EXISTENCE OF A GAP IN THE LOW ENERGY DENSITY OF STATES OF VITREOUS SILICA

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EXISTENCE OF A GAP IN THE LOW ENERGY DENSITY OF STATES OF VITREOUS SILICA

J. Lasjaunias, R. Maynard, M. Vandorpe

To cite this version:

J. Lasjaunias, R. Maynard, M. Vandorpe. EXISTENCE OF A GAP IN THE LOW ENERGY DEN-

SITY OF STATES OF VITREOUS SILICA. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-

973-C6-975. �10.1051/jphyscol:19786431�. �jpa-00217904�

JOURNAL DE PHYSIQUE Colloque C6, supplement au n" 8, Tome 39, aout 1978, page C6-973

EXISTENCE OF A GAP IN THE LOW ENERGY DENSITY OF STATES OF VITREOUS SILICA

J.C. Lasjaunlas, R. Maynard and M. Vandorpe,

Centre de Reoherohes sur les Tv&s Basses TempSratures, C.N.R.S.^ B.P. 166 X, 28042 Grenoble Cedex, France.

Resume.- Un calcul complet utilisant deux variables aleatoires dans le modele tunnel des systemes a deux niveaux conduit a un "gap" dans la densite d'etats des defauts et permet 1'interpretation si- multanee d'une variation en T de la chaleur specifique et de ^ T2 de la conduction thermique, variations deja observees dans certains systemes vitreux a tres basse temperature.

Abstract.- A complete calculation using two random variables in the two level tunneling model leads to a gap in the density of states of the defects and permits the simultaneous interpretation of a variation in T1 + v for the specific heat and 'VT2 for the thermal conductivity already observed in some glasses at very low temperatures.

The specific heat variations in T1 (v = 0.3

<v 0.5) observed in some vitreous systems/l ,2/ cannot be interpreted by a constant density of two level defects /3,4/ leading to a linear temperature depen- dence. Moreover, if one uses a variation of the den- sity of states n(E) <v E which can take in account such a temperature dependence of C , this implies for the thermal conductivity, with the hypothesis of the resonant scattering of the phonon by the two le- vel defects, a variation in T2~ which is in disagre- ement with the experimental results varying in T1•9

°"1 over more than one decade.

Here we assume the existence of a gap in the density of states and it is then possible to fit in good agreement both the specific heat and the ther- mal conductivity variations. The calculation is ap- plied here to the case of vitreous silica Suprasil W.

a) Expression of the density of states ; the gap at low energy.- We use the same parameters and make the initial hypothesis of the main double well tunneling model of A.H.V.-P. /3,4/ ; e is the asyme- try energy between the ground states of the two wells, C = C e~ the coefficient of tunnel coupling

* ° • ,_ i 1 ^2mVo1/2J

for the two states, with X = y (-g—) ' d, where m is the mass of the tunneling particule and d the dis- tance between the two wells.

For eigenstates of the hamiltonian the ener- gy splitting is E = /e* + Cz. The fundamental hypo- thesis of the model is the uniformity of the distri- bution P(e,X) = n„. But the potential barrier V has a maximum value of about 1 eV corresponding to the activation energy for diffusion process which im -

plies a X , therefore a minimum value C . =

r max min

- X

e m a x = A f o r the splitting, that we will call the gap A.

Instead of the variables e and X, it is bet- ter to use E and C. By using properly this cut-off in the distribution P(e,X), we get the following density of states :

tribution of a two-level system of energy splitting E.

Three regimes must be considered from this expression : - kT » C . , which corresponds to E

E

» C . ; so n(E) ^ n Log -z ^ n Log E. The

m m o Cm.n o

density of states varies very slowly and the spe- cific heat is almost linear in T.

- kT > C . : the density of states varies very ra-

^ m m

pidly, and the specific heat faster than linearly, typically in T1*3, which fits the data as shown by figure 1.

- kT 4 C . , where the specific heat decreases ex- mxn

ponentially.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786431

c)

Thermal conductivity.- The expression of the relaxation time for the phonon $w of polariza- tion

j

resonantly scattered by the two-level systems is given by

:

for Hu

²

Cmin

=

0 for $w < Cmin,

where B is the coupling energy between the phonon j

and the defects,

p

the density of the material. At the lowest temperatures the relaxation time limited by the dimensions of the sample becomes predominant

v.

:

TS'

=

2 , ^{with L}

^%

0.75 D (D

=

diameter of the

J X L

cylinder in our case) following Berman et a1.161.

If we consider the coupling energy B being isotropic and varying slowly with E, and a mean Debye sound velocity < vD >, we get for the expres- sion of the thermal conductivity K

:

Two extreme regimes exist

:

- ^if T-4~)

<<

Ti1

:

we get for K the T3 law, cor- responding to the Casimir regime.

Fig. 1

:

Specific heat of Suprasil W, once sub -

tracted the phonons contribution fi!I3=8.0 ~ ~ e r ~ . ~ - ' . K-' indicated by the dashed line and almost negli- gible below

%

0.3 K. The solid line corresponds to the fit with the tunnel model.

- ^{If . r - ' ( w ) >} zl, as .r-'(w)

%

w, we recover the

ksual T2 variation.

The physical reason for which the thermal conductivity remains as T2 in the domain of the gap Cmin, while specific heat depaats strongly from the T-law, is due to the fact that the density of sta- tes figuring in the relaxation time of the phonons has been multiplied by a coupling factor proportion-

nal to C2/E2. This weighting factor minors the contribution of the weakly coupled two-level de- fects as compared to the specific heat.

The effect of the delay time has not been taken in account here

:

an alternative discussion of these effects will be presented at the same con- ference 171.

We have compared these expressions to our experimental results (figure 1 and

2)

obtained for the vitreous silica Suprasil W. One gets a good fit with a value of the gap Emin

=

16 mK, a density of two level systems n

⁼

1.45 x 10~~Ierg.cm~ and a

0

mean coupling constant

=

0.55 eV. An analysis of other glasses is in progress and gives gap energy of the same order of magnitude.

Fig. 2

:

Thermal conductivity of Suprasil W. The solid line corresponds to the fit with the tunnel model.

For Suprasil W, such a value of the gap seems to be compatible with the ultrasound data.

Indeed it will predict a departure from the stan- dart law of resonant interaction at frequencies less than 300 MHz

(2

16 mK), domain in frequency which has not been investigated so far.

This model of gap is directly related tothe

random tunneling variable-h and in our opinion

must be considered as a proof of the tunneling na-

- t u r e of t h e two l e v e l d e f e c t s i n g l a s s e s .

AKN0WLEDGMENTS.- We thank R. Ramma1 f o r h e l p f u l d i s c u s s i o n s .

References

/I /

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