THE EFFECT OF DEFECT SCATTERING ON PHONON TRANSMISSION THROUGH INTERFACES

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THE EFFECT OF DEFECT SCATTERING ON PHONON TRANSMISSION THROUGH

INTERFACES

F. Sheard, G. Toombs

To cite this version:

F. Sheard, G. Toombs. THE EFFECT OF DEFECT SCATTERING ON PHONON TRANSMIS- SION THROUGH INTERFACES. Journal de Physique Colloques, 1972, 33 (C4), pp.C4-61-C4-64.

�10.1051/jphyscol:1972413�. �jpa-00215090�

JOURNAL DE PHYSIQUE

Colloque C4, supplkment au no 10, Octobre 1972,

(24-61

THE EFFECT OF DEFECT SCATTERING ON PHONON TRANSMIS SIBN THROUGH INTERFACES

F. W. SHEARD and G. A. TOOMBS

Department of Physics, University of Nottingham, Nottingham, N G 7 2RD, England.

R6sum6. - La transmission partielle des phonons

travers la surface de contact de deux solides, ou celle d'un solide et de I'helium liquide, donne une resistance de contact thermique

La thkorie d'klasticite classique est normalement utilisee pour obtenir le coefficient de transmission. Nous avons decrit cette transmission de phonons, comme dans la theorie de tunnelling des electrons

travers la jonction metal-isolant-metal, par un hamiltonien de transfert. L761Cment de matrice de transfert est pris tel que le courant de chaleur trait6 en perturbation donne le m6me resultat que dans le cas classique du traitement de desadaptation acoustique. La diffusion de phonons par les defauts au voisinage de la surface est donc incorporke dans la theorie

I'aide des termes appro- pries dans I'hamiltonien. Le calcul de

partir du processus de transmission de phonon en presence des defauts, montre que

est rkduit en magnitude. Ceci peut &tre interprkte par la destruc- tion du c8ne critique qui restreint les directions des phonons refractes dans le milieu ayant

plus faible vitesse du son. Puisque le large effet de desadaptation acoustique A la surface de contact solide-4He liquide resulte d'un c8ne de tres petit angle solide, cet effet peut 6tre important pour la resistance de Kapitza.

Abstract. - The partial transmission of phonons across the surface between two solids or

solid and liquid helium gives rise to an interfacial thermal resistance

Classical elastic theory

normally used to obtain the transmission coefficient. We have described this phonon transmission by means of a transfer Hamiltonian as in the theory of tunnelling of electrons through metal- insulator-metal junctions. The transfer matrix element is devised such that a perturbation calcu- lation of the phonon heat current gives the same result as the classical acoustic mismatch treatment.

Phonon scattering by defects near the surface may then be incorporated into the theory by including appropriate scattering terms in the Hamiltonian. Calculation of the defect assisted phonon trans- mission processes is found to give a reduction in the magnitude of

This may be interpreted as arising from the destruction of the critical cone which restricts the directions of the refracted pho- nons in the medium of lower sound velocity. Since the severe acoustic mismatch at the solid- liquid 4He interface results in a very narrow cone angle this effect may be particularly important for the Kapitza resistance.

1. Introduction.

The origin of the thermal contact resistance R between two solids or a solid and liquid helium is thought to lie in the partial transmission of phonons across the interface between them [I], [2]. Classical elastic theory is nor- mally used to obtain the transmission coefficient

[2], which is then independent of frequency and leads, at low temperatures, to the dependence R cc T - 3 .

In the case of the solid-liquid helium interface the severe acoustic mismatch results in a very small trans- mission coefficient. At normal incidence

4 pulp, v,, which may typically be - ^{10- to}

here p and v are the density and sound velocity for liquid helium while p, and

are the corresponding quantities for the solid. For a simplified model in which only longi- tudinal phonons are considered, the Kapitza conduc- tance h,

R i l is given by [2]

45

'

Ps

(1)

liquid parameters. However, experimental measure- ments of the conductance are always greater than this theoretical prediction (I), often by more than a n order of magnitude [3], [4]. There is in addition serious disagreement with the dependence o n the material parameters contained in eq. (1).

The disparity in the sound velocities, as well as bearing partial responsibility for the small transmis- sion coefficient, restricts the directions of the phonons transmitted into the liquid helium t o lie within a narrow cone of semi-angle 0, - v/u,. The existence of this critical cone follows from the boundary condi- tions imposed on the elastic waves a t the interface.

Using q and k to denote the phonon wave vectors in the solid and liquid helium respectively, the boundary conditions demand that

411

k l l me

mk

(2) where q l l , k l l are the components parallel t o the

(1)

Footnote. The disagreement

strictly with the more where k~ 1s Boltzmann's constant, which correctly general formulae derived

Khalatnikov

and Little

displays the theoretical dependence on the solid and which differ from eq.

only

numerical factor.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972413

C4-62 F. W. SHEARD A N D G. A. TOOMBS

interface and w,, wk are the phonon frequencies. Since

o,

qv,, wk

kv these conditions taken together are equivalent to the basic law of refraction and hence to the existence of the critical cone.

If the conservation laws (2) are relaxed by phonon interaction processes in the vicinity of the interface then we have the possibility of an increased conduc- tance since the kinematic restrictions imposed by the critical cone will be partially lifted. In this paper we consider the effect of elastic scattering by a layer of defects in the solid which relaxes the wave vector conservation condition in (2). The importance of sur- face preparation in the Kapitza resistance has been emphasized in recent experiments by Johnson and Anderson [5] who also suggested that a damaged layer might provide such an enhanced acoustic coupling.

2. Transfer Hamiltonian formalism.

It is diffi- cult to calculate directly from classical elastic theory the modification of the transmission coefficient due to phonon scattering processes in the vicinity of the interface. Haug and Weiss [6] have calculated an enhanced transmission arising from attenuation of the acoustic waves in the solid but their attenuation mecha- nism is not confined to a thin surface layer.

In order to make the problem tractable we shall describe the phonon transmission by means of a transfer Hamiltonian as in the theory of tunnelling of electrons through metal-insulator-metal junc- tions

The tunnelling Hamiltonian formalism has been particularly useful in describing many-body effects due to interactions between electrons in the metal electrodes and with impurities in the insulating barrier [8]. In a similar way phonon scattering in the solid may be incorporated into the theory of phonon transmission by including appropriate scattering terms in the Hamiltonian for the solid.

We therefore write the total Hamiltonian for the solid-liquid system as H

Hs + HL + HsL, where

express solid and liquid Hamiltonians in terms of phonon creation and annihilation operators. The scattering by surface defects is represented by V.

The solid-liquid interaction is given by a transfer Hamiltonian

which clearly gives rise to processes in which a pho- non is absorbed from the solid and emitted into the liquid helium and vice versa.

We calculate the phonon heat current JsL by trea- ting HsL in first-order perturbation theory. For a temperature difference AT across the interface

where n: is the Bose-Einstein distribution function.

In order for this expression to give the same result as the classical acoustic mismatch theory, the transfer matrix element Tqk must conserve the parallel compo- nent of wave vector. Since the delta function conserves phonon frequency, the perturbation approach pre- serves the basic conservation conditions (2). Compa- rison of eq. (4) with the acoustic mismatch heat flow only specifies the matrix elements for equal energies h o ,

hak. The complete wave vector dependence of Tq, has however been obtained in a microscopic derivation of the solid-liquid interaction by Bowley, Sheard and Toombs [9]. If the interface is chosen to be the plane

then

where V and

are the volumes of the liquid helium and the solid respectively, A is the interfacial area and eq is the phonon polarisation vector. It may be readily verified that evaluation of (4) using this matrix element (5) leads to the conductance quoted in eq. (1).

3 . Effect of surface defects on phonon transmis- sion. -It is important to know the wave vector depen- dence of the transfer matrix element since the effect of defects occurs in second order. We take a simplified defect scattering term of the form

v

1, Vqq, a,,

.

99'

A phonon q in the solid may now be scattered into the mode q' by the defect and then transmitted into the mode

in the liquid helium. Eq. (4) for the heat current is still valid provided the first-order matrix element is replaced by

We note that the solid-liquid interaction (3) does not by itself give rise to second-order terms. Such two- phonon transmission processes have identically zero probability.

The precise nature of the solid surface is not well understood. Phonon scattering in the surface layer may be due to random strain fields, dislocations or point defects. In order to gain some insight into the effect of surface defects we shall study two simple models which correspond to limiting cases. The first case is that of point defects which in the bulk solid result in a phonon relaxation time

cc w M 4 typical of Rayleigh scattering. The second case corresponds to a frequency-independent phonon relaxation time as might occur for clusters of defects or other extended perturbations whose size is comparable with or greater than the phonon wavelength.

If the defect scattering is to enhance the heat current

the second-order matrix element must dominate the

THE EFFECT O F DEFECT SCATTERING ON PHONON TRANSMISSION THROUGH INTERFACES C4-63

transition probability. We therefore compute this

contribution separately and compare it with the direct phonon transmission contribution (1). For these defect assisted phonon transmission processes the parallel component of wave vector is no longer conserved, since the defect matrix element does not conserve any components of wave vector. Thus the critical cone is destroyed and the phonons are emitted diffil- sely into the liquid helium.

T o estimate the effect of point defects we have taken a matrix element appropriate to mass defects and will regard the mass ratio A M / M as a parameter 5 1.

We omit details of the calculation except to remark that approximate evaluation of the sum over inter- mediate states yields a logarithmic term. The ratio of the second-order conductance to the first-order acoustic mismatch result is found to be

where Ni is the number of defects, which we have assumed to act independently, 0 is the Debye tempe- rature and a3 is the atomic volume of the solid. A consequence of the frequency dependence of the defect scattering is a T7 dependence for hf', giving the small ratio - lo-* in (7) (the logarithmic term is slowly varying over any temperature range of inte- rest). The factor ( V , / V ) ~ - 10' corresponds to the destruction of the critical cone since phonons are now emitted into a solid angle 2

rather than a narrow cone of solid angle - ~ ( V / V , ) ~ . Associating an area a 2 with each defect, the factor f

N i a2/A corresponds to the fraction of interfacial area presented by defects.

We thus see that f would have to be unrealistically large for point defects to be effective in the Kapitza resistance.

T o describe frequency-independent scattering by defects we have devised an effective matrix element, from which the scattering cross-section obtained by the

golden-rule

is just zD2. The parameter D hence corresponds to the linear dimension of a defect.

Although such

hard-sphere

scattering cannot strictly be obtained in low-order perturbation theory, some justification may be given by considering the scattering by resonant defects which is frequency- independent above resonance and Rayleigh-like below resonance.

For such defects the second-order conductance varies as T 3 and the ratio to the mismatch value

is

Here To is the temperature at which the dominant thermal phonon wavelength is - D. Clearly for

T < To this conductance must approach the previous

T7 law. We again obtain the factor (V,/V)~ from the destruction of the critical cone and the fractional coverage is now f

N i Z D ~ / A . In this case therefore

we have the possibility of a considerable enhancement of the conductance for a moderate defect coverage.

4. Conclusions.

Desp~te the simplicity of our description of the defect scattering we are able to conclude that surface defects with a frequency-inde- pendent scattering cross-section may considerably enhance the Kapitza conductance. In recent experi- ments, Johnson and Anderson [5] observed that sandblasting a copper surface reduced RK by a factor 10 to 20, suggesting that a damaged surface layer can enhance the acoustic coupling. However, the tempe- rature dependence was more rapid than T - 3 . Our model indicates that defect assisted phonon transmis- sion processes can indeed reduce RK by this order of magnitude by destroying the conservation require- ments of the critical cone. We might also speculate that phonon scattering by dislocations, which is weakly frequency dependent, would produce a similar reduction, but a temperature dependence more rapid than T - 3 .

The result (8) for

hard-sphere

surface defects indicates that hp' cc p/p, v, v, which is a different dependence on material parameters to (1). Thedepen- dence on uL1 implies RK

8, rather than O3 for the mismatch model. In an analysis of data on 14 diffe- rent materials with IHe at 1.5 OK, Challis [lo] obtained the variation RK

8°.99, though there is of course a considerable spread in these data. Another possible test of the importance of defect scattering is the pres- sure dependence of RK. For the first-order transmis- sion (1) this is given by the acoustic impedance pv of the helium. The ratio of the conductance at 20 atm to that at s. v. p. is then 1.7 and almost temperature independent. Experimentally this ratio is 1.1 + ^{0.1 [3].}

Our result h: ' cc p/v gives a ratio 0.8. Although this implies the conductance decreases with increased pressure rather than the small increase observed, it is nevertheless much closer to the observed ratio than the mismatch value.

It is clear that further calculations of the effect of defects would be valuable, particularly of the effect of dislocation scattering, though it is difficult to devise a realistic but tractable scattering matrix ele- ment. Since our theory and the recent experiments of Johnson and Anderson [5] suggest that destruction of the critical cone may play an important role in Kapitza conductance, experimental observation of phonons emitted from a heated solid surface in liquid helium would be most interesting.

Acknowledgments.

Part of this work was

carried out when one of us (FWS) was a t the

Centre de Recherches sur les Tr&s Basses TempC-

ratures, Laboratoires CNRS, Grenoble. He would

like to thank Professor B. Dreyfus for the hospi-

tality of the laboratory and Professor J. D. N. Cheeke

for many helpful discussions.

F. W. SHEARD AND G. A . TOOMBS

References

[I] KHALATNIKOV (I. M.), Zh. Eksp.

1952,

[7] COHEN (M. H.), FALICOV

M.) and PHILLIPS (J. C.),

687.

Rev.

121 LITTLE (W. A.),

[8] DUKE ?c. B.),

eds.

Seitz,

[3] POLLACK (G. L.),

Turnbull and

Ehrenreich, Academic Press, New York, 1969, Suppl. 10, Tunnelling in Solids.

141

D. N-1,

Paris, Colloque S ~ P P ~ . _[g] BOWJJE~ (R. Me), SHEARD (F. W.) and T o o ~ ~ s

A*), C 3, 1970, 31, 129. to be published. See also SHEARD

and

JOHNSON (W. L.) and ANDERSON (A. C.),

TOOMBS

A.),

1971, 37A, 101. 1972, in press.

161 HAUG (H.) and W~rss

to be published. [lo] CHALLIS (L. J.),

1968, 26A, 105.

DISCUSSION

W. EISENMENGER. - DO

think that non-linear acoustic properties of the helium-solid boundary or for instance the anharmonocity in the helium does lead t o inelastic processes in which a phonon entering the boundary from the solid is split into two phonons propagating into the helium. Since this process prin- cipally opens new transition channels on k, I would like to know whether you think such a process si important

F. W. SHEARD.

In our derivation of the transfer Hamiltonian using a microscopic theory of liquid helium [9] such anharmonic terms appear in which

a phonon in the solid couples to two excitations in the helium. Thus we may have the creation of a pair of phonons (or rotons) or processes in which a phonon (or roton) is inelastically scattered at the solid surface.

Inelastic scattering has been considered by Khalat-

nikov [I], although in a more phenomenological

fashion, and its contribution to the Kapitza conduc-

tance is small at 2OK. Of more importance is the

conversion of a phonon in the solid into a single roton

in the helium. A preliminary calculation [9] shows this

contribution may be comparable with the Khalatnikov

phonon conductance at 2 OK.