Oscillations of a continuous two-component disordered medium

Solid State Communications, Vol. 68, No. 8, pp. 807-810, 1988. Printed in Great Britain.

0038-1098/88 $3.00 + .00 Pergamon Press plc

O S C I L L A T I O N S OF A C O N T I N U O U S T W O - C O M P O N E N T D I S O R D E R E D M E D I U M L.I. Deich and E.V. Marchuk Kirensky Institute of Physics, Krasnoyarsk 660036, USSR

(Received 23 June 1987 in revised form 29 March 1988 by A.A. Maradudin) The dispersion law and attenuation of the optical phonons are investigated in the model of a disordered two-sublattice medium. It is shown that disorder induces extremum on the dispersion curve at k ~ k0, k0 is the correlation wave number of inhomogeneities. The increment of attenuation at k ,~ k0 has the form: v ,-~ (klko) 2"+~ where 2n is the order of zero of the spectral density of inhomogeneities S(k) at k = O. U S I N G N E W probes to characterize the structure of randomly-inhomogeneous materials is an actual problem today. In [1] it has been suggested for the first time to use the dispersion law of spin waves to study disordered ferromagnets. Further experimental investigations [2, 3] show the effectiveness of this method: the spatial fluctuations of the exchange and magnetization parameters with the correlation radius r0 >> a (a is the mean interatomic distance) have been discovered in the T M - M and T M - T M amorphous alloys. Further, analogous calculations of the dispersion law of the electromagnetic and plasma waves were made for the inhomogeneous metals and semiconductors [4], and calculations of the elastic waves in the inhomogeneously-deformed medium [5]. In this paper the dispersion law of the optical phonons in the disordered medium is investigated. Our goal is to elucidate the possibility to use this mode for the investigation of the disordered material's structure. Let us assume the medium consist of two different atoms A and B arranged so that Voronoy cells [6] contain exactly two atoms and thus, the medium is two-sublattice. Disorder in the system is connected with the local fluctuations of the cell's volume and their form which are shown in the fluctuations of the interatomic interaction and local densities of sublattices calculated in a certain small but macroscopic volume AV: a 3 <~ AV ~ r03 (as well as in [1-5] only the large-scale inhomogeneities with r 0 >> a will be considered). The energy of atom interaction write down in the form: 1

1 k,m

s#f

k,m

(1) where index s numbers the sublattices and ~ defines

the place of the k-th atom ofs-th sublattice. Such form of the potential means the neglect of the topological disorder [6]. Consideration of the applicability of this approximation for amorphous solids has been done in [7]. Averaging the potential (i) over the above mentioned volume A V by usual for elastic theory manner obtain in the continuum limit AV ~ 0 the next expression for thi,t q~i,t =

~IdV

~'(i~j)

,

}

+ g Y~ 4-rs(¢~j, ¢~, g: - if) •

(2)

.f#s

Here /}" and ~js are the potentials of the innersublattice and intersublattice interaction averaged over

Av, go =

(N N)/2 kax, + ax,}/

is the deformation tensor of S-sublattice. Availability of several sublattices has led to appearance of an additional dynamic variable - relative displacement of sublattices if - ~{. Equations of sublattice motion deduce from (2) by a standard manner; they have the following form (in harmonic approximation): 0A(r)~/A eO(r)~f

A A _ nAB .~n. - - ijrrm " - m , j n

--

nAB¢~ B -- 2Am)= 0 --im v-rn

s

-

RAB(~A --ira ~--rn -

--

Ci;mn~m,nj

-

c,7_&,,

s

j

_ DsA ~a -- qnm--rn,jn

,~)

=

O.

(3) Here C, D and B are force tensors of inner- and inter-sublattice interactions; c o m m a before indices means differentiation over corresponding coordinates, summation is implied under repeating indices. Writing down (3) we have assumed that interactions in the system are centrally symmetric and neglected the

807

8O8

CONTINUOUS TWO-COMPONENT DISORDERED MEDIUM

fluctuations of force tensors. Grounds for this will be done below. The density of sublattices 0A(r) and 0B(r) are supposed to be stationary random functions of coordinates, which are defined by their mean values 0~{~) =

(0A(B)(r))

=

mA(B)nA(S)

=

½mA(B)n

mean square fluctuations e2 = (g2(r)), 0(r) -- 00 and correlation functions KA(81(r -- r)

=

KAB(r -- r') =

(4) g(r) =

(gA(n)(r)gA(8)(r)) (5)

(gA(r)gs(r))

with arbitrary radii of correlation; mA(B)and nA(8) in (4) are the masses of the corresponding atoms and their concentrations respectively, rtA ns = in. Notice that if we neglect fluctuation parts of densities in equation (3) we obtain using (4) the equation of motion of homogeneous medium deduced in [8] by another manner. For the sake of simplification of calculation the medium is assumed to be isotropic and force tensors will be written in the form: =

c,:~, = c~6,j6~, D,.~ = Bkl =

+

C~(G6/,

+

6,,G)

dABb,jbk, + a'~Bb,~bj, + dBAbg,6;,

(6)

Cff6kl

which is determined by the symmetry properties of the tensors with respect to the transition of indices. Carry out Fourier transformations of the equation (3) with an account of (6) and average them over the ensemble of realizations of densities 0A and On by the method of papers [1]. Equations for average amplitudes deduced with an accuracy up to members of the order e2 write down in the matrix form:

Vol. 68, No. 8

indices. Functions S ( k ) are Fourier transforms of the corresponding correlation functions. Dispersion laws of optical and acoustic oscillations are obtained from the condition of existence of nontrivial solution of the system (7). Later on we shall confine ourselves only to the discussions of transverse modes as longitudinal waves will behave qualitatively in the same way. Having supposed e~ = e~ = 0 we obtain the known dispersion laws of long-wave optical and acoustic oscillations in the homogeneous medium: D.~±(k) = ~o+

=

O.~ + ~c±k2

(10)

s±k

where f~0, x± and s± are expressed through force constants and masses of the atoms A and B in a usual form. An account of inhomogeneities modifies expressions (10). The characteristic frequencies of optical phonons as a rule are much higher than the maximum acoustic frequencies. With an account of this remark we can neglect the scattering processes of optical waves into acoustic ones. Then the dispersion law of acoustic phonons wholly coincides with the corresponding expression of the [5], and the dispersion of optical waves is defined by the following expression: ~l(k)

=

~0{1 + ~~g i d 3 q [ ( 1 - q~'] q 2 j k 2 -1 q2

,]

(C A + o)4¢~RA)(A~) + (D s -- 094eA¢BRBA)(A s ) = 0 (D A

--

~48AeBRAS)(AA ) + (C ~ + o)4e2RS)(A B) = O.

(7) Here ~o is the frequency of oscillations and the matrices have the following form: CPm

=

( k 2 C A .3!- (~ _

D,~

=

(k2¢7~A -

(_020A)(~im

~- ( C A -3V

(~)(~im + k i k m ( d q- ~AB)

~A)k,k., (8)

where k~ is the component of a wave vector k; matrices C a and D ~ are obtained by substitution of indices A and B from (8). R ~, R ~, R ~n and R ~A represent themselves integral convolutions of the form: RA = R BA =

f d 3 q ( D B G B A ) S A ( k -- q) fd3q(CnGSA)SAS(k

-- q)

(9)

where G s A = (C B) ~ D A ( D B ) - ~ C A. Expressions for R n and R An are obtained from (9) by the transition of

+ 2~A£e (0AQgQg + J ) 2 SAS(k -- q) + ~(0~+

j)2 SS(k-q))}.

(ll)

Here ~ is the relative mean square fluctuation of the density of the corresponding sublattice ~ = e/00, a = to±/xll; Z-axis of the coordinate system is directed along the wave vector k. For the analysis of all possible types of modifications of the dispersion laws of optical phonons let us firstly consider the case when only one of the sublattices, for example A-th sublattice, contributes to modification. Then in (l l) the term with S A ( k ) will only remain. Consider asymptotic forms of the expression obtained at k ,~ k~ and k >> k0A where k0A is the correlation wave number entering function S A ( k ) , k~ ~ l / ~ . Estimating the corresponding integrals we find that in the region k ~ k0A the dispersion law

809

CONTINUOUS TWO-COMPONENT DISORDERED MEDIUM

Vol. 68, No. 8

changes from the form: n~(k) ~ n o l - ~2A Og + ~0a

× [(0{-t'-2)o'1+

tc±(kg) 2

5~2 + 2~ 1 3 ~+

a2 ( k ~ ) : ] } ~2c

{12) at k ,~ k~ to the asymptotic form n±(k) ~ no

f 1 - ~2A( ~ +

)2 o0

{}g

8~¢±k2j

(13)

at k >> k~. Here {r~ and {r2 are positive numbers determined by the expressions: a~

=

2n(k0A)2

S A(k)

k*'-

Fig. I. Modified dispersion law of optical waves. Full curve corresponds to the case ~c~ > 0; dotted line - to Kl < 0; dashed line - to the unperturbed dispersion law.

S A(k) dk;

i

ko

dk.

zc

From the expression (12) and (13) it is seen that the behaviour of dispersion curves considerably depends on the sign ~¢± and doesn't depend on the concrete form of the spectral density of inhomogencities S ( k ) . In the case when ~± > 0 the gap in the spectrum decreases and the dispersion curve has a minimum at k ~ k0A; at K± < 0 the gap increases and there is a maximum in this region (Fig. 1). The value of the shift of the optical gap A~(0) in comparison with the homogeneous medium case and the value of modification at k ~ k~ by the order of magnitude are determined by the expression: flo

f~o

tc;(kg)2

eA mA ~ m ~ f "

(14)

In general case the modification of the dispersion curve can be determined by any of three functions figuring in (9) or by a number of them. In the last case when the corresponding correlation radii are sufficiently far separated from each other one can observe a number of modifications of the type depicted in Fig. 1. Values of these modifications will be determined by the expressions of the form (13) where the value f~2/~&_k~ is multiplied by the corresponding multiplier which determines partial contribution of a given spectral density. The modification of the dispersion law of excitations in the inhomogeneous medium is accompanied by their attenuation. The analysis showed that the increment of attenuation behaves in the following way: v ~ (k/ko) 2"+1 at k ,~ k0 and v ~ k o / k at k >> k0. Here 2n is the zero order of the function S ( k ) at k = 0. The case when n = 0 corresponds to

the usually applied functions; for these functions S(0) ~ 0. In this situation the optical phonons at small k are determined badly. However, there exist physical situations which must be modelled by such functions S ( k ) that S ( 0 ) = 0 [9] (for example, if the inhomogeneity is connected with the random internal deformations). Then the region of anomalous attenuation is absent and the dispersion law modification obtained in the paper has the sense [10]. Note that the analogous situation appeared when considering spin waves in the ferromagnets with random anisotropy [11]. The dependence of the increment of attenuation on the wave vector has also maximum in the region k ~ k0 and can be used for the investigation of inhomogeneities. These results were obtained by the neglect of the fluctuations of force constants. The analysis showed that the contribution of the fluctuations of any of force constants into dispersion law modification is less than that of density fluctuations (Sk/f'zo) 2 times. This fact justifies the mentioned approximation. In conclusion discuss the model of medium considered in the work. The fact that every Voronoy's polyhedron contains equal number of atoms p is its principal feature. It means the presence of composition order in the system. The concrete calculation has been done for the case p = 2. The increase of a number of sublattices gives rise to the additional modes and correspondingly new canals of scattering on inhomogeneities appeared. However, if these modes are sufficiently separated frequencies so that their dispersion curves are not intersected then the new canals could be neglected as well as we neglect the

810

CONTINUOUS TWO-COMPONENT DISORDERED MEDIUM

scattering into acoustic waves. In this case the dispersion law of optical modes is described by the expressions (12)-(14) as before. It is only necessary to substitute instead of masses of atoms A and B certain effective masses corresponding to the modes under consideration. Breaking of composition order essentially complicates the classification and analysis of oscillations. However, in this situation there is a mode which is connected with the relative oscillations of groups of identical atoms. In such conditions as before this mode could be considered independently from other modes as oscillation of the effective two-sublattice medium. "Atoms" of this medium would have the mass depending on the concentration of alloy components. The dispersion of this mode is also described by the expressions (12)-(14). Acknowledgements - Authors are thankful to Prof. V.A. Ignatchenko for attention to the paper and useful discussions.

3. 4. 5. 6. 7. 8. 9. 10.

REFERENCES 1. 2.

V.A. Ignatchenko & R.S. Iskhakov. Zh. Eksp. Teor. Fiz. 72, 1005 (1977); 74, 1386 (1978); 75, 1438 (1978). V.A. Ignatchenko & R.S. Iskhakov, in Physics of Magnetic Materials (Proc. of the 2nd Intern.

11.

Vol. 68, No. 8

Conf. on Physics of Magnetic Materials, Jadwisin (Poland), Sep. 17-22, 1984) Part I Invited Papers. Singapore - Philadelphia: World Scientific, 1985, p. 527. L.J. Maksymowicz, D. Temple & R. Zuberek, J M M M 58, 303 (1986); 62, 305 (1986). V.A. Ignatchenko, Yu.I Mankov & R.V. Rachmanov. Zh. Eksp. Teor. Fiz. 81, 1772 (1981); 87, 228 (1984). L.I. Deich & V.A. Ignatchenko, Fiz. Tverd. Tela 27, 1883 (1985). J.M. Ziman, Models of Disorder. Cambridge University Press, Cambridge, 1979. L.I. Deich & V.A. Ignatchenko, Fiz. Tverd. Tela 29, 7825 (1987). A.M. Kosevich, Fundamentals of Mechanics of Crystalline Lattice, Nauka, Moscow (1978), p. 88. V.A. Ignatchenko & R.S. Iskhakov, Fiz. Met. i Metalloved, 65, (1988), N5 (in print). If two types of inhomogeneities with different correlation radii r0 "> ~0 and spectral densities with n = 1 and n = 0 respectively are present in the system there exists the region of the wave numbers ~0ro 3 '~ k 2 "~ ro 2 where attenuation is determined by long-wave inhomogeneities and the possibility of observing the modified dispersion law remains. I.V. Bogomaz & V.A. Ignatchenko, Spin Waves In Ferro- and Ferrimagnets with Random Anisotropy. Preprint N363F, Krasnoyarsk (1985).