Circular birefringence in zinc-blende-type semiconductors

Thin Solid Films, 233 (1993) 137-140

137

Circular birefringence in zinc-blende-type semiconductors P. Etchegoin and M. Cardona Max-Planck-lnstitut fiir Festkrrperforschung, Heisenbergstr. 1, W-7000 Stuttgart 80 (Germany)

Abstract

The existence of two different indices of refraction for left and right circularly polarized light (optical activity) depends, from the macroscopic point of view, on the point group of the lattice which fixes the symmetry of the macroscopic dielectric tensor e,,a(to, k). The point group of zinc-blende-type materials (To) does not allow the existence of optical activity. However, by lowering the symmetry by means of an external uniaxial stress along [ 100] (To--* D2d) optical activity becomes allowed for light propagating with k along [010] or [001], at least in spectral regions where the k independent piezobirefringence is negligible. We have performed experiments, based on transmission of light through crossed polarizers, on GaAs and InP. We show some results and discuss experimental aspects for an ellipsometry-type set-up.

1. Introduction and overview

Optical activity is a fascinating physical phenomena with a long history within the field of classical optics. It was observed as early as 1811 by Arago in quartz and subsequent work by Biot led to the discovery of optical activity in chemical solutions. The classical electrodynamics of these phenomena can be understood in terms of the simultaneous frequency and wavevector dependent dielectric tensor. The appearance of two circularly polarized modes of propagation with different indices of refraction is a natural consequence of Fresnel's equations when the dielectric tensor e~,~(og,k) includes offdiagonal terms proportional to k [1]. However, the microscopic origin of the optical activity takes place in the details of the electronic band structure and may require sophisticated methods for an explicit evaluation. The linear terms in k in e~,~(to,k) should arise from linear terms in the dispersion relation of the electronic bands. In ref. 2 a comprehensive treatment of the microscopic origin of the optical activity as well as the role played by excitons is given. The optical activity can be natural or induced by external perturbations. In any case, the symmetry of the lattice plays an important role and, even if the crystal possesses bands with dispersion relation proportional to k, the total contribution to the dielectric tensor may cancel out by symmetry in e~,~(og, k). The point groups that allow optical activity were listed in ref. 1. Except for T and O, which are cubic and lead to k-linear terms independent of the direction of k, and optical activity, the other groups that can sustain k-linear terms involve the presence of a preferential direction (axial groups). In addition to such terms, these point groups are likely to present natural

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linear birefringence which may quench the optical activity generated by the k-linear terms in ea,p(og,k). The occasional existence of an isotropic point, i.e. a particular frequency at which the extraordinary and ordinary indices of refraction have the same value, may restore the optical activity (also called rotatory power; the basic manifestation of optical activity is the rotation of the plane of polarization of a linearly polarized beam). The natural birefringence does not quench the optical activity for propagation along the optical axes in the C4, D4, (73, D3, C 6 and D 6 groups [3]. Stress-induced optical activity in the zinc-blende-type semiconductors has been proposed and demonstrated in refs. 4-6. In this case, measurements of optical activity are performed near isotropic points by measuring transmission through crossed polarizers. The incident radiation has to be linearly polarized along one of the optical axes of the linear birefringence [7] (ordinary or extraordinary axes). In this way, if optical activity were not present, the transmission would be zero (see next section). Here we show the applicability of an ellipsometric set-up to this problem as well as some results obtained for GaAs and InP.

2. Experimental method and results

In order to measure transmission of light through crossed polarizers under stress in the near-IR range (i.e. about 0.7-1.4 eV), we employed an ellipsometric set-up as follows. The analyser arm is set straight and the sample is positioned making use of a hydraulic-stress machine with a mobile sample holder that was described elsewhere [8]. The set-up is shown schematically

© 1993 - - Elsevier Sequoia. All rights reserved

138

P. Etche~oin, M. Car&ma / Circular hirqfHngence h* semiconductors

K Fig. 1. Experimental set-up. The figure shows part of a rotating analyser ellipsometer where the analyser arm is set straight with the polarizer arm. The sample is mounted in a hydraulic-stress machine [8] and the uniaxial stress applied vertically. The analyser is rotated during alignment and is stopped and crossed with respect to the polarizer afterwards (see text for details).

in Fig. 1. The sample is placed with its longest side vertical. The uniaxial stress is also applied along this direction. Typical dimensions of the sample are 18 × 1.8 × 1.5 m m 3. Measurements were performed in G a A s and InP, below the lowest (indirect) gap at room temperature, with the optical path kept in air. These two semiconductors are known to have an isotropic point of the linear birefringence below the gap [9, 10]. As a light source we employed a tungsten lamp and a 3/4 m length single-path m o n o c h r o m a t o r with a 600 lines m m -I grating and slits set to 0.5 mm. We used a cooled Ge detector to record the signal which is extremely sensitive in this photon energy range. The measurements have to be normalized to take into account the spectral dependence of the light source. If An = n e - n o represents the difference between the indices of refraction for light polarized along the extraordinary and ordinary axis, respectively, and an = nc - nR the difference between the indices for left and right circularly polarized light, then the transmission in the experimental set-up of Fig. 1 is given by [4-7] I0 - (An) 2 + (6n) 2 sin2

[(An)2 + (6n)2] i/2

(1)

where looc]E~l 2 (see Fig. 1), d is the thickness of the sample, and 2 the light wavelength in vacuum. This expression holds if the incident electric field is properly polarized along one of the optical axes of the linear birefringence (parallel or perpendicular to the stress in our case). Then if 6n were zero then I/Io = 0 as desired. Experimental results near the isotropic point of G a A s under stress were presented in refs. 4 and 6. To overcome alignment problems once the stress is applied, we used an ellipsometric procedure to align the polarizer: once the isotropic point (i.e. the energy at which An = 0) is detected, the m o n o c h r o m a t o r is placed far from this point at a fixed energy so that the optical activity is negligible in comparison with the linear bire-

fringence and the latter dominates. The material is well described under these conditions as a uniaxial crystal with two axes (ordinary and extraordinary) parallel and perpendicular to the applied stress. Then the analyser is rotated at a fixed angular frequency that depends on the detector being used. In our case (for G a A s and InP) for the results reported here and those of ref. 6, we used 600 rev rain ~ for the Ge detector. An analogous procedure to that followed during the calibration of a rotating analyser ellipsometer is then employed [11, 12]. The polarizer in Fig. 1 is moved back and forth around its vertical position O = 0 ~' and the transmitted light is analysed in terms of the Fourier coefficients of the angle-dependent polarization. It is clear that, far from the isotropic point (where interference with optical activity is negligible), the light is linearly polarized if O coincides with the direction of the applied stress (or is perpendicular to it), and elliptically polarized otherwise. The degree of ellipticity of the light, and in particular the point at which the light is passing through one of the optical axes, can easily be evaluated by measuring the Fourier coefficients for a range of O angles about the vertical, and then fitting the residuals of the polarization to a parabola [11-13] near O = 0 (see Fig. 1) or near O = 90 °. Expressing the detector signal in terms of the Fourier coefficients a~ and a, as l = constant x {1 + q[a: cos(2co~,,,t) +

a2

sin(2¢o~,,, t)] I (2)

where co.... is the angular velocity of the analyser, we can define the residuals of the polarization as R ( O ) = I - q ( a , + a~) for each angle O of the polarizer. The parameter r/in eqn. (2) has the same meaning as in a rotating analyser ellipsometer (see refs. 11 and 12 for details) and is essentially related to the speed of the detector. For a typical photomultiplier in the visible range r/is always close to unity (about 1.015 for an $20 photomultiplier at about 2000 rev min 1). For a slow detector, like the Ge-detector we are working with, ~t can be as large as 2.5. Even assuming the extreme case q = 1, R ( O ) (or I - R ( O ) ) is parabolic about O ~ 0' or O ~ 9 0 ~ and has a minimum (or maximum) about those angles where the light is perfectly aligned with one of the optical axes. An example of this kind is shown in Fig. 2 for GaAs. The two arrows indicate the position where the polarizer is aligned with the stress direction (ne) or perpendicular to it (no). The horizontal axis corresponds to the angle O of Fig. 1 with a conventional origin that is close to the vertical axis in our set-up. The rest of the experimental conditions are detailed in the figure. Note that the isotropic point of G a A s at X ~ 250 MPa is around 1.27 eV [6], while the measurement was performed at 1.1 eV. In this region the linear birefringence dominates and, at the same time, the Ge detector is still within its range of opera-

139

P. Etchegoin, M. Cardona / Circular birefringence in semiconductors 0.12

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e [deg] Fig. 2. Alignment of a GaAs sample. The two peaks show the positions at which the polarizer is parallel (ne) or perpendicular (no) to the stress. The alignment has been performed at hco ~ 1.1 eV, i.e. far from the isotropic point. At this photon energy the optical activity is quenched by the linear birefringence.

tion. Once the sample has been aligned, the analyser is stopped and crossed with respect to the polarizer. Measurements of the intensity ratio given by eqn. (1) are shown for the case of InP and G a A s at two similar stresses in Fig. 3. The similarity of eqn. (1) to Young's formula which represents the diffraction pattern of a single slit should be noted. Except at the isotropic point, where the prefactor of eqn. (1) is equal to unity, in general An >> 6n and the intensity maxima fall off rapidly. Maximum transmission is obtained at the isotropic point and its intensity at low stresses is given by ( I / I o ) i s = sin2[(rcd/2)lfn]]. Above a certain stress this is no longer true and the intensity at the isotropic point is almost stress independent. This is due to fluctuations in the internal stress at the isotropic point. Several physical parameters can be obtained from the experiment. The dispersion relation of 6n can be obtained by fitting eqn. (1) to the experimental data (provided that An has been previously measured by transmission along a [110] direction). The energy position of the isotropic point is also well defined and can be obtained as a function of the stress. In Fig. 4 we show the energy of the isotropic point of G a A s as a function of the stress for the experiments reported in ref. 6. The slope of the measured linear dependence on stress corresponds to what is expected from the hydrostatic shift o f the

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Fig. 3. Transmission through crossed polarizers for GaAs and InP at two similar stresses. The curves are represented by eqn. (1). It should be noted that the light is crossing the sample along [001] while the stress is applied along [100]. For light crossing the sample along [110], or equivalent directions, no ~ffect should be present [1, 6]. The arrows indicate the isotropic point of the linear birefringence. Its intensity at these high stresses arises mainly from unpolarized light generated by stress inhomogeneities.

fundamental edge (4.6 x 1 0 - S M P a -1) [14]. A microscopic model for the optical activity based on the appearance of stress-induced linear terms in k at F has been proposed and compared with the experiment (for GaAs) in ref. 6. More details of the microscopic mechanism can be found in refs. 6 and 15.

3. Conclusions

We have demonstrated the use o f a modified ellipsometer set-up for the study of stress-induced optical activity. Small corrections due to optical activity of the polarizers or other optical elements can be taken into account by calculating the total transmission in the Jones or Miiller matrix formalism [12] instead of eqn. (1). We have recently employed the Jones matrix formalism to explain the additional features in the transmission spectra that were observed in ref. 6 for light crossing along [110] and that were caused by internal stress. An extensive study of the stress-induced optical activity of G a A s and InP using this experimental method, as well as a complete microscopic version

P. Etehegobl. M. Cardona / Circular bire/ringence m semicom&ctors

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obtained with the empirical pseudopotential method is also in progress [15]. Acknowledgments

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Thanks are due to H. Hirt, M. Siemers and P. Wurster for technical help during experiments as well as D. S. Citrin for a critical reading of the manuscript.

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References

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Fig. 4. Position of the isotropic point of G a A s as a function of the external uniaxial stress. The slope of the curve agrees with predictions from the hydrostatic shift of the fundamental gap. In fact, the presence of the isotropic point is a consequence of the competition (with different signs) of the fundamental gap and higher energy transitions (dispersionless background) [14].

(including spin-orbit coupling) of the calculation presented in ref. 6, will be published elsewhere [15]. A microscopic calculation of the off-diagonal elements of the dielectric tensor based on the full band structure

I L. D. Landau and E. M. Lifshitz, Electro~tvnanlics o! Contim,m.v Media. Pergamon, Oxford, 1963, p. 342. 2 V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial DLv~ersion and Excitons, (Springer, Heidelberg, 1984). 3 W. Imaino, A. K. Ramdas and S. Rodriguez, Phys. Ret:. B, 22 (1980) 5679. 4 L. E. Sotov'ev and M. O. Cha]'ka, Sot. Phya. Solid State, 22 (4) (1980) 568. 5 L. E. Solov'ev, Opt. Spectrosc. (USSR), 48 (5) (1979) 323. 6 P. Etchegoin and M. Cardona, Solid State Commun., 82(19921 655. 7 M. V. Hobden, Acta. ('rystallogr. A, 24 (1968) 676. 8 J. Kircher, W. B6hringer. W. Dietrich. P. Etchegoin and M. Cardona, Ret:. Sci. lnstrum., 63(1992) 3733. 9 C. W. Higginbotham, M. Cardona and F. H. Potak, Phys. Rer., 184 (1969) 821. I(1 F. H. Pollak and M. Cardona, Phys. Ret. B, 172(1968) 816. 11 D. E. Aspnes and A. A. Studna, Rev. Sei. lnstrum., 49 (1978) 292. 12 D. E. Aspnes, J. Opt. Soe. Am., 70(1980) 1275. 13 J. Barth, R. L. Johnson and M. Cardona, in E. D. Pallik (ed.), Handbook of Optical (bnstants of SolMs H, Academic Press, Boston, MA, 1991. 14 M. Cardona, in E. Burstein (ed.), Atomic Structure and PropertM~ of Solids, Academic Press, New York, 1972, p. 513. 15 P. Etchegoin, J. Kircher, M. Cardona and C. Greim, Phys. Rer. B, 45(1992) 11721.