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Submitted on 1 Jan 1978
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LOW TEMPERATURE THERMAL CONDUCTIVITY
OF TWO NATURAL DIAMONDS : ANISOTROPIC
HEAT CONDUCTION IN THE BOUNDARY
SCATTERING REGIME
J. Vandersande
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-1017
LOW TEMPERATURE THERMAL CONDUCTIVITY OF TWO NATURAL DIAMONDS ; ANISOTROPIC HEAT CONDUCTION IN THE BOUNDARY SCATTERING REGIME
J.W. Vandersande
Department of Physios, University of the Witwatersrand, Johannesburg, South kfvioa.
Résumé.- La conductivité thermique de deux diamants naturels, de type lia, et d'axes respectif <100> et <110>, a été mesurée entre 0,5 K et 20 K. La conductivité pour le diamant avec l'axe <100>
a été trouvée être environ deux fois plus grande que celle de l'autre diamant.Cette anisotropie s'accorde qualitativement, avec les prédictions théoriques publiées,basées sur la focalisation des phonons.
Abstract.- The thermal conductivity of two natural type Ila diamonds with rod axis <100> and <110> respectively has been measured between 0.5 K and 20 K. The conductivity for the diamond with the <100> rod axis was found to be about double that for the other diamond. This anisotropy is in quali-tative agreement with published theoretical predictions based on phonon focusing.
The thermal conductivity of pure and perfect crystals of dielectric materials at low temperatures does not have a unique value for the material but depends upon the crystal dimensions. Casimir /1/was the first to develop a theory of the thermal con-ductivity applicable to this temperature range and he found the conductivity, in the case of rough sur-faces, to be proportional to the temperature cubed and to the diameter. Berman and co-workers /2,3/ computed modifications of Casimir's theory for crystals of finite length with various amounts of
specular reflection. McCurdy, Maris and Elbaum1*
des-cribed another way in which the conductivity may differ from the predictions of Casimir. They showed the conductivity in the boundary regime to depend on the orientation of the crystal rod axis. That anisotropy was accounted for in terms of phonon fo-cusing due to the fact that in elastically aniso-tropic cubic crystals the phonon phase and group ve-locities are, in general, not collinear. They con-firmed their predictions with experiments on sili-con and calcium fluoride.
The purpose of this work was to confirm the predictions of McCurdy et al. for.phonon focusing in diamond. According to them, phonon focusing will only occur in the <100> direction in diamond and the thermal conductivity will be about 38 percent greater than that predicted by Casimir. In the <110> and <111> directions the thermal conductivity will be as predicted by Casimir.
The two diamonds used were both type II a.'One
had a <i00> rod axis (IIa-9) while the other had a <110> rod axis (IIa-8). Both were annealed in an air strem at 800° C for 30 minutes and came out completely opaque. The surfaces were very pitted and rough and this damage could clearly be seen with the naked eye.
The results of thermal conductivity experiments on both diamonds are shown in figure 1. To clearly show any deviation from the behaviour predicted by Casimir for diffuse boundary scattering, K/T has been plotted against T.
Fig. 1 : A plot of K/T3 vs T for two different
diamonds with very rough surfaces. Diamond IIa-8 has a <110> rod axis whereas diamond IIa-9 has a <100> rod axis.
The theoretical Casimir value, corrected for the finite length of the diamonds, has been calculated
and drawn in for each diamond. K / T ~ is a measure of the phonon mean free path and should be a constant for perfectly diffuse boundary scattering (up to about 10 K in diamond). However, the phonon mean free path would be expected to increase at the lowest temperatures because of some specular reflec- tion and to decrease at higher temperatures because of the onset of point defect impurity scattering
(proportional to w4). Results of measurements on diamond IIa-8 show a mean free path with this be- haviour (see Run 1 in fig. 1). However, the results for diamond IIa-9 (Run 2) gave a K / T ~ value ( a phonon mean free path) much greater than that pre- dicted by Casimir. Specular reflection can defini- tely not account for this much larger than expected conductivity because of the very rough pitted sur- face. Also, specular reflection increases with de- creasing temperature whereas here the conductivity is nearly temperature independent between 2 and 10 K. It has only been in diamonds with polished surfaces that a nearly temperature independent boun- dary scattering with a phonon mean free path 2 to 3 times the Casimir value has been observed 151. To explain that behaviour, it was suggested than in polished diamonds the phonons interact with the strain field associated with the network of micro- craks underneath the abraded surface. The rough surface was actually polished off diamond IIa-9 and the results show a nearly temperature indepen- dent mean free path that is about 2.5 times the Casimir value between 2 and 10 K (see Run 3 in figure 2). It is interesting to note here that in diamonds with polished surfaces the conductivity in the <loo> and <]lo> directions is approximately the same. This diamond was then thinned from a mean side dimension of 1.30 mm for Run 2 to a side di- mension of 0.84 mm and the surface was roughened thermally. This resulted in a complete removal of the strain field. Run 4 in fig. 2 shows the measu- red K / T ~ curve. The similarity between Run 2 and Run 4 can clearly be seen. In both cases the phonon mean free path is about 70 percent larger than the Casimir value at 4 K. This compares with a meanfree path that is actually less than the Casimir value at 4 K for the diamond (IIa-8) with the <110> rod axis.
lished surface can account for the observedincrease in conductivity in Runs 2 and 4 for diamond IIa-9, the McCurdy et al. hypothesis of phonon focusing in the <loo> direction in diamond is very probably the correct explanation.
KIT' ~ I O ~ W C ~ ' K - ' ~ Ila- 9 0 run 2
8, a run 3
3
Fig.2 : A plot of KIT vs T for diamond IIa-9.For Runs 2 and 4 the diamond had a very rough surface and for Run 3 a polished surface.The side dimension of the diamond for Runs 2,3 and 4 were 1.30 mm,
1.07 mm and 0.84 mm respectively. mo- 600-• 500- 4 . -300 200
However, quantitavely there does seem to be a dis- crepancy between the experimentally observed enhan- cement of the conductivity and that theoretically predicted. This difference (70 percent compared to 38 percent) is too largeto be accounted for by experimental errors, which are estimated to be only about 7 percent. h O A run 4
Paa
a A A A A O A 8 0 A A.
.
oQ' O % 0 0 0 A A 0 0 A 0 0 0\..*
0 0 A A A*....*
0Cosirnir value for run. "- 2'
0
for run 3 / for run -4
References
111
Casimir, H.B.G., Physica (Utr.)5
(1938) 495. /2/ Berman, R., Simon, F.E. and Ziman, J.M., Pr0c.RSoc.Lond.
A220
(1953) 171./ 3 / Berman, R., Foster, E.L. and Ziman, J.M., Proc. R.Soc.Lond.
A231
(1955) 130.141 McCurdy, A.K., Maris, H.J. and Elbaum, C., Phys Rev. B
2
(1970) 4077./5/ Vandersande, J.W., Phys. Rev. B
