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Submitted on 1 Jan 1978
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THE THERMAL PROPERTIES OF AMORPHOUS ARSENIC AT LOW TEMPERATURES
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.
JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-978
THE THERMAL PROPERTIES OF AMORPHOUS ARSENIC AT LOW TEMPERATURES
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)
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 http://dx.doi.org/10.1051/jphyscol:19786433
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
2,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
lo3m-'~-' 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.
/2/ Jzckle, J., Pich6, L., Arnold, W. and ~unklin- ger, S., J. Non Cryst. Solids
2(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.
/4/ Greaves, G.N. and Davis, E.A., Philos. Mag.
/ 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.
/ 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.
/8/ Wu, C.T. and Luo, H.L., J. Non Cryst.Solids
/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