressionfor the longitudinal (adiabatic) sound velocity c,Q : c;Q

=

(c$;')~

+ ; y&r2TCV9 - 2

TSthermyGrr2.

(5.22)

68

SANJXTRGM

The scaling factor r2 [Eq. (2.59)] is equal to unity above liquid helium temperatures. The Grlineisen parameter varies between 1 and 3 for most metals (see Kittel [24, p. 1841). The sound velocities in the harmonic approximation can be obtained from the long wavelength limit of the phonon frequencies w(Qj). We will not go through the details of the derivation (see, e.g., Wallace [25]). The following results are found for the elastic constants, in terms of which the sound velocities can be expressed. S8rZ2e + c’ (VW(~) + 2g”K%(g) -I- i (g”)” K%Cg))l, B (5.23a) cl2 = (--&-)”

I-&

- S8rZ2e (5.23b)

8

where (5.24)

In Eq. (5.23), the x, y and z axes are assumed to be along the (1, 0, 0), (0, 1,O) and (0, 0, 1) directions which is a different choice from the one in the rest of this article. Verr(q) is given by Eq. (1.3.3), S by Eq. (3.39). ~(0) is the polarization for Q = 0 and on comparing Eqs. (1.7.5) and (3.29) one finds that k‘& 4?re2 rr(o) = 1 - yCk&/kF2

’

where yG is defined in (1.7.5b) (and has, of course, nothing to do with the Grlineisen parameter yGr). From (5.23) we can directly get the longitudinal sound velocities in the high-symmetry directions (see Kittel [24, p. 1211): (1, 030)

60

(1, 1,O)

CEO = HG,

+

(1, 1, 1)

4*

+ 2Gz

= GlP~

= Jdc,,

Cl2

+ 2C*J/P, + 4GdP.

(5.26)

The second and third terms in (5.22) may be interpreted in the following way. Both terms appear due to the local temperature variation accompanying the sound wave. The second term has the same form as in an insulator (except for the r2 factor)

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69

and it represents the ordinary difference between the adiabatic and the isothermal sound velocities. However, in metals the local temperature variation also gives an electric current which according to Eq. (5.12) affects the sound velocity. This is the reason for the appearance of the third term and it also explains the presence of the thermopower. The second and third terms in (5.22) have essentially the same temperature dependence and they are of the same order of magnitude, the latter being the smaller one. To see this, we insert (4.22) for the thermopower in the third term in (5.22). Also using Eqs. (2.58-61), we can write it as -(Z/W

mrr2b

d/(qz I qr>/(~ I w> TW’/ve

d(kz I kJ(l

I lW,,/r,,).

(5.27)

With the estimates (5.28)

2/<&T I q.>/(w I w> = 1/3c,1, and v, v% Equation

I km

I 1) - Gw@F)/~

5%n,

(5.29)

(5.27) takes the form (5.30)

--U/P) ydGBr2(~d~g~).

Thus the third term in (5.22) is a factor 1r,,/r,, I smaller than the second. From the magnitude of the thermopower this factor can be estimated to be about l/10 [cf. the discussion below Eq. (4.24)]. d. Attenuation of Longitudinal

Sound Waves

The attenuation of longitudinal sound waves OLDmay be obtained from the imaginary part -2M’,(Q@ of the retarded self-energy M&Q&?):

olo = ro(Q, Q = GoQ>. CSO

(5.31)

Before we write down the expression for aD we notice from Eqs. (3.8) and (4.31-33) that to first order in Q and Sz

KIO(‘& 0)

K,o

1

-K 00 = Koo(O, 0) ’ 1 _ i* (42 I 42) ( rgq 1 *

&o&40) 1 + Koo@, 0) (

$J <4x! I 42’) ). r,,

(5.32)

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SANDSTR6M

Now from (5.21), (5.31,32) we get the following expression for the attenuation cue:

(5.33)

Here Eq. (3.10) for the heat conductivity K and Eq. (3.11) for the thermopower Sthermhave been used [seealso the argument below Eq. (5.9)]. Of the three terms that contribute to aQ , the first term is usually referred to as the thermoelastic term. Its value is discussedand tabulated in Refs. [26]. The size of vPhonin the second term is difficult to estimate since it depends sensitively on the variation of r:Jq) through the zone. The phonon viscosity 71 nhonhas contributions both from the phonon-phonon and the electron-phonon collision terms, the latter being dominant at low temperatures. A detailed comparison between the structures of the first and second terms in (5.29) suggeststhat the contribution from the viscosity 9Phonis of the same order of magnitude as the contribution from the phonon part of the thermal conductivity in the first term. If the condition (5.8b) is relaxed, the thermoelastic term in (5.33) must be written in the following way: thermoelastic aQ

_

The thermoelectric corrections have here been neglected, which meansthat we have assumedthat K,2, < KoQK2Q [seealso Eq. (5.9)]. The second term in the curly brackets has essentially the same form and temperature-dependence as the thermoelastic term and is smaller by a factor 1r,,/F,, /. This follows from precisely the sameargument which led to Eq. (5.30). The second term in the square bracket in (5.33) has the sameform as the phonon contribution to the heat conductivity [see Eqs. (3.14) and (4.25b)], which is, as we have mentioned before, smaller than the electron contribution. It will thus only give small corrections to the thermoelastic term. The first term in the curly bracket gives the dominant contribution at low temperatures. We shall refer to this term as the electron contribution to the attenuation. In analogy with the second term in (5.33), the electron contribution can be interpreted as coming from the longitudinal viscosity 7: of the electron gas ez

TQ

=

*mbu,

2

rll

Q

,

(5.34)

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where the viscosity

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71

II

tensor is given by

&.vs = i(
(5.35)

and (5.36a) If the component of the electron gas viscosity tensor is unspecified, one is usually referring to the transverse part 7:’ (see, e.g., Ref. [27, p. 2051) where

Within the single-relaxation-time approximation the electron contribution to the attenuation was first derived by Pippard [6], whose result for a free electron gas in the collision dominated regime was (5.37) 7 is the single relaxation time which also appears in the electrical conductivity. Pippard obtained his result through calculating the Joule heat dissipated by a sound wave. His result has been generalized to nonzero magnetic field by Cohen, Harrison and Harrison [28]. In a later publication [29] Pippard also took into account band structure effects. Several authors have reviewed and extended these results (see, for example, Morse [30], Kittel [31, Chap. 171, Spector [32], Steinberg [33]). Many experimental results on ultrasonic attenuation [34] have confirmed that the temperature-dependence of the relaxation time appearing in the attenuation agrees with that of the relaxation time in the electrical conductivity. Also their absolute values have usually the same magnitude (within a factor of two) in simple metals. Sometimes, however, they differ up to a factor of five. In our treatment we have not made the single-relaxation-time approximation, and as a result we have found that two different relaxation times 722 and 72 appear in the electrical conductivity and in the electron contribution to the attenuation. This fact has been noticed before by Steinberg [7] and Bhatia and Moore [8]. We shall next discuss how 722 and ~,“1are related to each other. Using (4.43) and the approximation (4.2) with the distribution function q$ = (1 - P) 1k,E,‘) enables Q as us to write. 711

l(k,&‘Or) I (1 - P) I W,‘(k))12 o,2T’(o’ ‘) = (k,E,’

I (1 - P) 1k,E,‘)

(k, I k,)(k,E,‘Q

= (k,E,’

I k,E,‘)

j I’ I &E,‘(k)) -

@tl’ ,“;;‘”

’ .

(5.38)

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SANDSTRaM

For a system with scattering only close to the Fermi surface, which is spherical, we get @a&,’ I k&l!?

(5.40)

= Q&?YkmI J&J,

&cE,’I (1- p>Ib%‘)= (Q- 1)QYkl:Iw.

(5.41)

Here we have used the definition of u, given by Eq. (2.61). When writing an explicit formula for (k&,’ ( R;’ ( kJ&‘), we consider only the electron-phonon contribution, which is usually dominant. Using (2.7) and (2.14), we can write the matrix elements appearing in T&O, 0) and ~19in the following way: @a! I r I kz) I.k+%’ I r I b%‘) !

When deriving (5.42) we have used a method described, for example, in Ref. [35]. We will now explain the connection between the work of Bhatia and Moore [8] and our results. Bhatia and Moore calculate the ultrasonic attenuation for an isotropic electron gas. In analogy with the elastic constants, the components of the viscosity tensor for an isotropic model satisfy the following relation (see, e.g., Ref. [22]): q&m

(5.43)

= rl2.w + ~7l%a, *

They furthermore assume-as we have done in (5.42)-that place on the Fermi surface, which means that (k,E,’

+ k,E,’

+ k,E,’

I R;l I k,E,‘)

the scattering takes

= 0.

(5.44)

k,E,’ + k,E,’ + k,E,’ is constant on the Fermi surface, and can be taken outside the integral. Further, (1 1R,” 1k&‘(k)) = 0 since (1 - P) I 1) = 0 according to Eq. (2.26), and Eq. (5.44) follows directly. From (5.44) we find that

El qrr.rx + 271::,,, = 0. Thus, for an isotropic model, there is only one independent component electron gas viscosity. From (5.36, 37), (5.43) and (5.45), it follows that q”o” = jqy = +fl.

(5.45)

of the (5.46)

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Bhatia and Moore use the two relaxation and rg by 71

=

722(0,

T2

=

qTj@)

0)

=

CL

= (k,&’

I wkz

j (1 -

METAL.

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II

times TV and TV which are related to 722

I l-1

(5.47)

kz),

P) / k,E,‘)/(k,E,’

1r 1k,E,‘).

(5.48)

In (5.48), Eqs. (5.38) and (5.41) have been used. In a recent paper, Rice and Sham [9] have used the model of an isotropic solid in Ref. [S] and performed detailed calculations for the temperature variation of the ratio T~/T~ in potassium. Potassium is a suitable substance to investigate because band structure effects are negligible. They found that the ratio T2/T1 varies between 0.5 and 1.5 in the temperature interval 2” - 20°K. TJT~ was found to depend sensitively on the choice of the pseudopotential. The result that T2/T1 is so close to unity explains why Pippard’s approach of using a single-relaxation-time approximation works so well. Rice and Sham also extracted T.JT~ due to the electron-phonon scattering from the measurements of Natale and Rudnick [lo], and a ratio between 1.5 and 3.0 was obtained. This discrepancy between the theoretical and experimental values is not fully understood. One problem is to determine how much the results are affected by the anisotropy of the phonon system. For an anisotropic system, the formulas for the ultrasonic attenuation used by Bhatia and Moore [S] and Rice and Sham [9] are clearly not valid. Instead, the ultrasonic attenuation must be obtained from Eq. (5.42) without making use of (5.43). The effects of the anisotropic electron-phonon matrix elements on different averages in sodium have been discussed by Grimvall [36]. He found that the changes due to anisotropy can vary between a few per cent up to several hundred per cent in extreme cases, depending on the type of average. This makes it difficult to draw any general conclusions. We have so far assumed Umklapp-dominated scattering, which is the ordinary case. Are T&O, 0) and ~fi(O) still closely connected for the scattering in which normal processesdominate ? From (4.25) and (4.26) we find that

ado, 0) = [& I k3/rd ~~~(0,0) = [(k, / k,)/r,$

Wmklaw scattering,I r,“, I > I J?$ I>, + I’,“, + 2rzJ

(normal scattering, I r,“, I < I r,“, I). (5.49)

From (5.49) we see that T&O, 0) is given mainly by the number of Umklapp processesboth for Umklapp and normal scattering. T,",(O) on the other hand gets its contribution mainly from normal processeswhen these dominate. Since the Umklapp and normal contributions have quite different temperature behavior (seebelow), there is no reason to expect that T&O, 0) and ~$0) have values which are close to each other where normal processesdominate. Where can such scattering appear? In metals where the Fermi surface cuts the

74

SANDSTRijM

zone boundary it has been shown by explicit calculation that Umklapp processes are completely dominant at low temperatures (see, e.g., Ref. [37]). At higher temperatures the Umklapp contribution dominates the normal contribution in general. The only situation in which we can expect normal processes to dominate is in the alkali metals at low temperatures. In these metals, the Fermi surface does not cut the zone boundary, and the electron-phonon Umklapp processes vanish exponentially as T goes to zero. The normal processes on the other hand behave like T” where 01 = 5. If nothing else interferes, this means that Umklapp processes are negligible for low enough temperatures. At what temperature does the transition from Umklapp to normal scattering take place ? Experimentally this transition has only been observed in connection with thermoelectric coefficients (in potassium around a few degrees Kelvin [38]). At the same temperatures the electrical conductivity is dominated by defects. The reason for this difference is that (k, 1 r / k3c>and (qe 1r j q2) are changed by more than (4% I I’ / k,) by defect scattering. Hasegawa [39] has calculated the Umklapp and normal contributions to P,, in sodium and potassium. He found that these two contributions had about the same magnitude at 3°K in both metals and that the Umklapp contribution vanished rapidly at lower temperatures. In a defect-free metal we must also take into account the electron-electron Umklapp processes which become dominant at very low temperatures and which give a T2 contribution to the electrical resistivity, [12, p. 4121. This T2 dependence has been observed in free electron metals [40]. However, defect scattering can give the same temperature-dependence and it is unclear at present as to which of these is the source of the T2 contribution. Anyhow, we expect there to be a temperature interval where normal scattering is dominant, and which is limited from above by electron-phonon Umklapp processes and limited from below by electron-electron Umklapp processes. Also the anharmonic phonon-phonon scattering involves Umklapp processes. However, these are expected to be “frozen out” at temperatures higher than those for which the electron-phonon Umklapp processes are frozen out. e. Attenuation

of Transverse Sound Waves

Transverse phonons are simpler to handle due to the vanishing coupling to the local temperature deviation [see the discussion below Eq. (3.52)]. The equation of motion for the lattice deformation (u(QG)) takes, in this case, the form

Q' . QZ 4rre . -_ N - I C2M -1Q2 leL -2

sZ,Or MN ( k, I

I y@,

Q, 4w,

vie(Q) Q2 w] 4n-e

QQ)-

QQn) k * *

(5.50)

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X,(QL?) is given by (5.13), where the polarization index denotes modes. In the same way as in Eq. (5.26), we can directly write In sound velocities cSI in the harmonic approximation. directions the following results are well known (see, e.g., Wave in the (1, 0,O) direction:

one of the transverse down the transverse the high-symmetry Kittel [24, p. 1211):

ions moving in the (0, 1,0) or (0, 0, 1) directions,

Wave in the (1, 1,O) direction: ions moving in the (0, 0, 1) direction,

2 CSI

= G,iPY

ions moving in the (- 1, 1, 0) direction,

2 CSL

=

tdC,l

-

G,>lP.

Wave in the (1, 1, 1) direction: ions moving in the (1, - 1, 0) or (1, 1, -2) directions, 2 cs,

=

%G,

-

Cl2

+ GJP.

(5.51)

The elastic constants C,, , C,, , and C,, are given by Eqs. (5.23). In analogy with (5.16) and (5.18), we can write the second term in (5.42) as

= f r)~honQz(ti,),

(5.52)

where (5.53) ?&I)

=

r3-l).

Using (3.49), the third term in (5.50) may be written as (5.54)

76

SANDSTRGM

eI(Qln) [Eq. (3.48)] can be expanded to the two lowest nonvanishing and 52 in the following way: 1

c2Q2 4i7eZN(&)

m=

orders in Q

c2Q2 ve2 (

Q4rre2N(EF)

u,“ri2 )

. (5.55)

In (5.55), Q2 has been neglected compared with c2Q2, which is many orders of magnitude larger. Furthermore, the factor (3.40) has been put equal to unity in the same way as in the longitudinal case. If we use the same type of expansion for r as in (4.2) and take into account the distribution functions 1k,) and 1 qy), the fourth term in (5.50) can be rewritten as -2

Q

=-

23&

[

r,,

(kv I Y) + r (41/ I a> (k, I 0 kq (41/ I 4r)

-

r,k+,>]-

(5.56)

The second term on the right hand side of (5.56) takes the following form if the solution (2.51) of the transport equations is inserted: -2-L

Q MN

rk, (qv


+ (qu I R-l 1I”

I R-l

I k,)

I

0

Eeit,l mb

oIt&-d ) Q(cl> +
(5.57)

In the first term in (5.57) we substitute Eqs. (4.12) and (3.44) for R;’ and EL , recalling that EM = E for transverse modes. For the second and the third terms we use (4.35) and (4.47): -2

Q,

rr%k, I k,)

MN rkkrqq

c2Q”

L

- l-‘iq 4n-e2N(EF) ve2 TA

(til>

+

rkgrkk

2 52, MN

rkkrq,

-

‘ii

(5.58)

We note that the terms in (5.57) cancel each other to a large extent. The other two terms in (5.56) can be written in the following way:

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77

where Eqs. (2.41), (3.7) and (3.48) have been used. Next we substitute (5.52), (5.54), (5.58) and (5.59) back into the equation of motion (5.51), and with the aid of Eqs. (4.14) and (5.55) the following result is obtained:

$44

Zmb -[

-iQ

+

&Q2ue2 + w

C2Q2 9

22

1CC,>.

(5.60)

The first term in the square brackets in (5.60) gives a completely negligible contribution to the sound velocity. From (5.60) we get the following expression for the attenuation 01~of transverse sound waves [cf. also (5.21) and (5.31)] phon

aA

=

’ -

2

71 -

(22

PCSL

+2’ aMC,,

Q” [t’,“+i

+ --+I.

In the same way as for the longitudinal case, we can regard the two first terms in (5.61) to come from the viscosity of the phonon gas and the last one from the viscosity of the electron gas. We expect the phonon contribution to be dominant at higher temperatures and the electron contribution at low temperatures (T < 20°K). There is no thermoelastic term in the transverse attenuation since there is no local temperature deviation associated with a transverse wave. Otherwise, most of the discussion can be taken over directly. How does the inclusion of the last term in (5.50) modify the attenuation ? Firstly, it gives rise to the second term in (5.61). Secondly, it gives such a contribution that the sign of the second term in the square brackets is changed. The second term in (5.61) is smaller than the first roughly by a factor ] r,,/r,, I. The only exception is for such low temperatures that normal processes dominate the scattering. But at such temperatures the two first terms are negligible anyhow.

78

SANDSTRGM

The second term in the square brackets in (5.61) is usually negligible which can be seen in the following way. The second term in the brackets is smaller than the first approximately by a factor

c2/wp2v,2T;2.

(5.62)

It is only for temperatures below 20°K that the electron contribution to the attenuation is important and for such temperatures 722 w 1O-g - IO-12s(as estimated from measurements of the electrical conductivity). The other parameters in (5.62) have the following approximate values: the light velocity c w 3*108m/s, 21, w lo6 m/s, the plasma frequency wp SW 10-16~-1. Inserting these values into the factor (5.62), one finds that it is smaller than 10-3. From (4.50) and (5.35) we find that T:, is related to the transverse viscosity of the electron gas 7:’ in the following way [see also (5.34)]: rl:

=

rl&w

=

21 T1l

nmbve

(5.63)

.

In analogy with (5.38) we obtain

IkG’(W

I W,‘(k))12

v,2Tkdo) = (k, / k,)(k,E,‘(k) where, for a spherical system, (k,&‘(k) (b%,‘(k)

I W,‘(k))

1 r j k,&‘(k))

/ k,&‘(k))

’

(5.64)

can be estimated to be

= &Ykr

I k,).

(5.65)

It follows from (5.45) that if T,“, is closely connected to 722 , then so is ~1: . The Pippard theory is thus justified also for transverse phonons. In fact, for an isotropic system the ratio between the longitudinal and transverse electron contribution to the attenuation is given by 4~,

= (GJCsa)

- (4/3).

(5.66)

The elastic constants in Eqs. (5.23), and thus also the sound velocities, depend on the form of the electron-ion potential vie(q) via 6 [(Eq. (3.39)] and V&q) [Eq. (1.3.3)]. This dependence on v,,(q) is well known to be quite sensitive. Also, the detailed form of the correlation field strongly affects the sound velocities [Eqs. (1.7.5) and (5.25)], i.e., the value of yc is important. In the expressions (5.33) and (5.61) for the ultrasonic attenuation on the other hand, neither yG nor vie(q) enter explicitly. The only way the attenuation depends on the particular forms chosen for the polarization r(q) and vie(q) is via the matrix elements in the collision integral. Throughout this paper bandstructure effects have been neglected. This means

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79

that, for the sound velocities, terms up to second order in Vie(q) have been taken into account (see, e.g., Harrison [41]). In some recent papers [42-441, it has been investigated as to how much the sound velocities are affected by higher order terms in vie(q). For the alkali metals sodium and potassium, the change is about 5 % [43]; for aluminium, l&20 % [44].

6.

CONCLUSIONS

In this paper the ultrasonic attenuation in metals in the collision-dominated regime has been investigated. Three kinds of contributions to the attenuation have been discussed: the thermoelastic damping, and the damping due to the viscous properties of the phonon gas and of the electron gas. We have found that the thermoelastic damping has the same form as in insulators except for the factor r4 discussed below. The thermoelastic damping is larger in metals than in insulators, because it is proportional to the heat conductivity, and metals are better conductors than insulators. Above liquid helium temperatures, r2 is equal to unity, [Eq. (2.59)] but at temperatures lower than these, r2 vanishes as T4. Since the thermoelastic damping is negligible at such low temperatures, the factor r4 is not very important. The second contribution to the attenuation comes from the viscosity of the phonon gas r) rho” . The phonon viscosity looks formally the same as in insulators, but both the anharmonic phonon-phonon and the electron-phonon scattering contribute. The electron-phonon scattering dominates at low temperatures (T GZ T,/lO) and the anharmonic scattering at higher ones. Since the anharmonic part is described in the same way as in insulators, 7 phon has the same temperaturedependence at higher temperatures but it increases more slowly at the lower ones. Estimating the contribution from 7 Phonto the attenuation, we have found that it has about the same magnitude as the contribution to the thermoelastic damping from the phonon part of the thermal conductivity. Since the phonon part is smaller than the electron part, one can expect the phonon viscosity damping to be masked by the thermoelastic damping for longitudinal sound waves. But this is not so for transverse waves since for these there is no thermoelastic damping. The most important contribution at low temperatures comes from the viscosity of the electron gas @. This contribution behaves as T-5 when T+ 0 in an ideal system, while the thermoelastic term is proportional to T-l. This behavior appears only in a small temperature interval because at still lower temperatures the attenuation is dominated by defect scattering. In the single-relaxation-time approximation the electron contribution is proportional to the conductivity. If this approximation is dropped, two different relaxation times appear in these quantities. In this paper the results of Steinberg [S] and

SANDSTROM

80

Bhatia and Moore [9] have been generalized to an anisotropic system. For such a system the transverse and the longitudinal attenuation are not necessarily proportional to each other any longer. Rice and Sham [9] have shown in an explicit calculation for potassium that the relaxation times in the electron contribution and the electrical conductivity are quite close, thus verifying Pippard’s result. However, if normal processes dominate the scattering we have shown that this result will not be true anymore because the scattering by normal processes gives a contribution to the electron attenuation but not to the electrical conductivity, and the contributions from normal and Umklapp processes have different temperature dependence. Scattering in which normal processes dominate can be expected to appear in sodium and potassium below 3°K [39]. We have found that the only way the electron polarization and the electron-ion potential enter the electron contribution to the attenuation is via the screened electron-phonon matrix element in the collision integrals. In the expressions for the sound velocities, on the other hand both, the electron polarization and the electronion potential appear explicitly. The lowest order anharmonic contribution to the isothermal sound velocities has been shown to have the same form as in insulators, the only difference being that the atom-atom potential has been replaced by the effective ion-ion potential. Certain thermoelectric corrections to the thermoelastic damping and the difference between the adiabatic and the isothermal sound velocities were found. These are expected to be about 10 %. APPENDIX:

EQUATION OF MOTION

The Hamiltonian H = T g + $

FOR THE LATTICE

(u(Rt))

for the ions is2 [cf. Eq. (1.2.1)]:

+ ; c’ Vi,(R - R’ + u(Rt) - u(R’t)) RR’ z I dr uie(R + u(Rt) - r) p&t)

- c y R

-3t

DEFORMATION

cr

2c=N= RR’

jidR)

. jdW

IR-R’I

$$ s je6) h IR-rl

- ; u(Rt) 1 J(Rt).

Vii is the direct ion-ion interaction. jion and j, are the currents of the ions and electrons, respectively. fe is the electron charge density. c is the velocity of light. * For the fourth and fifth terms in (A.l), we have assumed the ions to be point charges. It is straightforward to relax this condition, as in the third term. However, it complicates the formalism considerably and one can show that the error involved is negligible.

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It is straightforward (see, e.g., Ref. [3]): A4 $

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to obtain the equation of motion

(u,(Rt))

= - c (V,V,,(R

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for (u(Rt))

in the form

- R’ + u(Rt) - u(R’t)))

R’

+

$

j

+ &tRt)

dr


- s

+

j

u(Rt)

-

r)

p&t)>

a
The right hand side of Eq. (A.2) is not in a very convenient form. We would like to expand the anharmonic part in terms of the effective ion-ion potential Verr [Eq. (1.3.3)] instead of Vi, , We therefore rewrite the first two terms on the right hand side of Eq. (A.2) as - c (V~V’eff(R - R’ + u(Rt) - u(R’t))) R’

+ 1
KJ(R

- R’ + u(Rt) - u(R’t)))

R’

- k j dr (VaTvi,(R + u(Rt) - r) p&t)).

(A.3)

According to Sandstrom and Hogberg [19] all anharmonic terms can be referred to the effective ion-ion potential as it appears in the first term in (A.3). Thus we need to keep the anharmonic terms only in the first term in the expression (A.3), and the other two may be treated in the harmonic approximation. The third term in (A.3) may thus be rewritten as j dr Va,‘dR

=

I

- WdrtD

- C j dr V&@ 0

- MdW

p&))

dr Vei,“ui,(R - r)
-a 0

dr VkdR

- r)hW

- MW))

p&D

(A-4) 595/71/r-6

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SANDSTR6M

because higher order terms in u(Rt) give anharmonic contributions. The only contribution to (p,(rt)> in the last term in (A.4) comes from the static charge distribution. Other contributions would give nonlinear terms in the equation of motion. To the lowest anharmonic order the first term in (A.3) may be written (see Ref. [3]):

We have here used the same notation as in Ref. [3], i.e.,

a,,, is the Kronecker symbol. Since we are only interested in linear terms in
whereAs+a,+a, = otherwise.

1 if q + q, + q2 is equal to a reciprocal

vector, and zero

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- e-"qz*(R2-R3)e,(qj)

e,,Wl)

e&&k,(q&)

83

II

~q+q1+q2+q3

.

(A.71

To obtain the underlined evaluated:

term in (A.4), the following

propagator

has to be

because p&t)

= -2e#+(rt)

#(rt).

(A.91

The three-point propagator (A.8) is complicated to calculate. The result for the corresponding term in the equation of motion is given in Eq. (A.20). The reader who is not interested in the details of the derivation can proceed to that equation. Rather than considering the general case at once, we will first treat a simple example which is going to be generalized later. We start by rewriting (A.8) as the time-ordered product

) #+(rt+) W)>,

(A.10)

because we are going to use Wick’s theorem [45]. Taking the third term in (A.11 into account, that theorem gives to lowest order

The significance of the L integral is explained in Appendix A.a in I. Using the definitions of the propagators (1.2.2-3) and the analogy of Eq. (I.A.2) for the Green functions, (A.1 1) can be rewritten as

- rd{%+(Rt, Rltl) G(rltl ,rt>GM, rltl) 2L &z sdrlV,‘:u&G - &(Rt, Wl) G’(rltl , 4 G%t, rltl>>.

(A.12)

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We linearize (A. 12) and substitute it back into the underlined term in (A.4), which as a result takes the form

+ D,<,,(Rt,WI) G>(vl , r0 @(rt, rdd>.

(A.13)

Next we Fourier transform (A.13) using (Eqs. 1.2.13-2.16). For the time being, we assume that Q < qD and Q < kT to avoid very unwieldy expressions

In the same way as in I [see the discussion below Eq. (1.2.21)], we choose a representation that makes the harmonic part of the phonon self-energy M,,(qo) diagonal. In this representation, the nondiagonal elements of D,Jqw) and Dj$(qw) can be neglected. Also the contributions from the nondiagonal elements of the nonequilibrium propagators will be neglected, because they are nonsingular for small Q and J2. Now using the following symmetry properties of the phonon propagators, Q(--Q, I&y-q, &C--Q, Dj>(-q,

-0)

= mlw),

-0)

= &<(qw),

-w,

Qf4 = D&w,

-w,

QL’) = &<(qw,

(A.15) QQ, Qii’),

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we find, by exchanging k3W3 and k,w, and by changing the signs of q and o in (A.14), that it can be written in the following way:

- {Bj:(qw,QQ) G%w,) G%,wd - @vJ, QJ-4 GOrs+J @W, ’ &+kp-kg

s(w

+

w4

-

3QQ) - @(q4 G%wd (%wJ, 9Qsz>l (A.16)

%)*

The expression (A.16) is illustrated schematically in Fig. la-c. So far we have only considered one single term in the expansion of Wick’s theorem. Some higher order terms must be taken into account. Investigating the diagrams in Fig. la-c, one can easily convince oneself that the vertices that appear there must be screened.

(a)

(b)

(cl

(4 FIG. 1. Illustration

of Eq. (A.16).

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SANDSTF&iM

Certain vertex corrections are also considered, which is indicated by the black sectors. These two modifications imply the following replacement in (A. 16):

(k,- k4). eWdh - k4)# 2) --t%& , k4),

(A.17)

where F&k3 , k.J is given by Eq. (I.A.13). We are also going to take into account the diagrams in Fig. Id-e, which give a collision-drag contribution. Such contributions are discussed and evaluated in Section (1.5). With these extensions, (A.16) takes the following form:

do,(k4 - k3, I %& , k,Y A4+a--lrs WJJ+

~4

- 4

QQ)G%wJ~ G%,4 - a; - &&4 @(k,o, , QQ)G'h-4 + 6&J) @WJJ~ G-%w, 3QQ)- ~$w) G%,wJ 3QQ) - @$w4 (A.18) G-%,4G'(b4) - W - l. . {E&w

G%,w,)

‘3&4w4

B(k4

Comparing (A.18) with Eq. (1.6.3) and again using the symmetry between k3w3 and k4W4 , (A.18) can be written as k4,JepOw4

3

Q-Q).

(A.19)

With the help of Eqs. (1.10.31), (1.10.16) and (2.14), one can show that (A.19) is equal to 2%

(ta 1p”(QQ)

QQ)

dqw, / y(k, QQ) - k * (ti(QsZ))

’

(A.20)

where the matrix notation (2.19) is used. The indices e and p on P’ indicate that only the electron-phonon contribution to the collision operator enters (A.20). This is obvious from the structure of (A.8). We will find in a forthcoming article that (A.20) gives rise to that component of the energy transfer from the sound wave to the combined electron-phonon system which comes from the electronphonon collisions [46, Section 41. It is well known that this term is most important at higher frequencies [28]. According to the discussion in Section 1.5, it is only possible to neglect the Q-dependence of the collision operator for Q < kT. Usually, terms involving the collision operator are negligible for higher frequencies, but (A.20) is a possible exception. We have assumed that IR < kT when deriving (A.20). As to what happens at higher frequencies, that will be discussed in the forthcoming paper.

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With the help of the expressions (A.47) Eq. (A.2) and get

METAL.

87

II

and (A.20), we can Fourier transform

--QW~(QQU -~2(QWdQ-QD PL+ -(q-T) -Qsg &&w, i-f&paJ1'2 Qsz> ( . (Q e(Qj)(Q (u(QQ)) e(Qj> 4Q =

.i2

Jl

+ ; ;

+ $2

c 0 (7 ah%

;2q

+ 8) -

2) SI

+ g> 4Q

. v? (Q 2e

TIQ

j

J;) j- e

4,(w4

+ g) *

(- 1) ‘“,ze2”‘”

+ g, 0) - 1 + g, 0)

g -

(-1) t,fe(g) E&L O) - l %fk 0) - i T (Q + g) * 4Q.d UQ + g) dQ + g, Q) (- i) . g * (u(Q.n>>&

+ ;’ g - 4Qj) g - (u(Q.n)> dd

+ [iQg

dg> (- f,

j,(QQ) $$--

iQZ ‘@

Q

(ii(Ql2))

Gw+

$1

* (1 -

QQ)

+2MN --JL (”ki 1rep(QQ’ / y(k, 452) - k - (ti(Q.n))

’

F)

e(Qj)

(A.21)

D+$, is given by (1.8.8). According to the discussion in Section 1.2, such nonsingular contributions as the one coming via DiXj, (j, # j,) can be calculated with the help of perturbation theory. Thus Dili, (j, # j2) can be obtained in analogy with (1.8.14).

G,i,(w

QQ

= ---M{&,(l)

QJl’)

- G$l)

D:(l’N

~~,~,(l, 1’1,

where l=(q+&J+$);

l’=

(

q-++).

(A.22)

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SANDSTRiiM

ZjljB(l, 1’) is given by Eq. (1.7.14). The Umklapp terms ps(Q + g, J2) (g # 0) and p,(g) give nonsingular contributions as well which can thus be obtained with the help of perturbation theory. In analogy with (3.31) and (3.24), we find the following form of the Umklapp component of the charge response of a deformation wave

v,(q) = 4re2/q2 is the Coulomb interaction. To distinguish between the dielectric function derived from the transport Eqs. (3.53), and the one derived with perturbation methods [(Eqs. (1.7.5), (A.23)] we have denoted the latter Eof(qW) since it is only valid in the collision-free regime. The static potential in the lattice

(A.24) that the electrons see gives rise to the following according to Eq. (3.25):

electron charge distribution,

(A.29 Substituting (A.22), (A.23) and (A.25) into (A.21), we get our final result for the equation of motion for the lattice deformation (u(QsZ)):

* Q *
- &

c’ !Q + 8) .

E

e(QNQ + g> - >

(A.26)

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where

MQQ) = w”(Qj) Sjjr -

-Q q+$ j

-Q j

-(q++j

q-+Q

jl

jl

jl

q-T

-(q++)

‘I++ jl

.il

Q J2

* 1 g {Dj,(l)D,,(l’)- Dj’(l) Dz(l’)l,

j’

.i’

(A.27)

and F,*(qjp,

QQ) = ; +, @ fQj;

q

-aQ

“91’

qj

(A.28)

(the star indicates complex conjugation). The interpretation of the different terms in Eq. (A.26) is discussed in Section 5. In the collision-free regime, p,(QsZ) can be calculated with the help of perturbation theory and then the second and the third terms from the end in (A.26) take the following form:

(A.29) We note that g = 0 is included in the summation in (A.29). From (A.29) one gets the standard expression for the electron contribution to the attenuation of phonons in the collision-free regime (see Refs. [20,47,48]). How such a calculation is performed is shown in Section 5. Since the imaginary part of EJQ + g, Sz) (g # 0) is proportional to Q for small Q and LJ, (A.29) leads to a ratio between the phonon line width and the phonon frequency, which goes to infinity when Q goes to zero. One can also see from the numerical results of both (20) and (47) that the line widths

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SANDSTR6M

of the transverse phonons go to zero slower than Q. This result cannot be correct because phonons are well-defined quasi particles also for small Q. This problem does not appear in the calculations for Li done by Schneider and Stall [46]. There are no Umklapp contributions for small Q values in the alkali metals since the zone does not cut the Fermi surface. This means that there is no attenuation of the transverse phonons for small Q in the high-symmetry directions, which agrees with what Schneider and Stoll found. The reason for these unphysical results is that we have used the one OPW model for the electron-phonon matrix element. The one OPW model is not always a good approximation close to reciprocal lattice vectors. Similar problems appear in connection with Umklapp contributions to the electrical resistivity in the one OPW model, which behave incorrectly in the low temperature limit (see Sham and Ziman [49]). Pytte [37] and others have explicitly shown that in a two-OPW model these problems disappear. We expect that the one-OPW model for the electron-phonon matrix element SQk, k’) [Eq. (I.A.13)] is satisfactory except when k - k’ is close to a reciprocal lattice vector. We write k - k’ as k-k’=g+q,

assuming that q lies in the first Brillouin zone. When &(g/2)q < W(g), where W(g) is the size of the bandgap (N/G’,) z&(g), S$.(k, k’) should be proportional to q instead of g + q. Thus we replace (I.A.13) for Faj(k, k’) by (see Pytte [37]): %ik

k’) = (o19-sq+ (1 -

sJ(q

+ $9) &(k

a Qg= l/(1 + u2)lj2 with u = i

- k’) 4f(k,

E’0g-24 I

k’) ;

*

,

(A.30)

(A.31)

On making the same replacement in the last term but one of (A.26), it takes the following form:

(A.32)

The replacement in (A.30) does not affect the attenuation vectors, but it gives a correct limit when Q + 0.

of phonons for larger Q

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91

REFERENCES

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