Spatial dispersion in the dielectric constant of GaAs

Solid State Communications, Vol. 9, pp. 1421—1424, 1971.

Pergamon Press.

Printed in Great Britain

SPATIAL DISPERSION IN THE DIELECTRIC CONSTANT OF GaAS* Peter Y. Yu and Manuel Cardona Department of Physics, Brown University, Providence, Rhode Island, 02912

(Received 21 May 1971 by E. Burstein)

Birefringence in single crystal GaAs due to the ~ dependence (or spatial dispersion) of the dielectric constant has been measured from 0.95~to 1.8g. Considerable enhancement was observed as the absorption edge was approached. The observed temporal dispersion in the birefringence was found to agree well with the prediction of a 4-band model involving no adjustable parameters. Within a model a direct relation was found to exist between the frequency dependence of the spatial dispersion and the warping of the J = 3/2 valence bands.

2nAn

1. INTRODUCTION RECENTLY Pastrnak and Vedam 1 found that a

=

A

2= aq2.

~(a

1111

a1122

—

a1212)q (2)

cubic crystal like Si was birefringent when light propagated along the [110] direction. They point-

The coefficients

13k1 are in general functions of w and as a result one would expect An to show dispersion (in this paper dispersion will mean temporal dispersion unless stated otherwise) in the vicinity of dispersion centers such as an energy gap or an exciton line. The indirect absorption edge of Si, too weak to produce significant dispersion in a, prevents one from approaching any direct energy gaps. Thus Pastrnak and Vedam did not find any dispersion in An other than that due to the q dependence in equation (2) (a being constant in this case). We chose GaAs for our experiment because has a lowestHence directstrong E0 gap at 1.43 eV ~(at room ittemperature).

ed out that this phenomenon was due to the fact that the dielectric constant was ~ dependent (~being the wave vector of light). In this letter we present spatial-dispersion-induced birefringence data for GaAs. This effect shows strong temporal dispersion near the direct gap E0. The results can be interpreted on the basis of a previously used tour band model of the E0 — E0 + A0 gap of GaAs, slightly modified so as to include the warping of the valence bands. Expanding the dielectric tensor ~ (w, ~) as a function of ~ to 2 seconc! order in ~ one obtains: ~~(a,~)

=

~3(w) + i7~~k(W)qk + ~

q~ q1.

(1)

dispersion should appear in a when approaching this E0 gap.

The notation used in equation (1) is that of reference 2. The third rank tensor vanishes in crystals with inversion symmetry and also in crystals with the zincblende structure. The birefringence observed by Pastrnak and Vedam for [110] can be related to the a’s by the following equation: *

a

Supported by the National Science Foundation

The apparatus we used was similar to that of Pastrnak and Vedam. The sample is a parallelepiped cut from an undoped, single crystal of GaAs. The orientation of the sample faces was [001], [110], and [110] and each dimension was about one centimeter. All measurements were performed at room temperature. The birefringence

and the Army Research Office, Durham.

n(1TO) 1421

—

n(001) measured forZ

I [110]

and

1422

SPATIAL DISPERSION IN THE DIELECTRIC CONSTANT OF GaAs

28 I.? .6 1.5

Wavelength (micron) .4 1.5 1.2 1.1

20-

Wavelength

I

22

1.0

.8 17 .6 5

I

~

I6~

IS

+

2

0

~

~

~vng

[oc I

—

OIOflQ

looT)

--

++

II [110]

~ £

3

4

~

l2~

16L

Vol. 9, No. 16

Aft?’ Co,,eCIJvn to, ~P9oto.io~ticEft,c, II [ho]

•

(0

0

42— 2

.~

e~

0

8].—

‘~

—~

0

~~[00l]

2(—

+

0

06

. .j

4~

07

06

09

0

2 I

FIG. 2. The birefringence n(llO) 07

0

-

,i~1l0

in

GaAs measured for two opposite directions at

*

06

3

Photon Energy leo I

08

09

Photon

0

Energy

H

12

13

14

e~

FIG. 1. The observed birefringence 5(110) n(001) in GaAs for two directions of propagation along the [1101 axis before and after correcting for the contribution of the built-in 11111 stress, The solid curve was obtained theoretically (see text).

propagation of light along the [001 I axis. The solid curve is obtained from reference 4 for i [111 I tensile stress of 7.9 - 100 dyne cn~. The insert shows the orientation of the sample faces. dispersion alone. This correction does not substantiallv- alter the strong dispersion in the birefringence of Fig. 1 near E~.

q [hO] is shown in Fig. 1. A minor complication in the interpretation of our results is that for light propagating along the 1001] axis the sample was found to be birefringent, contrary to the predictions of equation (1). The birefringence o (110) - 11(110) for two opposite directions of propagation along the [001] axis is shown in Fig. 2. The shape of these curves suggests that this birefringence is caused b a built-in stress 4 in the crystal along the [1111 direction, which is the direction of growth of the crystal. The solid curve in Fig. 2 was obtained from reference 2 along 4 for a tensile stress of 7.9 106 dyne ‘cm the [ill] direction, This explanation was further substantiated by measuring the direction of the principal axes of for cj [110]. The directions of the principal axes were found to be as predict-

To explain this enhancement in spatial dispersion near E. we use a model consisting of four parabolic bands, as suggested by Higginbotham er al. These bands, assumed parabolic, are the lowest conduction band, the two top J - 3/2 valence bands, and the associated I = 1/2 spinorbit split valence band. This model has been very successful in explaining the dispersion in the piezo-birefringence 4. 5 and in the Raman tensor6 of a number of Ill—V semiconductors, inciuding GaAs. It has been pointed out ~ that a simple way of calculating the dielectric constant for a finite wave vector ~ is to translate the valence bands by an amount q’ in k-space and to calculate the dielectric constant for direct transitions (cj - 0) between the resulting valence and conduction bands. By applying this method to the

ed on the basis of the combined effect of spatial dispersion (Fig. 1) and the built-in [111] stress (Fig. 2). We have, therefore, corrected o(1l0) -n(001) for the contributions of the built-in stress. The resulting experimental points shown in Fig. 1 should give the birefringence due to spatial

model of Higginbotham el a!. we find the following expression for (110) - e (001) for ~ [110]:

4

(1i0)

-

(001)

Dq2

-

—

~

16

0y.

o~,) \c~ / 131

Vol. 9, No. 16

SPATIAL DISPERSION IN THE DIELECTRIC CONSTANT OF GaAs

In equation (3) C~is a parameter which

1423

Higginbotham et al., 6w

characterizes the strength of the E0 contribution to C; it can be either calculated from known band parameters or4obtained from ag(x) fit to the dispersion The function = x_2[2 — in near E0. (1 — — (1 + x)~2] is responsible for the temporal dispersion of the spatial dispersion near E0. The constant D includes any additional nondispersive contributions from the E0 and E0 4 A0 gaps and from higher gaps: As a result of the shift bandsfor by the ~ the gaps and do not occurofat the thevalence same energy heavythe

100 = 6w,,, and the dispersive term in equation (3) will vanish. Since the effect of warping on e is small, we can modify the model of by 6w,~— introducing warping only in of the lation 6w,,,. The values A, calcuB and C for GaAs are taken from ~ band calculations.8 A plot of equation (3), with C 0 = 6.6 obtained from reference 4, is shown as the solid curve in Fig. 1. The only adjustable parameter used for this plot was D,experimental taken to be +data 59 (Bohr 2 sothe asvalue to fit ofthe at lu radii) We note that changes in this parameter do not sig-

light-hole bands. 3w~ 00and &o~, are, respectively, the splitting between the gaps for the heavy and light hole bands for ~ [111]. For an arbitrary ~ the splitting of the heavy and light hole bands is given by: 1

2

~L. 2

1 -~

m~-n ~

—

(q)

__________

.

(4)

m~.+ fli?h (q)

In equation (4) rn~.rn15, and mu are tht effective masses of the conduction, heavy hole and light hole bands, respectively. Using the well-known expression 2 ± [B2k4= C2(k~k:-nk-~k~ — k~k~)]~(5) 2w(k) = — 1k

riificantly change the agreement between the experimental and calculated temporal dispersion of the spatial dispersion near E0. Our theory represents the data very well except in the long wavelength region where the experiment uncertainty is large. As pointed out earlier D includes possible nondispersive contributions of E0, E0 + A0 not contamed in g(x) and the effects of higher gaps, namely E1, E, + A, and, E . With the same model forE, and E, + A1 as used in reference 4 we have estimated the contribution of E, and E1 + A, to D to be +24. This value suffices to explain the sign and the magnitude of D given above (D = + 59). In the spirit of the isotropic Penn model theE 2 contribution to D should be negligible.

for the warped valence bands of the groups IV and Ill—V semiconductors we obtain : 1

In the case of silicon theE, and E,

1

forq~i[100]___=_,4+B, 1 and ~ lull —=

—

.1

=

.~=

rn1

(~

—

A,

contribution to the effect under discussion should

—~l-B(6)

in,,

+

~

C~t ~ /

C~t 2 3 / (7) 2

From equations (4)—(6) it is obvious that if warping is neglected, as in the parabolic band model of

beroughlythesameasinGaAssincethegapsinvolved are comparable. Thus the calculated value of this contribution is sufficient to explain the non-dispersive effect in the transparency region reported by Pastrnak and Vedam (D = + 34). 1 Acknowledgements — We are grateful to Professor Vedam for sending us a preprint of his work prior to publication. We would also like to thank Bell and Howell for the GaAs used in this experiment.

REFERENCE S

J.

1.

PASTRNAK

and VEDAM, Phvs. Rev. (in press).

2.

AGRANOVICH V.M. and GINZBURG V.L., Spatial Dispersion in Crystal Optics and the Theory of Excitons, Wiley, New York. (1966).

3.

LONG D., Energy Band in Semiconductors, Wiley, New York, (1968).

4.

HIGGINBOTHAM C.W., CARDONA M. and POLLAK F.H., Phys. Rev. 184, 821 (1969).

5.

YU P.Y., CARDONA M. and POLLAK F.H., Phys. Rev. B3, 340 (1971).

6.

CARDONA M., Solid State Commun. (to be published).

1424

SPATIAL DISPERSION IN THE DIELECTRIC CONSTANT OF GaAs

Vol. 9, No. 16

7.

ZEPPENFELD K., Optics Commun. lic 119 (1969).

8.

POLLAK F.H., HIGGINBOTHAM C.W. and CARDONA M., Proc. mt. Conf. Physics of Semiconductors, Kyoto, Journal of the Physical Society of Japan, Vol. 21, Supplement (1966).

Raumdispersionserzeugte natürliche Doppelbrechung ist an GaAs Kristallen im Durchl~ssigkeitsbereich beobachtet warden. In der N~he der direkten Bandkante zeigt diese Doppelbrechung elne starke zeitliche Dispersion. Die mikroskopische Theorie des Phänomens wird angegeben. Es wird gezeigt, dass der Hauptanteil dieser Doppelbrechung von der Richtungsabh~ngigkeit der Massen im Valenzband bestimmt wird.