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A quantitative study of the piezooptical activity in GaAs
Solid State Communications, Vol. 97, No. 4, pp. 261.-266, 1996 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/96 $12.00 + .OO
003s1098(95)00572-2
A QUANTITATIVE
STUDY OF THE PIEZOOPTICAL
B. Koopmans, Max-Planck-Institut
ACTIVITY IN GaAs
P. Etchegoin, P. Santos and M. Cardona
fi,ir Festkiirperforschung,
Heisenbergstrasse
1, D-70569 Stuttgart, Germany
(Received 8 August 1995 by J. Kuhi)
The optical activity induced by uniaxial stress in GaAs is shown to be an order of magnitude smaller than previously reported. The former overestimation is explained by birefringence effects related to tiny misalignments in the stress, which we properly separated from the true optical activity using phase-sensitive transmission experiments. The new experimental results are compared to a simple critical point analysis and to full band structure empirical pseudopotential calculations. We find good agreement for the sign, the absolute value and the trends in the dispersion of the optical activity. Attention is paid to the role of the interband deformation potentials which determine the sign of the gyrotropic effects. A sign conflict, present in former calculations, is solved. Keywords: A. semiconductors, pressure.
B. optical properties, C. strain, C. high
1. INTRODUCTION ZINCBLENDE semiconductors become optically active if a uniaxial stress (X) is applied along one of its S4 axes, lowering the symmetry from Td to D2d. Evidence of this nonlocal phenomenon, linear in the wavevector q, was claimed to have been found in [l-3]. Here we will show that previous reports strongly overestimated the optical activity, and none of them measured the pure gyrotropic effect. Besides being of fundamental interest, the optical activity yields information about the stress-induced coupling of the valence and conduction bands, described by interband deformation potentials like b,.,. Etchegion et al. introduced a microscopic model for the piezooptical activity [3], demonstrating its relation to b,,,. Such a valence to conduction band coupling has also been applied to describe other stress-induced phenomena, like the dependence of the cyclotron resonance mass on stress [4,5] and the stress-induced spin relaxation [6,7], and is of importance in quantum wells and superlattices [8]. A particularly subtle point is the sign of the piezooptical activity, determining the direction of rotation of linear polarized light. This sign is only meaningful if one uses a consistent definition of the anion and cation sites in the lattice. In a calculation, one has to 261
be careful with the definition of the phases of the wave functions involved. In an experiment, the absolute orientation of the crystal has to be determined, which can be achieved, e.g. by preferential etching [9-131, or from resonant Raman scattering [ 12,131. The literature on these subjects is often ambiguous. The aim of this paper is to review the optical activity and related issues, and present the parameters involved in a correct way. For a quantitative interpretation of the piezooptical activity, one has to consider linear birefringence effects induced by the T, -+ DZd symmetry lowering as well. Including terms up to linear in q, the nonlocal dielectric tensor E(W,q) reads e;&w,q) = f+)(W) -t L$(W)qk + 1.
.
(1)
The linear birefringence is described by e!;‘(w), and the optical activity by ti$(w). So, one expects a ratio of circular to linear effects of the order of qa N 0.01, where a is the lattice constant. Hence, in order to resolve the piezooptical activity, measurements are restricted to frequencies close to an isotropic point, i.e. the frequency wiseat which the linear birefringence changes sign and thus vanishes. In contrast, previous papers on the piezooptical activity in GaAs reported surprisingly large ratios of circular to linear birefringence, Up t0 0.1 at Wise[2,3].
262
PIEZOOPTICAL
ACTIVITY IN GaAs
Here we show that the overestimation and the scatter in the reported data can be explained by tiny misalignments between the pressure axis and the S, axis. This gives rise to off-diagonal strain modes which, just like the optical activity, introduce offdiagonal elements in the dielectric tensor, though of different symmetry [2]. The significance of this misalignment effect comes from the difference between the isotropic point of the diagonal- and the offdiagonal strain modes. As an example, for GaAs the off-diagonal linear birefringence becomes as important as the pure optical activity for a misalignment of only a few tenths of a degree (Section 4). An additional complication is that in a simple transmission experiment through two crossed polarizers (CP) the off-diagonal dielectric tensor elements due to true optical activity cannot be distinguished from those due to off-diagonal strain-induced birefringence. To separate the two sources, one needs a phase sensitive detection technique, as we discuss below. We conclude that previous reports of piezooptical activity in GaAs, all using the CP configuration, must have suffered severely from the misalignment . 2. MACROSCOPIC
OPTICS OF GYROTROPIC CRYSTALS
Let us take the S, axes to be (10 0), and consider light propagating along i. The dielectric tensor, restricted to its 1 and j components, can then be written as +,q)
= n
fi+Anloo An,,,+i&
Anll,-iSn (4
Since we neglect absorption, all parameters in equation (2) are real. fi is the mean refractive index. For pure [loo] stress, the linear birefringence is given by An I o. = n, - n,, and the circular birefringence (gyrotropy) by 6n = nL - nR, where the circular polarized plane waves are defined as E(r, t) = [if i~]eiq~“-‘“‘, with + and - for left and right, respectively. The parameter An, 1o = IZ~_~,- H,+~ describes the linear birefringence due to off-diagonal [I IO] strain. In the low stress and small q limit, and for a small misalignment 0, with respect to pure [lo 0] stress, we can write: Sn 0: iq,X, An, 1o a X0,, and AnI 0o a X. The symmetry requirements are such that An, 1o changes sign for a 180” rotation of the crystal about [loo]. Under this transformation Anloo and Sn are invariant. Sn changes sign while rotating the crystal by 90”, and vanishes for q along [0, f 1, *l]. Substitution of equation (2) into the wave equations yields the Jones matrix A4 for propagation
Vol. 97. No. 4
along i 6nsian,,,sin~ an
cos4+*sin+ M=
-Sn + iAn, 1(1 sin4 an
cos$-*sin@ (3)
where An = An:,, + An;, o + 6n2, q5= (d/A)&, -t’ d is the sample thickness, and X the wavelength in vacuum. For Anloo = An, I ,, = 0, the Jones matrix reduces to M=
(
cos 4
-&sin4
&sin4 lbnl cos@
1
(4)
’
i.e. describes a rotation of the linear polarization. A positive Sn corresponds to an anti-clockwise (or left handed) rotation when viewed following the light. We discuss two types of transmission measurements of Sn, denoted as the crossed polarizer (CP) and the explicit polarization (EP) experiment. We define a polarizer angle Q and analyzer angle p, both positive for a clockwise rotation looking along the propagation direction i. The light transmitted through the strained crystal will generally be elliptically polarized, described by a rotation of the main axis of the ellipsoid 8,, = ,&,, - Q and an ellipticity E. In the CP experiment, the transmission Zcp is measured as a function of w for fixed cr = 0 and ,0 = 90”. The main drawback of this method is that the phase information is lost, so that the measurement does not resolve Bgyr and E separately. The detected signal corresponds to the squared modulus of the off-diagonal elements of the Jones matrix. For arbitrary cy (though fixing p - cr.= 90”), the transmitted intensity is given by Icp -_= IO
6n2 + [An* 10cos(20) + Anloo sin(2a)12
2 sin
4,
an2
(5) where Z, is the transmission for parallel polarizers (CX= @= 0). Equation (5) reduces to equation (2) of [3] in the limit a = 0, = 0. Note that the CP experiment cannot distinguish between An, 1o and bn, since both contributions show the same dependence on 4, i.e. a sin* 4. In the EP experiment egyr and 6 are measured explicitly vs polarizer angle a, thus retaining the phase information. In the absence of linear birefringence, one has e,,, = sin 4 and E = 0, independent of CX.In the presence of both circular and linear birefringence, application of equation (3) allows for a unique fit of AnlOO, An1 1o and Sn, provided one knows the argument 4 to within f7r/2. An elegant phase
Vol. 97, No. 4
PIEZOOPTICAL
sensitive alternative to measure the dispersion of Sn uses acoustooptic modulation (AOM) techniques which will be described in detail in a future publication [14]. 3. MICROSCOPIC DEFORMATION
263
ACTIVITY IN GaAs
MODELS AND POTENTIALS
We discuss next the microscopic model of [3], and, by a simple analytic derivation, solve the sign conflict of former calculations of the relevant deformation potential 6,.,, [ 151. The model describes contributions due to electronic excitations close to the F-point, which are strongly dispersive just below the E0 gap, and thus dominate the optical activity. We use the following conventions. An anion is situated at the origin and a cation at a(; f 4’). In addition, we use the phase conventions of [16] for the wavefunctions. In this definition the (1 1 1) faces are anion-like, and the antisymmetric form factors appearing in the pseudopotential calculations are positive. We assume h = m = e = 1, unless otherwise specified. We neglect the spin-orbit coupling for simplicity. Then. the highest valence band at l? has I’is symmetry, the first conduction band I’, symmetry, and the second conduction band I’is symmetry. These unperturbed states are denoted by {lx,,, ), Iy,), Iz,,)}, Is), and stress along 110 0] intro. {I%), IV,)?I&J]. U maxial duces an energy splitting between Ix,) and {i.vJ, I+,)} equal to 3b(Sii - Si2)X, where b is the deformation potential, Si, and Si2 are the elastic compliance constants, and X < 0 for compressive stress. The linear birefringence An, o0 is proportional to this energy splitting. In order to understand the optical activity, we have to consider the uniaxial strain-induced coupling of the IIs valence with the Ii5 conduction band, potential described by the deformation izt, = (x,.IHxlx~)/[2(sii - Siz)X], where Hx is the strain Hamiltonian. We denote the perturbed valence band states at I? by IxI),Iy:), and [z:). The states away from I along k,i are mixed states. Using k -p perturbation theory we find I+,,, k,) x Ix:,) + iP]&,).
(6)
I-1?, k;) = P/x:,) - i/y:,).
(7)
The mixing coefficient /? = (b,,/b)Qk,/Eb is linear in kr. Q = i(x~,/~,.(~~)IS real and positive within our phase conventions [7], and Eb is the valence to second conduction band gap at F. Sn has its origin in the different effective gaps for left and right circularly polarized light, caused by the different energies of the If,, k, - q,) -+ Js, kr) excitations. If we neglect stress-induced changes of the optical matrix elements,
and treat the linear and circular effects on equal footing, we find for the ratio of the two [17]:
-2LQs;
Wbl
dx)X
XIRoXI
(8)
for [lOO]-stress, and q)] 100 11. For q]) [01 01, or reversing our convention of the atomic sites, the sign of Sn changes. The spectral function g(x) (with x = w/E0 < I) results from the stress-induced change of the gap, and is negative [IS]. Substitution of some typical values for GaAs, jb,,/bJ z 0.7 [15], Q x 0.5a.u. [7], EL x0.016 Hartree, and qz M 0.0012a.u. (at wise = 1.28 eV), yields ]Sn/Anlool x 0.005, of the same order of magnitude as the handwaving approximation qa N 0.01. Equation (8) shows the linear dependence of 6n on b,,. The latter has been calculated for a large number of III-V and II-VI compounds, using the empirical pseudopotential method (EPM) and the empirical tight binding method (ETBM) [15]. It was claimed that the two methods give an opposite sign of b,,,. In particular, b,, was found to be positive from the EPM calculation. To check this result, we repeated the numerical calculation, and also performed an analytic calculation using 15 plane waves. In the latter basis set, the highest valence and second conduction bands are given by [18] II,) = %I[1 1‘lri, +I%[11 ll$
25’+YnPW,~25’
(9)
for n = w,c, and I= x,y,z. The [I 1 llr and [2001r states are linear combinations of (1 1 1) and (200) plane waves, respectively, symmetrized for the diamond type lattice. Note, that LY,= p, = 7c = 0 for the diamond structure, i.e. if the antisymmetric pseudopotentials are zero. The full results will be published elsewhere. Here we merely indicate the most important contributions to bcl,:
where R = 4 (2*/a)‘, I$( k’,“)are the (anti)symmetric components of the pseudopotential form factors (with g = 3,4,8, II,...) [18], and G is the magnitude of the reciprocal lattice vector. The first term (kinetic energy term) in equation (10) is dominant for typical parameters, and is negative since “iv and car are both positive within our conventions. The second term in equation (10) is the leading one arising from stressinduced changes of the pseudopotential form factors, since it is the only one which contains a combination of a,/3 and ys that does not vanish in the diamond structure. Using /?, = 1, a, M -i, and dV,“/d In G < 0 we find that this term is also negative. Hence b,, is
264
PIEZOOPTICAL
ACTIVITY IN GaAs
negative in this EPM estimation. We confirmed this by a full numerical calculation using 59 and 89 bands. We thus conclude that the sign of [ 151is wrong, and, unlike reported there, the signs obtained from EPM and ETBM are the same, provided the same conventions for the wavefunctions are used. The above calculation of Sn only includes the dispersive contributions. In the next section we present the result of full band structure calculations, showing that the nondispersive contribution is small at wisO,and that it has the same sign as the dispersive one. We would like to make a final comment on the observation that Jb,J N lb1 [15], whereas one might have expected lbflj < Ibl for a weakly ionic compound like GaAs. This large ratio IbJbj can be traced back to the opposite sign of the kinetic energy and pseudopotential contributions in b, and their equal sign in b,, causing an enhancement of the piezooptical activity (b,) relative to the linear birefringence (b) . 4. RESULTS
FOR GaAs
We measured the piezooptical activity of GaAs at room temperature in the EP and CP configuraGaAs crystals were cut to tions. Undoped 2.1 x 2.4 x 18 mm3 bars. All faces correspond to { 10 0) within f0.3”, as checked by X-ray diffraction. Uniaxial stress up to 300 MPa was applied along the longest axis of the bar. The optical setup consisted of a halogen light source, a $m double monochromator, collimating optics, and quartz Rochon prisms as polarizer and analyzer. The chopped signals were detected with a silicon photodiode and a lock-in amplifier. The faces of the crystals were oriented using preferential etching in a HF (48%) +Hz02 (30%) +H20 (1 : 1 : 1) solution for 10 min [9], and comparing the etch patterns with those on a standard GaAs (00 1) wafer of known orientation. The morphology of the patterns on the sample and on the reference were identical. Thus, we found the elongated structures to be along [ 1 1 O] for the [0 0 II-face, and along [1 i 0] for the [0 0 II-face within our convention. We checked our assignment by resonant Raman scattering [12,13]. Using the orientation from the etching, our Raman experiments are consistent with [13]. The top panels of Fig. 1 show EP measurements on two samples for light incident on a [0 0 i] face, and at a relatively low stress of X = - 109 MPa. Sample A shows the expected behaviour for an optical active medium; a rotation Bsyr almost independent of the polarizer angle Q, and a small ellipticity E. For the
Vol. 97, No. 4
sample A
sample E?
90
l&l
60
0.6 0.3 o..
-60 t
polarizer angle a (deg.)
photon energy (eV)
Fig. 1. Results of EP (top) and CP (bottom) measurements on two GaAs samples (A and B) at X = - 109 MPa. Top panel: experimental rotation es,,, (filled symbols) and ellipticity 1~(*(open symbols) at Wiseas function of polarizer angle a. Least square fits are indicated by full and dotted curves, respectively. Bottom panel: the corresponding measurement of the dispersion of Zcp close to wise = 1.28 eV. The maximum transmittance is 0.03 and 0.7, respectively. The dotted curves are magnified by a factor of 10. [00 1] and [00 i] face, we found an anti-clockwise rotation of the linear polarization, as defined looking along the propagation direction. For the [0 lo] and [0 i 0] faces, the rotation was clockwise. No average rotation was found for [0 1 l]-cut crystals. Although sample B was from the same batch, it shows a strong dependence of @syron cr, and a large 6. The completely different behaviour can be attributed solely to the large difference in internal misalignment. Both measurements could be fitted well using equation (3). The derived parameters are listed in Table 1, and show a difference by a factor of 5 in AnI , o for the two samples. The strong dominance of the [l 1 OJ-strain in sample B is also easily seen from the minimum of [e(* at N 45”. Unlike An, 1o, the fitted values of 6n agree within f5%, as we found for all samples we measured. This 6n was found to depend linearly on X in the low stress regime. This proves unambiguously that the intrinsic piezooptical activity of GaAs is positive for compressive stress (X < 0) and equal to
PIEZOOPTICAL
Vol. 97, No. 4
Table 1. Fitted parameters in units of TPa-’ of the measurements of Fig. 1. The column labelled “EP” (“CP”) lists the results from the EP (CP) experiments. The discrepancy between the values labelled 1 and 2 may be due to the complete reduction of the stress between the CP and EP measurement for sample B
EP
An,
I O/X
Jli-6n + Anr,,/X
EP
(6)
-
-0.202
1.296 (6)
-
0.293 (6) -
1.315
1.27
0.356’
1.0
fi
1
1.1
1
1.2
photonenergy
o.252
%yr. The measured values of An, 10 give some idea of the involved internal misalignment. The angle 0; for which An, ,,(19,) = Sn is found from AnI I 0(45”) ’
0.0
-0.4 I
(6) -
for [l OO]-stress and q along [0 0 11. Previously reported values of Sri/// at wisO as obtained from CP experiments are 0.6 TPa-’ [I], 2.2 TPa-’ [2], and - 1.3 TPa-’ [3], where the sign was determined only in the last reference. Thus, all of them overestimate the optical activity by up to an order of magnitude. The bottom panels of Fig. 1 show CP spectra for sample A and B. Both spectra show a main maximum at Wiseand decaying oscillations at both sides, as described by equation (5). However, as expected from the difference in An, I o found from the EP experiment, we see a large difference in the absolute Zcp for the two samples. The values of d6n2 + A$, o fitted from the main transmission maximum are m reasonable agreement with the EP parameters (Table 1). Actually, [2] considered the possibility of offdiagonal strain, but concluded that it was not the main mechanism in the CP data. The authors main argument was the almost constant value of Osyrvs o. However, it can be derived that the ratio of variation to mean value of 0gyr goes to zero in the small stress limit, independent of An, 1o. Thus, a reliable conclusion can be drawn only from a quantitative analysis of
Sn
‘2 t
20
I"'1
CP
Sri/// = -0.21 f 0.01 TPa-‘,
sin 20; =
--
1
. -0.2 &
CP
-0.225
Sri///
0.2 I’
X
Sample B
Sample A
265
ACTIVITY IN GaAs
(11)
where An, , o(45”) is the anisotropy in n for pure [l lo] stress (0, = 45”). This parameter can be found from [ 1 1 l]-stress birefringence experiments, since An, , o for q 1)[0 0 1] is equal to AnI 1, 1193. Substitution of An, io(45”)/X = 18TPa-* [20] yields 0; = 0.3”! For a really quantitative CP experiment, one needs 0, -K 0.3”, which is very difficult to achieve.
’
1.3
81 u -40 1.4
(eV)
Fig. 2. EPM calculations of Sri/// (thick lines) and An, 00/X (thin lines) of GaAs for frequencies close to wisO. Full curves are calculated with an excitonic enhancement of the dispersive contributions of 3.1, the dashed ones without enhancement. The dotdashed line shows the “nondispersive” 6n. The data point represents the averaged Sn(qW) from the EP experiments. Since 6n is so much smaller than previously thought, it becomes difficult to measure its dispersion. A CP experiment will surely not suffice. As an example, in [3] Sn was overestimated by a factor 6. Due to the Sn2 and An: i. dependence on the Zc, [equation (5)] only l/36 of the measured signal came from pure optical activity. The reported isotropic point for 6n at 1.33 eV was probably due to the isotropic point of An110 at 1.38 eV [20] shifted somewhat by a misalignment of the polarizer [o # 0 in equation (5)]. Preliminary AOM transmission spectra clearly exclude such an isotropic point at 1.33 eV, and show only a weak (and probably positive) dispersion of 6n [14]. Let us finally compare the experimental data with the theoretical estimates. At the isotropic point, the dispersive part of An, 0. exactly cancels its nondispersive part, which can be found from the low frequency value of Anloo/X = 17TPa-* [20]. This yields 6n/lAn 1oo( = 0.012, within a factor of two from the value 0.005 estimated from equation (8). Note that the simple estimation also predicts the proper sign: b,, < 0 in equation (8) yields 6n/X < 0. In addition to the simple analysis, we performed a full EPM band structure calculation, also neglecting spin-orbit interaction. To treat contributions close to the I point correctly, we used a dense distribution of k points close to I’, and a sparser mesh throughout the rest of the Brillouin zone. The results discussed here are obtained using a basis set of 59 plane waves, and an inter-polation to empirical pseudopotential form factors [21] to obtain d V,/d In G. Since the calculation neglects exciton effects, it underestimates the dispersive contributions close to the E. gap. Therefore, we scaled the dispersive contribution by a factor, such
266
PIEZOOPTICAL
ACTIVITY IN GaAs
that the calculated isotropic point in Anloo agreed with the experimental value of 1.28V. An enhancement of 3.1 was needed for this. In comparison, a factor 2 was found in [20] from the discrepancy between the experimental birefringence data and the one-electron theory. Figure 2 shows the dispersion of the calculated 6n and Antoo just below the E0 gap, with and without exciton enhancement, using the same enhancement for linear and circular birefringence. Also plotted is the experimental value of fin, which is - maybe somewhat fortuitously - in surprisingly good agreement with the band structure result. One should bear in mind that the neglect of spin-orbit interaction close to the E0 gap is a crude approximation. Furthermore, the final result is rather sensitive to the choice of the pseudopotential form factors, since the stress-induced effects scale with the derivatives d V,/d In G which were obtained by interpolation of the empirical Vg values. Despite these uncertainties, we can draw two rigorous conclusions from our calculations. (i) The nondispersive and dispersive contributions are of equal sign, and (ii) at Wisethe dispersive contributions are dominant, especially if one includes the excitonic enhancement. The first point explains the absence of an isotropic point in Sn as suggested by our preliminary AOM experiments, the second why the simple critical point analysis presented in Section 3 gives satisfactory agreement with the measurements. 5. CONCLUSION A detailed analysis of the piezooptical activity of GaAs at the isotropic point has been presented. We have shown that previous reports strongly overestimated the optical activity. The intrinsic value is found to be &I/X = -0.21 f 0.01 TPa-‘, for [loo] stress and q along [00 11, and with the anion to cation bond pointing in the positive [l 1 l] direction. Both, the sign and magnitude of Sn, agree with a simple critical point analysis and with a full EPM band structure calculation. The experimentally determined sign of 6n yields b,, < 0, confirming our correction of the previously reported sign of b,,,. If one succeeds in getting more reliable dispersion data, piezooptical activity may become an accurate probe of - in particular the sign of - these interband deformation potentials.
Vol. 97, No. 4
Acknowledgements - We would like to thank S. Zehender for the etching experiments, and L.C. Lew Yan Voon for a careful reading of the manuscript. B.K. acknowledges the Alexander von Humboldt Stiftung for financial support.
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H.R. Trebin, M. Cardona, R. Ranvaud & U. Rlissler, in Physics of Narrow Gap Semiconductors (Edited by J. Rauluskiewicz, M. Gorsk 8c E. Kaczmarck). p. 227. Polish Scientific Publishers, Warsaw (1978). R. Ranvaud, H.R. Trebin, U. Riissler & F.H. Pollak, Phys. Rev. B20, 701 (1979). A.T. Gorelenko et al., Izv. Akad, Nauk SSSR, Ser. Fiz. 50,290
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Phys. Rev. B31,
003s1098(95)00572-2
A QUANTITATIVE
STUDY OF THE PIEZOOPTICAL
B. Koopmans, Max-Planck-Institut
ACTIVITY IN GaAs
P. Etchegoin, P. Santos and M. Cardona
fi,ir Festkiirperforschung,
Heisenbergstrasse
1, D-70569 Stuttgart, Germany
(Received 8 August 1995 by J. Kuhi)
The optical activity induced by uniaxial stress in GaAs is shown to be an order of magnitude smaller than previously reported. The former overestimation is explained by birefringence effects related to tiny misalignments in the stress, which we properly separated from the true optical activity using phase-sensitive transmission experiments. The new experimental results are compared to a simple critical point analysis and to full band structure empirical pseudopotential calculations. We find good agreement for the sign, the absolute value and the trends in the dispersion of the optical activity. Attention is paid to the role of the interband deformation potentials which determine the sign of the gyrotropic effects. A sign conflict, present in former calculations, is solved. Keywords: A. semiconductors, pressure.
B. optical properties, C. strain, C. high
1. INTRODUCTION ZINCBLENDE semiconductors become optically active if a uniaxial stress (X) is applied along one of its S4 axes, lowering the symmetry from Td to D2d. Evidence of this nonlocal phenomenon, linear in the wavevector q, was claimed to have been found in [l-3]. Here we will show that previous reports strongly overestimated the optical activity, and none of them measured the pure gyrotropic effect. Besides being of fundamental interest, the optical activity yields information about the stress-induced coupling of the valence and conduction bands, described by interband deformation potentials like b,.,. Etchegion et al. introduced a microscopic model for the piezooptical activity [3], demonstrating its relation to b,,,. Such a valence to conduction band coupling has also been applied to describe other stress-induced phenomena, like the dependence of the cyclotron resonance mass on stress [4,5] and the stress-induced spin relaxation [6,7], and is of importance in quantum wells and superlattices [8]. A particularly subtle point is the sign of the piezooptical activity, determining the direction of rotation of linear polarized light. This sign is only meaningful if one uses a consistent definition of the anion and cation sites in the lattice. In a calculation, one has to 261
be careful with the definition of the phases of the wave functions involved. In an experiment, the absolute orientation of the crystal has to be determined, which can be achieved, e.g. by preferential etching [9-131, or from resonant Raman scattering [ 12,131. The literature on these subjects is often ambiguous. The aim of this paper is to review the optical activity and related issues, and present the parameters involved in a correct way. For a quantitative interpretation of the piezooptical activity, one has to consider linear birefringence effects induced by the T, -+ DZd symmetry lowering as well. Including terms up to linear in q, the nonlocal dielectric tensor E(W,q) reads e;&w,q) = f+)(W) -t L$(W)qk + 1.
.
(1)
The linear birefringence is described by e!;‘(w), and the optical activity by ti$(w). So, one expects a ratio of circular to linear effects of the order of qa N 0.01, where a is the lattice constant. Hence, in order to resolve the piezooptical activity, measurements are restricted to frequencies close to an isotropic point, i.e. the frequency wiseat which the linear birefringence changes sign and thus vanishes. In contrast, previous papers on the piezooptical activity in GaAs reported surprisingly large ratios of circular to linear birefringence, Up t0 0.1 at Wise[2,3].
262
PIEZOOPTICAL
ACTIVITY IN GaAs
Here we show that the overestimation and the scatter in the reported data can be explained by tiny misalignments between the pressure axis and the S, axis. This gives rise to off-diagonal strain modes which, just like the optical activity, introduce offdiagonal elements in the dielectric tensor, though of different symmetry [2]. The significance of this misalignment effect comes from the difference between the isotropic point of the diagonal- and the offdiagonal strain modes. As an example, for GaAs the off-diagonal linear birefringence becomes as important as the pure optical activity for a misalignment of only a few tenths of a degree (Section 4). An additional complication is that in a simple transmission experiment through two crossed polarizers (CP) the off-diagonal dielectric tensor elements due to true optical activity cannot be distinguished from those due to off-diagonal strain-induced birefringence. To separate the two sources, one needs a phase sensitive detection technique, as we discuss below. We conclude that previous reports of piezooptical activity in GaAs, all using the CP configuration, must have suffered severely from the misalignment . 2. MACROSCOPIC
OPTICS OF GYROTROPIC CRYSTALS
Let us take the S, axes to be (10 0), and consider light propagating along i. The dielectric tensor, restricted to its 1 and j components, can then be written as +,q)
= n
fi+Anloo An,,,+i&
Anll,-iSn (4
Since we neglect absorption, all parameters in equation (2) are real. fi is the mean refractive index. For pure [loo] stress, the linear birefringence is given by An I o. = n, - n,, and the circular birefringence (gyrotropy) by 6n = nL - nR, where the circular polarized plane waves are defined as E(r, t) = [if i~]eiq~“-‘“‘, with + and - for left and right, respectively. The parameter An, 1o = IZ~_~,- H,+~ describes the linear birefringence due to off-diagonal [I IO] strain. In the low stress and small q limit, and for a small misalignment 0, with respect to pure [lo 0] stress, we can write: Sn 0: iq,X, An, 1o a X0,, and AnI 0o a X. The symmetry requirements are such that An, 1o changes sign for a 180” rotation of the crystal about [loo]. Under this transformation Anloo and Sn are invariant. Sn changes sign while rotating the crystal by 90”, and vanishes for q along [0, f 1, *l]. Substitution of equation (2) into the wave equations yields the Jones matrix A4 for propagation
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along i 6nsian,,,sin~ an
cos4+*sin+ M=
-Sn + iAn, 1(1 sin4 an
cos$-*sin@ (3)
where An = An:,, + An;, o + 6n2, q5= (d/A)&, -t’ d is the sample thickness, and X the wavelength in vacuum. For Anloo = An, I ,, = 0, the Jones matrix reduces to M=
(
cos 4
-&sin4
&sin4 lbnl cos@
1
(4)
’
i.e. describes a rotation of the linear polarization. A positive Sn corresponds to an anti-clockwise (or left handed) rotation when viewed following the light. We discuss two types of transmission measurements of Sn, denoted as the crossed polarizer (CP) and the explicit polarization (EP) experiment. We define a polarizer angle Q and analyzer angle p, both positive for a clockwise rotation looking along the propagation direction i. The light transmitted through the strained crystal will generally be elliptically polarized, described by a rotation of the main axis of the ellipsoid 8,, = ,&,, - Q and an ellipticity E. In the CP experiment, the transmission Zcp is measured as a function of w for fixed cr = 0 and ,0 = 90”. The main drawback of this method is that the phase information is lost, so that the measurement does not resolve Bgyr and E separately. The detected signal corresponds to the squared modulus of the off-diagonal elements of the Jones matrix. For arbitrary cy (though fixing p - cr.= 90”), the transmitted intensity is given by Icp -_= IO
6n2 + [An* 10cos(20) + Anloo sin(2a)12
2 sin
4,
an2
(5) where Z, is the transmission for parallel polarizers (CX= @= 0). Equation (5) reduces to equation (2) of [3] in the limit a = 0, = 0. Note that the CP experiment cannot distinguish between An, 1o and bn, since both contributions show the same dependence on 4, i.e. a sin* 4. In the EP experiment egyr and 6 are measured explicitly vs polarizer angle a, thus retaining the phase information. In the absence of linear birefringence, one has e,,, = sin 4 and E = 0, independent of CX.In the presence of both circular and linear birefringence, application of equation (3) allows for a unique fit of AnlOO, An1 1o and Sn, provided one knows the argument 4 to within f7r/2. An elegant phase
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PIEZOOPTICAL
sensitive alternative to measure the dispersion of Sn uses acoustooptic modulation (AOM) techniques which will be described in detail in a future publication [14]. 3. MICROSCOPIC DEFORMATION
263
ACTIVITY IN GaAs
MODELS AND POTENTIALS
We discuss next the microscopic model of [3], and, by a simple analytic derivation, solve the sign conflict of former calculations of the relevant deformation potential 6,.,, [ 151. The model describes contributions due to electronic excitations close to the F-point, which are strongly dispersive just below the E0 gap, and thus dominate the optical activity. We use the following conventions. An anion is situated at the origin and a cation at a(; f 4’). In addition, we use the phase conventions of [16] for the wavefunctions. In this definition the (1 1 1) faces are anion-like, and the antisymmetric form factors appearing in the pseudopotential calculations are positive. We assume h = m = e = 1, unless otherwise specified. We neglect the spin-orbit coupling for simplicity. Then. the highest valence band at l? has I’is symmetry, the first conduction band I’, symmetry, and the second conduction band I’is symmetry. These unperturbed states are denoted by {lx,,, ), Iy,), Iz,,)}, Is), and stress along 110 0] intro. {I%), IV,)?I&J]. U maxial duces an energy splitting between Ix,) and {i.vJ, I+,)} equal to 3b(Sii - Si2)X, where b is the deformation potential, Si, and Si2 are the elastic compliance constants, and X < 0 for compressive stress. The linear birefringence An, o0 is proportional to this energy splitting. In order to understand the optical activity, we have to consider the uniaxial strain-induced coupling of the IIs valence with the Ii5 conduction band, potential described by the deformation izt, = (x,.IHxlx~)/[2(sii - Siz)X], where Hx is the strain Hamiltonian. We denote the perturbed valence band states at I? by IxI),Iy:), and [z:). The states away from I along k,i are mixed states. Using k -p perturbation theory we find I+,,, k,) x Ix:,) + iP]&,).
(6)
I-1?, k;) = P/x:,) - i/y:,).
(7)
The mixing coefficient /? = (b,,/b)Qk,/Eb is linear in kr. Q = i(x~,/~,.(~~)IS real and positive within our phase conventions [7], and Eb is the valence to second conduction band gap at F. Sn has its origin in the different effective gaps for left and right circularly polarized light, caused by the different energies of the If,, k, - q,) -+ Js, kr) excitations. If we neglect stress-induced changes of the optical matrix elements,
and treat the linear and circular effects on equal footing, we find for the ratio of the two [17]:
-2LQs;
Wbl
dx)X
XIRoXI
(8)
for [lOO]-stress, and q)] 100 11. For q]) [01 01, or reversing our convention of the atomic sites, the sign of Sn changes. The spectral function g(x) (with x = w/E0 < I) results from the stress-induced change of the gap, and is negative [IS]. Substitution of some typical values for GaAs, jb,,/bJ z 0.7 [15], Q x 0.5a.u. [7], EL x0.016 Hartree, and qz M 0.0012a.u. (at wise = 1.28 eV), yields ]Sn/Anlool x 0.005, of the same order of magnitude as the handwaving approximation qa N 0.01. Equation (8) shows the linear dependence of 6n on b,,. The latter has been calculated for a large number of III-V and II-VI compounds, using the empirical pseudopotential method (EPM) and the empirical tight binding method (ETBM) [15]. It was claimed that the two methods give an opposite sign of b,,,. In particular, b,, was found to be positive from the EPM calculation. To check this result, we repeated the numerical calculation, and also performed an analytic calculation using 15 plane waves. In the latter basis set, the highest valence and second conduction bands are given by [18] II,) = %I[1 1‘lri, +I%[11 ll$
25’+YnPW,~25’
(9)
for n = w,c, and I= x,y,z. The [I 1 llr and [2001r states are linear combinations of (1 1 1) and (200) plane waves, respectively, symmetrized for the diamond type lattice. Note, that LY,= p, = 7c = 0 for the diamond structure, i.e. if the antisymmetric pseudopotentials are zero. The full results will be published elsewhere. Here we merely indicate the most important contributions to bcl,:
where R = 4 (2*/a)‘, I$( k’,“)are the (anti)symmetric components of the pseudopotential form factors (with g = 3,4,8, II,...) [18], and G is the magnitude of the reciprocal lattice vector. The first term (kinetic energy term) in equation (10) is dominant for typical parameters, and is negative since “iv and car are both positive within our conventions. The second term in equation (10) is the leading one arising from stressinduced changes of the pseudopotential form factors, since it is the only one which contains a combination of a,/3 and ys that does not vanish in the diamond structure. Using /?, = 1, a, M -i, and dV,“/d In G < 0 we find that this term is also negative. Hence b,, is
264
PIEZOOPTICAL
ACTIVITY IN GaAs
negative in this EPM estimation. We confirmed this by a full numerical calculation using 59 and 89 bands. We thus conclude that the sign of [ 151is wrong, and, unlike reported there, the signs obtained from EPM and ETBM are the same, provided the same conventions for the wavefunctions are used. The above calculation of Sn only includes the dispersive contributions. In the next section we present the result of full band structure calculations, showing that the nondispersive contribution is small at wisO,and that it has the same sign as the dispersive one. We would like to make a final comment on the observation that Jb,J N lb1 [15], whereas one might have expected lbflj < Ibl for a weakly ionic compound like GaAs. This large ratio IbJbj can be traced back to the opposite sign of the kinetic energy and pseudopotential contributions in b, and their equal sign in b,, causing an enhancement of the piezooptical activity (b,) relative to the linear birefringence (b) . 4. RESULTS
FOR GaAs
We measured the piezooptical activity of GaAs at room temperature in the EP and CP configuraGaAs crystals were cut to tions. Undoped 2.1 x 2.4 x 18 mm3 bars. All faces correspond to { 10 0) within f0.3”, as checked by X-ray diffraction. Uniaxial stress up to 300 MPa was applied along the longest axis of the bar. The optical setup consisted of a halogen light source, a $m double monochromator, collimating optics, and quartz Rochon prisms as polarizer and analyzer. The chopped signals were detected with a silicon photodiode and a lock-in amplifier. The faces of the crystals were oriented using preferential etching in a HF (48%) +Hz02 (30%) +H20 (1 : 1 : 1) solution for 10 min [9], and comparing the etch patterns with those on a standard GaAs (00 1) wafer of known orientation. The morphology of the patterns on the sample and on the reference were identical. Thus, we found the elongated structures to be along [ 1 1 O] for the [0 0 II-face, and along [1 i 0] for the [0 0 II-face within our convention. We checked our assignment by resonant Raman scattering [12,13]. Using the orientation from the etching, our Raman experiments are consistent with [13]. The top panels of Fig. 1 show EP measurements on two samples for light incident on a [0 0 i] face, and at a relatively low stress of X = - 109 MPa. Sample A shows the expected behaviour for an optical active medium; a rotation Bsyr almost independent of the polarizer angle Q, and a small ellipticity E. For the
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sample A
sample E?
90
l&l
60
0.6 0.3 o..
-60 t
polarizer angle a (deg.)
photon energy (eV)
Fig. 1. Results of EP (top) and CP (bottom) measurements on two GaAs samples (A and B) at X = - 109 MPa. Top panel: experimental rotation es,,, (filled symbols) and ellipticity 1~(*(open symbols) at Wiseas function of polarizer angle a. Least square fits are indicated by full and dotted curves, respectively. Bottom panel: the corresponding measurement of the dispersion of Zcp close to wise = 1.28 eV. The maximum transmittance is 0.03 and 0.7, respectively. The dotted curves are magnified by a factor of 10. [00 1] and [00 i] face, we found an anti-clockwise rotation of the linear polarization, as defined looking along the propagation direction. For the [0 lo] and [0 i 0] faces, the rotation was clockwise. No average rotation was found for [0 1 l]-cut crystals. Although sample B was from the same batch, it shows a strong dependence of @syron cr, and a large 6. The completely different behaviour can be attributed solely to the large difference in internal misalignment. Both measurements could be fitted well using equation (3). The derived parameters are listed in Table 1, and show a difference by a factor of 5 in AnI , o for the two samples. The strong dominance of the [l 1 OJ-strain in sample B is also easily seen from the minimum of [e(* at N 45”. Unlike An, 1o, the fitted values of 6n agree within f5%, as we found for all samples we measured. This 6n was found to depend linearly on X in the low stress regime. This proves unambiguously that the intrinsic piezooptical activity of GaAs is positive for compressive stress (X < 0) and equal to
PIEZOOPTICAL
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Table 1. Fitted parameters in units of TPa-’ of the measurements of Fig. 1. The column labelled “EP” (“CP”) lists the results from the EP (CP) experiments. The discrepancy between the values labelled 1 and 2 may be due to the complete reduction of the stress between the CP and EP measurement for sample B
EP
An,
I O/X
Jli-6n + Anr,,/X
EP
(6)
-
-0.202
1.296 (6)
-
0.293 (6) -
1.315
1.27
0.356’
1.0
fi
1
1.1
1
1.2
photonenergy
o.252
%yr. The measured values of An, 10 give some idea of the involved internal misalignment. The angle 0; for which An, ,,(19,) = Sn is found from AnI I 0(45”) ’
0.0
-0.4 I
(6) -
for [l OO]-stress and q along [0 0 11. Previously reported values of Sri/// at wisO as obtained from CP experiments are 0.6 TPa-’ [I], 2.2 TPa-’ [2], and - 1.3 TPa-’ [3], where the sign was determined only in the last reference. Thus, all of them overestimate the optical activity by up to an order of magnitude. The bottom panels of Fig. 1 show CP spectra for sample A and B. Both spectra show a main maximum at Wiseand decaying oscillations at both sides, as described by equation (5). However, as expected from the difference in An, I o found from the EP experiment, we see a large difference in the absolute Zcp for the two samples. The values of d6n2 + A$, o fitted from the main transmission maximum are m reasonable agreement with the EP parameters (Table 1). Actually, [2] considered the possibility of offdiagonal strain, but concluded that it was not the main mechanism in the CP data. The authors main argument was the almost constant value of Osyrvs o. However, it can be derived that the ratio of variation to mean value of 0gyr goes to zero in the small stress limit, independent of An, 1o. Thus, a reliable conclusion can be drawn only from a quantitative analysis of
Sn
‘2 t
20
I"'1
CP
Sri/// = -0.21 f 0.01 TPa-‘,
sin 20; =
--
1
. -0.2 &
CP
-0.225
Sri///
0.2 I’
X
Sample B
Sample A
265
ACTIVITY IN GaAs
(11)
where An, , o(45”) is the anisotropy in n for pure [l lo] stress (0, = 45”). This parameter can be found from [ 1 1 l]-stress birefringence experiments, since An, , o for q 1)[0 0 1] is equal to AnI 1, 1193. Substitution of An, io(45”)/X = 18TPa-* [20] yields 0; = 0.3”! For a really quantitative CP experiment, one needs 0, -K 0.3”, which is very difficult to achieve.
’
1.3
81 u -40 1.4
(eV)
Fig. 2. EPM calculations of Sri/// (thick lines) and An, 00/X (thin lines) of GaAs for frequencies close to wisO. Full curves are calculated with an excitonic enhancement of the dispersive contributions of 3.1, the dashed ones without enhancement. The dotdashed line shows the “nondispersive” 6n. The data point represents the averaged Sn(qW) from the EP experiments. Since 6n is so much smaller than previously thought, it becomes difficult to measure its dispersion. A CP experiment will surely not suffice. As an example, in [3] Sn was overestimated by a factor 6. Due to the Sn2 and An: i. dependence on the Zc, [equation (5)] only l/36 of the measured signal came from pure optical activity. The reported isotropic point for 6n at 1.33 eV was probably due to the isotropic point of An110 at 1.38 eV [20] shifted somewhat by a misalignment of the polarizer [o # 0 in equation (5)]. Preliminary AOM transmission spectra clearly exclude such an isotropic point at 1.33 eV, and show only a weak (and probably positive) dispersion of 6n [14]. Let us finally compare the experimental data with the theoretical estimates. At the isotropic point, the dispersive part of An, 0. exactly cancels its nondispersive part, which can be found from the low frequency value of Anloo/X = 17TPa-* [20]. This yields 6n/lAn 1oo( = 0.012, within a factor of two from the value 0.005 estimated from equation (8). Note that the simple estimation also predicts the proper sign: b,, < 0 in equation (8) yields 6n/X < 0. In addition to the simple analysis, we performed a full EPM band structure calculation, also neglecting spin-orbit interaction. To treat contributions close to the I point correctly, we used a dense distribution of k points close to I’, and a sparser mesh throughout the rest of the Brillouin zone. The results discussed here are obtained using a basis set of 59 plane waves, and an inter-polation to empirical pseudopotential form factors [21] to obtain d V,/d In G. Since the calculation neglects exciton effects, it underestimates the dispersive contributions close to the E. gap. Therefore, we scaled the dispersive contribution by a factor, such
266
PIEZOOPTICAL
ACTIVITY IN GaAs
that the calculated isotropic point in Anloo agreed with the experimental value of 1.28V. An enhancement of 3.1 was needed for this. In comparison, a factor 2 was found in [20] from the discrepancy between the experimental birefringence data and the one-electron theory. Figure 2 shows the dispersion of the calculated 6n and Antoo just below the E0 gap, with and without exciton enhancement, using the same enhancement for linear and circular birefringence. Also plotted is the experimental value of fin, which is - maybe somewhat fortuitously - in surprisingly good agreement with the band structure result. One should bear in mind that the neglect of spin-orbit interaction close to the E0 gap is a crude approximation. Furthermore, the final result is rather sensitive to the choice of the pseudopotential form factors, since the stress-induced effects scale with the derivatives d V,/d In G which were obtained by interpolation of the empirical Vg values. Despite these uncertainties, we can draw two rigorous conclusions from our calculations. (i) The nondispersive and dispersive contributions are of equal sign, and (ii) at Wisethe dispersive contributions are dominant, especially if one includes the excitonic enhancement. The first point explains the absence of an isotropic point in Sn as suggested by our preliminary AOM experiments, the second why the simple critical point analysis presented in Section 3 gives satisfactory agreement with the measurements. 5. CONCLUSION A detailed analysis of the piezooptical activity of GaAs at the isotropic point has been presented. We have shown that previous reports strongly overestimated the optical activity. The intrinsic value is found to be &I/X = -0.21 f 0.01 TPa-‘, for [loo] stress and q along [00 11, and with the anion to cation bond pointing in the positive [l 1 l] direction. Both, the sign and magnitude of Sn, agree with a simple critical point analysis and with a full EPM band structure calculation. The experimentally determined sign of 6n yields b,, < 0, confirming our correction of the previously reported sign of b,,,. If one succeeds in getting more reliable dispersion data, piezooptical activity may become an accurate probe of - in particular the sign of - these interband deformation potentials.
Vol. 97, No. 4
Acknowledgements - We would like to thank S. Zehender for the etching experiments, and L.C. Lew Yan Voon for a careful reading of the manuscript. B.K. acknowledges the Alexander von Humboldt Stiftung for financial support.
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