THERMAL DIFFUSIVITY OF VITREOUS SILICA (16 mK < T < 2.3 K)

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THERMAL DIFFUSIVITY OF VITREOUS SILICA (16 mK < T < 2.3 K)

J. Lewis, J. Lasjaunias

To cite this version:

J. Lewis, J. Lasjaunias. THERMAL DIFFUSIVITY OF VITREOUS SILICA (16 mK < T < 2.3 K). Journal de Physique Colloques, 1978, 39 (C6), pp.C6-965-C6-966. �10.1051/jphyscol:19786428�.

�jpa-00217900�

JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-965

THERMAL DIFFUSIVITY OF VITREOUS SILICA (l6 mK < T < 2.3 K)

J.E. Lewis and J.C. Lasjaunias

Centre de Reoherahes sur les TrSs Basses Temperatures, C.N.R.S., Cedex 166, 38042 Grenoble, FRANCE.

Résumé.- La diffusivité thermique de Si02 vitreux (< 1,5 ppm OH) entre 16 mK < T < 2,3 K est en excellent accord avec les valeurs dérivées de la conduction thermique et la chaleur spécifique du même échantillon obtenues par des techniques indépendantes, et correspond pour T < 200 mK à : C ce x1'3 5 - °»05. Ceci implique l'existence d'un gap à très basse énergie dans la densité d'états des systèmes à deux niveaux.

Abstract.- The thermal diffusivity of vitreous Si02 (<1.5 ppm OH) between 16 mK < T < 2.3 K is in excellent agreement with values derived from thermal conductivity and specific heat data on the same sample by independant techniques, and indicates (T < 200 mK) C <*..T ~ °'05. This implies the existence of a gap at very low energy in the density states of the two level systems.

The excess specific heat in vitreous systems can be attributed to two level systems having a broad distribution in their energy splittings /l/.

The initial assumption that their density of states is constant at low energies leads to a linear spe- cific heat at low temperatures. Instead, recent /2/

measurements follow T1 power law for T < 0.5 K with VH1.3 (Si02) and ^0.45 (B203). Our results confirm this behavior, which, however, are not in contradiction with the general tunnelling model where the existence of a cut-off of the coupling between wells leads to a gap at low energy in the density states /3/, and hence a non-linear specific heat.

The sample (Suprasil-W) and experimental set up have been fully described /4/, only the following changes being made. The heat pulse CVJ-10 ms) was now delivred via an automatic timing circuit and, instead of the carbon resistors, doped silicon resistance thermometers of small thermal mass and negligibly short response time were used, one ther- mometer serving to measure the transient, the second any steady state gradient in the sample. Noise levels required at AT/T ^ 5-10 %. Temperature transients to and beyond the temperature maximum (t < 1 s) were displayed on a storage scope, a chart recorder following the slower exponential decay to the base- line, whose drift was negligible, due to the large thermal mass of the sink.

The solution /5/ for one dimensional heat flow in a sample with a "radiative" heat leak at x = L only, heated by a pulse at x = 0 is

CO

8 = H f («2 + h, 2) ( a2 + h* + h '2) "1 c o s a x / L exp

MC I n n n

- a2 D t / L2 (1)

where a are the roots of a tan a = h', the heat n n

leak factor defined by sample thermal resistance/

link thermal resistance = (Lg/AgKg) (A.K / L ) . For x < 0.5L and 0 < Dt/L2 < 0.1, the heat leak can be neglected and (1) reduces to :

6 = 9 (xe°*5/v^D)t"1/2 exp-x2/4Dt (2) m

i.e. ln(0i/t) is linear in l/t with slope m = -x2/4D.

Various criteria /6/ indicate the finite heating pulses and addenda have negligible effect on the profiles. At all temperatures In 6i/t is linear in l/t over about a decade of time (typically at 123 mK for 0.1 < t < 1 s ) . The upper limit is due to the increasing contribution of the leak, the lower because In Q\/i is initially superlinear, i.e. the profile rises faster than given by (2). This effect was definitely established and possibly indicates a time dependent specific heat. This exponential de- cay of the transient for t > 1 s can be computer analysed using (1) and gives h'. Eq. (1) well des- cribes the profiles (t > 0.1 s) using these values.

Figure 1, which includes several different experimental runs, shows the results for D. The curve is the behavior of K/pC derived from thermal conductivity and specific heat data of Lasjaunias et al. Ill on the same sample and where the thermal conductivity of Spectrosil B, good to within 10 %, Permanent Address : Department of Physics, SUNY

Plattsburgh, NY 12901, U.S.A.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786428

as K is not very sample dependent (Zeller and Pohl /7/), has been used above 0.5 K, because data was not available for our sample. The two sets of data agree closely, and for T < 200 mK, D

TO'^' '

which implies C

T" " ' , i.e. v

0.35 + ^0.05.

A linear variation for C below 200

is exclued.

10 20 40 100 1000

Temperature (mK)

Fig.

+ I

Figure 2,~hows In h' vs In T. The T rise in h' for T

100

and the putative T- drop above

K can be best understood as a heat link that is dominated by Kapitza contact resistance at low tem- peratures (KS

T~

%

^{T ~ )} and by normal metallic heat flow in the higher range (KS

T~

%

^{T) as}

h'

%/KS. This'reasonable behavior indicates that

correctly describes the transient. A naive ana- lysis of this link gives a contact area of 5

cm2 and a value 0.28 watt1cm.K for KCU, a factor 2- 4 smaller than accepted but possible, as the wire is heavily cold worked.

0.1

I

I

10 20 50 100 1000

Temperature (mK)

Fig. 2

with a constant density of states. However the gene- ral analysis of the two level model 131 results in a specific heat that deviates from the lineat T law when a complete calculation is done using double

integration over the parameters E (asymmetry energy) and X (coupling energy) varying from zero to infi- nity. This deviation is due to existence of a cut -

off in the high values of the potential barrierV max and consequently a gap in the low energy part of the density of states. The existence of the gap is pro-

1

+v ven by the observation of such a dependance in T both in this study

0.1 s), and in earlierstu- dies

several seconds) where the results have now been fitted, together with the thermal conducti- vity, using the generalized two level model leading to a value of about 15

for the gap 181.

We wish to thank R. Maynard, D. Thoulouze, and R. Ramal for their permanent interest in this topic, G. Schumacher for the use of his dilution refrige- rator, and B. Picot for his experimental assistance.

One of the authors (J.E.L.) would like to thank the Centre de Recherches sur les TrSs BassesTempSraCures for the hospitality extended to him during his sabba tical year 1976-77.

References

/I/ Anderson P.W., Halperin B.I. and Varma C.M., Philos. Mag. _11 (1972)

121 Lasjaunias J.C., Ravex A., Vandorpe M. and Hunklinger S., Solid State Commun 17 (1975) 1045 Lasjaunias J.C., Thoulouze

and Pernot F., Solid, State Commun 5 (1974) 957.

131 Phillips W.A., J. Low Temp. Phys. 1 (1972) 351 Rammal R., Thesis Grenoble (1977)(unpublished) Maynard R. and Rammal R., to be published.

141 Lasjaunias J.C., Picot B., Ravex A., ThoulouzeD.

and Vandorpe M., Cryogenics 17 ⁽¹⁹⁷⁷⁾

151 Carslaw H.S. and Jaeger J.C., Conduction of Heat

in Solids, 2nd Education, Oxford (1959).

I61 Alterovitz S., Deutscher G. and Gershenson M., J. Appl. Phys. 46 (1975)' 3637

Cape,J.A. and L=an,G.W.,J.Appl.Phys. 2-(.1963) 1909

171 Zeller R.C. and Pohl R.O., Phys. Rev.

(1971) 2029.

181 Vandorpe M., Lasjaunias J. C., to be published.

Clearly the present results together with

the previous results of Lasjaunias s., ^confirm

that C

with v 0.3, a result not compatible