Determination of the phonon density of states from specific heat data

Solid State Communications, Vol. 74, No. 8, pp. 843-845, 1990. Printed in Great Britain.

0038-1098/90 $3.00 + .00 Pergamon Press plc

D E T E R M I N A T I O N OF THE P H O N O N DENSITY OF STATES F R O M SPECIFIC HEAT DATA W.C. Kok Department of Physics, National University of Singapore, Lower Kent Ridge Road, Singapore 0511, Republic of Singapore

(Received 26 June 1989 by M.F. Collins; in revised form 18 January 1990) The phonon density of states is calculated from specific heat data using a minimization procedure for which the density of states is nonnegative at all frequencies. The method is applied to amorphous germanium and annealed selenium glass.

1. I N T R O D U C T I O N THERE have been extensive studies relating to the extraction of information about the density of states from specific heat measurements [1-3]. The linear integral equation connecting these two physical quantities is a Fredholm type equation of the first kind. Hovi and Pautamo [4] have employed a simple method of discretizing the integral equation and solving the resulting set of linear equations to calculate the density of states of ionic crystals from measurements of the specific heat. Since such specific heat data inevitably contain experimental errors, the investigation reduces to an extreme value problem. A solution of the corresponding set of linear simultaneous equations obtained using the method of Lagrangian multipliers leads to a density of states which sometimes takes on negative values at certain frequencies. In Section 2, we modify their basic technique to ensure that the density of states is non-negative for the whole range of frequencies and the minimization problem is solved using a steepest descent iterative procedure. We have applied the method to crystalline copper which has a well-known density of states [5]. The extraction of the density of states from specific heat values computed using this density of states of copper shows a peak at around 5.7 THz which is in the neighbourhood of the broad peak occurring in the original density of states used in the calculations. However, it is not possible to resolve the more detailed features of the spectrum such as the presence of a sharp second peak. The reason is that sharp peaks in the density of states encompass very narrow domains in frequency space; with the present method based on integration (essentially a smoothing process), the integral representing the specific heat is discretized by sampling the density of states at regular frequency intervals, therefore sharp peaks occurring over small

frequency ranges make relatively small contributions and one would not be able to reproduce these peaks in the density of states nor the cusps associated with van Hove singularities by inversion of specific heat data. Accordingly, in Section 3, the density of states is calculated for two noncrystalline materials (amorphous germanium and annealed selenium glass) which are expected to have smoother densities of states with less structure arising from structural disorder. 2. M E T H O D FOR THE N U M E R I C A L INVERSION OF SPECIFIC HEAT DATA The specific heat due to lattice vibrations is given by the usual formula

c(T)

= kB

i

p(v)

(hv/kB T) 2 ehv/kBr (ehV/kBr- 1)2 dv

(1)

0

where p(v) is the number of oscillators in the frequency interval v and v + dr. This is a linear integral equation for p(v) when experimental data on the specific heat c(T) are available. Applying the trapezoidal rule to the RHS, the above equation becomes

c(T)

m (hvi/k ~ T) z ehV,/kBr = ks L p(vi) (ehV,tk,r Av 1)2 i=l

(2)

where the frequency range is divided into m equal intervals of size Av. If cj = c(Tj)/kB and x~ = p(vi)Av, then

ajkXk = Cj ( j = 1,2 . . . . .

n)

(3)

k=l

or, in matrix form, A X = C where

(hvk/kB Tj) 2 ehVk/kn~ (ehVk/kArj_ 1)2 (3) is a set of linear equations in xk, which is the

843

844

D E T E R M I N A T I O N OF T H E P H O N O N D E N S I T Y

Vol. 74, No. 8

density of states apart from a constant factor. Since the specific heat measurements are subject to experimental errors, we define V =AX-C

and minimize V r V. A direct calculation of the xk's from known values of the specific heat sometimes yields negative values of xk for some k values [4] and in order to ensure non-negative values in such cases, we may replace the elements of X by

y~,

=

0

x~

50

100

150

200

250

300

350

ENER6Y (cm -1 )

and then minimize V r V with respect to real values of Y~, Y2, . • • , Ym by the standard method of steepest descent. (b)

3. R E S U L T S To illustrate the kind of results that can be expected from the above technique, we apply it to deduce the density of states (DOS) of amorphous germanium, for which the specific heat data are available and the DOS is also known. As the difference between the specific heats at constant pressure and at constant volume is of a few per cent at room temperature and smaller at lower temperatures, the data on Cp has been used in the calculations. The density of states of amorphous germanium has been computed in the frequency range from 0 to I = 500 c m - 1at intervals of 25cm -~ using data on the specific heats from two sources. Specific heats Cp in the high temperature range from 230 K to 450 K (at intervals of 20 K) for amorphous (vapor-deposited) germanium are obtained from measurements by Chen and Turnbull [6] and Cp in the low temperature range from 2.5 K to 3 0 K at invervals of 2.5 K have been taken from the data of King and Phillips [6]. To satisfy the conditions that the density of states vanishes in the limits of zero frequency and high frequency, the starting values for Yk are obtained from the curve sin2(2rCVk/l) for which p(v) = 0 at v = 0 and at v = l, and at low frequencies p(v) oc v 2. The steepest descent iterative procedure gives values ofyk from which the corresponding values xk are obtained as shown in Fig. l(a). The overall

I

50

I

I

100

I

I

,

I

I

1

150 200 25o ENERGY (cm -1 )

I

I

3o0

t

350

Fig. I. (a) x Calculated density of states of amorphous germanium; - Density of states of crystalline Ge is broadened by a convolution with a Gaussian factor of half-width 25 cm t [8]. (b) The reduced R a m a n spectrum of amorphous Ge from room temperature data [8].

shape of the computed DOS curve is in agreement with a Gaussian-broadened DOS of germanium [8]. The figure shows that the peaks in the DOS occur at almost the same frequencies in both cases. The reduced Raman spectrum of amorphous germanium is shown in Fig. l(b) for comparison. Because of the exponential factors in the integrand in equation (1), the contri-

Table 1. Peaks in the density o f states Calculated frequencies o f annealed Se glass 1.1THz (37cm - I ) 4 . 1 T H z (137cm i) Observed frequencies in amorphous Se [10] 27cm -~, 74cm -~ 127cm -1

6.2 T H z and 6.6 T H z (207 c m - i and 220 c m - i ) 235cm -j and 255cm (peak with shoulder)

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D E T E R M I N A T I O N OF THE PHONON DENSITY

to various vibrational modes are [9] 0-70 cm ~ (torsion), 85-110 cm ~ (liberation), 115-160 cm- ~ (bending) and 220-225cm -~ (stretching). It is generally recognised that amorphous Se contains the different allotropic forms of crystalline Se. The details of the present numerical calculations on annealed Se glass reveal peaks in the density of states which lie in the above frequency ranges for vibrational modes of crystalline Se. In Table 1, the frequencies of the calculated peaks of annealed Se glass are compared with the observed peaks of Raman spectra of amorphous Se; it is evident that some correlation exists as the frequencies characteristic of rings and chains of Se are susceptible to the local environment [8]. It should be noted that although some information on the density of states may be extracted from specific heat measurements, this is ultimately dependent on the experimental data available.

•

0.60

0.55

0.50

"~

0.45

.•-'D

0.40

o.3s

o

0.30

:=

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••,,.•.

o 0.25

020

Acknowledgement - T h i s work is supported by a grant from the National University of Singapore•

0.15

0.10

REFERENCES

0.05

1. 0[

0

t

,

,

,

,

2

~,

6

8

10

•

,

12

Frequency (THz}

2. Fig. 2. Calculated density of states of annealed Se glass•

3. 4.

bution to the summation in (3) from ajk is insignificant when the exponent

5.

hvk/kB Tj >> 1.

6.

(4)

This implies that at low temperatures, the specific heat is largely determined by the low frequency part of the vibrational spectrum and it is only at high temperatures that a substantial portion of the spectrum comes into play• Hence the availability of the relevant specific heat measurements in the appropriate temperature range (preferably from the same study) is essential in order to deduce the detailed features of the entire spectrum• .We next consider the DOS of annealed Se glass obtained with experimental data [7] on the specific heat Cp from 5 K to 200 K. The density of states is calculated in the frequency range from 0 to l = 12 THz at intervals of 0.1 THz. The results are shown in Fig. 2 with V TV = 0.269 j2K-2mol 2 (after 350 iterations). In crystalline Se, the frequency ranges corresponding

7. 8.

9. 10.

E.W. Montroll, J. Chem. Phys. 10, 218 (1942); ibid 11, 481 (1943)• J.W. Loram, J. Phys. C: Solid State Physics 19, 6113 (1986) and references therein. R.G. Chambers, Proc. Phys. Soc. 78, 941 (1961)• V. Hovi & Y. Pautamo, Annales Academiae Scientiarum Fennicae, Series A VI 68, 3 (1961). R.M. Nicklow, G. Gilat, H.G. Smith, L.J. Raubenheimer & M.K. Wilkinson, Phys. Rev. 164, 922 (1967). H.S. Chen & D. Turnbull, J. Appl. Phys. 42, 4214 (1969); C.N. King & W.A. Phillips, Phys. Rev. Lett. 32, 538 (1974). S.S. Chang & A.B. Bestul, J. Chem. Thermodynamics 6, 325 (1974). M.A. Brodsky, in Light Scattering in Solids L (Edited by M. Cardona), Springer, Berlin (1983); J.E. Smith, Jr., M.H. Brodsky, B.L. Crowder, M.I. Nathan & A. Pinczuk, Phys. Rev. Letts. 26, 642 (1971). H. Bilz & W. Kress, Phonon Dispersion Relations in Insulators, Springer Series in Solid-State Sciences 10, Springer, Berlin (1979). I. Watanabe & T. Yamamoto, Jpn J. Appl. Phys. 24, 1282 (1985); K. Nagata, T. Ishikawa & Y. Miyamoto, Jpn J. Appl. Phys. 24, 1171 (1985); V.N. Bogomolov, V.V. Poborchy, S.G. Romanov & S.I. Shagin, J. Phys. C: Solid State Phys. 18, L313 (1985); G. Lucovsky & C.K. Wong, J. Non Crystalline Solids 75, 51 (1985).