The theory of the sound attenuation in one-dimensional metals

Solid State Communications, Vol. 49, No. 8, pp. 779-782, 1984. Printed in Great Britain.

0038-1098/84 $3.00 + .00 Pergamon Press Ltd.

THE THEORY OF THE SOUND ATTENUATION IN ONE-DIMENSIONAL METALS L.V. Chebotarev and E.A. Kaner Institute of Radiophysics and Electronics, Academy of Sciences of the Ukrainian S.S.R., 310 085 Kharkov, USSR

(Received 19 September 1983 by V.M.Agranovich) The elasticity theory equations are obtained for a ld conductor. The frequency dependence of the sound attentuation is analysed, the spatial dispersion being strong or weak. The effect of oscillations of the attenuation is predicted which is due to a jumping nature of electronic motion in a non-uniform field of the sound wave, with a fixed jumping length. That is why the oscillations are of the geometric resonance type. Because of absence of Landau's damping, the frequency dependence of attenuation in a region of strong spatial dispersion is quadratic rather than linear one, as in 3d metals. This dependence is determined by a quantum nature of electronic scattering on separate impurities which move with an oscillating lattice.

1. IT HAS BEEN SHOWN in the previous paper [ 1] that the interaction of electrons with long-wavelength phonons in a ld conductor is essentially changed compared to the three-dimensional casd, due to a onedimensional nature of electronic motion. For one thing, the usual deformational interaction of electrons with a sound is nearly completely screened thanks to the fact that the deformational potential on the Fermi surface is a constant, while energy bands do not overlap. As a result, the deformational interaction and the Coulomb one cancel each other in the limit q -->0 (q is a wave vector). Secondly, in a ld conductor the collisionless Landau damping does not contribute to attenuating the sound by electrons, since they move along a fixed axis with a constant speed v. Finally, a one-dimensional electronic system is highly disordered even by weak scattering [2], the localization of all electron states being of importance [3]. For these reasons, the principal mechanisms for electrons to interact with a sound are the so-called "cross-deformational" interaction resulting from modulation of a random scattering field by the sound wave, and Stewart-Tolman's effect (an inertial mechanism). In the present communication the elasticity equations are written for a ld conductor and the frequency dependence of the sound attenuation by electrons is found including both the strong and weak spatial dispersion cases. 2. The Hamiltonian of electrons interacting with the sound in a ld conductor is [1]

It=

dx

+(x) --ivo3 a--~

+ l j'd3r(Mnd~ + KiklmUikUlm)

). Here 4÷(x) and 4(x) are the two-component operators of creating and annihilating an electron with momentum -+Po at a point x (the electrons move along x-axis); r is the mean free time of electron with respect to the backscattering; mo is the free electron mass, two-row matrixes

The parentheses embracing operators 4 ÷ and 4 denote summation over ionic chains as well as trace over "spinor" indexes attached to 4 ÷ and 4. The phonons are taken three-dimensional, M is the ionic mass, N is the number of ions in unit volume, u is a displacement vector, 6 = ~u/at, usa is the strain tensor, Ksmm are the adiabatic elasticity moduli, Ask is the dimensionless complex tensor of "cross-deformational" potential. The latter arises in a deformed crystal due to electronimpurity interaction in the same way as the deformational potential does in consequence of electron-ion interaction. Explicit expressions for Kiktm and Ask are given in [ 1].

+l(~+(x)a+~(x)~+)14(x) ) 779

THEORY OF THE SOUND ATTENTUATION IN ONE-DIMENSIONAL METALS

780

× e -iqx Tr Po [A(x, t), B],

The random field ~'(x) describing elastic scattering of electrons by impurities is Gaussian and f-correlated

q-(x)V(x,))

= ta(x-x,);

q'(x)~'(x,)> = (V(x)V(x,)> = 0,

(3)

where l = vr is the mean free path of electron when scattered backwards, the brackets denote averaging over configurations of ~'(x). In equation (1) the first term represents the kinetic energy of electron measured from the Fermi level, including the electron-impurity interaction energy. The second term is the sound wave energy. The third term describes cross-deformational (~ uln ) and inertial (~ u~) interactions of electrons with a sound. 3. The equations of lattice motions are obtained from the Hamiltonian (1) according to general rules of mechanics. To do this, in H, strictly speaking, ti must be replaced by P/M before writing Hamilton's equations of motion. Since in equation (1) there is a term of the first order in lattice momentum (due to Stewart-Tolman's effect), the relation between momentum and velocity is Mfi = P--movSp(~J+o3~J) = P

--

--j, mo. e

(4)

where Sp includes averaging over the random field ~(x), the trace over "spinor" indexes and the thermodynamical average;j = j(r, t) is the electric current density (j is parallel to x-axis). The force F in elasticity equations (f~ = F) is evaluated using the formula F = -- 6H/Su. The result is

32urn Mniii = Kiklm OXkOXl

mo Oji e Ot

1 ~Tik a: Oxn

(5)

Tin(r, t) = Sp [~J+(X)tin(x)ff(x)]; tin(x) = 1 [~.(x)d+Ai n + ~.+(x)aA,n]. T

(6)

The current density j and the tension tensor Tin may be found on the lines of the linear response theory [4] mo I x ( q , c o ) ---- - - - - o ( q , co)fix --(fTlm)qwUlm, (7) C

Tin(q, co) -

m° (Tinj)qwft x -- (TinTtm)qwUtra. e

Here o(q, co) is the conductivity, the brackets (AB)qw designate the Fourier-transform of retarded correlation functions having the following structure

0

-L]2

(9)

where Po is the equilibrium density matrix with the Hamiltonian (1) taken at u -- 0;A(x, t) is the Heisenberg operator, Tr denotes trace over the quantum states of electron in a fixed configuration of ~'(x) and subsequent averaging over such configurations. Substituting equations (7) and (8) into equation (5) gives the elasticity equation for a l d metal:

qnqturn Mnco2ui = I ginlr n + al(TikTlm)qt°] 2

mo co [Six( /Tl.,)q~oqlu., a2 e

+ (Tikf)qcoqnUx].

(10)

4. To evaluate correlators of the type (9) including reconstruction of electron states by the scattering field ~(x), we have developed a new technique which enables to take into account a tensor nature, matrix structure, complexity as well as the dependence of the vertex operators Tin on the random field ~'(x), all this in a straightforward manner. A detailed discussion of this technique will be published later. Here we communicate the results of calculation o f the correlators in question. The cross correlators obey the symmetry relationship (/TMq~

(11)

= -- ( T ~ k j ) - q ~ .

This leads to cancellation in equation (10) o f the two last terms when the wave vector q is directed parallel or transverse to x axis. This means that the two major mechanisms of electron-phonon interaction, the crossdeformational and inertial ones, do not interfere, i.e. they act independently in elasticity equations. Such a conclusion is a consequence of different symmetries which have related terms in equation (1) in respect to transformation v -~ -- v. The correlator (TikTlm)tlto is

(TmT'rn)q°~=

AikA~ra+/X*k/X'ra(l+i~)4nar Co

(8)

+ 8 ~ i l (/'~k - a * k ) ( ~ m

- ~'r~)

x [S(co, q) + S(co, -- q)]

dx

(12)

The function S(co, q) is given by an integral 1

LI2

(AB)qo~ = - - i f dt ei~t f

Vol. 49, No. 8

S(co, q) = ~- e ~ ; d~ e -~ ~(~ -- 13)y (~), where y(~) is the solution of the equation:

13 = -- 2icor, (13)

Vol. 49, No. 8

THEORY OF THE SOUND ATTENTUATION IN ONE-DIMENSIONAL METALS

d [P -- i(66 -- qv) 7-]y (~) = 3f(~) + 2(~ -- 3) ~ [~f(~)] ;

(14) f(~) = - - e ~ E i ( - - ~ ) ;

0 =--d- ~

+3~

+ ~

--1

,

(lS)

which obeys the duly boundary conditions being finite at ~ = 3 and vanishing as ~j ~ + ~. A differential operator 0 may be treated as the operator of dimensionless collision frequency, since the left-hand side of equation (14) resembles a linearized kinetic equation. In the correlator (1 2), the first term does not depend on the localization effect and may be obtained in the Born approximation with respect to the crossdeformational interaction. The second term in equation (12) proportional to Im Aik Im Alto is affected by localization, as the multifold scattering effects result in a characteristic equation (14). Let us calculate S(66, q). Consider first the highfrequency region 6o7->> 1. According to equation (13), y(~) is then to be evaluated with large ~. One finds from equation (1 4) y (~) = A ~-1, while

1 S(66, q) = Y(3) = 1 --i(66--qV)r"

(16)

5. The general formulas (10) and (1 2) allow to investigate the propagation of long-wavelength sound with arbitrary polarization in any direction in a ld metal. Here we will confine ourselves to studying the longitudinal sound with wavevector q along the high-conductivity direction. Introduce the adiabatic sound speed s: Kxxxx = Mns 2 . Thanks to smallness of the relative attenuation I" of the sound, the following expression may easily be obtained from equation (10).

P = (66"r)Z

~ rnv ] Re - - O o

1 + 2(pol)~

[[A[2-- A~ ReS+(66, q)]}

(21)

Here Oo = nee2r/m; e and m are, respectively, the charge and effective mass of an electron; Axx = A = A~ + i2X2, S+(66, q) = S(w, q) + S(66, -- q). In the righthand side of equation (21) the wave number is to be replaced by co/s. The conductivity 0(66, q) has been found in [5] with account for the spatial dispersion. Let us analyse the frequency dependence of the sound attenuation in diverse ranges. Low frequencies, wz "~ 1. There are several intervals depending on the value of the spatial dispersion parameter • = ql = tolls. (a) When coz[ln corl
The low-frequency region, 607-~ 1, is more complicated. One can try the solution of equation (1 4) as an expansion in powers of 3:

1 S(66, q) = ~ S o ( q ) + S,(q)

78 1

t\mv/ 1

(17)

+ 2(P0l)2 (I A

i2 -- n11 A2~,/ "2)/

(22)

/

(b) In the range where Corlln ¢orl > slY, but COT<

The leading term in S(66, q) is

s/v,

,~ 7 d u Sh(rrulZ)v3(u)[l + v(U)/2] So(q) = --~ j

P c

~

~)

+ iql

'

I (mo,7

(18)

0

while the correction S~(66, q) is given by expressions

Sl(q) = ~,

ql~(2[ln66TI)-l;

(19)

X r]1

[66r\2~'

S

oo

7r2 ( d p / a Sh 0rp/2) $1 (q) = -~- : ch 3 (TrU/2)

IAr 2 + zx~/41

+ S-£17' J

0

v(u) [v3 (u) + 2v~(u) + 3v(u)- 21 x

v(p) + iql

-- u3(P) [v(--~)+-2]I

(20)

2[v(p) + iql] z J" The formula (20) holds for 2q/lln 66r1 >> 1. In contrast to the conductivity 0(66, q), the expansion of S(66, q) in terms of 66r "~ 1 does not contain powers of In 66r even at q = 0.

(23)

Here the attenuation due to Stewart-Tolman's effect oscillates as a function of frequency with a slowly varying period ~rs/vrlln (66r/2)1. These oscillations are due to the geometric resonance in Re 0(66, q) in a nonuniform field of the sound wave, which arises in consequence of the jumping nature of conductivity with a fixed jumping length 2 l lln (66r/2) 1. (c) In the range of a strong spatial dispersion, where w r > s/v, but 667-,~ 1,

782

THEORY OF THE SOUND ATTENTUATION IN ONE-DIMENSIONAL METALS

parameter tolls becomes comparable with unity, all this leading to a very small value (~ 10 -4) of the oscillating term in equation (23). Then, in the range coils > 1, the contribution of the inertial mechanism decreases as co-I and the attenuation takes up its minimum value

60-

4s-

Frnin = JO-

I$-

0

'/0 -t

,"

¢o'¢

!

Z

5

9t0

ta t

Fig. 1. The frequency dependence of the relative attenuation P of the longitudinal sound in a ld conductor. F is measured in terms of Fo = Z(mv/2Ms) (I A I/pol) 2 ; A2 = 0; The value K = rolls is plotted as abscissa. (I) a = 10, v/s = 103 ; (2) a = 4, v/s = 0.5 x 10 3.

(rnv2) l I mos3 ~2 I IAI~2 I F = (mr)Z ~ ( [ ~ ] + [2-~o/] J "

(24)

High frequencies, mr N 1. Here the sound attenuation is given by the same equation (24). Figure 1 represents the calculated frequency dependence of the relative attenuation P in the most interesting case when parameter a = IAIm

Vol. 49, No. 8

( p o l ) > 1.

(25)

The value a measures a relative weight of StewartTolman's effect compared to the cross-deformational interaction. When a > 1 the two mechanisms are in competition with each other. Figure 1 does not show the range (a) of extremely low frequencies, where F depends linearly on m, the inertial mechanism being negligible. The reason is that by choosen values o f a the range (a) corresponds to tolls < 10 -2 . With growing frequency, up to rolls ~ a, the main contribution to the sound attenuation gives the inertial mechanism. At first, P increases as m 3 In 2 mr. Then an oscillation of P developes in the range 0.15 < rolls < 0.25. Subsequent oscillations do not appear, since for s/v ~ 10 -3 the value 2 Iln (mz/2)[ ~ 20 already in the second period, the

Z m°slAl Mv po I

S

a t m = c~--.

(26)

v

Such a minimum results from the competition of the two damping mechanisms and is possible only provided a >~ 1. After the minimum the attenuation P increases in direct ratio to m, controlled by the cross-deformational interaction. The weight o f Stewart-Tolman's effect is reduced as a 2 with diminishing a. Let us discuss the physical nature of that part of impurity attenuation which is proportional to I A 12 and does not depend on spatial dispersion. This attenuation has quantum nature, taking place at the moment an electron is scattered by a separate impurity oscillating with the lattice. At this moment the electron absorbs a quantum taw. A local nature of such scattering leads to its independence of macroscopic lengths characterizing electronic motion in the crystal. In particular, this type of scattering does not depend on the localization of electron states in a l d conductor. It is just for this reason that the corresponding term is not affected by spatial dispersion. Such an interpretation is also backed by a formal calculation showing the related contribution to the correlator (12) to be determined by electron's Green functions with coinciding spatial arguments, due to a short-range nature of impurity potential. Meanwhile, the scattering proportional to A] in equation (21) results from the multiple scattering by different impurities, thus being affected both by localization effects and by spatial dispersion. REFERENCES 1.

L.V. Chebotarev & E.A. Kaner,

Solid State

Commun. 49, 357 (1984). 2.

I.M. Lifshitz, S.A. Gredescul & L.A. Pastur,

3.

Introduction into the Theory of Disordered Systems. Nauka, Moscow (t 982). N.F. Mott & W.D. Twose, Adv. Phys. 10, 107

4. 5.

(1961). L.D. Landau & E.M. Lifshitz, The Statistical Physics. Nauka, Moscow (1976). L.V. Chebotarev & E.A. Kaner, Solid State Commun. 4 8 , 3 3 3 (1983).