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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. The quality of this reproduction is dependent upon the quality of the copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand com er and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 THEORETICAL AND EXPERIMENTAL STUDIES ON QUANTUM WELL SPATIAL LIGHT MODULATORS by Man-Fang Huang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Electrical Engineering) December 1995 Copyright 1995 Man-Fang Huang UMI Number: 9621714 UMI Microform 9621714 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90007 This dissertation, written by M w -J:flng..H ua.ag.......................................................... under the direction of h.& > ...... Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of re quirements for the degree of DOCTOR OF PHILOSOPHY Dean o f Graduate Studies Date §ep_ t era b e r 1 5 , 19 9 5 DISSERTATION COMMITTEE Chairperson ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor, Professor Elsa Garmire, for her continuous support and guidance throughout this work. I would like to thank Professor Murray Gershenzon and Professor Edward Goo for serving on my dissertation committee, Professor Milton Bimbaum and Professor William H. Steier for serving on my guidance committee. I wish to thank Dr. Yen-Kuang Kuo for his friendship and valuable discussions during my graduate studies. I would like to thank Dr. Yi-Jen Tsou, Dr. Ching-Mei Yang and Dr. Steffen Koehler for discussions and experiment assistance. I am grateful to Dr. Mingwhei Hong of AT&T Bell Laboratory for growth of the GaAs/AlGaAs MQW samples, to Dr. Afshin Partovi of AT&T Bell Laboratory for fabrication of the GaAs/AlGaAs MQW samples and valuable discussions, to Dr. Tom Hasenberg of Hughes Research Laboratory for growth of the InAs/GaAs SPSLS samples and discussions. It has been a great experience working at the Center for Laser Studies of USC. Thanks also go to Dr. Gary Wang, Ergun Canoglu, Dr. Jean Yang, Hermine Fermanian, Kim L. Reid and all the members of the center for their help and friendship. Finally, this work would not be possible without the support of my parents. I wish to share this achievement with my mother, Shen-Yu Hsueh and my father, Feng-Tu Huang. TABLE OF CONTENTS Page ACKNOWLEDGMENTS .......................................................... ii LIST OF TABLES ........................................................................ v LIST OF FIGURES ...................................................................... vi ABSTRACT ....................................................................................... xi Chapter 1. INTRODUCTION .............................................................. 1 2. LO SS-IN D U CED PH A SE-SEN SITIV E SPA TIA L LIGHT MODULATOR WITH WIDE FIELD OF VIEW 9 Introduction .............................................................. 9 Approximate Theoretical Analysis ........................ 13 Exact Analysis .......................................................... 18 Discussion ................................................................. 21 Conclusion ................................................................ 25 3. OPTICAL CHARACTERISTICS OF GaAs/AlGaAs M ULTIPLE QUANTUM W ELL STRU CTU RES UNDER EXTERNAL ANISOTROPIC STRAIN ........ 28 The Fabrication Process and Transmission Measurements ............................................................ 30 Data Analysis and Discussion .................................. 37 Application in Intensity Modulation ....................... 50 L ii 4. THEORETICAL ANALYSIS ON MQW STRUCTURES UNDER ANISOTROPIC BIAXIAL S T R A IN ................ 61 The Band Structure of a Quantum Well .................... 63 Effect of Strain ............................................................. 66 Absorption Anisotropy ................................................ 79 Discussion ...................................................................... 86 5. FABRICATION PROCESSES OF MQW MODULATORS 111 Processing of the p-i-n MQW modulators ................ 112 GaAs Substrate Removal ............................................ 117 6. InA s/G aA s SHO RT-PERIO D STRAINED-LA YER SUPERLATTICES GROWN ON GaAs AS SPATIAL L IG H T M O D U L A T O R S : U N IF O R M IT Y MEASUREMENTS .............................................................. 129 Sample Structure .......................................................... 132 Experiments ................................................................... 140 Discussion ...................................................................... 154 Appendix A. Derivation of the strain Hamiltonian under biaxial anisotropic strain ................................................................. 91 B. Derivation of dispersion relation and effective mass when the compression strain is along [110] and the tension strain is along [110] ................................................. 99 C. Evaluation of the transition matrix elements for the MQW under anisotropic strain ........................................... 102 iv LIST OF TABLES Table Page 4.1 Parameters used in plotting Fig. 4.2 ............................... 76 5.1 Etching rates and selectivity of citric acid/hydrogen peroxide for GaAs and AlGaAs at different volume ratios for different aluminum concentration .................. 123 6.1 Summary of the SPSLS MQW parameters determined from TEM images ............................................................ 135 6.2 A bsorption coefficient-linew idth product, aAB, for several different samples ................................................. 146 6.3 Average values of a(0) and A a (2, 4, 6, 8 and 10 volts) and distribution range for 15 selected pixels of TCH1364, TCH1365 and TCH1366. The absorption coefficients have a unit of pm -1. The operating wavelength is 961.6 nm for TCH1364, 964.0 nm for TCH1365, and 962.7 nm for TCH1366 ........................ 156 v LIST OF FIGURES Figure Page 2.1 Configuration of the phase modulator that can give a continuous change in phase consists of a three-mirror cavity with lossy slabs inserted between mirrors ......... 11 2.2 Phase diagram explaining the concept of the three- mirror-cavity phase modulator ...................................... 14 2.3 Total phase 0 as a function of the round-trip absorption in each slab region when a phasor analysis is used. The param eters chosen are: ri= 0 .3 , r2=0.46, r3 = l , (|)l=yi=0.7rt, <>2=y2-y 1 =0.62tc, and Er / Ej = 0.1 .......... 17 2.4 (a) The relation between round-trip absorptions Ai and A 2 when the am plitude of the reflected light is maintained constant, (b) Total phase < |> as a function of round-trip absorption in each slab region when Fabry- Perot resonances are included. The parameters used are: ri= 0.3, r2=0.49, r3 = l, 0 1 = 0 .7 177t, and 02=0.71771;, E r/Ei=0.1 20 2.5 (a) The amplitude of the total reflected light as an function of Aj when the length of each slab is longer than the designed length by 15 A. (b) the total phase 0 as a function of round-trip absorptions under the same condition ............................................................................... 23 2.6 (a) The relation between A \ and A2 when the round-trip phases 0i and 02 are different from that in Fig. 4 by 0.05;c. (b) The total phase 0 as a function of round-trip absorption under the same situation .............................. 24 3.1 Schematic diagram of the MQW device under study ...... 34 vi 3.2 Experimental setup for the transmission measurements .. 35 3.3 Transmission spectrum of the MQW sample bonded to sapphire at room temperature without applied voltage ... 35 3.4 Transm ission spectra of the strained MQW with polarization parallel to (a) compression and (b) tension directions for various applied voltages ........................... 36 3.5 Transmission spectra of (a) unstrained MQW (Fig. 3.3) and (b) strained MQW [Fig. 3.4 (a)]. The dashed curves are fitted Fabry-Perot transmission functions .............. 40 3.6 Absorption spectrum of the MQW sample assuming that all absorption is due to the whole MQW region. The solid line is for the sample bonded to sapphire at room temperature (i.e. no strain). The dash-dotted and dotted lines are for the sample bonded to x-cut LiTaC>3 at 150 °C with polarizations parallel to the compression and tension directions, respectively ....................................... 41 3.7 Absorption spectra of the MQW sample bonded to LiTaOj at 150 °C for various applied electric field. The polarization of the incident light is parallel to the (a) compression (y-axis) and (b) tension directions (z-axis of the LiTaC>3 substrate) ....................................................... 45 3.8 Calculated birefringence, which is the difference of the refractiv e index betw een the tw o orthogonal polarizations, as a function of wavelength ..................... 46 3.9 Calculated refractive index changes for the polarization along the compression and tension directions as a function of w avelength with an applied voltage of 3.9 V ............................................................................... 46 vii 3.10 Theoretical HH exciton shift as a function of applied field for both strained and unstrained MQW. Experimental data from Fig. 3.7 are also plotted as a comparison ..... 49 3.11 D ifference betw een the absorption coefficients corresponding to the two different polarizations for various applied electric fields as a function of wavelength .................................................................... 52 3.12 Absorption coefficient difference as a function of applied field at the zero-bias HH exciton peak ........................... 52 3.13 Illustration of polarization rotation caused by an anisotropic absorption and alignment of the analyzer state in order to reduce the off state transmission ........ 56 3.14 On-state transmission for the strained MQW modulator based on polarization rotation as a function of MQW thickness for several An .................................................. 57 3.15 On-state transmission as a function of MQW thickness for several Aa (off) (in unit of p n r 1) ........................... 57 4.1 Compressive strain and tensile strain as a function of bonding temperature. The solid lines are calculated by taking only the first order of the thermal expansion into account. The dashed lines are calculated by considering both the first order and second order thermal expansion coefficients ......................................................................... 70 4.2 Energy band gaps between the lowest conduction band and the HH and LH bands as a function of bonding temperature for bulk GaAs ............................................. 75 viii 4.3 Dispersion curves of GaAs under biaxial anisotropic strain for (a) k//[ 110], (b) k//[J.10] and (c) k//[001] directions ............................................................................. 78 4.4 Polarization selection factors, as given in Eq. (4.40)- (4.43), as a function of bonding temperature for (a) the HH and (b) the LH when the polarization direction of the incident light is along either compression or tension direction. A E q w = 18 meV was used ........................... 85 4.5 Heavy hole polarization selection factors as functions of bonding temperature for two different A E q w when the polarization directions of the incident light are parallel to compression and tension .................................................. 89 4.6 HH polarization selection factors of two polarizations as a function of A E qw for three different AT ................... 89 4.7 Energy difference between HH and LH bands as a function of w ell w idth for several alum inum concentration ..................................................................... 90 5.1 Processing procedure for p-i-n MQW modulator .......... 114 5.2 Processing procedure for image reversal ....................... 115 5.3 Metallic mount for polishing of the MQW sample ......... 119 5.4 Black wax is applied onto the thin film side of the MQW sample. The AlAs release layer can be etched off with dilute HF acid .................................................................... 125 6.1 Schematic diagram of TC H 1364 MQW Structure ......... 136 6.2 TCH 1364 outline .............................................................. 138 6.3 TCH 1365 outline .............................................................. 139 ix 6.4 TCH 1366 outline ............................................................... 141 6.5 I-V curves for TCH 1364, TCH 1365 and TCH 1366 ....... 141 6.6 Experimental setup for transmission measurements. The white light source was focused to a spot of 50 pm x 50 pm ........................................................................................ 142 6.7 Transmission spectra of (a) TCH 1364 (b) TCH 1364 (c) TCH 1366 for various applied voltages .......................... 148 6.8 Absorption spectra of (a) TCH 1364 (b) TCH 1364 (c) TCH 1366 for various applied voltages .......................... 149 6.9 Absorption change for (a) TCH 1364 (b) TCH 1365 (c) TCH 1364 at various applied voltages ............................ 150 6.10 Absorption coefficient changes for various applied voltages for the selected pixels of sample TCH 1364 ..... 151 6.11 Absorption coefficient changes for various applied voltages for the selected pixels of sample TCH 1365 ..... 152 6.12 Absorption coefficient changes for various applied voltages for the selected pixels of sample TCH 1366 ..... 153 6.13 O n/off ratio and dynamic as a function of applied voltages for TCH 1364 159 6.14 Insertion loss for (a) TCH1364 (b) TCH1365 (c) TCH 1366 163 6.15 Excitonic peak wavelength of selected pixels for (a) TCH1364 (b) TCH 1365 (c) TCH 1366 ........................... 164 X ABSTRACT Three topics, which include the design of multiple quantum well (MQW) phase and intensity spatial light modualtors (SLM) and the study of the uniformity of spatial light modulators, are investigated in this dissertation. First, the feasibility of achieving practical phase modulation by only changing the absorption of an MQW device is investigated. A SLM that consists of a three-mirror cavity with lossy slabs inserted between the mirrors is presented. By changing only the absorption of these slabs, this device can have a gradual change in phase, but no intensity modulation. Theoretical analysis shows that the total phase tuning range can be as large as 2k with an absorption coefficient change less than 9,000 cm-1; the length of each MQW slab is around 1.5 pm . The fact that the overall transmission is 1% make this a challeging approach. Also explored in this dissertation are the room temperature anisotropic absorption characteristics of a GaAs/AlGaAs multiple quantum well structure under anisotropic biaxial strain. This was achieved in a sample that was bonded to an x-cut LiTaC>3 substrate at 150 °C, with the [110] and [110] directions of the multiple quantum well along the y and z axes of LiTaC>3 , respectively. A 25% stronger heavy hole excitonic feature can be observed when the polarization of the incident light is in the compression direction, which thus results in larger quantum confined Stark effect. Theory on MQW structures under anisotropic biaxial strain, which considers both spin-orbital xi Hamiltonian and strain Hamiltonian, is developed and used to explain the experimental results. The uniformity of two-dimensional transmission-type spatial light modulators using InAs/GaAs short-period strained-layer superlattices (SPSLSs) in wells, which operate near 960 nm, are investigated. The absorption characteristics of the individual pixels across a -2 0 0 pixellated wafer are measured. The results show that the variation of the absorption coefficient change due to nonuniformity is around ±8% at an applied voltage of 10 volts for a fixed operating wavelength, which causes a ±2% deviation in on/off ratio and ±0.1 dB change in dynamic change. Chapter 1 Introduction Optical modulation is an important aspect of photonic circuits since it impresses information on an optical beam. This may be done either by direct modulation of the optical source or through a separate, external device. Basically, optical modulators change some property of the lightwave, such as intensity, phase, frequency, or polarization. Also, they operate in either guided-wave form or free-space form. Modulators that are integrated in guided-wave form typically introduce an optical change and transmit the optical beam. These devices should be compatible with optical fibers, which are the mainstay of optical communications. Optical modulators that are implemented in free-space form may be operated in either transmissive or a reflective mode. Arrays of modulators are basic elements of spatial light modulators for use in optical computing and optical signal processing. Because semiconductor modulators can be performed at very high speed using small structures with low power consumption [1] they are good candidates for optical modulators. Recently, m odem crystal growth techniques make it possible to grow alternate ultra thin layers of different compounds to form multiple quantum well (MQW) structures. Because of the band gap difference between the two components, the electron-hole pairs, or excitons, are confined in the low gap layers. Due to this confinement, exciton binding energies are greatly enhanced and excitonic states become stable even at room temperature [2], The 1 excitonic absorption is strongly influenced by an applied electric field, in both strength and position of the excitonic absorption peak. The red shift of the excitonic electroabsorption, when an electric field is applied perpendicular to the layers, has been known as the quantum confined Stark effect (QCSE) [3]. A number of MQW modulators based on QCSE are developed. Potential applications of MQW modulators in all- optical logic devices [4], optical interconnects [5,6] and all-optical signal processing [7,8] have been reported. In this work, the characteristics of both optical phase modulators and MQW intensity modulators are investigated. The uniformity of the MQW devices in an array is also studied. Optical phase modulators are of interest for applications in fiber optic communications or in a highly parallel multi-element geometry for image processing [9-11]. The phase of an optical beam can be tuned by varying the optical path length. Since the electro-optical effects are usually weak, such phase modulators use waveguide geometries or rather long length. However, for some applications phase modulators operate on light normal to the plane of the device rather than in a waveguide configuration [12]. Recently, a binary phase modulator in a vertical geometry using a Fabry-Perot cavity was presented [13]. However, this device is unsuitable for applications where a gradual change in phase is required [14]. In chapter 2, a phase modulator that can give a continuous change in phase is investigated. Electric field induced absorption change in GaAs/AlGaAs MQW structures, i.e., the quantum confined Stark effect (QCSE), has received 2 considerable attention for modulator applications. However, the on/off contrast ratio of GaAs/AlGaAs transm ission modulators has been limited by the finite absorption coefficient change due to broadening of the exciton peak [15]. In order to enhance the contrast ratio, a MQW structure can be integrated in an asymmetric Fabry-Perot (AFP) [16]. Although an AFP modulator can provide a high on/off ratio, fabrication of these devices requires strict control over composition and thickness [17]. For large wafers, the nonuniformity of growth across the wafer degrades performance. Quite recently, optical anisotropy in GaAs/AlGaAs multiple quantum well (MQW) subject to thermally induced uniaxial and biaxial strain, and optically addressed high contrast m odulation due to polarization rotation in transmission through such thermally strained MQW were investigated by Shen et al. [18-22]. In Refs. 18-22 the MQW thin films were bonded to the substrates in such a way that the strain was along the crystalline axes of the MQW. Calculations show that if the strain is applied along the [110] and [110] directions of the MQW, for the same MQW structure, the zero-field excitonic absorption anisotropy is 25% larger than those obtainable from a MQW with biaxial strain along the crystalline axes of the MQW and a MQW with uniaxial strain. In chapter 3, the room temperature optical absorption characteristics of a GaAs/AlGaAs MQW structure under anisotropic biaxial strain is investigated. The strain was produced by bonding the MQW to a LiTaC>3 substrate at 150 °C with [110] and [110] of the MQW along the y and z axes of LiTaC>3, respectively. The difference in the 3 thermal expansion coefficient of the MQW and the LiTaC>3 produces the com bined compressive and tensile strain on the device at room temperature. The field dependence of the optical absorption of the resulting strained GaAs/AlGaAs multiple quantum well is also studied. Quantum confined Stark effect m easurem ents made in the two orthogonal polarizations at various fields up to 7.4 V/pm are reported. Some fabrication processes of such MQW devices are described in detail in chapter 5. In chapter 4, the anisotropic absorption properties of the GaAs/AlGaAs MQW structures under thermally-induced biaxial strain are theoretically investigated. The analyses are based on a theoretical model which considers both the spin-orbital Hamiltonian and the strain Hamiltonian. The wave functions, which can be derived from solving the total Hamiltonians, are used to calculate the dipole matrix elements for interband transitions and evaluate the anisotropic absorption properties. Effect of variation of parameters such as well width, barrier height on the performance of the strained GaAs/AlGaAs MQW electroabsorption modulators are discussed. In the past few years, most of the electro-absorption modulators have been made in the GaAs/AlGaAs material system. However, all these devices work at wavelength between 800 and 880 nm and require substrate removal. Quite recently, promising results have been obtained in the fabrication of surface emitting lasers with the use of strained layer growth of InGaAs on GaAs [23]. These lasers typically emit in the wavelength region between 900 and 1100 nm, a region where the 4 GaAs substrate is transparent, which is useful both for lasers and for modulators. Fabrication of optical modulators which are compatible with these lasers can have an important contribution to the realization of the new optical computing structures. All-binary InAs/GaAs short-period strained-layer superlattices (SPSLSs) have received much attention recently [24-29]. These all binary InAs/GaAs SPSLSs are an ordered counterpart for the ternary InGaAs strained alloys. The uniform ity of tw o-dim ensional transmission-type spatial light modulators using InAs/GaAs SPSLSs in wells, which operate near 960 nm, is studied in chapter 6. The absorption characteristics of the individual pixels over the entire semiconductor wafer are investigated. This study is im portant for parallel optical signal processing, in which the light beam is incident perpendicular to the two-dimensional arrays of modulators. The change in the performance of the SLM due to nonuniformity is discussed. 5 R eferences [1] N. Peyghambarian and H. M. Gibbs, "Optical nonlinearity, bistability, and signal processing in semiconductors", J. Opt. Soc. Am. B, vol. 2, pp. 1215-1227, 1985. [2] H. Okamoto, "Semiconductor quantum-well structures for optoelectronics -recent advances and future prospects-, Jap. J. Appl. Phys., vol. 26, pp. 315-330, 1987. [3] D. A. B. Miller, "Band-edge electroabsorption in quantum well structures: the quantum-confined Stark effect", Phys. Rev. Lett., vol. 53, pp. 2173-2176, 1984. [4] A. Stohr, "InGaAs/GaAs multiple-quantum-well modulators and switches", Opt. and Quan. Elec., vol. 25, pp. S865-S883, 1993. [5] C. C. Barron, "Design, fabrication and characterization of high-speed asymmetric Fabry-Perot modulators for optical interconnect applications", Opt. and Quan. Elec., vol. 25, pp. S885-S898, 1993. [6] A. L. Lentine, "Photonic switching nodes based on self electro-optic effect devices", Opt. and Quan. Elec., vol. 24, pp. S443-S464, 1992. [7] T. Y. Hsu, "Multiple quantum well spatial light modulators for optical processing applications", Opt. Engr., vol. 27, PP. 372-383, 1988. [8] R. G. Walker, "High-speed II1-V semiconductor intensity modulators", IEEE J. Quantum Electron., vol. QE-27, pp. 654-667, 1991. [9] D. R. Weight, J. M. Heaton, B. T. Hughes, J. C. H. Birbeck, K. P. Hiton and D. J. Taylor "Novel Phased Array Optical Scanning Device Implemented Using GaAs/AlGaAs Technology," Appl. Phys. Lett., vol. 59, pp. 899-901,1991. [10] T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, "Joint transform Correlator Using a Phase Only Spatial Light Modulator," Jap. J. Appl. Phys., vol. 29, pp. L1293-L1296 (1990). [11] T. H. Barnes, T. Eiju, K. Matsuda, H. Ichikawa, M. R. Taghizadeh, and J. Turunen "Reconfigurable Free-space Optical Interconnections with a Phase-only Liquids-crystal Spatial Light Modulators," Appl. Opt., vol. 31, pp. 5527-5535, 1992. [12] J. A. Neff, R. A. Athale and S. H. Lee, "Two-Dimensional Spatial Light Modulators: A Tutorial," Proc. IEEE, vol. 78, pp. 826-855,1990. [13] B. Pezeshki, G. A. Williams, and J. S. Harris Jr., "Optical Phase Modulator Utilizing Electroabsorption in a Fabry-Perot Cavity," Appl. Phys. Lett., vol. 60, pp. 1061-1063, 1992. [14] T. H. Barnes, K. Matsumoto, T. Tiju, K. Matsuda, and N. Ooyama, "The Application of Phase-only Filters to Optical Interconnects and Pattern Recognition," J. Modem Opt., vol. 37, pp. 1849-1863, 1990. [15] T. Y. Hsu, "Amplitude and phase modulation in a 4 |im-thick GaAs/AlGaAs multiple quantum well modulator", Elec. Lett., vol. 24, pp. 603-604,1988. [16] See, for example, R. H. Yan, R. J. Simes, and L. A. Coldren, IEEE Photon. Technol. Lett., vol. 1, p. 273, 1989. [17] A. Obeidat, J. Khurgin, S. Li, Opt. and Quan. Elec., vol. 25, p. S917, 1993. [18] H. Shen, M. Wraback, J. Pamulapati, P. G. Newman, M. Dutta, Y. Lu, and H. C. Kuo, "Optical anisotropy in GaAs/Al,Gai.,As multiple quantum wells under thermally induced uniaxial strain," Phys. Rev. B, vol. 47, pp. 13933-13936, 1993. [19] Y. Lu, H. C. Kuo, H. Shen, M. Taysing-Lara, M. Wraback, J. Pamulapati, M. Dutta, J. Kosinski and R. Sacks, "Creation of in-plane anisotropic strain in GaAs/AljGa^As multiple quantum well structures," Mat. Res. Soc. Proc., vol. 300, pp. 537-542, 1993. [20] H. Shen, M. Wraback, J. Pamulapati, M. Dutta, P. G. Newman, A. Ballato, and Y. Lu, "Normal incidence high contrast multiple quantum well light modulator based on polarization rotation," Appl. Phys. Lett., vol. 62, pp. 2908-2910, 1993. [21] H. Shen, M. Wraback, J. Pamulapati, M Dutta, J. Kosinski, W. Chang, P. Newman and Y. Lu, "A novel high contrast multiple quantum well spatial light 7 modulator based on polarization rotation," OSA Tech. Digest, vol. 6, pp. 161- 164, 1993. [22] H. Shen, J. Pamulapati, M. Wraback, M. Taysing-Lara, M. Dutta, H. C. Kuo, and Y. Lu, "High contrast optical modulator based on electrically tunable polarization rotation and phase retardation in uniaxially strained (100) multiple quantum wells," IEEE Photonics Technology Letters, vol. 6, pp. 700-702,1994. [23] See, for example, C. J. Chang-Hasnaian, M. W. Maeda, N. G. Stoffel, J. P. Harbison, L. T. Florez and J. Jewell, "Surface emitting laser arrays with uniformly separated wavelengths", Electron. Lett., vol. 26, pp. 940-942, 1990. [24] T. C. Hasenberg, D. S. McCallum, X. R. Huang, A. L. Smirl, M. D. Dawson and T. F. Boggess, "Optical studies of InAs/GaAs on GaAs short-period strained-layer superlattices grown by MBE and MEE", J. Crystal Growth, vol. Ill, pp. 388-392, 1991. [25] F. J. Grunthaner, M. Y. Yen, R. Fernandez, T. C. Lee. A. Madhukar and B. F. Lewis, "Molecular beam epitaxial growth and transmission electron microscopy studies of thin GaAs/InAs (100) multiple quantum well structures," Appl. Phys. Lett., vol. 46, pp. 983-985, 1985. [26] J. M. Gerard, J. Y. Marzin, B. Jusserand, F. Glas and J. Primot, "Structural and optical properties of high quality InAs/GaAs short-period superlattices grown by migration-enhanced epitaxy," Appl. Phys. Lett., vol. 51, pp. 30-32,1989 [27] Takafumi Yao, "A new high-electron mobility monolayer superlattice," Jan. J. Appl. Phys., vol. 22, pp. L680-L682, 1983. [28] H. Toyoshima, K. Onda, E. Mizuki, N. Samoto, M. Kuzuhara, T. Itoh, A. Okamoto, T. Anan and T. Ichihashi, "Molecular-beam eptiaxial growth of InAs/GaAs superlattice channel modulation-doped field-effect transistor," J. Appl. Phys., vol. 69, pp. 3941-3949, 1991. [29] J. Lopata, N. K. Dutta and N. Chand, "(InAs)l/(GaAs)n superlattice quantum well lasers," Mat. Res. Soc. Symp. Proc., vol. 281, pp. 287-292, 1993. 8 Chapter 2 Loss-Induced Phase-Sensitive Spatial Light Modulator with Wide Field of View In this chapter the feasibility of achieving practical phase modulation by only changing the absorption of the device is investigated [1]. We present here a novel spatial light modulator (SLM) that consists of a three-mirror cavity with lossy slabs inserted between the mirrors. By changing only the absorption of these slabs, this device can have a gradual change in phase, but no intensity modulation. Theoretical analysis, using numbers appropriate to multiple quantum wells (MQW), shows that the total phase tuning range can be as large as 2n with an absorption coefficient change less than 9,000 cm-1; the length of each MQW slab is around 1.5 pm. This gives an expected 16° field of view with the nanosecond speed typical of semiconductors. 2.1 Introduction Optical phase modulators are of interest for applications such as in either single element or few element geometries for fiber-optic communications or in a highly parallel multi-element geometry for image processing [2-4], Conventionally, the phase of an optical beam is tuned by changing the optical path length. Since the electro-optical effects are usually weak, such phase m odulators use waveguide geometries or rather long length. However, for some applications such as image processing and optical computing, phase modulators operate on 9 light normal to the plane of the device rather than in a waveguide configuration [5]. Phase modulation usually requires a long path length. Recently, a binary phase modulator in a vertical geometry using a Fabry-Perot cavity was presented [6]. However, this device is unsuitable for applications where a gradual change in phase is required [7]. A phase modulator that can give a continuous change in phase consists of a three-m irror cavity with lossy slabs inserted between mirrors. The configuration is shown in Fig. 2.1; the lengths of the lossy slabs are Li and L 2 , and ri, r2 and r3 are the electric field reflection coefficients of the mirrors 1, 2, and 3, respectively. The idea is to control the phase of the reflected light from all three mirrors by varying only the absorption of the slabs, holding the reflection coefficients of the mirrors fixed. As we will show later, the phase of the reflected light can be tuned by changing only the absorption coefficients within the slabs; the intensity of the reflected light can be maintained constant if the absorption of the two slabs is varied in a prescribed way, determined by numerical calculation, as we will discuss below. 10 Er e * SLAB 1 Mirror 1 SLAB 2 Mirror 2 N Mirror 3 Fig. 2.1. Configuration of the phase modulator that can give a continuous change in phase consists of a three-mirror cavity with lossy slabs inserted between mirrors. 1 1 The optimization of such a phase modulator depends on several param eters. In order to find where in the param eter space the modulator will operate most efficiently, an approximate phasor analysis can be used to estimate values of the relevant parameters. These approximate solutions are then used as initial values for the exact Fabry- Perot resonance analysis. In this chapter, the analysis of this phase- sensitive SLM using both the approximate phasor and exact Fabry-Perot resonance approaches is presented. We show that there is a tradeoff between overall reflectivity and the amount of required absorption change. We find that, using numbers typical of semiconductor MQW, we can achieve the m odulator with A a =9,000 cm - 1 and overall reflectivity of 1%. Higher reflectivity can be achieved if larger A a is available (provided that the values of the absorption coefficients are within the possible range of the semiconductor material utilized.) In real QW, there is usually an index change An whenever there is an absorption change A a [8 ]. By proper cavity design this index change can be take into account. It is not the purpose of this study to present a complete design for such a MQW-SLM, but only to investigate the feasibility of our initial concept: can absorption changes alone be sufficient to achieve practical phase modulation? We will find, finally, that the fabrication tolerances for this device are very stringent on Fabry-Perot length. 1 2 2.2 Approximate Theoretical Analysis We first demonstrate how to choose relevant param eters to achieve a full 2k phase-sensitive SLM by using an approximate phasor analysis. In this approach, we consider only first order reflected light E i, E2 and E3 , which are the fields reflected from mirrors 1, 2 and 3, respectively. The electric fields of the reflected waves can be treated as three phasors and expressed as follows: E 3 = P 3eiy2 = Ei r3 ( l- r i 2 )(l-r22) e-(Ai+A2 ) e i(< t> i+ < l >2) (2.3) where < } > 1 and <>2 are the round-trip phases and A i and A 2 are the round-trip absorptions of first and second slab, respectively, and where O ti is the intensity absorption coefficient of the corresponding lossy medium and T is the filling factor of the lossy medium within the slab (the fraction of the slab that contains, for example, QW). These equations are valid in the limit that multiple reflections can be ignored. The phase diagram in Fig. 2.2 shows that, by fixing length Pi and varying the lengths, P2 and P3, of the two vectors describing reflection from r2 and r3, a total phase tuning range of 2 k can be obtained without changing the amplitude of the resultant final vector. It indicates that we Ei = Pi = Ei ri E 2 = P 2 e»yi = Ei r2 (1-ri2) e-Ai e i(t > i (2.1) (2.2) Aj — (X j L; Tj, i — 1, 2 (2.4) 1 3 can have a pure phase modulator by varying only absorption A i and A2 (i.e. lengths of vectors) in a prescribed way. The direction of each vector is fixed since only absorption modulation is considered here. Fig. 2.2. Phase diagram explaining the concept of the three-mirror-cavity phase modulator. 14 The total reflected field for this model is given by Er e i(t > = Ei + E2 + E3 (2.5) where Er and < } > are the amplitude and the phase of the reflected light. Solving Eq. (2.5) gives Er = V [Pl+P 2COs(yi)+P3Cos(y2)]2+[P2sin(yi)+P3sin(y2 ) ] 2 (2.6) In order to ensure no intensity modulation, let Er be a constant and then solve Eq. (2.6) for P3 as a function o f P2 . However, parameters Er, P i, y i and y i have to be properly chosen so that a 2 tu phase shift can be obtained. Equations (2.7) and (2.8) indicate that only when yi and y2 are located in quadrants I and III or II and III or II and IV, is it possible that cos< |> and sin ({ > can be either sm aller or larger than zero. Only then is it possible that < | > can be located in any quadrant, i.e. an entire phase tuning range of 2k is possible. Figure 2.2 shows an example when yi is equal to 0.7tc, which is in quadrant II, and y2 is equal to 1.32tc, which is in quadrant III. Further analysis shows that the ratio Er / Pi has to be less than unity in order to have 2tc phase change. This implies that the total reflectivity of the phase m odulator m ust be sm aller than the reflectivity of the first mirror. Er cos ( J ) = Pi + P 2 cos yi + P 3 cos y2 Er sin < > = P2 sin yi + P3 sin y i . (2.7) (2.8) 1 5 The values of P2 and P 3 , which are calculated from the phasor analysis, can then be applied to the following equations, which are derived from Eqs. (2.2) and (2.3), to give the values of the absorption A i and A2 A i = log [ n ( l-ri 2 )] -lo g < | | ) (2.9) A 2 = log [ ^ (l-r 2 2 )] - log ( P | ) (2 .1 0 ) We intentionally set the reflection coefficient r3 equal to unity to give the highest reflectivity possible. Since ri is related to P i, r2 can then be arbitrarily varied until practical values of A i and A2 are obtained. Figure 2.3 shows that phase < ) > can vary from -n to k when the absorption Ai is changed from 0.8 down to 0.1 and back to 0.95 while A 2 is changed from 0.6 to 1.1, back down to 0.1 and back to 0.6. The param eters chosen were ri= 0 .3 , r2=0.46, r 3 = l, < |> i = yi=0.7rc, <l>2=y2-yi=0.627t, and Er / Ei = 0.1. Note that when Ai is changed, A 2 is varied accordingly so that the intensity of the reflected light can be maintained constant. 16 C/3 2 < D c n a X O n o MQW1 ^ MQW2 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 1.2 Round-Trip Absorption Fig. 2.3. Total phase < |> as a function of the round-trip absorption in each slab region when a phasor analysis is used. The parameters chosen are: n=0.3, r2=0.46, r3 = l, <t>i=yi=0.7jc, < ( > 2 =y2 -y l=0.627t, and Er/ Ej = 0.1. 17 2.3 Exact Analysis Using the above first order approximation, we have demonstrated that phase-modulation can be controlled while m aintaining constant amplitude. However, the phasor analysis is approximate because it ignores multiple reflections. We now show that we can still have a pure phase modulation by doing the exact Fabry-Perot analysis including multiple reflections. For the three-mirror Fabry-Perot cavity shown in Fig. 2.1, the overall reflection coefficient is Er ^ _ r3 e~(A i+ A 2) + r2 e~A i+i<l)i + r i T2 n e~A 2+i<l> 2 + r i C2 11) Ei n r3 e‘(A+A2) e'Wi+fo) + rj r2 e'Ai+i'fri + r2 r3 + 1 The procedure to solve this problem is similar to the one that we discussed before. We use the results of section 2.2 as initial guesses for the proper parameter values. Since the contribution coming from higher order reflections is small compared to the first order reflection, we can expect that the parameters to be used in the exact result calculated from Eq. (2.11) will be close to those determined from the approximate phasor analysis. To ensure no intensity modulation, let Er/Ei be a constant and then numerically solve A2 as a function of A 1 by using Eq. (2.11). The parameters that we came up with in the first order approximation can be used as the initial values. Calculation shows that a total tuning range of 2 k can be easily obtained by gradually changing < j )2 with < J > i fixed at its initial value. We can then vary r2 in order to derive operating conditions which use absorptions A i and A 2 18 which are practical for fabrication. Figure 2.4(a) shows the relation between A i and A2 when the intensity of the total reflected light is maintained constant. As shown in Fig. 2.4(b), a total phase tuning range of 2tc is possible. In this particular design, Ai varies with a minimum of 0.17 to a maximum of 1.06 and A2 varies from 0.21 to 1.12 with param eters Er/E i=0.1, ri= 0.3, r2 =0 .4 9 , r3 = l, <j>i=0.7177c, and <}>2=0.71 7ic. To this point, the calculation has been general, with parameters chosen to give reasonable maximum and minimum values of absorption. 19 A2 1 3 u C 3 ■ C a. 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.2 1 AI ( a ) O MQW1 MQW2 1 ! ' ' 'A * * * * * 0.5 -0.5 $ B > , 1 0 0.4 0.2 0.6 0.8 1 1.2 Round-Trip Absorption (b) Fig. 2.4. (a) The relation between round-trip absorptions Ai and A2 when the amplitude of the reflected light is maintained constant, (b) Total phase < } > as a function of round-trip absorption in each slab region when Fabry-Perot resonances are included. The parameters used are: ri= 0.3, r2 =0 .4 9 , r 3 = l, <|)i=0.717ji, and <j>2=0.717ti, Er/Ej=0.1. 20 Suppose that this device is operated using a MQW as described by Lengyel [9]. In order to have practical absorption coefficients, we calculate the required length of each MQW slab, assuming it composed o f a 96-period 105A GaAs well and a 96-period 50A Alo.32Gao.6 8 As barrier, and find a result of L = 1.5 J im for each slab. The required ranges of the absorption coefficients are from 1705 c m 1 to 10,360 cm -1 for MQW1 and from 2076 c m 1 to 11,000 c m 1 for MQW2. These calculations assume a wavelength set at 853 nm (~ 0.012 eV to the long wavelength side of the peak of the zero-field excitonic absorption) as indicated in Fig. 5 of reference 9. A suitable mirror between two MQW slabs consists of 5 pairs of Alo.3 2Gao.6 8 As/AlAs (583A/666A) quarter-wave layers, providing sufficient reflection. 2.4 Discussion In this design we have ignored the refractive index change which occurs in MQW when there is an absorption change (i.e. electro reflection). If we consider this refractive index change in MQW, clearly the shape of Fig. 2.4 will change. Also, the < J > 1 & < J >2 for optimized performance will be somewhat different. This can be understood by imagining that the phasors P2 and P3 in Fig. 2.2 become somewhat tilted with field. We could easily adjust our analysis to include this change, but it would be very device-specific. Rather than design a specific MQW structure as exactly as possible, we choose to here investigate how sensitive the design is to fabrication tolerances. As an example, assume that the length of each MQW slab is longer than the designed length by 15 A. This fabrication error corresponds to an increase of 0.057c for both < |> i and < ( > 2 . If the relation between Ai and A 2 calculated from the original parameters is still followed, then the amplitude of the total reflected light w ill no longer be constant. Figure 2.5 shows the am plitude of the total reflected light as a function of Ai and the total phase < |> as a function of round-trip absorptions A i and A2 for this case. If the thickness of each MQW slab can be measured, we can easily find out the new relation between Ai and A2 to keep the total intensity of the reflected light constant. Alternatively, the relation between Ai and A 2 to keep the total intensity constant can be learned through a feedback teaching circuit after the device is built. As a demonstration o f this adjustment, Fig. 2.6(a) shows how A i varies with A2 when < ( )i and §2 change to 0 .7 6 7 c and Fig. 2.6(b) shows the total phase as a function of A i and A2 . This is the limiting case, because it requires Ai — > 0. 22 & T 3 2 a. B < c /3 T3 5 ® 4 ) C/3 a JS C U 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 AI ( a ) O MQW1 a MQW2 1 0.5 0 -0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 Round-Trip Absorption (b) Fig. 2.5. (a) The amplitude of the total reflected light as a function of Aj when the length of each slab is longer than the designed length by 15 A. (b) the total phase < |> as a function of round-trip absorptions under the same condition. 23 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.2 1 AI ( a ) o MQW1 A MQW2 0.5 "O 2 -0.5 0.2 0.4 0.6 0.8 0 1.2 1.4 1 Round-Trip Absorption (b) Fig. 2.6. (a) The relation between Aj and A2 when the round-trip phases < ( > i and < j >2 are different from that in Fig. 4 by 0.0571. (b) The total phase $ as a function of round-trip absorption under the same situation. 24 Finally, the field of view of a phase m odulator operating in reflection is given by the requirement that nkL(sec0 -1) « 7t, (2.12) When the maximum index change is 8 n, the traditional phase modulator requires that L be equal to A,/28n. So after making a sm all angle simplification, Eq. (2.12) becomes Often fast nonlinearities, such as the electro-optic effect, have 8 n on the order of 10 4, so 0 « 0.01 rad. The phase modulator proposed here has total length ~ 3 pm, n ~ 3.5, X ~ 0.85 pm, so that the field o f view is given by 0.28 rad (16°), or almost 30 times increase in the field of view. The MQW nonlinearities can have nanosecond speeds, providing, for the first time, the possibility of fast, wide field of view phase modulators. 2,5 Conclusion A new wide field of view phase modulator has been suggested using the principal that large absorption changes can be used in a three mirror cavity to achieve large phase changes. The proposed device can act purely as a phase modulator without intensity modulation. W e have shown that the total phase can be tuned from -n to n gradually by (2.13) 25 changing the absorption coefficients of two slabs within a three-mirror cavity. The proposed configuration could be a practical design since GaAs/AlGaAs MQWs required to achieve the desired performance can be fabricated. However, it appears to be necessary to control the tolerances very closely. 26 R eferences [1] Man-Fang Huang and Elsa Garmire, "Loss-induced phase-sensitive spatial modulator with a wide field of view," Appl. Optics vol. 33, pp. 2856-2860, 1994. [2] D. R. Weight, J. M. Heaton, B. T. Hughes, J. C. H. Birbeck, K. P. Hiton and D. J. Taylor "Novel Phased Array Optical Scanning Device Implemented Using GaAs/AlGaAs Technology," Appl. Phys. Lett., vol. 59, pp. 899-901, 1991. [3] T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, "Joint transform Correlator Using a Phase Only Spatial Light Modulator," Jap. J. Appl. Phys., vol. 29, pp. L1293-L1296, 1990. [4] T. H. Barnes, T. Eiju, K. Matsuda, H. Ichikawa, M. R. Taghizadeh, and J. Turunen "Reconfigurable Free-space Optical Interconnections with a Phase-only Liquids-crystal Spatial Light Modulators," Appl. Opt., vol. 31, pp. 5527-5535, 1992. [5] J. A. Neff, R. A. Athale and S. H. Lee, "Two-Dimensional Spatial Light Modulators: A Tutorial," Proc. IEEE, vol. 78, pp. 826-855,1990. [6 ] B. Pezeshki, G. A. Williams, and J. S. Harris Jr., "Optical Phase Modulator Utilizing Electroabsorption in a Fabry-Perot Cavity," Appl. Phys. Lett., vol. 60, pp. 1061-1063, 1992. [7] T. H. Barnes, K. Matsumoto, T. Tiju, K. Matsuda, and N. Ooyama, "The Application of Phase-only Filters to Optical Interconnects and Pattern Recognition," J. Modern Opt., vol. 37, pp. 1849-1863, 1990. [8 ] J. I. Pankove, Optical Processes in Semiconductors. (Dover, New York, 1975), Chap. 4, p.89. [9] G. Lengyel, W. Jelly and R. W. H. Engelmann, "A Semi-empirical model for Electroabsorption in GaAs/AlGaAs Multiple Quantum Modulator Structures," IEEE Quan. Elec., vol. 26, 296-304, 1990. 27 Chapter 3 Optical Characteristics of GaAs/AlGaAs Multiple Quantum Well Structures Under External Anisotropic Strain Electric field induced absorption change in multiple quantum well (MQW) structures, i.e., the quantum confined Stark effect (QCSE) [1], has received considerable attention for modulator applications in the past few years. Generally, these modulators can be classified as following two major categories: "norm ally-off modulators operate in the wavelength region with strong excitonic absorption at zero bias and reduce its optical absorption w ith applied voltages; "normally-on" modulators, on the other hand, operate at the long wavelength side of the exciton with a low zero-bias absorption. The self-electro-optic effect devices (SEED) are typical norm ally-off modulators which require a reduction in optical absorption with applied field to yield a negative differential photoconductivity and thus, bistable characteristics [2]. Normally-on modulators use essentially the same absorption change, but have a low residual absorption in the on-state. No m atter which type of modulator is used, the on/off contrast ratio of a transmission-type GaAs/AlGaAs modulator has been limited to be less than 10:1 by the finite absorption coefficient change [3]. Effort has been made to improve the performance of these modulators. Some effort has been focused on increasing the electroabsorption of the MQW by altering its structure, for example, by optimizing the well width [4-6], barrier height [6 -8 ] and barrier thickness [6,9]. Several 28 QW structures such as graded-gap [10], coupled asymmetric [11] and three-step asymmetric [ 1 2 ] have been investigated in an effort to enhance the on/off ratio of the modulators. To further increase the m odulation perform ance, a MQW structure can be incorporated with an asymmetric Fabry-Perot (AFP) [13]. This device uses the absorption change due to QCSE to change the reflectivity of a Fabry-Perot etalon at a wavelength corresponding to a cavity mode. Although an AFP modulator can provide a high on/off ratio, fabrication of these devices requires strict control over composition and thickness [14]. For large wafers, the nonuniformity of growth across the wafer degrades performance. Further, to be low- loss, they m ust be used in reflection, which limits their system capabilities. Recently, transm ission-type G aA s/A lG aA s MQW modulators based on polarization rotation with high on/off contrast ratio were demonstrated [15,16]. These modulators em ployed the anisotropic excitonic absorption created by a thermally induced uniaxial strain to achieve the polarization rotation and improve the on/off ratio. In this chapter, the enhancem ent o f heavy-hole excitonic absorption and quantum confined Stark effect for a GaAs/AlGaAs multiple quantum well structure under simultaneous compressive and tensile strain along two perpendicular directions is discussed. This anisotropic strain was produced by bonding the MQW to a LiTaC>3 substrate at 150 °C with [110] and [110] of the MQW along the y and z axes of LiTaC>3, respectively. The difference in the thermal expansion coefficient o f the MQW and the LiTaC>3 produces com bined 29 compressive and tensile strain on the device at room temperature [15- 17]. The modulator performance for this biaxially strained MQW structure is also investigated. The fabrication process and transmission measurements for this strained GaAs/AlGaAs MQW structure are described in section 3.1. The removal of the oscillation in transmission spectra due to Fabry- Perot effect and the experimental results of transmission measurements are discussed in section 3.2. The birefringence due to the absorption difference between the two orthogonal polarizations will also be addressed. Finally, in section 3.3, the prospective application of this strained MQW as an intensity modulator is discussed. 3.1 The Fabrication Process and Transmission Measurements A m ultiple quantum well p-i-n structure was grown by Dr. Minghwei Hong of AT&T Bell Laboratories using molecular beam epitaxy on (lOO)-oriented GaAs substrate for this study. A n-doped GaAs layer of 1000A was grown on the n+ GaAs substrate first, with doping concentration of lxlO 18 cm 3, followed by a 5000A layer of n- doped Alo.5Gao.5As. An intrinsic MQW structure consisting of 30 periods of alternating intrinsic 75 A GaAs and 100 A Alo.3Gao.7As layers was then grown on top of the n-doped layer. The p-i-n structure was completed by growing a 500A p-doped Alo.5Gao.5As on top of the i layer with doping concentration of l x l 0 1 8 cm-3, followed by a 100A highly-p-doped GaAs layer. 30 The p-type contact was deposited to the p-doped layer of the MQW sample. This MQW device was then bonded to an x-cut LiTaC>3 substrate at 150°C with the p side of the MQW sample attached to LiTaC>3 , using a thin layer of transparent, high-strength epoxy [18]. This MQW was bonded to LiTaC^ in such a way that the [110] and [110] directions of the MQW are along the y and z axes of LiTaC>3 , respectively. This sample was then left in the oven for a few hours until the epoxy was cured. Subsequently, the GaAs substrate was removed using chemical etching (detail of chemical etching is described in chapter 5). The n-type contact was then deposited on the n-doped layer of the MQW sample. Finally, this sample was etched to reveal the p- type contact. Fabrication of the device was completed by bonding both n and p-type contacts with metallic wires. The schematic diagram of the device is shown in Fig. 3.1. All fabrication was carried out by Dr. Afshin Partovi of AT&T Bell Laboratories. The reason to choose x-cut (or y-cut) LiTaC>3 is because the thermal expansion coefficient along the z-axis, £z, is different from that along the x (or y) axis, £// [15]. Moreover, the thermal expansion coefficient of the MQW, Cmqw. is larger than £z but smaller than £//. Therefore, when the temperature was lowered from 150 °C down to room temperature, the MQW suffered compressive strain, calculated to be -0.116% along the [110] direction, and tensile strain of 0.035% along [ 110]. Since the thickness of the MQW epitaxial thin film (a few pm) is much smaller than that of the LiTaCb (0.45 mm) and the epoxy is very 3 1 thin, it is reasonable to assume that the strain is added mostly upon the MQW. For comparison purposes, another MQW thin film with the same structure was bonded to a piece of transparent sapphire at room temperature. This sample suffered no strain. The experimental setup for the transmission measurements is shown in Fig. 3.2; which used a white light lamp as a source and an optical m ultichannel analyzer (OMA) with 0.5 nm resolution as a detector. The intensity of the incident light on the sample was about 10' 4 W/cm2. A polarizer with extinction ratio 105:1 was used to choose the polarization direction of the input light. The I-V curve of this p-i-n MQW sample was measured prior to the transmission measurement. As indicated from the I-V curve, the sample is leaky when the reverse bias is applied to the sample. In order to avoid the heat problem due to leakage of the sample, a single-pulse voltages with 0 . 2 second duration at various amplitudes were applied to the MQW to study the field dependence of the optical absorption. This signal was also used to synchronize the OMA and the electrical shutter which was placed before the MQW sample. The transmission spectrum was obtained by dividing the light intensity in the presence of the sample to that without sample. Figure 3.3 shows the transmission spectrum for the unstrained MQW sample at room temperature without applied field. The room tem perature transmission spectra of the strained MQW with the polarization along the compression and tension directions for various applied voltages are shown in Fig. 3.4. As shown in Figs. 3.3 and 3.4, oscillation in the transmission spectra due to the Fabry-Perot resonance 32 is observed. The transmission equation of a lossy Fabry-Perot cavity can be used to obtain the inherent absorption spectra of the strained MQW structure. 33 n type contact 5000A 5250A 500A 100 A E p o x y £ n-doped Alo.5 Gao.5 As i-layer (MQW) 75AGaAs/100A A lQ :jG ao 7 As x 30 p-doped Alo.5 Gao.5 As p-doped GaAs LiTaO, t [001] P type contact z [HO] 4 A MQW x-cut LiTaOj > [110] - > - Y Bonded at 150 °C Fig. 3.1 Schematic diagram of the MQW device under study. 34 white source Sample Filter lens lens Aperture Optical Mulitchannel Analyzer (OMA) Polarizer Electrical Shutter pulse generator Fig. 3.2. Experimental setup for the transmission measurements. 80 70 60 50 40 30 20 700 750 800 850 900 950 1000 Wavelength (nm) Fig. 3.3. Transmission spectrum of the MQW sample bonded to sapphire at room temperature without applied voltage. 35 'w ' e o e o c c d 70 0 V 60 2 V 3 V 3.9 V 50 40 30 20 800 810 820 830 840 850 860 870 880 Wavelength (nm) (a) polarization // compression 70 0 V 60 2 V 3 V 3.9 V 50 40 30 20 800 810 820 830 840 850 860 870 880 Wavelength (nm) (b) polarization // tension Fig. 3.4. Transmission spectra of the strained MQW with polarization parallel to (a) compression and (b) tension directions for various applied voltages. 36 3.2 Data Analysis and Discussion Owing to the surface reflections from the air-sample and sample- substrate (L iT a 0 3 , sapphire or epoxy) interfaces, Fabry-Perot resonances are observed in the transmission spectra shown in Figs. 3.3 and 3.4. Using the transmission equation for a lossy Fabry-Perot cavity, the absorption spectra, a (A,), can be derived from the transmission spectra as follows [19]: 1 , r C - V C 2 - 4 R f R b T 2 a = - where a ~' L 111 [ 2 R f R b T (3,l) 5 C = 2 V Rf Rb [ 1 - 2 sin2 ^ ] T + (1 - Rf )(1 - Rb ), (3.2) 5 = 2 y n d. (3.3) T : transmission spectrum obtained from the experiment; 8 : round-trip phase; Rf, Rb : front and back surface reflectivities, respectively; L : MQW i-region thickness (absorption length); d : total sample thickness (cavity length); A,: photon wavelength; n ; average linear refractive index of the sample. Since the MQW sample consists of multi-layers, it is not easy to determ ine the theoretical refractive index of the cavity and the roundtrip phase. The average linear refractive indices, n, can be 37 obtained by fitting the refractive indices at the local maxima and minima of the transmission spectra with a linear function and setting 8 to be either an even or odd multiple of 2n. At long wavelength there is no absorption and therefore the transmission spectrum can be fit to a transmission function for a Fabry- Perot cavity. The reflection coefficient of the sam ple-substrate interface, Rb, is left as a free parameter. The transmission spectra of the unstrained MQW (Fig. 3.3) and strained MQW [Fig. 3.4 (a)], as well as the corresponding fits to the Fabry-Perot transmission functions are shown in Figs. 3.5 (a) and (b). Figure 3.6 shows the resultant absorption spectra for the unstrained and strained MQW samples at zero applied voltage after the Fabry-Perot effect is removed. The solid line in Fig. 3.6 shows the absorption spectrum of the MQW sample bonded to sapphire at room temperature without applied voltage. This absorption spectrum does not depend on the direction of the polarizer. The dash-dotted and dotted lines are the room temperature absorption spectra of the MQW sample bonded to x-cut LiTaCb at 150 °C for the polarization directions along com pression ([110]) and tension ([110]), respectively. For this calculation, the absorption coefficients are averaged over the entire MQW thickness. As shown in Fig. 3.6, both the heavy hole (HH) and light hole (LH) exciton peaks are shifted to longer wavelength when the MQW is under biaxial strain. Notice that when the polarization of the incident light is along the compressive strain direction, i.e. the y axis of LiTaC>3, 38 the HH absorption coefficient is larger than when the polarization is along the tensile strain direction, i.e. the z axis of LiTaC>3 . In contrast to the HH, the relative absorption for the LH exciton peak is reversed. 39 90 80 70 60 50 40 transmission FP resonance 30 20 700 800 750 850 900 950 1000 Wavelength (nm) (a) 80 70 § 60 C / i I 40 H 30 20 700 750 800 850 900 950 1000 Wavelength (nm) (b) Fig. 3.5 Transmission spectra of (a) unstrained MQW (Fig. 3.3) and (b) strained MQW [Fig. 3.4 (a)]. The dashed curves are fitted Fabry-Perot transmission functions. The fitted reflection coefficients of the sample-substrate interface, Rb, are 0.085 for the unstrained MQW and 0.094 for the strained MQW. 40 FP transmission p//compression (0 V) 1.4 -T 1-2 £ § 1.0 e is 0.8 V - i D < 3 0.6 e o & 0.4 o 5 / 5 < 0.2 0.0 820 825 830 835 840 845 850 Wavelength (nm) Fig. 3.6. Absorption spectra of the MQW sample averaged over the whole MQW region. The solid line is for the sample bonded to sapphire at room temperature (i.e. no strain). The dash-dotted and dotted lines are for the sample bonded to x- cut LiTaC>3 at 150 °C with polarizations parallel to the compression and tension directions, respectively. HH Tension LH Compression no strain — j> 41 Moreover, when the polarization o f the incident light is in the direction of the compressive strain, the absorption o f the strained MQW at the HH exciton peak is even larger than without strain. The HH excitonic absorption is enhanced by about 25 %. This relation is reversed for the LH absorption exciton peak. Since the excitonic absorption peak becomes higher for the HH and lower for the LH, with the presence of an applied field, the shift of the excitonic feature to longer wavelengths due to QCSE can create a larger absorption change at the HH exciton wavelength compared to the unstrained MQW. This indicates that the on/off ratio o f a normally-off transmission modulator can be enhanced by utilizing this strained M QW structure which operates in the wavelength region with strong excitonic absorption at zero bias where self-electro-optic effect devices work [2 0 ]. The anisotropic strain added upon the MQW modifies both the HH and LH zero-field transition energies and the exciton oscillator strength, thus affecting the optical properties o f the MQW . The anisotropic absorption spectra o f this strained MQW can be understood by evaluating the polarization selection rules. Since the off diagonal terms of the strain Hamiltonian [21] become nonzero for the MQW under anisotropic strain, the strain-split states will consist of mixtures of basis functions for the HH and the LH at k// = 0. Therefore, the MQW becomes biaxial and the intensity of transitions between HH (or LH) and conduction bands w ill depend on the polarization direction of the incident light [22]. Under this circumstance, because of the mixing between HH and LH bands, the HH oscillator strength is enhanced at the 42 expense of reduced LH oscillator strength w hen the polarization of the incident light is along the compression direction. More discussion on polarization selection rules will be given in chapter 4. Analysis also shows that the HH exciton peak has a redshift of 5 nm when the sample is bonded at 150 °C. From the experimental results, the redshift of the HH exciton peak is 3.2 nm, corresponding to a strain release of about 20%. This strain release could be due to the finite thickness of the epoxy which was utilized to bond the MQW thin film to the LiTa0 3 substrate. The absorption spectra for various applied voltages, when the polarization of the incident lig h t is along the direction of the compressive and tensile strain, are shown in Fig. 3.7(a) and (b), respectively. The expected QCSE is observed for both polarizations. A maximum absorption coefficient change of ~ 0 . 6 p m '1, averaged over the total width of the MQW structure, was obtained at 837 nm (HH exciton peak at zero bias) with polarization parallel to the y axis for a moderate applied electric field ( ~ 7.4 V /pm ). This result can be further improved by optimizing the QW structure, as discussed in chapter 4. Since the absorption spectrum becomes anisotropic for MQW under anisotropic strain, the am ount of the absorption coefficient difference between the two orthogonal polarizations can be used to calculate the birefringence of the strained MQW, according to the K ram ers-K ronig relations. Figure 3.8 shows the calculated birefringence, which is the difference of the refractive index between 43 the two polarizations, as a function of wavelength. The arrow shown in Fig. 3.8 indicates the position of the zero-field HH exciton peak. At this wavelength, the birefringence is zero at zero bias but it increases to be ~ 0.01 at an applied voltage of 3.9 V. This refractive index change will result in a field-dependent phase retardation. It means that for linearly- polarized incident light, the polarization of the transmitted light will turn into ellipsoidal polarization at higher applied voltage. The refractive index change (relative to zero field) with an applied voltage of 3.9 V for the two polarizations, calculated using the Kramers-Kronig relation, is shown in Fig. 3.9. As indicated in Fig. 3.9, the index change due to an applied voltage is higher for polarization along the compression than the tension direction. 44 Absorption Coefficient ( |im ) Absoiption Coefficient ( ) im ) 1.4 HH LH 0.8 .* r * ' ' . s ' 0.6 0.4 ■ 0.2 3.9 V 0.0 820 825 830 835 840 845 850 Wavelength (nm) ( a ) LH HH 0.8 0.6 0.4 0.2 3.9 V 0.0 820 825 830 835 840 845 850 Wavelength (nm) Fig. 3.7. Absorption spectra of the MQW sample bonded to LiTa0 3 at 150 °C for various applied electric field. The polarization of the incident light is parallel to the (a) compression (y-axis) and (b) tension directions (z-axis of the LiTa0 3 substrate). 0 .0 2 0.01 3.9 V 0 0V - 0.01 - 0.02 820 825 830 835 840 845 850 Wavelength (nm) Fig. 3.8. Calculated birefringence, which is the difference of the refractive index between the two orthogonal polarizations, as a function of wavelength. 0.02 0.01 s > o 0.00 - c l - 0.01 Tension - 0.02 c » -0.03 < -0.04 Compression -0.05 820 825 830 835 840 845 850 Wavelength (nm) Fig. 3.9 Calculated refractive index changes for the polarization along the compression and tension directions as a function of wavelength with an applied voltage of 3.9 V. 46 As discussed before, the strain-split states of the MQW will consist of mixtures of basis functions for the HH and the LH; therefore, the effective masses of the HH and LH bands are modified by the external anisotropic strain. The effective masses of the HH and LH can be obtained from the dispersion curve by solving for the eigen-energy of the total Hamiltonian, which is the sum of the Luttinger Hamiltonian and the strain Hamiltonian. The details will be discussed in chapter 4. From the envelope function approach in heterostructures [23,24], the wavefunction for the conduction (valence) band can be expressed as where <|>c(v)(z) is the envelope function for the one dimensional potential problem, and uc(v),o(r ) is the zone center conduction (valence) band Bloch function for the bulk material. The effective Hamiltonian for the envelope function in a single quantum well is where F is the magnitude of the electric field and Ho is the zero-field single quantum well Hamiltonian, which is given by H #c(v)(r) = <t>c(v)(z) elk//'r uc(v),o(r) (3.4) H = H0 + lei F z (3.5) for Izl < ^ Lw for Izl > ~ y (3.6) where mw and mb are the z direction strain-related effective masses of the electron (or hole) in the well and barrier, and Vc,v is the depth of 47 the QW for the conduction band (Vc) or the valence band (Vv). The QW depths are related to the total band offset AEg = Egstrained AlGaAs - g g S tr a in e d GaAs V c = Q c A E g , V v = Q y A E g ( 3 . 7 ) where Qc and Qv are conduction band and valence band offset ratios, respectively. The eigenvalues Ec,v and eigenfunctions 0c»v for the conduction and valence bands can be found using the transfer matrix method [25-27]. The transition energy between the nth conduction band and /nth heavy (light) hole confined level is expressed as Ene-mh(lh) = Egstrained GaAs + Ene + Emh(lh) - EQXne'mh(lh) (3.8) where Egstrained GaAs is the band gap of strained GaAs, which will be discussed in chapter 4; Eexne~ mh(lh) js ^ excjton binding energy for the transition, and n, m are the subband indices for the conduction and valence band, respectively. Figure 3.10 shows the theoretical wavelength shift o f the HH exciton peak as a function of applied electric field for the strained and unstrained MQWs. The experimental wavelength shift of the HH exciton peak for the strained MQW, which is obtained from Fig. 3.7, is also shown in Fig. 3.10 for comparison purposes. For this calculation, we assume that there is no strain release for the strained MQW. As indicated in the figure, the experimental results are close to those obtained from theory for both strained and unstrained MQWs. The similarity between strained and unstrained results implies that the shift 48 in excitonic wavelength depends mainly on the MQW structure, such as well width, instead of external strain. > £ •S c 2 o x V 8 ■ © — e x p e rim e n ta l: strain ed MQW E8- - th e o r e tic a l: strain ed MQW ■W-— th e o r e tic a l: u n stra in e d MQW 7 6 5 4 3 2 1 0 2 0 4 6 8 10 12 Field (V/pm) Fig. 3.10 Theoretical HH exciton shift as a function of applied field for both unstrained and strained MQW, assuming that the strain is due to the differential thermal expansion between the LiTa0 3 and MQW when the difference between the bonding and operating temperature is 130°C. Experimental data from Fig. 3.7 are also plotted as a comparison. 49 3.3 Application in Intensity Modulation As discussed by Shen et al. [15,16], the anisotropic absorption characteristics of a thermally-strained MQW can be used to change the polarization state of the incident light. The on/off ratio of a modulator can be improved by utilizing such polarization rotation. A polarizer is inserted in the path of the transmitted beam and oriented perpendicular to the polarization of the off state. By doing so, the transmitted light in the off state is significantly reduced and the on/off ratio of the modulator can be greatly enhanced. A contrast ratio of 330:1 has been demonstrated for such thermally strained MQW by optically addressing [15]. They also reported high contrast electrically-addressed optical modulation utilizing electrically tunable polarization rotation, with an observed 5000/1 contrast ratio and a 10% on-state transmission [16]. In this section, the performance of such strained MQW modulator is analyzed in detail. Figure 3.11 shows the difference betw een the absorption coefficients corresponding to the two orthogonal polarizations, [1 1 0 ] and [ 1. 1 0 ], for various applied electric fields as a function of wavelength. As indicated in figure, the absorption difference becomes smaller with increasing applied voltage. This phenomenon is more obvious for the HH exciton peaks, as compared to the LH. Figure 3.12 shows the absorption coefficient difference as a function of applied field at the zero-bias HH exciton peak. The monatomic decrease of absorption coefficient difference with increasing field implies that a tunable polarization rotation can be realized. If 50 choosing the maximum polarization rotation state as the off state and inserting an analyzer aligned perpendicular to the rotated polarization, the transmission of the off state can be set to zero. Application o f an electric field reduces the rotation angle and provides transmission through the polarization analyzer. Even though the polarization rotation may be small, the on/off ratio can be greatly enhanced. This device would operate in the normally off condition. Figure 3.13 illustrates the concept of this modulator. 5 1 _r- 0.3 'E 3 0.2 V O C a o . i 4) 1 3 Q 0.0 C 'V S -0 -1 s u -0.2 0V c . 2 & s x > < 2 V -0.3 -0.4 820 825 830 835 840 845 850 Wavelength (nm) Fig. 3.11 Difference between the absorption coefficients corresponding to the two different polarizations for various applied electric fields as a function of wavelength. 0 . 0 - o '35 S . Q » - 0 . 1 - 8 c 1> » X - 0 . 2- -0.3 0 2 3 4 1 Applied Voltage (volt) Fig. 3.12. Absorption coefficient difference as a function of applied field at the zero- bias HH exciton peak. 52 Theoretically the on/off ratio for this modulator can be extremely large. The most important issue here is to maximize the on-state transmission. The transmission o f this modulator can be calculated using the Jones matrix representations [28]. Referring to Fig. 3.13, the electric field of the transmitted light projected on the analyzer direction can be expressed as E yana E in -id> c o s 2 ©' - sin Q 'c o s© ' e ° y d/2e -i r /2 0 1 J t? ana * -’2 _ _ _ _ _ p *9 = V T - s in Q 'c o s © ' s i n 2 ©' 0 g ttz c V 2 e ir/2 1 ( 3 .9 ) where < ) > = ^ ( n y + nz ) ko d, (3.10) T = (ny - nz ) k0 d, (3.11) 0 ' = 4 5 ° - G o f f , ( 3 . 1 2 ) Ein : input electric field G off : polarization rotation angle at the off state a y, a z : absorption coefficient along compression and tension directions, respectively ny, nz : refractive index along compression and tension directions, respectively d : MQW i-region thickness The transmitted light intensity with insertion of the analyzer is expressed as I ( V ) = IE a n a ,y l2 + I E a na,zl2 - The input light intensity is equal to Io = IE in l2 . Therefore, the on-state transmission o f this modulator is derived as 53 I ( V ) e ' “ y' {sin2 [0(V) - 0off] I0 “ 2 cos [0(V ) +45°] + cos [20(V)] cos [20off] sin2[-^-]} (3.13) where 0(V ) is the polarization rotation angle at the on state. As discussed before, the refractive index difference, An, between two orthogonal polarization may not be zero. When An * 0, the transmitted light is elliptically polarized with the major axis of the ellipse rotated away from 45° due to the anisotropy in absorption [29]. W ith consideration the effect of An, the polarization angle 0(V) is given as Figure 3.14 shows the on-state transmission of this strained MQW m odulator as a function of MQW i-region thickness for several different birefringence, An, with a y = 1 p m '1. As indicated in Fig. 3.14, the on-state transmission is enhanced with the increase of the MQW thickness. However, the increase of the transmission as a function of MQW thickness is limited by the absorption of the MQW when the MQW thickness is large. Also, the transmission is enhanced if the birefringence is increased. This is due to the fact that when the transm itted light becomes elliptically polarized, the electric field component projected on analyzer can be increased. Notice that the [28,29] tan [2[0(V)+45°]}= tan [2 ta n ^ e 1 ^ 0 0 ' a ^(V)] d]} x cos (27tAnd/A,). (3.14) 54 transmission is small for this device because only 30 periods of MQW layers were used(the MQW thickness is around half micrometer). Figure 3.15 shows the on-state transmission as a function of MQW thickness for several off-state absorption differences. As shown in Fig. 3.15, the transmission increases when the off-state absorption is increased. This is caused by the larger polarization rotation angle obtainable from a device with higher off-state absorption. In conclusion, for application in intensity modulation based on polarization rotation, larger absorption anisotropy and birefringence are preferred. Also, more QW layers are preferred when the on-state transmission is not limited by the absorption o f the MQW. More discussion on enhancement of absorption anisotropy for the strained MQWs is given in chapter 4. 55 Analyzer Z ▲ in off ► Y ana Fig. 3.13. Illustration of polarization rotation caused by an anisotropic absorption and alignment of the analyzer state in order to reduce the off state transmission. 56 0.025 0.020 c o 0.015 Vi c S o.oio 1 3 Vi 0.005 0.000 0 0.5 1 1.5 2 2.5 d (Jim) Fig. 3.14 On-state transmission for the strained MQW modulator based on polarization rotation as a function of MQW thickness for several An. Aa(off) = 0.3 /pm An=0.05 An=0.03 An=0.01 ' An=0.00 c .2 ’ { 7 5 Vi Vi i 2 1 3 Vi t C o 0.015 0.010 0.005 0.000 An = 0.01 Aa=0.7 A a=0.5 A a=0.3 A a = 0 .1 0 0.5 2.5 1 1.5 2 d(pm ) Fig. 3.15 On-state transmission as a function of MQW thickness for several Aa (off) (in unit of lin r1 ). 57 References [1] D. A. B. Miller, D. S. Chemla and T. C. Damen, "Electric field dependence of optical absorption near the band gap of quantum-well structures," Phys. Rev. B, vol. 32, no. 2, pp. 1043-1060, 1985. [2] D. A. B. Miller, "Quantum-well self-electro-optic effect devices," Optical Quan. Elec., vol. 22, pp. S61-S98, 1990. [3] T. Y. Hsu, W. Y. Wu and U. Efron, "Amplitude and phase modulation in a 4 pm thick GaAs/AlGaAs multiple quantum well modulator," Elect. Lett., vol. 24, no. 10, pp. 603-604, 1988. [4] K. W. Jelley, R. W. H. Engelmann, K. Alavi and H. Lee, "Well size related limitations on maximum electroabsorption in GaAs/AlGaAs multiple quantum well structures," Appl. Phys. Lett., vol. 55, no. 1, pp. 70-72, 1989. [5] M. Whitehead, P. Stevens, A. Rivers, G. Parry, J. S. Roberts, P. Mistry, M. Pate and G. Hill, "Effects of well width on the characteristics of GaAs/AlGaAs multiple quantum well electroabsorption modulators," Appl. Phys. Lett., vol. 53, no. 11, pp. 956-958, 1988. [6 ] N. Susa and T. Nakahara, "Design of AlGaAs/GaAs quantum wells for electroabsorption modulators," Solid-State Electronics, vol. 36, no. 9, pp. 1277- 1287, 1993. [7] B. Pezeshki, S. M. Lord, T. B. Boykin and J. S. Harris, Jr., "GaAs/AlAs quantum well s for electroabsorption modulators," Appl. Phys. Lett. vol. 60, no. 22, pp. 2779-2781, 1992. [8 ] K. W. Goossen, J. E. Cunningham and W. Y. Jan, "Excitonic electroabsorption in extremely shallow quantum wells," Appl. Phys. Lett. vol. 57, no. 24, pp. 2582-2584, 1990. [9] K. W. Goossen, J. E. Cunningham and W. Y. Jan, "Electroabsorption in ultranarrow-barrier GaAs/AlGaAs multiple quantum well modulators," Appl. Phys. Lett., vol. 64, no. 9, pp. 1071-1073, 1993. 58 [10] T. Hiroshima and K. Nishi, "Quantum-confined Stark effect in graded-gap quantum wells," J. Appl. Phys. vol. 62, no. 8 , pp. 3360-3365, 1987. [11] M. N. Islam, R. L. Hillman, D. A. B. Miller, D. S. Chemla, A. C. Gossard and J. H. English, "Electroabsorption in GaAs/AlGaAs coupled quantum well waveguides," Appl. Phys. Lett. vol. 50, no. 16, pp. 1098-1100, 1987. [12] N. Susa, "Improvement in electroabsorption and the effects of parameter variations in the three-step asymmetric coupled quantum well," J. Appl. Phys. vol. 73, no. 2, pp. 932-942, 1993. [13] See, for example, R. H. Yan, R. J. Simes, and L. A. Coldren, IEEE Photon. Technol. Lett. vol. 1, 273 (1989). [14] A. Obeidat, J. Khurgin, S. Li, Opt. and Quan. Elec. vol. 25, S917 (1993). [15] H. Shen, M. Wraback, J. Pamulapati, M. Dutta, P. G. Newman, A. Ballato and Y. Lu, "Normal incidence high contrast multiple quantum well light modulator based on polarization rotation," Appl. Phys. Lett., vol. 62, no. 23, pp. 2908- 2910, 1993. [16] H. Shen, J. Pamulapati, M. Wraback, M. Taysing-Lara, M Dutta, H. C. Kuo and Y. Lu, "High contrast optical modulator based on electrically tunable polarization rotation and phase retardation in uniaxially strained ( 1 0 0 ) multiple quantum wells," IEEE Photo. Tech. Lett. vol. 6 , no. 6 , pp. 700-702, 1994. [17] M. J. Joyce and J. M. Dell, "Strain effects in chemically lifted GaAs thin films," Phys. Rev. B vol. 41, no. 11, pp. 7749-7754, 1990. [18] Epo-tek 314 was used for this study. [19] M. Kawase, E. Garmire, "Single-pulse pump-probe measurement of optical nonlinear properties in GaAs/ AlGaAs multiple quantum wells," IEEE Quantum Electron, vol. 30, pp. 981-988, 1994. [20] D. A. M iller, D. S. Chemla and T. C. Damen, "The quantum well self electrooptic effect device: optoelectronic bistability and oscillation, and self linearized modulation," IEEE Quantum Electron, vol. 21, pp. 1462-1467,1985. 59 [21] G. E. Pikus and G. L. Bir, "Effect of deformation on the hole energy spectrum of germanium and silicon", Soviet Phys. Solid State, vol. 1, pp. 1502-1917, 1959. [22] F. H. Poliak and M. Cardona, "Piezo-electroreflectance in Ge, GaAs, and Si", Phys. Rev., vol. 172, no. 3, pp. 816-837, 1968. [23] G. B astard and J. A. Brum , "Electronic states in sem iconductor heterostructures", IEEE J. Quantum Electron., vol. QE-22, no. 9, pp 1625-1644, 1986. [24] Gerald Bastard, Wave mechanics applied to semiconductor heterostructures, Halsted press, New York, 1988. [25] P. J. Stevens, "Computer modeling of the electric field dependent absorption spectrum of multiple quantum well material", IEEE J. Quantum Electron., vol. QE-24, no. 10, pp 2007-2015, 1988. [26] A. K. Ghatak, K. Thyagarajan and M. R. Shenoy, "A novel numerical technique for solving the one-dimensional Schrodinger equation using matrix approach ■ -application to quantum well structures", IEEE J. Quantum Electron., vol. QE- 24, pp 1524-1531, 1988. [27] B. Jonsson and S. T. Eng, "Solving the Schrodinger equation in arbitrary quantum-well potential profiles using the transfer matrix method", IEEE J. Quantum Electron., vol. QE-26, no. 11, pp 2025-2035, 1990. [28] A. Yariv and Pochi Yeh, Optical waves in crystals. Chap. 5,1984. [29] H. Shen, M. Wraback, J. Pamulapati, P. G. Newman, M. Dutta, Y. Lu and H. C. Kuo, "Optical anisotropy in GaAs/Alx Gai_xAs multiple quantum wells under thermally induced uniaxial strain," Phys. Rev. B, vol. 47, no. 20, pp. 13933- 13936, 1993. 60 Chapter 4 Theoretical Analysis on Multiple Quantum Well Structures Under Anisotropic Biaxial Strain In this chapter, the anisotropic absorption properties of the GaAs/AlGaAs MQW structure under thermally-induced biaxial strain are investigated. The analyses are based on a theoretical model which considers both the spin-orbital Hamiltonian and the strain Hamiltonian. The wave functions, which can be derived from solving the total Hamiltonians, are used to calculate the dipole matrix elements for interband transitions and evaluate the anisotropic absorption properties. The effect of variation of parameters such as well width and barrier height on the performance o f the strained GaAs/A lG aA s MQW electroabsorption modulators will be discussed. Studies on physical and optical properties of strained MQW structures have been focused on the lattice-mismatched MQWs in the past few years [1-3]. These strained MQWs are of great interest recently because they allow transparent substrates and the growth of new materials to help extend the working wavelength of the optical devices. The strain in lattice-mismatched materials also modifies the electronic structure. As a consequence, the perform ance of FET devices [4] or lasers [5] is improved due to enhanced hole mobility and lower density of states in strained materials. Theoretical analyses on the lattice-mismatched MQW structures are well-documented [2,6]. The built-in biaxial strain of pseudom orpically-grow n strained MQW 6 1 contributes only to the diagonal terms in the strain Hamiltonian and thus does not change the cubic symmetry of the materials [2,6]. Therefore, the absorption spectra remain isotropic for the lattice-m ism atched MQW structures. The effect of external stress on GaAs/AlGaAs MQW structures has also been investigated in the literature. The stress is usually applied along either the [001] or [100] direction [7-9], or along the [110] direction [ref. 10] of the MQW structure. These theoretical and experimental studies have concentrated primarily on the determination o f the strain-induced variation in electronic energy levels. Recently, anisotropic optical properties in GaAs/AlGaAs MQWs under thermally- induced external strain was studied [11,12]. Either uniaxial or biaxial strain was applied along the [100] or [010] direction of the MQWs. In this chapter, the effect of anisotropic biaxial strain that is applied along the [ 1 1 0 ] and [1 1 0 ] directions is studied as it affects the optical properties of GaAs/AlGaAs MQW structures. In section 4.1, the Hamiltonians used to characterize the electron and hole states in a quantum well are reviewed. The strain Hamiltonian for anisotropic biaxial strain (which is achieved by bonding the GaAs/AlGaAs MQW thin film to an x-cut LiTaC>3 substrate as described in chapter 3) is derived and discussed in section 4.2. In section 4.3, the dipole transition matrix elements are evaluated and used to explain the anisotropic absorption feature of the strained MQWs. Finally, the effect of parameter variations on the performance of the MQW devices is discussed in section 4.4. 62 4.1 The Band Structure of a Quantum Well To investigate the anisotropic optical properties in a strained MQW, several properties, such as wave functions and effective masses of electron and hole states for the MQW structures under biaxial strain, need to be studied. The characteristics of these properties can be investigated by solving the eigenvalue equation. Both conduction band and valence band Hamiltonians, which are used to characterize the band structures, are discussed in this section. The band structure near the band edge is of great significance, since the optical characteristics in a MQW heterostructure are closely related to the electronic structure near the conduction band minimum and the valence band maximum. In bulk semiconductors, the electronic states near the conduction band edge can be characterized by the S-like states and the electron effective mass. In a quantum well, in the framework of the effective mass approximation, the electron problem can be characterized by the following equation [13,14] - V2V c(k,z) + V c(z ) T c(k,z) = E >Fc(k,z) (4.1) where Vc(z) represents the potential barrier seen by the electrons. For the direction parallel to the z axis, the momentum vector k is a good quantum number and the wave function, 4/ (k,r), for the conduction band can be expressed as 63 = 0c(z) e,k//'P uc,0(r) (4.2) where < J > c(z) is the electron envelope function for the one dimensional potential well problem, k// is the in-plane two-dimensional wave vector, p is the in-plane radial coordinate, z is the coordinate in the growth direction and u c,o(r) is the zone center conduction band Bloch function for the bulk material. The electron envelope functions can be solved numerically by the transfer matrix method [15-17]. The hole states are more difficult to solve because of the fourfold degeneracy at zone center. In bulk III-V compound semiconductors, at Brillouin zone center, the eigenstates have pure angular-momentum 3 3 3 1 character corresponding to I 2 ’ ~ 2 > anc* * 2 ’ ± 2 > ^ states- I* 1 GaAs, the split-off band (I ^ , ± ^ >) is far away from the HH and LH bands and thus its effect on m ost hole-related properties can be ignored. It has been known that the valence band structure can be explained by the Luttinger Hamiltonian [18,19], which is given as 1^ !> 2 ’ 2 2 ’ 2 .3 1 3 3 2 ’ ’ 2 2 ’ ’ 2 > H = S+T L M 0 L* S-T 0 M M* 0 S-T -L 0 M* -L* S+T (4.3) 64 where a 2 s = -yi( 2 ^ ) (kxz + ky + kz ), (4.4) a 2 T = - n ( 2 ^ )(}‘x2 + k y2 - 2kz2)’ (4-5) r— *R 2 L = 2V3 7 3 ( 2 ^ ) (kx - iky) kz, (4.6) * fi2 M = V3 ( ) [Y2(ky2 - kx2) ' 2 7 3 ikxky], (4.7) and yi, y2, and Y 3 are Luttinger parameters. Some of the off-diagonal terms are non-zero, which indicates that mixing between the HH and LH states exists when k is away from k=0. In the presence of a quantum well, the z component of the momentum in the Luttinger Hamiltonian can be treated as an operator (kz = - i (d/dz)) [2]. Thus, the 4 x 4 Luttinger Hamiltonian is given as Hih L’ M 0 L'* Hih 0 M ♦ M 0 H,h -L’ 0 * M -L’* % 65 where |(kx 2 + ky2)(yi + Y2) -(71 - 2y2)gj2 1 + v h < z). (4 -S > ) [(kx 2 + ky2)(yi - 7 2 ) - (71 + 2 y 2 ) J | ] + Vh<z), (4.10) (4.11) (4.12) and Vh(z) is the potential for the hole states. When k// = 0, the off- diagonal elements L' and M are zero. The problem thus reduces to the simple one dimensional quantum well problem, which is similar to the electron problem as shown in Eq. (4.1). Therefore, the wave function of the hole states can be written in the following form: where j is the index o f the valence bands. For a non-zero k//, the valence band problem consists o f four coupled second order partial differential equations. These differential equations can be solved via a variational method [20] or a finite-difference method [2,21]. The Hamiltonian discussed above can then be incorporated with the strain Hamiltonian to solve for the wave functions and dispersion relations. 4.2 Effect of Strain In general, hom ogeneous strain can be divided into two contributions: hydrostatic components, which give rise to a volume 66 4 'v(k,z) = X 4>,j(z) eik//‘P uj,„(r), (4.13) change without disturbing the crystal symmetry, and shear components, which normally reduce the symmetry present in the strain-free lattice. These strain components in turn produce changes in the electronic band structure and vibrational modes [6,22], which thus affect the optical properties of the material [23]. The strain dependence of electronic levels can be characterized by deformation potentials, i.e. the energy shift per unit strain, which are typically in the range from 1 to 10 eV [6]. The effect of strain can be incorporated into the k p calculation by introducing a strain Hamiltonian into the spin-orbital Hamiltonian. It has been shown that the strain Hamiltonian He can be written as [23,24] Hg (c’v) = -a(c’v) (e xx+ e yy+e zz) ■ 3 b(v) [(Lx^ - ^ ) e xx + c.p.] - 2V3 d(v) [{Lx,Ly] e xy + c.p.], (4.14) where e ij denotes the components of the strain tensor, L is the angular momentum operator, c.p. denotes cyclic permutations with respect to the indices x, y, and z, and the quantities in the curly brackets indicate the symmetrized product: [Lx,Ly] = (LxLy+LyLx)/2. The parameter a(c’v) is the hydrostatic pressure deform ation potential for either conduction bands (c) or valence bands (v). The quantities b(v) and d(y) are uniaxial deformation potentials appropriate to strains of tetragonal and rhombohedral symmetries, respectively, and are both zero for the S-like conduction band. The total Hamiltonian is thus given by 67 H = Hs.o. + He . (4.15) At the Brillouin zone center the matrix for Hs.o. is diagonalized. Therefore, it is possible to construct the conduction band and valence band Hamiltonians separately. As described in chapter 3, the MQW structure under investigation is bonded to an x-cut LiTaC>3 at an elevated temperature, T, with the [110] and [110] directions of the MQW along the y axis and z axis of LiTa0 3 , respectively. Hence, at room temperature, Troom, the induced compressive strain e i is along the [110] direction, and the tensile strain G2 is along the [110] direction. The compressive and tensile strains can then be expressed as where £ m q w (6.2x1 0"6 /°C at room temperature) is the thermal expansion coefficient of the MQW structure, £z (16.1xl0‘6 /°C at room temperature) and £// (4.1xl0‘6 /°C at room temperature) are the thermal expansion coefficients of the LiTaC>3 along the z axis and the x (or y) axis, respectively. The sign of e j is negative for compressive strain and positive for tensile strain. Figure 4.1 shows the compressive and tensile strains as a function of bonding temperature. The solid lines are calculated by taking only the first order of the thermal expansion into account. The dashed lines are calculated by considering both first order and second order thermal expansion coefficients [25]. W hen the e l = ( CMQW - £ //) x ( T - Troom ), e 2 = ( CMQW - Cz ) X ( T - Troom ), (4.16) (4.17) 68 bonding temperature is less than 150 °C, the effect due to higher order expansion coefficient is small and thus can be neglected [26]. Since stress is applied along the [110] and [110] directions of the MQW, the strain tensor in the plane of MQW layer, which is perpendicular to the growth direction, can be found by means of a coordinate transformation. The strain along the growth axis of the MQW, which is produced by the Poisson effect [27], can be found by minimizing the strain energy. The derivation of this strain is shown in Appendix A and the results are given as follows £ i+£2 £ xx = £ yy = 2 ’ (4*18) £ l-£2 £ xy = 2 ~^ ’ (4-19) £ zz = - q Y 1 + G 2^’ (4 ' 2 ° ) where e i and G2 are defined in Eq. (4.16) and Eq. (4.17), respectively, and C 12 and C n are the elastic stiffness constants. 69 tensile strain - 0.1 w £ - 0.2 compressive strain -0.3 first order only include higher order -0.4 -0.5 0 50 100 150 200 250 300 350 400 bonding temperature (°C) Fig. 4.1. Compressive strain and tensile strain as a function of bonding temperature. The solid lines are calculated by taking only the first order of the thermal expansion into account. The dashed lines are calculated by considering both the first order and second order thermal expansion coefficients. 70 The strain Hamiltonians for the conduction band and valence band can be derived by plugging the strain tensors, Eqs. (4.18)-(4.20), into the strain Hamiltonian, Eq. (4.14): Ci? Hg (c) = -a(c) (1 - )(e i + e 2 ) = - 5EH< C ) (4.21) He (v) = -a(v) (1- ) ( e i + G 2) + 3 b(v) ( | + ^ ) ( g i + g 2)(LZ 2 - j L2) - 2^3 d(v)~ 1 2-e ~ {Lx,Ly } = - 8EH(v) + 5ET (Lz2 - 1 L2 ) - 2 8Er {Lx,Ly} (4.22) C n where 8Eh(c) = a(c) (1 - ) ( g i + g 2 ) (4.23) C l 9 SEhW ^ a(v) (1 - ^ 7 )(g 1 + g 2) (4.24) 1 C l 9 8Et = 3 b(v>( i t ^ ) ( € i + € 2 ) (4.25) S E r s 'Jl dW e 1 ' £ 2 (4.26) As indicated in Eq. (4.21), the minimum of the conduction band is shifted with respect to the average valence band energy by SEh (c). The change in the effective mass is also attributed to the change in the conduction and valence band energy separation [6]. However, since the effective mass of the conduction band is small and the band gap for GaAs is large compared to the change in energy gap due to strain, the 71 change in effective mass of the conduction band can be neglected. Calculation shows that the variation in effective mass of the conduction band changes the excitonic properties of the MQW by less than 5% [28]. Therefore, the eigenfunction of the electron state can be solved using Eqs. (4.1) and (4.2). The effect of strain on the valence band is much m ore complicated. The representation of the angular momentum for a four- 3 3 1 1 fold p3/2 multiplet (J= ^ ,M j=± ^ ,± ^ anc* a Pl/2 doublet (J= ^ ,M j=± ^) needs to be calculated. The details are described in Appendix A. As discussed in Appendix A, it is possible to diagonalize the 6 x 6 valence band Hamiltonian into two degenerate 3 x 3 matrices, written in the order of 13/2,3/2>, 13/2,-l/2> and 11/2,-l/2> , as: - 8Eh< v *+-] 5E j r ; 5ER f j s E n I I - 5 E h (v)~ ] 8E t - ^ S E r (4.27) ■ 5 E r f~2 - 8E -A« where A0 is the spin-orbit splitting of the valence band. As discussed before, the coupling of the spin-orbital band into the HH and LH bands can be neglected for GaAs. Therefore, the strain Hamiltonian can be reduced to a 2 x 2 matrix. The energy shift of the electron state, as given in Eq. (4.21), can be incorporated into the diagonal terms of Eq. 72 (4.27). Thus, the change in the energy bandgap between the conduction and valence bands due to the hydrostatic strain is given as 8Eh = 8 E h ^ - 8Eh(v) = (a(c) - a(v)) (1 - )(e i + e 2 ) = a(l- § ^ ) < £ l + e 2). (4.28) and the energy band gap between the conduction and valence band for materials under biaxial anisotropic strain can be calculated by solving the eigenvalue for Eq. (4.21) and Eq. (4.27) analytically as: Eg,strained = Eg)U n stra in ed + SEh i [(“3 ) + ( ^ ^ ^ ^ (4-29) where the and "+" signs are for HH and LH, respectively. Figure 4.2 shows the energy band gap between the lowest conduction band and the heavy hole (HH) and the light hole (LH) bands for bulk GaAs as a function of bonding temperature. The parameters used in this calculation are listed in Table 4.1. Figure 4.2 indicates that the HH and LH bands are split and shifted toward the conduction band under biaxial anisotropic strain. Notice that the HH is more sensitive to bonding temperature as compared to the LH. As indicated in Fig. 4.2, the HH exciton peak has a redshift of around 5 nm when the bonding temperature is 150°C. From the experimental data in chapter 3, the redshift of the HH exciton peak is 3.2 nm. This is corresponding to a strain release of about 20%. 73 In addition to the strain-induced changes in energy gaps and splitting due to the lowering of symmetry, to fully describe the electronic energy levels of the strained MQW it is important to know information about the variations in effective masses. 74 1.425 LH 1.420 > r t i £ 1 4 1 5 1-410 X 1.405 HH 1.400 1.395 bonding temperature (°C) Fig. 4.2. Energy band gap between the lowest conduction band and the HH and LH bands as a function of bonding temperature for bulk GaAs. 75 Table 4.1. Parameters used in plotting Fig. 4.2. GaAs AlAs Eg(eV)<a> 1.424 3.4 Ao(eV)M 0.34 0.28 kslp, lx>l2 (eV) (a) 25.7 2 1 . 1 7i (a) 6 . 8 4.04 \ < a> 2 . 1 0.78 Ji (a) 2.9 1.57 a(eV) < » > ) 8 . 6 8 7.96 b(eV) ( • » ) -2 . 0 -1.5 d(eV)(b) -4.55 -3.4 Cu ( x lO 11 dyn/crri2 )(b) 1 1 . 8 8 1 2 . 0 2 Ci2 ( x lO 11 dyn/cm2 )(*’) 5.38 5.7 C4 4( x 1 0 1 * dyn/cm2)^ ) 5.94 5.89 C (°c-1) (c) 6 . 2 x 1 0 6 4.9 x 10' 6 (a) P. Lawaetz, Phys. Rev. B 4, 3460 (1971) (b) S. Adachi, J. Appl. Phys. 53, 8775; J. Appl. Phys. 58, R1 (1985) (c) V. Swaminathan and A.T. Macrander, Materials Aspects of GaAs and InP Based Structures. Prentice Hall, 1991. 76 As discussed in section 4.1, the valence band structures can be characterized by the Luttinger Hamiltonian. Thus, under strain, the total Hamiltonian is the sum of the Luttinger Hamiltonian, Eq. (4.3), and the strain Hamiltonian, Eq. (4.27). The dispersion relations of the valence bands can be found by solving the eigenenergies of the total Hamiltonian. The details are discussed in Appendix B. From Appendix B, the effective masses along k i :[ 110], k2:[l_10] and k3:[001] can be expressed as follows: -[Y 1± t U 72±V 3 t 12Y 3]> (4-30> mki = -iyi±T\ 1Y 2 + V3T12Y3L (4.31) mju ^ J = -(Y1T 2 n m ) , (4.32) where Y 1 > Y 2 and Y 3 are the Luttinger parameters, as given in Table 4.1. The param eters t |i and r|2 are strain-related and are defined in Appendix B, and "+" and signs are for ±3/2 and ±1/2 bands, respectively. The dispersion curves for these three directions are shown in Fig. 4.3 (a), (b) and (c) for GaAs. The results indicate that the effective mass in the plane of the MQW layer (which is perpendicular to the growth axis) is no longer a constant. This is different from the situation for lattice-mismatched MQW that has a constant effective mass in the plane under biaxial strain [4,6]. 77 HH (a) k//[ 110] ............--------------------- — - ^ H H j n J LH j \ (b) k//LL10] HH > > b J j u . LH u e U J (b)k//[001] Fig. 4.3. Dispersion curves of GaAs under biaxial anisotropic strain for (a) k//[1 1 0 ], (b) k//[1.1 0 ] and (c) k//[0 0 1 ] directions. 78 4.3 Absorption Anisotropy The oscillator strength, which is proportional to the absorption coefficient and can be used to evaluate the optical response, is given as [29]: 2(E’D cv)^ f = E ^ ’ < 4 -33> where E is the transition energy and gcv, defined as <clp.lv>, is the crystal momentum matrix element; and, (£-£cv) is the square of the transition m atrix element. In the framework of the effective mass approximation [14], which is valid for k//=0, the wave functions for the electron and valence bands can be expressed by Eqs. (4.2) and (4.13). Thus, the transition matrix element is given by [14] < C l £ J3 lv > = < U c l £ J L l u v ><<t>cl<l>V> + < uc I uv >< < t > c I £ • £ I < t> v >, (4.34) where £ is the unit polarization vector of incident light and j) is the electron momentum. For interband transitions the first term in Eq. (4.34) is more important than the second term. The second term becomes crucial for intersubband transitions where the dipole moment comes from the envelope function part of the wavefunction. Thus, the square of the transition matrix becomes 79 I< C I £ • £ I V > l 2 = ( k Uc I £ • £ I Uv > l 2 ) I < <t>c I < t> v > I2 = (miM 2 ) Mcv2 , (4.35) where mi is the polarization selection factor which depends on the polarization direction of the incident light. The second term, M2, is the average transition matrix for k=0 Bloch states and is a constant for a given material (as listed in Table 4.1). The third term, Mcv2, describes the wavefunction overlap probability between conduction and valence bands. In order to understand the anisotropic absorption properties of the MQW under strain, we have to find out the polarization selection rule, i.e. to calculate the value of k u cl£-pluv>l2. As discussed in section 4.2, the zone center Bloch function of the conduction band, uc, is still S- like. However, the off-diagonal terms of the valence band Hamiltonian, as indicated in Eq. (4.27), are non-zero and therefore ±m j are no longer good quantum numbers. The strain-split states will consist of mixtures of basis functions for mj=±3/2 and m j=±l/2 [23]. Hence, the crystal becomes biaxial and the intensity of the transitions will depend on the azimuthal angle of the incident light. To find out the mixed valence band Bloch functions, the strain Hamiltonian [Eq. (4.27)] can be treated as a small perturbation of the Luttinger Hamiltonian [Eq. (4.8)] for the unstrained MQW structure. This Hamiltonian is given by a 4 x 4 matrix, comprised of 2 x 2 block diagonal matrices, which is 80 (4.36) w here AEq w = Elh - Eh h » Elh and Ehh are solutions o f the unperturbed Hamiltonian. The zero of energy has been chosen at 5Eh + (Elh+Ehh)/2, where 8Eh is the shift in the energy levels due to the hydrostatic component of the strain, as given in Eq. (4.28). The resulting mixed valence band wave functions can be solved analytically as: i h h > = n = a + V l+ t|2/3 1 V2 Vl+T|2/3 ^ * • (4.37) I LH > = i y c V 1+tj2/3 2 ^ ^ y * i ■ | / ^ (4.38) where T | SEr (4.39) ^AEqw + 3^Et 8 1 W ithout induced strain, r\ is equal to zero. In this case, the HH 3 3 state reduces to a pure I 2 * - 2 > state at ^// = anc* ^ state ■ 3 ^ 1 reduces to a pure ^ 2 ’ 2 > state‘ As derived in Appendix C, the transition m atrix elem ent l<Ucl£.-£>Juv>l^ can be calculated for the situation when the light is normally incident to the MQW structure with polarization direction parallel to either the compressive strain (i.e. [110]) or the tensile strain (i.e. [110]) direction. The results are kcl£l 10.J>IHH>I2 = y - (1 + \ ) = HHcompression M2 (4.40) kcl£1 1 0 .clHH>l2 = ^ ( l + HHK„sio„M2 (4.41) for the transition between the heavy hole (HH) band and the conduction band, and l< cl£llO .£lL H > l2 = (1 - 2 ^ = ^^com pression M 2 (4 .4 2 ) k cl£ 1 1 0 .ElLH>l2 = ^ (1 - ^ ^ 2 / 3 ) = LHtension M2 (4.43) for the transition between the light hole (LH) band and the conduction band. Since t| is larger than zero, by comparing Eq. (4.40) with Eq. (4.41), the polarization selection factor HHC O m pression is larger than 82 H H t e n s i o n . which indicates that the zero-field H H excitonic absorption peak is larger when the polarization of the incident light is along the [110] direction, when compared to the [110] direction. In contrast to the H H , as shown in Eqs. (4.42) and (4.43), the relative absorption for the L H exciton peak is reversed. This explains the anisotropic properties of the absorption spectra. Notice that when there is no strain or band mixing, r| is equal to zero. Thus, the polarization selection factor H H c o m p r e s s i o n is equal to H H t e n s i o n , and both factors reduce to ^ . This is the case for the H H - conduction band transition in unstrained MQW or pseudomorphically- grown strained MQW [2,21], independent of polarization direction. A similar result is concluded for the LH-electron transition. When q is equal to zero, L H COm p r e s s i o n is equal to L H te n s i o n , and both factors reduce to ^ . Figure 4.4 shows the HH and LH polarization selection factors, as given in Eq. (4.40)-(4.43), as a function of bonding temperature for the two orthogonal polarization directions: [110] and [110] (AEq w = 18 meV was used for the calculation). As indicated in Fig. 4.4, although the increasing or decreasing rates of change of the polarization selection factors with respect to the bonding temperature are different for HH and LH (for the same polarization), the difference in the polarization selection factors between the two orthogonal polarizations for HH is equal to that for LH. This indicates that the absorption anisotropy, which is proportional to the difference in the polarization selection 83 factor between the two orthogonal polarizations, should be the same for HH and LH. However, as shown in Fig. 3.11, the experimental data show that the absorption difference of HH between the two polarizations is 0.3 pm -1 which is larger than 0.12 pm -1 of LH. This may be due to the fact that only the transition at k// = 0 is considered. As indicated in Eq. (4.8), when k// * 0, there is a coupling between HH and LH bands due to the non-zero off-diagonal terms. This means that the transition matrix elements or the polarization selection factors also depend on k// [2]. 84 0.65 < 4 - 1 c 0 • a 8 ■ § C m o e d N 1 £ 0.60 0.55 0.50 0.45 0.40 0.35 0.30 HH compression — - HH tension 50 100 150 200 250 300 350 Bonding Temperature (°C) ( a ) 0.35 0.30 <2 0.25 e I 0.20 0.15 LH compression — - LH tension c o '8 N 1 °-10 "o c u 0.05 0.00 50 100 150 200 250 300 350 0 Bonding Temperature (°C) (b) Fig. 4.4 Polarization selection factors, as given in Eq. (4.40)-(4.43), as a function of bonding temperature for (a) the HH and (b) the LH when the polarization direction of the incident light is along either compression or tension direction. AEqw =18 meV was used. 85 Quite recently, optical anisotropy in GaAs/AlGaAs MQW subject to thermally induced uniaxial and biaxial strain, which is obtained by bonding MQW thin film to LiTaC>3 in such a way that the strain was along the crystalline axes of the MQW, was studied by Shen et al. [11,12]. Theory was developed by these authors and used to explain the experimental results with satisfactory performance. The derivation of the strain Hamiltonian and transition matrix element fo r this specific situation is shown in Appendices A and C. Using the results of Appendices A and C , for the same MQW structure (assuming A E q w = 18 meV), the calculated zero-field excitonic absorption anisotropy for our sample is ~ 60% larger than that obtainable from a MQW with biaxial strain along the crystalline axes of the MQW, and ~ 30% higher than that obtainable from a MQW with uniaxial strain. 4.4 Discussion When anisotropic stress is applied to a MQW structure, the absorption anisotropy is mainly due to the interaction between HH and LH at k// = 0. Therefore, the energy level difference between HH and LH will also affect the magnitude of the absorption anisotropy. This means that, for the same amount o f strain, different M QW structures may have different absorption anisotropy. Figure 4.5 shows the HH polarization selection factors as a function of bonding temperature for the two orthogonal polarization directions, [110] and [110], with two different A E q w ( A E q w = E l h - E h h ) - A s indicated in Fig. 4.5, at the same bonding temperature the 86 absorption anisotropy, which is proportional to the difference of the HH polarization selection factor between the two polarizations, is larger when AEq w is smaller. The reason is that when the energy difference between HH and LH levels is smaller, the interaction between the HH and LH is stronger; consequently, a larger absorption anisotropy is obtained with the same amount of strain. Therefore, to maximize the absorption anisotropy, a M QW structure with a sm aller AEq w is preferred. Notice that if only the enhancement of the HH excitonic absorption is concerned with polarization in the compression direction, as indicated in Fig. 4.5, a MQW structure with a smaller AEq w is also preferred. Figure 4.6 shows the HH polarization selection factors of the two polarizations as a function o f A E q w for three different bonding temperature. As shown in Fig. 4.6, absorption anisotropy is decreased when A E q w is increased. Figure 4.6 indicates that the relation between absorption anisotropy and bonding temperature is not a linear function. The absorption anisotropy saturates gradually when the bonding temperature is increased. The energy difference between HH and LH levels, A E q w , is mainly determined by the well width and barrier height of a single quantum well. Figure 4.7 shows A E q w of GaAs/AlxG ai-xAs QW as a function of well width for several different aluminum concentrations. The effective mass difference between well and barrier has been taken into account. A band offset ratio of 0.35 for the valence band is used for this calculation. As indicated in Fig. 4.7, a QW structure with 87 wider well width and smaller barrier height has a smaller AEq w - It means that, in order to have a larger absorption anisotropy or higher HH excitonic absorption, a GaAs/AlGaAs MQW structure with wider well width and smaller aluminum concentration is preferred. However, for a wider well width, the oscillation strength is smaller because electrons and holes are loosely confined in the GaAs well. This implies that the absorption coefficient, and thus the absorption anisotropy, is smaller for the MQW with wider well width. Moreover, for a shallow QW, the exciton absorption decreases abruptly due to dissociation of excitons caused by the shallow barrier when an electric field is applied, resulting in a severe reduction in the on/off ratio [30]. Therefore, there exists an optimal well width and barrier height that will yield the best performance. This issue can be further studied in the future. 88 0.65 S 0.60 g 0 - 5 5 | 0.50 1 3 a 0.45 _o 3 0.40 'S 1 0.35 5 0.30 0.25 0 50 100 150 200 250 300 350 Bonding Temperature (°C) Fig. 4.5. Heavy hole polarization selection factors as functions of bonding temperature for two different AEqw when the polarization directions of the incident light are parallel to compression and tension. Assume Troom = 20°C. p // compression AE QW = 10 meV - - AEQW = 18 meV p // tension 0.70 £ 0.60 c o JS 0.50 % c o | 0.40 o C l. X X 0.30 0.20 f p // compression p // tension T = 120 °C T = 170 °C T = 220 °C 0 30 35 5 10 15 20 25 AEqw (meV) Fig. 4.6. HH polarization selection factors of two polarizations as a function of AEqw for three different bonding temperatures. 89 > I U J 1 Si 60 50 x= 0 . 1 x= 0 . 2 - x=0.3 - - x=0.5 - - x = 1 . 0 40 30 20 10 0 0 150 50 100 200 w ell w idth, d (A ) Fig. 4.7. Energy difference between HH and LH bands as a function of well width for several aluminum concentration. 90 Appendix A Derivation of the strain Hamiltonian under biaxial anisotropic strain The effect of strain can be taken into account by introducing a strain Hamiltonian into the spin-orbital Hamiltonian. It has been shown that the strain Hamiltonian H. for a given band at k=0 can be written as [23,24]: He(c .v > = -a*v> (G „+e n+e J - 3 b « » » [(L,2 - 1 L 2 ) e „ + c.p.] -2V3d«»>[{Ll,Ly} e ^ + c .p J (A .l) where e denotes the components of the strain tensor, L is the angular momentum operator, c.p. denotes cyclic permutations with respect to the indices x, y, and z, and the quantities in the curly brackets indicate the symmetrized product: {L„Ly} = (L,Ly +Ly LJ/2. The parameter a( c v | is the hydrostatic pressure deformation potential for a given band. The quantities bl,) and d( v | are uniaxial deformation potentials appropriate to strains of tetragonal and rhombohedral symmetries, respectively, and are both zero for the S-like conduction band. The total Hamiltonian is then given by H = Hso. + He (A.2) At Brillouin zone center the matrix for Hs o. is diagonalized. Thus, the strain contribution to Hs o. will not be considered at this point, and it is possible to construct the conduction band and valance band Hamiltonians separately. The strain Hamiltonians for two different cases are derived as follows: (A) Compressive strain along [110] and tensile strain along [HO] 91 x' [110] > x Fig. A.I. Coordinate systems. The strain components in the (x\y’,z') system are given by e „v = e , e .1 = Y e ,y = G y 'z * = G = 0 (A.3) (A.4) (A.5) where e , is the compression, e 2 is the tension, and y is the unknown strain along z'. W e can express the strain components in the (x,y,z) system by doing the following transformation: g = a_e aT (A.6 ) where a = h ' h 0 7 i T i 0 o o 1 and aT is the transpose of a. From (A.6 ), 92 € .. = € yy= ^ ^ , (A.7) (A.8 ) 6 « = Y • To find the strain component g 12, i.e. the strain along the growth axis, we need to minimize the strain energy. The strain energy for the cubic crystal is given by [26] U = j C,, (e „ 2 + g „2+ e „2 ) + C 12 (e „e „ + € „e , + e J + 2 C 4 4 ( € 2 y 2 + g y 2 2 + g „ 2) (A.9) where C M , C,2 > and C„ are elastic stiffness coefficients. Differentiating (A.9) with respect to g a results in = C 1 1 G 2 2 + C,2( G „ + G yy) = 0 G II = - & 1 ( G „ + G yy) (A. 10) '-" II Plug (A.7) and (A.8 ) into (A. 10), then e II = - S u ( G , + e 2)=Y . (A. 11) '-"11 Since only g „, g ^ , g „, and g „ y are not equal to zero, (A.l) becomes He‘ «>=-a'*'(G„+€y y + G j (A. 12) Ht,v >= - ^ ’(e ,,+ g„+<= J - 3 b'*’[(L,2- ^ L2 ) g „+ (Ly 2- 1 L 2 ) g w + (L, 2 - \ L2 ) g „ ] - 2V3 d-> [ {L„Ly} g J (A. 13) 93 The next step is to find the representation for the angular momentum. In the absence of strain, the spin-orbit interaction splits the six-fold degenerate p-like valence-band 3 3 1 1 multiplet into a four-fold p3/2 multiplet (J= ^ ,M,=± 2 ^ 2 ^ anc* a doublet (J= j 1 3 3 3 1 ,M,=± 2 )- The eigenfunctions for these six valence-band states, I ^ i 2 >’ * 2 ’ ^ T*' and I ± j> , can be obtained by finding linear combinations of products of the three spherical harmonics Y, „ Y, 0 , and Y,,.„ and the two spin wave function r and 1 . These combinations can be obtained from the Clebsch-Gorden coefficients as follows [31]: l§ § > = Y,., T (A. 14) l2 2 > = 1 0 T + 1 !l - i >=V F y'°i+ V P ||T . 3 3 1 2 ' 2 > = Y|-'1 4 \ > = - ^ J T y ' ° ' + aI Y y -•'i 1 2 = V T * " ’ Define L,= L,+ iLy and L.= L,- iLy, then {L,,Ly} = ^ (L ,L y+ L y L , } = - ^ ^ . In addition, we can also make use of following relations: L,YlB = m Y,„ LJYljn = 1(1+1) Yljn L± Yljn = y j (l+m )(l±m +l) ^ljnll (A. 15) (A. 16) (A. 17) (A. 18) 94 From (A. 14) to (A. 18), each 6 x 6 matrix for either (L,2- ^L2 ) or {L,,Ly} can be diagonalized into two degenerate 3 x 3 matrices. After some manipulations, the matrices for (Lz 2- ^L2 ) and {L„Ly} can be expressed as 0 3 0 - - 3 f 2 f 2 0 0 3 I-3 - 1 * 2 ~2 > (A. 19) 1 ~2 > U ^ I y ) = 2<ri 2 - f} /~6 0 0 0 0 I 3 -3 * 2 2 > ,3 . 1 2 2 > i_! - 1 2 ~2 > (A.20) 6J+62 ■ » -yy 2 1 .* . „ , 1 Since e „ = e w = ' , so (L,J- ^ „ + (Ly 2- ^ L2) e , = (L.2 + L y 2- f L 2 ) - ! r i = (-LI2+ L2 - f L 2 ) ^ = -(l , 2- I l 2 ) ^ (A.21) From (A.19) to (A.21), (A.12) and (A.13) can be written in following form He« = -a‘ c > (1 - )(e , + e 2 ) = - 5E„«> (A.22) '-'I I 95 Ht < v ) = -a( v >(l - )(€, + e J + 3 b * > )(€, + e 2)(L,2- i L2 ) - 2V3 {L„Ly} = - 8 E „ < * > + 8 Et (L,2- ^ L2 ) - 2 8 Er {L„Ly} 5Eh (V + | 5 E t V f 5ER V j s t k / f 5e r -5E h(V - j 8 ET / 2 3 8eT (A.23) ' i i * Er / 2 - 3 8 ET - 8 E( ^ } - A o where A » is the spin-orbital splitting. Since a( c | and a( v ) cannot be measured separately, we have 6 E„ = 8 E„( C ) - 8 Eh( v > = (a( c ) - a( v ) )(l - S 1 1 )(e , + e 2 ) '-II = a ( l - p Ji)(e1 + 6 2 ) v -n This is the change in energy bandgap between the conduction and valence band due to the hydrostatic strain. (B) Compressive strain ( e ,) along [100] and tensile strain ( e j along [010] In this situation the strain components can be written as e „ = e , 6yy=e2 e u = y (unknown) (A.24) (A.25) 96 Following the same procedure as in part (A), e „ can be found by minimizing the strain energy and is given by e a = -pu (e„+€y y ) = -&2(e,+ e 2 ) (A.26) v-n v -m i Hence Eq. (A .l) can be re-written as follows: He = -a (1 - )(e , + e 2 ) - 3 b {(Lx2 - \ V) e , + (Ly : - ^ L2 ) e 2 + (LI 2- 5 LJ) [ - § ^ ( e 1 + 6 2 )]} (A.27) where Lx - 4 L l l 6 2-/1 ^6 1 1 1 2 / 3 ~ 6 3 / 2 1 1 f e 3 - f i 0 (A.28) 2-f3 i~6 L 2 - — L2 = Ly 3 L - 1 1 2. f 3 6 1 1 i~6 3 / 2 6 1 3 / 2 0 (A.29) From (A. 19), (A.28), and (A.29), Eq. (A.27) can be expressed as 97 H€ = -S E hI - Define 5ET ' = b (g H e = b - - ( e i + e 2-2y) 2. f 3(e i~e 2) " f 6^e i ' G2) ^ = 3 (e , - e 2) ~6 ( e i + e 2 - 2 y ) ' ^ ( G i + e 2 -2Y ) 1 1 T 6( e i ' e 2 ) ^ ( e i + e 2 - 2^ 0 , + e 2- 2y) and 5E,,’ = b (e 1 - e 2 ), then 1 , ^ 3 , n . 5E h + T SEp - T 8 e r Y 7 5ER ^ 3 . 1 1 " 2 8er f™ SEh " T 5et 1 ' / 2 5eT y - s B R ‘/ 2 5ET - S E „ -A . (A.30) 98 Appendix B Derivation of dispersion relation and effective mass when the compression strain is along [110] and the tension strain is along [110] It has been shown that the valence band structures can be explained by the Luttinger Hamiltonian [23,24]. Thus, under strain, the total Hamiltonian is equal to the Luttinger Hamiltonian plus the strain Hamiltonian, which is derived in Appendix A. As an approximation, the 6 x 6 matrix can be reduced to a 4 x 4 matrix assuming Ao» 8 Er, 8 E„, so that the heavy hole and light hole bands are decoupled from the split-off band. In this situation the strain-dependent Hamiltonian can be written as .3 3 1 2 2 > ,3 1 1 2 2 > ,3 1 2 '2 > , 3 3 i 2 - 2> S + T + e- 8 EH L M +ie' 0 * L S-T-e - 8 Eh 0 M +ie' M*- i e ' 0 S-T-e - 8 Eh -L 0 * M - ie ' -L* S+T+e-8E„ where 6 =^8Er 1 (B.2) and y„ y2, and y, are Luttinger parameters. The eigenenergy can be found by subtracting the eigenvalue for each diagonal term and setting the determinant of Eq. (B .l) to zero. This results in the following expression: (S-T-e -E) 2 (S+T+e -E) 2 - 2 [M2 +e ,2 +ie ’ (M*-M)] (S+T+e -E)(S-T-e -E) + 2L2 [M2 +e ,2 +ie '(M*-M)] + [M2 +e ,2 +ie ’ (M*-M ) ] 2 + L4 = 0. (B.8 ) Since we are interested in the effective masses along [110], [110], and [001], we can make a transformation from the crystal system, (k„ky ,k,), to the strain coordinate system, (k„k 2,k3 ), by setting k ,= v 5 k ,‘ v f k 2 k , ^ k , + i k , (B.9) K = k3 where k„ k2 , and k3 are along [110], [110], and [001], respectively. Direct substitution of (B.9) into (B.7) to obtain the transformed Hamiltonian is extremely tedious. We can, instead, determine the masses by diagonalizing the Hamiltonian for specific directions in k space, i.e. k,, k2 , and k3 successively. If we set k2=k3=0, then Eq. (B.8 ) has the following solution: E(k,) = -8EH -Y,k,2±{(Y22+3Y32)k,4 -2(6Y2+V36 'Y 3)k,2 +6 2+e ,2},/2, (B.10) Since we are concerned with the bands near k=0, we can expand E(k,) for small k,. Eq. (B.10) then becomes “ fi2 E(k,~0)=-8EH ±(e 2 +e ,2)i/ 2+[-y,±(g 2 +e ,2 ) l/2 [-(e y2+V3€ Y 3)]k , 2 ( ^ (B .ll) 100 The effective mass mk l* can be found by setting ^ = ^ 0 = - to ± tl,7,± (B.12) where rj, = [ 1 + (^-)2] l/2 , ti 2 = £ - ) ti„ (B.13) e € 3 1 when "+" is for m , = ± 2 bands, and is for m, = ± 2 bands. Following the same procedure, the dispersion relations and effective masses for the other two situations can be expressed as follows: « fi2 E(k 2)=-5EH ± ( e 2 +e '2)l/ 2+[-7 ,± (e 2+ e , 2)‘ 1/2 (V3e 'y 3- e 7 2)]k 22 ( ^ ) (B.14) ^ = -[7,111,7,+ V3ll273 ] (B.15) jjg 2 E(k 3)=-8 E„±(e 2 +e , 2)i/ 2+[-7 ,± 2 (g 2 +e *y'a G7 2]k32 ( ^ (B.16) - ( 7 ,+ 2T|,y*)- (B.17) 101 Appendix C Evalution of transition matrix elements for the MQW under anisotropic strain (A) Compression along [110] and tension along [110] The strain Hamiltonian is given by Eq. (A.23). Since the off-diagonal terms are not 3 3 3 1 1 1 zero, the three valence bands, 12 . 2 * 2 ’ " 2 anc* ^2'~2>’aTe m‘xe<^ together. If we ignore the I ^ , ± ^ > band, then the eigenfunction can be found analytically by setting when HH and LH represent heavy and light hole bands, respectively. The eigenfunctions IJ,m,> represented by Yl f m are given by (A. 14), and Yljn are related to the p orbitals by [32] I LH > = i (C.l) where 2^Eq w + 102 ly>T = ^ Y „ T - Y „ T (C.2) lz> i = Y101 After some manipulations, we obtain .3 3 1 . . '2 2> = ^ y>T l l X> = - p (2 lz> t + lx-iy> r} (C.3) , 1 1 1 ,, i • I j ~ 2 > = {lz> i - lx-iy> r} Plugging (C.3) into (C.l), IHH> and ILH> can be represented by lx>, ly>, and lz>. Also, the conduction band, lc>, is the S orbital and is given by ls>. The oscillation strengths of the optical absorption between the HH and LH bands and the conduction band are proportional to the square of the optical transition matrix, which are given by kcl£.plHH>l2 and kcl£.plLH>l2 , respectively, where £ is the unit polarization vector of the electric field of the incident radiation and g is the linear momentum. From symmetry considerations, it can be shown that the only nonzero matrix elements of M are [4] M = <slp,lx> = <slpy ly> = <slpjz>. (C.4) (i) For incident polarization parallel to the compression direction, i.e. [110], the polarization vector £ . , 0 can be written as (C.5) For transitions between the HH band and the conduction band, 103 '•» 'at| = < cl^ | p,IHK> + <c pyIHH> = ^ a + T = ^ ttM + >^ k <,- T = l} " M 2 Vl+n2/3 2V6 V l+r|2 /3 + (1+ , 1 ) 1/2 M + - ^ (1- , 1 - t a M V l+Tl2 /3 2^ V 1+tiV3 = — r — n + —= = i ! /2 + — n . — — — v g i 2 ^ V l+r|2 /3 ^ V l+T|2 /3 + i ^ [ j = (l+ - = J ^ n + j = ( l - - = J = = ) l n l v 2 y i+ t]2 /3 v 6 V l+ riV 3 M2 1 1+n Therefore, I<cI£M 0.eIHH>I2 = [ 1 + £ 1 ]. (C.6 ) Vl+n2 /3 On the other hand, for transitions between the LH band and the conduction band, <cl£ll0.EILH> = - y [ ± (1- )m - - 4 (1+ T ' 1 -— ) l /2 1 ^ 2 V l+il2 /3 V l+Ti2 /3 + i y [ 4 ( l - - p L = T - ^ ( 1 + - = L = ) ! /2 ]. ^ 2 V l+ri2 /3 Vl+Ti2 /3 Therefore, kcl£U 0-ulLH>l2 = ^ [ 1 - i ■ 1+T]— ]■ (C.7) V l+r|2 /3 (ii) For incident polarization parallel to the tension direction, i.e. [110], the polarization vector £ ll0 can be written as 104 _ 1 _ fiu« = "7s ( - £ .+ £ ,) • (C.8) Following the same procedure as in part (i), we can obtain kcl£il0.EIHH>l: = - ^ [ - ^ (1+ j K (l— j J = V l+T|2 /3 V l+T|2/3 X*] 1 )l/2-4= O -T -^ )l/2] 2 ^ V l+n2 /3 ^ V1+T173 M 1 kcl£ll0.EILH>l2 = - ^ [ - ^ (1- 1 zXa +-\= (1+- 1 Xn] •Mr 1 n • l ? [ ^ (1 ^ V l+if/3 ^ V1+T173 ^ ]. 21 v2 v* vt^ t ^ v l+tl2 /3 Therefore, we have kcl£il0.EIHH>l2 = ^ [ 1 + ^ - -] V1-T1V3 kcl£ll0.EILH>l2 = x t 5 - ] ■ V 1-r|2 /3 (C.9) (C.10) (B) Compression along [100] and tension along [010] The total Hamiltonian, which is the sum of the Luttinger Hamiltonian and the strain Hamiltonian, is given as H = 1 2 + 8 e t ) / 3 - 2 8Er /3 2 SEr 1 2 (AEQW+ 8 E t ) (c.ll) 10 5 The mixed eigenfunctions are given as I HH > = (1 + ...... . + T 2 ( ' 1 V 1+3k2 )1/2lf ,+ -U> I LH > = -j= (1 - V 1+3k2 )1/2I§ ,± § > — 1 ^ V 1+3k2 \1 /2 |3 + i > > 12 ’ 2 ’ - 5E r ' where k = A E qw + 6 E j '' Following the same procedure as in part (A), we obtain i i m u 1 2 M2r , 1 1+3k l < c l £ 10o . j ) I H H > l _ _ [ 1 + 2 i-------------1 V 1+ 3k 2 l<cte0 1 0 .BIHH>l2 = ^ [ i + ± -7-'-— — ] V 1+3k2 . . „ „ l2 M2 r , 1 1+3k , l<cl£10o .plLH>l — o [ 1- - 1--------- ] “ V 1+3k2 kcl£0l 0.EIHH>l2 = ^ [ 1- \ / - 3-K ]. V 1+3k2 (C.12) (C.13) (C.14) (C.15) (C.16) (C.17) 106 References [1] E. P. O'Reilly, "Valence band engineering in strained-layer structures," Semicon. Sci. Tech. 4, pp. 121-137, 1984. [2] S. C. Hong, M. Jaffe and J. Singh, "Theoretical studies of optical modulation in lattice matched and strained quantum wells due to transverse electric fields," IEEE QE-23, no. 12, pp. 2181-2195, 1987. [3] T. E. Van Eck, P. Chu, W. S. C. Chang and H. H. Weider, "Electroabsorption in an InGaAs/GaAs strained-layer multiple quantum well structure," Appl. Phys. Lett. vol. 49, no. 3, pp. 135-136, 1986. [4] M. E. Favaro, G. E. Fernandez, T. K. Higman, P. K. York, L. M. Miller and J. J. Coleman, "Strained layer InGaAs negative-resistance field-effect transistor," J. Apple. Phys. vol. 65, no. 1, pp. 378-380, 1989. [5] E. S. Koleles, "The use of strain to optimize quantum well device performance," Mat. Res. Soc. Symp. Proc., vol. 281, pp. 141-152, 1993. [6 ] See, for example, Fred Poliak, in Semiconductors and semimetals, edited by T. P. Pearsall (Academic, New York, 1990), vol. 32, Chap. 2, and reference therein. [7] C. Jagannath, E. S. Kotels, J. Lee, Y. J. Chen, B. S. Elman and J. Y. Chi, "Uniaxial stress dependence of spatially confined excitons," Phys. Rev. B, vol. 34, no. 10, pp. 7027-7030, 1986. [8 ] C. Mailhiot and D. L. Smith, "Effects of compressive uniaxial stress on the electronic structure of GaAs-Gai_xAlxAs quantum wells," Phys. Rev. B, vol. 36, no. 5, pp. 2942-2945, 1987. [9] G. D. Sanders and Y.-C. Chang, "Effects of uniaxial stress on the electronic and optical properties of GaAs-Alx Gai.xAs quantum wells," Phys. Rev. B, vol. 32, no. 6 , pp. 4282-4285, 1985. 107 [10] B. Gil, P. Lefebvre, H. Mathieu, G. Platero, M. Altarelli, T. Fukunaga and H. Nakashima, "Reflectance spectroscopy on GaAs-Gao.5 Alo.5 As single quantum wells under in-plane uniaxial stress at liquid-helium temperature," Phys. Rev. B, vol. 38, no. 2, pp. 1215-1220, 1988. [11] H. Shen, M. Wraback, J. Pamulapati, P. G. Newman, M. Dutta, Y. Lu and H. C. Kuo, "Optical anisotropy in GaAs/Alx Gai_xAs multiple quantum wells under thermally induced uniaxial strain," Phys. Rev. B, vol. 47, no. 20, pp. 13933- 13936, 1993. [12] Y. Lu, H. C. Kuo, H. Shen, M. Taysing-Lara, M. Wraback, J. Pamulapati, M Dutta, J. Kosinski and R. Sacks, "Creation of in-plane anisotropic strain in GaAs/AlxG ai.xAs multiple quantum well structures," Mat. Res. Soc. Symp. Proc. vol. 300, pp. 537-542, 1993. [13] G. Bastard and J. A. Brum , "Electronic states in sem iconductor heterostructures", IEEE J. Quantum Electron., vol. QE-22, no. 9, pp. 1625- 1644, 1986. [14] Gerald Bastard, Wave mechanics applied to semiconductor heterostructures, Halsted press, New York, 1988. [15] P. J. Stevens, "Computer modeling of the electric field dependent absorption spectrum of multiple quantum well material", IEEE J. Quantum Electron., vol. QE-24, no. 10, pp. 2007-2015, 1988. [16] A. K. Ghatak, K. Thyagarajan and M. R. Shenoy, "A novel numerical technique for solving the one-dimensional Schrodinger equation using matrix approach -application to quantum well structures", IEEE J. Quantum Electron., vol. QE- 24, no. 8 , pp. 1524-1531, 1988. [17] B. Jonsson and S. T. Eng, "Solving the Schrodinger equation in arbitrary quantum-well potential profiles using the transfer matrix method", IEEE J. Quantum Electron., vol. QE-26, no. 11, pp. 2025-2035, 1990. [18] J. M. Luttinger and W. Kohn, "Motion of electrons and holes in perturbed periodic fields", Phys. Rev., vol. 97, no. 4, pp. 869-883, 1955. 108 [19] J. M. Luttinger, "Quantum theory of cyclotron resonance in semiconductors: general theory", Phys. Rev., vol. 102, no. 4, pp. 1030-1041, 1956. [20] J. Y. Marzin, in Heterojunction and semiconductor super lattices, ed. by G. Allan, G. Bastard, N. Boccara, M. Lannon and M. Voos, Spring, Berlin, 1986. [21] S. Hong, J. Singh, "Theoretical studies of polarization dependent electro-optical modulation in lattice matched and strained multi-quantum well structures," Superlattices and Microstructures, vol. 3, no. 6, pp. 645-656, 1987. [22] E. O. Kane, "Strain effects on optical critical-point structure in diamond-type crystals", Phys. Rev., vol. 178, no. 3, pp. 1368-1398, 1969. [23] F. H. Poliak and M. Cardona, "Piezo-electroreflectance in Ge, GaAs, and Si", Phys. Rev., vol. 172, no. 3, pp. 816-837, 1968. [24] G. E. Pikus and G. L. Bir, "Effect of deformation on the hole energy spectrum of germanium and silicon", Soviet Phys. Solid State, vol. 1, pp 1502-1917, 1959. [25] Y. S. Kim and R. T. Smith, "Thermal expansion of lithium tantalate and lithium niobate single crystals," J. Appl. Phys. vol. 40, pp. 4637-4641, 1969. [26] When the bonding temperature is 150 °C, with consideration of the second order thermal expansion, the tensile strain is decreased by 20% and the absolute value of the compressive strain is increased by 5%, as shown in Fig. 4.1. These strain differences correspond to a 3% increase of HH exciton peak redshift as compared to that when the second order thermal expansion coefficient is not taken into account. [27] J. F. Nye, Physical properties of crystals, Oxford Science Publications, 1985. [28] J. P. Loehr and J. Singh, "Nonvariational numerical calculations of excitonic properties in quantum wells in the presence of strain, electric fields, and free carriers," Phys. Rev. B, vol. 42, no. 11, pp. 7154-7162, 1990. [29] W. T. Masselink, "Absorption coefficients and exciton oscillator strengths in AlGaAs-GaAs superlattices", Phys. Rev. B, vol. 32, no. 12, pp 8027-8034, 1985. 109 [30] N. Susa and T. Nakahara, "Design of AlGaAs/GaAs quantum wells for electroabsorption modulators," Solid-State Electronics, vol. 36, no. 9, pp. 1277- 1287, 1993. [31] L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York, pp 218 and 438, 1968. [32] R. L. Liboff, Quantum Mechanics 2nd Edition, Chap. 9, Addison Wesley, 1992. 110 Chapter 5 Fabrication Processes of MQW Modulators In addition to structure design and crystal growth, several processing steps are required to fabricate a useful p-i-n GaAs/AlGaAs M QW m odulator, w hich include photolithography, etching, metalization, and substrate removal. It is possible that some defects may exist in semiconductor materials; in order to avoid short circuit formed in p-i-n structures due to defects, each semiconductor wafer has to be processed into several small mesas [1]. In those situations where arrays are required, e.g. in intensity m odulation applications, the sem iconductor wafer has also to be processed into small m esas. M oreover, to ensure that the series resistance presented at the metal/semiconductor interfaces is very small and can be ignored during device operation, p- and n-type ohmic contacts of high quality have to be made [2]. In section 1 of this chapter the processes of making mesas and ohmic contacts are described in detail. Because the GaAs substrate is highly absorptive at the excitonic wavelength of the GaAs/AlGaAs MQW modulator, it needs to be removed from the G aAs/AlGaAs thin film in order to m ake a transmission-type intensity modulator. There are two different ways which are commonly used to remove the GaAs substrates. The first method involves the use of either wet etching (chemical etching) [3][4] or dry etching (plasma etching) [5] to directly "eat" the GaAs substrate. The substrate will be etched off with this method. Another approach is to use the lift-off technique to separate the epitaxial thin film from the 111 GaAs substrate. This is done by growing a release layer (or so-called sacrifice layer) between the thin film and substrate, which will then be etched away with appropriate chemicals (e.g., dilute hydrofluoric acid) [6]. In this situation, the GaAs substrate will remain reusable after the process. In section 5.2 chemical etching and lift-off technique are discussed; both of them were used to fabricate the devices investigated in this work. 5.1. Processing of the p-i-n MQW modulators The MQW/GaAs wafer is first cleaved into a few pieces. The exact size or shape of each cleaved sample depends on the specific application. Before further processing the cleaved w afer has to be cleaned by trichloroethylene (TCE), acetone and m ethanol, then follow ed by de-ionized (DI) water. The procedure for further processing is illustrated in Fig. 5.1 and discussed as follows. 5.1.1 Photolithography The first step is to pattern the photoresist onto the p-type side of the sample for future p-type metalization. AZ 5214 photoresist is applied onto the sample at a spin rate of 5000 rpm for 30 seconds. This results in a photoresist layer of about 1.2 pm thick. Photoresist AZ 5214 can be used for both positive and negative photoresist processing. For our application, the patterned photoresist has to provide a mushroom-shaped lip or retrograde wall-angle, creating a discontinuity in the metal for liftoff. There are two ways to create a mushroom 112 shaped lip. One is to use positive processing and then soak photoresist in chlorobenzene for 90 seconds after exposure and before development. The other is to do image reversal using negative photoresist [7]. The use of the latter method results in a pattern which is reversed to the mask; however, it provides a better resolution than the former method does. In this work the negative processing method was chosen due to the limitation of the available masks. After the photoresist is applied, the wafer is put on a hot plate and baked at a constant temperature of 120 °C for 45 seconds to drive out all traces of solvent from the photoresist. The wafer is then exposed to UV light with an intensity of 10 mJ/cm2 for 6 seconds through the mask [Fig. 5.2(a)]. As shown in Fig. 5.2(b), the exposed areas are wider at the top and narrower at the bottom due to the decay of UV light inside the materials. The wafer is then post-baked at 120 °C for another 60 seconds. After post-baking, the exposed areas become harder and insensitive to UV light; however, the unexposed regions are still sensitive to any significant exposure to m id-UV radiation and subsequent development in conventional positive photoresist developer. This wafer is then blanket-exposed (without mask) to UV light of 10 mJ/cm2 for 1 minute, as shown in Fig. 5.2(c), followed by developing in 1:4 AZ400K/D1 water for 20 seconds. The cross-section of the resulted pattern is shown in Fig. 2(d). This pattern has a retrograde wall-angle shape and is ready for the metalization process. The photoresist pattern consists of rectangular openings of 100 x 50 pm2 with center-to-center distance of 550 pm. 113 Photolithography j Au/Zn Deposition (Metalization) then Lift-off (remove photoresist) n-doped GaAs ^ — photoresist V ' p-dopedGaAs i-MQW n-doped GaAs Au/Zn electrode n-doped GaAs J Photolithography followed by Chemical Etching Photolithography followed by Au/Sn Deposition and Lift-off ■ wet etching ■ i y I n-doped GaAs n-doped GaAs n-doped GaAs Annealing then W ire Bonding photoresist p-doped GaAs i-MQW n-doped GaAs n-doped GaAs n-doped GaAs Fig. 5.1 Processing procedure for pin MQW modulators. 114 UV light ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ p i n M Q W s a m p l e p i n M Q W s a m p l e mask photoresist (Patterned Exposure) unexposed area exposed area (Post Bake) UV light ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ p i n M Q W s a m p l e (Blanket Exposure) p i n M Q W s a m p l e (Development) Fig. 5.2 Processing procedure for image reversal. 115 5.1.2 Metalization, Liftoff and Annealing A practical approach to make low resistance ohmic contacts for semiconductors has been realized by forming a highly doped layer between the semiconductor and the metal, using alloys. The Au-Ge/Ni [8 ] and Sn-Au [9] alloys are commonly used to contact n-type GaAs. As for contacts to p-type GaAs, Au-Zn [9] [10] is often used. In our processing, Au/Sn and Au/Zn are used as n-type and p-type ohmic contacts, respectively, for GaAs/AlGaAs MQW modulators. The electron beam evaporation system is used to deposit Au/Sn, and the thermal evaporation system is used to deposit Au/Zn. Prior to ohmic metal deposition, the patterned wafers are dipped in a 1:5 HC1/DI water solution for 1 minute to remove oxide layers which hinder MQW modulators from forming good ohmic contacts. A 1 0 0 A thick layer of Sn (or Zn) is deposited first, followed by 3000 A of Au layer. Next, the photoresist is removed from the wafers by immersing the samples in acetone. It takes less than 1 minute to completely lift off the unwanted metal. Finally, the samples are annealed at 440°C for 1 minute [11] so that Sn (or Zn) can diffuse into GaAs and form a heavily doped layer. 5.1.3 Etching Next step is to process each wafer into small mesas. A mask with squares of 250 pm x 250 pm, which has a center-to-center distance of 550 pm , is used to pattern the photoresist. The mask is aligned so that the distance from the edge of p-type contact to the edge of photoresist square is around 10 pm to avoid short circuit. The chemical solution used to do etching is prepared by mixing 85% phosphoric acid (H3PO 4 ), 116 30% hydrogen peroxide (H 2 O 2 ) and de-ionized water (DI H2 O) at volume ratios of 3:1:50, which has been found to have a slow and stable etching rate for GaAs/AlGaAs materials [12] and is suitable for our application. The etching rate of undoped GaAs is measured to be about 800 to 900 A/min. The etching rates for doped GaAs or AlGaAs are about 20 to 50 A/min higher than that of undoped GaAs. The required etching depth is the sum of the thickness of p- and i- type regions and 1 0 0 0 A ~ 2 0 0 0 A of n-type materials. The etching time can then be determined based on the required etching depth and etching rate. The final etching depth can be measured with the Alpha step profile meter. 5.2 GaAs Substrate Removal 5.2.1 Chemical Etching Three steps are required to remove the GaAs substrate with chemical etching. Firstly, mechanical polishing is used to remove a major portion of the GaAs substrate until the sample has a thickness of about 100 pm. Secondly, nonselective chemical etching is used to further reduce the thickness of the sample by 60 ~ 70 pm. A minimum sample thickness of 20 ~ 30 pm is required to avoid accidental etching of the AlGaAs MQW thin film. Finally, selective etching is used to remove the GaAs substrate completely. Some detailed information about the procedure to remove the GaAs substrate with chemical etching is given as follows. 117 For convenient handling each sample is permanently mounted on a piece of transparent substrate using either transparent wax or epoxy with the MQW thin film facing the transparent substrate. This transparent substrate can be a piece of glass, sapphire or LiTaC>3 , depending on specific applications. If epoxy is used, the mounted sample has to be cured either at room temperature for more than 24 hours or at a certain elevated temperature for a few hours. The thickness of the sample, before and after attaching the sample to the transparent substrate, needs to be measured in order to determine the thickness of wax or epoxy. The thickness of the GaAs substrate needs to be monitored during polishing to ensure that the sample is not overly polished. The polishing metal mount, which consists of a ring mount with a cylindrical mount inside as shown in Fig. 5.3, is specially designed so that the sample can be polished evenly in all directions. As many as four pieces of MQW samples can be mounted near the center of the cylindrical mount using double-sided tape, as shown in Fig. 5.3. A few dummy samples, e.g. pure GaAs wafers glued to glass, are mounted surrounding the MQW samples. A radially symmetric configuration o f the pieces helps minimize the amount of wedge after grinding. Five different grain sizes of aluminum grits are used in sequence to polish the MQW samples. The adherent polishing cloth for each grain size of aluminum is attached on a flat glass plate. The aluminum grit is first mixed with clean water in a plastic bottle and then fed onto the polishing cloth. To polish, the operator holds the ring mount and 118 moves it as writing a figure "8 " so that the samples can be polished evenly. The cylindrical mount should be rotated by a few degrees (e.g. 90°) once in a while to get better polishing quality. ring mount cylindrical mount Dummy sample C 3— □ □ MQW sample Fig. 5.3 Metallic mount for polishing of the MQW sample. 119 For the polishing of the MQW samples used in this thesis, a grain size o f 22.5 pm was first used to reduce the thickness of the GaAs substrate to about 130 pm. The samples were then polished to about 110 pm thick using 9.5-pm grit with Buehler polishing cloth "Texmat". These samples were further polished down to 103 pm using 5 pm grit with the similar cloth. The 3-pm grit and the microcloth were then used to polish the sample down to about 100 pm . Finally the 1-pm grit and the microcloth were used to polish the MQW samples until they have shiny surfaces. The Starret dial-m icrom eter, which has a resolution of 0 . 2 pm, was used to measure the thickness of the sample during polishing. The sample should be rinsed thoroughly and dried with N 2 gas before measuring. Next step is to reduce the thickness of the sample to about 20 pm using nonselective chemical etching. A de-ionized w ater (DI H 2O) /hydrogen peroxide (H2O2) /sulfuric acid (H2SO 4) etching solution with a volume ratio of 3:1:1 was used for this purpose. The etching rate of this etchant is around 3 pm/min at 0 °C [13]. This etching solution is prepared as described below. Firstly, put 150 ml of DI water in a 500 ml beaker. This beaker stays in an ice bath with a thermometer inside the beaker to monitor the temperature. Then, 50 ml of sulfuric acid is slowly poured into beaker while the solution is stirred by a magnetic bar with the ice bath on a stirring plate. It takes about half an hour to pour sulfuric acid into the beaker completely. Finally, add 50 m l of 30% hydrogen peroxide into the solution. 120 Notice that hydrogen peroxide can not be added to the solution before the H2SO4. Otherwise, heat resulting from H2O/H 2SO4 solution will cause the H 2O 2 to decompose into w ater and oxygen. Final temperature of the prepared solution is around 10 °C. The etching rate of this solution is measured to be about 3.1 pm/min. To ensure that MQW samples are not overly etched, a piece of GaAs sample (which was used as a dummy sample when polishing the MQW samples) was dipped into the etchant 5 minutes before dipping the MQW sample and another piece of GaAs dummy sample inside the etchant. Both the MQW sample and the second GaAs dummy sample have a thickness of about 15 pm when the first GaAs dummy sample disappears due to chemical etching. The second GaAs dummy sample, which was dipped into the etchant with the MQW sample, can be used as a timer later in selective etching. As a final step, selective etching is used to remove the rest of the GaAs substrate. Hydrogen peroxide /ammonium hydroxide solution has been commonly used as a selective etchant for GaAs/AlGaAs structure [14]. However, this etchant suffers the disadvantage of low selectivity of around 10 and it attacks the AZ 5214 photoresist (the selectivity is defined as the ratio between GaAs etching rate and AlGaAs etching rate). Many efforts have been devoted to improving the selectivity by agitating the etchant solution, by adjusting the PH values, or by providing a steady jet stream of the etching solution. Recently, citric acid /hydrogen peroxide solution was found to be a preferable selective etching in many ways including: better selectivity, sharper edges and 121 smoother etched surfaces. It is also compatible with photoresists [15]- [17]. However, the optimal volum e ratio between citric acid and hydrogen peroxide varies for different conditions. A lso, the useful ratio is limited to a small range. Since this is a critical stage in the w hole process, the optim al volum e ratio and selectivity fo r GaAs/AlGaAs have been studied and are described in detail as follows. There are two kinds of citric acids which can be used for selective etching: anhydrous and monohydrate. Since the etching rate depends on the amount o f water inside the etchant, in order to control the etching rate precisely, anhydrous citric acid was chosen in this study. To prepare for the solution the anhydrous citric acid crystals (C6H8O 7) are first dissolved in de-ionized water at a ratio o f 1 g citric acid to 1 m l DI w ater (i.e., 50% by weight). Since the reaction is endothermic and hinders quick dissolution, the citric acid has to be mixed with de-ionized water at least a few hours in advance to ensure complete dissolution and room temperature stability. Then, this 50% citric acid is mixed with 30% hydrogen peroxide at a given volume ratio x (i.e., x parts of 50% citric acid to one part o f 30% hydrogen peroxide by volume). The etching rates of GaAs and AlGaAs at different volume ratios for different Aluminum concentrations are listed in Table 5.1. As indicated in Table 5.1, the optimal volume ratio is 3.5. The etching rates for this volume ratio are 3300 A/min for GaAs and 35 A/min for Alo.3Gao.7As. The selectivity is calculated to be about 94 for this specific situation. W hen the volume ratio increases or decreases, the etching rate o f Alo.3Gao.7As approaches to that of GaAs, and thus the 122 selectivity dramatically decreases. Notice that when the aluminum concentration is higher than 0.3, the selectivity increases by a large amount. Table. 5.1 Etching rates and selectivity of citric acid/hydrogen peroxide for GaAs and AlGaAs at different volume ratios for different aluminum concentration volume ratio 50% citric acid /30% H202 GaAs Al0 3 Ga0 7As etching rate etching rate (pm/min) (|j.m/min) . . ^ 0 £ p a0.5^S . . selectivity etching rate selecuvity (ptm/min) 2 : 1 0.008 0 . 0 0 2 4 3 : 1 0 . 2 0.003 67 3.5 : 1 0.33 0.0035 94 4 : 1 0.36 0.0043 84 0.001 360 1 0 : 1 0 . 2 0.19 1 123 To do the selective etching for the MQW sample, both the MQW sample and the GaAs dummy sample, left from the nonselective etching, are dipped into the selective etchant. When the GaAs dummy sample is etched away, the MQW sample should contain no GaAs substrate. If indeed the GaAs substrate of the MQW sample is etched away completely, a shining surface of the thin film should be observed due to the slowdown of the etching rate. When this finished MQW sample is viewed with a microscope, a red color can be seen through the MQW thin film. 5.2.2. Lift-off Epitaxial thin films of high quality can be obtained using the lift off technique. Another advantage of the lift-off technique, as compared to the method of chemical etching discussed above, is that after the MQW thin film is undercut from the GaAs growth substrate, the substrate is reusable. Yablonovitch et al. [5] measured a selectivity of more than ten million between Alo.4Gao.6As and the AlAs release layer. These authors also found that the etching rate increased by many orders of magnitude when the aluminum concentration increased from 40% to 50%. Therefore, lift-off technique can be applied to separate the AlGaAs/GaAs MQW thin film from the GaAs growth substrate if the aluminum concentration is lower than 40% and an AlAs release layer is grown between the thin film and substrate. A small amount of Apiezon black wax dissolved in TCE (5 g black wax in 20 ml TCE) has to be prepared in advance. The black wax 124 provides the tension required for lift-off process; it can also be used to support the MQW structure after lift-off since the film is too thin to be alone. To start the lift-off procedure the MQW sample is first cleaned with TCE, acetone, then followed by methanol. After rinsing the sample with DI water, the sample is dipped into an ammonia solution (ratio of ammonia to water is 1 to 5) for 15 seconds to remove the oxide layer. The sample is then rinsed with DI water and dried with N2 gas. The Apiezon black wax can then be applied onto the thin film side of the sample (several times to build up the thickness), as shown in Fig. 5.4. Black Wax MQW Thin Film AlAs Release Layer G a A s S u b s t r a t e Fig. 5.4 Black wax is applied onto the thin film side of the MQW sample. The AlAs release layer can be etched off with dilute HF acid. Let the wax dry for 10 minutes. In the meantime, the oven can be turned on and kept at a temperature of about 120 °C. The sample is then put in a holder and kept in the oven for 15 minutes to smooth the black wax. After the sample is removed from the oven and cooled for 1 0 minutes, the four sides o f the sample (not the top and bottom surfaces) should be polished gently to remove any black wax, as the wax will prevent the HF acid from reaching the AlAs release layer. After rinsing with the DI water, dip the MQW sample into 10% HF acid. Usually it takes 10 to 48 hours to release the MQW thin film from the GaAs growth substrate. The sample should be sitting in the HF acid with the black wax facing up; under this situation, the thin film (with the black wax, of course) will separate from the substrate automatically after the lift-off process is accomplished. If the sample has to be kept in the HF acid overnight, the container should be placed in the refrigerator to slow down the etching rate. After lift-off of the thin film, rinse the thin film with DI water followed by methanol. The methanol will dry in the air in a short time. The thin film can then be attached to a transparent substrate (which can be LiTaC>3, sapphire, or glass) using epoxy. The black wax can be washed off the sample with TCE. If the lift-off is successful, the two surfaces of the MQW thin film are smooth and shining. 126 R eferen ces [1] D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood and C. A. Burrus, "Electric field dependence of optical absorption near the band gap of quantum well structures", Phys. Rev. B 32(2), pp. 1043-1060, 1985. [2] M. J. Howes and D. V. Morgan, Gallium Arsenide materials, devices, and circuits, John Wiley & Sons, New York, 1985. [3] J. J. LePore, "An improved technique for selective etching of GaAs and Gai- xAlxAs", J. Appl. Phys. 51(12), pp. 6441-6442, 1980. [4] R. A. Logan and F. K. Reinhart, "Optical waveguides in GaAs-AlGaAs epitaxial layers", J. Appl. Phys. 44(9), pp. 4172-4176, 1973. [5] Sorab K. Ghandhi, VLSI fabrication principles - silicon and gallium arsenide, John Wiley & Sons, New York, 1994. [6 ] E. Yablonovitch, T. Gmitter, J. P. Harbison, and R. Bhat, "Extreme selectivity in the lift-off of epitaxial GaAs films," Appl. Phys. Lett. 51(26), pp. 2222-2224, 1987. [7] Eric Aling and Craig Stauffer, "Image reversal photoresist", Solid State Tech., pp. 38-39, June 1988. [8 ] Yih-cheng Shih, Masanori Murakami, E. L. Wilkie, and A. C. Callegari, "Effects of interfacial microstructure on uniformity and thermal stability of AuNiGe ohmic contact to n-type GaAs", J. Appl. Phys. 62(2), pp. 582-589, 1987. [9] G. H. Dohler, G. Hasnain, and J N. Miller, "In situ grown-in selective contacts to n-i-p-i doping superlattice crystals using molecular beam epitaxial growth through a shadow mask", Appl. Phys. Lett. 49(12) pp. 704-706, 1986. [10] Tatsuyuki Sanada and Osamu Wada, "Ohmic contacts to p-GaAs with Au/Zn/Au structure", Jap. J. Appl. Phys. 19(8), pp. L491-L494, 1980. 1 2 7 [11] K. W. Goossen, J. E. Cunningham and W. Y. Jan, "GaAs-AlAs low-voltage refractive modulator operating at 1.06 pm", Appl. Phys. Lett. 57(8), pp. 744- 746, 1990. [12] Yoshifumi Mori and Naozo Watanabe, "A new etching solution system, H 3 PO4 - H 2 O2 -H 2 O, for GaAs and its kinetics", J. Electrochem. Soc., vol. 125, No. 9, pp. 1510-1514, 1978. [13] Shinya Iida and Kazuhiro Ito, "Selective etching of gallium arsenide crystal in H2 SO4 -H 2 O2 -H2 O system", J. Electrochem. Soc. 118(5), pp. 768-771, 1971. [14] Kelly Kenefick, "Selective etching characteristics of peroxide/ammonium- hydroxide solutions for GaAs/Alo.i6 Gao.8 4 As", J. Electrochem. Soc. 129(10), pp. 2380-2382, 1982. [15] C. Juang, K. J. Kuhn and R. B. Darling, "Selective etching of GaAs and Alo.3Gao.7 As with citric acid/hydrogen peroxide solutions", J. Vac. Sci. Technol. B 8(5), pp. 1122-1124, 1990. [16] M. Tong, G. D. Ballegeer, A. Ketterson, E. J. Roan, K. Y. Cheng and I. Adesida, "A comparative study of wet and dry selective etching processes for GaAs/AlGaAs/InGaAs pseudomorphic MODFETs", J. Electrochem. Materials 21(1), pp. 9-15, 1992. [17] Gregory C. DeSalvo, Wen F. Tseng, and James Comsa, "Etch rates and selectivities of citric acid/hydrogen peroxide on G aA s, Alo.3 G ao.7 As, Ino.2 Gao.8 As, Ino.5 3 Gao.4 7 As, Ino.5 2 Alo.4 8 As, and InP", J. Electrochem. Soc. 139(3), pp. 831-835, 1992. 1 2 8 Chapter 6 InAs/GaAs Short-Period Strained-Layer Superlattices Grown on GaAs as Spatial Light Modulators: Uniformity Measurements Semiconductor electro-optical modulators have received much attention during the past decade. These devices exist both in a guided wave and a free space configuration. The guided wave configuration is being investigated especially for application in fiber-based communication systems and the free space configuration is found to be extremely useful for applications in optical information processing, optical computing and neural networks. Until now, most of electro-absorption modulators have been made in the GaAs/AlGaAs material system. Most of them use the quantum confined Stark effect (QCSE) by which the absorption of the wells shifts towards longer wavelengths when an external voltage is applied across the quantum well. With GaAs/AlGaAs quantum wells as the active region, devices have been made with good characteristics. However, all these devices work at wavelength between 800 and 880 nm. In the past few years, multiple quantum well technology has been exploited in pseudomorphic layer structures, such as InGaAs/GaAs MQWs. The absorption edge of these MQWs can be adjusted via the composition and thickness of the wells, with energies below the band gap of GaAs. This leads to increased flexibility for bandgap engineering in a wide wavelength range around 1 pm. 129 The ternary alloy InGaAs grown on GaAs is of particular interest for device application because its high electron mobility, large T -L valley separation and small band gap make it attractive for high-speed electronic devices and low-threshold lasers [l]-[3]. This wavelength region is of importance owing to the possibility of using diode lasers near 980 nm to pump the Nd:YAG and Nd:YLF solid state lasers which operate near 1.06 pm. Although the lattice constants of InGaAs and GaAs are different, the lattice mismatch can be accommodated as elastic strain if the InGaAs layer is thinner than the critical thickness for dislocation formation. The introduction of strain results in new features that are important for device design. Quite recently, promising results have been obtained in the fabrication of surface emitting lasers with the use of strained layer growth of InGaAs on GaAs [4], These lasers typically emit in the wavelength region between 900 and 1100 nm, a region where the GaAs substrate is transparent, which is useful both for lasers and for modulators. Fabrication of optical modulators which are compatible with these lasers can have an important contribution to the realization of the new optical computing structures. The fabrication of devices containing short-period strained-layer superlattices (SPSLSs) of all-binary (InAs)m /(GaAs)n compositions have received much attention recently [5-9]. The low temperature electron m obility of the field-effect transistor (FET) using InAs/GaAs monolayer superlattices is found to be one order of magnitude higher than that of InGaAs with the same impurity concentration due to the 130 suppression of alloy scattering [8]. A 15% improvement in electron Hall mobility for modulation-doped FET (MODFET) using InAs/GaAs superlattices in place of InGaAs alloy with equivalent In concentration was also observed [9]. Recently, six-period (InAs)i/(GaAs)4 SPSLSs laser was fabricated and exhibited a broad area threshold current density of 100 A/cm2. This is improved over 140 A/cm2 for lasers with random alloy Ino.2Gao.8As/GaAs single QW active regions whose total thickness is comparable to that of six-period I(InAs)i/(GaAs>4 SPSLS laser [10]. This may due to reductions in the number of impurities and interface roughness. Therefore, it may be possible that all-binary InAs/GaAs short-period strained-layer superlattices (SPSLSs) become a ordered counterpart for the ternary InGaAs strained alloys. In this chapter, optical properties of tw o-dim ensional transmission-type spatial light modulators using InAs/GaAs SPSLSs in wells, which operate near 960 nm, are investigated. The absorption characteristics of the individual pixels over the entire semiconductor wafer, i.e., the uniformity of the devices, are discussed. This study is important for parallel optical signal processing, in which the light beam is incident perpendicular to the two-dimensional arrays of modulators. Studies on the uniformity are not well-documented in the literature, except in the field of two-dimensional MQW asymmetric Fabry-Perot (ASFP) modulators [11-13]. According to K. -K. Law et al. [12], the optimum operating wavelength of an ASFP modulator can shift as much as 5 nm over a 4- mm-diameter semiconductor wafer. This is due to the fact that the 1 3 1 characteristics of the high finesse structures are sensitive to the precise change of the layer thicknesses. The requirements on the growth to obtain large arrays of these devices with sufficient uniform ity are stringent, although the high contrast ratio and low insertion loss of the ASFP m odulator are prom ising for practical applications. The characteristics of the transmission-mode InAs/GaAs SPSLS modulators are studied in this chapter. These modulations are attractive due to the attraction of inherent cascadability from a system perspective. Three different sem iconductor wafers were used for this investigation. The growth conditions, fabrication processes, and the structures of the three SPSLS MQW modulators are described in section 6.1. In section 6.2, the results of the transmission measurements are presented. Discussion of the experimental results is given in section 6.3. 6.1 Sample Structure The three MQW samples (designated as TCH1364, TCH1365 and TCH1366), which contain InAs/GaAs SPSLS wells, used in this work were grown by Dr. Tom C. Hasenberg at Hughes Research Laboratories with molecular beam epitaxy (MBE) method. A computer-controlled digitized RHEED system, which is very sensitive to spatial intensity variations along the specular streak in RHEED [14], was used to monitor the growth in real-time. The RHEED pattern was monitored with a video camera and the analog video was fed to a Macintosh computer and an S-VHS VCR where it was recorded. As thin as one monolayer can be grown with MBE. 132 Different growth conditions were investigated at the Hughes Research Laboratories [14]; the optimum growth procedure which produced a MQW modulator of best optical quality (narrowest excitonic absorption peaks and sharpest decay in the absorption tail) was utilized to grow the T C H 1364-1366 MQW samples used in this work. The SPSLS wells were grown at a constant substrate temperature (~ 500 °C) and different interruption stages (15 second pause stage followed by a 10 second arsenic soak before the In As layer) were introduced in order to obtain smoother interfaces for the InAs deposition on GaAs because it is grown at low temperature. The InAs layer w as grown by conventional MBE (i.e. arsenic and indium shutters open). Figure 6.1 shows the schematic diagram of the TCH 1364 p-i-n MQW structure. A silicon doped GaAs layer o f 500 nm was grown on the n+ GaAs substrate first, with doping concentration of 1 x 1018 cm-3, followed by 30 nm undoped GaAs buffer layer. An intrinsic MQW structure consisting of 50 periods o f 138 A GaAs barriers and 51 periods of SPSLS wells was then grown on top of the GaAs buffer layer. The p-i-n structure was completed by growing another GaAs buffer layer 30 nm thick and a beryllium doped GaAs layer 500 nm thick with doping concentration of 1 x 1018 cm*3. This SPSLS structure is labeled as 1x7x5, where 1 and 7 are the num ber of InAs and GaAs monolayers, respectively, constituting the unit cell in the SPSLS, and 5 is the number of layers of InAs that alternate with 4 layers of GaAs to form each SPSLS well. For this specific situation, the indium content is calculated to be about 15 % in the wells. Sample TCH 1365 has the same 133 structure with TCH 1364. For sample TC H 1366, the i region contains forty periods of MQWs that consist of 1x7x5 InAs/GaAs SPSLS wells and 122 A GaAs barriers. Some physical parameters of the three MQW samples which were determ ined from the transm ission electron microscopy (TEM) images (data provided by Tom. C. Hasenberg) are summarized in Table 6.1. Theoretically, the thickness of one monolayer for the zinc-blende structure along the (001) direction is one half of the unit cell lattice constant in that direction, i.e. the thickness of one monolayer is 3.3 A for InAs and 2.83 A for GaAs. Thus, the theoretical well width for the 1x7x5 SPSLS is 95.74 A. As shown in Table 6.1, the well widths determined from TEM are very close to the theoretical values, which indicates that the crystal growth is highly reliable. 134 Table. 6.1 Summary of the SPSLS MQW parameters determined from TEM images. sample SPSLS well width (A) GaAs barrier width (A) MQW periods Total well width (A) Total MQW thickness (A) TCH1364 96 138 50 4896 11796 TCH1365 99 136 50 5049 11849 TCH 1366 98 122 40 4018 8898 135 GaAs (Be) 500 nm p = 1x10 cm GaAs (undoped) 30 nm InAs/GaAs SPSLS well GaAs barrier 13.8 nm periods InAs/GaAs SPSLS well GaAs (undoped) 30 nm GaAs (Si) 500 nm n = IX 10 cm GaAs (Si) Substrate InAs 1 ML GaAs 7 ML InAs 1 ML GaAs 7 ML InAs 1 ML GaAs 7 ML InAs 1 ML GaAs 7 ML InAs 1 ML Fig. 6.1. Schematic diagram of TCH1364 MQW Structure. 136 These three samples were then processed to achieve two- dimensional p-i-n modulators following the procedures described in chapter 5, except that the conductive GaAs substrates are transparent for the wavelength region of interest for current situation and thus do not have to be removed. Figures 6.2 to 6.4 show the position of each sample which was cleaved from the original wafer and the final structures of the two-dimensional p-i-n modulators. For all MQW modulators, the dimension of each mesa is 250 pm x 250 pm with a center-to-center distance of 350 pm in one direction and 550 pm in the other direction. There are about 200 mesas fabricated on each sample. The Au/Zn p-type contact, which is grown on top of each mesa with the thermal evaporation method, has a dimension of 50 pm x 150 pm. Thus, the optical window has a size of 200 pm x 250 pm. The Au/Sn n- type contact was fabricated via electron beam deposition near the edge of each sample, as indicated in the figures. 137 Wafer TCH 1364 u n it: mm 7 pixel (2,2) Portion of wafer used in this work pixel (1,17) n-type contact □ □□□□ □□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □□ □□□ □□ □□ □□□□ □ □□□ □ □ □□□□ □□ □□ □ □ □ □□ □ □ □ □ □ □□□□ □□ □□ □ □ □ □□ □ □ □ □ □ □□□ □ □□ □□ □ □ □ □□ □ □ □ □ □□□□ □□ □ □'fTBv] □ □ □ □ □ □ □ □ □ □ □ □ □ □ < - - 8 — > 6.5 550 pm 250 L tm p-type contact 250 pm 350 pm Fig. 6.2 TCH 1364 outline. 138 99 t 6 I 1 .3 i - □□□□□□□□□□□□□ □□□□□□□□□□□□□□ □□□□□□□□□□□□□□ a a a d o □□□□□□□□□ □□□□□□□□□□□□□□ □□□□□□□□□□□□a aaaoaaaaaaa □□□□□□□□a □□□□□□□□ --------- □□ □ □ □ O D D □ □ □ □ □ □ □ □ D DOaODDDO □□□□□□DODD 6 7 □□□□□□□□□□a □□□□□□□□□□a □□□□□□□□□□□' □□□□□□□□□□a 10 ^ [Oil] u n it: mm Fig. 6.3. TCH 1365 outline. [Oil] 1.3 t 5 * i < 8 > ■ □□□□□□□□□□□□a □□□□□□□□□□□□□□□a □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ a □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ O D D □ □ □ D U O □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □□□□□□□□□□□□□□ t 5 I u n it: mm Fig. 6.4. TCH 1366 outline. 139 6.2 Experiments I-V characteristics curves were m easured p rio r to the transmission measurements. For each sample, several pixels were arbitrarily chosen to test the I-V characteristics. Figure 6.5 shows the typical I-V curves for TCH 1364, TCH 1365 and TCH 1366 (one pixel per sample). The I-V curves for other pixels have similar outlook as those shown in Fig. 6.5. As shown in Fig. 6.5, the forward turn-on voltage for each M QW modulator was about 0.5 volt and no breakthrough phenomenon was observed when 10 V reverse bias was applied to the sample. The dark current at a reverse bias o f 5 volt was less than 10 (iA. The main difference among the three samples are the magnitudes of the forward-bias resistances (TCH 1364 > TCH 1365 > TCH 1366). From the I-V curves we conclude that both growth and fabrication processes were very successful. The experimental setup for transmission measurements is shown in Fig. 6.6. A microscope lamp with a regulated power supply was used as a white light source for this experiment. The incident light was focused to a spot of 50 pm x 50 pm on the sample with a lens of 40 mm focal length and a lOx microscope objective. Without inserting the MQW sample, the intensity stability of the white light source was tested to be about ± 2%, which places a lower bound on the accuracy for transmission measurements. A RG850 color filter was used to block the light whose wavelengths are below 850 nm. A computer-controlled optical multiple-channel analyzer (OMA) with resolution of 0.5 nm was used as a detector. 140 0.5 0.4 0.3 I(mA) 0 2 0.1 0.0 - 0.1 -10 -8 -6 -4 -2 0 2 4 V (volt) Fig. 6.5. I-V curves for TCH1364, TCH1365 and TCH1366. TCH 1365 TCH1366 TCH1364 14 1 Lens (f = 4 cm) v / . . R G 8 5 0 K H filter 8 c m probe for 7 5 c m probe for n-type contact 10X microscope objective power supply _+ □ OMA - 10X m ic ro s c o p e objective Fig. 6.6. Experimental setup for transmission measurements. The white light source was focused to a spot of 50 pm x 50 pm. 142 About fifteen pixels were tested for each sample. These 15 pixels were randomly chosen from the devices roughly in every four columns and every two or three rows. Each selected pixel was individually addressed using external electrical probes. The transmission spectra were taken by varying the applied voltage from 0 up to 10 volts. The typical transm ission spectra for TCH1364, TCH 1365 and TCH 1366 are shown in Fig. 6.7 (a), (b) and (c). The sharp dips near 963 nm are attributed to the heavy hole (HH) resonances (n=l e-hh), which shift to longer wavelength when the applied voltage is increased. The maximum transmission changes occurred at heavy hole (HH) exciton peak and were 12.1%, 11.6% and 9% for TCH1364, TCH 1365 and TCH 1366, respectively. The corresponding absorption spectra for TCH 1364, TCH 1365 and TCH 1366 are shown in Fig. 6.8 (a), (b) and (c). The absorption coefficients were normalized to the entire MQW region. The peak values of the HH excitonic absorption are comparable to each other for these three samples. Notice that the HH exciton peak disappeared completely at about 8 V for TCH 1364 and TCH 1365, and at about 6 V for TCH 1366. The rounded shoulders near 930 nm are attributed to the light hole (LH) resonance (n=l e-lh). This feature was not observed in the InGaAs/GaAs alloy structures because the strain removes the valence band degeneracy in the InAs layers and LH is not confined in the InAs. However, in the SPSLS structure, the LH can tunnel through the SPSLSs so that LH resonances can be seen [15]. 143 As reported by K. Goossen et al. [16], the product o f absorption coefficient (a ) and linewidth (AB) of the exciton is roughly independent of material system from 850 nm to 1064 nm. The material which they used for the investigation was InGaAs/GaAsP (wavelength range between 920 nm to 1064 nm ). The absorption coefficient of InGaAs/GaAsP MQW at 960 nm roughly equaled 1.33 pm -1 if it was normalized to the total well width only and the bandwidth was about 9.3 meV. The absorption coefficients for TCH 1364, TCH 1365 and TCH1366 (selected pixels are indicated in Fig. 5.7) are 1.17 pm-1 , 1.08 p m 1, and 1.07 pm -1 , respectively, if they are also normalized to the total well width. The absorption coefficients obtained from TCH1364- 66 are smaller than that obtained from the InGaAs/GaAsP material. This might be due to the fact that a smaller well width (~ 95 A) was used for K. Goossen's sample, which provided more confinement for the excitons. To determine the bandwidth, a Lorentzian lineshape was used to fit the long wavelength portion of the exciton absorption curves. The fitted bandwidths for TCH 1364, TCH 1365 and TCH 1366 are 11.2 meV, 11.6 meV and 13.9 meV, respectively. As shown in Table 6.2, the absorption-linewidth products, aAB, for these three samples are comparable with that of the InGaAs/GaAsP MQW except TCH 1366. The linewidth of TCH 1366 is larger than those of TCH 1364 and TCH 1365, which might be due to different growth temperatures (as indicated by Dr. Hasenberg). However, the absorption-linew idth product for TCH 1366 is still in the same order of magnitude with that 144 of the InGaAs/GaAsP MQW. It indicates that the quality of these samples are alike. M. Jupina et al. [15] also studied the InAs/GaAs SPSLS MQW structure, except that their samples contained 32% indium and the HH peak was close to 975 nm. The absorption coefficient and the fitted bandwidth (from Fig. 1 in [15]) were 1.1 jin r 1 and 11 meV, respectively. These values are also close to those of the InGaAs/GaAsP MQW, as shown in Table 6.2. The absorption coefficient-linewidth product for the InGaAs/GaAs structure, which operates at 977 nm [17], is also listed in Table 6.2 for comparison purpose. 145 Table 6.2 Absorption coefficient-linewidth product, aAB, for several different samples. Sample a (pm '1) AB(meV) a A B Ref InGaAs/GaAsP @960nm 1.33 9.3 12.4 [16] TCH 1364 @ 962 nm 1.17 11.2 13.1 ours TCH 1365 @ 964 nm 1.08 11.6 12.5 ours TCH 1366 @ 963 nm 1.07 13.9 14.8 ours InAs/GaAs SPSLS @975nm 1.10 11.0 12.1 [15] InGaAs/GaAs @977pm 0.82 15.2 12.6 [17] 146 The typical absorption changes A a = a(0) - a(V ) for TCH 1364, TCH 1365 and TCH 1366 are shown in Fig. 6.9 (a), (b) and (c), where oc(0) and a(V ) are the absorption coefficients without and with applied voltage, respectively. As indicated in Fig. 6.9, the maximum absorption change is about 0.26 p m 1 near HH exciton peak. At low applied voltages, the absorption changes of TCH1366 are comparable with those of TCH 1364 and TCH 1365. However, at higher applied voltages, the absorption change of TCH1366 saturates faster than TCH 1364 and TCH 1365. This may be due to the broader exciton bandwidth of TCH 1366. The absorption changes of all selected pixels at HH peak for TCH 1364, TCH 1365 and TCH 1366 are shown in Figs. 6.8 to 6.12, where the position of each pixel is identified by its row and column numbers. As shown in these figures, the absorption changes of the selected pixels do not show specific relation for each wafer. Notice that the wavelengths of the HH exciton peaks depend not only on the structures of the MQWs, but also on the specific position on the wafers. These data will be analyzed and how the variation o f the data affect the performance of the modulators will be discussed in next section. 147 Transmission Transmission Transmission 0.6 0.5 0.4 10 v 0.3 TCH1364 row2 column 18 0.2 960 980 1000 900 920 940 Wavelength(nm) (a) 0.6 0.5 0.4 10 v 0.3 TCH1365 row9 column 16 0.2 960 980 1000 900 920 940 Wavelength (nm) (b) 0.6 0.5 0.4 0.3 TCH 1366row5 column3 10 v 0.2 900 920 940 960 1000 980 Wavelength (nm) (C) Fig. 6.7. Transmission spectra of (a) TCH1364 (b) TCH1364 (c) TCH 1366 for various applied voltages. 148 ( U i T l/ l ) » TCH1364 row2 columnl8 OV 2 V — - 4 V 6 V 8 V 10 V 0.8 n=2 e-hh 0 .6 n=l e-hh 0.4 0 .2 0 .0 960 1000 980 900 920 940 Wavelength(nm) (a) TCH 1365 row9 column 16 0.8 2 V - -4 V 6 V 8 V 10V 0 .6 0.4 0 .2 0 .0 960 980 1000 940 900 920 Wavelength (nm) (b) TCH1366 row5 column3 OV 0 .8 4 V 6 V 8 V 10 V 0 .6 0.4 0 .2 0 .0 940 960 980 1000 900 920 Wavelength (nm) (C) Fig. 6.8. Absorption spectra of (a) TCH 1364 (b) TCH 1364 (c) TCH 1366 for various applied voltages. 149 ( u j t I / i ) (a) »- ( o ) » = » V (^/D (A ) n-(0)n = o v (uiri/[) (A) »- (0 ) » = 0 . 4 TCH 1364 row2 column 18 2 V 4 V 6 V 8 V 10 V 0.3 0 .2 0 .1 0 .0 -0 .1 -0 .2 960 980 1000 920 940 900 Wavelength(nm) (a) 0.4 TCH1365 row9 columnl6 - 2 V -4 V - 6 V - 8 V - 10 V 0.3 0.2 0 .0 -0 .1 -0 .2 960 1000 980 900 920 940 Wavelength (nm) (b) 0.4 TCH 1366 rowS column3 2 V 4 V 6 V 8 V 10 V 0.3 0.2 0.1 0.0 3 -o.i - 0.2 960 980 900 920 940 1000 Wavelength (nm) (c) Fig. 6.9 Absorption change for (a) TCH 1364 (b) TCH 1365 (c) TCH 1364 at various applied voltages. 150 = a (0)-a (V)(/|im) A a = a (0) - a (V) (/pm) A a = a (0) - a (V) (/pm) 0.35 R ow 2 0.30 2V 3 V 4 V 0.25 0.20 «V 7 V »V 0.15 0.10 10 V 0.05 0.00 c o lu m n n u m b e r (a) 0.35 R ow 5 0.30 2 V 0.25 4 V 5 V 6 V 2V 8 V 9V 10 V 0.20 0.15 0.10 0.05 0.00 c o lu m n n u m b er (b) 0.35 R ow 8 0.30 2 V 3 V 4 V 5 V 6 V 0.25 0.20 0.15 8 V 9 V 10V 0.10 o- 0.05 0.00 c o lu m n n u m b er (c) Fig. 6.10. Absorption coefficient changes for various applied voltages for the selected pixels of sample TCH 1364. 151 = a (0)-a (V)(/nm) A a = a (0) - a (V) (/nm) A a = a (0) - a (V) (/nm) 0.30 Row 6 0.25 O- 0.20 -O 0.15 0.10 0.05 10 V 0.00 column number (a) 0.30 Row 7 0.25 -O 0.20 0.15 0.10 0.05 10 V 0.00 column number (b) 0.30 Row 9 0.25 - O - 0.20 0.15 0.10 < 3 0.05 10 V 0.00 0 5 1 0 15 2 0 25 30 column number (c) Fig. 6.11. Absorption coefficient changes for various applied voltages for the selected pixels of sample TCH 1365. 152 0.30 Row 3 / — s £ § 0.25 0.20 a I 0.15 /- S O B 0.10 I I 0.05 10 V 0.00 column number (a) 0.30 Row 5 0.25 0.20 a 0.15 / * > o B 0.10 I I •S 0.05 10 V 0.00 column number (b) 0.30 Row 7 0.25 0.20 a ' 0.15 . — — V o B 0.10 I I 0.05 10 V 0.00 5 15 0 1 0 2 0 column number (c) Fig. 6.12. Absorption coefficient changes for various applied voltages for the selected pixels of sample TCH 1366. 153 6.3 Discussion A number of parameters can be utilized to characterize the performance of a single electro-absorption optical modulator: the on/off contrast ratio, the insertion loss, the bandwidth, the operation speed and the power consumption [18-20]. Moreover, parameters required to characterize two-dimensional spatial light modulators also include the uniform ity of absorption coefficient, the uniform ity of operation wavelength and the cross-talk between adjacent pixels [13,21]. In this section, we focus on discussing the uniform ities of absorption coefficient and excitonic wavelength from experimental results. The average values of a(0) and Aa(V) for all selected pixels and the variation from their average values for the three different samples are tabulated in Table 6.3. A fixed operating wavelength was chosen for each sample, which is the average wavelength of the HH peaks of all selected pixels, since a two-dimensional modulator array should be operated at the same wavelength. As indicated in Table 6.3, in terms of a(0), the maximum and minimum values are around 10% away from the average values. At an applied voltage of 2 V, A a has a deviation of about 18%. However, for 4 V and above, the deviations range from 6% to 12%. At higher biased voltages, the deviations are even smaller. Since the experimental error came mainly from the power variation of the white light source, as mentioned before, which is about ± 2 %, the experimental results indicate that the variation of A a among different pixels may be the inherent characteristics of the samples. 154 Notice that although the MQW structures of TCH1364 and TCH1365 are the same, the performance of the TCH1364 is better than that of the TCH 1365. Moreover, the deviations of a(0) and A a(V ) are larger than those of TCH 1365. Part of this effect may be attributed to the fact that the position of sample TCH 1364 was close to the center of the wafer, while sample TCH 1365 was cut from the edge of the wafer, as shown in Figs. 6.2-3. It seems that the center portion of the wafer has a better quality than the edge of the wafer. This may also explain why the bandwidth of the TCH 1364 is narrower than that of the TCH 1365 as mentioned before. 155 Table 6.3 Average values of a(0) and Aa (2,4, 6, 8 and 10 volts) and distribution range for 15 selected pixels of TCH 1364, TCH 1365 and TCH1366. The absorption coefficients have a unit of pm-1. The operating wavelength is 961.6 nm for TCH1364, 964.0 nm for TCH1365, and 962.7 nm for TCH 1366. \ a b s o r p - ^vtion sam ple's^ a(0) Aa(2) Aa(4) Aa(6) Aa(8) Aa(10) TCH 1364 0.499±9% 0.084±12% 0.192±9% 0.252±9% 0.263±9% 0.265±8% TCH 1365 0.454±7% 0.061±18% 0.150±12% 0.211±11% 0.235±10% 0.241±10% TCH 1366 0.478±11% 0.107±18% 0.204±11% 0.236±7% 0.239±8% 0.250±6% 156 For a transmission-type electro-absorption modulator, the on/off ratio (R) is defined as the ratio of the output power at on-state over that at off-state, i.e., R = exp(Aad), where A a is the absorption change between the on and off states and d is the MQW thickness. Assuming the fractional deviation of Aa, due to the non-uniformity is ±8, then the corresponding fractional variation of R is given by a d ( A a ± A a - 8) d „ A a d A R _ e__________________- e t A a - 8) d , ~r = e AoTd = e - 1 ~ ±5-Aad (6.1) For TCH 1364 and TCH 1366, both of them were cleaved from the center o f the wafers, as shown in Table 6.3, the distribution range of absorption change at 10 V applied voltage [Aa(10 V)] is about ±6~8%. The fractional variation in the on/off ratio caused by this absorption change non-uniformity is calculated to be about ±2%. In analog applications, the system performance is usually defined in terms of dynamic range, which is contrast ratio expressed in dB. This is given as D = 10-log(Ton/T off) = 10-log[exp(Aad)] (dB), where Ton and T 0ff are the on and off state transmission, respectively. The difference of the dynamic range (AD) due to a non-uniformity of A a is given by AD = 10-log [ e (Afx±Afx 6)d] . 10-log [eAad] = 10-log [e±5Aad] (6.2) For a 7% deviation of A a, the dynamic range is changed by ±0.1 dB. For TCH 1364, the dynamic range is 1.36 dB; therefore, the 157 available gray levels are about seven. If the deviation of A a is reduced to be half of the original value, the available gray levels will be increased to be twice of the original value. Therefore, for applications in analog systems, the non-uniformity of Aa does affect the available gray levels and limit the system performance. From Table 6.3 one can calculate the upper and lower bounds for both the on/off ratio and the dynamic range. As an example, Fig. 6.13 shows the on/off ratio and dynamic range as a function of applied voltages for TCH 1364. As indicated in Fig. 6.13, on/off ratio and dynamic range increase as applied voltage increases. For this specific example, both curves saturate when the applied voltage is near 8 V. 158 0 •a S 1 o 1.45 TCH 1364 1.40 1.35 1.30 1.25 0.8 1.20 -e— On/Off (upper bound) -•— On/Off (lower bound) 0.6 1.15 o- - - - Dynamic Range (upper bound) - Dynamic Range (lower bound) 0.4 1.10 1.05 0.2 0 8 10 2 4 6 12 o b O c Q Voltage (V) Fig. 6.13. On/off ratio and dynamic range as a function of applied voltages for TCH 1364. 159 Another important factor which affects the system performance is the insertion loss. For an electro-absorption modulator, the insertion loss is defined as L = 1 - Po/Pin, where P0 represents the output power at the off-state and Pjn is the input power. For a normally-off modulator, the insertion loss of the modulator can be expressed as l-exp(-tx(0)-d). Therefore, the non-uniformity of the absorption coefficient results in non-uniformity of the insertion loss. T. H. Wood et al. reported that the insertion loss of a 2 x 2 GaAs/AlGaAs MQW spatial light modulator could be changed from -2.9 dB to -6.2 dB for adjacent pixels [22]. This could be mainly due to the incomplete removal of the absorbing substrate. This will not occur in an InAs/GaAs SPSLS MQW since the substrate is transparent. Figure 6.15 shows the insertion loss of all selected pixels for the three MQW samples. As shown in Fig. 6.15, the different insertion loss among all pixels for each sample are within 0.75 dB. This value is much smaller than in GaAs/AlGaAs MQW [22]. Ideally, all pixels in a two-dimensional spatial light modulator need to be operated at the same wavelength. If the operating wavelength shifts away from the bandwidth regime in which the modulator can m aintain the same on/off ratio, the system perform ance will be degraded. As discussed before, part of the non reproducibility in the measurements is due to the inherent material non-uniformity. We may expect that the wavelength of the exciton peak will shift accordingly, which in turn will affect the system performance. Figure 6.16 shows the excitonic wavelength as a function of applied voltages for all selected pixels of the three samples. As indicated in the figure, the 160 deviations of the exciton wavelength are within 2 nm for TCH1364 and TCH1365 and 3 nm for TCH1366. These numbers are small compared to the ~ 10 nm spectral width of the HH exciton peaks (Fig. 6.9). Based on the non-uniformity of the absorption coefficients, the lim itation in the perform ance of the InAs/GaAs SPSLS MQW modulators can be predicted. It might be helpful if comparison between the binary InAs/GaAs SPSLS MQW structure and the ternary InGaAs/GaAs MQW structures can be made. Although similar studies on ternary MQWs were not well-documented in the literature, the uniformities of InGaAs vertical cavity surface emitting laser (VCSEL) 2-D array were discussed [23,24]. Both VCSEL 2-D arrays contained three 80 A In0. 2Ga0.sAs/100 A GaAs quantum wells. The distribution of the threshold current, which was related to the variation of the insertion loss and the gain coefficient, was about 18% for the 1024-element VCSEL array (the size of the device was 3.2 mm x 3.2 mm) [23] and 25% for the 64-element VCSEL array (the size of the device is 2 mm x 2 mm ) [24]. It implies that the uniformity o f the three-period InGaAs/GaAs MQW is similar to those of our 50-period InAs/GaAs SPSLS MQWs. In conclusion, we have studied the uniformity at various pixels across a ~ 200 pixellated transmission-type InAs/GaAs SPSLS MQW spatial light modulator. The variation of the absorption coefficient change is around ±8% at an applied voltage of 10 volts for a fixed operating wavelength, which causes a ±2% deviation in on/off ratio and ±0.1 dB change in dynamic range. For applications in digital systems, 161 the perform ance is not significantly affected by this deviation. However, for applications in analog systems, the signal-to-noise ratio is significantly affected due to the non-uniformity, which in turn affect the available gray levels. The variation of the absorption coefficient at HH exciton peak without applied voltage is around ±10% for all three samples used in this work. This non-uniformity results in a variation in the insertion loss of less than 1 dB, which is better than those available from GaAs/AlGaAs MQWs. 162 Insertion Loss (dB) Insertion Loss (dB) Insertion L oss (dB) -2.0 TCH1364 Row 2 Row 5 Row 8 -2.5 -3.0 -3.5 -4.0 -4.5 5 15 20 10 column (a) - 2.0 TCH1365 Row 6 Row 7 Row 9 -2.5 -3.0 -3.5 -4.0 -4.5 0 5 15 20 25 10 column (b) -2.0 TCH1366 Row 3 Row 5 Row 7 -2.5 -3.0 -3.5 -4.0 -4.5 -5.0 0 5 10 15 20 column (c) Fig. 6.14. Insertion loss for (a) TCH1364 (b) TCH1365 (c) TCH1366 163 Wavelength o f exciton p eak (nm ) Wavelength o f exciton p eak (nm ) Wavelength o f exciton p eak (nm) TCH1364 o r2 c6 □ r2 c9 © r2 cl2 X r2 cl5 + r2c!8 A rS c6 0 rS c9 5 r5 cl2 ■ r5cl5 □ r5 cl8 • r8c6 V r8c9 ■ r8c!2 ♦ r8 cl5 — Average 1 2 3 Voltage (V) (a) 968 967 966 965 964 963 974 1 2 3 Voltage (V) (b) TCH1366 T C H 1 3 6 5 D o r6 c2 ■ □ 1 6 c6 « rf> clO A B x 1 6 c l 6 + r6 c2 0 A a r7 c9 ■ 9 o r7 cl3 s r7 cl7 * ■ r7 c21 B S ■ 9 • r7 c25 ■ A • r9c8 • • 1 B ^ ^ ^ B v i9 c l 2 B .. 0 ■ i9 c l 6 a i9 c2 0 O x i9 c24 b a Average a . . i . . . . » . . . . i . . . . i . . . . i . . . . » . . o r3 c2 □ r3 c6 © r3 clO X r3 cl4 + r3 cl8 A r5c3 o r5c7 B r5 cl 1 B r5 c l5 • r5 cl9 9 r7 c3 9 r7c7 B r7 e ll A r7cl5 X r7 cl9 ■ " Average 1 2 3 Voltage (V) (c) Fig. 6.15. Excitonic peak wavelength of selected pixels for (a) TCH1364 (b) TCH1365 (c) TCH1366. 164 References [1] S. D. Offsey, W. J. Schaff, L. F. Lester, L. F. Eastman and S. 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