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Submitted on 1 Jan 1972
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PHONON TRANSMISSION ALONG SAPPHIRE PLATES
W. Day
To cite this version:
W. Day. PHONON TRANSMISSION ALONG SAPPHIRE PLATES. Journal de Physique Colloques,
1972, 33 (C4), pp.C4-65-C4-67. �10.1051/jphyscol:1972414�. �jpa-00215091�
JOURNAL DE PHYSIQUE
Colloque C4, supplkment au no 10, Octobre 1972, page C4-65
PHONON TRANSMISSION ALONG SAPPHIRE PLATES
W. DAY (*)
Royal Society Mond Laboratory University of Cambridge
Rbsumb.
-Nous rendons compte d'experiences dans lesquelles des phonons de haute frequence,
ala fois excites thermiquement et monochromatiques, sont cr&s et detect6s
ala surface d'un monocristal de saphir a des temperatures de
0,3B
1 OK.Des processus inelastiques dans le systkme (mis en evidence par I'usage de jonctions tunnel supraconductrices comme detecteurs passe-haute de phonons monochromatiques) ainsi que le contact thermique du saphir et d'un bloc de cuivre sont etudies.
Abstract. - Experiments are reported in which high frequency phonons, both thermally distri- buted and monochromatic, are generated and detected on a sapphire single crystal substrate at temperatures between
0.3OK and 1
O K .Inelastic processes in the system (revealed by using super- conducting tunnel junctions as high pass detectors of the monochromatic phonons) and the ther- mal bonding of the sapphire to a copper block are investigated.
Introduction. - Experiments have been carried out in which monochromatic and broad band thermal phonons are generated and detected in thin films eva- porated on synthetic sapphire single crystals.
Copper heaters were used to generate the thermal phonons and superconducting aluminium tunnel junctions to generate the monochromatic phonons.
When a junction is biased at or above the gap edge (2 A, the energy gap in aluminium) the main current carrying process is the breaking of ground state pairs which injects excitations into both films. The recombi- nation of these excitations to Cooper pairs releases energy as phonons at the gap frequency, 2 Alh.
The tunnel junctions were also used to detect both thermal and gap frequency phonons. When the junc- tion is biased at less than 2 A the current is due solely to the tunnelling of excitations present in the films. The number of excitations increases with temperature as exp(- AlkT) and thus the cc within-gap current
))(which is approximately proportional to the number of excitations) can be used as a thermometer. Mono- chromatic phonons of energy 2 2 A entering the junction films can break Cooper pairs, creating extra excitations and hence extra current which will be a measure of the number of phonons arriving. The number of extra excitations in the films when mono- chromatic phonons are incident is proportional to the lifetime of excitations to recombination. This is inversely proportional to the number of thermal excita- tions and therefore proportional to exp(A/kT). The signal therefore increases with decreasing temperature.
The generator was fed with a constant current chopped at a frequency of 80 Hz and the detected
(*)
Supported during this work
by aScience Research Council Grant.
signal was measured using a Princeton HR 8 cr lock-
in
)>amplifier. For most of the experiments the relevant
relaxation and propagation times in the system were shorter than the chopping period, so that detected signals correspond to steady state conditions.
The experiments were performed in a He3 cryostat, the sapphire being bonded to a copper block one end of which formed the bottom of the He3 pot. A minimum temperature of .29 OK was attainable.
The signals obtained using copper heaters are a measure of the local temperature change and this is made up to temperature drops across all thermal resistances to the He3 bath. The monochromatic phonons (which have an equivalent temperature T
=hvlk > 4
OK)are rapidly thermalized by the electrons in the copper, so only thermal resistances before the copper block will affect the signal. Because the tunnel junction is a high pass detector, any inelastic processes will show up as a loss of signal.
For monochromatic phonons each junction on the crystal can be used as generator or detector. Thus with an array of n junctions, a rr transfer sensitivity
))(detected current + generator current) can be measur- ed for each pair. These results can be most easily displayed in an nxn matrix, where the element (i, j ) is the transfer sensitivity with the ith junction as detector and the
jthjunction as generator.
The detector sensitivity is inversely proportional to the normal resistance of the junction, so that matrix is normalised to 1 ohm detectors. Once normalized the matrix should be symmetric
:this can be seen by considering the phonon flow
-for the case of diffuse flow, we can compare the system to a passive electrical network where two nodes have a transfer resistance (independent of direction). Considering direct flight of phonons, the reversibility of flight paths must give
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972414
C4-66 W. DAY
the same signal. Thus the two normalized signals i
+j and j
+i should be equal, giving a symmetric matrix. This provides a check on the accuracy of any measurements made.
Thermal b~ndiog.
-The bonding of the sapphire to the copper block is important in these experiments.
Apiezon r( N
)>grease was used initially, but the bonding resistance was found to be more than two orders of magnitude greater than expected (Vilches and Wheatley [I]). This was because the grease had pulled away from the sapphire on cooling, leaving contact only at the points where the sapphire was pressed down For good thermal contact a bonding agent needs to be plastic to a low tempxature so that it can follow the different thermal contractions within the rigid system we are attempting to bond. Bostik (*) thinned with acetone was found to make a better bond.
Monochromatic phonons.
-The expected form of the signal variation in the sapphire can be found by solving the Boltzmann equation for the phonon popula- tion. This has been done by Anderson and Sabisky [2]
but the form of the result is easily seen by considering an electrical analogue. In one dimension, with sapphire to substrate resistance RB per unit length, and sapphire resistance Rs per unit length, the equations for the nth element of the resistive network (Fig. 1) are
tance was observed, fitting approximately a decay length of 8 mm. This shows that with Bostik and acetone the phonons escape easily from the sapphire (and are subsequently thermalized either in the Bostik or in the copper).
With cr N
>)grease bonding the major thermal resis- tance is the poor bond to the copper - so one might expoct the results with monochromatic phonons to be the same as those with thermal phonons. With thermal phonons, there was no variation in signal along the crysial and a thermal time constant of some milli- seconds was measured (by finding the phase shift between the chopping signal and the detected signal).
This is consistent with heating up of the sapphire.
However, with monochromatic phonons there was significant variation of the signal along the crystal and the time constant was much smaller (less than .5 milliseconds). This indicates rapid thermalization of the monochromatic phonons. One might suppose that this was associated with the presence of the bond- ing material. However it was still present when only the last three millimetres at one end of the crystal were bonded with Bostik and acetone (as in Fig. 2).
8 acetone Junctions
1 2 3 4 5
I " " " " "
FIG. 1. - An electrical analogue of the bonded sapphire plate.
As 6x tends to zero, these equations tend to dildx
=V/R, and dV/dx
=iRs (where i and V are the phonon current and phonon population respecti- vely). These equations give an exponential solution with decay length (R,/R,)% ; R, will also include the rate of inelastic scattering of phonons, when such processes are present. This equation only applies at distances large compared to the phonon mean free path (which will be approximately the thickness of the crystal) as does the solution of the Boltzmann equation by Anderson.
The sapphire crystals used were 38 x 25 x 2.5 mm.
The length to width ratio being small, the form of the solution will not be exactly as in the one dimensional case but it will be similar.
With Bostik and acetone bonding one face of the crystal to the copper, a rapid loss of signal with dis-
(*) Bostik is a nitrile rubber based adhesive in a ketone solvent, manufactured by Bostik Limited, Leicester, England.
FIG. 2. -Experimental configuration to show loss of mono- chromatic phonons along the plate.
If there were no loss processes, moving from the source towards the unbonded end would show a uniform phonon population except within a few tnean free paths of the generator. Tables I and I1 and figures 3 and 4 show the results of two experi- ments with the crystal bonded in this way. It is clear that there is still loss of monochromatic phonons.
The thermalization is not reproducible from sample to sample and does not correlate with phonon fre- quency, nor is there any measurable temperature dependence.
Matrix for experiment 1
Junction I 2 3 4 5
- -
- -- -
1 - 3.7 2.6 1.8 1,45
2
-3.1 2.2 1.55
3 - 2.65 1.75
4 - 2.45
5
-Transfer sensitivity ( x lo5)
(2 A
=440 peV, T
=.31
OK).PHONON TRANSMISSION ALONG SAPPHIRE PLATES
Matrix for experiment
2
Junction I 2 3 4 5
- - - - -
-1 - 2.5 1.5 1.05 .7
2 - 2.45 1.4 .9
3
-2.4 1.2
4 - 1.9
5
-Transfer sensitivity ( x lo5) (2 A
=440 peV, T = .31
O K ) .3 0
0
10
L
1 2 3 4 5 Junct~onsFIG. 3. - Results of experiment 1-showing phonon population gradient towards the bonded end (beyond 5) and an almost
uniform population at the unbonded end.
Transfer 105
i
ensitivityI . I
1 2 3 4 5
Junctions
FIG. 4. - Results of experiment 2-note the increased loss at the unbonded end compared to experiment 1.
Anderson [3] has also observed loss of mono- chromatic phonons in his experiments, but the loss was much more rapid, and was probably connected with phonon transmission into the helium in which his samples were immersed. This cannot apply to our experiments in which the samples were in a vacuum.
The irreproducibility indicates that the loss we observe cannot be associated with intrinsic material properties of the sapphire (phonon-phonon interac- tions, for example). In very recent. experiments it has been shown that the pressed indium electrical contacts to the junctions are a source of loss of the mono- chromatic phonons, and that this inelastic loss also occurs for evaporated indium films. This is being further investigated.
References
[I]
