Phonon contribution to the absorption of ultrasound in amorphous solids at moderately low temperatures

Physica B 254 (1998) 92—98

Phonon contribution to the absorption of ultrasound in amorphous solids at moderately low temperatures Yu.P. Krasny!, N.P. Kovalenko", J. Krawczyk!,* ! Opole University, Inst. Mathematics, Oleska 48, 45-052 Opole, Poland " I.I. Mechnikov State University, Dvorianskaya 2, 270100, Odessa, Ukraine Received 11 April 1997; received in revised form 6 January 1998; accepted 8 May 1998

Abstract Sound absorbtion in amorphous solids is considered to be due to the scattering of sound waves from the thermal phonons. It is shown that the dependence of the absorption coefficient on the temperature displays a maximum in the interval 10[¹[100 K. The frequency dependence of the absorption coefficient is investigated. Numerical calculations for amorphous Mg and Zn illustrate the theoretical results. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Phonons; Ultrasound; Amorphous solids

1. One of the most important experimental results of the last decades was the discovery of a set of universal properties of amorphous solids that are practically independent of the specific chemical composition of the compounds. These properties are not typical for crystalline solids and are often called “anomalous”. They concern the low-temperature behaviour of the heat capacity [1—3], thermal conductivity [1—3], thermal expansion [4], propagation and absorption of ultrasound [5—8], and other properties governed by specific features of atomic dynamics. A typical temperature and frequency dependence of acoustic absorption in amorphous solids is presented in Fig. 1 [5]. These experiments indicate a different behaviour of the absorption coefficient (AC) in different temperature intervals. For 1(¹(10 K the AC is proportional

to ¹/ (2(n(3). For 10(¹)100 K it displays a maximum; at the same time the AC decreases with increasing frequency. At very low temperatures (¹(1 K) the AC decreases with increasing intensity, and then it stabilizes [6—8]. A theoretical explanation for this behaviour of the AC at very low temperatures was made in Refs. [7—10] taking into consideration an interaction between the sound wave and localized atomic states (“tunneling states”). In this paper the anomalous behaviour of the acoustic AC at moderately low temperatures (1(¹(100 K) is explained as a result of the interaction of ultrasound wave with thermal phonons. 2. The Hamiltonian of a monotonic amorphous dielectric or ion subsystem in an amorphous metal can be written in the following form:

* Corresponding author. Tel.: #48 54 58 41; e-mail: [email protected].

N P2 H K " + j #º(R ,2, R ), 1 N 2M j/1

0921-4526/98/$ — see front matter ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 4 1 9 - 0

(1)

Yu.P. Krasny et al. / Physica B 254 (1998) 92–98

93

Here the indices a, b denote the Cartesian components of the vectors R and m and mQ means the derivative of m with respect to time. We assume the anharmonic terms are small in comparison with the quadratic ones. Therefore, they can be treated as small perturbations that cause transitions between the states of the unperturbed Hamiltonian H K . The probability of a transi2 tion of the system from the state DsT with energy E into the state D f T with energy E under the s f action of the perturbation H K is 3 2p P " DSsDH K D f TD2d(E !E ). s?f 3 f s +

Fig. 1. Temperature dependence of absorption of longitudinal acoustic waves in glassy, SiO [5]: (a) u"930 MHz; 2 (b) u"1000 MHz. The dashed line, corresponds to crystalline SiO (quartz). 2

where the first term is the kinetic energy and º(R ,2, R ) the potential energy of interaction of 1 N N atoms (ions) with masses M in a volume ». Let R ,(R ,2, R ) be the coordinates of the 0 01 0N quasiequilibrium positions of the atoms (ions) which correspond to one of the minima of º. The case when the positions R do not form a crystal 0 lattice is considered in this paper. At ¹O0, the atoms (ions) oscillate near the equilibrium positions. One can expand º in a Taylor series in powers of the atom displacements nK "nK (R ) and j 0j confine the expansion to terms of the second and the third order in the oscillation amplitudes. Then Hamiltonian (1) assumes the following form: H K "H K #H K , 2 3

(2)

where N M ­2º 1 0 , H K "º # + mKQ 2# + + mK a mK b 2 0 j1 j2 ­Ra ­Rb 2 j 2 0j1 0j2 j/1 j1j2 a,b

(3)

(6)

As the system can be found in any of the quasiequilibrium states it is necessary to average expression (6) over all such states [11—13]: 2p P " DSsDH K D f TD2d(E !E ). s?f 3 f s +

(7)

Then Hamiltonian H K acquires the following form: 2 K H K "H M #*H K . (8) 2 2 2 The first term H MK is the Hamiltonian correspond2 ing to the averaged configuration (see Ref. [11]); the second term describes the fluctuations of the Hamiltonian with respect to the mean equilibrium configuration. The energy E and the eigenstate D f T can be & expressed in analogous forms E "EM #*E , f f f D f T"D fM T#D*f T,

(9)

where EM and D fM T are the energy and eigenfunction f of the operator H MK , respectively, and *E and D*f T 2 f arise as a result of phonon scattering from density fluctuations. These terms are of the fourth order in the displacements (see Refs. [11—13]) and that is why one can omit them in Eq. (7). Then

º "º(R ,2, R ), 0 01 0N

(4)

2p PM + DSsN DH K D fM TD2d(EM !EM ). s?f 3 s f +

1 ­3º 0 H K " + + mK a mK b mK c . j1 j2 j3 ­Ra ­Rb ­Rc 3 3! 1 2 3 0j 0j oj j1,j2,j3 a,b,c

(5)

In the second quantization representation the Hamiltonian and displacements nK take the j

(10)

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Yu.P. Krasny et al. / Physica B 254 (1998) 92–98

form [11] 3 KM "ºM #+ + +u (k)[CK k` C` #1], (11) H j j kj 2 0 k j/1 1@2 + [CK k #CK `k ] nK "+ j k 2NMu (k) j ~j j ,j ]e(k,j)exp(ikR ), (12) 0j where e(k, j) is the polarization vector of a j-type phonon with momentum +k, +u (k) is the energy of j the phonon. They are defined by [11,13]

A

B

P

k2 v 0 u2(k)"u2(k)" U(k)# d3k k2 U(k ) 1 1 1z 1 M (2p)3M ][a(k ; k ; k #k)!a(k )], 1x 1y 1z v 0 u2(k)"u2(k)"u2(k)" d3k k2 U(k ) t 3 2 1 1x (2p)3M

P

][a(k ; k ; k #k)!a(k )]. (13) 1x 1y 1z The indices “l” and “t” refer to the longitudinal and transverse phonons, respectively, the function U(k) represents the Fourier transform of the interparticle interaction potential, v "»/N, and a(k) is the 0 structure factor. Formulae (13) were used to find dispersion curves for amorphous Mg and Zn in Refs. [11,13]. They have a characteristic roton-like minimum in the short-wavelength region (see Fig. 2). It follows from Eq. (10) that the averaged transition probability PM has a nonzero value s?f only in the case of energy conservation EM "EM . It s f is easy to show that energy conservation does not take place when three phonons are created or annihilated simultaneously: the probability of such pro-

cesses is equal to zero. Only processes with the creation of two phonons and a simultaneous annihilation of a third phonon (and vice versa) are permitted. The probability of such a process is equal to [12,14,15] Pq

1 nq nq (1#nq ) 1j1 2j2 3j3 " I j § j ?q3j3 N2 u (q )u (q )u (q ) q1j1§q2j2§q3j3 j1 1 j2 2 j3 3

q 1 1 2 2

]dM+u (q )#+u (q )!+u (q )N, j1 1 j2 2 j3 3 1 (nq #1)(nq #1)nq 1j1 2j2 3j3 I Pq q q " q q q 3j3? 1j1§ 2j2 N2 u (q )u (q )u (q ) 1j1§ 2j2§ 3j3 j1 1 j2 2 j3 3 ]dM+u (q )#+u (q )!+u (q )N. j1 1 j2 2 j3 3 (14) Here Iq

j§ j§ j

A B

"

q q 1 1 2 2 3 3

+ 3 1 2p Kq q q K* , q q q 1j1§ 2j2§ 3j 1j1§ 2j2§ 3j3 2M N +

" + + ea(q j )eb(q j )ec(q j ) 1 1 2 2 3 3 j1j2j3 a,b,c ­3º 0 ] ­Ra ­Rb ­Rc 0j1 0j2 0j3 ]expMi(q R #q R #q R )N, 1 0j1 2 0j2 3 oj3 when nq is the number of phonons with moj mentum +q and polarization j. In the general case the phonon distribution function depends on time and coordinates, i.e., Kq

j§ j§ j

q q 1 1 2 2 3

nq "nq (r, t). j j If the phonons form a dilute gas, the kinetic equation for a certain kind of phonon takes the

Fig. 2. Dispersion laws for amorphous Mg (a) and Zn (b) u : longitudinal oscillations, u ; transversal oscillations. 5

Yu.P. Krasny et al. / Physica B 254 (1998) 92–98

following form [12,15]:

In this case the condition

P P

­nq v2 ­u (q ) ­nq 1j1# 1j1" + 0 d3q d3q j1 1 2 3 (2p)6 ­t ­r ­q 1 j2j3

G

] [(1#nq )(1#nq )nq !nq nq (1#nq )] 1j 1 2j 2 3j3 1j1 2j2 3j 3 ]

Iq

j§ j§ j

d[+u (q )#+u (q )!+u (q )] j1 1 j2 2 j3 3 u (q )u (q )u (q ) j1 1 j2 2 j3 3

q q 1 1 2 2 3 3

1 # [(1#nq )nq nq !nq (1#nq )(1#nq )] 1j 1 2j2 3j3 1j1 2j2 2j 2 2 ]

Iq

j§ j§ j

H

d[+u (q )!+u (q )!+u (q )] j1 1 j2 2 j3 3 u (q )u (q )u (q ) j1 1 j2 2 j3 3 (15)

q q 3 3 2 2 1 1

One has to solve this equation to find the temperature dependence of the AC. 3. Consider a sound wave with wave vector q, frequency u, and polarization j that propagates through a solid along the x-axis. From the quantum mechanical point of view the wave considered represents the one-dimensional motion of phonons. As a result of their interaction with thermal phonons, the density n of sound phonons depends on x only and obeys the kinetic Eq. (15). In the stationary case, this equation simplifies to nq (r, t)"n (x), (16) 1j1 j ­nq (r, t) ­n (x) 1j1 " j "0, ­t ­t ­u (q ) ­nq (r, t) ­n (x) 1j1 j1 1 "u j , (17) j ­x ­q ­r 1 where u is the velocity of sound phonons. j Let us consider temperatures such that +u@k ¹. B For sound waves (see Fig. 2) u&109 s~1 and for 10(¹[100 K, +u&10~11 J@k T&10~8—10~7 J. B The corresponding wave length is j"(2p/u)u & j 2]10~5 m Aa, where a is the distance between particles. Therefore, the dispersion relation can be approximated for the sound waves considered as u (q)"u q, j j where u is a constant. The thermal phonon energies j have an order of magnitude +u (q ); +u (q )&k ¹. j2 2 j3 3 B

95

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(19) +u (q ),+u@+u (q )#+u (q ) j2 2 j3 3 j1 1 is satisfied. This means that the second term in the kinetic Eq. (15) vanishes, i.e., processes of sound phonon decay into two thermal phonons are impossible. Hence, the main role in sound absorption is played by processes of absorption of sound quanta by thermal phonons. For a macroscopic sound wave n (x)A1 and j nq #1"n (x)#1+n (x). (20) 1j1 j j For thermal phonons one can use the equilibrium distribution function

C A B D

~1 +u (q) j !1 . (21) nq +n(0) q " exp j j k ¹ B The potential energy º(R , 2, R ) is explicitly 01 0N characterized by a parameter r that defines the 0 radius of interparticle interaction. Utilizing dimensionless quantities, that is r "R /r , is more conve0j 0j 0 nient. For a homogeneous and isotropic amorphous system the function Iq q q has to be real and to 1j1§ 2j2§ 3j3 depend on the moduli of the q only after configuraj tional averaging: Iq

"I(q r ; q r ; q r ). j§ j§ j 10 20 30 In Eq. (15), the limits of integration lie in the region q q 1 1 2 2 3 3

A B

6p2 1@3 . (22) 0)q)q " D v 0 The integrand I(q r ; q r ; q r ) can be expanded in 10 20 30 a Taylor series because r [1/q . According to Ref. 0 D [12] we have

A B

+ 3 2p r6 0 I(q r ; q r ; q r )+I(0, 0, 0)"I " 10 20 30 0 2M + 27N ­3º ­3º 0 0 ] + + ­ra ­rb ­rc ­ra ­rb ­rc 1 2 3 1 2 j1,j2,j3 a,b,c 0j 0j 0j 0j 0j 0j3

(23)

After integration over angles and taking Eq. (16)—(23) into account, Eq. (15) takes the final form 1 dn (x) j "! n (x). l(u) j dx

(24)

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Yu.P. Krasny et al. / Physica B 254 (1998) 92–98

Here l(u) is the mean free path for a sound phonon:

P

P

qD qD 1 v2 0 " + q2 dq q2 dq W(u (q ); 2 2 3 3 j2 2 l(u) (2p2)2u j j2j3 0 0 (25) ]u (q ))d(u#u (q )!u (q )) j2 2 j3 3 j3 3 with W(u (q );u (q ))"!W(u (q );u (q )) j2 2 j3 3 j3 3 j2 2 I 1 "0 + uu (q )u (q ) j2 2 j3 3 ~1 +u (q ) j2 2 !1 ] exp k¹ B ~1 +u (q ) j3 3 !1 . (26) ! exp k¹ B The value of l~1(u) is twice the AC [8,15]. Numerical calculations show (see Fig. 2) that the dispersion curve u (q) has a roton-like minimum. Near this j minimum the approximation

GC A B D C A B D H

+(q!q )2 0j (27) u +u(0)(q)"D # j j j 2m j can be used. For small q we have the approximation u (q)+u q. (28) j j The term +2(q!q )2/2m is a second-order infini0j j tesimal in comparison with +D &k ¹. Then we take j B +2(q!q )2 0j @+D &k ¹. (29) j B 2m j To obtain a numerical estimate of the integrals in Eq. (25) we use the following approximation. Let the function u (q) have its maximum value at q"q . j 1j Then, it is possible to put u (q)"u q in the integrand j j in integrating within the limits (0, q ), and one can 1j put u (q)+u(0)(q) within the limits (q , q ). In such j 1j D j a way we obtain for l~1(u):

GP

P

q1j2 q1j3 1 v2 0 "+ q2 dq q2dq 2 2 3 3 l(u) (2p2)2u j 0 0 j2j3 ]W(u q ; u q )d(u#u q !u q ) j2 2 j3 3 j2 2 j3 3 q1j2 qD # q2 dq q2 dq 2 2 3 3 0 q1j3 ]W(u q ; u(0)(q ))d(u#u q !u(0)(q )) j3 3 j2 2 j3 3 j2 2

P

P

P

#

qD

P

q2 dq 2 2

qD

q2 dq W(u(0)(q ); j2 2 3 3

qj qj ]u(0)(q ))d(u#u(0)(q )!u(0)(q )) . j2 2 j3 3 j3 3 1 2

1 3

H

(30)

The last term in this expression vanishes because of the d-function. In the first and second terms, integrations are performed over q . In the third term it is 3 done over q . One can obtain 2 I ­n(0) 1 ­n(0) ~1 j 1$ j $W(u ; u $u)"! 0 j j +u2 ­u 2 ­u j j j ­2n(0) 1 ­n(0) ~1 1 j # j ] u# ­u2 6 ­u u j j j ­3n(0) 1 ­n(0) ~1 j # j ] ­u3 2 ­u j j ­2n(0) 1 j ] u2#2 , ­u2 u j j taking the inequality +u@+u into account. j Let us use an implicit form for the function n(0) q (21) and take into account the rapid decrease of j the integrands. Therefore, the upper limits of integrations can be removed to infinity in the first two integrals. In the third integral the upper and lower limits can be taken as plus infinity and minus infinity, respectively. Then, all the results are presented in the analytical form

G CA B A B D CA B A B A B A B D H

A B GG A BH

A B

A B

1 k ¹ 2 k ¹ 3 +u B B ! "+ A 1#f j1 +D j1 +D l(u) 2p2 k ¹ j1 j1 B j1 +2u2 #O , (31) k2¹2 B where 1 = f (e)"B exp ! + e~5@2Jn#1 j1 j1 e n/0 !n D j1 e~3@2 exp # , (32) e Jn#1

A B C D A B

p2 D2 v j1 , 0 I + A " j1 (2p2)2u 0 3 +u3 u3 1 j j2 j j2 3 D 2 u q j1 0j1 j1 , 2m B " j1 +q2 j1 p3@2 D j1 0j1 m D D " j1 j1. j1 +q oj1

A

BS

(33)

Yu.P. Krasny et al. / Physica B 254 (1998) 92–98

97

One can see from Fig. 2 that +D +D 5 ' - Z1. (34) k ¹ k ¹ B B Consequently, the main contribution to the sound absorption arises from the interaction of sound phonons with thermal longitudinal phonons, i.e.,

A BG A B A BH

A B

1 k ¹ 2 k ¹ 3 +u "A B 1#f B ! - +D - hD l(u) 2p2 k ¹ B +2u2 #O , k2¹2 B where f (e) is, according to Eq. (32) and (34), 1 f (e)+B exp ! [e~5@2#D e~3@2]. e

A B

(35)

(36)

As follows from Eq. (35), l~1(u) decreases with increasing u in accordance with the experimental results (see Fig. 1). The temperature dependence of l~1(u) is determined by the behaviour of the function f (e) which in turn determines the contribution to the sound absorbtion from the interaction of the sound wave with thermal phonons for wave vectors in the vicinity of the minimum on dispersion curve (i.e., with “rotons”). This function vanishes for eP0 and ePR. It takes its maximum value at

A

B S

+D 25 1 1 5 1 - " e~1" !D # # D # D2. m k ¹ 16 4 - 4 2 2 B . The present stage of the theory does not allow performing ab initio calculations of the phonon dispersion curves for such a complex compound as SiO . That is why ab initio investigation of the 2 ultrasound absorption in glassy SiO according to 2 Eq. (35), and a subsequent comparison with the available experimental data [5] is impossible at the present time. Another way of solving this problem is by fitting to the experimental data, as has been done in Ref. [16] in the consideration of the thermal conductivity of amorphous materials. However, such a pure fitting procedure would not clear up and reflect the real “physical picture” of the phenomena under investigation according to the opinion of the authors of Ref. [17]. A good many different amorphous materials, independent of their chem-

Fig. 3. (a) Temperature dependence of f (e)"(+D /k ¹)2 - B 1/D l(u) for amorphous Zn: (1) l"0, (2) l"1013 (Hz), (3) l" 5]1013 (Hz); e"k ¹/+D ; l"u/2p (Hz). (b) Temperature deB pendence of f (e)"(+D /k ¹)2]1/D l(u) for amorphous Mg: l B l (1) l"0, (2) l"1013 (Hz), (3) l"5]1013 (Hz); e"k ¹/+D ; B l"u/2p (Hz).

ical composition, display a similar temperature dependence of the heat capacity, ultrasound absorption, and electrical conductivity in the moderately low temperature region. Therefore, amorphous Mg and Zn, for which rigorous theoretical calculations could be performed, are chosen to illustrate the approach presented. For amorphous Mg and Zn (see Fig. 2) D +0.5 and e +0.43. This means that . f (e) reaches its maximum at the following temperatures: for Mg (D &2]1013 s~1) ¹ &60 K, . for Zn (D &3]1013 s~1) ¹ &100 K. . Hence at low (¹@¹ ) and high (¹A¹ ) temper. . atures l~1&¹2. At moderately low temperatures

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Yu.P. Krasny et al. / Physica B 254 (1998) 92–98

(¹&¹ ) the function l increases faster than . ¹2 with increasing ¹ at first, and then this growth becomes slower. In Fig. 3 the dependence

A B

+D 2 1 k ¹ on e" B k ¹ A l(u) +D B is shown for amorphous Mg (B &50) and Zn (B &25). Qualitatively, these curves reproduce the experimental dependence of sound absorbtion on frequency and temperature. In conclusion one should emphasize that at very low temperatures (¹(1 K) the anomalous thermophysical properties of glasses are explained by the interaction of phonons with tunneling states; then at moderately low temperatures (10(¹(100 K) these properties are explained by an “unusual” behaviour of the dispersion relation of the phonon excitations (with a minimum in the short-wavelength region). Other “anomalous” properties can be also explained on the basis of the quasiphonon model without any additional hypothesis [11—13]. F(e)"

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