JOURNAL OF COMPUTATIONAL ELECTRONICS Proceedings of the 8th INTERNATIONAL WORKSHOP ON COMPUTATIONAL ELECTRONICS (IWCE-8) Beckman Institute, University of Illinois October 15-18, 2001 L 20030213 107 j DISTRIBUTION STATEMENT A Approved for Public Release Distribution Unlimited Volume 1-2002 2002 KLUWER ACADEMIC PUBLISHERS Boston/U.S.A. Dordrecht/Holland London/U.K. 4. 4 Journalo f ComputationalE lectronics EDITORS D. K. Ferry K. Hess Department of Electrical Engineering Beckman Center Arizona State University University of Illinois Box 875706 405 North Mathews Avenue Tempe, AZ 85287-5706 Urbana, IL 61801 [email protected] [email protected] EDITORIAL BOARD Narayan Aluru Paolo Lugli Beckman Center Dipartimento di Ingegneria Elettronica University of Illinois II Universita di Roma tor Vergata Asen Asenov Peter Markowich Department of Electrical Engineering Insitute for Mathematics University of Glasgow University of Vienna John R. Barker Wolfgang Porod Department of Electrical Engineering Department of Electrical Engineering University of Glasgow Notre Dame University Glasgow, United Kingdom Umberto Ravaioli Robert Eisenberg Beckman Center Bard Professor and Chairman University of Illinois Department of Molecular Biophysics Rush Medical Center Christian Ringhofer Deparment of Mathematics Stephen Goodnick Arizona State University Department of Electrical Engineering Arizona State University Gerhard Wachutka Institute of Physics of Chihiro Hamaguchi Electrotechnology Department of Electrical Engineering Technische Universittits Moinchen Kochi Inst. Tech. Wolfgang Windl Joseph W. Jerome Department of Material Science Department of Mathematics Ohio State University Northwestern University Columbus, Ohio JOURNAL OF COMPUTATIONAL ELECTRONICS Volume 1, Numbers 1/2, July 2002 Proceedings of the 8th INTERNATIONAL WORKSHOP ON COMPUTATIONAL ELECTRONICS (IWCE-8), Beckman Institute, University of Illinois, October 15-18, 2001 Editorial ................................................................................ Um berto Ravaioli 7 Eigenstate Selection in Open Quantum Dot Systems: On the True Nature of Level Broadening ............... ........................................................................... R. Akis, D .K. Ferry and J.P Bird 9 On the Completeness of Quantum Hydrodynamics: Vortex Formation and the Need for Both Vector and Scalar Quantum Potentials in Device Simulation ................................................... John R. Barker 17 On the Current and Density Representation of Many-Body Quantum Transport Theory ....... John R. Barker 23 A Space Dependent Wigner Equation Including Phonon Interaction ......................................... ........................................................ M .Nedjalkov, H. Kosina, R. Kosik and S. Selberherr 27 RTD Relaxation Oscillations, the Time Dependent Wigner Equation and Phase Noise ........................ ............................................................................. H .L. G rubin and R. C. Buggeln 33 Modeling of Shallow Quantum Point Contacts Defined on A1GaAs/GaAs Heterostructures: The Effect of Surface States ............................................................. G. Fiori, G. lannacconea nd M. Ma cucci 39 Study of Noise Properties in Nanoscale Electronic Devices Using Quantum Trajectories ..................... ............................................................... Xavier Oriols, FerranM a rtin and Jordi Sufie 43 Monte-Carlo Simulation of Clocked and Non-Clocked QCA Architectures .................................. .................................................... L. Bonci, M. Gattobigio, G. lannacconea nd M. Macucci 49 A Wigner Function Based Ensemble Monte Carlo Approach for Accurate Incorporation of Quantum Effects in Device Simulation ............................................................... L Shifren and D.K. Ferry 55 The Effective Potential in Device Modeling: The Good, the Bad and the Ugly ............................... ................................................................ D.K. Ferry, S. Ramey, L. Shifren and R. Akis 59 Wigner Paths for Quantum Transport ................................... Paolo Bordone and Carlo Jacoboni 67 Parallelization of the Nanoelectronic Modeling Tool (NEMO I-D) on a Beowulf Cluster .... GerhardK limeck 75 Towards Fully Quantum Mechanical 3D Device Simulations ................................................ ....................................... M. Sabathil, S. Hackenbuchner,J .A. Majewski, G. Zandler and P Vogl 81 Simulation of Field Coupled Computing Architectures Based on Magnetic Dot Arrays ....................... ........................................................................ Gyt rgy Csaba and Wolfgang Porod 87 Numerical Acceleration of Three-Dimensional Quantum Transport Method Using a Seven-Diagonal Pre-Conditioner ................................... David Z-Y Ting, Ming Gu, Xuebin Chi and Jianwen Cao 93 Numerical Investigation of Shot Noise between the Ballistic and the Diffusive Regime ....................... .............................................................. M .M acucci, G. lannaccone and B. Pellegrini 99 On Ohmic Boundary Conditions for Density-Gradient Theory .............................................. ........................................................... M .G. Ancona, D. Yergeau, Z Yu and B.A. Biegel 103 Molecular Devices Simulations Based on Density Functional Tight-Binding .......... A/do Di Carlo, Marieta Gheorghe, Alessandro Bolognesi, Paolo Lugli, Michael Sternberg, Gotthard Seifert and Thomas Frauenheim 109 Role of CarTier Capture in Microscopic Simulation of Multi-Quantum-Well Semiconductor Laser Diodes ................................................. M.S. Hybertsen, B. Witzigmiann, M.A. Alain and R.K. Smith 113 Numerical Study of Minority Canrier Induced Diffusion Capacitance in VCSELs Using Minilase ............. .............................. ................. Yang Liu, Fabiano Oyafuso, Wei-Choon Ng and Karl Hess 119 Quantum Transport Simulation of Carrier Capture and Transport within Tunnel Injection Lasers ............. ....................................... Waneqiang Chen, Xin Zheng, Leonard F Register and Michael Stroscio 123 Modeling of Semiconductor Optical Amplifiers .............................. Andrea Reale and Paolo Lugli 129 Hybrid LSDA/Diffusion Quantum Monte-Carlo Method for Spin Sequences in Vertical Quantum Dots ....... ........................................................ P Matagne, T Wilkens, J. P Leburton and R. Martin 135 Theoretical Investigations of Spin Splittings and Optimization of the Rashba Coefficient in Asymmetric AlSb/lnAs/GaSb Heterostructures ................................ X. Cartoix4,D .Z.-Y Ting and T.C. McGill 141 Modeling Spin-Dependent Transport in InAs/GaSb/AlSb Resonant Tunneling Structures ..................... ....................................... D.Z-Y Ting, X. Cartoixam, T.C. McGill, D.L. Smith and J.N. Schulman 147 Tunneling through Thin Oxides-New Insights from Microscopic Calculations .............................. ............................................................... M .Stiddele, B. Tuttle, B. Fischera nd K. Hess 153 Full Quantum Simulation of Silicon-on-Insulator Single-Electron Devices ................................... ............................... Frederik Ole Heinz, Andreas Schenk, Andreas Scholze and Wolfgang Fichiner 161 A 3-D Atomistic Study of Archetypal Double Gate MOSFET Structures .................................... ...................................................... Andrew R. Brown, Jeremy R. Walling and Asen Asenov 165 3-D Parallel Monte Carlo Simulation of Sub-0.1 Micron MOSFETs on a Cluster Based Supercomputer ....... ........................................................................ Asim Kepkep and Um berto Ravaioli 171 Hole Transport in Orthorhombically Strained Silicon ............................ FM. Bufler and W. Fichtner 175 Empirical Pseudopotential Method for the Band Structure Calculation of Strained-Silicon Germanium Materials ............................................ Salvador Gonzalez, Dragica Vasileska and Alexander A. Demkov 179 A Computational Exploration of Lateral Channel Engineering to Enhance MOSFET Performance ............ .................................. ............................ Jing Guo, Zhibin Ren and M ark Lundstrom 18 5 Monte Carlo Simulations of Hole Dynamics in Si/SiGe Quantum Cascade Structures ........................ .................................................................... Z. Ikoni(, P Harrison and R. W. Kelsall 191 Calculation of Direct Tunneling Current through Ultra-Thin Gate Oxides Using Complex Band Models Fo r Si O .............................. 2 .... ....................... ..................... .................... .... Atsushi Sakai, Akihiro Ishida, Shigeyasu Uno, Yoshinari Kamakura, Masato Morifuji and Kenji Taniguchi 195 Comparison of Quantum Corrections for Monte Carlo Simulation ........................................... ................................................... Brian Winstead, Hideaki Tsuchiya and Umberto Ravaioli 201 Monte Carlo Based Calculation of the Electron Dynamics in a Two-Dimensional GaN/AlGaN Heterostructure in the Presence of Strain Polarization Fields ........................... Tsung-Hsing Yu and Kevin F Brennan 209 Parallel Approaches for Particle-Based Simulation of Charge Transport in Semiconductors ................... ............................................................ M. Saraniti,J . Tang, S. Goodnick and S. Wigger 215 Full-Band Monte Carlo Simulation of Two-Dimensional Electron Gas in SOI MOSFETs .................... ..................................................................... H .Takeda, N. M ori and C. Hamaguchi 219 Band-to-Band Tunneling by Monte Carlo Simulation for Prediction of MOSFET Gate-Induced Drain Leakage Current ............................................ Edwin C. Kan, Venkat Narayanan and Gen Pei 223 A Computational Technique for Electron Energy States Calculation in Nano-Scopic Three-Dimensional InAs/GaAs Semiconductor Quantum Rings Simulation ... Yiming Li, 0. Voskoboynikov, C. Lee and S.M. Sze 227 Fully Numerical Monte Carlo Simulator for Noncubic Symmetry Semiconductors ........................... ........... Louis Tirino, Michael Weber Kevin E Brennan, Enrico Bellotti, Michele Goano and P Paul Ruden 231 Theoretical Study of RF Breakdown in GaN Wurtzite and Zincblende Phase MESFETs ...................... .................................................. M. Weber L. Tirino, K.F Brennan and Maziar Farahmand 235 Quantum Mechanical Model of Electronic Stopping Power for Ions in a Free Electron Gas ................... ... Yang Chen, Di Li, Geng Wang, Li Lin, Stimit Oak, Gaurav Shrivastav, Al E Tasch and Sanjay K. Banerjee 241 An Analytical 1-D Model for Ion Implantation of Any Species into Single-Crystal Silicon Based on Legendre Polynomials ............ G. Shrivastav, D. Li, Y Chen, G. Wang, L. Lin, S. Oak, A.F Tasch and S.K. Banerjee 247 On the Electron Transient Response in a 50 nm MOSFET by Ensemble Monte Carlo Simulation in Presence of the Smoothed Potential Algorithm .................. Gabriele Formicone, Marco Saranitia nd David K Ferry 251 Quantum Corrections in the Monte Carlo Simulations of Scaled PHEMTs with Multiple Delta Doping ....... ................................................................................. . K. Kalna and A. Asenov 257 Thermally Self-Consistent Monte Carlo Device Simulations .......... N.J. Pilgrim, W Batty and R. W Kelsall 263 3D Monte Carlo Modeling of Thin SOI MOSFETs Including the Effective Potential and Random Dopant Dis- tribution ....................................................................... S.M .Ramey and D .K. Ferry 267 Low-Field Mobility and Quantum Effects in Asymmetric Silicon-Based Field-Effect Devices ................ .............................................. L Knezevic, D. Vasileska, X. He, D.K. Schroder and D.K. Ferry 273 Quantum Potential Corrections for Spatially Dependent Effective Masses with Application to Charge Confinement at Heterostructure Interfaces ............................... J.R. Watling, J.R. Barker and S. Roy 279 Comparison of Three Quantum Correction Models for the Charge Density in MOS Inversion Layers ......... ........................................................................... Xinlin Wang and Ting-wei Tang 283 Can the Density Gradient Approach Describe the Source-Drain Tunnelling in Decanano Double-Gate M OSFETs? ........................................................ JR. Watling, A.R. Brown and A. Asenov 289 A Particle Description Model for Quantum Tunneling Effects ....... Hideaki Tsuchiya and Umberto Ravaioli 295 Journalo f ComputationalE lectronics is published quarterly. SUBSCRIPTION RATES The subscription price of Journalo ComputationalE lectronics for 2002, Volume 1 (4 quarterly issues), including postage and handling is: Print OR Electronic Version: EURO 350.00/US $350.00 per year ORDERING INFORMATION/ SAMPLE COPIES Subscription orders and requests for sample copies should be sent to: Kluwer Academic Publishers or Kluwer Academic Publishers 101 Philip Drive P.O. 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Manufacturedi n The Netherlands. Editorial It is a great privilege for me to serve as guest editor of the first issue of the Journal of Computational Electronics, and it seemed only natural to make the kick-off of this new journal coincide with the publication of the Proceedings of the 8th International Workshop on Computational Electronics (IWCE), held at the Beckman Institute of the University of Illinois on October 15-18, 2001. Over the last decade, the IWCE has grown into the main forum where new results and ideas in computational electronics are presented and discussed. A national Workshop on Computational Electronics was first held in 1990 at the Beckman Institute, under the auspices of the National Center for Computational Electronics and the National Science Foundation. The goal of the meeting was to foster interdisciplinary interaction between scientists in electrical engineering, physics, applied mathematics and computer science. The experiment proved to be very successful and in 1992 the first IWCE was held, again at the Beckman Institute, followed by events in Leeds, U.K. (1993); Portland, OR (1994); Tempe, AZ (1995); Note Dame, IN (1996); Osaka, Japan (1998); Glasgow, U.K. (2000), and again Urbana, IL (2001). The next workshop will be for the first time in Italy, in 2003. The format of IWCE creates many opportunities for interaction and discussion among the participants, always with a large representation of graduate students who are particularly encouraged to attend and present papers or posters. Many lasting collaborative interactions have resulted from discussions initiated at an IWCE. The Journal of Computation Electronics fills the need for a publication dedicated to physical simulation of devices and processes, with a focus on interdisciplinary work and large scale supercomputing applications. The community typically attending IWCE best represents the audience addressed by the Journal of Computational Electronics, but the composition of this community has grown over the years to include even more discipline areas. The emphasis of the first workshops was on classical device simulation approaches (drift-diffusion and hydrodynamics models) and particle Monte Carlo methods, while other areas have gained increasing importance at following meetings. These areas include quantum transport and quantum device simulation, opto-electronics, process simulation and, more recently, molecular devices, MEMS and transport in biological ion channels. Rapid technological advances in new directions of research and the widespread availability of high performance computers and clusters, have clearly challenged the computational electronics community to address simulation problems of increasing complexity in the nano-technology area. These efforts require even more contributions from other fields of expertise, from heat transfer and micro-fluidics to computational chemistry and computational biology. I believe that the Journal of Computational Electronics has the potential to become the pre-eminent publication on multidisciplinary aspects of electronics simulation, with the editorship in the capable hands of David Ferry and Karl Hess, some of the most respected scientists in computational electronics (and incidentally the two people who have been most influential on my own professional career). The membership of the editorial board includes international leaders, covering an impressive range of expertise in all relevant areas. While working on the preparation of the IWCE-8 proceedings issue, I was also very impressed by the high quality of the contributions and I am confident that the Journal of Computational Electronics is off to a good start. If the quality of future submission to the regular issues will continue to be on this level, the success of this new journal is assured. The quality of the papers submitted for publication on the IWCE proceedings also reflects the commitment by funding agencies and institutions that have continued to support the workshop over the years. IWCE-8 would not have been possible without the direct support of the National Science Foundation, the Beckman Institute of the University of Illinois, the US Office of Naval Research, the Distributed Center for Advanced Electronics Simulation (DesCArtES), and the technical sponsorship of the IEEE. Umberto Ravaioli University of Illinois at Urbana-Champaign Journal of Computational Electronics 1: 9-15, 2002 © 2002 Kluwer Academic Publishers. Manufactured in The Netherlands. Eigenstate Selection in Open Quantum Dot Systems: On the True Nature of Level Broadening* R. AKIS,t D.K. FERRY AND J.P. BIRD Centerf or Solid State Electronics Research and Departmento f ElectricalE ngineering, Arizona State University, Tempe, AZ 85287-5706, USA [email protected] Abstract. We show that transport in open quantum dots can be mediated by single eigenstates, even when the leads allow several propagating modes. The broadening of these states, generally localized in the interior, can be virtually independent of lead width. As such, the Thouless argument, invoked to suggest that all states should be unresolvable under these conditions, can in fact fail. Thus, any transport theory based on such assumptions (in particular, random matrix theory) must be called into question, as the fluctuations produced by these states can in fact dominate the conductance. These trapped states also produce interesting and potentially useful effects in coupled dot systems as well. Keywords: device modeling, quantum transport, quantum dots 1. Introduction used to imply that their discrete DOS is unimportant for an understanding of transport. A key feature of the First applied to disordered conductors, the Thouless Thouless argument is an assumption of uniform level argument relates the conductance of a system to the broadening, independent of the specific details of the diffusion-induced broadening of its energy levels. Ac- energy states. While this seems reasonable for diffusive cordingly, a metal may be viewed as a system with conductors, in open quantum dots we demonstrate here strongly-broadened energy levels, while an insulator is that the level broadening is highly non-uniform and that one whose density of states (DOS) consists of isolated single eigenstates may remain resolved, thus demon- peaks (Thouless 1977). While the Thouless argument strating that the Thouless argument does not generally provides an understanding of the origins of localization hold. These results have important implications for the- in disordered conductors, it has recently become possi- oretical analyses of such structures. ble to study electron transport in ballistic quantum dots This paper is organized as follows. In Section 2, the (Jalabert, Baranger and Stone 1990, Baranger, Jalabert Thouless argument is summarized. In Section 3, our and Stone 1993a, b, Lin and Jensen 1996, Wirtz, Tang method of calculation is briefly described. A discus- and Burgdbrfer 1997, Marcus et al. 1992, Chang et al. sion on conductance resonances in open dots is found 1994, Bird et al. 1996, 1999, Sachrajda et al. 1998). in Section 4. In Section 5, we discuss decomposing the These open structures consist of a central scattering open dot wave functions in terms of closed dot eigen- cavity that is coupled to external reservoirs by means of states. In Section 6, the focus is shifted to coupled dot quantum point contacts (QPCs). Since the conductance systems. Conclusions are drawn in Section 7. of these structures (measured in units of the dimension- less conductance e2/1h) is typically larger than unity, it is often thought that the Thouless argument may be 2. The Thouless Argument *Work supported by the Office of Naval Research. The Thouless argument follows by noting that the tTo whom correspondence should be addressed, energy levels in a conductor of length L should be 10 Akis uncertain by an amount F -_ hD/L2, where D is the 15.5 . '' diffusion constant and L2/D is the time required to dif- -'i.T A fuse across the sample. Since the average level spac- ing in the conductor may be written as A = I1/NL0, where NE is the DOS and d is the dimensionality, the ratio F/A may be written as (Lee, Stone and Fukuyama (b) 1987): 140 F hD d 155. (1) A -2- NEL. Using the Einstein relation (a e2N-D) to relate > - the DOS to the conductivity (a), Eq. (1) may be sim- p lifi e d to y ie ld : r h. F =-aLd-2 = 9, (2) -0.25 B (T) 0.25 -0.25 B (T) 0.25 Figure I. In (a). a portion of the spectrum is plotted as a function where g is the dimensionless conductance, with units of of E and B for a 0.3 ptnm square dot. The conductance. G, is plotted e2/J. Equation (2) is the crux of the Thouless argument vs. energy, E, and magnetic field. B. for open quantum dots with leads allowing (b) one mode. (c) four modes and (d) nine modes. and suggests that, in a metallic conductor (g > 1), the The lighter regions of shading correspond to higher values of the level broadening, F, is always comparable to, or greater conductance G. The dot schematics are shown in the insets. The than, the average level spacing. In a dot whose point labels a and b correspond to the positions of Fano resonances. contacts each support N propagating modes then, by assuming Ohmic addition of the two point contacts, the conductance may be written as g = N. Since N > 1 4. Conductance Resonances in Open Dots is required for the dot to be open, it is therefore often argued that Eq. (2) proves that the energy levels of open We begin by showing the correspondence between the dots can never be resolved (F >_ A for N > 1). energy spectrum of a closedsquare dot with the conduc- tance features exhibited by the open system. Figure 1( a) shows a portion of the energy spectrum as a function of 3. Method of Calculation magnetic field for a 0.3 pm quantum dot. Figure 1(b) shows what happens when the dot is now opened and Our simulations are performed on a discrete lattice connected to external waveguides by QPCs that are using a numerically stabilized variant of the transfer at the top edge of the dot as shown in the inset. In matrix approach (Usuki et al. 1995). The dot is this case the width of the QPCs have been adjusted enclosed inside a waveguide which extends a finite so that a single mode propagates. What is plotted is number of lattice sites in the transverse (y) direction. G(E, B) with lighter shading corresponding to higher The structure is broken down into a series of slices conductance. For the entire energy range shown in this along the longitudinal (x) direction. Imposing an picture, G < 2e2/h. The picture shows resonant be- electron flux from the left, one translates across havior, as indicated by the striations that are super- successive slices and, on reaching the end, one imposed on the conductance. Comparing this picture obtains the transmission coefficients which enter the with the spectrum shown in Fig. l(a), G(E, B) clearly Landauer-Baittiker formula to give the conductance, shows the influence of the closed dot DOS, as the ba- In cases where we examine closed dots, to obtain the sic pattern is reproduced. However, certain resonance spectrum and the eigenstates, we solve a finite dif- lines appear to be shifted in comparison to their spectra ference Schrodinger equation with Dirichet boundary counterparts and there are certain features in the con- conditions. This sparse matrix eigenvalue prob- ductance that apparentlyd o not have a spectral analog. lem is done numerically by using ARPACK routines In particular, there are linear resonance features that (www.caam.rice.edu/software/ARPACK/index.html), actually cross at B = 0 T. In contrast, the spectrum which use Lanczos/Arnoldi factorization. shows lines that appear linear for much of the range Eigenstate Selection in Open Quantum Dot Systems 11 shown, but bend over in the region near B = 0 T. Thus, 1 (a) rather than crossing, they appear to form a type of anti- 0.9 4)l crossing. This line shifting and line creation illustrates another effect that the QPCs have-they act as a pertur- ,0.8 r= 0.00469 eV bation that results in the creation of new eigenstates not CY 0.7 present in the perfectly square system. In Fig. 1 (c), the 0.6 QPCs have now been adjusted to permit four modes to 0 propagate. The conductance here ranges from -2e 2/ h 0.5 to -8e 2/h. Despite the fact that the dot is far more 0.4 "open" than in the previous case there is still resonant 0.3 behavior. However, the picture is somewhat simplified 14.64 14.65 14.66 14.67 14.68 14.69 14.7 compared to Fig. l(b). What remains are a series of E( meV) parabolic curves as well as sets of almost parallel res- 7.6 onance lines, tilted to the left and the right, forming a very regular cross hatched pattern. These patterns yield --------. characteristic fingerprints in the conductance fluctua- 7.5 (b) tions that have in fact been observed experimentally 7.45 (Bird, Akis and Ferry 1999). In Fig. l(d), the QPCs 0 04 7.4 r= 0.00283 rneV support nine modes. Here the parallel lines have van- q -0.0371 ished, leaving only the parabolic striations. Clearly 7.35q the broadening introduced by the QPCs is highly 7.3 nonuniform. 7.25 In Fig. 2(a) and (b), respectively, we plot G(E) vs. 14.38 14.4 14.42 14.44 14.46 14.48 14.5 E, focusing on the conductance resonances labeled "a" E( meV) and "b" in the previous figure. The asymmetric line- Figure 2. Conductance, G vs. energy, E, for the single mode dot shape of these features is characteristic of Fano res- (a) and the nine mode dot (b). Fits to the Fano resonance formula are onances, which occur in systems where quasi-bound also plotted as dotted lines. states are coupled to a continuum (see Gtres et al. (2000) and references therein). These may be repre- sented by the functional form (Gores et al. 2000): energy because the "effective" dot size is larger. This point is explored in further detail in the next section. G = Gh + Go82( ,++ q1 )2 (3) 5. Eigenstate Decomposition where E = (E - ER)! F, ER is the energy on reso- The relationship between the open dot resonances and nance, q is an asymmetry parameter that depends on the eigenstates of the corresponding closed system can the background phase shift, Gb the background con- be quantified by doing a spectral decomposition. Since ductance that the resonance sits upon, and Go deter- closed-dot eigenstates form an orthogonal basis set, mines the magnitude. The dashed lines are the fits. the wave functions of the open dot can be expressed Significantly, the resonance in (b) is sharper than in (a), even though the QPCs are much wider. The in- asjailinearecombinaiono sets show the resonant wave functions which are both projection in the dot region: of the "bouncing ball" variety. That is, the standing C 0 where G - cioo sI (4) waves trapped between the upper and lower bound- n...' r -= '°n/ (4) aries appear to be aligned with the orbital trajectory that a classical billiard would take if it were bouncing Figure 3(a) shows G(E) vs. E for a nearly square dot between them. This behavior is reminiscent of the scar- (the dimensions are 0.3 gm by b = 0.307 Itm, the non- ring of the wave function by classical orbits observed commensurate shape was chosen to insure that the lev- in chaos theory (Heller 1984). The two resonant states els of the rectangular dot were not degenerate). The shown here can be thought of as being largely equiv- QPCs allow 2 modes in the energy range displayed. At alent. The resonance in (b) however occurs at a lower the top are markers that indicate the positions of the