Inplane optical anisotropy of GaAs/AlAs multiple quantum wells ...

Inplane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona Citation: Appl. Phys. Lett. 69, 782 (1996); doi: 10.1063/1.117890 View online: http://dx.doi.org/10.1063/1.117890 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v69/i6 Published by the American Institute of Physics.

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In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

~Received 8 April 1996; accepted for publication 22 May 1996! We present a technique, microscopic reflectance difference spectroscopy ( m RDS!, for the measurement of optical anisotropy with sub-micron resolution. The technique is applied to the determination of the in-plane anisotropy of GaAs/AlAs multiple quantum well structures in a phase resolved way, both below and above the fundamental gap. Confinement and local field effects are discussed, and a comparison is made with microscopic calculations based on a tight-binding Hamiltonian for the electronic states. © 1996 American Institute of Physics. @S0003-6951~96!00232-X#

Confinement and local field effects in quantum well ~QW! heterostructures leads to optical anisotropies for inplane ~along the layers! light propagation. Such anisotropies have been studied extensively in ~001!-grown zincblende QW’s using a variety of techniques, like absorption,1,2 photoluminescence,3 birefringence4,5 and time of flight6 spectroscopy. In this letter we present a reflection study of the full complex anisotropy in the dielectric response of QW’s, using a new reflectance difference technique7 denoted as microscopic reflectance difference spectroscopy ( m RDS!. A fundamental difficulty in the former in-plane transmission approaches is the requirement of establishing transmission along the thin QW’s. Usually this is achieved in a waveguide geometry. The waveguide core consists either of a multiple quantum well4,5 ~MQW! or of a single quantum well1,2 ~SQW! embedded within a thick barrier. In the case of a MQW core the absorption restricts measurements of «' 2« i 5D«5D« 1 1 iD« 2 , the difference of the dielectric response perpendicular and parallel to the growth axis, to its real part D« 1 ~birefringence! and only below the fundamental gap (E 0 ). For a core consisting of an embedded SQW the low filling factor of the guide allows a measurement of D« 2 ~absorption! across the gap, however, at the cost of loosing all sensitivity to D« 1 . Our new reflection approach accesses D« 1 and D« 2 simultaneously, and without the need of a waveguiding geometry. Required are a high sensitivity and a spatial resolution below the total MQW thickness, which were achieved by incorporating an RDS-spectrometer in a confocal microscope. An application to GaAs/AlAs MQW’s is discussed in this letter. A schematic drawing of our m RDS set-up is presented in Fig. 1. Chopped light from an Ar1 pumped tunable Ti:sapphire laser is spatially filtered and passed through a polarizer. A microscope objective ~1003, NA50.95! focuses the beam to a ~sub!micrometer spot. The ~110!-cleaved MQW is mounted with its growth axis at 45° with respect to the polarizer on a scanning piezoelectric device. The position of the spot on the sample can be monitored using a CCD camera. A photoelastic modulator ~PEM! with its main axis parallel to the polarizer is mounted just above the objective to reduce depolarization effects from other optical components 782

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to a minimum. Upon reflection from the MQW the polarization is slightly modified, and thus the reflected light is modulated at its second pass through the modulator. Finally, the beam is focused on a confocal pinhole, passes an analyzer ~at 45°), and is detected by a Si-photodiode. The resulting signals at the chopper frequency (V dc) and the first- and secondharmonic of the PEM frequency (V 1 f and V 2 f , respectively!, are analyzed using lock-in amplifiers. The complex reflectance difference (r' 2r i ), the anisotropy (D«52nDn), and the modulated signals are related through:7 r' 2r i V 2f V 1f 2Dn 5a 2 f 5 1 ia 1 f , 1/2 ~ r' 1r i ! ¯ « 21 V dc V dc

~1!

where ¯ « is the mean dielectric constant of the anisotropic medium, a n f 5 A2/J n (A 0 ), J n (A 0 ) is a Bessel function, and the retardation of the PEM is set to A 0 52.405 @as to make J 0 (A 0 )50]. The polarization effects of optical components between the PEM and the analyzer can be corrected for by multiplying the coefficients a n f by an appropriate complex phase factor. Spurious contributions due to misalignment and imperfections of the optical components ~e.g., strain in the objective! are surpressed in lowest order by calibration of the set-up using the isotropic substrate adjoining the MQW. A point which might need some special attention is the use of strongly focused beams—necessary to achieve the

FIG. 1. Experimental arrangement of the m RDS measurements. The inset ~bottom right! shows the ~110!-cleaved multilayer ~MQW: GaAs/AlAs, CL: ¯A2 # . Ga0.3Al0.7As cladding!, and the incident polarization eˆ along @ 11

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© 1996 American Institute of Physics


FIG. 2. Measured anisotropy (V 2 f /V dc) and mean reflectance (V dc) at \ v 51.57 eV as a function of the spot position on the multilayer structure ~MQW: 3003 50 Å/50 Å GaAs/AlAs, CL: Ga0.3Al0.7As cladding!.

needed spatial resolution—which inevitably leads to a significant depolarization and to a wide spread in incident angles. It should be realized, however, that the signals in the PEM configuration are actually proportional to the field amplitudes averaged over the reflected beam profile, not the intensities. As a consequence the induced depolarization cancels out. This is in sharp contrast with, for instance, the rotating analyzer method in which a depolarization is not separable from a true ellipticity. We checked experimentally that our quantitative results were hardly affected by the aperture of the objective. A related effect shows up at interfaces. If the laser spot overlaps two regions with a different dielectric response, the cylindrical symmetry is broken, and the depolarization experienced by different parts of the beam do not cancel any longer, leading to interface-related modulated signals. Molecular beam epitaxy was used to grow a multilayer composed of 300 periods of ~100!-GaAs/AlAs (50 Å well and barrier width and a total thickness of 2.8 m m!, sandwiched between two 1.5 m m Ga0.3Al0.7As cladding layers, on top of an undoped ~001!-GaAs substrate ~see inset of Fig. 1!. For the purpose of m RDS measurements, performed at room temperature ~RT!, the multilayer structure was cleaved along ~110!. Transmission experiments on identical samples have been published in Ref. 5. Figure 2 shows a typical example of the m RDS signals while scanning the laser spot across the various layers of interest. The mean reflectivity (V dc) images the difference in the mean dielectric response of the respective layers. The zero of V 2 f , determined by the calibration of the polarizer, is obtained by assuming the GaAs substrate to be isotropic. A finite anisotropy of the MQW is observed, approximately constant throughout the layer. In addition, peaks are found at each of the interfaces due to the effects described above. They are most clearly seen at the GaAs/cladding and cladding/air interface, where ¯ « has its largest discontinuities. The broadening of the features suggests a resolution of ;0.7 mm. Figure 2 clearly shows that the modulated signals are constant at the central 1–1.5 m m region of the MQW, and reflect its intrinsic anisotropy. The anisotropy of the MQW is Appl. Phys. Lett., Vol. 69, No. 6, 5 August 1996

strongly dispersive, as is illustrated in Fig. 3 which shows the measured spectral dependence of the real and imaginary part of D«. The dominant spectral features can be easily understood from the character of the valence bands at the zone center (G). The confinement along the growth axis zˆ acts as a uniaxial perturbation, which splits the heavy hole ~hh! and light hole ~lh! bands. The hh and lh bands at G transform like u 3/2,63/2& 5 1/A2 u (x6 iy) 61/2& and u 3/2,61/2& 5 1/A6 u @ (x6 iy) 71/212z 61/2# & , respectively ~indices 61/2 represent the spin!. The hh band is coupled to the fully symmetric conduction band ~e! only by light polarized perpendicular to the growth axis ~resulting in a positive peak in D« 2 ). Contrary, lh–e excitations, which by the confinement are shifted to a somewhat higher frequency, are favored by light polarized along zˆ ~yielding a negative D« 2 ). The positive peak in D« 2 at 1.563 eV, and the negative feature at 1.603 eV are assigned to the hh–e and lh–e excitation, respectively. The frequency of the hh–e excitation agrees well with the observed RT luminescence gap ~1.567 eV!.5 Note that at RT the excitons have a large lifetime broadening, and cannot be distinguished from transitions to the continuum, which are also renormalized by e–h interactions. From the character of the hh and lh wave functions at G, one would expect (D« 2 ) lh /(D« 2 ) hh5(1/622/3)/ (1/220)521. Experimentally we find (D« 2 ) lh / (D« 2 ) hh;20.2. The lower ratio agrees with a reduced anisotropy of the lh exciton as observed for 50 Å GaAs SQW’s.6 In addition, the simple estimate neglects the contribution of kW Þ0 transitions and excitonic effects. The hh peak corresponds to an absorption coefficient for the GaAs well of a hh52.83104 cm21 , corrected for the fact that only 50% of the MQW consists of GaAs ~i.e., using a filling factor of 0.5!. This value is in proper agreement with SQW absorption experiments which yielded 1.2 ~5.0! 3104 cm21 for 100 ~20! Å SQW’s.2 From the data of Fig. 3 a low frequency limit of the birefringence Dn(0)5(n' 2n i ) v →0 50.05 is found, whereas Dn(0)50.20 was reported5 from waveguided transmission experiments on an identical MQW structure. This discrepancy arises from an erroneous indexing of the interference fringes in the transmission spectra of Ref. 5. One should realize that if the first-order transmission fringe ~at which the phase difference experienced by the ordinary and extraordinary waves is 2 p upon transmission! is not within the experimental frequency range, additional information is required to access the absolute index of the fringes. In Ref. 5 it was assumed that Dn is approximately dispersionless far below E 0 , so that ] Dn/ ]v 50 could be used at v ;1.4 eV. Although on first sight this seems a reasonable assumption, we emphasize that even a small but finite dispersion can lead to drastic errors in the estimated Dn(0).8 Note that such an ambiguity is not met in our reflection approach, and, actually, the data of Fig. 3 rather accurately reproduce the interference fringes presented in Ref. 5. The low frequency limit (D«50.35) obtained by m RDS agrees well with a simple effective medium approach for the local field effects:4 D«5 ^ 1/« & 21 2 ^ « & '0.4, where ^& indicates the compositional GaAs/AlAs average. The efKoopmans et al.

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FIG. 3. Dispersion of D«5«' 2« i ~top: D« 1 , bottom: D« 2 ) obtained for a GaAs/AlAs (50 Å/50 Å! MQW. The m RDS data ~fat lines and symbols! are compared with the tight-binding calculation ~thin lines!.

fective medium estimate is only strictly valid for v →0, and does not take into account the resonant structure near E 0 . It does, however, correctly reproduce the magnitude and sign of the off-resonant D« 1 ~below hh–e and above lh–e!, which is dominated by higher energy transitions around the Penngap. Figure 3 also shows the results of a tight-binding ~TB! calculation of D«, discussed in more detail in Ref. 5. An sp 3 s * basis was used and spin-orbit coupling was taken into account. Only the fundamental gap frequency and a slight broadening of the features have been adjusted in the curves of Fig. 3. A reasonable overall agreement is found with the experiment. In particular, the lh-hh splitting matches well. The good correspondence of the hh–e amplitude is somewhat surprising, since e–h interactions are neglected in the TB approach, and these enhance the oscillator strength at the band edge significantly. The neglect of local field effects leads to an overall underestimation of the off-resonant D« 1 . An interesting aspect of the calculation is the relatively strong feature at 1.65 eV due to hh2 –e transitions. The presence of these excitations, forbidden at G, evidences the strong mixing away from the zone center. In some of our experimental spectra a very weak structure is found around 1.65 eV, which may be tentatively assigned to such a ‘‘forbidden’’ transition. These higher sub-band features will be a subject of future investigations.

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m RDS experiments, as presented in this letter, might be easily extended to higher critical points in the band structure, at which the anisotropic effects should be considerably enhanced due to the larger oscillator strength. One may then speculate whether it would be possible to measure confinement induced optical gyrotropy. Such gyrotropic effects are allowed within the D 2d point-group symmetry of ~001!QW’s for propagation along @110#, and have been observed in bulk zincblendes under uniaxial stress.9,10 Although we estimated the gyrotropy at the E 0 gap to be just below the noise level in our m RDS experiments, it might be well measurable at higher frequencies. In conclusion, we presented a new technique for studying in-plane anisotropies in the dielectric response of quantum well structures. m RDS offers phase sensitivity, and can be applied both below and above the fundamental gap. Our application to GaAs/AlAs MQW’s showed a good correspondence with previous transmission approaches and with a tight-binding calculation. An anomalous high birefringence found in previous waveguide experiments could be traced back to an ambiguity in that configuration. m RDS shows interesting prospects for anisotropy studies at higher critical points, of which the study of the confinement induced gyrotropy is especially challenging. Finally, the m RDS technique provides a microscopic probe of effects that reduce the crystal symmetry, such as stress11 and electric fields. The authors would like to thank P. Etchegoin and A. Fainstein for helpful discussions. Further thanks are due to H. Hirt and M. Siemers for technical support, and T. Ruf for a critical reading of the manuscript. B. K. acknowledges the Alexander von Humboldt Stiftung for financial support. 1

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