Acoustic matching of superconducting films to substrates

Journal o/"Low TemperaturePh~,'sics,VoL 37, Nos. 3/4, 1979

Acoustic Matching of Superconducting Films to Substrates* Steven B. Kaplan Electromagnetic Technology Division, National Bureau of Standards, Boulder, Colorado (Received May 7, 1979)

Acoustic mismatch theory is used to estimate phonon transmission coefficients for various superconductor/substrate interfaces. It is shown that the conventionally employed substrates offer the largest acoustic mismatch to many of the commonly studied superconductors, thereby leading to unnecessarily large phonon-trapping and other nonequilibrium effects. Most available experimental results are shown to be in reasonable agreement with the theoretical estimates. 1. INTRODUCTION It has long been established that the resistance to the flow of phonons from superconducting thin films has a noticeable effect on film nonequilibrium properties. Rothwarf and Taylor 1 were the first to point out that interactions between the quasiparticles and phonons of energies greater than twice the BCS gap parameter A could not be neglected in determining the nonequilibrium steady-state quasiparticle density. They predicted that repeated pair-breaking and recombination processes involving these excitations result in an increased population of quasiparticles. This causes an increase in the apparent quasiparticle recombination lifetime by the socalled "phonon-trapped factor" (1 + ~'ff~-s), where ~'B is the phonon lifetime against pair-breaking and ~-~ is the lifetime for the high-energy phonons to be lost by other processes. Gray et aL 2 explained that the larger effective quasiparticle recombination lifetimes of AI films on sapphire versus those on glass substrates arise from the larger acoustic mismatch between AI and sapphire, which hampers phonon escape and thereby increases ~-,. The dominant role of r in the recombination process in Sn films was first established by Sai-Halasz et al. 3 Eisenmenger et al. 4 have pointed out that in *Work supported by a National Research Council Postdoctoral Fellowship and Office of Naval Research Contract No. N00014-78-F-0040. 343 0022-2291/79/1100-0343503.00/0 9 1979 PlenumPublishingCorporation

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films where the thickness is less than the phonon pair-breaking mean free path A, the effective lifetime is sensitive to the critical angles for total internal reflection of acoustic waves to the degree that such reflection is specular. For films where d ~>A, the effective recombination time z~e~ocd. For films where d >>A, ten becomes independent of d and yields information about the degree of phonon mismatch between film and substrate. Langenberg 5 has shown that the phonon-trapping effect can be minimized by optimizing the acoustic match from superconducting film to substrate. An early application of this method was the use of a BaF2 substrate in a measurement of the effective quasiparticle lifetime in A1, 6"3~ which was therefore believed to be nearly equal to the "intrinsic" recombination time (that is, it was assumed that zv/z~ 1. Because of the "hardness" of A1, the calculations for A1 films show a tendency to match well to all substrates except A1203 and Si. A near-perfect match is found for calcite, but this material is extremely anisotropic. The matching of the "softer" superconductors In and Pb is not as easy. Using the criteria outlined above, we see from Table I that metal halide salts should be likely prospects. Table II shows that, indeed, AgBr, T1Br, and T1C1 are the best acoustic matches. Selenium also appears to be a fair match, although single-crystal material may not be available.~ There is a variety of materials that provide good matches to Nb and Sn besides the materials mentioned in connection with In and Pb: in particular, PbF2 appears to be the best match. PbTe and similar materials show encouraging transmission coefficients, but these I V - V I compounds are not easily found in the intrinsic state. 21 Finally, because of its large density, the transmission coefficients for Ta are found to *The anisotropy factor was determined by Eq. (2) for other crystal structures as well. For hexagonal crystals A 1/2 equals the ratio of the acoustic velocity for transverse waves traveling perpendicular to that of those traveling parallel to the sixfold axis. For trigonal crystals, the ratio of wave velocities in the [100] and [001] directions is described by A l/2. See Ref. 19. COne m a y choose to evaporate a film of Se on a suitable substrate before depositing the metal film in order to maximize the acoustic match.

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be lower than those of Nb, but the same subset of substrate materials provide the best acoustic matches. Because the acoustic coupling of superconducting films is of current interest, 22"23 calculations of the transmission coefficients at metal/metal interfaces are given at the bottom of Table II. The softer superconductors are again found to be poorly matched to the other materials. 4. C O M P A R I S O N

WITH EXPERIMENTS

4.1. Thermal Boundary Resistance

The experimental determination of phonon transmission coefficients was first accomplished by measuring the thermal boundary resistance. While this method contains the difficulty of measuring a small temperature discontinuity across the interface, the results are in reasonable agreement with the isotropic acoustic mismatch model, although a small T 2 contribution accompanies the T 3 term expected for Debye solids. (The T 2 term not only is characteristic of a non-Debye-like phonon density of states, but also may arise from defects and surface roughness.) Several authors 24-28 have reported reasonable agreement with the isotropic model even when the substrate is anisotropic. Neeper and Dillinger 24 found that the thermal conductance at an In/sapphire boundary (perpendicular to the c axis of the substrate) is approximately 0 . 0 2 8 T 3 + 0 . 0 0 8 T 2 W cm -2 K -1. By extrapolation of Little's figures (see Ref. 9) to values of c2/cl > 0.5, the coefficient of the T 3 term was expected by the authors of Ref. 24 to lie between 0.022 and 0.036. As mentioned in Section 2 of this paper, the values given by Little for large c2/c~ ratios underestimate the correct theoretical values by as much as 30%. Thus, it is not surprising that I calculate the thermal conductance within an isotropic Debye model to be 0 . 0 4 7 T 3 W cm -2 K -1. In view of the fact that the critical angles for total reflection are small (6c - 8 ~for transverse and 24 ~ for longitudinal waves), one could do an improved calculation assuming only c-axis velocities of the sapphire substrate are sampled, but this results in a negligible change in the calculated conductance. G o o d agreement between theory and experiment for Pb, In, and Sn films on sapphire substrates was also observed by Cheeke et al. 28 for the lowest heater powers used. (For heater temperatures in excess of --20 K, dispersive effects were probably observed. 26-28) In the case of Sn films on Z - c u t quartz substrates, the isotropic model predicts a thermal conductance only 15% larger than that reported by Cheeke etal. in Ref. 27. An adjustment to the SiO2 acoustic velocities corresponding to an average over a 60 ~ cone 27 results in a value nearly equal to that observed. In the same manner, good agreement is found with the results of Ref. 27 for In and Sn on Z - c u t quartz. In general, the isotropic model predicts a

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larger conductance than is actually observed. This may be an artifact of this model, since the agreement for interfaces having larger critical angles appears to be better than for those that sustain more total reflection. However, there may be other reasons as well. For example it was shown in Ref. 25 that the thermal conductance is sensitive to the quality of adhesion of the film to the substrate; it was suggested that an imperfect bond may be able to allow the transmission of heat only via longitudinal waves. In view of the crude nature of these calculations, it is believed that the agreement between theory and experiment is certainly good enough to draw simple conclusions concerning the appropriate application of particular materials for use as substrate materials. However, it should be noted that Herth and Weis 26 have at times observed a factor two increase in the thermal conductance between Pb and sapphire. They attribute this to an improved acoustic match due to gases trapped beneath the Pb film. This suggests methods for improving phonon removal from thin films. 4.2. Quasiparticle Recombination and Phonon Trapping

As noted in Section 1, the experimentally measured quasiparticle recombination time refr is longer than the intrinsic time rr by a phonontrapping factor due to repeated pair-breaking and recombination processes. The size of this factor is related to the ratio of the phonon mean free path against pair-breaking A and the film thickness d as well as the phonon transmission coefficients ~Tt and r/t. Closed-form solutions are available for three distinct regions of film thickness4: (a) For d