Sound propagation in an anharmonic metal

ANNALS

OF PHYSICS:

71, 93-128 (1972)

Sound Propagation

in an Anharmonic

III. The Collision-Free

Metal.

Regime*

ROLF SANDSTROM Department

of Theoretical Physics, Royal Institute S-100 44 Stockholm 70, Sweden

of Technology,

Received July 6, 1971

The electron contribution to the phonon damping is calculated by solving the generalized transport equations derived in I. In particular, the collision-free regime and the transition from the collision-dominated to the collision-free regime are investigated. The solution of the transport equations is a generalization of the one given in II and it is valid also outside the collision-dominated regime. No single-relaxation-time approximation is used and the collision operator is taken into account in full. Certain higherorder relaxation times are taken into account only approximately. It is shown, however, that this approximation only affects the attenuation in a narrow frequency range around the transition from the collision-dominated to the collision-free regime. Even there, its influence is weak as demonstrated by some numerical results for potassium. For phonon wavevectors much smaller than the Debye wavevector, the generalized transport equations take a form which is similar to the semiclassical equations. For larger wavevectors, the generalized transport equations give the same results as perturbation methods. This is explicitly verified for the attenuation. We will also show that these two methods of solving the generalized transport equation give overlapping results for certain wavevectors.

1. INTRODUCTION This paper is the third in a serieson sound propagation in metals. The purpose of these papers is to investigate sound waves without using semiclassicalmethods. In the first of these papers (Ref. [I], hereinafter referred to as I) generalized transport equations for the electron-phonon system were derived, which are valid for all phonon frequencies. These equations were solved in the second paper (Ref. [2], hereinafter referred to as II). With the help of this solution the ultrasonic attenuation was obtained in the collision-dominated regime. In the present paper the ultrasonic attenuation for higher frequencies will be calculated. * Research sponsored by the Swedish Natural Science Research Council.

93 0 1972 by Academic Press, Inc.

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SANDSTRiiM

It was shown in I that for wavevectors of the sound wave Q, such that Q < qD , kF the generalized transport equations could be reduced to a form which apart from some many-body corrections was identical to the corresponding semiclassical equations. For larger wavevectors, the indirect part of the collision operator could be neglected and then the generalized transport equations give the same results as perturbation methods. We will show here that these two methods of solving the generalized transport equation give overlapping results for the attenuation for certain wavevectors. To calculate the sound velocities and the ultrasonic attenuation the equation of motion for the lattice deformation, (u(Rt)), was derived from first principles in II. Earlier, this equation of motion had been obtained by semiclassical methods (see e.g., Ref. [3]). In the equation of motion, certain quantities appear, such as the charge density and current, which must be calculated by solving the generalized transport equations. As mentioned above for Q Q qD , kF , this means solving equations which have the same form as the semiclassical transport equations. For larger Q, the solution is already available from the perturbation theory. We are going to solve these transport equations without making a single-relaxation-time approximation. Bhatia and Moore [3] have earlier solved the electron transport equation for an isotropic solid. The solution was obtained in terms of a difference equation. If their method is generalized to anisotropic solids, the difference equation becomes much more complicated and it is difficult to solve. We will, instead, start from the formal solution to the transport equations which is given in II. This solution was obtained by a method originally suggested by Krumhansl and Guyer [4]. In this method the formal solution is expressed in terms of an operator R-l. R-l is related to the inverse of the scattering operator r in a space consisting of all phonon and electron distribution functions except for certain eigenfunctions of I’. To get the explicit solution, some matrix elements of R-l must be evaluated. This was done in II, however, by a method restricted to the collision-dominated regime. To be able to calculate the matrix elements of R-l also for larger wavevectors a new, more general, method will be developed. With the help of this method, the electron contribution to the ultrasonic attenuation will be calculated. When discussing the interaction between the sound wave and the electrons, the wavevector regions Q& < 1 and Q/l, > 1 will be referred to as the collision-dominated and the collision-free regimes respectively. /1, is the mean-free path of the electrons. The transition from the collision-dominated to the collision-free regime will also be treated. In this paper only the electron contribution to the attenuation will be discussed. We expect the electron contribution to dominate the anharmonic contribution for T 5 20°K. At such temperatures, the electron contribution can often be extracted directly from ultrasonic experiments by subtracting a temperature-independent

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95

background. Unfortunately, one part of this background, namely the dislocation damping [5], is not quite temperature independent [6]. Since the electron contribution to the attenuation in superconductors goes rapidly to zero when T -+ 0 (see e.g. Ref. [7]) this gives a possibility of obtaining the background in normal conductors. Sometimes, however, the difference between the dislocation dampings of the normal conductor and the superconductor must be taken into account [6]. Within the single-relaxation-time approximation, the electron contribution to the attenuation for Qlle 2 1 was first obtained by Pippard [8]. Bhatia and Moore [3] took the full scattering operator into account and they expressed the attenuation in terms of the solution of a difference equation. Explicit solutions to this difference equations were given for Q& < 1 and QL& > 1. In the collisiondominated regime they found that the attenuation could be expressed in terms of a specific relaxation time. This has also been shown in II. However, when approaching QA, = 1 the number of relaxation times involved in the attenuation increases essentially to infinity. In Ref. [3] only the case when all relaxation times were assumed to be equal was treated for Qfle = 1. Here a more general situation will be investigated. A few relaxation times will be taken into account exactly and the rest with the help of a relaxation-time approximation. This generalization is very essential, because the relaxation-time approximation for the higher-order relaxation times influences the solution for neither QL$ Q 1 nor for Q& > 1, but only in a narrow wavevector region around QLI, w 1. The significance of this relaxation-time approximation will be discussed in detail. In Section 2, expressions for the ultrasonic attenuation will be derived. In these expressions matrix elements of R-l appear. These will be evaluated in Section 3. In Section 4, the matrix elements of R-l will be investigated in the collisiondominated regime and they will be compared with the corresponding expressions derived in II. The collision-free regime will be studied in Section 5, and explicit expressions for the ultrasonic attenuation will be obtained. For the region in between, i.e., Q/l, = 1, the expressions for the attenuation are so involved that numerical methods must be used. Finally, in Section 6, the expressions for the attenuation will be summarized and discussed. Some numerical results for potassium in the transition region will also be presented.

2. ELECTRONIC

CONTRIBUTION

TO THE ATTENUATION

We are going to calculate the ultrasonic attenuation using the equation of motion for the lattice deformation (u(Rt)>. The following expression was derived in the appendix in II [Eq. (II.A.26)]:

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SANDSTRijM

’ (a~gQ + (1 - %&g -I- Q))(~Q~Q -I- (1 - aQa)(g -1 1 . &(Q + g) vc(Q + g> [ v(Q + g, Q) + 4Q + g, 0) !J

-2MN--z

(’ kj j re”(QQ’

4w 1 y(&, QQ)

Q-Q) -

k . (ii) > ’

+

Q)) * (u(QQ))

1 (2.1>

The same notation as in II will be used. X,,(QG) and Fj are defined in (II.A.27-28). J(QG) is the external force on the ions, which have the mass M and charge Ze. p = MN/Q, is th e mass density of the system. ai(qw, 452) and r(k, QLn> are the deviations from equilibrium of the phonon and electron distributions induced by the sound wave [Eq. (11.2.3)]. The first and third terms together give the phonon frequencies and the anharmonic contribution to the attenuation. The lowest-order anharmonic contribution to the phonon frequencies is also included. In the collision-dominated regime, the first term in (2.1) gives the isothermal sound velocities, and the third the difference between adiabatic and the isothermal sound velocities, and also the thermoelastic and the phonon contribution to the attenuation (see II). jAQQ1 and p,(Qsz) are the charge current and density. c is the light velocity. e(Qj) is a phonon polarization vector. vie(q) is the ion-electron potential and v,(q) the Coulomb potential. Ecf(qW) is the ordinary dielectric function [Eq. (1.7.5)] which is valid in the collision-free regime. F?p is the electorn-phonon part of the collision operator r [Eq. (11.2.14)]. The factor [Eq. (II.A.30)] is introduced to take care of the deficiencies in the one-OPW model for the electron-phonon matrix elements. In the one-OPW model = 0. The four last terms in (2.1) give the electron contribution to the attenuation. The last term in (2.1) will be referred to as the collision-absorption term. The reason for this is that in the semiclassical theory, it gives rise to that component of the energy transfer from the sound wave to the combined electron-phonon system which comes from the electron-phonon collisions (Ref. [9, Section 4)]. The collision-absorption term is only important aQg

aQg

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97

for transverse modes at high frequencies, 8 2 loll rad/sec [lo]. The last term but one in (2.1) consists of a number of Umklapp terms. We will find later that these need not be taken into account until we reach still higher frequencies. In this paper only the electron contribution to the attenuation will be discussed. The electron contribution dominates the anharmonic contribution at low temperatures (below l/10 of the Debye temperature) and our discussion will be restricted to such temperatures. We will furthermore assume that the phonons are in equilibrium, thus putting aj(qw, QL?) = 0. We know from II that this assumption has two consequences. Firstly, some thermoelectric corrections to the phonon contribution of the attenuation are dropped. Since these corrections have the same type of temperature dependence as the rest of the phonon contribution, they must of course also be skipped. Secondly our discussion here will only be valid for the scattering dominated by Umklapp processes. For the scattering where normal processes dominate, we found in II that the nonequilibrium phonon distribution plays a very important role. The alkali metals below a few degrees Kelvin are thus excluded [l I]. In the same way as in II we will assume that the direction of propagation is along one of the high-symmetry directions in a cubic solid. The x axis is chosen to be parallel to that direction and they axis is assumed to lie in the vibration plane of one of the transverse modes. We will consider either pure longitudinal or pure transverse modes. Using the fact that the phonons are assumed to be in equilibrium and thus olj(qw, QL?) = 0, the last term in (2.1) can be written as (2.2) j denotes the longitudinal mode or one of the transverse modes. Strictly speaking a complete set of functions should have been introduced in (2.2) (cf. the discussion in Section 3). However, one can estimate that the error involved is less than 10 %. Around the Fermi surfaces kj = &‘(k) mb , where E(k) is the electron energy [Eq. (1.10.14)] and mbthe effective electron mass,[Eq. (11.3.7)]. Also, using (11.3.38), (2.2) can be rewritten as

According to Eq. (11.3.40), the factor [- 4rre22/u,,(Q) Q2] is very close to unity for Q < kF . When the last term in (2.1) was derived in the appendix in II, only terms to lowest order in Q and J2 were considered. Higher-order contributions will be discussedin the appendix. According to Eqs. (11.3.49-50) the longitudinal and transverse components 595/71/I-7

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SANDSTRtiM

of the electron current jet and j,, can be expressed in terms of the lattice deformation (u(QsZ)) in the following way (2.4) (2.5)

where the ionic charge current ji,, i

Ion =

is given by

Ne Q%e(Q)

- LRP)

(ii(

(2.6)

4Te2

Since we are neglecting the coupling to the local temperature deviation, E&Q@ is given by (11.3.37) and ~,(Q1;2) by (11.3.48). If the condition Q < kF is not satisfied, eo(QJ2) has to be replaced by G&Q@. ~~(Q1;2>, on the other hand, is so close to unity for such wavevectors that the electron charge current j&Q@ induced by the sound wave can be neglected. The induced charge density pe(QSZ) can be obtained with the aid of (2.4) and the continuity equation (11.3.18). To evaluate the attenuation contribution from the last term but one in (2.1), the imaginary part of ccr(qw) is needed. As long as w < quF , where vF is the Fermi velocity, the imaginary part of the dielectric function is much smaller than the real part. If the expression (1.7.5) is used for Ecr(qW) we find that J&-.----z-1 %rGP)

Gd

J%t 17,(P)

[l + GM9 QN

- G(d)12 ’

where II, is the polarization part of the Lindhardt dielectric function. w < qoF , J&V &(qw) is given by (see e.g. Schrieffer [12, p. 1391): 9&t a&P)

= W/24(4q)

@WF - 41,

(2.7)

For (2.8)

where 0 is the step function O(x) = 1 O(x) = 0

x 3 0, x < 0.

9h~l7~(qw) in (2.8) is evaluated with the help of the free-particle propagators. Strictly speaking, 9&~17,(qw) should be evaluated with the full propagators

(cf. the discussion in Section 1.7). Unfortunately, it is not possible to do this analytically except in the limit w Q w D , q < kF . Performing the integration in (I.A.17) and using (1.7.13), one finds that the only modification of (2.8) is that the electron mass m should be replaced by the effective mass mb . One can show that this replacement does not affect the attenuation. If we write the equation of motion for (u(QsZ)) [Eq. (2.1)], in the form

Q2

(2.9)

170

ALBEVERIO

conditions on the behavior of the potential for large distances (“conditions at co”) pathologies like positive-energy bound states or singular-continuum spectrum can actually arise. V. Neumann and Wigner [V. N-W 115constructed an example of a 2-body potential V, radial symmetric in the distance r of the 2 particles and going to 0 for r + co, for which the Hamiltonian for relative motion of the 2 particles has a positive-energy eigenvalue embedded in the “continuum” [0, co). The potential is asymptoticallysr7 V(r)

= -

for

F+O(-)j-)

r--+02,

and the eigenvalue E = 1 (in the case where units are chosen so that fi = 1 and the reduced mass of the particles is l/2). Roughly speaking, the solution of Schrbdinger equations for E = 1 becomes square integrable because the wellarranged “wiggles” of the potentials at large distances give rise to interference phenomena. V. Neumann-Wigner example can be generalized. In fact, for any given finite set of nonnegative numbers El ,..., Em , it is possible to find,* by the method of Gelfand-Levitan (e.g. [Ge-Le I], [Jos-Ko 11) “all” radial symmetric potentials V(r) giving rise, for a given angular momentum Z, say I = 0, to given phase shifts 6, (e.g. 6, = 0) such that the Hamiltonian is uniquely defined as a self-adjoint operator on the domain of the kinetic energy and has eigenvalues in El ,..., Em (and continuous spectrum [0, co)) (see footnote (17) for explicit examples). On the other hand, no criterion is nowadays available to detect, in general, the existence of positive-energy eigenvalues purely from the behavior of the potential at infinity,6 but in the case of 2 particles it is known that when 5 In the original paper of V. Neumann and Wigner there is an algebraic mistake; the correct version of the example is given by Simon [Si 11. 6 The potential for all r is given by V(r)

=

-32

sin r[g(r)3

cos r -

sins r + g(r) cos r + sin3 r]

3g(r)a

11 + gW1”

where g(r)

and the corresponding

= 2r -

sin 2r

radial part of the eigenfunction, belonging to the eigenvalue 1 i.e., such that

a2

(-- W

-:G+V)$=#,

is

#(r) =

c sin r rU + g(r)“)

(c being a normalization constant). Note the L, but “slow” decrease of the eigenfunction at ~0 (see also footnotes”*‘*)in contrast to the eigenfunctions belonging to isolated negative eigenvalues. See Appendix VI). ’ It is easy to generalize the example to the case of the spectrum of the operator --d + V in n-dimensional euclidian space, where d is the generalized Laplace operator. * I am grateful to Professor

R.

Jost for this remark.

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SANDSTRiiM

where (2.13)

As in II, the longitudinal quantities have an index Q, and the transverse ones an index 1. co(Q) and e,(Q) for example denote a longitudinal and a transverse polarization vector, respectively. wg is the plasma frequency for the electrons. To simplify the notation somewhat, we have introduced the quantity F(Q) = - Q2vi,(Q)/4m2Z.

(2.15)

According to Eq. (11.3.40), J’(Q) is close to unity for Q < kF . R-l is defined in Eq. (11.4.1). For 8 = Q = 0, R-l is the inverse of the collision operator in a space where certain eigenfunctions have been projected away. Equations (2.11) and (2.12) are derived under the assumption that Q Q kF , qD . For larger wavevectors, the perturbation expression for the longitudinal current j,,(QQ) must be used instead of (2.4) and j,,(QD) should be neglected according to the discussion above. The following result is then obtained

G(Q) = &

T {e(Q.d. [Q~Q + (1 - Q,>(Q + g)1)2&
1

XT’

Ig+QI

’ ((1 - G(Q + g))17,(Q + g, 0) n,(Q + g) + 11” 0(2kr -

I g + Q I)’

a&Q) = q?(Q); (2.16) We note that g = 0 is included in the summation in (2.16). The collision-absorption term has been neglected in (2.16) for the longitudinal case, which is easily justified. We will verify later that (2.1 l)-(2.12) and (2.16) give overlapping results in a certain wavevector range. According to Eqs. (2.1 l)-(2.12), the following matrix elements of R-l

(k, I R-W I k,), (k, I R-l I k) (k, I R-W I k,), (k, I R-l I W must be evaluated to get the attenuation Section II.4 for the collision-dominated

(2.17)

of the phonons. This was done in regime. The results are given in

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101

Eqs. (11.4.40) and (11.4.48). On substituting these equations into (2.1 l)-(2.12) and replacing F(Q) by unity, we find the following expressions for the electron contribution to the attenuation in the collision-dominated regime

Zmp Q20, 711I 20

ao = c,g2M

ue2( ~2

_ 25202(Q) CL I rep I kc) 1 (kc I kd WP2 (2.18)

[l + 17,(g) dg)12

0(2kF - g). (2.19)

~1”,and 7t1 are defined in Eqs. (11.4.43) and (11.4.50). A simple estimate shows that all terms in (2.18) and (2.19) except the terms that are proportional to 7: and T:. are quite negligible. This fact has been used in II, and one seesthat (2.18) and (2.19) agree with the results of II [cf. Eqs. (TI.5.33) and (11.5.61)]. 3. EVALUATION

3a. Evaluation of the Relaxation Times In this section we are going to evaluate the matrix elements of R-l by a method which is valid also outside the collision-dominated regime. In particular, the matrix elements (2.17) will be calculated. This method is thus more general than the one used in Section II.4, but it is also considerably more complicated. To obtain a formal solution of the electron and phonon transport equations (11.2.1)-(11.2.2), which is the form the generalized transport equations take for Q < go, kF , the operator R-l was introduced. In the compact notation of Section 11.2, the transport equations can be written as (Q - Q . V + ir) I Y> = I @At>,

(3.1)

where (Sz - Q . V) j v) is the flow term and r I v) is the collision term. The solution I y) is given by Eq. (11.2.51) together with Eqs. (11.2.57). We showed in

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SANDSTRiiM

Section II.3 that the different transport coefficients such as the electrical and thermal conductivities and the thermopower can be expressed as matrix elements of R-l. As we have seen in II and in Section 2 in this paper, certain matrix elements of R-l must be evaluated to obtain the ultrasonic attenuation. It was explained at the end of Section II.4 why the method of that section does not work outside the collision-dominated regime. The flow term is then no longer small compared with the collision term and it is not possible to describe the solution of the transport equations by a few Q-independent distribution functions. In fact it is clear from Ref. [3] that an essentially infinite set is needed, if the Q-dependent part of the flow term is comparable in magnitude with the collision term. We start by rewriting Eq. (11.4.1), namely

in the following way by adding (i/~) R-l to both sides (Q - Q . V + ;) Equation

R-’ = ($

-

iI’) R-l + (1 - P) - PQ . VR-l.

(3.3)

(11.2.50) has been used here. The diagonal matrix (I/T) is defined as

(l/T)= pa)

(IiT.)).

(3.4)

and T , are so far arbitrary positive constants, but they will later be chosen as relaxation times for the phonon and the electron systems respectively. Dividing (3.3) by (JJ - Q * V + i/T) gives us

T=

R-l=

[l/(Q-Q.v++)](l

[l/(Q-Q*V++)](&iF)R-‘+ -

[l/(”

- Q . V + f)3

-p>

PQ . VR-l.

(3.5)

One advantage in writing (3.2) in the form (3.5) is that for large wavevectors Q, when J2 - Q . V is much larger than the collision operator ir the dominating contribution to the matrix elements (2.17) comes from the last two terms in (3.5). Thus, for large Q, the last two terms in (3.5), which are easy to handle, automatically give the correct expression for R-l. Next we rewrite (3.5) in the following way R-l = c [l/(52 :

[l/(g

-

Q *V

+ f)](I

- Q . v + f)]

hI,><& I + (1 - P) -

[l/(Q

ir I y$) $$

- Q . V + $)]

R-l) PQ . VR-1, (3.6)

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where {$j} is a complete normal set of distribution

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III

functions

in the notation of Section 11.2. olj(qj,w) is a phonon distribution electron distribution function. I,!$~are the normalisation factors

and ri(k) an

v&j =
(3.7)

In Section 11.4, matrix elements like (k, I R-l / k,) were calculated. Those expressions are correct to second order in Q. From Eqs. (11.4.40) and (11.4.51), one can see that the result involves the two relaxation times 722 and T$ . When evaluating (k, 1R-l I k,) only the distribution functions j k,) and I 4%) were considered in (11.4.2) and (11.4.39). If also other distribution functions had been taken into account, corrections of the order Q4 and higher would appear. In fact one can show that for each order of Q2 by which one wants to extend the results for (k, ] R-l I k,) at least one new distribution functions must be considered and hence new relaxation times appear. For the electron system, the expansion parameter is roughly QufraV where 7&v is some average relaxation time. This means that for Qu~T~~~~ a very great number of relaxations times must be taken into account. This fact has been noticed before [3]. In practice it is not possible to take into account more than a few relaxation times exactly. We are going to take into account the rest with the help of a relaxation-time approximation. Thus for the higher-order distribution functions, we make the approximation

= (dSbJ)I

(l/T,) 1 %tqjP))

+ tyj(k)l (l/T&) / yt@)),

(3.8)

where l/7, and l/Te are assumed to be average relaxation times of the phonon and the electron systems. The significance of the approximation (3.8) will be discussed below. The purpose of writing (3.1) in the form (3.6) is now clear. With the help of the approximation (3.8), only a finite number of terms in the sums overj and I will appear in (3.6). 3b. Longitudinal

and Transverse Attenuation

According to Eqs. (2.11)-(2.14), the matrix elements (k= I R-W I k,) ’ (k, I k4

(k, 1R-W I k,) (k, I kd

(3.9a and b)

(kc I R-l I kz) tkz I kd ’

tk, I R-l I k,) tk, I kJ

(3.10a and b)

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SANDSTRiiM

appear in the electron contribution to the attenuation. We are going to evaluate these matrix elements with the aid of Eq. (3.6). The problem is to find a suitable set ] &j> and pick out the functions that should be taken into account exactly and the ones for which the relaxation-time approximation (3.8) can be applied. First of all, we are going to neglect the coupling between the phonon and the electron distribution functions, which means that matrix elements like (k, 1r 1qz) will be neglected. We found in Section 11.4 that this coupling is important only when normal processes dominate the scattering, which can only happen in the alkali metals at very low temperature (see Section 11.5). It would be straightforward to include this coupling but since it makes the formalism much more complicated we will disregard that special situation. This assumption implies that all phonondrag effects will be neglected and that the phonon distribution functions will not enter the problem at all. We further assume that the scattering of the electrons takes place on the Fermi surface. Thus only the values of the different yj(k) on the Fermi surface are relevant. The corrections would be of the order (kT/eF)2. We can then express yj(k) in terms of spherical harmonics YLM . For the sake of convenience, we define YLM(O, v) in the following way: YLM(&o) = Ppycos e) * cos Mp, Y&By)

= PP(cos

O
~9)- sin I M I v

where PLM(cos 0) are the associated Legendre polynomials. The polar axis is assumed to be along the x-axis and q~is measured from the y-axis. If, for the time being, we restrict ourselves to an isotropic phonon system (the electron system has already been assumed to be isotropic) we can show that the electron-phonon part of the scattering operator r has the property [3]:

(YL, I r I YL’M’) = 0

if

L # L’

or

A4 # M’.

(3.11)

For the electron-electron part of r, (3.11) is also true. The following relation for the spherical harmonics will turn out to be useful (2L + 1) j+

Y,,(k) F

= (L + 1 - M) Y L+w(k)

-

(L + M) Yw,dk).

(3.12)

It follows from (3.12) that (Ydk)

1[l/(Q

- Q - E'(k) + $11

Y,*,,(k))

# 0

only if M = M’

(3.13)

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because (Y,,(k) 1(k,)” j Y,,,(k)) must be nonzero for some n. From (3.11) and (3.13) and the fact that Y,,(k) = k,/kF and Y,,(k) = kg/k, , one can see that when evaluating the matrix elements (3.9) and (3.10) only functions of the type Y,,(k) enter (3.6) where M = 0 in the longitudinal case and A4 = 1 in the transverse case. If we, however, drop the assumption of an isotropic phonon system, we find that (3.11) is not satisfied. In fact r is then not diagonal either in L or in M. At first sight it seems that we have to take into account all Y,,(k). But if we use the cubic symmetry of the scattering operator r, we can show that that is not the case. To demonstrate the symmetry properties of r, it is convenient to use the cubic harmonics KLM . One finds, after some manipulations with the expressions for the collision operator, that

h,(k) Ir IKc.dW= 0

for

L # L’

or

M # M’.

(3.14)

M is an arbitrary numbering of the different cubic harmonics for a particular L. The only possible exceptions appear when K LM and KLtM’ behave in the same way for all cubic symmetry operations. To find the form that (3.11) takes, we expand the spherical harmonics in terms of cubic harmonics. The result is presented in Table I for L up to 4, in the notation of von der Lage and Bethe [13]. Two different cubic harmonics with the same L-value (L < 4) change signs in different ways at least for some symmetry operation. Using (3.14), the following conclusions can immediately be drawn from Table I: (1) there is no nonvanishing matrix element of r between different Y LM for L = 1. (2) For L = 2, the only possible coupling would be between Y,,, and Y,, . But even (Y,, 1r 1 Y,,) is zero as one can easily see by exchanging the y and z axes when evaluating the matrix element. (3) For L = 3, there is coupling between Y,, and Y,, and between Y,-, and Y,-, . (4) For L = 4 there is coupling between Y,, and Y4a , between Y,-, and Y,-; and between Y,, and Y,, . However, as can easily be shown, there are no nonvanishing matrix elements between Y,, and, respectively, Y,,, and Y,, . The reason for these two exceptions for L = 2 and L = 4 is that [rJ1 and [‘y& as well as [y& and [y& can be transformed into each other by cubic symmetry operations. When evaluating the longitudinal attenuation, we are interested to know which functions that couple to Y,,, , L = I, 2, 3,4. From the discussion above we find that the only new function that enters up to L = 4 is Y,, . In the corresponding transverse case, we see that Y,, and Y,, couple to Y,, and Y,, and thus these functions must also be considered. Equation (3.6) gets rapidly more complicated with an increasing number of I,$ taken into account. We are only going to consider functions up to L = 2 exactly. For the higher-order distribution functions for L > 2 the relaxation-time approximation (3.8) will be used.

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SANDSTRGM TABLE

I

Expansion of Spherical Harmonics in Terms of Cubic Harmonics@

%W’m = +[A - &II &kFeYzl

=

+kkFaYz-l

= [ca], = kJc,

$W’ze = -{MI $kp2Yz--2 ;Y&F~

= k2 - (k2/3)

[Q& = k,k,

= [r,]x = [S,],

+ &IS)

= k,’ - k,’

= k Yk z = k.* - %kFek,

&k2Ya = -$tc,‘lv - i[&lv = kJkz2 - (k2/5)1 &kp2Ys--1 = -@&le + &;lz = k,ke - (k,“P>l i’gkp’ Ya, = k’lz = k&,2 - k,‘) &VY,-, = [Al = k&k, i’skp”Y32

= -~[ES’]V

gskFsY3-3

= -$[~a’],

$&p4Y,o

= +QI 3

&kF4Yt1

=

&kF4Y,-,

+ $[S,], -

+ &[S;]z

-&[S,‘],

-

3kse)

= k,(3kve

- [Y& + &J&

-&]z =

= kq(k,’

Q[S,],

= km* - Rk,* - (k,Wl

= k,k,{ks2

- +[E&

- k**)

-

= k,k,{ksa

+kkpa} -

+kpa}

+pb4Y4z = MI + +[A = (k,’ - kz2Wz2- (k,V)l @P/105) Y,-, = M, = kMLa - (kr*/Vl (kF4/105)Y4, = -&‘I, (kF4/105)Y4-,

(kd/WY,, (kF4/420)Y,-,

=

- $[<,I, = kJcv(kv2

-+[S;],

= &,I

+ &],

+ &ah

= [S,‘],

= k&,(3k,2

-

3k,2) -

kze)

- &~rla) = kv4 + k,” - 6kv2k,e

= kvk,(kv8

- k,2)

a The notation of von der Lage and Bethe [13] is used. The square brackets indicate that the corresponding function is not normalized.

The significance of the approximation times of the type - 1 rLM

(3.8) is now clear. Firstly, all relaxation

= (YLM@) I r I Ldk~ (YLMOC)

are assumed to be equal to l/r8 .

I YLMorN

L = 3, 4,...

(3.15)

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Secondly we put (YLh4t.w I r I YL,M@N = 0

LfL'

or

Mf

L>2

or

L’>2.

M' (3.16)

Some information about the size of these nondiagonal matrix elements can be obtained from the numerical calculations of the resistivity made with the help of the variation principle [14]. If for example a distribution function of the form a,Ydk)

+ azY&)

is used, where a, and a2 are the parameters that should be varied, one can show that the following result is obtained for the relaxation time appearing in the electrical conductivity (Y,,Or) I r I Y&M J’,,(k) I Y&N (r,,or) I r I ~,,or))( Y,,Q I r I ~~~0) - v,,(k)

I r I yl,w

*

The value of this relaxation time has for example been evaluated for sodium at 40°K [15]. It was found that the value was only 4 % different from

(Y,,(k) I Y&MY&)

I r I Y&N

which means that (Y&k) I r I Y,,(k)) must be essentially smaller than the diagonal matrix elements. In general the relaxation times have to be evaluated numerically. In the simple Bloch model it is, however possible to make this evaluation analytically in the extreme low-temperature limit. The following result is obtained for the electronphonon scattering [3] 71/Q = (l/2) L(L + 1). In the high-temperature

(3.17)

limit (T > 0) [3] Tl/TL R3 2.

(3.18)

The only more detailed evaluation of TL has been performed by Rice and Sham [ 161. They calculated TV and 72 for potassium and they found that 72/71 varied roughly between 0.4 and 1.5, between 2°K and 20°K. Their result was rather sensitive to the choice of the pseudopotential. It is clear that the relaxation-time approximation (3.8) is not entirely satisfactory, but we expect it to better at slightly higher temperatures than at very low temperatures. According to the discussion above, we are only going to take into account electron distribution functions with nonvanishing values on the Fermi surface for

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SANDSTRijM

the set (&} in (3.6) when evaluating the matrix elements (3.9) and (3.10). If spherical harmonics are used for (&}, (3.6) can be rewritten in the following way R-1 =

c Tl2

1

I YL,,(W

Q - Q . E’(k) + $

(YL+&)I$ - ir I YL,&W) (YLe~,(k>l R-l (YLIM1 I YLIMIXYLeMe I YL,MJ 1

+

1 _ I J’ooW’ooI _ (YInI I Ycm)1

I(

ii - Q - E’(k) + $

I Yoo)QW’m-. I R-l e&l I YOO)(YI,I YIO)I (3.19)

In (3.19), Eqs. (X2.61) and (11.3.7) have been used as well as the definition [Eq. (11.2.26)], which in this case takes the form

of P (3.20)

P = I Yo,@Wdk) I/( Y,,(k) I YodW

Applying the relaxation-time approximation (3.15) and (3.16) to the higher-order relaxation times and multiplying (3.19) from the left by (YLM(k)[ and from the right by I LdW~ we get [R-‘I$

=

c

r&

i

( *e

L1=1.2

-

irEL,)

[R-l]&

+ r% - r~Qu,[Rvl]~~ L = 1, 2,

A4 =‘O, 1.

(3.21)

It follows from Table I and (3.14) that only terms in (3.19) for which L, = L, and M1 = M, make contributions when L1 < 2 and L, < 2. A4 = Ml as can be shown from (3.13). It follows also from (3.13) that the last term in (3.21) is only nonzero for M = 0. The following notation has been introduced in (3.21)

[R-l]Ey’ z

VLMI

4cYLM

MM’ rLLT

=

R-l

I YL,M,)

’

(“AJ 1 Q - Q - E’(k) + (i/Te)

I yL~M*>l

2/[(Y,JrnYL’M’

rMy’ LL

(3.22)

I YLX

I YLdYL’h4

-

VLM

-

d/c< yLM

If M = M’ in (3.22)-(3.24),

I r I YL’M’)

1 yLM)(

YL’ M’ I yL%‘>~

YL’ M’)

(3.23) ’

(3.24) ’

only one M is written out. Thus for example,

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FEY = r,M,, . From Table I we find that the matrix elements (3.10) are given by [R-l]:: for A4 = 0 and M = 1. The two equations (3.21) have the following solution: [R-l]:

I(1 - (+ f (+ =

[R-l]:

i

1-

- ir:)

- irg)

rK + r?Q~~)(l

df (-

($-

- (2

- ir:)

- ir;)

rE)

r$ + r$‘u,Q) 1 (3.25)

i

](l - ($ + ($

- Z’x) r: + rFQo,)(l

-

ir.$)

= (1 + rlyQv,)

To obtain (k, I irR-l

1: (-

(-&

-

- (-&

- Cz) rg)

r.$ + rzu,Q)

ir:)

f

r,? - r,~Qu,r,~.

(3.26)

I k,) and (k, I iT’R-’ I k,) we rewrite (3.2) as

irR-l

= 1 - P - (1 - P)@ - Q * V) R-l.

(3.27)

In (3.27) we substitute (3.6) for R-l and obtain

- Cl - ‘1 J-J _

. V + (jlT) (l - ‘1

Q

Sz-Q.V + (1 - p) Q _ . v + (+) Q

PQ * VR-l.

(3.28)

Proceeding in the same way as we did while deriving (3.21) we have from (3.28) that [iTR-‘]g

= I-- $+ -$

+ irg (+

+ $-

- ir$)

($ rg[R-‘]g

-

iI’:> + 5

rz - -$ r,“: ;

rTueQ/ M=

0,l.

[R-l]: (3.29)

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SANDSTRijM

With the aid of Table I we find that the matrix elements (3.9a) and (3.9b) are given by [PR-l]g for A4 = 0 and A4 = 1, respectively. Using the following identity (Q is parallel to the x-axis)

D-

G’(k)

Q . E’(k)

(3.30)

+ We)

and Eq. (3.12), we can obtain a relation between the different rE, a,,, Qve

,--

=

CL+

1 -Ml

+w+w

(;;

J

21

1;

(;;';

;I

;

y-

$EI,L,

L = 0, l,...

;;g)&L*

(3.31)

L’ = 0, l,...

M = 0, I,..., where

Qve ae- z J-2+ (i/Te) v, = +/d/3

(3.32)

[cf. (11.2.61)].

If L or L’ is less than Min rj$$, the corresponding quantity in (3.31) is assumed to be zero. The values of the normalization factors of the spherical harmonics have been used in (3.31) 1 M=O WLMCW

YLMW)

=

W%)

2L

=w4

+

2L1+

1

(3.33) 1 y:;j

M # 0.

iV(E,) is the electron density at the Fermi level. With the aid of (3.31), rz”, 18, rr, and r$ can be expressed in terms of rz rloO- - -ia,rfl

,

(3.34) (3.35)

rlo = -ia,&

,

(3.36) (3.37)

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10 10 rlo = rZo = 0,

2/o I

r,‘, =

5.- 1 ~++-rL]y i 3

r,‘, = n -5.1 z 3

By applying Eqs. (3.34)-(3.40) (3.25), (3.26) and (3.29) as

M

.M 4

= rll - zrzl

(3.38)

6.

(3.39) (3.40)

rZ1 1.

we can, after some simple manipulations,

1-!- + @+ iC3ril4

5 4--IMI

a,

5

rewrite

I

4-[MI

1 eve;

M = 0, 1, (3.41)

M

* r21

5 4--IMI

1~-Qu6~[4-~M’]r~/=r$; Qua

M = 0, 1, (3.42)

M=

0, 1.

(3.43)

According to Eqs. (3.34)-(3.40), rg” and rg can be expressed in terms of rE. For zero temperature, rg can be evaluated analytically and the following result is obtained (cf. Kittel [17, Eqs. (17. 22-23)]} 0 1 rll = 52 + (i/7J rll1 = r$ =

L(l

ae2

1 Q + We)

3

tan-l( 10 a,) dJa,

1

Z (VJ ad3

(3.44)

)’

((1 + 3ae2) tan-l(q/3

a,) -

6

a,}.

(3.45)

Corrections due to nonzero temperature are smaller by a factor of the order (W/E,)* and are thus negligible. If (3.44) and (3.45) are used, the desired expressions for the matrix elements (3.9) and (3.10) are given by (3.41)-(3.43). Mis equal to zero for the longitudinal modes and unity for the transverse modes.

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4. THE COLLISION-DOMINATED

REGIME

In this section we are going to compare the form the matrix elements (3.41)-(3.43) takein the collision-dominated regime with the corresponding expressions derivedin Section 11.4. In the collision-dominated regime, the wavelength of the sound wave is large compared with the mean-free path of the electrons (QvF7, << 1). This means that Qvr is much smaller than the involved inverse relaxation times l/~~ , r& = Fir , r.$ , P.& which are usually quite close in magnitude. For Qur7, < 1, a, [Eq. (3.32)] is small (I a, I < 1) and it can be used as an expansion parameter. The following result is obtained for rg [Eqs. (3.44) and (3.45)]

r,“, =

l J-2+ We) 1

1

[1-

r11 = sz + (i/TJ

On substituting

(1 - i

ae2

+

7

~22

+

35ae4

+

. ..I.

+

*..

27

3

5ae2

I

(4.2)

.

(4.1) and (4.2) into (3.35) and (3.39) respectively, we find that p21 --

-

r121

-

=

[

1 -

A!&2

1 -

$2

+

...

+

...

[

1 ,

(4.3)

.

(4.4)

I

From Eqs. (4.1)-(4.4) we get rfl-jrzl

i

'

$=

40

e

?Iae2+...,

(4.5)

Sz + (i/TJ 35 ' Q

+

?ta,2+.... (ibe)

(4.6)

35

On inserting (4.3)-(4.6) into (3.41) and (3.42), these equations take the following form, if one retains terms to second order in Q: [R-l]:

On+

=

irg

(Q+iI'~)@+i~~)-

[R-‘1: = v,Q d(

4 Pi M ’ )/Cl2 +

(4.7) vzQ2

4-iM'

iI'$@

-I-

irg).

'

(4.8)

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On substituting (4.1)-(4.4) and also (4.7)-(4.8) into (3.43), we find the following expression, after some straightforward but rather lengthy manipulations:

[ilT1j:

= (1 - sz +“lrr:

1+

(44.9) M=O,

1.

In (4.9) again only terms to second order in Q have been retained. Introducing quantities-&Q) and ~~(52) in the same way as in II:

the

T1”l

1 I2 + iTi

3 3’

(4.10) (4.11)

and using (2.13) and (2.14), (4.8) can be written as 722w4

T22(QQ’ =

1 + u~*Q~T,~,(Q) T,,(OGJ) ’

TA(QQ)

1 + u,~Q~T:(D)

=

T~~(OQ)'

(4.12)

(4.13)

for M = 0, 1. If (4.12) and (4.13) are substituted in (4.9) for M = 0, 1 we find that (k, 1iI’R-l / kJ = 1 + iT22(QQ)(Q + iu,“Q”Tl”l@>>, 6% I 4

(4.14)

(k, I irR-l 1k,) = 1 + ~T,‘,(QSZ)(Q (k, I kd

(4.15)

+ iu,“Q”&(sZ)).

We see directly that (4.12), (4.14) and (4.15) agree respectively with Eqs. (4.40), (4.49a) and (4.49b) in II. The expression corresponding to (4.13) is not explicitly given in Section 11.4, but one can show that it is consistent with the results of that section. We have thus verified that (3.6), with the distribution functions {&I up to L = 2 taken into account exactly, gives the correct result to second order in Q. This is sufficient for obtaining the attenuation in the collision-dominated regime. The higher-order terms are treated approximately since the relaxation-time approximation (3.8) is applied for them. To get also the higher-order terms exactly, distribution functions of higher L must be taken into account in (3.6), with L increasing by unity for each extra order Q2. 595/71/1-8

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114

5. THE HIGH-FREQUENCY

LIMIT

In this section we are going to calculate the ultrasonic attenuation for high frequencies (QuF7, > 1, D M c, Q). In this limit, the expressions for the matrix elements that enter the attenuation have a simple form. In these expressions, rT1 and r: appear, which for Qv~T, > 1 can be expanded in orders of l/a, as [Eqs. (3.44)-(3.45)] r”11

=

-

Q+ We?) Q2ve2 I’ + 1

.L ’ 2 63

Q + we> _ (Q + P/7eD2+ 3Q2va2

Qve

-j,

3

-If “’ = Sz + (i/r&) 2 2 G3 a, I

On substituting

(5.1) (5.2)

(5.1) and (5.2) into (3.35) and (3.39) we find that (5.3) (5.4)

Next we insert (5.1)-(5.4) into (3.41) and (3.42)

(5.5)

(5.6) [R-l]:,

=

3Tf + (37d4)2Gl iQvs 4 d3 3Q2ve2

WeI> --

5 3

w-t2

- P17el)+ ... , Q2ve2

(5.7)

(5.8)

Using (5.1)-(5X), we find that the leading terms of (3.43) are given by

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Next we are going to insert these results into (2.11) and (2.12) to obtain the ultrasonic attenuation. The equations we find in this way look rather unwieldy. But since we want the matrix elements of R-l only for Q < kF , we can neglect terms which are smaller than the leading term by a factor (Q/kF)“. Thus all terms except k”TF can be disregarded in the denominators in (2.1 l), because kTF is of the same magnitude as kF . Furthermore, 8’(Q) can be put equal to unity [cf. Eqs. (11.3.39) and (2.15)]. With these simplifications, the first term in (2.11) for the longitudinal attenuation can be written as c+(Q) = i&-&L so

(1 e

TF

T22

(k, I R-W / k,) >’ (k I k,)

(5.11)

The only simplification in (2.12) is that F(Q) can be replaced by unity. On inserting (5.5), (5.7), (5.9) and (5.10) into these expressions we find that 1 Zmb Que., so TTT’

(5.12)

1 Zmb Qu, 4 43 3n

(5.13)

4Q) = 5 cM

a,(Q) = - 2 MC,,

Since we have assumed QvF to be much larger than all the involved inverse relaxation times, only terms to lowest order in ~/Qv~T, have been retained in (5.12) and (5.13). In fact, this means that only the lowest-order terms in (5.5)-(5.10) need to be taken into account. The inclusion of the factors

(

1 _ (k, I R-W I k,) )’ (k, I k,)

(5.14)

(

1 _ (k, I R-W I k,) 1’ 6% I &,I

(5.15)

in (2.11) and (2.12) is a result of the collision-drag in (5.12) and (5.13) correspond to

effect. (5.14) and (5.15) would

(5.16) (5.17)

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SANtSTR6M

according to (5.9) and (5.10). Since we have assumed that Qu, > r& = rtl , both (5.16) and (5.17) can be replaced by unity as we have done in (5.12) and (5.13). olc(Q) in (5.12) is simply linear in Q. The transverse component aI on the other hand has a more complicated behaviour. Three different wavevector regions can be distinguished: Qao <

3rr Qve4

~4

(5.18b)

+Q+,

o

Q>t

In (5.18) the anomalous introduced.

(5.18a)

1,

Cl

(5.18~)

skin depth &, and the classical skin depth 6,t have been 8 3 = c2vt?4 43 0 i2co,237r ’

(5.19)

(5.20) From (5.18) we can conclude that the longitudinal and transverse attenuations have the same type of Q-dependence for Q&, < 1. However, for larger wavevectors their behaviour is quite different, Qs, being equal to unity for Sz = lOlo rad/sec. For Q??, 2 1, a,(Q) is proportional to Q-3 and for still larger wavevectors it is Q-independent. With the help of Eqs. (2.9) and (2. lo), one can show that (5.18~) comes from the collision-drag contribution to the last term in (2.1), namely the collision-absorption term. In the derivation of the collision-absorption term, only terms to lowest nonvanishing order in Q and L? were considered. The higher-order contributions are discussed in the appendix. We find that the lowest-order corrections are proportional to @2/H’)” and (c,Q/~T)~, respectively, with a multiplying factor that is smaller than the lowest-order term. It is also possible to estimate what to replace (5.18~) by for Sz > kT. For such frequencies aI(Q) is independent of the temperature and is proportional to an, where n is the same exponent that appears in the low-temperature behaviour of the electron-phonon contribution to the electrical resistivity. In the Bloch model, n is equal to 5. For example, at T = lO”K, kT/fi m 1012rad/sec which means that the higher-order corrections to (5.18~) are not important until very high frequencies.

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Neither the last term in (2.1 l), nor the last term in (2.12) have so far been taken into account.

m2

((1 -

G(Q+

Using Eq. (II.A.30), bandgaps, Eq. (5.21) For the longitudinal compared with (5.12)

1

lg+QI &(Q + g, 0) v,(Q+

4~ g))

g) + I}” 0(2kF - ’ g + Q ‘) . (5.21)

one finds that for E’(g/2)Q/2 < / W(g)l, W(g) being the takes the form given by the last terms in (2.18) and (2.19). case, one can easily estimate that (5.21) is negligibly small as long as

Gd2) Q < 2

I Wg)l.

According to (2.19) the transverse component of (5.21) is proportional to Q4 for E’( g/2)Q/2 < 1 W(g) 1.The value of the exponent in this Q-dependence may depend on the particular model chosen for the eletron-phonon matrix element (II.A.30). Comparing (5.18~) and (5.21) one finds that they are of the same magnitude for 2 ctLkF4[r”s]:,/eF or with values inserted 52 M 1012. This means Q4 = I ~kY&)l that (5.21) is significant only at very high frequencies. For 52 > kT higher-order terms derived in the appendix must be taken into account. Then (5.21) should be compared with (A.12) instead of (5.18~). One finds that (A.12) is smaller than (5.21) by a factor sZ/cF < 1. The higher-order contributions to the collision-absorption term seem to be hidden by the larger contribution (5.21). The ultrasonic attenuation has been obtained by inserting expressions for the charge density and current into the equation of motion for the lattice deformation (2.1). For Q < qD , kF the charge density and current were obtained by solving the transport equations (11.2.1)-(11.2.2), and for larger wavevectors by perturbation theory. The result which was obtained by the first method for QuF7, > 1 is given in Eqs. (5.12), (5.13 [or (5.18)]) and (5.21) and by the second method in (2.16). Since both these methods can be regarded as special cases of a more general one, namely the solution of the generalized transport equations (1.8.1) and (1.8.2) according to the discussion in Section I.1 1, one can expect that they give overlapping results in some wavevector region. This is straightforward to verify. First of all we note that all terms in (5.21) are included in (2.16). Apart from the collision-absorption term, the only term left in (2.16) is the one for g = 0. This

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SANDSTRGM

term only gives a contribution it takes the following form

Q(Q)

to the longitudinal N

= ~ c.s&&f

attenuation

Z2Qm2 4’TTv70(0,w

and for Q < kF/10

.

(5.22)

The corrections to (5.22) are smaller by a factor (Q/kF)“. Using the fact that

I&(0, 0) = $

(5.23)

and that mbv, = k,/1/3, one finds that (5.22) agrees with (5.12). The collisionabsorption term is negligible in the longitudinal case and consequently it has been skipped in (5.12). In the transverse case we see directly that (5.18~) and (5.21) agree with (2.16). We can thus conclude that the two methods to obtain the attenuation give the same results for the longitudinal modes in the range l/0,7, < Q < k,/lO and for the transverse modes for Q&i > 1. 6. SUMMARY OF RESULTS AND CONCLUSIONS

In Section 2, general expressions for the electron contribution to the ultrasonic attenuation were derived. In these expressions, matrix elements of the operator R-l are involved. R-l appears in the formal solution of the transport equations. A method is developed in Section 3 to evaluate these matrix elements. This method is more general than an earlier one which was established in II.4 and which is only valid in the collision-dominated regime (QL& < l), II, being the mean-free path of the electrons. It was shown in Section 4 that these two methods give identical results in the collision-dominated regime. For Q& M 1 the solution of transport equations is a difficult problem and thus also the evaluation of the matrix elements of R-l. The reason is that it is neither possible to expand the solution in terms of Q, nor in terms of l/Q. One finds that a great number of relaxation times influence the solution. Tn Section 3, the approximation is made of putting all relaxation times equal except for a few of the most relevant ones. When evaluating the matrix elements of R-l appearing in the formulas for the attenuation, the coupling to the phonon distribution functions can be neglected, and the electron distribution functions can be expanded in terms of spherical harmonics YLM . In an isotropic solid only functions for M = 0 enter for longitudinal modes and for M = 1 for transverse modes. In a solid with cubic symmetry, this property is essentially kept for modes in the high-symmetry directions. However, some exceptions appear; thus, for example the functions Yd4 and Yz3have to be considered. In Section 3, functions up to L = 2 were taken into account exactly, and the rest in a relaxation-time approximation.

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Thus only the following inverse relaxation times are involved in the expressions for the attenuation: r&, r$,, , r:, , r,l, l/~~ [cf. Eqs. (3.25), (3.26), (3.29)]. Due to the cubic symmetry r& is equal to r:, . For an isotropic model r& is also equal t0 r& . l/~~ should be estimated from the other relaxation times, if possible from the higher-order ones. It is clear from Sections 4 and 5 that the choice of 7, does not affect the attenuation for either Q/l, Q 1 or for QL& > 1, but only for Q& = 1. It is of course straightforward to extend the equations (3.25) (3.26) and (3.29) to take into account higher-order distribution functions than those for L=2. However, it is apparent that the complexity of the equations will increase very quickly for increasing L. rg can be obtained from the experimental values of the resistivity. When calculating the attenuation for higher frequencies, the values of r& and I’& can be estimated from the low-frequency values of the attenuation. The relaxation time, (r&-r has been evaluated numerically for potassium by Rice and Sham [ 161. Some of the higher-order relaxation times must have been calculated when the resistivity has been evaluated with the help of the variational principle. Unfortunately in these papers only the final results for the resistivity have been presented from which it is not possible to get these relaxation times. The results for the attenuation in the collision-dominated regime (Qne < 1) were derived already in II. The attenuation in the collision-free regime (Q(le > 1) was obtained in Section 5. For QA, = 1, the expressions (3.25) and (3.29) for the matrix elements (2.17) are so complicated that they must be evaluated numerically. Some results for the longitudinal attenuation in potassium are shown in Figs. 1 and 2. The values for r& and r& have been taken from measurements of the electrical resistivity and low-frequency attenuation [18]. l/re is chosen as the average value of r& and r& . It is seen in Fig. 1 that the transition from the collision-dominated to the collision-free regime happens quite smoothly and the result for the attenuation is qualitatively the same as in the single-relaxation-time approximation. In Figs. 1 and 2 it is also illustrated how the attenuation is changed when T, is varied within a factor of two. One sees that this always affects the attenuation by less than 30 %. The result is thus rather insensitive to the precise value of T, . As was anticipated, the attenuation only depends on the value of 7, in a narrow range around Q/l, = 1. The results for the attenuation are summarized in Figs. 3 and 4. When the wavelength of the sound wave is greater than the anomalous skin depth (Qs, < I), the longitudinal and transverse attenuation behave in essentially the same way. In the collision-dominated regime (Q& < l), the attenuation is proportional to Q2 [Eqs. (2.18) and (2.19)], and for Q(l, > 1, it is linear in Q. For Q&, > 1, the longitudinal attenuation is still linear [Eq. (5.12)], while the transverse attenuation has a more complicated behaviour [Eq. (5.18)]. Thus the latter to start with is proportional to Q-3 (Qs < l), and then becomes, for still higher frequencies,

120

SANDSTRijM

independent of Q. This Q-independent term comes from the collision-absorption term in the equation of motion for the lattice deformation. When this term was derived in II only terms to the lowest nonvanishing order in Q and Sz were taken into account. The higher-order terms are discussed in the appendix. It is found there that the lowest-order corrections have the form @/k7J2 (assuming that SZ/kT < 1). The situation for Q/kT > 1 is also briefly discussed. In the Bloch model, this term is found to be proportional to LF. At such high frequencies at which these higherI

I

I

1

I

I

I

I

I

I

I

--

to2

-2 2 10’ -0 6 r 3 z loo w E Q

10-l

1o-2 I Mob 2

I

5

I

I

ld lob FREQUENCY ( c/s I

I

I

10”

FIG. 1. Longitudinal attenuation in potassium as a function of frequency around the transition region at 10 K. l/7, is chosen as the average of I’FI and I’& . 7s is varied a factor of two to see how much it affects the attenuation. The arrow shows where Qon, = 1. SRTA is the corresponding curve in the single-relaxation-time approximation.

SOUND

IN

AN

ANHARMONIC

METAL.

FIG. 2. Longitudinal attenuation in potassium as a function is chosen as in Fig. 1. The arrow shows where Qv~T. = 1.

121

III

of temperature

at 50 MC. 7s

order terms become important, (5.21) too must be taken into account. In fact, it follows from the discussion in Section 5 that (5.21) dominates the higher-order corrections to the collision-absorption term. For E’(g/2)Q < I W(g)l, W(g) being the bandgaps, the transverse part of (5.21) is proportional to Q4. This part of the transverse attenuation is uncertain, because it depends sensitively on the corrections to the one-OPW expression for the electron-phonon matrix element. And it is difficult to get a reliable model for these

Q=q,/lO FIG.

3. Schematic plot of the electron contribution

to the longitudinal

qD attenuation.

122

SANDSTRiiM

FIG.

4. Schematic plot of the electron contribution

to the transverse attenuation.

corrections. The longitudinal part of (5.21) appears as Q2 for E’(g/2)Q < 1 W(g)1 and it is always dominated by the linear term (5.22). For e’(g/2)Q 2 ] W(g)], the Q-dependence of (5.21) has more structure. Together with (5.22) it gives the ordinary expression for the electron contribution to the attenuation of thermal phonons. In the expressions for the electron contribution to the attenuation, there appear the charge density p,(QsZ) and current j,(QsZ) induced by the soundwave. These two quantities have been obtained by solving the generalized transport equations. For Q < kF, qD , this means solving an equation which is close in form to the semiclassical transport equations, and for higher frequencies using perturbation methods. It has been verified in Section 5 that these two different approaches give the same results for the attenuation in an overlapping wavevector region. Experimental information about the attenuation of phonons in metals is available in two frequency regions: below about lOlo rad/sec by ultrasonic methods and above 1012rad/sec by inelastic neutron diffraction. With ultrasonic methods numerous experiments have been performed some of which are mentioned in II. On the other hand, measurements by neutron diffraction have been performed for only the following metals: Al [19], K [20] and Pb [21]. The theoretical and experimental results for the attenuation in the collisiondominated regime have been discussed in II. When going from the collisiondominated to the collision-free regime the Q-dependence of the attenuation changes from the quadratic to the linear. This change has been observed in many

SOUND IN AN ANHARMONIC METAL. III

123

experiments for longitudinal soundwaves (cf. [22-261). The transverse attenuation has only been measured for Q&, < 1 and the drop for Q&, > 1 has so far not been observed. If bandstructure effects are included, the size of this drop is considerably reduced and the Q-dependence of the attenuation becomes linear instead of constant for Q&i > 1 (Ref. [27, p. 4701). Due to bandstructure effects (nonspherical Fermi surface and a varying deformation parameter over the Fermi surface), the attenuation shows considerable anisotropy in the collision-free regime (Q(le > 1) (see [22, 25, 26, 28-301). In freeelectron metals like Al, the anisotropy is less pronounced [24]. The electron contribution to the linewidths of thermal phonons has been evaluated numerically by Bjiirkman, Lundquist and Sjolander [31] and by Johnson [32] for Al, by Buyers and Cowley for one-symmetry direction in K [20], and by Schneider and Stoll for Li [33]. For K the electron contribution to the linewidths is small compared with experimental uncertainties, which makes a comparison between the theoretical and experimental values difficult. For Al the situation is more favourable. Already at such a high temperature as 80”K, the electron contribution is comparable in magnitude with the anharmonic contribution, the former being dominant for small wavevectors and the latter for wavevectors closer to the zone boundary [34]. One finds that the theoretical values for small Q are somewhat too small to fit the experimental values, particularly for the transverse modes. The reason for this discrepancy is at present unclear, but it is probably due to experimental uncertainties. Experiments at lower temperatures than 80°K would probably clarify this point considerably.

APPENDIX: HIGHER-ORDER TERMS IN THE ENERGY TRANSFER FROM THE SOUND WAVE DUE TO THE ELECTRON-PHONON COLLISIONS (COLLISION ABSORPTION) We have found in Section 5 that the collision-absorption sz -‘MN L

(’ kj 1rep(QQ)

term in (2.1), namely

dw QQ) 1y(k, QL’) - k . (it)

(A.0

gives an important contribution to the transverse attenuation for Q&i > 1 where 6,1 is the classical skin depth. When deriving this term in the appendix in II, only terms to lowest order in a were considered. Here we are going to estimate the magnitude of the higher-order terms. In most cases it is not necessary to consider the Q and Q dependence of the collision operator r. Thus, for example, when calculating the charge and temperature deviations induced by a sound wave by solving the transport equations, it is enough to take into account the Q and Sz dependence to lowest order. The reason is

124

SANDSTR6M

that the flow term dominates the collision term at high frequencies (see the discussion in Section 1.4). (A.l), on the other hand, gives a contribution to the attenuation involving the collision operator which is important at so high frequencies that Q&r > 1. It is then no longer possible to neglect the Q and Sz dependence of the collision operator. We have seen in Section 5 that it is the collision-drag part of (A.l)

which is dominant for Q&i > 1. Thus only the Q and L? dependence of this part will be discussed. We must now go back and look at the derivation of (A.2) in the appendix in II. (A.2) is given by the last term in (II.A.18) [apart from a factor (-l/M)]. This collision-drag contribution is illustrated in Fig. l(d-e) in II. From Fig. l(d-e), one finds that the last term in (II.A.18) changes to the following expression, if higher-order terms in Q and Sz are taken into account

. @a- k,h (ka- k3 *
- {Dj(qw)

G(k,w,)

G(k,wd

-

4<(v)

w3

+ cfJ- J-34,---L,+s-01

G’(k3o3)

G<@w&.

(A.3)

We consider only the imaginary part of (A.3). The real part gives a negligible contribution to the sound velocity. Using the explicit form for Pqj(k3 , k4) Eq. (II.A.30) and also Eq. (I.A.17), we can write (A.3) in terms of the polarizability WWJ) :

- Nw4 + (1 - ~,,Mi + g>). f&h) &(9 + gN2 * P&J) WI + g, WI - 4-%PJ) fl’h

+ g9aI1

- Ks,q + (1 - +,)(q + gN - 4cr + Q, A) &(a + &I2 * VUq + Q, w + J-317(q+ g, w>- 4% + Q, OJ+ Q) n’( q + g, w>ll. G4.4)

SOUND

IN

AN

ANHARMONIC

Using Eq. (1.2.3a) and the corresponding cations can be made in (A.4):

METAL.

III

125

relations for 17, the following modifi-

+ Di<(q + Q, w + J-4f17”(q+ g, ~11 = 1’ 2 ; KDiYq + Q, w + fz) - Dj”(s + Q, w + Q)) -tg + q, 4 - 4% + Q, w + QXfl’(q + g, w>- -(g + q, w))l =

s $ ; En(w)- 4~ + WW(q

+ Q, w + J-3- Dj”(q + Q, w + f2>>

* Qwq + g, WI - fl”(q + g, WI).

(A.3

For the second equality in (A.5), we have used the fact that D,W has poles only in one half-plane which means that the integration over this term gives a vanishing result. We have also used the fact that 17< and D< are purely imaginary and that the complex conjugates of Dr and 17’ are Da andLra. For the third equality, (1.2.3a) and (1.2.30) have been applied. The polarization part P(qw) is directly related to the ordinary dielectric function E&qw): %f(QW> = 1 + (4rre2/q2) qqwj. The difference Djr(qw) density function Ai

(A.6)

- D,“(qw) can be expressed in terms of the phonon spectral[see Eqs. (1.2.3a) and (1.2.30)] D/(qw)

-

D/(qw)

= (2ri/M)[Aj(qw)/w].

(A.71

On applying (AS) in (A.4) with (A.6) and (A.7) inserted we find that (A.4) takes the following form

aas)(q+ &I * dcr + Q, i) qfe(q+ g>>”

* [n(w)

-

n(w + Q)] AAq

+ Q3 w + D) 9kz E&Q + g, w).

w+Q

(A.8)

126

SANDSTRGM

We assume that Ai can be taken in the harmonic approximation [see Eq. (1.2.33)]. For +%z cCf, we use Eqs. (2.7) and (2.8). (A.8) will then give the following contribution to the attenuation:

* e(q + Q, 3 uie(q+ g))" h + dj (s + g)j @W, - I g + q I) [l + J&h + g, 0) oc(q+ g)(l - G(q + gM2 =Wi + Q) - {b(4q + Q>- Q) - 4-4 + QNlG4 + Q) - J4 - [n(o(q + Q) + J-4- +-4q + QDl(4-4 + Q) + J-4)g

,q : g , 3 64.9)

where X2 = CsjQ. We note that (A.9) is even in both Sz and Q. Expanding (A.9) in powers of 52, we find that

o~j(Q)= &83 II a& {bpgq+ (1 - %J(q + dl * 4s + Q, 3 dq + da @W,

- I g + QI>

- (’ + g)i” 11 + l&,(q + g, 0) v,(q + g)(l - G(q +

m2

1 ‘271 Is+81 *

I

w

g))12

1 1 32 wtq+Q,.A) (A.lO)

(

To estimate the order of magnitude of the higher-order terms in Sz, we use a Debye model for the phonon frequencies wj(q). In the same way as the Bloch formula for the electrical resistivity is derived (see Ref. [14]), we find the following result for (A.lO) in the low-temperature limit ai

= ---

1

m2

’ “‘(‘)”

M=csj 4 - 3,rr3 c$f?

(S! c(5) + oj2(Q) ,f3”3! ((3) + .--I,

(A.ll)

where 5! C(5) = 124 and 3! C(3) = 7.21. The derivation of the Bloch formula involves several steps which are difficult to justify, such as for example the handling of the Umklapp processes. All the same, in many cases it gives the right order of magnitude for the resistivity. It is reasonable to expect that the situation is about the same for (A. 11). From (A.1 1) we see that

SOUND

IN

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ANHARMONIC

METAL.

III

127

the higher-order terms are (O/H)2 times the lowest-order term. (A. 11) is clearly only valid for Sz < kT,. It is also possible to estimate (A.9) in the limit Gr > kT using the same type of procedure. The following result is then obtained: (A.12) (A.12) is not valid unless Q,(Q) > 10 kT. The exponent 5 has the same origin as the T5 dependence of the electrical resistivity in the Bloch formula at low temperatures. We note that the factors in front of the powers of Qj(Q)/3 are small both in (A. 11) and (A. 12) compared with the factor in front of the lowest-order term. This means that the higher-order terms are not important until fairly high frequencies. The higher-order terms in Q in (A.9) can also be estimated. One finds that in the same way as in (A.1 1), the lowest-order correction is proportional to (cSjQp)“. The situation when cSjQ/3 > 1 is difficult to handle because Q appears in a very complicated way in (A.9). Without making any precise estimates, a similar type of behaviour as in (A.12) can be expected.

REFERENCES 1. R. SANDSTR~~M, Sound Propagation in an Anharmonic Metal I. The Generalized Transport Equations, Ann. Phys., to appear. 2. R. SANDSTR~~M, Sound Propagation in an Anharmonic Metal II. The Collision-dominated Regime, Ann. Phys., to appear. 3. A. B. BHATIA AND R. A. MOORE, Phys. Rev. 121 (1961), 1075. 4. J. A. KRUMHANSL, Proc. Phys. Sot. (London) 85 (1965), 921; R. A. GIJYER AND J. A. KRUMHANSL, Phys. Rev. 148 (1966), 766, 778; R. A. GUYER, ibid., 148 (1966), 789. 5. A. GRANATO AND K. MCKE, J. Appl. Phys. 27 (1956), 583, 789. 6. W. P. MASON, Phys. Rev. 143 (1966), 229. 7. H. J. WILLIARD, R. W. SHAW, AND G. L. SALINGER, Phys. Rev. 175 (1968), 362; J. E. RANDORFF AND B. J. MARSHALL, Phys. Rev. B2 (1970), 100. 8. A. B. P~PARD, Phil. Mug. 46 (1955), 1104. 9. T. HOLSTEIN, Phys. Rev. 113 (1959), 479. 10. M. H. COHEN, M. J. HARRISON, AND W. A. HARRISON, Phys. Rev. 117 (1960), 937. 11. A. HASEGAWA, J. Phys. Sm. Japan 19 (1964), 504. 12. J. R. SCHRJEFFER, “Theory of Superconductivity,” Benjamin, New York, 1964. 13. F. C. VON DER LAGE AND H. A. BETHE, Phys. Rev. 71 (1947), 612. 14. J. M. ZIMAN, “Electrons and Phonons,” Oxford, 1960. 15. M. P. GREENE AND W. KOHN, Phys. Rev. 137 (1965), A513. 16. T. M. RICE AND L. J. SHAM, Phys. Rev. Bl (1970), 4546. 17. C. KITTEL, “Quantum Theory of Solids,” Wiley, New York, 1963. 18. G. G. NATALE AND I. RUDNICK, Phys. Rev. 167 (1968), 687. 19. R. STEDMAN AND G. NILSSON, Phys. Rev. 145 (1966), 492. 20. W. J. L. BLJYERS AND R. A. COWLEY, Phys. Rev. 180 (1969), 755.

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21. A. FURRER AND W. HIu;, “Experimental Phonon Frequencies and Widths of Lead at 5,80 and 290 K,” Technical Report AF-SSP-38, Eidg. Institut fur Reaktorforschung, Wiirenlingen, Switzerland. 22. R. E. MACFARLANE AND J. A. RAYNE, Phys. Rev. 162 (1967), 532 (Cu, Ag, Au). 23. W. A. FATE, Phys. Rev. 172 (1968), 402 (Pb). 24. K. C. HEPFER AND J. A. RAYNE, P/zys. Letters 28A (1968), 163 (Al). 25. E. Y. WANG, R. J. KOLOUCH, AND K. A. MCCARTHY, Phys. Rev. 175 (1968), 723 (Cu). 26. E. S. BLINK AND J. A. RAYNE, Phys. Lett. 23 (1966), 38; Phys. Rev. 177 (1%9), 673 (In). 27. J. MERTSCXING, Phys. Status Solidi 37 (1970), 465. 28. R. E. MACFAIRLANE, J. A. RAYNE, AND C. K. JONES,Phys. Len. 19 (1965), 87; 19 (1965), 354, (Ag, Au). 29. K. Foss= AND J. R. LEIE~WITZ, Phys. Len. 22 (1966), 140. 30. H. J. WILLIARD, Phys. Reo. 175 (1968), 367. 31. G. BJ~RKMAN, B. I. LUNDQUIST, AND A. SJ~LANDER, Phys. Reu. 159 (1967), 551. 32. R. JOHNSON, “Electronic Contributions to the Phonon Damping in Metals,” Technical Report AE-328, AR Atomenergi, Studsvik, Sweden. 33. T. SCHNEIDERAND E. STOLL, Phys. Kondens. Mater. 9 (1969), 32. 34. T. H~GBERG AND R. SANDSTR~M, Phys. Status Solidi 33 (1969), 169; R. SANDSTR~M AND T. H&BERG, J. Phys. Chem. Solids 31 (1970), 1595.