Journal ArticleWe calculate the exact thermal conductivity and the heat capacity of an insulator for which the lattice dynamics are given by a phonon gas in the presence of frozen-in disorder, in the special case of the "backward-scattering'' model of impurity scattering

Mattis, Daniel C.

Molina, Mario I.

English

Journal ArticleWe calculate the exact thermal conductivity and the heat capacity of an insulator for which the lattice dynamics are given by a phonon gas in the presence of frozen-in disorder, in the special case of the "backward-scattering\u27\u27 model of impurity scattering

The University of Utah: J. Willard Marriott Digital Library

PHYSICAL REVIEW B VOLUME 46, NUMBER 13 1 OCTOBER 1992-1 Thermal conductivity and thermodynamics of phonons for an exactly soluble model of disorder Mario I. Molina· and Daniel C. Mattis Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 4 November 1991; revised manuscript received 20 April 1992) We calculate the exact thermal conductivity and the heat capacity of an insulator for which the lattice dynamics are given by a phonon gas in the presence of frozen-in disorder, in the special case of the "backward-scattering" model of impurity scattering. The thermal conductivity of an insulator is difficult to calculate theoretically; transport theory applied to pho-nons is notoriously inexact and inconclusive. Recently, however, the Kubo-type formulas for such systems have been formulated. 1 Nevertheless, these formulas are rath-er complicated and in general require summation of an infinite number of diagrams, so that in the process of evaluating them a certain number of simplifications and approximations have been found useful or necessary. 1 The effects of scattering on the thermodynamics proper-ties are also of physical interest. One would like to know, in general, how the changes in thermodynamics proper-ties due to scattering effects correlate with the transport properties. In this paper we examine an extremely simple and ex-actly solvable "backward-scattering model", in which stabilizing anharmonicities are included. For this model, we can obtain the one- and two-body Green functions without any approximation. A similar model has been used successfully in the study of classical transport2 and in the electron gas. 3 We believe this to be the first appli-cation to phonons. The thermal conductivity and specific heat (or arbitrary other thermodynamic functions) are found exactly, in terms of normalized distribution func-tions P q (v q ) of the random scattering matrix elements v q' The result for the latter is c (T) = k B{32(2Vol)-1 X 1: J dvqPq(vq ) q," X {il,,(vq )/[2 sinh({3il,,(vq )/2)] J 2 , (1) where (3= 1 IkB T, 1i= I, a = ±, and wqy2(2Ivql-l), vq <-I/2 il+(vq)= wqyl +2v q, -1/2 <Vq < 1/2 2wqYvq, vq > 1/2 , j2WqYlvql, Vq < -1/2 il_(vq)= WqY~, -1/2<vq <l/2 wqY2(2vq-l), vq>1/2. (2) (3) Here Wq is the frequency of the qth normal mode, in 46 the absence of disorder. For perfect crystals, P q (v q ) ap-proaches a delta function 8(vq), and (1) reduces to the textbook formula for the lattice specific heat. The specific heat associated with disorder is c( T) -co( T), where co( T)=kB{32(Vol)-1 1: [wq/2 sinh({3wq/2)]2 . q The high-temperature limit of c (t) is obtainable in a straightforward manner. In the limit (3-'>-0, [wq/2 sinh({3wq/2)f-'>-{3-2, with f dv P(v)= 1 indepen-dent of the choice of P (v) and the sum over q restricted to the first Brillouin zone, Eq. (1) yields the usual Dulong-Petit value of kB per site in this limit. To analyze the low-temperature behavior, more infor-mation concerning P q (v q) is required, it is convenient to examine P near the two critical points, Ivi =0 and Ivi =t. For Ivi near 0, the normal modes are close to those of a perfect crystal, so that the resulting contribution in three-dimensions is a: T 3, the Debye law. But if there is a finite probability that Ivi ~ t, we find that under fairly general circumstances it is this region that dominates the low-temperature specific heat. For suppose P(v):=:: A (2Ivl-l )-Y, independent of q, in the neighborhood of Ivi :=::t. (The exponent r has to be < 1 in order for P to be integrable). With this ansatz it is a matter of some simple algebra to evaluate Eq. (1) in the low-temperature limit, and obtain c(T)a: AT2(1-y). For any - 1 < r < 1 this contribution dominates over the T3 contribution from the region I v I :=:: O. A particular choice, r = t, yields a linear specific heat at low temperature, in conformity with experimental evidence covering a wide variety of glasses. 4 Thus, it appears that strong random-scattering matrix elements, Ivl > t, are somehow related to the thermodynamics properties of glasses. We shall see why and how this occurs in more detail, below. For the configurationally averaged thermal conductivi-ty (K) we find a somewhat similar-looking, yet funda-mentally different, result: (K)=kB {32(21T/3)(Vol)-1 1: Pq(O)wq q*O X {Sq 1[2 sinh({3wq/2) W , (4) where Sq =awq/aq is the sound velocity. The shape of 8173 © 1992 The American Physical Society 8174 MARIO I. MOLINA AND DANIEL C. MA TIIS 46 P q (u q ) at 1 U q 1 =1=0 is irrelevant to the value of K. The for-mula (4) takes on a more conventional look if we identify P q (0) with the lifetime 1" q.5 In the glass phase, P q (u q ) = 0 for 1 U q 1 < t; Eq. (4) predicts (K) = O. While it is true that amorphous solids have a far lower thermal conductivity than crystals, and this result goes in the right direction, it is truly extreme and is an artifact of the model. The significance of the critical points at 1 U q 1 = 0 and t can be understood as follows. For any 1 U q 1 > 0 the degen-eracy between standing waves a: sin(q'r) and a: cos(q'r) is lifted. Traveling wave packets, which transport energy coherently, can be constructed only if these waves are de-generate. Therefore, the (configurationally averaged) thermal conductivity must come about entirely from the unperturbed fraction of normal modes, those for which 1 U q 1 = O. This explains the proportionality of (K) to Pq(u q =0) found in Eq. (4). The model and its motivation are described as follows. In any solid, the distribution of defects (vacancies, inter-stitials, foreign atoms, broken bonds, etc.) scatters plane-wave normal modes k into k', with matrix elements M k,k' Y wkwk" The M k,k' are governed by the nature, and spatial distribution, of the defects. Although a number of approximate theories such as the coherent potential ap-proximation have been developed to obtain the thermal properties of defective solids by averaging over the na-ture, and spatial distribution, of the defects, these methods have proved incapable of yielding the delicate phase relationships among eigenstates needed to evaluate the Kubo formulas. Now, the principal contributions in thermal transport come from the transmission and reflection coefficient of each plane-wave normal mode. Our simplified model as-sumes, essentially, that such coefficients may be treated as independent random variables. Specifically, we consider just th~ne backscattering matrix element Mk,-k Y wkw-k=Ukwk with the random matrix quantity Uk governed by a statistical distribution function Pk(Uk)' Because of the possibility that some 1 Uk 1 is too large, such that the normal modes k and - k become overcoupled and the renormalized frequency becomes complex, we in-troduce an additional quartic term into the model. The quartic term comes with a very small coefficient (compa-tible with a small Griineisen constant). Therefore, if 1 Uk 1 is small, the quartic term has no effect but when 1 Uk 1 is sufficiently large that the modes become overcoupled, the quartic term allows a new equilibrium state to be achieved. Either way the thermodynamic and transport properties can be computed exactly for arbitrary values of the Uk'S and the results averaged over the P k ( Uk )'s. This is precisely what we now do. Because the procedure may be unfamiliar to those readers who are most con-cerned with the properties of glassy materials, we show all the steps in the calculations. The derivation now follows, and is quite straightfor-ward. The model Hamiltonian is H = H 0 + HI + H 2 where, Ho= ~ (Wk/2)[-PkP- k+QkQ-d, (5) kz >0 HI = ~ WkUk[Q~ +Q~k] , (6) kz >0 H 2=r ~ Q2Q2 ~ ~ wk k -k' (7) kz >0 and Pk=a-k-at and Qk=a-k+at. The Uk are in-dependent random potentials defined for half the k's. As H is hermitean, Uk = U _ k defines Uk over the remaining half. We will take Uk as real. The quartic term in the Hamiltonian has been added to insure stability when 1 U q 1 > t· The anharmonic parameter S will be taken to be arbitrarily small or even zero, at the end of the calcula-tion. For values of 1 U q 1 < t, the anharmonic term is just a irrelevant perturbation and the limit s~O can be taken safely, leaving H as a quadratic form. However, when 1 U q 1 > t, the anharmonic term proves crucial by shifting the position of each normal mode to a .!.lew, random, equi-librium position at distance of 0 (1 /v s). The "glassy phase" in the present model requires not just IUq 1=1=0, but IUq I> t. Equations (2) and (3) reflect the profound change in the spectrum as 1 U q 1 passes through this critical point. At the critical point, one set of normal modes softens critically. Beyond it, there is found a sym-metry breaking in each normal mode, and disorder be-comesJrozen-in at random in the amount Ifql a: l/v~. Let us recast the Hamiltonian in a useful form. First, rewrite H in terms of the bare operators I ak l: H = ~ wd a ta k + a ~ ka - k + 1) kz >0 +( 1/2)uk[(at +a_k)2+(a ~k +ak )2] +s(at+a_k)2(a~k+ak)2l . (8) Now per0rm a rotation: ak=(1/v2)(b k+b_k ) and a -k =( 1/v2)(bk -b -k)' The Hamiltonian becomes H= ~ wk!btbk+b~kLk+l kz >0 +( 1/2)ud(bk +bt )2+(b_k +b ~k )2] +(s/4)[(bk +bt )2-(Lk +b ~k )2]2l. (9) Now define Qk=(bk +bt)/v2; P k=(-i/V2)(bk - b t ). In terms of these, H becomes H= ~ wd(1/2)Pk2+(1/2)(1+2uk)Qk2 k z >0 +( 1/2)( 1-2uk )p'2 k +( 1/2)Q'2k +S[Qk2+p'2kfl. (10) Now interchange P and Q in the negative k sector: P '- k ~ - Q '-k; Q '- k - P'- k to restore symmetry. The Hamiltonian becomes H = ~ wk! (1 /2)P k2 + (1 /2)(1 +2uk )Qk2 k z >0 +( 1I2)p'2 k +( 1 /2)( 1-2uk )Q~k +S[Qk2+Q'2k ]2l. (11) 46 THERMAL CONDUCTIVITY AND THERMODYNAMICS OF ... 8175 Following Ref. 4, we perform the symmetry-breaking transformation Qk-+Qk+/k and Q~k-+Qk+l-k· This introduces a number of new contributions to H: (a) constant terms, (l/2) l: lUkl (l +2vk )/~ +(l- 2vk )/~k kz >0 (b) linear terms, (112) l: lUkl [2( 1 + 2Vk )/k +SSftf~k lQk kz >0 (c) quadratic terms, (l/2) l: lUd[I+2vk+12Sf~+4Sf~klQk2 kz >0 + [1- 2vk + 12Sf~k +4Sf~ lQ~k + 161tf -kQkQ~k 1 , and (d) cubic terms; (l/2) l: lUklSSfkQk3 +SSf -kQ~k +SSfkQkQ~k kz >0 with only the quartic terms remained unaffected. (14) We determine the Uk,! -kJ by the stability require-ment that all linear terms vanish. This leads to [2(l +2Vk)+SSf~ +SSf~k l/k =0 , [2(l-2vk)+SSf~k+SSfUI -k=O. (16) (17) (l) For -+ < vk < +, the only solution is Ik =0= I -k' (II) for Vk<-t, we have l-k=O and Ik=V-(l+2vk)/4S, and (III) for Vk>+' we have Ik=O and I -k=Y( 2vk-1)/4S· Casel: -112<Vt <112 Since all I are zero, the Hamiltonian reduces to H = l: lUkl (l 12)Pk2 + (l /2)( 1 + 2vk )Qk2 k z >0 which is easily diagonalized by a contraction-expansion transformation: Pk-+AtPk; Qk-+Ak1qk. Similarly for P~k,Q~k. The Hamiltonian becomes H = l: (1/2)lUk[A~~ +(l/A~)(l +2v k )q~ k z >0 +A~tP~k +(1 /A~k)(l-2vk )q~t.l . (19) We choose A~=Yl+2vk and A~k=YI-2vk and the Hamiltonian becomes H = l: I (l/2)lUkyl +2Vk(P~ +q~) kz >0 +(l/2)lUkYI-2vk(p~k+q~k)J. (20) Finally, if we define qk = (l /v2)( ak + at) and Pk =( -j /v2)(ak -at), then H takes the form +(l/2)(yl +2vk +yl-2vk)J . (21) CaseII: Vt < -1/2 In this case I -k =0 and Ik =y -( 1 +2vk )/4S. After inserting these into Eqs. (12)-(15) and in the limit S-+O, the Hamiltonian becomes H = l: lUd -( 1 116S)(l-2Ivkl )2+(1 /2)[Pk2 +P/~k +2(2Ivkl-l)Qk2 +SIVk IQ~k lJ (22) kz >0 As in case I, we perform a contraction-expansion transformation, getting A~ = Y 2( 21 vk 1-1) and A~k=2-v'fvJ. The Hamiltonian now reads H= l: lUtf-(1/16S)(I-2Ivklf kz >0 +( 1 /2)y2(2Ivkl-l )(p~ +q~) +YIVkl(p~k +q~k >l . (23) H = l: lUtf y2(2Ivkl- l )(atak+ 1/2) kz >0 +2YIVk I(a~ka_k + 112) -( 1I16S)(21vk 1-1 )2J Case III: Vt > 1-(24) In terms of the diagonal operators I ak J, H takes the Following an analogous procedure to that of case II, ~ ~-8176 MARIO I. MOLINA AND DANIEL C. MArrIS 46 H= l: wk[2yvk(atak+1/2) H= l: [n+(vk)atak+fL(vk)a!..-ka-d , (26) kz > D kz > D + y2(2vk -1 )(a!..-ka-k + 1/2) -(1/16S')( 2vk-l)2l . (25) where n+(vk) and !L(vk) were defined in (2). Equation (1) follows directly from Eq. (26) and from c(T)=(Vol)-I(d(H)ldT) where (H) is the total inter-nal energy averaged over the (random) v's. In summary, omitting the constant term, the dynamic portion of the Hamiltonian is For the thermal conductivity we start from the Kubo formula for phonons: I kB (3 f 00 f/3 K= 3V I l: l: Sk'SqWkWq dt dA(nk(O)nq(t +iA) , o b"D q*D D 0 (27) where nk ==atak and ( ... ) indicate both thermal and configurational averages. After some algebra, K can be ex-pressed as ikB (3 f 00 K= 3V I l: l: sk'SqWkWq dt t< [nk(t),nq(O)]} . o k*D q*D D (28) For the backward-scattering model, it is convenient to rewrite Eq. (28) as (29) and after averaging over configurations, (30) We need to expand the commutator in terms of the diagonal operators [ak l. Taking kz > 0 for definiteness, the trans-formation that diagonalize H takes the following form: where A± =( I±A~)/(23/2Ak) , B± =i (I±A~k)/(23!2A_k) . The A'S are 2ylvkl vk<-1/2(CaseIl) A~= y~ IVkl < 1/2 (Case I) Y2(2vk-1) vk> 1/2 (Case III), Y2(2Iv k l-l) vk < -1/2 (Case II) A~k= yl-2vk IVk 1<1/2 (Case I) 2YVk vk> 1/2 (Case III). The integral over vk can be divided into three portions: From - 00 to t, from t to t, and from t to + 00. Case I: In the middle region, where I vk I < t we get ([nk(t)-n_k(t),nq(O)-n_q(O)]) =5qk(i/2)[c +( (atak) - (a!..-ka-k) )sin(W+ -n_)t) -c _ (1 + (atak) + (a!..-ka-k) )sin(W+ +n_)t)] , where c± == [(A~±A~k)/AkA-kf and (a~kak) = [exp«(3n±)_l]-l. Use of the identity limE~of;; dt t exp( -Et)sin(wt)= -1T5'(w) reduces Eq. (30) for K to (31) (32) (33) (34) (35) (36) (37) 46 THERMAL CONDUCTIVITY AND THERMODYNAMICS OF ... 8177 (38) But, {(d /du)( (atuk) - (a~ka_k») Vir. =0= -2Prok exp[prok]l( exp[prok]_1)2 , (39) so (K) reduces to ( ) = 2rrkBp2 2 Pk(Uk=O)rokexp[prok] K 3Vol ~ Sk 2 . k >0 ( exp[prok]-l) z (40) Cases II and III: Iukl> l/2. Here Qk -Qk + fk and Q'---k -Q'---k + f -k' which implies, ak-ak +(I/2)(fk + f -k) and a -k-a-k +( 1 12)(fk - f -k)' With this, the number operators nk transform as (41) We must use this transformed nk =atak in Eq. (30). Let us examine one (four) of them, say [nk(t),nq(O)]. Clearly, [nk(t),nq(O)] will give no contributions, since it is proportional to 6'(,0,+ -,0,_). Because uq*O, (,0,+ -,0,_) can only vanish at q = 0, which has no support. Also terms like [n k' a q] are zero in thermal equilibrium, since they are linear in ( a k ) , therefore, linear in (ak)' The surviving terms are of the form (1 /4)(fk + f -k)(f q + f -q )[atU),a:(O)]+(I /4)(f k + f -k )(fq + f -q )[at(t),aq(O)] +( 1 /4)(fk + f -k)(f q + f -q )[at(t),a: (0)] +( l/4)(f k + f -k)(f q + f -q )[ak(t),aq(O)] . We have used the fact that the f are real. Expanding the bare operators ak in terms of the eigenoperators set ak' ac-cording to Eqs. (31) and (32) and using the fact that A ± are always real, while B ± are always imaginary we get, after some algebra, the following for the surviving terms: (-1/2)6kqUk + f -k )2{ (A + + A _ )2sinW+t)-(B + - B _ )2sinW_tl) . (42) When replacing this into Eq. (30), we must evaluate the integrals (-1 12)6kq(fk + f -k )2( A + + A _ )2 f 00 dt t sinW+t)+(I /2)6kq(f k + f -k )2(B + - B _ )2 f 00 dt t sinW_t) . o 0 (43) Each one of these integrals is proportional to a 6', with support at 'o'+(Uk)=O and fi_(uk)=O. For Uk < -t, only'o'+ can be zero, namely at Uk = t, while ,0,_ cannot be zero. For Uk> t, the converse is true. Let us suppose, Uk < - t. Then, (43) reduces to (44) where ,0,+ =rok V -2(1 +2uk)' Then, Eq. (44) is essentially of the form 'o'~6'('o'+), which vanishes identically. Similarly, the other three terms arising from Eq. (30), after the symmetry-breaking transformation, give no contribu-tion. For the case Uk > 1/2, we also find no contribution. So far we have only investigated the ro=O term conductivity in the context of the present exactly solvable model. However, it is also possible to generalize to finite ro, by inserting a factor exp(irot) in Eq. (27). This allows to compute ultrasonic attenuation as a function of ro. With the argument in terms such as (44) being ('o'±±ro), one sees that the in-tegrand leading to (K(ro» ceases to vanish identically, for it now includes factors as Pq[I12±(ro/roq)2]ro2• While the present model is thus capable of being extended to the study ultrasonic attenuation, we leave such extensions for future investigation. ·Present address: Department of Physics, University of North Texas, P.O. Box 5368, Denton, TX 76203. Iph. Choquard, Helv. Phys. Acta 36, 415 (1963); J. M. Lut-tinger, Phys. Rev. 135, A1505 (1964); B. Deo and S. N. Behera, ibid. 141,738 (1966). 2M. Dresden, Rev. Mod. Phys. 33, 265 (1961); P. M. Bakshi and E. P. Gross, Ann. Phys. (N.Y.) 49, 513 (1968); D. C. Mattis, A. M Szpilka, and H. Chen, Mod. Phys. Lett. B 3, 215 (1989). 3Hua Chen and Daniel Mattis, Mod. Phys. Lett. B 5, 309 (1991); Hua Chen, Daniel Mattis, and Sergey Rudin (unpublished). 4Mario Molina and Daniel Mattis, Phys. Lett. A 159, 337 (1991). 5E.g., P q (0) ex: q - 3 for the Rayleigh law. 

Thermal conductivity and thermodynamics of phonons for an exactly soluble model of disorder

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Thermal conductivity and thermodynamics of phonons for an exactly soluble model of disorder

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