Interfacial three-phonon processes and anomalous interfacial energy transport

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395—397. Solid State Communications, Vol. 29, pp. Pergamon Press Ltd. 1979. Printed in Great Britain.

INTERFACLAL THREE—PHONON PROCESSES AND ANOMALOUS INTERFACIAL ENERGY TRANSPORT W. H. Saslow and M. E. Lumpkin

*

Department of Physics, Texas A&M University, College Station, Texas 77843, USA (Received 6 October 1978 by A.A. Maradudin)

Interfacial three—phonon processes require only a moderately strong matrix element (- lO~erg/cm2) to explain the anomalous energy transport observed at high phonon frequencies across the interface between ordinary and “quantum” materials. This mechanism is qualitatively consistent with a wide variety of experiments, and has a large number of symmetry—allowed couplings with which to explain the details of particular exper:iinents. Near “onset”, this mechanism contributes anomalous reflection and transmission varying as the sixth power of the incident phonon frequency.

The problem of Kapitza conductance (hk) and related high—frequency phonon pulse phenomena may briefly be summarized as fo1lows)~~LInter— facial energy transport, either in steady—state (hk) or pulse measurements, is reasonably well understood in terms of simple acoustical match— 1mg when the characteristic frequency is low enough. However, for interfaces between ordi— nary and “quantum” materials (liquid or solid 3He, 4He, H 2, or D2), when the characteristic frequency grows large enough (T ~ 0.1 K for hk measurements and f ~ 20 GHz for phonon pulse experiments), a new “anomalous” mechanism mani— 3 about fests Itself. Although much the is anomalous known, in in— a terfacial energy transport, there is no univer— qualitative fashion, sally true quantitative characterization of it. It is therefore difficult to construct a detailed theory of this effect. Nevertheless, one can at least expect of a theory that it satisfy the following three criteria. First, it must be able to explain, on the whole, the qualitative features of the existing experiments. Second, it must have sufficient generality and flexi— bility to be consistent with the quantitative features of any particular experiment. Third, it must suggest new experiments for which the theory makes clear—cut qualitative predictions, and new experiments for which the theory makes definite quantitative predictions. Interfacial three—phonon processes, mediated by interfacial cubic anharmonicity, provide a mechanism which satisfies these criteria, It should be noted that the interfacial three—phonon processes have already been studied theoretically and discounted. In his seminal paper on the hk problem, Khalatnikov considered Interfacial three—phonon processes using a semi— classical matrix element.4 More recently, Sheard et al and Sluckin et al,6 using a semi—micro— scopic approach, came to conclusions similar to those of Khalatnikov. Both approaches assumed *

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that liquid 4He is so light that it vibrates as if against an immovable wall, and the ordinary solid vibrates as if against a vacuum. Both approaches obtaine4 a matrix element which varies as (pq/k)~-/2 in the long—wavelength limit, ~or a process in which a phonon of wave— vector k from the ordinary÷solid,,decaysinto two phonons of wavevector p and q in liquid 4He. Refs. 4 and 5 obtained the result that such interfacial three—phonon processes contri— bute negligibly to hk, and Ref. 6 that they con— tribute negligibly to the transmission of energy by high—frequency phonons. due to bulk three—phonon be surWorkers in the area grocesses of phonon would attenuation prised by such a (pq/k)~-’~ matrix element. Bulk cubic anharmonicity in solids, based as it is on an interaction energy dens~tywhich is trilinear in the strain 3nuB (where u is the displace— meflt)~gives a matrix element which varies as (kpq) ,2 The prefactor B of this term has a value which is typically about ten times the values of the elastic con~t~nts,8 which are gen— erally lo~~]~ — 1012 erg/cm . Hence B is of the order 1012 — 1013 erg/cm3 for ordinary solids. For liquid 4He, B is much smaller, since it varies as pc2 (p is the mass density and c is the sound velocity); at pres~uresnear 20 atm, B is of the order 108 erg/cm3. We believe that a (pq/k)l/2 matrix element is unphysical. We trace the k’~-’2dependence to the fact that the surface interaction has been taken to depend upon the displacement of the solid at the interface, rather than the gradient of the displacement, which gives k1-12. The lat— ter should appear, since cubic anharmonic terms in bulk, and at a solid—solid interface, vary as the third power of differences of ionic coordi— nates, which in the long—wavelength limit trans— lates to the third power of gradients of the ionic displacements. It is plausible that this form should hold, in the long—wavelength limit,

Boeing Commercial Aircraft Company, P. 0. Box 3707, Seattle, Washington 98124. 395

396

INTERFACIAL THREE—PHONON PROCESSES

for a solid—liquid interface. To phenomenologically characterize the inter— facial cubic anharmonic interactions, one keeps the usual bulk form but multiplies by a function f which is unity over a thickness a of the order of the atomic separation at the interface. Re— placing f by a 6(z), where z = 0 locates the mathematical interface, we find an interfacial cubic harmonic interaction which is cubic in the surface strain, and is multiplied by Ba 6(z). With a 10—8 cm we find C 5 Ba of the order l0~— lO~erg/cm~at an interface between two ordinary materials, and of the order 100 erg/cm2 at an interface between two light, “quantum” ma— terials. For the interface between a “quantum” and an ordinary material, C should lie between 100 and lO~erg/cn2. Using the first Bo~na~pro~imation,we have considered the process k -* p + described above. Three estimates of C were made.’-° In the first two cases, the total transmission from NaF to 4He due to such three—phonon prpcesses was oh— tamed and equated to the direct acoustic trans— mission, for a l 2K phono~pulse. Values of 3~uz) symmetry and about lO~erg/cm and 10 erg/cm~were obtained for one with element (~.i~)3symmetry. In the third a matrix with ( case we attempted to fit the angular emission distribution data (more precisely, the frac— tional transmission within the “critical cone”) of Wyatt et al,~-~-as a function of pulse tern— perature. The fit was only qualitative, since most of the data was taken for pulse tempera— tures so high that the first Born approximation breaks down (this point will be discussed later), but it i~of interest that our value for C, about 10 erg/cm2, is consistent with the other estimates. Hence, our three estimates of C, al— though on the high side of what might be expect— ed apriori, are reasonable. We now consider how this mechanism satis— fies each of the three criteria enumerated at the beginning of the paper. Qualitative Features: 1) The mechanism does not depend upon whether the “quantum” ma— terial is liguid or solid, in agreement with experiment.1-~-4 It is not the quantum nature per sec of such materials that matters, but the fact that such materials have low densities and low sound velocities. As a consequence, direct acoustic transmission from classical materials is inefficient and, more important, the phase space for certain interfacial three—phonon processes becomes enormously large 2) the mechanism per— silts inelastic (frequency shifting) phonon scat— tering, as observed experimenta1ly;~--~’~-6 3) the mechanism permits mode—conversion, as observed experimentally;1316 4) the mechanism permits energy transport outside the “critical cone” in liquid 4He, both for phonons incident from the solid and from the 4He, as observed experimen— tally;~-~5) the mechanism, in the Bor~i approxi— nation, varies as f6 (where f is the incident phonon frequency), so that it is negligible at low frequencies, and comes to dominate at high 1-7 Since an effective transmission varying as f6 exceeds unity for large f, the frequencies. matrix element must be renormalized, which will produce saturation. Just such a saturation ef— fect, as f is varied, has been observed experi— mentally;18 6) the mechanism is consistent with the fact that 4He films only 3 layers thick

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(— 12 of was bulk liquid 4He)-3’ ~give In behavior Ref. 15, like this that effect found for 290 GHz phonons normally incident from the solid. The cubic anharmonic interaction converts this phonon into two phonons in the 4He of charac— teristic frequency 145 GHz at 45 to the normal. Such phonons have a 16 ~ wavelength and see 17 ~ effective thickness. It is plausible that bulk behavior can occur in this situation. Generality and Flexibility: There are six— teen possible interfacial cubic anharmonic terms consistent with rotational invariance about the interf ace. With z being the normal to the in— terface, the cubic anharmonic terms can be ob— tamed as products of ~, ~‘~x + ~ 3~u — ayux~ (~~uz)2 + (3yUz)2 and (3zux)2 + (~zuy) Such a large number of possibilities can give a wide variety of angular distributions for pho— nons emitted in three—phonon processes, and should make it possible to fit experimentally observed distributions. Unfortunately, the an— gular distribution experiments which have been performe&-1- were done so before it was realized

how strict surface preparation must be, so that contaminated. In addition, the crystal surfaces employedthese wereexperiments probably were done with heat pulses having a broad fre— quency spectrum which peaked at frequencies too high for our first Born approximation calcula— tions to be valid. The only pulse experiments which, to date, have been performed on “clean” crystal surfaces, are those of Weber et al on LiF and NaF cleaved in liquid 4He)-9 Assuming that the interfacial anharmonic interactions have the same symmetry for both cases, and that renorinalization effects are not important, it is possible, from their reflectivity measurements, to deduce that: 1) ~u (a = x,y) does not have a large coeffi— cient ~ (from the 100% reflectivity of “fast transverse” pulses in LIF); 2) ~zu~ and/or ~ + 3~u) has an appreciable coefficient C (from the 9~%reflectivity of the “slow trans— verse” pulses in L1F); 3) ~zUz does not have a large coefficient C (from the 96% reflectivity of the longitudinal pulses in NaF). The re— flectivity of “transverse” pulses in NaF add no new information. It is clear that the inter— facial cubic anharmonicity has enough freedom to be consistent with these experiments on “clean” surfaces. For “dirty” surfaces, it would be difficult to perform a meaningful analysis of the symmetry of this anharmonicity. New Experiments: 1) As the pressure is in— creased, both p and c increase, thus decreasing the relative importance of the interfacial three—phonon processes. At frequencies high enough that such processes dominate the direct acoustic transmission, the effect of an increase in pressure should be to decrease the effective transmission and to increase the effective re— flectivity (but mode—conversion and frequency— conversion processes in the reflectivity should be decreased); 2) For interfaces between two “quantum” materials, such as solid H liquid 411e, the relatively good acoustic 2 and p11mg, the relatively small matrix element, and the relatively small amount of phase space for interfacial three—phonon processes should conbine to give a negligible “anonalous” inter— facial energy transport; 3) As the frequency f is increased, the 4He film thickness d at which

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INTERFACIAL THREE—PHONON PROCESSES

397

bulk behavior occurs should decrease, with d-f~ (unless d is so small that the interfacial an— harmonic interaction has not leveled off’ such leveling has probably occured l2~films); 4) In the off “onset” frequency regimefor where inter—

malous mnterfacial energy transport across in— terfaces between “ordinary” and “quantum” ma— terials, a universal behavior 6) for and the which onset predicts of the interfacial three— (f phonon processes responsible for this anomalous

facial three—phonon processes are just becoming important, but are not so important that satu— ration effects are observed, they should contri— bute as f6 to the excess effective transmission (above that due to direct acoustic transmission), to the node—converted reflectivity, to the fre— quency—converted reflectivity, and to the change in the ordinary reflectivity (no mode—conversion and no frequency—conversion) from its acoustic value. This f6 behavior in the “onset” fre— quency regime should be universal. It is pos— sible that the “excess” Kapitza conductance shows an “onset” behavior varying as T6 times the ordinary T3 value,1’2 yielding an “excess” hk varying as I9. However, due to thermal averaging it may be difficult to observe such behavior unless hk measurements of extremely high precision can be made, To summarize, we have found a mechanism which is consistent with existing data on ano—

energy transport. Note: After preparation of this manuscript, we became aware of the experiments of Wyatt and Crisp.20 These authors used two types of phonon detector (one sensitive to all frequencies, and one sensitive only to frequencies above a certain cut—off) to study phonon emission from NaP to liquid 4He. They found that phonons outside the critical cone have a lower frequency spec— trum than those inside the critical cone. This is easily understood in terms of interfacial three—phonon processes, using the fact that phonons produced by this mechanism are lower in energy than the initial phonons. Thus, phonons outside the “critical cone”, having been produced by the interfacial three—phonon process, are lower in energy (on the average) than phonons inside the critical cone, which are dominantly produced by direct transmission (with no change in energy).

REFERENCES 1. For reviews of the hk problem alone, see POLLACK, G.L, Rev. Mod. Phys. 41, 48 (1969); SNYDER, N.S., Cryogenics 10, 89 (1970). 2. For reviews of the hk problem which also discuss high—frequency phonon transmission, see CHALLIS, L.J., J. Phys. C7, 481 (1974); ANDERSON, A.C., Phonon Scattering in Solids, p. 1, Plenum, New York (1976), edited by CHALLIS, L.J., RMIPTON, V.W., and WYATT, A.F.G.. 3. See ANDERSON, A.C. (Ref. 2). 4. KILALATNIKOV, I.M., Zh. Eks. i Teor. Fiz. 22, 687 (1952). 5. SHEARD, F.W., BOWLEY, R.M., and TOOMBS, G.A., Phys. Rev. AS, 3135 (1973). 6. SLUCKIN, T.J., SHEARD, F.W., BOWLEY, R.M., and TOOMBS, G.A., J. Phys. CS, 3521 (1975). 7. LANDAU, L.D. and LIFSHITZ, E.M., Theory of Elasticity, Sect. 26. Pergamon, New York (1975). 8. SHIREN, N.S., Phys. Rev. Lett. 11, 3 (1963). 9. KITTEL, C., Introduction to Solid State Physics, Table 4.1. Wiley, New York (1956) 2nd edition. 10. Details of these calculations may be found in LUMPKIN, M.E. and SASLOW, W.M. (submitted for publication), preprints of which are available from WMS. 11. WYATT, A.F.G, PAGE, G.J., and SHERLOCK, R.A., Phys. Rev. Lett. 36, 1184 (1976), 12. FOLINSBEE, J.T. and ANDERSON, A.C., Phys. Rev. Lett, 31, 1580 (1973). 13. GUO, C.J. and NARIS, H.J., Phys. Rev. Lett. 29, 855 (1972). 14. BUECHNER, J.S. and MARIS, H.J., Phys. Rev. Lett. 34, 316 (1975). 15. KINDER, H. and DIETSCHE, W., Phys. Rev. Lett. 33, 578 (1974). 16. DIETSCHE, W. and KINDER, H., J. Low Temp. Phys. 23, 27 (1976). 17. The frequency—dependence for the rate at which the process it -‘+ occurs may be seen as follows. Fermi’s Golden Rule tells us to sum the square of the matrix element over allowable final states, subject to energy conservation. Considering it to be a phonon of frequency f in the ordinary material, the unrestricted phase space for its decay into phonons ~ and ~ in the “quantum” material varies as f6. However, for a “clean” surface the matrix element introduces conservation of the two components of transverse momentum, which reduces f6 to f . Energy conservation further reduces f4 to f3. Since the square of the matrix element varies characteristica~lyas f3, the rate for this process, and the fractional energy transmission, goes as f . Only the details of the phase space analysis and of the matrix element provide the prefactor of f6. This prefactor turns out to be large only in the case of an ordinary—”quantum” interface. See Ref. 10 for details. Note that for “dirty’ surfaces, transverse momentum is not conserved, leading to an f8 dependence. This situation has not been considered further, since it clearly depends upon details of how “dirty” the interface is. Note that, for surface inhomogeneities which vary relatively slowly on the scale of thermal phonon wavelengths, the surface will appear to be relatively “clean”. 18. SABISKY, E.S. and ANDERSON, C.H., Solid State Commun. 17, 1095 (1975). 19. WEBER, J., SANDMANN, W., DIETSCHE, W., and KINDER, H.,Phys. Rev. Lett. 40, 1469 (1978). 20. WYATT, A.F.G. and CRISP, G.N., J. de Physique 39, Colloque C6, 244 (1978).