Infrared Physics, 1961, Vol. 1, pp. 11 l-l 15. Pergamon Press Ltd. Printed in Great Britaio

INFRARED ABSORPTION IN GALLIUM ARSENIDE

T. S. Moss and T. D. F. HAWKINS

Royal Aircraft Establishment, Famborough, Hams., England

(Received 14 December 1960)

Abstract-Measurements of absorption constant covering the range 1 cm-r to l(r cm-l have been made on single crystal gallium arsenide. The absorption edge is very steep up to 4000 cm-l, where there is a knee beyond which the absorption increases relatively slowly with photon energy. The energy bands have been calculated using Kane’s theory. From these we have obtained a theoretical absorption curve which shows very good agreement with the experimental data.

INTRODUCTION

GALLIUM arsenide is a semiconductor with many interesting properties and it has been studied quite extensively since the III-V compounds were first described by Welker. (l)

In recent months it has become of great technological interest for use in devices such as transistors, Esaki and parametric diodes. It is also a very promising solar battery material, @) and for use in this field in particular it is necessary to have detailed optical data up to high absorption levels.

Original measurements of absorption, near the edge, were made by Oswald and Schade. @ These results (which reached an absorption level of only 100 cm-l) have not been extended to any shorter wavelengths, although papers on long wavelength properties such as free carrier absorption@) and Reststrahlen,(5) have been published recently.

EXPERIMENTAL DETAILS AND RESULTS

The measurements were made on very pure single crystal material (3 x 1016 electrons cm-S) kindly provided by the Plessey Co. A tungsten lamp and Leiss double monochroma- tor (fitted with flint glass prisms) provided the radiation. The resolution was 5 x 1O-9 eV and great care was taken to ensure spectral purity. The detector was an infrared photomulti- plier. Satisfactory measurements could be made with specimen insertion losses up to lo3 : 1 in the region of rapidly varying absorption and up to 10 4 : 1 on the flat part of the absorption curve.

The material was cut, ground, and polished with diamond paste. Specimens 7 I_L to 1 cm in thickness were used; those over 50 p were self supporting and their thickness was measured by direct gauging, those below 50 p were glued to glass backing plates and were measured interferometrically using wavelengths in the 5-15 p region.

The results obtained for the absorption coefficient (K) are shown in Fig. 1. It will be seen that the absorption edge is very steep, K rising rapidly from 4 cm-l to 4000 cm-l. The main part of the absorption edge is exponential, as has now been observed for many other solids. @) The slope is 100 eV-l---i.e. three times steeper than would result from a simple l/kT Boltzman factor, and about ten times steeper than for silicon. (2)

111

112 7;. S. Moss and T. D. F. HAWKINS

At higher levels the absorption increases only slowly with energy. At long wavelengths there is a “tail” on the absorption curve and k: does not fall below ~1 cm-l within the range of measurement, although the computed free carrier absorption@) is only 0.04 cm-’ at 1 I*.

0 0 Experimental results -----Theoretlcol curve

1 I I I 1 I.4 I.5 I

Photon energy, eV

FIG. 1. Absorption in gallium arsenide.

CALCULATION OF ENERGY BANDS AND ABSORPTION COEFFICIENT

The energy bands have been computed following Kane. (‘1 The E-k curves are given by:

k2P2 = E&E, - E,) (ES + A)/(E, + 2413) 0)

where E, is the energy gap, A is the spin orbit splitting and

P= = E,& + AME, + U/3) 2m,* atomic unitst (2)

where m,* is the electron mass at the bottom of the conduction band. The parameters used are:

m,* = 0.072 rn,@t A = 0.33 eV (9)

t In atomic units, m = e = h/2rr = 1. The unit of length is the first Bohr orbit radius, 0.53 A.

Infrared absorption in gallium arsenide 113

and Eg = 1.4 eV (estimated room temperature value) giving P2 = O-38 atomic units.

Equation (1) gives results for the conduction band (EC) the light hole band (ES) and the split-off valence band (Es). They are plotted against k2 in Fig. 2. Also shown is the heavy hole band (E,), drawn simply to correspond to a mass m, = 0.68 m,. (lo) It will be seen that

e

2 4 6 8 IO 12

-0.3 -

94-

FIG. 2. E-k curves for gallium arsenide.

there is little curvature of the conduction band (in the plot against k2), but the light hole band is quite non-parabolic. The masses of the light hole and split-off bands can be found from the slopes of the curves as k +- 0. They are:

m2 = O-085 m, m3 = 0.25 m,

The absorption coefficient, assuming direct, vertical, transitions is given by: (‘)

114 T.S.Moss andT.D,F. HAWKENS

K = -4$ ~~~p~ atomic units ss’

where n is the refractive index, I&, the photon energy and the summation is over the three valance bands (j = 1, 2, 3).

The optical matrix element is:

MF = (2~~~3) ((A,Cf + C,Aj)2 + (A& - B&J’) (4)

where the bracketed term-which is always near unity-is computed fur each k vaIue from the coefficients :

where the s&ix i refers to bands I&, I?& and Es and N is a no~a~i2ing factor such that

A” + I32 4 C‘J = 1

For the parabolic El band, d, = 0, B, = 1 and C, = 0. The density of states pi is given in terms of the slopes of the E-k curves by

The absorption ~oe~cient plotted from equation (3) is shown in Fig. 1, The agreement with the experimental data is seen to be very good, particularly at short wavelen~hs where the calculated curve does not depend on the use of any arbitrary constants or adjustable parameters. In the nejgbbourhood of the absorption edge the tit has been improved by sliding the curve slightly sideways. This process gives a value for the energy gap at 292 “K of Eg = 1.41 eV, which is considered ta be rather more accurate than the value of 1.4 eV assumed in the analysis,

The above absorption theory is essentially that for a perfect Ga-As lattice-with no

perturbations due to thermal vibrations or crystalline irregularities.oQ The presence of these could well explain the slope of the edge observed under the experimentai conditions.

The steepness of the edge and the good agreement between the magnitudes of the theo- retical and experimental absorption coe~~ients co&m that the assumptions of vertical transitions and band extrema at kOO0 are correct.

REFERENCES

1. Wnmq H., Z. Naturf. 7a, 744 (1952) and &a, 248 (1953). 2. Moss, T. S., Solicf &ate Efectran. 2, 222 (196% 3. OSWALD Vm F., and R. S~XADE, 2. Naturf, 9a, 611 (1954).

Infrared absorption in gallium arsenide 115

4. SPITZER, W. G., and J. M. WHELAN, P/y. Rev. 114, 59 (1959). 5. FRAY, S. J., F. A. JOHNSON, and J. E. QUAIUUNGTON, Proc. Phys. Sot. 77, 215 (1961). 6. Moss, T. S., Optical Properties of Semiconductors, pp. 30, 39. Butterworths, London; Academic Press,

New York (1959/1961). 7. KANE, E. O., J. Phys. Chem. Solids 1, 249 (1957). 8. Moss, T. S., and A, K. WALTON, Proc. Phys. Sot. 14, 131 (1959). 9. BRAUNSTEIN, R., J. Phys. Chem. Solids 8,280 (1959).

10. q EHRENRWCH, H., Phys. Rev. (in press). 11. PANENTER, R. H., Phys. Reo. 97, 587 (1955).