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D. Jones, N. Thomas, W. Phillips

To cite this version:

D. Jones, N. Thomas, W. Phillips. THE THERMAL PROPERTIES OF AMORPHOUS ARSENIC AT LOW TEMPERATURES. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-978-C6-979.

�10.1051/jphyscol:19786433�. �jpa-00217906�


JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-978


D.P. Jones, N. Thomas and W.A. Phillips

Cavendish Laboratory, Madingley Road, Cambridge, England

Résumé.- Les mesures de la chaleur spécifique C de l'arsenic amorphe au-dessus de 0,3 K sont compa- tibles avec l'absence d'un terme linéaire, tandis que la conduction thermique K varie en T3 entre 0,1 et 0,25 K. Par contraste avec d'autres verres, ces résultats indiquent l'absence possible de systèmes à deux niveaux. Cependant, jusqu'à 15 K l'arsenic amorphe se comporte comme d'autres maté- riaux vitreux : C/T présente un maximum à 5 K et K montre une dépendance faible en T.

Abstract.- Measurements of the specific heat C of bulk a-As down to 0.3 K are consistent with the absence of a linear term, whilst the thermal conductivity K varies as T3 between 0.1 and 0.25 K.

These results are in contrast with other amorphous solids, and point to the possible absence of two- level systems in a-As. Up to 15 K, however, a-As behaves like other glasses : C/T3 shows a peak at 5 K and K has a very weak T dependence.

Measurements of the thermal conductivity K and specific heat C of amorphous insulators below 1 K have revealed deviations from crystalline beha- viour /l/. C has an additional term linear in tem- perature T, and is proportional to T2 . Both these properties, and ultrasonic measurements /2/, have been explained by the existence of a constant den- sity fo states of two-level systems which contribute to C and scatter sound waves. These states are thou- ght to arise by the tunnelling of atoms through a barrier between two potential minima /3/:a uniform dis-

tribution of barrier height and of the asymmetry of the double potential well leads to a slowly increa- sing density of states in the appropriate energy range. C and K are similar for nearly all bulk glasses investigated so far, and this may be because their structures are broadly similar, containing 2-fold co-ordinated atoms. Amorphous arsenic, howe- ver, is 3-fold co-ordinated Ik/ which may imply less freedom for tunnelling.

Samples of bulk a-As, made by the sublimation of crystalline As, were obtained from Mining and Chemical Products Ltd. C was measured from 0.3 to 20 K using a pulse technique /5/, and K, from 0.1 to 15 K and at 77 K, using a two-heater technique/6/.

The heat capacity results are plotted in fi- gure 1 as C/T3 against T. These are in reasonable agreement with those published by Lannin et^ sd 111 above 2 K. The curve shows a pronounced maximum at 5 K ; this is taken as evidence for a deviation of the phonon density of states g((o) from an u) depen- dence at about 10 cm . The results below 1 K are consistent with no linear term in C /9/, and such

Fig. 1 : Specific Heat of As plotted as C/T3 versus T. (a) a-As, this work ; (b) thin film a-As, (c) rhombohedral As (lattice contribution) ; both from IS/.

a term, if it exists, must be smaller than 0.1 T UJ g K . I f all the thermal energy is contained in sound waves, v, the average sound velocity, ta- kes the value 1.6 x 10! i s " .

The thermal conductivity results are plotted in figure 2. Below 0.25 K, K i s proportional to T3'0 - 0'2 but at higher temperatures shows the pla- teau observed in other amorphous materials /I/.

Assuming a Debye model, with cut-off frequency to_, K can be written as

K(T) = ^ J C(o),T) v H(u,T) do) (1)

J o

C(o),T) is the contribution of phonon of frequency oi to the total heat capacity at temperature T, and Z is the phonon mean free path.

As K = I3 below 0.25 K, this points to a cons- tant mean free path L of about 25 ym. This is much smaller than the sample thickness (1 mm) and so

Article published online by EDP Sciences and available at


must be due to microscopic features. Inspection of 'cleaved' surfaces of a-As with a scanning electron microscope revealed small holes about 0.5 pm across.

The number was estimated as 1017


giving a mean free path in reasonable agreement with experiment.

The thermal conductivity of vitreous silica below 100 K has been explained in terms of the scat- tering of phonons by two-level systems with a density of states of the form n(~) = no + n2E2


where E is the energy splitting of the state 161. In a-As, no (from K) must be less than 3 x 10'~e~-'cm-~ (assu- ming the coupling constant is the same as in Si02), consistent with the absence of a linear term in C.

Figure 2 a calculation with no = 0 and n2 = 8 x


e ~ - ~ c m - ~ : n2 would contribute at most 10% to the measured heat capacity.

Fig. 2 : Thermal conductivity K of a-As. Full cir- cles


experimental results. Lines



(see text) : solid curve


modelling dispersion, short dashes


structure scattering, long dashes


scattering from two-level systems with a quadratic density of states.

Alternatively, the data can be fitted by a mean free path given by

-1 61.0 -4

(2) The second term (structure scattering) arises from spatial variations in density and elastic constanLs /lo/. The calculated k with D = 1.9 x


m-'~-' and R,, = 1.5 nm is shown in figure 2.

Equation I takes no account of the real form of the density of phonon states g(w) or of any varia- tion of the propagating character of phonons with frequency. The eEf ect of this "dispersion" can be

modelled by using a cut-off frequency for the trans- verse modes at 10 em-', and splitting equation 1 in- to longitudinal and transverse contributions. Taking the ratio of sound velocities to be 2 and R = 25 l.~m, the curve shown in figure 2 is obtained. This indi- cates that dispersion alone cannot account for the plateau in K, but it does mean that any scattering can be less strong. However, if equation 2 is used with a cut-off at 10 cm-l, the value of D required to fit the data is reduced by only 20X. Dispersion is therefore not of great importance.

It is clear from the curves in figure 2 that almost any form of .9 which increases sufficiently rapidly with w will fit the data : hence the scatte- ring mechanism in the plateau region cannot be spe- cified uniquely. However, the results are consistent with the total absence of two-level systems, and if any such states are present at low energies (&/k %

0.5 K) then they are far fewer in a-As than in, say, vitreous silica.

We are grateful to the Science Research Council for supporting this research, and for the award of two Research Studentships (N.T. and D.P.J)


/I / Zeller, R.C. and Pohl, R.O., Phys.Rev.


(1971) 2029

/2/ Jzckle, J., Pich6, L., Arnold, W. and ~unklin- ger, S., J. Non Cryst. Solids


(1976) 365 /3/ Phillips, W.A., J. Low Temp.Phys. 7 (1972) 351

Anderson, P.W., ~alperin, B.I. and-varma, C.M., Philos. Mag.


(19721 1

/4/ Greaves, G.N. and Davis, E.A., Philos. Mag.


(1974) 1201

/ 5 / Bachmann, R., Desalvo, F.J., Geballe, T.H., Greene, R.L., Howard, R.E., King, C.N., Kirsch, H.C., Lee, K.N., Schwall, R.E., Thomas, H.U.

and Zubeck, R.B., Rev.Sci.Instrum.


(1972) 205

/ 6 / Zaitlin, M.P. and Anderson, A.C., Phys.Rev.

B g (1975) 4475

/7/ Lannin, J.S., Eno, H.F. and Luo, H.L., Solid State Commun.


(1978) 81

/8/ Wu, C.T. and Luo, H.L., J. Non Cryst.Solids


(1974) 437

/9/ Jones, D.P., Thomas, N. and Phillips, W.A., to be published

/lo/ Jzckle, J., Proc. Int. Conf. on "The Physics of Non-crystalline Solids IV", Clausthal-Zeller- feld, 1976 (ed. G.H. Frischat), p. 568




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