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
344
Steven B. Kaplan
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). The role of the acoustic
mismatch at superconductor/substrate interfaces in the elongation of the
quasiparticle diffusion length was recognized by Wong and Langenberg to
be detrimental to the operation of controllable weak links. It was shown that
a better choice of substrate increased the observed number of if-induced
Josephson steps in their device. 7 The acoustic mismatch that results in a
large phonon-trapping enhancement of the quasiparticle density in superconducting tunnel junctions under nonequilibrium current injection has
been shown to contribute significantly to the heating of these devices. 8 It is
therefore of practical importance to improve the acoustic match for highcurrent-density devices so that excessive heating does not occur.
In this paper a simple acoustic theory is used to estimate phonon
transmission coefficients for various superconductor/substrate interfaces.
As first noted by Langenberg, 5 the most often used substrates are ~hown to
result in poor matching to many superconducting films. Suggestions for
particular film/substrate combinations as well as general guidelines are
given. The results of experiments are shown to be consistent with the theory
in most cases, but there remain some questions concerning the validity of
applying the theory to A1 films on glass substrates. In addition, inaccurate
results of previous work are described and explained.
2. M E T H O D OF C A L C U L A T I O N
The acoustic mismatch theory described by Little 9 was used to calculate
the transmission coefficients for longitudinal and transverse (SV and SH)
acoustic waves incident on the substrate from a superconducting film. These
media were assumed to be semi-infinite in extent and in perfect planar
contact. Although recombination phonons in Pb films may have
wavelengths of order 1.5 nm, it was assumed that dispersion and other
corrections to this classical continuum theory are negligible (in the worst
case of Pb, dispersion corrections to the acoustic velocities amount to less
Acoustic Matching of Superconducting Films to Substrates
MEDIUM I
345
z
MEDIUM 2
Fig. 1. A longitudinal wave propagating in the x - z plane is incident on the
boundary of media 1 and 2 (the x - y plane) with an angle of incidence 0x.
Reflected and transmitted longitudinal and SV transverse waves are
generated at the interface. Incident SV waves also generate pairs of
~eflected and refracted longitudinal and SV waves. Incident SH waves
generate only reflected and refracted SH waves.
than 10%). No attempt was made to include the effects of phonon attenuation.* The directions of the refracted waves were determined by Snell's law.
The angle-dependent transmission coefficients r/(0) were calculated by
constraining the parallel and perpendicular components of both the displacement velocity and the stress tensor to be continuous across the boundary. The calculation for incident SH waves was straightforward, since no
waves with displacement vector parallel to the plane of incidence are
generated at the interface. For either longitudinal or SV waves, a set of four
coupled equations for the complex displacement velocities of the four waves
generated at the boundary was solved. (See Fig. 1.) These calculations are
outlined in the Appendix. The acoustic velocities of the substrates often
exceeded those of the metals. U n d e r these conditions it was observed that
for incident longitudinal waves a critical angle exists beyond which only
transverse SV waves are transmitted. A discontinuity in the slope of the
longitudinal-to-SV wave transmission coefficient was observed at this angle.
A larger critical angle was observed beyond which total internal reflection
occurs. Similar behavior was found to occur for incident SV waves. The
critical angles for total reflection of SH and SV waves were observed to be
*It has been suggested that when the p h o n o n wavelength approaches the length of its m e a n free
path, surface waves m a y increase the total p h o n o n transmission coefficient. 1~
346
Steven B. Kaplan
identical. T h e s e c h a r a c t e r i s t i c s a g r e e with t h e c a l c u l a t i o n s of W e i s . 11 T h e
c a l c u l a t i o n s of 77(0) w e r e c h e c k e d b y e n s u r i n g t h a t t h e s u m of e n e r g y fluxes
of r e f l e c t e d a n d t r a n s m i t t e d w a v e s e q u a l e d t h a t of t h e i n c i d e n t wave. A
s a m p l e c a l c u l a t i o n of ,/(0) for Pb on A1203 was f o u n d to b e in e s s e n t i a l
a g r e e m e n t with t h e results d i s p l a y e d in Ref. 11. T h e a n g u l a r a v e r a g e of t h e
t r a n s m i t t e d e n e r g y fluxes was t a k e n to d e t e r m i n e t h e t r a n s m i s s i o n
coefficient for e a c h of the t h r e e p o l a r i z a t i o n s of i n c i d e n t waves, with the
a s s u m p t i o n t h a t t h e acoustic velocities c~ a n d ct a r e isotropic. It can b e s h o w n
t h a t if o n e a v e r a g e s o v e r all p o s s i b l e p o l a r i z a t i o n s , the a v e r a g e t r a n s v e r s e w a v e t r a n s m i s s i o n coefficient r/t is s i m p l y t h e a v e r a g e of t h e S V - w a v e a n d
S H - w a v e transmissivities. T h e c a l c u l a t i o n was a g a i n c h e c k e d using t h e
reciprocity relation 9
=
c~i c,i-li# lCl2S c~Ji,
1n
Ct 2,/Cl 1
0.~
0.:
0,.
0.1
0.2
0.5
1
2
5
10
Fig. 2. Contours of longitudinal phonon transmission coefficient rh for the
transmission of acoustic waves from medium 1 to medium 2 with respect to the
ratios of longitudinal wave velocities and mass densities in the two media. This
figure is presented to illustrate inaccuracies in the previous calculation of Little
(see Fig. 4 of Ref. 9 and note that tit is twice Little's l"t).
Acoustic Matching of Superconducting Films to Substrates
347
where the subscripts denote the values of r/l and rh for transmission from
medium i to medium i, and r/l is the transmission coefficient for longitudinally polarized waves. Although the results of these calculations agree
with the tabulated results of Cheeke e t al., 12 discrepancies as large as 30%
were found between the present results for r/~ and r/t and those of Little. 9 As
an example, we calculated transmission coefficients for c~2 = 2Ctl and ct2 =
2 c t l with the densities pl and p2 being equal and c~ = 2ct in both media. We
find (r/t; r/,)= (0.408; 0.205). From the contours of equal transmission in
Ref. 9 these values are found to be ( ~ 0.29; - 0 . 1 5 ) . It is believed that this
arises from the small number of points used in Ref. 9 to generate these
curves. 13 Figure 2 is a plot of contours of r/l for longitudinal acoustic waves
incident on medium 2 with respect to the ratio of longitudinal velocities and
densities of medium 2 to medium 1, where ct = 2ct in both media. Several
features of these curves are absent from the similar plot of Fig. 4 of Ref. 9.
The discontinuities in the slope of the contours as one crosses the line
ct2 = Ctl arise from the fact that total internal reflection in medium 1 occurs
outside the cone formed by the critical angle above this line. As one crosses
the line cz2 = 2ct~, the slope changes because of the stronger conversion of
longitudinal to transverse SV waves as Cll approaches c~2.
3. R E S U L T S
It has been assumed for simplicity that the superconducting films and
their substrates may be approximated by isotropic solids. Most thin films
may be characterized by randomly oriented crystallites, although it has been
found, for example, that Sn films evaporated onto sintered alumina substrates are highly oriented in the [100] direction. 14 For large acoustic
mismatch, only a small solid angle of incidence about the [100] direction is
sampled. For a polycrystalline alumina substrate, the calculated transmission coefficients for the slow and fast transverse modes in this direction
differ by only 10 and 14%, respectively, from the value of r/t calculated for
polycrystalline Sn. The discrepancy drops to about 1% for an AgBr substrate. One must realize that inaccuracies also occur because of anisotropy in
the single-crystal substrates. The smallest errors arising from the assumption
of isotropy generally accompany the largest transmission coefficients. It is
believed that the accuracy of the calculations is thus sufficient to guide one in
obtaining a suitable match between film and substrate.
Table I is a compilation of data for the direction-averaged acoustic
velocities and mass densities for six superconductors and various substrate
materials, accompanied by the temperatures associated with these data. (All
data are from Ref. 15, except those for BaF2, Si, and soda-lime glass, which
were estimated from the elastic moduli given in Refs. 16-18, respectively.)
348
Steven B. Kaplan
TABLE !
Physical Properties of Six Superconductors and Various Substrate Materials
Temperature,
K
Density,
g/cm 3
ct,
c~,
105 cm/sec
105 cm/sec
Material ~
Crystal type
AI
In
Nb
Pb
Sn
Ta
AgBr
AgC1
A1203
BaF2 b
Calcite
CdSe
CdTe
CsBr
CsI
GaAs
GaSb
HgTe
InAs
InSb
KBr
KCI
KI
LiI
LiF
PbF2
PbS
PbTe
RbBr
RbCI
RbI
Se
Sie
SiO2
Soda-lime
glassa
SnTe
8rF2
TiO2
TIBr
TICI
Cubic
Hexagonal
Cubic
Cubic
Tetragonal
Cubic
Cubic
Cubic
Trigonal
Cubic
Tri~onal
Hexagonal
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Cubic
Trigonal
Cubic
Trigonal
Amorphous
0.
4.2
4.2
0.
4.2
0.
room
0.
0.
0.
160.
298.
77.
4.2
4.2
77.4
298.
4.2
4.2
0.
room
4.2
4.2
room
4.2
300.
room
0.
4.
4.2
4.
300.
0.
77.2
room
2.73
7.47
8.59
11.59
7.38
16.71
6.47
5.69
3.99
4.94
2.72
5.68
5.86
4.60
4.71
5.31
5.61
8.10
5.67
5.97
2.75
2.03
3.17
4.06
7.32
7.79
7.56
8.37"
3.43
2.82
3.67
4.84
2.33
2.66
2.48
6.65
2.70
5.14
2.35
3.50
4.17
2.84
3.37
10.90
4.43
6.75
3.61
3.44
2.65
2.36
5.16
4.26
2.90
4.29
3.72
3.04
4.09
2.61
2.85
4.52
3.43
3.65
3.12
2.74
3.25
2.37
2.27
8.98
6.09
5.73
3.26
0.904
2.17
0.997
1.82
2.08
1.17
1.34
6.45
2.29
3.47
1.63
1.57
1.52
1.38
3.00
2.47
1.38
2.32
2.03
1.68
2.29
1.45
1.61
2.64
1.72
1.99
1.79
1.46
1.70
1.24
1.21
5.34
4.09
3.48
1.088
1.024 e
0.720
1.959
1.472
1.271
0.789
0.631
0.945
0.976
0.850 e
0.956 e
1.455
0.928
0.921
1.352
1.342
1.569
1.410
1.415
0.591
0.556
0.483
1.365
1.258
1.086
0.884
0.495
0.491
0.510
0.445
1.603 e
1.250
1.243 e
1.000
Cubic
Cubic
Tetragonal
Cubic
Cubic
0.
4.2
room
0.
293.0
6.48
4.32
4.26
7.75
7.00
3.48
2.88
9.20
2.34
2.27
1.93
1.34
5.13
1.20
1.15
0.455
0.903
1.394
0.888
0.783
aAll data from Ref. 15 unless otherwise noted.
bData estimated from elastic constants given in Ref. 16.
CData from Ref. 17.
aData estimated from elastic moduli given in Ref. 18.
eSee Ref. 19.
A 1/2
Acoustic Matching of Superconducting Films to Substrates
349
The anisotropy factor A is usually given in terms of the elastic constants
by19.*
A
= 2 c 4 4 / ( C l l -- C12)
(2)
The square root of A is the ratio of the two transverse-wave velocities in the
[100] direction for cubic and tetragonal crystals ( A - 1 in an isotropic
material such as BaF2). It appears in Table I as a guide to the variation of
sound velocities in these materials. 2~An attempt was made to use data taken
near 0 K. If only higher temperature data were available, no adjustments
were made. The acoustic velocities may increase by as much as 15%
between 300 and 0 K for soft materials, and less for hardS; density increases
are typically 1% over this temperature range and thus may be neglected.
The calculated transmission coefficients are given in Table II. One can
immediately see that the most commonly used substrates (A1203, Si, SiO2,
and soda-lime glass) are the poorest acoustic matches to the six metals. This
is not surprising, since, in general, these substrate materials have much
larger acoustic velocities than the superconductors under study. This is a
particularly acute problem for the Pb/A1203 and in/A1203 interfaces,
where rh is less than 2%. In order to obtain a better match, one must strive to
obtain the smallest ratio c2/cl possible, where the subscripts 1 and 2 refer
respectively to the film and substrate. Since the available phase space favors
transverse waves, it is important to obtain c2Jcl, < 1. This condition allows
one to avoid phonon trapping in the film due to total internal reflection. It is
less important (see Fig. 1) to have 02/01 > 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.
350
Steven B. Kaplan
e~
o~
0
~
.
~
~
~
~
~
Acoustic Matching of Superconducting Films to Substrates
[..
351
352
Steven B. Kaplan
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
Acoustic Matching of Superconducting Films to Substrates
353
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 << A and if phonons are reflected specularly at the substrate,
Te-d = ~.~1(1 - cos Od) + rr-t I (1 - cos Oct)
(3)
where ~'r~ and rrt refer to recombination processes with the emission of
longitudinal and transverse phonons, respectively, and 0d and 8 , are the
critical angles for total reflection of these waves. If diffuse scattering occurs,
phonon trapping is less severe.
(b) For d ~>A and if bulk phonon scattering processes are negligible,
one may use formulas for specular 29 and diffuse 3~ reflection of phonons at
the film/substrate interface. It is shown in Ref. 30 that for this range of
thicknesses the phonon-trapping factor may be approximated by
Tefr/rr = 4 d / r l A
(4)
where r/is the average value of r/t and r/t. The approximation is surprisingly
good for the case of specular reflection. One expects an error of less than
354
Steven B. Kaplan
25% within the range of the formula's validity. For diffusive reflection, the
accuracy of Eq. (4) is considerably worse for small r/. 30
(c) For larger thicknesses where bulk phonon losses are dominant, the
phonon-trapping factor becomes independent of both ~7 and d.
It must be remembered that the validity of these approximations is
dependent on several assumptions, which are discussed in depth elsewhere3~ (a) Phonons are treated as ballistic point particles. (b) The phonon
distribution is considered uniform and isotropic in space. (c) Multiphonon
processes are neglected in Eqs. (3) and (4). Specific situations where these
assumptions break down will be discussed later in this section.
It is clear from Eq. (4) that comparisons of calculated and measured
quasiparticle and phonon lifetimes 31'32 are capable of yielding information
about the phonon-trapping effects and thermal boundary resistances. This
knowledge is useful in designing experiments which eliminate some of the
complications due to poor acoustic matching. Some of the available data
shall be discussed here with a view to suggesting ways of reducing thermal
boundary resistance and internal reflections with the use of more appropriate substrate materials. Since the Kapitza conductance of metals in
contact with liquid helium is rather large, 33"~41shall refer whenever possible
to experiments in which the samples were not in direct contact with the
helium bath.
Once a value of ~'en is obtained, one may determine the accuracy of the
acoustic mismatch theory using Eq. (4) if one is able to estimate A as well as
~'r. The estimates of rr for Sn and Pb found in Ref. 32 are based on
a2(fl)F(f~) tunneling data and are believed to be good to better than 30%,
but the value given for AI is considerably less certain. The values of ~'B for
most metals given in Ref. 32 are not as accurate as the zr values, due to the
crude nature of the approximations made in their calculation. Therefore,
independent estimates of A for Sn and A1 will be used in order to calculate
phonon-trapping factors.
4.2.1. Tin
Two different types of experiments have been done With Sn films in
vacuum. 3'30'31"34"35The data from Ref. 3 were measured by the microwave
response of thin Sn films on Z - c u t quartz substrates. It was found that
ten = 3.22 • 106~'rd for a 90-nm film, where d is measured in cm. If the
transverse and longitudinal mean free paths are taken from Ref. 31 to be
A , = 7 0 n m and A t = 160nm, it is found that 7/=0.175, where ,/ is the
appropriate average of r/1 and 7/,:
I-2r/,
r/~'lf 2
1 "1-2/3
= I---r-+--2-//--r + ~/
L Ct
Cl I L C t
CI J
(5)
Acoustic Matching of Superconducting Films to Substrates
355
From Table II, one calculates rl for the Sn/SiO2 interface to be 0.17 for an
isotropic substrate; making corrections to the calculations for the anisotropy
of the quartz substrate as described earlier results in a value of ~ .= 0.15. In
view of the approximation made and since A is not known very accurately,
the agreement between theory and experiment must be considered fortuitous. The decay-time measurements for Sn on Si presented in Refs. 31 and
35 may be represented as re~ = 1.89 • 106~-rd. This results in an estimate of
r / = 0.28; the calculated average from Table II and Eq. (5) is found to be
0.10. Within this theoretical framework, the agreement between theory and
experiment is not good, but may be acceptable for forming gross conclusions
about the nature of the film/substrate boundary.
One may also calculate re, (or equivalently, rl) by invoking detailed
balance 36 to eliminate rr and A from Eq. (4) when the phonon-trapping
factor is much greater than unity:
-1
N o r F2~t 7/~1F 2
1 ]-t
re. = 2-N r d t c-t- 7 + -cT, ] t c--T+--T
,
c,J
(6)
where N a t and N r are the respective densities of phonons with energies
f~ > 2A and quasiparticles that are present in thermal equilibrium. If NaT is
estimated within the D e b y e approximation and N(0) is given by 37 1.4 x
1022 eV -1, re, is estimated from the data of Tables I and II to be 4.5 •
1 0 - 6 s e c for a 300-nm Sn film on a Si substrate. This is to be compared with
the measured value of 4.2 • 10 -6 sec given in Refs. 31 and 35. In Ref. 34,
decay-time measurements were also analyzed using detailed balance to yield
r / = 0.115 for the Sn/Si interface, which compares well with the calculated
average value 0.10. The origin of the discrepancy between the two methods
is not clear.
In order to limit the phonon-trapping factor for reasonably thick Sn
films or tunnel junctions, one may place the samples in liquid helium. It was
shown in Ref. 34 that ~/ for the Sn/helium interface is about 0.25, which
reduces re, for Sn and Si in vacuum by a factor 3.8. Furthermore, the data of
Table II show that if one chooses a PbF2 or TIBr substrate, one could hope
for another reduction factor of 3.5 for a thick film. For reasonably thin films
(d - 15 nm) on these substrates, the microwave reflectivity experiments of
Sai-Halasz et al. 3 would be able to measure zr without dressing due to
phonon trapping.
For the thinner films used in Ref. 3 (d - 40 nm < A), reasonably good
agreement with theory was found by Eisenmenger et al. 31 using Eq. (3). By
neglecting longitudinal phonons in their analysis, they calculated a phonontrapping factor of 15. This emphasizes the point that in order to minimize
phonon trapping due to total internal reflection, one must use substrates
such as T1Br where c 2 < c 1.
356
Steven B. Kaplan
4.2.2. A l u m i n u m
Although many experimental determinations of ~'en in clean A1 films
have been done, 6'38-4~ the interpretation of these results is still controversial. 31'32 Long has measured A, to be approximately 350 nm in clean AI
films,41 leading one to expect the linear thickness dependence of ~'efrto break
down at d - 200 nm. 39 However, the experiments of Gray et al. show a
linear dependence of ~~fr on d down to 100 nm. By using Eq. (5) and
neglecting the longitudinal contribution to recombination, Gray's data were
reanalyzed in Ref. 39 to yield ?7 = 0.11 + 0.01 for the A1/sapphire interface.
This is to be compared with the average value of 0.21 as calculated from Eq.
(5) and the data of Tables I and II. Long argued 39 that the poor agreement
between theory and experiment is consistent with elastic phonon scattering
from grain boundaries and tunnel barriers.t This effect would increase the
effective phonon escape path and thus increase the measured z~n.This would
explain the linear thickness dependence for thin films as well as the low value
of the deduced phonon transmission coefficient. This is disputed by Eisenmenger et al., 31 who suggest that the actual A may be as low as 100 nm.43'~:I
have recalculated 71 for the pulse-decay measurements of Gray et al. using
Eq. (6) and N ( 0 ) = l . 7 5 x 1 0 ~2 states/eV to yield ~7-0.27 for the
A1/sapphire interface, which compares more favorably with the results of
the acoustic mismatch theory without modification.w However, a similar
calculation for the steady-state ~-~nmeasurements for A1 on sapphire by
Gray et al. yields a lower figure** of ?7- 0.096. One can only say that the
value predicted by the acoustic mismatch theory lies between these two
values. Because of the enhanced electron-phonon coupling in these films, 38
the estimate of ~r given in Ref. 32 is not appropriate for a calculation of ?7
using Eq. (4). Further experiments on clean A1 films should be done to
resolve the differences between the results of the pulsed and steady-state
experimental methods.
By comparing measured z~ for A1 tunnel junctions evaporated on glass
slides both immersed in liquid helium and in vacuum, Smith and Mochel4~
*Note that the experimental lifetimes reported in Ref. 40 have been increased by a factor 2.86,
per Smith. 41
t T h e transmission coefficient for transverse SH waves on a 20-/~ A1203 barrier in planar
contact between two A1 films has been calculated by the author to be only 71% for a 400-1xeV
phonon.
CThe data reported in Ref. 43 were for granular AI films of T~ ~ 2 K, which therefore are
expected to have a shorter p h o n o n m e a n free path due to the enhanced e l e c t r o n - p h o n o n
interaction in these samples over that expected for clean A1 films.
w
electronic density of states used in Ref. 36 to calculate NT in Eq. (6) was missing the
phonon-renormalization factor 1.43. The additional neglect of a factor 2 in the calculation of
N r resulted in the bulk of the discrepancy between the values of 0.11 and 0.27 for ~.
**The values of ~'~frof Ref. 35 have been increased by a factor 2.86 per Gray, 44 for the same
reasons given in the previous footnote.
Acoustic Matching of Superconducting Films to Substrates
357
determined the phonon-trapping factor in vacuum to be 8 . 6 • 1.0 for a
double-junction geometry with total film thickness of 90 nm. This large
factor was interpreted in Ref. 31 to arise from total internal reflection of
phonons at the substrate. A phonon-trapping factor of 7 was calculated by
assuming the acoustic properties of glass to be similar to those of A1203. A
glance at Table 1 shows that this comparison is not valid; in fact, total
reflection should not be a problem in AI films on most types of glass. One
would expect a phonon-trapping factor of - 2 for such a double junction,
but attention to phonon trapping at the tunnel barriers 4~ could raise that
figure significantly. In addition, elastic phonon scattering in the disordered
substrate near the film can hinder phonon escape. The uncertainties that
exist in a calculation that may include such additional phonon-trapping
effects do not favor a meaningful comparison with the acoustic mismatch
theory. Furthermore, it was assumed in Ref. 40 that electron-electron
interactions could be neglected in the data analysis; the deduced factor of
8.6 would be considered an overestimate if this were not a good approximation.* However, one can draw the conclusion that in order to measure a ~'e~
which is close to the intrinsic value, one must use a substrate where phonon
transmission is almost perfect for all angles of incidence.
Table I shows that the choice of BaF2 as a substrate avoids the
complication of total internal reflection in the case of AI, since c2t/c2,/c1~ are
less than unity. Table II shows that the calculated phonon transmission
coefficients are nearly unity for the A1/BaF2 interface. The almost perfect
isotropic nature of BaF2 (see Table I) gives increased confidence to these
calculations. Chi and Langenberg have reported 6"3~that for 30-nm A1 films
on BaF2 substrates reefmeasured for samples in vacuum were about the same
as the value measured for samples in contact with liquid helium. This is
consistent with a large value of r / f o r the A1/BaF2 interface. Moreover, the
measured value of r~e agrees with the estimate of r, reported in Ref. 32 to
within a factor 2. A more careful comparison between theory and experiment is not meaningful since the intrinsic recombination times in the dirty A1
thin films (To - 1.7 K) may be shorter than the value calculated for bulk A1.
4.2.3. L e a d
Jaworski et al. 23 have measured refr for 350 nm of Pb on a glass substrate
in contact with liquid helium to be approximately 100 times the theoretical
estimate given in Ref. 32. This is consistent with the estimate by Kaplan et
al. 32 of ra --3.5 x 10 -11 sec and the assumption of ~/--0.5 appropriate for
superconductor/superfluid helium surfaces. 33 The phonon-trapping factor
may be reduced to - 4 for a 35-nm film in contact with liquid helium on a
*There are indications that electron-electron interactions m a y be of importance in A1 (cf. Ref.
29). If this were so, the deduced trapping factor would be smaller.
358
Steven B. Kaplan
substrate of TIBr, T1C1, AgBr, or Se. In addition, the critical angles for total
internal reflection for these substrates are rather large because of their
relatively small acoustic velocities. However, it is clear from this analysis
that it is extremely difficult to devise an experiment for Pb films where
~'~ ~ zr, because of the short phonon mean free path in this metal.
4.2.4. Suggestions for In, Nb, and Ta
Although there are theoretical estimates of the intrinsic quasiparticle
recombination time 32 in In, Nb, and Ta, there is as yet no experimental
verification of these calculations. It is clear from Tables I and II that In
presents problems similar to those of Pb: low acoustic velocities and poor
phonon matches to most substrates do not allow one to design thin-film
experiments with negligible phonon-trapping effects, especially for transverse phonons. The substrates mentioned in connection with Pb are the best
choices.
The larger acoustic velocities in Nb result in reasonable acoustic
matching to a variety of substrates, including PbF2, which is an almost
perfect match. However, the short phonon pair-breaking lifetimes* in Nb
result in a phonon-trapping factor of - 4 even for films as thin as 5 nm.
Because of its large mass density, the acoustic matching of Ta is slightly
more difficult than for Nb, but Table II shows ~ / - 0.80 for a PbF2 substrate
and ~7~0.64 for an AgBr substrate. With the estimate of ~'B~
4.2 x 10 -11 sec from Ref. 32 it is estimated that a film thickness d of 23 nm is
needed to reduce the phonon-trapping factor to 4. It may indeed be difficult
to measure ~'r for In, Nb, and Ta without complications due to phonontrapping effects using conventional methods.
4.3. Controllable Weak Links
The length of the nonequilibrium region in a controllable weak link 7 is
believed to be dependent on the geometry and the quasiparticle diffusion
length ~ 0c (reef)1/2. This length should be short with respect to the coherence
length in order to observe good Josephson phase coupling. Wong and
Langenberg have shown that about 25% more rf-induced Josephson steps
are seen in the device's current-voltage characteristic if sapphire is used as a
substrate material in preference to glass. They attribute this to a large
probability of phonon reentry into the film if a disordered substrate is used
rather than to a possibly larger value of r/for sapphire. This results in a larger
phonon-trapping factor and a larger value of A. Table II indicates that the
use of a TICI or other suitable substrate rather than sapphire could decrease
*A value of rB ~ 1.9• 10 -12 sec was calculated using the method of Ref. 29, ~2(~'l)F(fl)
tunneling data, 45 and F(fl) from neutron data. 46
Acoustic Matching of Superconducting Films to Substrates
359
the value of A by as much as a factor 3 for Sn and a factor 5.5 for Pb. This
should be kept in mind in the design of other devices where one wishes to
minimize the size of the nonequilibrium region.
4.4. Other E x p e r i m e n t s
Kirtley et al. s have used an A I / A I O x / A I / A I O x / P b B i double junction
evaporated on soda-lime glass to measure the intrinsic quasiparticle relaxation time (without phonon trapping) and the nonequilibrium quasiparticle
energy distribution in their dirty A1 films ( T c - 2 . 4 K ) . An effective
temperature T* (which was greater than the helium bath temperature Tb
due to film heating) was measured as a function of the electrical power Pin
delivered to the structure by monitoring the current-voltage characteristic
of the A 1 / A I O x / P b B i tunnel junction. It was found that
Pin = 88
. 4 - T~ )
where A (4 m W K -4 for the l-ram 2 sample) can be related directly to the
boundary conductance by hk = A T 3. Although the sample was not in direct
contact with the helium bath because of a coating of photoresist, it is not yet
clear why the phonon transmission coefficients calculated from the
measured value of A are more than an order of magnitude lower than the
predictions of the acoustic mismatch theory. The repetition of this experiment with clean Al films in vacuum may yield some more easily interpretable
results free of possible complications due to phonon trapping at grain
boundaries and the necessity of considering two interfaces. Again, the use of
a glass substrate may be the cause of the rather large observed phonontrapping effect. The use of a more suitable substrate should minimize
heating effects in this and other experiments.
Finally, there exist some results for A1 films on various substrates which
seem to imply much smaller phonon transmission coefficients than are
predicted by the acoustic mismatch theory. 4T By measuring (a) the broadening of heat pulses, (b) the smearing of the rising edge of a "step function"
signal to a superconducting bolometer, or (c) the measured temperature rise
of a resistively heated film, the relaxation times in thin films have been
measured to be 30 nsec, while one would expect 1 nsec based on the
mismatch of film and substrate. Because the experimental details of these
studies are not known, I shall only comment that more work should be done
on well-characterized A1 films to be better able to compare theory with
experiment.
5. C O N C L U S I O N S
The acoustic mismatch theory of Little and Weis has been shown to be
in rough agreement with the results of various experiments with In and Sn
360
Steven B. Kaplan
films and (to a lesser extent) A1 films on Si, SiO2, and A1203 substrates.*
Corrections to the isotropic model do not yield substantially different
results. The observed phonon-trapping effects for films on glass substrates
appear to be much larger than those predicted by the theory. It has been
shown that the most often used substrates result in low values of phonon
transmission coefficients along with the added complication of total internal
reflection of phonons in the films. Suggestions have been made for the
selection of more appropriate substrate materials, which would help to
reduce phonon trapping. This would allow the better determination of
intrinsic quasiparticle recombination times, shorter quasiparticle diffusion
lengths, and a reduction of heating phenomena for thin-film geometries.
APPENDIX
Let us consider an acoustic plane wave that propagates in the x - z plane
of medium 1 and is incident on the x-y plane that represents the boundary
between the semi-infinite media 1 and 2 (see Fig. 2). If calculations are
restricted to isotropic media, the application of Snell's law results in
sin 02
sin01
- -
ell
-
- -
el2
sinyl
siny2
Ctl
Ct2
(A1)
where O~and yi are the respective angles of propagation of longitudinal and
transverse waves measured with respect to the z axis. The boundary
conditions require that the following quantities be continuous at the interface:
(a)
(b)
(c)
(d)
The sum of the normal components of displacement vectors (or
equivalently, displacement velocities).
The sum of the tangential components of displacement vectors (or
displacement velocities).
The sum of the normal components of the stress tensor.
The sum of the tangential components of the stress tensor.
The calculation of transmission coefficients rt,(yl) for incident SH waves is
particularly simple and will therefore be described separately from the
outline of the SV and longitudinal (L) wave calculations.
A.1.
Incident
SH Waves
SH waves are transverse (shear) waves with displacement vectors in the
y direction. (Thus, one may say that they have horizontal polarization.) It is
*Note added in proof: Discrepanciesbetween theory and experiment may arise from imperfect
adhesion of films to substrates.44
Acoustic Matching of Superconducting Films to Substrates
361.
therefore impossible for an incident SH wave to produce transmitted SV or
L waves which have displacement vectors in the x - z plane. We see that only
boundary conditions (b) and (d) are nontrivial. We define the displacement
vectors in the x, y, and z directions to be u, v, and w, respectively. Following
Zoeppritz, 48 the only nonzero c o m p o n e n t of the displacement vector of the
incident SH plane wave with amplitude B, wave vector K, position vector r,
and frequency o~ is given by
Vl=Bexp[i(Ki
"r-wt)]=Bexp[i(-~iz+orx-oJt)]
(A2)
where
•1 ~-" (O.)/Ctl) COS 3'1,
Or =
(0)/C,1)
s i n 3'1
are, respectively, the z and x components of K. Similar forms are assumed
for the reflected SH wave (with displacement amplitude D ) and the transmitted SH wave (with amplitude F). Boundary condition (b) leads to
B +D =F
(A3)
The y c o m p o n e n t of the tangential stress on the boundary Zy is given by
Z,, = pc 2,[0 V/Oz + 0 W/Oy ]
(A4)
and of course w = 0 for SH waves. Boundary condition (d) then leads to
B - D - p2ct2 cos ]/2 F
(A5)
plCtl COS T1
By multiplying Eqs. (A3) and (A6), we obtain
1 ~---( D ) 2 Jr- P2Ct2 COS 3'2 ( B ) 2
PlCd COS 3/1
(A6)
which will prove to be of interest shortly.
The energy flux of the incident acoustic wave is the product of its energy
density times the normal c o m p o n e n t of its acoustic velocity:
energy flux = (plwZB2ctl cos y l ) / 2
(A7)
We may at once recognize Eq. (A6) to represent the sum of the ratios of the
reflected and transmitted energy fluxes to the incident energy flux. The
acoustic transmission coefficient is found to be
r/,(yl) = 4x/(1 + x ) 2
(A8)
362
Steven B. Kaplan
where
X = (P2Ct2 COS 3`2)/(PlCtl COS 3`1)
Finally, one calculates the angle-averaged transmission coefficient:
r/t(SH wave) = 2
f
~v/2
7/(3`1) sin 3`1 cos 3`1 d3,1
aO
=2
(A9)
Io'
r/(3`1) sin 3'1 cos 3`1 d3`1
where 0c = sin -1 (Ctl/Ct2) is the critical angle for total internal reflection for
transverse waves.
A2.
Incident SV and L Waves
SV waves are transverse waves with displacement vectors in the x - z
plane. Therefore, an incident SV wave may result in reflected and transmitted L waves as well as reflected and transmitted SV waves (see Fig. 2). An
incident L wave also may split into four waves. Thus, the calculation of the
transmission coefficients results in four coupled equations, rather than the
two that were solved in the case of SH waves.
It is instructive to define the velocity potentials r and ~O, which
correspond to L and SV waves, respectively, where the time derivative of the
displacement vector or particle velocity is given by 49
v =Vd~+VxxO
(A10)
The boundary conditions (a)-(d) at the boundary of media 1 and 2 may then
be written as
19r
--
Oz
plC2 i
V261+201c21
+
I~11 19r
c~x
--
Oz
q
01112
or
o0a=or
o412
Ox
Oz
Oz
0 ~1
Ox
( A 11 a)
Ox
Tx ~ ] = o~c ,2 V
r
(Allb)
202c,2(~- z
~l
(Allc)
plc~ (2 o24,1 +o201 o~,l,1'~
OxOz
Ox2
2/.. o2r
o%: o~r
~ z 2 } = O 2 C a k Z o x o z q Ox2
~z2} ( A l l d )
For the case of an incident plane L wave, the incident, reflected, and
transmitted L-wave potentials r
Cren, and (~t . . . . may be written in the
plane-wave form of Eq. (2) with respective amplitudes A', C', E ' and wave
vector ki such that
Acoustic Matching of Superconducting Films to Substrates
r
ki 9 r = •
+o'x = +--cos
COX
0i + - - sin 0~
eli
363
(A12)
eli
w h e r e i refers to m e d i u m 1 or 2, and the positive sign refers only to the
reflected wave ~b~n. T h e potentials ,~]refl and I//t . . . . represent reflected and
transverse (SV) waves. T h e y m a y also be written in the f o r m of Eq. (2) with
respective amplitudes D ' and F ' and z c o m p o n e n t of wave vector
3i = +(to~ca) cos 3"i
(A13)
w h e r e i = 1 or 2 and the positive sign again refers to the reflected wave. It
should be n o t e d that o- defined in Eq. (2) m a y be written
O)
&)
60
0.)
or = - - sin 01 = - - sin 3'1 = - - sin 02 = - - sin 3"2
Cll
Ctl
Cl2
(AI')
Ct2
as a corlsequence of Snell's law. T h e b o u n d a r y conditions m a y n o w be
written as
a l ( A ' - C') + o'D' = - o t 2 E ' + o F '
(A14a)
-
o'(A' + C') - f l l D ' = a'E' + fl2F'
2
(A14b)
w
plc,1 [(A + C ' ) ( c o t 2 3/1 - 1) + 2 D ' cot 3'1]
2
~
2
= p2c~2[E (cot 3 ' 2 - 1 ) - 2 F ' cot 3'2]
(A14c)
2
plc tl [ - 2 ( A ' - C ' ) cot 01 + D ' ( 1 - cot 2 3"1)]
2
= p2c t2 [ - 2 E ' cot 02 + F ' ( 1 - cot 2 3'2)]
(A 14d)
T h e s e four coupled equations are solved to find C ' , D ' , E ' , and F ' in terms of
A ' . F o r angles of incidence
O1 > sin-1(Cll/ C21)
the application of Eq. (1) shows that cos 02 is imaginary, leading in general to
c o m p l e x amplitudes for the reflected and refracted waves except E ' , which is
zero. Finally, total reflection of the incident wave occurs w h e n
01 > sin -1 (Cll/CEt)
T h e e n e r g y flux of the incident L wave m a y be written
e n e r g y flux = (pltoEI A ' 2 c o s 01)/2Cll
(A15)
E n e r g y conservation therefore imposes the constraint that
Ic'l 2 c111D'[ 2 cos 3"14 02c111E'12 cos 02 p2cttlF'l 2 cos 3,2
1 = ~ - ~ - ~ c1,1A,12 cos 01 02c2,lm'12~osE+t:,,c2,la'l 2 cos 01
(A16)
364
Steven B. Kaplan
This was u s e d to c h e c k the v a l i d i t y of t h e s o l u t i o n s of Eqs. ( 1 4 a ) - ( 1 4 d ) . T h e
s u m of t h e last t w o t e r m s of Eq. (16) r e p r e s e n t i n g t h e a n g l e - d e p e n d e n t
t r a n s m i s s i o n coefficient ~Tt(O) for i n c i d e n t L w a v e s was i n t e g r a t e d o v e r all
angles of i n c i d e n c e as in Eq. (9) to d e t e r m i n e r/t.
E q u a t i o n s similar to Eqs. ( 1 4 a ) - ( 1 4 d ) a n d (16) m a y be set u p for an
angle t r a n s v e r s e S V w a v e with a m p l i t u d e B ' in o r d e r to c a l c u l a t e rh(SV
wave). F o r a r b i t r a r i l y p o l a r i z e d t r a n s v e r s e w a v e with a d i s p l a c e m e n t v e c t o r
t h a t m a k e s an angle x with r e s p e c t to the x - z p l a n e , it can b e s h o w n t h a t the
t r a n s m i s s i o n coefficient is
rh = rh(SV) cos 2 X + ~Tt(SH) sin 2 X
(A17)
T h e r e f o r e , w h e n all p o s s i b l e p o l a r i z a t i o n s are c o n s i d e r e d , t h e t o t a l t r a n s v e r s e - w a v e t r a n s m i s s i o n coefficient is s i m p l y t h e a r i t h m e t i c m e a n of r/,(SV)
a n d r/t(SH).
ACKNOWLEDGMENTS
I w o u l d like to t h a n k Prof. D. N. L a n g e n b e r g for m a n y e n l i g h t e n i n g
discussions of this p r o b l e m a n d for s h a r i n g t h e results of his u n p u b l i s h e d
calculations. I w o u l d like to f u r t h e r a c k n o w l e d g e fruitful discussions with
Prof. W . A . Little, Dr. H a s s e l L e d b e t t e r , a n d Prof. W . E i s e n m e n g e r .
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