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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.

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

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)

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 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 **

### lo-'

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### -

calculations(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 **

### lo3

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)

References

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

**Bi **

(1971)
2029
/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
**2**

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

*2 *

^{(19721 1 }

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

*9 *

(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.

### 43

(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.

*2 *

(1978) 81
/8/ Wu, C.T. and Luo, H.L., J. Non Cryst.Solids

### 12

(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