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Design, Fabrication and Characterization of a Suspended Heterostructure by Vincent Louis Philippe Leduc A thesis submitted to the Department of Physics, Engineering Physics & Astronomy in conformity with the requirements for the degree of Master of Science Queen’s University Kingston, Ontario, Canada September 2007 c Vincent Louis Philippe Leduc, 2007 Copyright Abstract This thesis presents the design and theoretical modeling of an aluminum gallium arsenide/gallium arsenide heterostructure from which suspended nanoscale mechanical resonators with embedded two-dimensional electron gas (2DEG) can be made. The mechanical characteristics of the resonator and the piezoelectric actuation scheme are investigated using finite-element modeling. For a 836 nm-long, 250 nm-wide and 164 nm-thick beam with gold electrodes on top, out-of-plane flexural vibrations are verified to be piezoelectrically excited at the beam’s fundamental frequency of 925.6 MHz. Fabrication recipes for the making of ohmic contacts to the 2DEG, Hall bars and suspended structures are developed using the designed crystal structure. Electrical properties of the 2DEG are evaluated in both large, unsuspended structures as well as in sub-micron size suspended structures. It is found that the 2DEG has a reasonable electron density of 7.04 × 1011 cm−2 and electron mobility of 1.72 × 105 cm2/V·s. i Acknowledgments To begin with, I would like to thank my thesis advisor Rob Knobel for his patience, generosity and invaluable comments. I also wish to thank Guy Austing and Zbig Wasilewski for their assistance in crystal growth and design. Special thanks to Olubusola Koyi for showing me so much of what I needed to know when I started. Thanks also for the many amusing discussions. Special thanks to Greg Dubejsky as well for all the help over the course of my master’s. My thanks also go out to all the other students who worked in our labs in the past two years : Jennifer Campbell, Mark Patterson, Allan Munro, Kyle Kemp and Ben Lucht. Without all your work this project would not have been possible. Thanks to all my office mates for the discussions and hockey pools : Greg, Busola, Jennifer, Steve, Aaron, Ben and Tom. The departmental staff also receives my acknowledgments. I want to especially thank Loanne Meldrum and Tammie Kerr for their help and for making administrative matters clear and easy enough for me to comprehend. Thanks to Kim MacKinder, ii for the many times she helped me find the parts I needed and for all the help with the cryogenics. Thanks to Gary Contant and Chuck Hearns for helping someone who’s never been very good with his hands in the machine shop. Thanks to Donna, John and especially Jennifer, who did all the driving, for the many nice hiking trips. Hope we get to do some even better ones. Thanks to Lenko for all the squash and badminton games. Finally I would like to send thanks to my family. Thanks to my uncle Pierre and my aunt Marthe for their warm welcome when they allowed me to stay at their home in Montréal for the occasional conference or summer school. Merci à mon père Robert, à ma mère Hélène et à ma soeur Évelyne et ma grandmère Lucille pour leur support tout au long de mes années d’université. Sans vous je ne sais pas comment j’aurais fait. Je vous aime tous. iii Contents Abstract i Acknowledgments ii Contents iv List of Abbreviations and Symbols ix List of Tables xvii List of Figures xix Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction to NEMS and Nanomechanics . . . . . . . . . . . . . . . 1 1.1.1 Characteristics of NEMS . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Applications of NEMS . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Scope of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1 iv 1.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Nanomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Vibrating Mode Shapes and Frequencies . . . . . . . . . . . . 11 2.1.2 Transient Behavior of a Vibrating Beam . . . . . . . . . . . . 19 The Quest for Displacement Detection Limit . . . . . . . . . . . . . . 23 2.2.1 The Quantum Harmonic Oscillator . . . . . . . . . . . . . . . 25 2.2.2 Limiting Factors for Displacement Sensing . . . . . . . . . . . 29 2.3 GaAs Usage in Mechanical Devices . . . . . . . . . . . . . . . . . . . 32 2.4 Piezoelectric Actuation in GaAs . . . . . . . . . . . . . . . . . . . . . 41 2.5 The Quantum Hall Effect in Two-dimensional Electron Gases . . . . 52 2.5.1 Inversion Layers and Modulation Doping . . . . . . . . . . . . 52 2.5.2 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . 55 2.6 Suspended Two-dimensional Electron Gases . . . . . . . . . . . . . . 60 2.7 The Piezoelectric, SET-based, Displacement Detector . . . . . . . . . 62 2.2 Chapter 3 Design and Simulation of Heterostructure . . . . . . . . 70 3.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2.1 73 Simulations of Electronic Properties . . . . . . . . . . . . . . . v 3.2.2 Mechanics simulations . . . . . . . . . . . . . . . . . . . . . . Chapter 4 Fabrication 77 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.1 Wafer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Ohmic Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Design of Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.1 Spin-coating of Resist Layers . . . . . . . . . . . . . . . . . . 87 4.4.2 Patterning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.3 Pattern Developing . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4.4 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.5 Lift-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.6 Pattern Alignment . . . . . . . . . . . . . . . . . . . . . . . . 94 Wet Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.5.1 Etching of the Wafer . . . . . . . . . . . . . . . . . . . . . . . 99 4.5.2 Mask Removal 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.6 Reactive Ion Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.7 Observation of Patterns and Structures . . . . . . . . . . . . . . . . . 107 Chapter 5 Experiments 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Testing of Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 vi 5.2 Testing of 2DEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2.1 Procedures for Quantum Hall Effect Measurements . . . . . . 111 5.2.2 Procedures for Illumination of 2DEG . . . . . . . . . . . . . . 114 5.2.3 Results for a Large Unsuspended Hall Bar . . . . . . . . . . . 115 5.2.4 Results for a Small Suspended Hall Bar and Beams . . . . . . 122 Chapter 6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.1 Heterostructure redesign . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.2 Improve Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.3 Continue Characterization . . . . . . . . . . . . . . . . . . . . . . . . 140 6.4 Experimental Test of Actuation . . . . . . . . . . . . . . . . . . . . . 140 6.5 Integration with Sensitive Amplifiers . . . . . . . . . . . . . . . . . . 140 Chapter 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Appendix A Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A.1 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A.1.1 He-3 refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . 157 A.1.2 Cooling Procedures . . . . . . . . . . . . . . . . . . . . . . . . 159 A.2 Wiring and Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 A.3 Mounting Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 vii A.4 Wire Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.5 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.5.1 Lock-in Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . 164 A.5.2 Pre-Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.5.3 Programmable Current Source . . . . . . . . . . . . . . . . . . 165 A.5.4 Current Source . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A.5.5 Multimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A.5.6 Temperature Controller . . . . . . . . . . . . . . . . . . . . . 166 A.5.7 Superconducting Magnet and Power Supply . . . . . . . . . . 166 Appendix B List of Programs . . . . . . . . . . . . . . . . . . . . . . . 167 Appendix C 1DPoisson Input Files . . . . . . . . . . . . . . . . . . . . 172 Appendix D Health and Safety Issues . . . . . . . . . . . . . . . . . . 175 viii List of Abbreviations and Symbols ∆xSQL Standard Quantum Limit ∆xzp Zero Point motion of an oscillator ǫij Component of strain ~ Planck’s constant divided by 2π µ Carrier mobility; Damping coefficient ν Poisson’s ratio ωn Angular frequency of mode n ω Angular frequency Â1 Real amplitude operator Â2 Complex amplitude operator Ĥ Hamiltonian operator ix N̂ Number operator P̂ Momentum operator X̂ Position operator φ Electrostatic potential ρ Mass density; Charge density ρSD Source-drain sheet resistance σij Component of stress θn Slope of deformation of a beam in mode n ǫ Strain σ Stress ξ Permittivity matrix c Elastic stiffness matrix d Piezoelectric coefficients matrix E Electric field vector Q Electric charge density displacement vector s Elastic compliance matrix x ξs Static dielectric constant ξ∞ High-frequency dielectric constant B Magnetic field strength cij Elastic stiffness component d Depletion length dij Piezoelectric coefficient En Bending energy of mode n E Young’s modulus; Energy e Electronic charge EF Fermi energy level G Bulk modulus g Spin degeneracy factor h Thickness; Planck’s constant I Current Iz Bending moment kB Boltzmann’s constant xi l Length lφ Temperature-dependent phase breaking length My Torque na Carrier area density Q Quality factor q Electronic charge RH Hall resistance RL Longitudinal resistance sij Elastic compliance component T Temperature TQL Minimum noise temperature of an amplifier un (x, t) Time-dependent displacement of a doubly-clamped beam in mode n U (x, t) Time-dependent displacement of a doubly-clamped beam VG Gate voltage VH Hall voltage VL Longitudinal voltage xii w Width wm Mechanical width wef f Effective width 2DEG Two-dimensional electron gas AC Alternating Current AFM Atomic Force Microscope Alx Ga1−x As Aluminum Gallium Arsenide in a mole fraction of x Al and (1−x) GaAs AlGaAs Aluminum Gallium Arsenide CAD Computer Assisted Design DC Direct Current EBL Electron Beam Lithography FEM Finite-Element Modeling FET Field Effect Transistor FFT Fast Fourier Transform GaAs Gallium Arsenide GPIB General Purpose Interface Bus xiii H2 Dihydrogen gas H2 O2 Hydrogen peroxide HF HydroFluoric acid I-V Current-Voltage I/O Input/Output IVC Inner Vacuum Can LaB6 Lanthanum hexaboride LED Light-Emitting Diode LT-GaAs Low-temperature grown GaAs MBE Molecular Beam Epitaxy MEMS Micro Electro-Mechanical Systems MF319 A PMGI solvent MIBK Methyl IsoButyl Ketone MMCX Type of RF coaxial connector MOCVD Metal Organic Chemical Vapour Deposition MOSFET Metal Oxide Semiconductor Field Effect Transistor xiv MSDS Material Safety Data Sheet N2 Dinitrogen gas Nano Remover PG A PMGI solvent NEMS Nano Electro-Mechanical Systems NPGS Nano Pattern Generation System PCD Probe Current Detector PG 101 A PMGI developer PID Proportional/Integral/Derivative (controller) PIN p-type/intrinsic/n-type (diode) PMGI Polymethylglutarimide PMMA Polymethylmethacrylate QND Quantum Non-Demolition QPC Quantum Point Contact RF Radio Frequency RIE Reactive Ion Etching RPM Rotation Per Minute xv RTA Rapid Thermal Annealer SdH Shubnikov-de Haas (oscillations) SEM Scanning Electron Microscopy; Scanning Electron Microscope SET Single Electron Transistor TTL Transistor-Transistor Logic (signal) XP 101 A PMGI developer xvi List of Tables 2.1 Material properties of GaAs and Alx Ga1−x As [35, 36] . . . . . . . . . 2.2 The six components of strain as defined for an infinitesimal cubic el- 34 ement (see figure 2.7). The first line gives the elongation, while the second line gives shearing strain [37, 12]. . . . . . . . . . . . . . . . . 4.1 37 Electron beam lithography settings. All patterning is done at an accelerating voltage of 40 kV. Magnification refers to the magnification factor. The feature size is usually the desired width of the smallest feature (e.g. the width of a beam). The center-to-center distance represents the spacing between the two centers of exposure points. The line spacing is the spacing between two lines of exposure. Offset is the pattern origin offset needed for good alignment between patterns at high magnification and patterns at low magnification. . . . . . . . . . 5.1 89 Results of 2DEG characterization and physical dimensions for sample A117 xvii 5.2 Observable plateaus in the Hall resistance of sample A, where i = h/e2 RH and RH = VH /I. I was taken to be constant at 0.500 V/10.093 MΩ = 4.9539 × 10−8 A. Fitted by averaging VH over plateaus. . . . . . . . . 119 xviii List of Figures 1.1 Quality factor of mechanical resonators varying in volume from macroscale to nanoscale. The maximum attainable Q seems to decrease linearly with the logarithm of the volume of the devices [2]. 2.1 . . . . . Example of a doubly-clamped beam made in GaAs/AlGaAs. The scale bar shows one micron. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 3 11 A beam of length l, width w and thickness h. A cross-sectional element of length dx and area A = wh is shown, while the displacement U (x, t) in the z direction is a function of x and time t only [13]. . . . . . . . 2.3 12 Definition of angle θ. R(x) gives the radius of curvature at point x. x′ represents the displaced neutral axis of a bent cantilever while x gives the original neutral axis. The angle θ is formed by the tangent at point x on x′ and x [13]. 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Mode shapes for the first four modes of flexural out-of-plane vibration for a doubly-clamped beam [13]. xix . . . . . . . . . . . . . . . . . . . . 17 2.5 Resonant frequency plotted against beam length for Euler-Bernoulli theory (plain curve) and experimental measurements (points) on piezoelectric Al0.3 Ga0.7 As doubly-clamped resonators. Notice that while Euler-Bernoulli theory seems to describe well the beam length dependence of resonant frequency it fails to predict the exact frequencies when the beam material is not isotropic nor homogeneous. In this case, the beams are made of three Al0.3 Ga0.7 As layers, two of which are heavily Si-doped. There is of course the expected deviation caused by irregularities in the beam shape due to the fabrication procedure [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 18 Expected amplitude of motion near the resonance of the fundamental flexural mode for a doubly-clamped beam in GaAs (ρ = 5.3 g/cm3 , E = 101 GPa). Here l = 3 µm, w = 0.8 µm, h = 0.2 µm with a Q of 2000. A force (F0 /l) cos ωt of magnitude F0 = 1 nN is distributed over the surface of the beam and its frequency ω varied within ±2% of the resonant frequency. The resonant frequency is ≈ 99.7 MHz. The amplitude of motion of the resonator’s mid-point is displayed on the vertical axis with a maximum of ≈ 2.64 nm at resonance. 2.7 . . . . . . 24 A cubic volume element undergoing deformation given by the displacement vector u of the origin O [12]. . . . . . . . . . . . . . . . . . . . xx 36 2.8 A GaAs tuning fork with electrode configuration for in-plane flexural vibrations [41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 42 Left : A typical GaAs wafer with primary flat on the bottom and secondary flat at a right angle on the left-hand side of the wafer. Crystal directions are indicated. Right : a possible configuration for a piezoelectric doubly-clamped beam designed for actuation (or sensing) of the of out-of-plane flexural motion. [31, 40]. . . . . . . . . . . . . . . 45 2.10 The optimal electrode placement on a beam oriented in one of x1 = h011i for out-of-plane flexural motion. Electrode polarities are indicated [40]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.11 Crystal design for the actuation mechanism described in [43]. The top drawing shows how an electric field applied in the piezoelectric layer (resistive intrinsic GaAs) causes longitudinal strain. However in this case the piezoelectric layer is centered on the neutral axis and so only longitudinal oscillations can be excited. The two remaining drawings show how for the same direction of electrical field, the cantilever can be made to go up or down simply by moving the piezoelectric layer below or above the neutral axis. xxi . . . . . . . . . . . . . . . . . . . . 48 2.12 The efficiency of the actuation method in [43] is demonstrated with applied AC signals of amplitude as low as 5 µV to a cantilever with ω0 ≈ 8 MHz; Q = 2700; l = 4µm; w = 0.8 µm; t = 0.2 µm. The position of the cantilever is monitored by optical interferometry. . . 49 2.13 A. Change in amplitude with respect to the DC bias voltage applied to the ground electrode. B. Drawing showing the change in the depletion region width when a DC bias voltage is applied. C. Three different PIN diode designs can provide increasing, constant or decreasing amplitude with applied DC bias [43]. . . . . . . . . . . . . . . . . . . . . . . . . 50 2.14 A. Scanning electron microscopy image of the doubly clamped beam. B. Change in frequency and amplitude caused by different applied biases. C. Opposite crystal orientations give opposite behaviors for the change in frequency with bias voltage. The inset shows steps in frequency caused by the addition of 10 mV bias voltage [43]. . . . . . . 51 2.15 Schematic view of a metal-oxide-semiconductor field effect transistor (MOSFET). An inversion layer is formed at the interface of the semiconductor, p-type silicon, and the insulator, silicon dioxide. The electric field is provided by a positive voltage applied on the aluminum gate deposited on the surface. Heavily doped regions near the source and drain provide the carriers [44]. . . . . . . . . . . . . . . . . . . . xxii 53 2.16 Electron energy levels diagram for a MOSFET. The electric field applied on the aluminum gate causes the bands to bend near the insulator layer. The conduction band falls below the Fermi level in this region. The electrons start by filling the hole states at the bottom of the valence band, however, when all these states are filled up to the Fermi level, the remaining carriers populate the conduction band. Thus, a conducting two-dimensional gas is obtained [44]. . . . . . . . . . . . 54 2.17 Electron energy levels diagram for a AlGaAs/GaAs heterojunction. Since pure GaAs remains slightly p-type, the electrons falling from the n-doped AlGaAs occupy first the hole states at the bottom of the valence band but eventually fill the potential well at the interface. There, a two-dimensional electron gas is formed [44]. . . . . . . . . . 2.18 A typical Hall bar [44]. . . . . . . . . . . . . . . . . . . . . . . . . . 55 57 2.19 Magnetoresistance measurements performed on a suspended 2DEG showing negative magnetoresistance at low magnetic field strengths. The inset show spin-splitting at higher magnetic fields [49]. xxiii . . . . . 63 2.20 (a) Nanomechanical resonator in GaAs in [110] orientation. (b) Circuit used for magnetomotive technique. (c) Mechanical response of the beam around the 115.4 MHz resonance peak (in-plane vibrations were used). The different curves indicate the response for magnetic fields ranging from 1 T to 12 T [49]. . . . . . . . . . . . . . . . . . . . . . . 64 2.21 (a) Micrograph of the GaAs/AlGaAs suspended structure showing circuit used for magnetomotive technique. Suspended quantum dot structures are coupled to the beam in order to investigate how they interact. The quantum dots are created by using the edge depletion effect : indentations are made at a 65◦ angle in a rectangular beam. (b) Mechanical response of the beam for different driving powers. Note that considerable nonlinearity appears as the power is increased. The inset show the response for varying magnetic fields [50]. . . . . . . . . . . . 65 2.22 (a) A micrograph of the piezoelectric QPC displacement detector. (1) The wire providing the out-of-plane force. (2) and (5) are the source and drain for the ohmic contacts to the 2DEG. (3) and (4) are the two QPC defined on the beam, but only one was used at a time [58]. xxiv . . 66 2.23 (a) Schematic of the experiment. A magnetic field actuates the beam up and down using the current provided by the local oscillator (LO). A lock-in amplifier is used to monitor the current through the QPC. (b) The current response of the QPC near the resonant peak [58]. . . 67 2.24 (a) Proposed heterostructure design. The sacrificial layer may be selectively etched by an HF dip. A 2DEG is formed at the interface of the GaAs and the Al0.3 Ga0.7 As using the well-known modulation doping technique. (b) Sketch of the NEMS device. The upper half of the diagram show the SET, whose island is connected to a detection electrode while the lower half shows the actuation electrode accompanied by two ohmic contacts to the 2DEG [10]. 3.1 . . . . . . . . . . . . . . . . . . . 68 Diagram of the designed 2DEG heterostructure for applications to suspended structures. The doping used in the donor layers was 1.5 × 1019 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv 72 3.2 Plot of the conduction band plotted against depth in the heterostructure as simulated in 1DPoisson [61] in the nonsuspended case. The Fermi level is set at zero. Near the surface, the conduction band boundary condition was set to a 0.6 eV Schottky barrier to account for the effect of surface states according to the numbers found in literature for GaAs [61, 64, 65]. A second well is seen just above the sacrificial layer, however, this must be considered as non-conducting since a low-temperature GaAs was grown there. An electron density of 5.238 × 1011 cm−2 is obtained in the 2DEG layer. 3.3 . . . . . . . . . 78 Plot of the conduction band plotted against depth in the heterostructure as simulated in 1DPoisson [61] for the suspended case. The Fermi level is set at zero. The conduction band boundary conditions is set to 0.6 eV Schottky barriers at the surface, to account for the exposed GaAs cap layer [61, 64, 65]. The bottom layer, however, is low-temperature GaAs and has was set to 0.47 eV as per the numbers found in reference [66]. The 2DEG layer has an electron density of 4.750 × 1011 cm−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi 79 3.4 Mosaic depicting the shape of a beam in time with the beam starting from rest and moving under the voltage applied to the actuation electrode on the right end. The time interval between each snapshot is ≈ 0.2 ns and the color bar gives the potential (blue is −3 V, red is +3 V). a) Positive voltage is applied to the beam at rest. The beam start to bend upwards. b) Beam has bent upwards while potential is reducing. c) Beam center reaches its apex; the voltage is near zero, becoming negative. d) Beam returns to its equilibrium position under the influence of a negative voltage. e) Voltage starts back once again towards zero; the beam moves down. f ) After the beam center reaches its minimum, the beam start moving up again under positive voltage. 4.1 81 SEM picture of an array of four beam patterns in close proximity. The beam ends show a curvature that was not defined in the original pattern design and is caused by the proximity effect. . . . . . . . . . xxvii 86 4.2 Diagram of a complete fabrication process. a) PMGI covered by PMMA are spin-coated on the wafer. b) Patterned is exposed in a SEM. c) Pattern is developed in a solution of MIBK:isopropanol in a 1:3 ratio. This develops the top layer of PMMA. d) The exposed PMGI on the bottom layer is removed under the opening made in the PMMA by use of a PMGI developer (XP101). Further ‘undercut’ is obtained by a dip in MF319. e) Metal film is evaporated (physical vapor deposition using an electron beam evaporator). f ) The remaining resist is lifted off by a proprietary solvent (Nano Remover PG). Two choices are available for the rest of the process. I. If RIE is used to create the mesa : 1) Evaporated Ni/Ge/Au ohmic contacts are first annealed at 415◦ C for 15 seconds. 2) A 60 nm thick Ni mask is applied and the mesa is created by RIE in BCl3 gas. 3) Finally the Ni mask is removed and the sacrificial layer removed by a solution of HF. II. If liquid etching is used to prepare the mesa : 2) The mesa is protected by a metal mask (Ti or Ni) and etched in a citric acid/hydrogen peroxide mixture. 1) The contacts are evaporated and annealed. 3) Mask is removed and removal of the sacrificial layer is made by dipping in HF. 95 xxviii 4.3 Schematic view of the fine alignment process. In the drawing, coarse alignment is already done and four ohmic contacts with outgoing electric leads are visible in dark gray under the four alignment windows. The black represents the area of the field of view that is not scanned by the SEM and includes portions of pattern A which are not desirable to be exposed. Finally, the “L”-shaped polygons are positioned over their corresponding squares of pattern A with the computer mouse in NPGS [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 SEM picture showing the holes etched unintentionally in the mesa of a large Hall bar structure that was protected by a titanium mask. 4.5 98 . 101 SEM picture showing the poor shape resulting from the definition of a mesa with the citric acid / hydrogen peroxide etchant for a ≈ 500 nm beam. The titanium mask was of the shape of a rectangular beam and was removed in HF. 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 SEM picture showing the undercut made by etching the Al0.7 Ga0.3 As sacrificial layer for 1 minute in a 5% HF solution. The heterostructure being somewhat transparent at 20 kV accelerating voltage, we are able to see the shape of the sacrificial layer underneath. The sacrificial layer is slightly over-etched, as indicated by the two ends of the beam being just above and below the remaining Al0.7 Ga0.3 As support. xxix . . . . . . 104 4.7 SEM picture showing mesa as defined by RIE in a BCl3 gas using a 60 nm thick nickel mask. The beams of this pattern are, in ascending order, ≈ 300 nm, ≈ 400 nm, ≈ 500 nm, ≈ 650 nm and ≈ 1.1 µm wide. 106 5.1 Circuit used for testing contacts to 2DEG. . . . . . . . . . . . . . . . 109 5.2 A typical I-V trace obtained for Ni/Ge/Au ohmic contacts with the microscope light turned off. The contacts were square-shaped, ≈ 200 µm of side and ≈ 400 µm apart, center-to-center. As can be seen, the curve is quite linear, indicating that the contacts are ohmic for this current range. A linear fit gives an intercept of −1.95 × 10−9 A and an overall resistance of ≈ 671 Ω (the inverse of the slope). Error bars are shown but difficult to see on this scale. 5.3 . . . . . . . . . . . . . . . . . . . . 110 Circuit used for magnetoresistance measurements on sample A. SR830 and SR850 refer to Stanford Research Systems lock-in amplifiers [81]. 5.4 112 Circuit used for magnetoresistance measurements on sample B. SR830 and SR850 refer to Stanford Research Systems lock-in amplifiers [81]. On the other hand, SR 5113 refers to the 5113 model pre-amplifier from Signal Recovery [84]. 5206 refers to the lock-in amplifier model by EG&G, which has since been bought by Signal Recovery. . . . . . 113 5.5 Circuit used for illumination of samples. A simple standard red LED was used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 xxx 5.6 The first pattern used in characterizing the 2DEG. The scale bar at the bottom indicates a millimeter. The source and drain pads are left and right while the remaining pads connected to thin transverse leads are used to measurement the longitudinal and transverse voltages. 5.7 . 116 Sample A : the Hall bar (not suspended) used for taking magnetoresistance measurements. The dimensions of the bar are shown : the spacing between two longitudinal leads was 391.1 µm, the width of the bar was 105.1 µm and the total length of the bar was 1373.6 µm. 5.8 . . 117 eVH plotted against IB for the illuminated sample for 0.017 ≤ B ≤ 0.2 T. The inverse of the slope of the linear fit gives an electron sheet density of (7.04 ± 0.01) × 1011 cm−2 (see subsection 2.5.2 for the theory concerning this). The intercept is non-zero because of the uncertainty 5.9 in the readings of our instruments for very small magnetic fields. . . 118 Magnetoresistance measurements after illumination of sample A. . . 120 5.10 Longitudinal resistance plotted against inverse magnetic field after illumination of sample A. Spin polarization is clearly visible at high magnetic fields. No evidence of a beat is present, which could indicate that two subbands are conducting in the 2DEG. Parallel channels as these would likely have different frequencies in 1/B and hence would produce a beat [47]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 xxxi 5.11 Suspended Hall bar used for characterization of the 2DEG. The mesa definition etch (in this case RIE) went deeper than the bottom of the sacrificial layer and thus a similar shape to that of the suspended Hall bar can be seen under it. . . . . . . . . . . . . . . . . . . . . . . . . 122 5.12 Effect of pulses from the LED on the two-wire resistance of a suspended micron-wide beam such as the one in figure 5.17 at T ≃ 77 K. The pulse duration was one second and a relaxation time of one minute is allowed between each of the 10 pulses. The current through the LED at each of the pulses is approximately 1 mA. . . . . . . . . . . . . . . . . . . 124 5.13 Effect of pulses from the LED on the longitudinal resistance of a suspended Hall bar such as the one in figure 5.11 at T ≃ 300 mK (a four-wire measurement). The measured resistance is the longitudinal voltage difference across the Hall bar, divided by the measured sourcedrain current. The pulse duration was one second and a relaxation time of one minute is allowed between each of the 10 pulses. The current through the LED at each of the pulses is approximately 1 mA. . . . 125 5.14 Longitudinal resistance and Hall resistance as a function of magnetic field for a ≈ 500 nm-wide suspended Hall bar, such as the one in figure 5.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 xxxii 5.15 Conduction band energy relative to the Fermi level (set at zero) for the heterostructure in the unetched case and for cases where it was etched 5 nm on all sides, and etched 10 nm on all sides. In all cases an electron density of ≈ 5 × 1011 cm−2 is predicted in the 2DEG (diminishing the more material is etched). . . . . . . . . . . . . . . . . . . . . . . . . 129 5.16 ‘Swiss cheese’ motif found in the sacrificial layer of a suspended Hall bar. The circular gaps in the sacrificial layer are caused by the HF solution reaching the sacrificial layer through holes left in the mesa by the citric acid etch step that defined the mesa. . . . . . . . . . . . . 131 5.17 A suspended one-micron-wide beam. The sidewall of the heterostructure shows no sign of attack by the HF suspension. . . . . . . . . . . 132 5.18 An undercut alignment mark with the sidewall of the heterostructure looking intact after an HF dip. 6.1 . . . . . . . . . . . . . . . . . . . . . 133 Conduction band energy relative to the Fermi level (set a zero) for the proposed new design of the heterostructure. The sacrificial layer is a full 1000 nm thick but is plotted only until a depth of 400 nm in the heterostructure in order to keep the figure clear. An electron density of 5.174 × 1011 cm−2 is predicted in the 2DEG. xxxiii . . . . . . . . . . . . 138 6.2 Conduction band energy relative to the Fermi level (set a zero) for the proposed new design of the heterostructure when the sacrificial layer has been removed. The plot assumes a barrier of 0.72 eV for the AlGaAs/vacuum interface. An electron density of 5.508 × 1011 cm−2 is predicted in the 2DEG. . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.1 Drawing of the HE-3-SSV He-3 refrigerator and cryostat [87]. . . . . . 158 A.2 Drawing of both sides on the mounting stage. On the left, the side facing down towards the superconducting magnet is where the sample is glued and wired bonded to the various DC and RF leads. On the right is the side facing up away from the magnet and towards the top of the cryostat where the wires exit. The black represents conductive metal. All of the backside of the board is a ground plane, as is the area under where the sample is glued. . . . . . . . . . . . . . . . . . . . . 163 xxxiv Chapter 1 Introduction 1.1 Introduction to NEMS and Nanomechanics While micro electro-mechanical systems (MEMS) have invaded the industry in a variety of fields [1], nano electro-mechanical systems (NEMS) have yet to do the same. If the former can be described as a class of devices using mechanical parts, usually several micrometers in size coupled to electronic transducers; the latter can be thought of similarly, but scaled down with dimensions measured in nanometers. 1 CHAPTER 1. INTRODUCTION 1.1.1 2 Characteristics of NEMS Even though the dimensions of NEMS devices are very small, they are still larger than the atomic scale, and the fundamental rules of mechanics remain a good approximation. The most important behavior of rigid bodies used in electro-mechanical systems is the resonant frequency of that body. Vibration modes come into play in every device making use of vibrating beams, cantilevers, membranes or any other resonating body. The main advantage in shrinking devices from MEMS to NEMS is that their smaller dimensions make it possible to reach higher frequencies while maintaining a high mechanical responsivity. In specific terms, that means that NEMS are able to respond to smaller forces, lower thermal gradients and require less driving power to operate than MEMS [2]. This in turn makes NEMS ideal candidates for becoming very sensitive high-bandwidth transducers operating in the microwave domain, i.e. the frequency range from ≈ 100 MHz to several GHz. These advantages are afforded to NEMS by not only their small size, but also by their potentially extreme surface-to-volume ratios; the scaling laws of the forces [3] and our ability to design and fabricate NEMS with intricate structure. While we are able to construct impressive structures, fabrication of such small devices remains a challenge. In particular, there are currently no mature and reliable parallel processes that allow for mass production of devices with very fine features. Imprinting and embossing are the finest parallel processes available to date, but are not as popular CHAPTER 1. INTRODUCTION 3 Figure 1.1: Quality factor of mechanical resonators varying in volume from macroscale to nanoscale. The maximum attainable Q seems to decrease linearly with the logarithm of the volume of the devices [2]. as other serial processes, for they have not been developed commercially. As such, most often a serial process such as electron beam lithography is chosen. Fabrication will be discussed more thoroughly later on. One might expect the quality factor, Q, for which 1/Q is roughly defined for now as representing the degree of internal dissipation of a resonator, to increase in smaller devices since there are likely to be fewer internal defects. In fact there seems to be an opposite trend as suggested by the plot in figure 1.1. This is due to the increasing importance of the surface of the device as it reaches nanometer size. Thus, the limiting factor for high Q becomes less of an internal dissipation problem, and more of a surface one. In fact, for the same dimensions, polycrystalline resonator CHAPTER 1. INTRODUCTION 4 present Q-factors very close to other resonators made from pure crystals [2]. Reliable ways of obtaining high Q presently remain elusive. 1.1.2 Applications of NEMS As expected, NEMS have a multitude of possible applications. A sample of them will be briefly described here. Firstly, NEMS resonators can be used for detecting a single spin using a powerful technique named magnetic resonance force microscopy (MRFM) [4, 5]. By attaching a magnetic tip to an ultra-sensitive cantilever and exciting electrons with an RF field, it is possible to detect single spins in a region close to the tip (a “resonant slice”). This can be thought of as a mechanically-detected magnetic resonance imaging. It is hoped that this technique may evolve so that direct molecular imaging becomes possible. This would allow, for example, the identification of unknown chemicals by directly looking at the atoms composing a molecule while simultaneously determining the spatial arrangement of the atoms. The molecular structure of complex proteins could be obtained in this manner. A different approach to this problem is one involving large arrays of mass sensing NEMS resonators. When molecules attach themselves to vibrating body, the body should experience a frequency shift, which is detectable in a NEMS resonator. The idea is that using arrays of beams of different chemical coatings will allow us to CHAPTER 1. INTRODUCTION 5 record the so-called signature of a chemical : expose the array to that chemical and the beams will each react differently to it. A simple example of this as a hydrogen detector is demonstrated in [6]. BioNEMS are what could be called one of the holy grails of nanotechnology. Indeed, they have been speculated about for quite some time in works of fiction. They would be devices small enough to be inserted inside cells, and advanced enough to perform reactions with the chemical species in that volume. Currently this field is more or less limited to using atomic force microscopy (AFM) to interact with biomolecules [2]. This is but a very limited overview of the applications for NEMS. However, it is exciting to imagine that NEMS devices can be designed to fulfill a multitude of tasks improving speed, accuracy and reducing size over current microtechnology. 1.2 Motivation One of the most exciting applications of NEMS is undoubtedly that they be used to investigate fundamental science. Specifically, with ultra-sensitive displacement sensing of NEMS resonators, one could probe the inner workings quantum mechanics in several ways. To start with, since Heisenberg’s uncertainty principle places limits on the precision one can measure amplitude and phase (or any non-commuting observables), it should be possible to experimentally test it on a macroscopic (relative CHAPTER 1. INTRODUCTION 6 to particles) nanomechanical resonating body [7]. Secondly, there is not complete agreement on how, or even if, such large objects should obey quantum mechanics [8]. Thirdly, if this is possible, then the most interesting experiment to perform would be to place a nanomechanical resonator in a coherent superposition of states, and carefully monitor its transition for quantum mechanics to classical mechanics. Indeed, to this day, no one has a clear idea of how the collapse of the wavefunction occurs; the explanation of the mechanism is very dependent on the chosen interpretation of quantum mechanics [9]. It is hoped that the quantum-to-classical transition of such a large mass may provide us with answers [8]. To achieve this goal, there are a priori two challenges to overcome. The first is to cool down and place a nanomechanical resonator in a quantum mechanical state. The second challenge, and the one concerning this work, is to design and implement sensitive, low-noise transducers that allow us to study a resonator placed in such a state. 1.3 Scope of Work This section will describe briefly the work that was done in order to introduce the next chapter in which background information is covered. In order to overcome the aforementioned challenges, a design was made of an AlGaAs/GaAs heterostructure that meets the requirements of the proposal for an CHAPTER 1. INTRODUCTION 7 ultra-sensitive piezoelectric displacement sensor described in [10]. This device, in short, consists of a piezoelectric resonating beam (with embedded two-dimensional electron gas) coupled to an actuation electrode and a single-electron transistor. Selfconsistent Poisson-Schrödinger simulations were made to ensure that the heterostructure yielded a two-dimensional electron gas (2DEG) along the middle of the beam and with appropriate electron density. Furthermore, finite-element modeling of the device was done to characterize fundamental modes of vibration and verify the actuation mechanism. Once the crystal was grown, a recipe for depositing and annealing ohmic contacts to the 2DEG was developed. Contacts were deposited using electron beam lithography and physical vapor deposition. The wafers were then annealed in a rapid thermal annealer. Current-voltage (I-V) characteristics of the contacts were tested in a probe station with data acquisition software. A wet etching recipe for creating mesas in the wafer (reliefs that confine the 2DEG laterally to some shape) was developed using a combination of citric acid and hydrogen peroxide. This, combined with the ohmic contacts recipe, allowed us to fabricate Hall bars, in which the quantum Hall effect can be observed when varying the perpendicular magnetic field at low temperatures. The obtained magnetoresistance measurements were used to calculate the electron mobility and density of the 2DEG in a microscopic Hall bar at T = 300 mK. CHAPTER 1. INTRODUCTION 8 The following logical step was to fabricate suspended Hall bars, so that electron density may be evaluated in suspended structures. Doubly-clamped beams of different widths were also built in the hope that they may be used for evaluating the 2DEG depletion length and to test piezoelectric actuation. Upon testing the transport properties in those Hall bars and beams however, it was found that the 2DEG was depleted, unless illuminated by a light-source at room temperature. Thus, further testing was done on the suspended structures to see the cause of this problem and a redesign of the heterostructure was proposed. It was originally envisioned that piezoelectric actuation of doubly-clamped beams, accompanied by some form of displacement sensing, would complete the work in this thesis. However, this could not be done in the end because of the unexpected depletion of the 2DEG in suspended structures. Full completion of the project, i.e. the building of a working ultra-sensitive piezoelectric displacement detector is a longerterm project which will likely require at least two more years. 1.4 Organization of Thesis Chapter 2 will present theoretical background information and reviews relevant publications. It explains much of what is needed to understand this project and its motivation. Chapter 3 details the design and numerical modeling of the heterostructure that CHAPTER 1. INTRODUCTION 9 is so crucial to the project. Chapter 4 discusses all the fabrication processes that were developed in order to fabricate suspended nanostructures out of the heterostructure. Chapter 5 reviews the experiments that were done to characterize the electrical characteristics of the two-dimensional electron gas in the heterostructure. Chapter 6 looks at future work that needs to be done in order to bring the project to full completion. Finally, chapter 7 summarizes the conclusions. Chapter 2 Background This chapter will review the theory and literature necessary for a full understanding of the work that was done and its underlying motivation. It will also discuss some concepts pertaining to future work on this project. 2.1 Nanomechanics Since this work deals with the design, fabrication and characterization of nanomechanical resonators and their displacement detectors, I will derive the equations determining their vibration, in particular in the out-of-plane flexural case. The derivation will be made according to what is commonly named Euler-Bernoulli theory, which ignores rotational inertia and shear. For simplicity and relevance, the modeling will deal with rectangular cross-section beams of homogeneous isotropic material, clamped 10 CHAPTER 2. BACKGROUND 11 Figure 2.1: Example of a doubly-clamped beam made in GaAs/AlGaAs. The scale bar shows one micron. at both ends [11, 12, 13, 14]. Figure 2.1 shows an example of what a doubly-clamped beam made in GaAs/AlGaAs looks like in a scanning electron microscopy image. 2.1.1 Vibrating Mode Shapes and Frequencies Consider first the simple one-dimensional problem of finding the out-of-plane flexural displacement U (x, t) = U (x)U (t) of a beam positioned as in figure 2.2. The x-axis is positioned along the beam’s neutral axis, i.e. an imaginary line passing through the exact center of the beam. A cross-sectional element of length dx and area A = wh would be subject to forces from the neighboring elements Fz (x + dx) and −Fz (x) on each of its faces and torques −My (x + dx) and My (x). Balancing the forces and 12 CHAPTER 2. BACKGROUND U(x,t) z y x h dx w l Figure 2.2: A beam of length l, width w and thickness h. A cross-sectional element of length dx and area A = wh is shown, while the displacement U (x, t) in the z direction is a function of x and time t only [13]. torques about one side of the element results in the following equations, where ρ is the mass density : ∂ 2 U (x, t) Fz (x + dx) − Fz (x) − ρA dx =0 ∂t2 (2.1) Fz (x + dx) dx − My (x + dx) + My (x) = 0 (2.2) For linear modeling purposes, we may expand the equations using Taylor series about the point x, and eliminate the higher order terms in dx, giving : ∂ 2 U (x, t) ∂Fz = ρA ∂x ∂t2 ∂My Fz (x) = − ∂x (2.3) (2.4) To calculate the torque, we will need to define several quantities. The first is Young’s modulus, E, defined as the ratio of stress to strain when a material is under 13 CHAPTER 2. BACKGROUND tension. Stress, strain and Young’s Modulus will be discussed in greater detail later in the thesis. Next, we need the beam’s bending moment of inertia defined as the moment of inertia about the z axis [13] Iz = Z 2 z dA = A Z w/2 −w/2 Z h/2 z 2 dz dy = −h/2 wh3 , 12 (2.5) with the result for Iz being for a rectangular beam of width w and thickness h. The Euler-Bernoulli theory states that the local radius of curvature at point x on the neutral axis x is equal to [13] R(x) = EIz . My (x) (2.6) We now define the bending angle θ as the angle formed by the local tangent to the displaced neutral axis x′ and the original neutral axis x. This is made clear upon examination of figure 2.3, where to keep the diagram as simple as possible the neutral axis of a cantilever was drawn. A small change ds along the displaced neutral axis will be accompanied by a change dθ of angle θ. Thus, ds = R(x)dθ and dθ(x) 1 My (x) = = . ds R(x) EIz (2.7) Two observations can now be made, both in the case of a small bending angle θ. The first is that ds ≃ dx so that My (x) dθ(x) ≃ dx EIz (2.8) 14 CHAPTER 2. BACKGROUND z x x+dx x ds θ θ+dθ dθ R(x) R(x + dx) Figure 2.3: Definition of angle θ. R(x) gives the radius of curvature at point x. x′ represents the displaced neutral axis of a bent cantilever while x gives the original neutral axis. The angle θ is formed by the tangent at point x on x′ and x [13]. 15 CHAPTER 2. BACKGROUND as per equation 2.7. The second observation is that the change in deflection with the x coordinate is given by dU (x) = tan θ(x) ≃ θ, dx (2.9) which represents the slope of the beam’s deformation. By combining equations 2.8 and 2.9, the final expression for torque appears as My = EIz ∂ 2 U (x, t) . ∂x2 (2.10) This makes the Euler-Bernoulli approximation valid only for small deformations. A wave equation results from equations 2.3, 2.4 and 2.10 : ∂2 ∂x2 ∂ 2 U (x, t) ∂ 2 U (x, t) EIz = −ρA . ∂x2 ∂t2 (2.11) For a uniform beam, EIz does not vary in x, so it becomes EIz ∂ 4 U (x, t) ∂ 2 U (x, t) = −ρA , ∂x4 ∂t2 (2.12) which can be satisfied by a solution of the form U (x, t) = U (x)e−iωt ; U (x) = eκx . (2.13) (2.14) 16 CHAPTER 2. BACKGROUND More accurately, κ will have to take on values of ±β or ±iβ, where β= √ ω ρA EIz 1/4 , (2.15) giving a real spatial solution of U (x) = a cos βx + b sin βx + c cosh βx + d sinh βx. (2.16) Now, boundary conditions must be applied. Since a doubly-clamped beam is considered here, we require that [U (x)]x=0 = [U (x)]x=l dU (x) = dx x=0 dU (x) = dx = 0, (2.17) x=l that is, the displacement and speed of both ends are always zero. The first two conditions impose that a = −c and b = −d, leaving us with a solution of U (x) = a (cos βx − cosh βx) + b (sin βx − sinh βx) . (2.18) The last two boundary conditions imply that b=a (sin βl + sinh βl) (cos βl − cosh βl) 0 = a (1 − cos βl cosh βl) . and (2.19) (2.20) Therefore a may take on any value if the values of β are confined to a discrete set of values determined by cos βn l cosh βn l = 1, (2.21) 17 CHAPTER 2. BACKGROUND Un(x) x Figure 2.4: Mode shapes for the first four modes of flexural out-of-plane vibration for a doubly-clamped beam [13]. where n gives the mode number. Solved numerically, the solutions are β1 l = 4.73004, β2 l = 7.8532, β3 l = 10.9956, β4 l = 14.1372 [13] and so on. β0 is not allowable because it would produce a singularity. The final expression of Un (x), the mode shape of mode number n, takes on this form : sin βn l + sinh βn l Un (x) = an (cos βn x − cosh βn x) + (sin βn x − sinh βn x) . cos βn l − cosh βn l (2.22) The resulting first four mode shapes are displayed in figure 2.4. Referring back to equation 2.15, it is seen that the mode frequencies are given by ωn = s EIz (βn l)2 . ρA l2 (2.23) A more complete description is given in [11], for here we have ignored both the beam’s rotational inertia and shear. However, for mechanical resonators where the variation CHAPTER 2. BACKGROUND 18 Figure 2.5: Resonant frequency plotted against beam length for Euler-Bernoulli theory (plain curve) and experimental measurements (points) on piezoelectric Al0.3 Ga0.7 As doubly-clamped resonators. Notice that while Euler-Bernoulli theory seems to describe well the beam length dependence of resonant frequency it fails to predict the exact frequencies when the beam material is not isotropic nor homogeneous. In this case, the beams are made of three Al0.3 Ga0.7 As layers, two of which are heavily Si-doped. There is of course the expected deviation caused by irregularities in the beam shape due to the fabrication procedure [15]. in fabricated dimensions is large, the Euler-Bernoulli approximation is a good enough guide. Figure 2.5 shows a comparison of the Euler-Bernoulli theory and experimental measurements realized with piezoelectric resonators in Al0.3 Ga0.7 As. 19 CHAPTER 2. BACKGROUND 2.1.2 Transient Behavior of a Vibrating Beam Energy of a Vibrating Beam Now that the mode shapes and frequencies of a vibrating beam are known, the transient behavior should also be discussed, that is, the time-dependence of the motion. The deflection of a beam vibrating in the manner previously discussed varies harmonically in time [11]: u(x, t) = X Un (x)Un (t) (2.24) X Un (x) (An cos ωn t + Bn sin ωn t) . (2.25) n = n This is simply a consequence of equations 2.13 and 2.16. The sum is carried over all the superimposed normal modes. However, we are mainly concerned in this work with high Q resonators so that modes are well separated in frequency. Thus the motion of the resonator near the mode at frequency ωn will not be influenced by contributions from other modes. Henceforth the sum will be dropped and the deflection labeled as un (x, t). The energy accumulated in bending the beam is given by the work done in order to mold the beam into a mode shape. This ‘strain energy’ is expressed mathematically as [14, 11] : 1 En = 2 Z hdθn Mn i (2.26) 20 CHAPTER 2. BACKGROUND for mode shape Un (x). Here, the integral runs over the length l of the beam and the h i delimiters indicate an average value in time over a period. Mn is the mode’s torque, defined as Mn = EIz ∂ 2 un (x, t) ∂x2 (2.27) and θn is the slope of the deformation of the beam : θn = ∂un (x, t) . ∂x (2.28) We now have dθn ∂θn ∂t ∂θn = + dx ∂x ∂x ∂t ∂θn = ∂x ∂ 2 un (x, t) = ∂x2 (2.29) (2.30) (2.31) and dθn = ∂ 2 un (x, t) dx. ∂x2 (2.32) This leaves us with EIz En = 2 Z l * 0 ∂ 2 un (x, t) ∂x2 2 + dx (2.33) and ∂ 2 un (x, t) = −βn2 un (x, t) , 2 ∂x (2.34) 21 CHAPTER 2. BACKGROUND therefore using equation 2.15, we obtain En = = Now, αn = Rl 0 ρA ωn2 2 ρA ωn2 2 Z 0 l u2n (x, t) dx Un2 (t) Z 0 (2.35) l Un2 (x)dx. (2.36) Un2 (x)dx will only be a function of (βn l) so it is a constant with dimensions of length solely dependent on mode number. Furthermore, let m∗n = ρAαn be an effective mass and kn = m∗n ωn2 an effective spring constant. It appears then that the bending energy of the beam in a given mode is simply that of an harmonic oscillator: 1 En = m∗n ωn2 Un2 (t) 2 1 = kn Un2 (t) . 2 (2.37) (2.38) If the normalization for U (x) is chosen properly, then hUn2 (t)i would be the mean square amplitude of the resonator’s maximum, length-wise. Driven-Damped Harmonic Oscillator In practice, one must also consider in a first approximation a vibrating beam as a driven-damped oscillator. Dissipation in nanomechanical resonators remains a poorly understood phenomenon. The available literature offers many different explanations for why it occurs 22 CHAPTER 2. BACKGROUND including thermoelastic loss, attachment loss and loss due to the measurement process itself (see [14] and references therein). There are also several ways of modeling dissipation including defining a complex Young’s modulus but including a simple velocity-dependent damping term in the wave equation is sufficient for our purposes. We assume here that damping has negligible effect on the mode shapes, as described by Un (x) - which implies low loss. Our previous wave equation was equation 2.12 and with the added terms for driving force and damping it now takes the form of ρA ∂ 2 un (x, t) ∂un (x, t) ∂ 4 un (x, t) + EI +µ = Fn (x, t), z 2 4 ∂t ∂x ∂t (2.39) where µ is the damping coefficient and F (x, t) the force per unit length applied on the beam, for a given mode. Note that the latter can be used both to represent the actual intended driving force of the beam, but also can include terms representing noise modeled as certain random forces. To obtain the equation of motion, we need to multiply by Un (x) and integrate over the length of the beam. It is also useful to remember that un (x, t) = Un (x)Un (t) and that ∂x4 Un (x) = βn4 Un (x). The integration proceeds as : ∂ 2 Un (t) ρA ∂t2 Z 0 l Un2 (x) dx + EIz βn4 Un (t) Z 0 l Un2 (x) dx Z ∂Un (t) l µUn2 (x) + ∂t 0 Z l = Un (x)Fn (x, t) dx. 0 (2.40) 23 CHAPTER 2. BACKGROUND Referring back to the definition of αn = ρAαn where γn = Rl 0 Rl 0 Un2 (x) dx, we obtain ∂ 2 Un (t) ∂Un (t) = fn (t), + EIz αn βn4 Un (t) + γn 2 ∂t ∂t µUn2 (x) dx and fn (t) = Rl 0 (2.41) Un (x)F (x, t) dx. With a few substitutions using equations 2.23, m∗n = ρAαn and kn = m∗n ωn2 , the equation simplifies to m∗n ∂Un (t) ∂ 2 Un (t) + γn + kn Un (t) = fn (t), 2 ∂t ∂t (2.42) which represents the equation of motion for a driven-damped oscillator vibrating in mode n. We may now define an effective quality factor Q = ωn m∗n/γn , valid in the small damping limit where γn ≪ ωn . The expected amplitude of oscillations for a doubly-clamped beam in GaAs is calculated using the theory shown in this section and plotted in figure 2.6. 2.2 The Quest for Displacement Detection Limit In recent years, a bit of a race has developed in the scientific community towards reaching the limits of position detection. Heisenberg’s uncertainty principle [16] is well known for imposing limits of simultaneous knowledge of position and momentum. Consequently, there is a fundamental limit on the precision with which one may repeatedly measure the position of an object. A lot of the underlying motivation for this comes from the push for gravitational wave detection. As of 2005, the Laser Interferometer Gravitational Wave Observatory CHAPTER 2. BACKGROUND 24 Amplitude of motion HnmL 3 2.5 2 1.5 1 0.5 98 99 100 101 Frequency of driving force HMHzL Figure 2.6: Expected amplitude of motion near the resonance of the fundamental flexural mode for a doubly-clamped beam in GaAs (ρ = 5.3 g/cm3 , E = 101 GPa). Here l = 3 µm, w = 0.8 µm, h = 0.2 µm with a Q of 2000. A force (F0 /l) cos ωt of magnitude F0 = 1 nN is distributed over the surface of the beam and its frequency ω varied within ±2% of the resonant frequency. The resonant frequency is ≈ 99.7 MHz. The amplitude of motion of the resonator’s mid-point is displayed on the vertical axis with a maximum of ≈ 2.64 nm at resonance. 25 CHAPTER 2. BACKGROUND (LIGO) has achieved a displacement sensitivity of about 30 times the fundamental limit using macroscopic masses [17]. In a different approach, R. G. Knobel and A. N. Cleland performed an experiment in 2003 involving a nano-scale resonator whose displacement is detected by a single-electron transistor that reached a factor of 100 from the limit [18]. The current known record stands at a factor of 3.9, a feat accomplished by Naik et al. [19]. Now, it is but an assumption that mechanical structures as large as nano-scale resonators should even obey quantum mechanics. Indeed, many believe a superposition of states for a body that large to be impossible or at least that such states may only exist for very short amounts of time (see for example references [9, 17]). On the other hand, if it is possible, then it would be reasonable to treat a resonator as a quantum harmonic oscillator, since it has already been established that the classical harmonic oscillator can describe its classical motion. Therefore, a brief overview of the quantum harmonic oscillator and a derivation of the limit of displacement sensing will now be made. 2.2.1 The Quantum Harmonic Oscillator When the classical Hamiltonian for an harmonic oscillator undergoes quantization, it becomes an operator given by Ĥ = 1 2 1 P̂ + mω02 X̂ , 2m 2 (2.43) 26 CHAPTER 2. BACKGROUND where P̂ and X̂ are the momentum and position operators respectively. According to Schrödinger’s time-independent equation, Ĥ |x, pi = E |x, pi, applying this operator to any eigenstate expressed in the position/momentum basis will give the energy of the oscillator. However, using the formalism of creation and annihilation operators, it is possible to rewrite the Hamiltonian in the basis of energy eigenstates. In this case, we have Ĥ = 1 N̂ + 2 ~ω0 , (2.44) where N̂ is the so-called number operator. When applied to an energy eigenstate, it will have the energy level for eigenvalue: N̂ |ni = n |ni. Thus, in the energy eigenstates basis, the energy of the oscillator is given by Ĥ |ni = 1 n+ 2 ~ω0 |ni . (2.45) A key point that must be understood is that n gives the occupation factor of the mechanical mode with frequency ω0 . In essence, the higher n will be, the greater the amplitude of the motion will be, for the mode at frequency ω0 . The Energy Basis and Zero-Point Motion By using the aforementioned creation and annihilation operators in the energy basis, it becomes simple [20] to show that n must be a non-negative integer, i.e. n ∈ {0, 1, 2, 3, ...}. It appears then that the energy of a quantum harmonic oscillator 27 CHAPTER 2. BACKGROUND may never be zero. This minimum energy is often called zero-point energy and is equal to 1 E0 = ~ω0 . 2 (2.46) Naı̈vely, one could then assume that the smallest detectable motion would be that of an oscillator with only zero-point energy. In that case, if the oscillator has mass m, the wave function is described by a Gaussian distribution of width 2 1/2 x = r ~ ≡ ∆xzp . 2mω (2.47) This is the zero-point motion of a quantum harmonic oscillator. In literature, that value is also defined as the limit of measurement, for it can be also obtained straight from Heisenberg’s uncertainty principle [21, 17]. The standard quantum limit, or ∆xSQL = r ~ , 2mω (2.48) is defined as this root mean square amplitude. In theory, it is the smallest displacement one can hope to detect with a given oscillator-detector pair, although in practice there are some additional limiting factors, as will be discussed later. Note that the bigger this figure is, the easier it will be to reach it with a detector. Aside on the Measurement of the Displacement Amplitude Several ways exist of measuring the displacement amplitude of an oscillator. The first and simplest, which has already been introduced, is the ‘amplitude and phase’ CHAPTER 2. BACKGROUND 28 method, in which the transducer will ask the resonator “what is your amplitude and phase ?”. The experimenter then gets values for a1 and a2 (the real and complex amplitudes) which are related to position and momentum by the following relation : X̂ + i P̂ = Â1 + iÂ2 e−iωt . mω (2.49) It is this method upon which the standard quantum limit is defined and does indeed represent the minimum amount of error given when using this ‘amplitude and phase’ method on a single measurement. However, other ways of measuring can improve on this. Furthermore, the ‘amplitude and phase’ technique does not avoid back-action, meaning that the values of a1 and a2 are influenced by the act of measuring. A ‘back-action-evading’ measurement [21] or ‘quantum non-demolition’ (QND) measurement [22, 23] is one which can be, in principle, repeated time after time without back-action perturbing the measured observable. In fact, back-action is redirected to unwanted observables. A first example of a QND measurement is quantum counting, in which a transducer asks the oscillator “What is your number of quanta ? but, do not tell me anything about your phase”. In theory, n can be known with arbitrary accuracy using this method. It is also a fact that quantum counting can determine the magnitude of the amplitude (a21 + a22 )1/2 with far more accuracy than the amplitude and phase technique when n ≫ 1. This is because for the amplitude 29 CHAPTER 2. BACKGROUND and phase method ∆a1 = ∆a2 ≥ ~ 2mω 1/2 , (2.50) 1 mω 2 Â1 + Â22 − , 2~ 2 (2.51) while we have for quantum counting that N̂ = with theoretically no lower bound on ∆n [21]. Another possible measurement would be a back-action-evading measurement of the real amplitude a1 , leaving a2 completely indeterminate [21]. In practice, however, a QND scheme is difficult to implement. An interaction Hamiltonian that commutes with N̂ is needed so that n is not perturbed by the measurement. This requires an strictly non-linear interaction (e.g. quadratic, quartic, and so on). For example, for a quantum counting measurement at microwave frequencies, it is known that quadratic coupling to a resonator by a transducer is hard to achieve [21]. 2.2.2 Limiting Factors for Displacement Sensing In order to reach the limits of displacement sensing, there are several factors that must be considered. Firstly, the thermal fluctuations of the motion of an object must be sufficiently low. Secondly, quantum fluctuations will place a lower limit on the motion of a body as well as the minimum disturbance caused by a displacement transducer. 30 CHAPTER 2. BACKGROUND Thermal Fluctuations Whenever a mechanical mode is allowed to exchange energy with a thermal bath, its will take on random fluctuations distributed according to the Bose-Einstein distribution so that, on average, the energy is given by hEi = ~ω 1 1 + ~ω/k T B 2 e −1 . (2.52) When kB T ≪ ~ω, the average energy drops to a fraction of ~ω. We then say that the mode is frozen-out. This is desirable so that the thermal fluctuations do not amount to be higher than the spacing between energy levels: this will guarantee that the energy level will not change randomly (hence, the term frozen-out). It seems, then, that one should choose low temperatures combined with high frequencies to reach the quantum limit of displacement sensing, however, doing so will augment the need for more and more sensitive transducers. This arises from the fact that usually higher frequencies mean lower amplitudes for resonators, and that ∆xSQL too gets lower with increasing frequency, meaning it will be harder to reach. Quantum Fluctuations Quantum fluctuations encompass many phenomena that can all be fundamentally attributed to different expressions of Heisenberg’s uncertainty principle. Firstly, there is the zero-point motion given by ∆xSQL , as explained earlier. In addition, the measurement apparatus itself will impart some motion onto a resonator, 31 CHAPTER 2. BACKGROUND a phenomenon dubbed back-action. For a phase-insensitive linear amplifier measuring the position of a quantum harmonic oscillator, it can be modeled as a “noise temperature”, calculated to be [24] TQL = ~ω0 . ln 3 mω (2.53) If this is translated into a position variance and combined with the zero-point motion, one obtains [21, 7] ∆xQL = r ~ ≈ 1.35 ∆xSQL . ln 3 mω (2.54) ∆xQL is a much more realistic benchmark than ∆xSQL for estimating the quantum limit for a given resonator-amplifier system because it incorporates both the resonator’s zero-point motion and accounts for the minimum possible back-action made by the amplifier. This represents the best one can hope for but in practice there will be once again more limiting factors that can be globally categorized as measurement noise. These factors may include shot noise [25], 1/f noise [26] or thermal noise interference. Note as well that most amplifiers impart more than the minimum amount of back-action of the system they are measuring. Many argue that single electron transistors are at a disadvantage in this respect while quantum point contacts [27, 28] represent truly quantum-limited amplifiers with minimum back-action [29, 30]. CHAPTER 2. BACKGROUND 2.3 32 GaAs Usage in Mechanical Devices Even though the overwhelming majority of the electronic chips that surround us are based on silicon materials and processing technology, it would be an oversight to ignore what the III-V semiconductors have to offer. Specifically, we are interested here in gallium arsenide (GaAs) and its related compounds. Methods are now available to create almost atomically perfect (with very few defects) crystals of successive layers of these compounds, such as metal organic chemical vapor deposition (MOCVD) or molecular beam epitaxy (MBE) [31, 32]. Combined with etching processes, they allow us to manufacture extremely thin structures of mixed composition, yet still of almost perfect crystallinity. Indeed, mono atomic layers are not uncommon [33]. Advantages of GaAs over Si include its direct energy gap, which allows GaAs devices to be interfaced easily with optical technologies. This is compared to Si, which has an indirect band gap, though there are ways around this such as porous silicon or superlattices [34]. The defect generation rate during growth is also lower for GaAs than Si, meaning that it will perform with better reliability in specialized markets where high doses of radiation are common [33]. Most important in this work is the fact that GaAs and its best developed alloy, the ternary aluminum gallium arsenide (AlGaAs), have piezoelectric and piezoresistive properties. GaAs, like many other III-V compounds arranges itself in a zincblende crystal, a crystal structure made up of two face-centered cubic lattices shifted by a CHAPTER 2. BACKGROUND 33 vector (1/4, 1/4, 1/4), representing a quarter of a diagonal in the crystal’s unit cube. In the case of GaAs, one sublattice is composed entirely of gallium, while the other is composed of arsenic. Unlike elemental semiconductor crystals like silicon, inversion symmetry is not respected at all lattice sites in GaAs because of the two different elements composing the crystal. As a consequence of the mixed composition, electrons tend to shift towards the arsenic atoms, which actually results in a dipole moment oriented in the crystal’s [111] axis. Thus a non-vanishing piezoelectric coefficient is present for GaAs. Additionally, each pair of the eight {111} surfaces will have a surface richer in As, and one richer in Ga meaning that growth rates and etch rates will be influenced by orientation of the faces [35]. Table 2.1 presents a few of the material properties of relevance for GaAs and Alx Ga1−x As, where x indicates the mole fraction of Al. For most quantities, a linear interpolation between the values for GaAs and AlAs is sufficient to obtain values for Alx Ga1−x As (this is sometimes called Vegard’s Law)[36, 32]. Note that in this work, the acronym Alx Ga1−x As is used to strictly describe the ternary compound, that is, 0 < x < 1. 34 CHAPTER 2. BACKGROUND Table 2.1: Material properties of GaAs and Alx Ga1−x As [35, 36] Property GaAs Alx Ga1−x As Crystal structure Zincblende Zincblende 3 Density ρ (g/cm ) 5.360 5.36 − 1.6x Elastic stiffness coefficients cij (GPa): c11 118.8 118.8 + 1.4x c12 53.8 53.8 + 3.2x c44 59.4 59.4 − 0.5x −12 −1 Elastic compliance coefficients sij (10 Pa ) : s11 11.7 11.7 + 0.3x s12 3.7 3.7 + 0.2x s44 16.8 16.8 + 0.2x Young’s modulus E (GPa) 85.3 85.3 − 1.8x Poisson’s ratio ν 0.31 0.31 + 0.1x Bulk modulus (GPa) G 75.5 75.5 + 2.6x Piezoelectric coefficient d14 (pm V−1 ) −2.69 −2.69 − 1.13x Static dielectric constant ξs 13.18 13.18 − 3.12x High-frequency dielectric constant ξ∞ 10.89 10.89 − 2.73x Stress and Strain In order to have clear definitions of the table’s quantities, several quantities must be defined and first come stress and strain. Stress is defined as the force per area acting across any given surface of a volume element of a material. In the completely general case, stress may act in any direction relative to a surface inside (or on) a solid, and thus is best described by tensor algebra rather than a simple vector. For a force F normal to an area A, stress σ = F/A. In the most general case, dFj = PN i=1 σij dAi , where N is the number of dimensions, dFj the components of the resulting force, σij the components of stress and dAi the areas of the sides of an infinitesimal volume element. CHAPTER 2. BACKGROUND 35 The direction of stress components can be decomposed in two: normal stress, where the force acts perpendicular to the surface and shearing stress, where the forces acts parallel to the surface. Therefore, at first glance for a cubic element, three components are required to describe normal stress (σxx , σyy and σzz ) and six components for shearing stress (σxy , σyx , σxz , σzx , σyz and σzy ). However, at equilibrium and for two perpendicular sides of a cubic element, the components of shearing stress perpendicular to the line of intersection of these sides are equal. Hence, σxy = σyx , σxz = σzx , σyz = σzy and only six components are necessary for a complete description of stress [12]. As for strain, a suitable definition must be made stating how to measure it, but suffice it to say that it represents the amount of deformation in a stressed material. For example, consider a small cubic volume of side dl in a beam. Under stress, this volume will deform and for small flexural vibration we may use the approximation that its cross-section will remain unchanged, while its length varies. The strain would then be the change in length over its original length otherwise known as elongation : strain ǫ = ∆l/dl [37]. However there is another type of strain called shearing strain which can be thought of as the distortion of the angle between the faces of the cubic element. Figure 2.7 shows how two points, A and B deform under a displacement of the origin O to O′ by a displacement vector u = (ux , uy , uz ). It is first understood from the figure that the displacement in the x-direction of the point A to the point 36 CHAPTER 2. BACKGROUND z dx O dy dx u dy uy ux x y uy + ∂u ∂x dx A’ dz B O x O’ A y B’ B A y x ux + ∂u ∂y dy Figure 2.7: A cubic volume element undergoing deformation given by the displacement vector u of the origin O [12]. A′ is ux + ∂ux dx, ∂x (2.55) where the first term gives the absolute displacement in the x-direction and where the second term corrects for the increase of the coordinate x. It is then said that the unit elongation at point O in the x-direction is ∂ux /∂x. The same reasoning may be applied to the y- and z-directions and these constitute the first three components of strain. To obtain the remaining components of strain, one needs to examine the change in angle between the planes of the cubic element. The right-hand side of figure 2.7 exemplifies by showing the situation between the planes xz and yz. The ydisplacement of from point A to A′ is uy + ∂uy dx ∂x (2.56) CHAPTER 2. BACKGROUND 37 Table 2.2: The six components of strain as defined for an infinitesimal cubic element (see figure 2.7). The first line gives the elongation, while the second line gives shearing strain [37, 12]. y x z ǫxx = ∂u ǫyy = ∂u ǫzz = ∂u ∂x ∂y ∂z y y x z z x + ∂u ǫxz = ∂u + ∂u ǫyz = ∂u + ∂u ǫxy = ∂u ∂y ∂x ∂z ∂x ∂z ∂y but the change in angle from the segment OA to O′ A′ is simply ∂uy /∂x. Similarly for OB and O′ B ′ the change in angle equals ∂ux /∂y. Thus the angle AOB, originally a right angle, is changed by a total of ∂uy /∂x + ∂ux /∂y, a quantity giving the shearing strain between the xz and yz planes. An equivalent logic may be used to obtain the remaining two components of strain, which give the shearing strain between the planes xy and xz and the planes yx and yz. To summarize, table 2.2 gives the six components of strain, as defined for the deformation of an infinitesimal cubic element [12]. Young’s Modulus and Bulk Modulus Young’s modulus (E) represents the ratio of stress to strain when a material is under tension, i.e. a force trying to pull apart a material. This very same ratio may be different when forces are compressive and pressure is applied to shrink the material. In this case, the ratio is named bulk modulus (G) [37]. Young’s modulus describes the relation between elongation and normal stress in 38 CHAPTER 2. BACKGROUND isotropic materials (materials for which elastic properties remain equal in all directions). For example, ǫxx = σxx . E (2.57) The bulk modulus gives the relation between shearing strain and shearing stress in isotropic materials [12]. Again, for example ǫxy = σxy . G (2.58) Poisson’s Ratio Poisson’s ratio is the quantity that specifies how much an element contracts laterally when extension is applied longitudinally. Simply put, Poisson’s ratio (ν) is defined from the following equations for an extension in the x-direction [12] : ǫyy = −ν σxx E ǫzz = −ν σxx E (2.59) Stiffness and Compliance Matrices As not every material can be considered isotropic, it might be desired to have stressstrain relations that include the varying elastic properties in the different directions. This is afforded to us by full 6 × 6 matrices that give the relation between stress and strain and vice-versa, the components of which are called elastic coefficients. A full definition of these for GaAs and Alx Ga1−x As follows. 39 CHAPTER 2. BACKGROUND Material Properties of GaAs and Alx Ga1−x As While it may seem redundant to included both the elastic stiffness coefficients, elastic compliance coefficients, Young’s modulus, Poisson’s ratio and bulk modulus, they could all be useful depending on the formulation desired. For quick and simple modeling, considering GaAs as an isotropic material might be sufficient. However, it should be noted that Young’s modulus and Poisson’s ratio do change with orientation within the GaAs crystal, as seen in reference [35]. The values given earlier in table 2.1 are for tension applied along [110] and contraction applied in a perpendicular direction. Although some of the properties are temperature dependent, temperature was not specified here nor in the references as it there is little data and consensus on the values near T = 0 [38]. The definition of material properties matrices for the crystallographic coordinate system in which (x1 , x2 , x3 ) = ([100] , [010] , [001]) follows. Because of the zincblende structure, the stiffness matrix c takes on this form : 0 0 c11 c12 c12 0 c c c 0 0 0 12 11 12 c c c 0 0 0 12 12 11 . c= 0 0 0 c44 0 0 0 0 0 c44 0 0 0 0 0 0 0 c44 (2.60) 40 CHAPTER 2. BACKGROUND The compliance matrix s resembles the stiffness matrix in form and can be obtained by replacing cij ’s by their corresponding sij ’s. The stress-strain relationship can be written as σ = cǫ or ǫ = sσ in strain-stress form, where σxx σ yy σ zz σ= σ xy σxz σyz (2.61) is the stress matrix, whose components have units of force per area and where ǫxx ǫ yy ǫ zz ǫ= ǫ xy ǫxz ǫyz (2.62) is the strain matrix, whose components are unitless. Note that s = c−1 . A set of coupled equations are required to describe the piezoelectric effect. In 41 CHAPTER 2. BACKGROUND strain-charge form, they are ǫ = sE=0 σ + dT E, (2.63) Q = cσ + ξ σ=0 E. (2.64) Here, E is the electric field vector of dimensions (3×1) and ξ is the dielectric constant (permittivity) matrix of dimensions (6 × 3). sE=0 indicates the strain matrix when no electric field is applied. Similarly, ξ σ=0 indicates the dielectric constant matrix when no stress is present. Q is the electric charge density displacement vector of dimensions (6 × 1), whose components have units of charge per area. The piezoelectric coefficient matrix d therefore needs to be of dimensions 3 × 6 and takes form as: 0 0 0 0 d14 0 d = 0 0 0 0 d14 0 0 0 0 0 0 d14 (2.65) Due to the zinc-blende structure and the chosen coordinate system, all components but d14 , d25 and d36 vanish and they are all equal in value. 2.4 Piezoelectric Actuation in GaAs In the early 1990’s, Soderkvist, Hjort and others conducted an extensive investigation of piezoelectric resonators and sensors made with GaAs [39, 40, 41]. They found CHAPTER 2. BACKGROUND 42 Figure 2.8: A GaAs tuning fork with electrode configuration for in-plane flexural vibrations [41]. that GaAs could be used in piezoelectric transducers almost as efficiently as quartz. Depending on the desired vibration modes, care had to be taken to have the resonator in the proper crystal orientation and the electrodes be properly placed. For example, figure 2.8 shows a GaAs piezoelectric tuning fork structure [41]. Since the most common GaAs wafer orientation is the (100) variety, it helps to rotate the coordinate system from (x1 , x2 , x3 ) = ([100] , [010] , [001]) to one in which x3 = [100], i.e. x3 is normal to the wafer’s surface. The transformations that need to be applied are described in reference [42]. Accordingly, the other relevant material properties will also have to be “rotated” for the equations to remain consistent. We may then choose arbitrarily the orientation of the x1 and x2 axes as long as they remain perpendicular to each other and both perpendicular to x3 . Thus, using a (x01 , x02 , x03 ) = ([01̄1] , [01̄1̄] , [100]) coordinate system as a reference, a general coordinate system (x1 , x2 , x3 ) obtained by a right-hand rule rotation of angle φ around 43 CHAPTER 2. BACKGROUND x03 = x3 will yield a piezoelectric coefficient matrix of 0 0 0 2 sin 2φ −2 cos 2φ 0 d14 . d= 0 0 0 2 cos 2φ 2 sin 2φ 0 2 − cos 2φ cos 2φ 0 0 0 2 sin 2φ (2.66) Referring back to the coupled piezoelectric equations, particularly the term dT E, expressed as 0 0 − cos 2φ 0 0 cos 2φ E 1 0 0 0 d14 E , 2 2 2 sin 2φ 2 cos 2φ 0 E 3 0 −2 cos 2φ 2 sin 2φ 0 0 2 sin 2φ (2.67) it is seen that the third column elements will be multiplied with the E3 component of the electric field. In essence, they show the response to an out-of-plane electric field 44 CHAPTER 2. BACKGROUND in our coordinate system. The result of the multiplication is −E3 cos 2φ E3 cos 2φ 0 d14 , 2 2E sin 2φ + 2E cos 2φ 1 2 2E sin 2φ − 2E cos 2φ 2 1 2E3 sin 2φ (2.68) where the first three components are contributions to elongation and the last three add to shearing strain. Therefore if out-of-plane flexural vibrations are desired and with an hypothetical beam oriented along x1 , φ = nπ/2 should be chosen in order to maximize the elongation, meaning that any of the h011i directions will fulfill the requirement. A possible orientation of a GaAs piezoelectric beam along with a typical GaAs (100) wafer are depicted in figure 2.9. Note that with this particular setup, the elongation will be caused in-plane, in the x1 and x2 directions. Because the beam is clamped, the oscillations in the elongation will couple to flexural vibrations when of appropriate frequency. In fact, Masmanidis et. al [43] demonstrate that this works for cantilevers as well as doubly-clamped beams as long as the piezoelectric layer is off-center with regard to the object’s neutral axis. No elongation can be directly caused in the x3 direction. Now that the proper orientation for a flexural resonator has been determined, care 45 CHAPTER 2. BACKGROUND [100] [100] [011] x3 x 2 [011] x1 [011] [011] Figure 2.9: Left : A typical GaAs wafer with primary flat on the bottom and secondary flat at a right angle on the left-hand side of the wafer. Crystal directions are indicated. Right : a possible configuration for a piezoelectric doubly-clamped beam designed for actuation (or sensing) of the of out-of-plane flexural motion. [31, 40]. must be given to how its motion will be detected, if one is to make a piezoelectric sensor. In fact, the same conditions will apply to an actuator, because of the duality of the piezoelectric effect : not only will an applied electric field produce strain, but an applied strain will conversely produce an electric field. The duality makes the piezoelectric effect particularly suited for applications where it is desired to detect the motion of a structure excited to resonate at its fundamental frequency. It has been established that a field normal to the wafer surface is needed for out-of-plane flexural motion, however, more accurately the electric field needs to be changed across the cross-section of the beam [40]. Detection of the displacement of a piezoelectric structure occurs as follows. A mechanical stress (or strain) will displace the dipoles of GaAs (and AlGaAs) such that a polarization field is created. Surface and volume bound charges appear and generate an electric field. It is precisely those 46 CHAPTER 2. BACKGROUND x3 − x2 + x1 + − Figure 2.10: The optimal electrode placement on a beam oriented in one of x1 = h011i for out-of-plane flexural motion. Electrode polarities are indicated [40]. charges which allow us to detect that stress occurs in a piezoelectric material. If the charges were neutralized, there would then be no effective way of detection. Good conductors will not make suitable piezoelectric materials for this reason. However, electrodes attached to piezoelectric materials will neutralize the bound surface charges and allow one to detect stress by monitoring the current (or total charge) drawn from the electrode during the neutralization process. The placement of the electrodes will be critical in determining what motion is detected or caused. The most suitable electrode configuration and their polarities for out-of-plane flexural motion is showed in figure 2.10 (the polarities are reversed depending on the direction of motion). This arrangement will allow the charges on all four surfaces of a doubly-clamped beam to be detected, however it is not an absolute necessity to have all four electrodes, two will be sufficient as will be seen later. The most important fact to keep in mind is that the electrical field must be varied in an asymmetric fashion inside the beam, with respect to the beam’s neutral axis. This is well demonstrated by the piezoelectric actuation mechanism described CHAPTER 2. BACKGROUND 47 recently by Masmanidis et al. [43]. Although they use only a pair of electrodes in both a GaAs cantilever and a GaAs doubly-clamped beam, they manage to excite vibrations because their piezoelectrically active layer is not centered on the neutral axis. An ingenious design (see figure 2.11) using a p-type/intrinsic/n-type (PIN) diode allows for a completely integrated actuation mechanism requiring no metal electrodes other than simple bond pads far removed from the resonator. A p-type substrate directly below the suspended structure connects to the p-type bottom layer of the PIN diode and acts as a ground electrode. The top n-type layer of the PIN diode serves as the driving electrode, where an AC signal is applied. If the voltage remains below the cut-in and above diode breakdown, no current flows in the diode and an electric field causes strain the the intrinsic GaAs layer. The advantages of this method are that it is integrated, efficient and tunable. The efficiency is demonstrated by the authors in figure 2.12. Amplitudes as low as 5 µV with powers as low as 5 fW are used to drive a cantilever. Finally, Masmanidis et al. show that their actuation technique is tunable in two ways by application of a DC bias voltage on the ground electrode. Firstly, one can change the amplitude of oscillation by changing the width of the depletion region in the PIN diode, thereby making the piezoelectrically active layer larger or thinner as seen in figure 2.13. The authors also demonstrate that different CHAPTER 2. BACKGROUND 48 Figure 2.11: Crystal design for the actuation mechanism described in [43]. The top drawing shows how an electric field applied in the piezoelectric layer (resistive intrinsic GaAs) causes longitudinal strain. However in this case the piezoelectric layer is centered on the neutral axis and so only longitudinal oscillations can be excited. The two remaining drawings show how for the same direction of electrical field, the cantilever can be made to go up or down simply by moving the piezoelectric layer below or above the neutral axis. CHAPTER 2. BACKGROUND 49 Figure 2.12: The efficiency of the actuation method in [43] is demonstrated with applied AC signals of amplitude as low as 5 µV to a cantilever with ω0 ≈ 8 MHz; Q = 2700; l = 4µm; w = 0.8 µm; t = 0.2 µm. The position of the cantilever is monitored by optical interferometry. PIN diode design can be made to have decreasing, constant or increasing amplitude with bias voltage. Secondly, they show that a frequency shift is caused in doubly-clamped beams when a DC bias is applied (see figure 2.14). The frequency shift is also accompanied by a change in amplitude. The frequency shift can be made positive or negative depending on the crystallographic orientation of the beam, which is a demonstration of piezoelectric anisotropy (meaning that the sign of the piezoelectric coefficient changes when orientations are reversed). CHAPTER 2. BACKGROUND 50 Figure 2.13: A. Change in amplitude with respect to the DC bias voltage applied to the ground electrode. B. Drawing showing the change in the depletion region width when a DC bias voltage is applied. C. Three different PIN diode designs can provide increasing, constant or decreasing amplitude with applied DC bias [43]. CHAPTER 2. BACKGROUND 51 Figure 2.14: A. Scanning electron microscopy image of the doubly clamped beam. B. Change in frequency and amplitude caused by different applied biases. C. Opposite crystal orientations give opposite behaviors for the change in frequency with bias voltage. The inset shows steps in frequency caused by the addition of 10 mV bias voltage [43]. CHAPTER 2. BACKGROUND 2.5 52 The Quantum Hall Effect in Two-dimensional Electron Gases 2.5.1 Inversion Layers and Modulation Doping The two-dimensional electron gas is a physical manifestation of a more general phenomenon named inversion layer, which can form at the interface of a conductor and an insulator once an electric field is applied. A well-known technique for creating inversion layers is that which is commonly used in the metal-oxide-semiconductor field effect transistor (MOSFET, see figure 2.15). As the name puts it plainly, a semiconductor is covered by an insulator (often silicon dioxide) and an electric field applied by a metal gate on the surface, attracting electrons to the semiconductor-insulator interface. There, they sit in a quantum well, causing their motion perpendicular to the interface to be quantized. As a result, they lose two degrees of freedom and are forced to propagate in a plane. This process is named ‘band bending’ for the effect the applied gate voltage has on the conduction band, as shown in figure 2.16. Near the interface, the conduction band can be seen to fall below the Fermi level, indicating that the material will conduct in this region. In a MOSFET, the carriers (electrons) are provided by heavily n-doped regions near the source and the drain. Impurities as they are, the dopants significantly increase the scattering of the electrons in the inversion layer. Additionally, the many 53 CHAPTER 2. BACKGROUND +VG +++++++++++++++++++++ 1111111111111111111111 0000000000000000000000 0000000000000000000000 1111111111111111111111 D 00000 11111 0000000000000000000000 1111111111111111111111 00000 11111 00000 11111 0000000000000000000000 1111111111111111111111 Al 000 111 000000000000000000000000 111111111111111111111111 000 111 00000 11111 00000 11111 0000000000000000000000 1111111111111111111111 000 111 000000000000000000000000 111111111111111111111111 000 111 00000 11111 00000 11111 000 111 000000000000000000000000 111111111111111111111111 000 111 00000 11111 00000 11111 SiO 2 000 111 000000000000000000000000 111111111111111111111111 000 111 00000 11111 00000 11111 000 111 000000000000000000000000 111111111111111111111111 000 111 00000 11111 00000 11111 S 11111 00000 n −−−−−−−−−−−−−−−−− n p−Si Figure 2.15: Schematic view of a metal-oxide-semiconductor field effect transistor (MOSFET). An inversion layer is formed at the interface of the semiconductor, ptype silicon, and the insulator, silicon dioxide. The electric field is provided by a positive voltage applied on the aluminum gate deposited on the surface. Heavily doped regions near the source and drain provide the carriers [44]. defects present in the insulator layer will also contribute to scattering. Thus it is generally desirable to remove the defects of the insulator and the dopants from the vicinity of the inversion layer. A solution to the first problem is to use not a simple oxide layer as the insulator, but rather a crystalline material in which there is nearly an absence of defects. Since AlGaAs has a wider band gap than GaAs, it can be used as the insulator at the semiconductor-insulator interface. By using molecular beam epitaxy (MBE), it is also possible to make atomically sharp transitions of these two materials. Therefore AlGaAs and GaAs make ideal candidates for obtaining inversion layers and have indeed been used in this manner for several years [45, 46, 44, 47]. Finally, in order to distance the dopants from the inversion layer, the modulation doping technique was developed and is afforded to us by MBE. By concentrating 54 CHAPTER 2. BACKGROUND EF 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 eVG 1111 0000 1111 0000 1111 0000 1111 0000 1111 SiO 2 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 Al 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 00001111 1111 0000 1111 Conduction Band Valence Band Figure 2.16: Electron energy levels diagram for a MOSFET. The electric field applied on the aluminum gate causes the bands to bend near the insulator layer. The conduction band falls below the Fermi level in this region. The electrons start by filling the hole states at the bottom of the valence band, however, when all these states are filled up to the Fermi level, the remaining carriers populate the conduction band. Thus, a conducting two-dimensional gas is obtained [44]. the dopants in thin layers, separated from the AlGaAs/GaAs interface by “buffer layers”, the dopants are moved hundreds of Angstroms away in “donor layers”. The donor layers are usually thin, but highly doped, and so this distribution of dopants is sometimes called “delta doping” in reference to a delta function. The buffer layers are layers of crystalline AlGaAs and remain undoped so that their defect density is low. In the AlGaAs/GaAs heterostructure, an electric field is provided by the ionized dopants found in the donor layers. The GaAs layer is of course made as pure as possible to reduce defects but will remain typically weakly p-type [44]. The electrons falling from the donor layers to the potential well first fill a few states at the bottom 55 CHAPTER 2. BACKGROUND AlGaAs GaAs Conduction Band Conduction Band EF Valence Band E z Valence Band Figure 2.17: Electron energy levels diagram for a AlGaAs/GaAs heterojunction. Since pure GaAs remains slightly p-type, the electrons falling from the n-doped AlGaAs occupy first the hole states at the bottom of the valence band but eventually fill the potential well at the interface. There, a two-dimensional electron gas is formed [44]. of the valence band but eventually occupy states in the conduction band below the Fermi energy, at the heterojunction (see figure 2.17). In this way, a two-dimensional electron gas is obtained there. 2.5.2 The Quantum Hall Effect The quantum Hall effect is commonly used in determining the characteristics of a 2DEG, as seen in the many examples already presented [48, 49, 50, 51]. Its discovery was announced in 1980 by von Klitzing, Dorda and Pepper and it was for this CHAPTER 2. BACKGROUND 56 discovery that von Klitzing was awarded the 1985 Nobel prize in physics. A short introduction to the effect will be made followed by an explanation of how it may be used to measure the characteristics of a 2DEG. Two-dimensional systems of electrons exposed to magnetic fields can display under certain conditions in the longitudinal direction infinite scattering time, meaning that no scattering occurs and the resistance vanishes. This is accompanied by quantized steps in the Hall resistance (transverse direction) defined as RH = VH/I and going in steps of h/e2 j where VH is the Hall voltage, I is the current flowing in the longitudinal direction and j is an integer. When exposed to magnetic fields, the energy levels of electrons in a 2DEG will further split into Landau levels. In an ideal case these are sharp, δ functions in a density of states diagram. In reality, because of impurities found in experimental samples, a broadening of the Landau levels occurs and the levels are highly degenerate. The degeneracy increases linearly with magnetic field and so as the field is varied, the Fermi level may jump from one Landau level to in between two Landau levels and vice-versa. This causes what are called Shubnikov-de Haas (SdH) oscillations in the longitudinal resistance, as whenever a Landau level is completely filled, the Fermi level must lie between Landau levels, and the resistance vanishes. 57 CHAPTER 2. BACKGROUND VL l y S VH w I D x Figure 2.18: A typical Hall bar [44]. Experimental Aspects of the Quantum Hall Effect Important information can be gathered from the observation of the quantum Hall effect. The carrier density along with its mobility can be determined using the magnetoresistance measurements. The experimental requirements for observing the quantum Hall effect are low temperatures, two-dimensional electron systems, strong magnetic fields perpendicular to the surface and a Hall bar. The Hall bar (see figure 2.18) is a multiple terminal mesa structure allowing one to observe SdH magnetoresistance oscillations and measure the Hall resistance. It uses a source and a drain for current and two or more perpendicular leads for measuring the longitudinal voltage drop. Opposite leads can be used to measure the Hall voltage. 58 CHAPTER 2. BACKGROUND Carrier density There are at least two ways of calculating the carrier density in a 2DEG. The first one involves only the classical Hall effect. In this simple case, the Hall voltage is given by [52] IB na e IB , ⇒ na = eVH VH = (2.69) (2.70) where na is the electron sheet density and B the perpendicular field strength. After magnetoresistance measurements are taken, one can use them in the near zero field region, i.e. well before SdH oscillations start showing, to get the density. For instance, by plotting IB against eVH , the density can be obtained from the slope. The second method of calculating the carrier density involves the SdH oscillations. SdH oscillations are periodic when plotted against 1/B . The period of the oscillations allows us to calculate the density according to [49, 47] na = ge , h∆ (1/B ) (2.71) where g is the spin degeneracy factor and ∆ (1/B ) is the period in 1/B of the SdH oscillations. The spin degeneracy factor depends on whether the energy levels are spin-split or not : at low magnetic fields no spin splitting of the energy levels is visible, thus g = 2 should be used since electrons are spin-half particles. At higher magnetic fields, once the Zeeman energy splitting is larger than the width of the 59 CHAPTER 2. BACKGROUND Landau levels, a factor of g = 1 should be included. Carrier Mobility The carrier mobility is a measure of how much dissipation occurs in the 2DEG when no magnetic fields are applied. In many ways it represents the quality of a 2DEG and the ability of the electrons within it to conduct. Note that for comparison purposes, the mobility must be evaluated at near zero magnetic field, for it becomes usually very different in value in the quantized Hall regime. It can be simply found from the zero-field resistivity and density [44]: µ= 1 , ρSD na e (2.72) where µ is the mobility and ρSD is the source-drain resistivity. One should keep in mind that for a two-dimensional system resistivity has the same units as resistance, Ω. For this reason, its units are sometimes labeled Ω/ instead of Ω · m in three dimensions. Equally often is the term sheet resistance used in place of resistivity. This can be calculated using the zero-field source-drain resistance with ρSD = W I l VSD (2.73) where W is the width of the 2DEG, l the length between the two leads used for measuring longitudinal voltage drop, I the source-drain current and VSD is the sourcedrain voltage. CHAPTER 2. BACKGROUND 2.6 60 Suspended Two-dimensional Electron Gases The first suspended structures with embedded 2DEGs came in 1998 by Blick et al. [48] and Beck et al. [53]. Because of the piezoelectric properties of the material used, GaAs, it was surmised that they would allow the realization of sensitive, microwave bandwidth, displacement detectors. Another possible application considered were bolometers for high-sensitivity calorimetry. The approach used at the time resembles what is being used in this work. It consisted of successive layers of GaAs and AlGaAs, arranged and doped in such a way that, by the modulation doping technique, a 2DEG was created and a sacrificial layer was present, thus allowing for the suspension of structures. The quantum Hall effect is commonly used to characterize suspended 2DEGs or otherwise more conventional 2DEGs. By performing some magnetoresistance measurements, one can estimate the carrier mobility and density and possibly other parameters, one of which will be discussed later. Several differences were found in these compared to the ones in conventional Hall bars. Since suspended structures are usually small, on the order of a micron in length, lest they collapse, their width is much smaller than a conventional Hall bar. Edge depletion then becomes a non-negligible factor. Surface states at the sidewalls of the structure will partially deplete the carriers of 2DEG on each side, reducing the effective width of the gas from the mechanical width, wm to w = wm − 2d, where d is CHAPTER 2. BACKGROUND 61 the depletion length. This length, for a single wm , will vary from sample to sample as it is dependent on several factors including electron density, mobility and sheet resistivity. The rate of cooling of a sample is also believed to affect depletion length [45]. It can be said that for any large mesa structures, where the mechanical width wm is many times that of the edge depletion length d, w ≃ wm . However this will not be true of smaller mesas where wm is of the same order as d. Sheet resistance can be determined from a large sample in which edge depletion is negligible. However, due to the fact that a wafer might be inhomogeneous, one cannot count on the fact that the sheet resistance of a small sample would remain the same. Edge depletion near a conducting channel may also create additional electron scattering, thereby increasing the sheet resistance. Electron density is simply not guaranteed to be the same in the large samples than in small ones either. Therefore Choi et al. developed a method based on localization theory [54] for determining the effective width which does not depend on the mobility nor sheet density. By measuring the magnetoresistance at a fixed temperature, it is possible to determine w uniquely since " # 2 2 2 −1/2 ∆R e w B R e2 lφ − lφ−2 + = R L π~ 3~2 (2.74) gives the magnitude of the magnetoresistance. Here, at a fixed temperature T , R is the zero field resistance of the channel, ∆R is the change in resistance, L is the length CHAPTER 2. BACKGROUND 62 of the channel, e the charge of an electron, B is the magnitude of the perpendicular magnetic field applied and lφ is the temperature-dependent phase breaking length [55]. They obtain for their samples d ranging from 300 nm to 800 nm [45], while Blick et al. estimate it to be from 175 nm to 230 nm in their suspended structures [56, 49]. The smaller depletion lengths probably result from the higher electron mobility and density of the latter case. Interestingly, edge depletion can be exploited to make electronic systems of even lower dimensionality, such as 1D channels or quantum dots [50, 57]. Therefore, when considering resonators with embedded 2DEGs a minimum mechanical width will be required in order for current to flow, thereby influencing resonant mode frequencies if the resonator is actuated for in-plane vibrations and placing a lower bound on mass of the resonator. 2.7 The Piezoelectric, SET-based, Displacement Detector While NEMS devices using GaAs/AlGaAs heterostructures have been realized before, there are few examples. The earliest work was done by Blick and Hoeberger et al. [49, 50] and Cleland et al. [58] in 2002. There was a cantilever-based device created in 1998 by Beck et al. [53] but is more in realm of MEMS than NEMS. A CHAPTER 2. BACKGROUND 63 Figure 2.19: Magnetoresistance measurements performed on a suspended 2DEG showing negative magnetoresistance at low magnetic field strengths. The inset show spin-splitting at higher magnetic fields [49]. micromechanical cantilever incorporating a 2DEG was also used by Harris et al. in 2001 [51]. In the first case, suspended structures were fabricated using an GaAs/AlGaAs heterostructure. Vibrational properties of the beams were characterized by driving them using the magnetomotive technique [13], but the real goal of the experiments was to investigate transport in suspended 2DEGs and quantum dots. Figures 2.20 and 2.21 show beams realized for those experiments with accompanying mechanical response. In the first example, a simple GaAs beam was used while in the latter the full GaAs/AlGaAs heterostructure was employed. Cleland et al., however, went further by directly making use of the 2DEG as part 64 CHAPTER 2. BACKGROUND Figure 2.20: (a) Nanomechanical resonator in GaAs in [110] orientation. (b) Circuit used for magnetomotive technique. (c) Mechanical response of the beam around the 115.4 MHz resonance peak (in-plane vibrations were used). The different curves indicate the response for magnetic fields ranging from 1 T to 12 T [49]. of a displacement sensing transducer. Quantum point contacts (QPC) defined on 2DEGs have been recognized since the work of van Wees et al. [28, 59] to show quantized plateaus in their conductance in steps of 2e2/h. Electrostatic potentials at the QPC’s electrodes restrict the flow of carriers in the 2DEG in this manner. Thus if a QPC is properly biased to in-between two conductance steps, very little change in electrostatic potential can lead to a large change in conductance. This makes the QPC a potentially very sensitive charge detector. In Cleland’s work, however, the temperature of the sample was not low enough for quantized conductance steps to be observed. The I-V characteristics of the QPC were simply characterized and that information used in detecting the beam motion. CHAPTER 2. BACKGROUND 65 Figure 2.21: (a) Micrograph of the GaAs/AlGaAs suspended structure showing circuit used for magnetomotive technique. Suspended quantum dot structures are coupled to the beam in order to investigate how they interact. The quantum dots are created by using the edge depletion effect : indentations are made at a 65◦ angle in a rectangular beam. (b) Mechanical response of the beam for different driving powers. Note that considerable nonlinearity appears as the power is increased. The inset show the response for varying magnetic fields [50]. CHAPTER 2. BACKGROUND 66 Figure 2.22: (a) A micrograph of the piezoelectric QPC displacement detector. (1) The wire providing the out-of-plane force. (2) and (5) are the source and drain for the ohmic contacts to the 2DEG. (3) and (4) are the two QPC defined on the beam, but only one was used at a time [58]. The resonator was actuated with the magnetomotive technique, however the working principle of the displacement detector was not based on this technique. When the beam is bent in an out-of-plane fashion by the magnetic forces in a gold wire running on top of it, electric fields generated by the piezoelectric effect will modulate the potential of the QPC on top. This in turn creates a detectable change in the current flowing through the QPC. Figure 2.22 shows a micrograph of the device, while figure 2.23 depicts a more explicative diagram of the experiment accompanied by the current response observed. The same year, Knobel and Cleland proposed a more ambitious device; a piezoelectric, single electron transistor (SET) based displacement sensor with a resonator made of a GaAs/AlGaAs heterostructure [10]. Instead of detecting the piezoelectric 67 CHAPTER 2. BACKGROUND Figure 2.23: (a) Schematic of the experiment. A magnetic field actuates the beam up and down using the current provided by the local oscillator (LO). A lock-in amplifier is used to monitor the current through the QPC. (b) The current response of the QPC near the resonant peak [58]. charge induced by the beam’s flexure with a QPC, they propose to use a SET. Since the I-V profile of the SET is extremely sensitive to the charge accumulated on its island, a whole detection electrode would be placed on top of the beam, and wired to the island of the SET (effectively becoming the island). Another electrode at the opposite end of the beam will provide actuation of the beam when excited by a RF signal. One can expect considerable back-action from the detection electrode as the circuit connected to it will also influence the charge on the island. The ground plane for the detection and actuation signals is provided by ohmic contacts made to the 2DEG passing through the beam. Figure 2.24 shows a schematic of the proposed experiment. √ For a 1 GHz resonator, it is estimated that a sensitivity of order 10−17 m/ Hz can be obtained with the measuring scheme, which represents a significant improvement 68 CHAPTER 2. BACKGROUND Figure 2.24: (a) Proposed heterostructure design. The sacrificial layer may be selectively etched by an HF dip. A 2DEG is formed at the interface of the GaAs and the Al0.3 Ga0.7 As using the well-known modulation doping technique. (b) Sketch of the NEMS device. The upper half of the diagram show the SET, whose island is connected to a detection electrode while the lower half shows the actuation electrode accompanied by two ohmic contacts to the 2DEG [10]. √ on simply using capacitive coupling (3.8 × 10−15 m/ Hz is achieved in one of the most recent experiments using capacitive coupling [7]). Included in this figure are the current and back-action noise of the SET [10]. Furthermore, accompanying such high frequencies as required to approach the regime where ~ω ≥ kB T , are small amplitudes of motion that clearly do not favor capacitive coupling while the piezoelectric scheme remains quite sensitive in small, stiff oscillators. Consider the following argument. Piezoelectric detection is sensitive to changes in the strain in the beam (specifically here to the elongation). Let ǫmin be the minimum detectable change in strain. Then both the minimum detectable force and the 69 CHAPTER 2. BACKGROUND minimum detectable displacement are [53] Fmin = Ewh2 6l ǫmin and 2 2l ǫmin . Umin = 3h (2.75) (2.76) (2.77) Both scale favorably (become smaller) as dimensions are reduced while keeping the same aspect ratio. For capacitive coupling, on the other hand, let the smallest detectable displacement be dmin . If we model the coupling as a parallel plate capacitor (say between the gate electrode of an SET and a metal film on the surface of a beam), then dmin ∝ A , Cmin (2.78) where A is the area of the side of the gate electrode and the side of the metal film and Cmin the minimum detectable change in capacitance. If A = lhf , that is, the product of the length of the beam and of the thickness of the metal film, then this also scales favorably with diminishing length, but slower than in the piezoelectric case. While this does not compare the absolute value of sensitivity, it demonstrates how piezoelectric coupling has its sensitivity raised faster than capacitive coupling with shrinking dimensions. As mentioned earlier, the absolute sensitivity if the SET-based piezoelectric detection scheme is already expected to be smaller than the best result obtained to date using capacitive coupling [10, 7]. Chapter 3 Design and Simulation of Heterostructure As part of this project, a 2DEG heterostructure was designed and simulated so that suspended resonators with embedded 2DEG may be built. This is similar to the work done or proposed in references [48, 49, 50, 10], as discussed in chapter 2. In this chapter, a review of the design process for the heterostructure will be made including any simulations that helped refine the design and its actual composition. 70 CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 3.1 71 Design The design of the device proposed in [10] had to respect three conditions : (1) ability to make suspended nanoresonators (2) presence of a 2DEG extending in both suspended parts and as-is parts of the wafer (3) use of piezoelectric materials. With the knowledge described in chapter 2, a design was conceived that would fulfill all conditions. The choice of an GaAs/AlGaAs heterostructure was an evident one, for it can provide a 2DEG while both intrinsic materials are piezoelectric. In order to suspend a beam, a sacrificial layer had to be included. To determine which material that layer would be, as opposed to the material of the beam itself, one has to consider which etchant will be used to remove it. This etchant needs to etch the sacrificial layer but not the remaining layers. The choice was made to go with hydrofluoric acid (HF) as an etchant, as it provides sufficient selectivity between low percentage and high percentage aluminum in AlGaAs layers [48, 49, 60] (see chapter 4). It was therefore decided that the AlGaAs layers surrounding the 2DEG would be composed of Al0.3 Ga0.7 As while the sacrificial layer would be made of Al0.7 Ga0.3 As, the latter being etched much faster in HF than the former. 5 nm thick GaAs cap layers would go both on top of the beam and on the bottom (i.e. just on top of the sacrificial layer) to prevent oxidation of the AlGaAs layers, as in the case of [48, 49]. Carriers would be provided by delta doping (n-type) the AlGaAs layers surrounding the 2DEG, CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 72 5 nm GaAs cap layer 30 nm undoped Al0.3 Ga0.7As 77 nm 2DEG 164 nm 2 nm δ-doped Al0.3 Ga0.7 As 40 nm undoped Al0.3 Ga0.7As 10 nm undoped GaAs 40 nm undoped Al0.3 Ga0.7As 600 nm 2 nm δ-doped Al0.3 Ga0.7 As 30 nm undoped Al0.3 Ga0.7As 5 nm LT-GaAs cap layer 600 nm Al0.7 Ga0.3As sacrificial layer GaAs substrate Figure 3.1: Diagram of the designed 2DEG heterostructure for applications to suspended structures. The doping used in the donor layers was 1.5 × 1019 cm−3 . using the modulation doping technique. The decision was made to use a quantum well in making a 2DEG for reasons of symmetry. Having AlGaAs/GaAs interfaces and identical doping layers on both sides of the 2DEG layer produces symmetrical band bending resulting in a symmetric quantum well, as opposed an asymmetric one as seen in chapter 2, figure 2.17. See figure 3.1 for complete details on the final heterostructure design. CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 3.2 73 Simulations To ensure that the design was sound, two types of simulations were done before the crystal was grown. First, simulations determining the conduction band profile of the heterostructure and the carrier density in the different layers were done. Then, to verify that the actuation method was sound, finite-element modeling (FEM) simulations of vibrating beams made out of the heterostructure were made. These mechanical simulations allowed us to see what fundamental frequency of vibration to expect with a beam covered with metal electrodes, giving a slightly different result than the simplified Euler-Bernoulli theory. 3.2.1 Simulations of Electronic Properties Self-consistent Poisson-Schrödinger simulations using Greg Snider’s 1DPoisson program [61, 62, 63] were made of the heterostructure. These calculations solve Poisson’s equation (relating the electrostatic potential to the charge density) and Schrödinger’s equation in a self-consistent manner. Before discussing the results, I will briefly explain how the two equations interact and how the simulations are performed. The simulations were made in one dimension and so we may write Schrödinger’s equation as 2 ~ d 1 d − + V (z) ψj (z) = Ej ψj (z), 2 dz m∗ (z) dz (3.1) CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 74 where z is the coordinate representing the depth in the heterostructure, m∗ (z) is the effective mass of electrons, Ej the energy and ψj (z) the wavefunction. The chosen potential energy is V (z) = −qφ(z) + ∆Ec (z), where q is the electronic charge, φ(z) is the electrostatic potential and ∆Ec (z) is a pseudopotential correction introduced by Snider et al. to account for the band offset at the interface of different materials [61]. To relate all of this to Poisson’s equation, we must find the charge density. To start with, the occupation number of state j near T = 0 K is given by the Fermi-Dirac distribution : m∗ (z) nj (z) = π~2 Z ∞ Ej 1 1 + e(E−EF )/kB T dE. (3.2) Now we can get the density of electrons by computing n(z) = s X j=1 |ψj (z)|2 nj (z), (3.3) where s is the number of bound states. Finally, the charge density will simply by given by the density of ionized donors, ND (z) (which are positive), subtracted by the electron density : ρ(z) = q (ND (z) − n(z)). Thus, within materials, Poisson’s equation takes the form of d dz d ξ(z) dz φ(z) = − q (ND (z) − n(z)) , ξ0 (3.4) where ξ(z) and ξ0 are the material permittivity and the permittivity of vacuum respectively. CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 75 The algorithm used in the simulation is the following (note here that the subscript indicates iteration number and not energy level) [61]: 1. Solve Schrödinger’s equation with a trial potential φi (z) giving an initial value of Vi (z) for potential energy. 2. Compute ni (z) using the obtained eigenenergies and wavefunctions from step 1. 3. i → i + 1. Calculate φi (z) from Poisson’s equation using ni−1 (z) found at step 2. 4. Insert φi (z) into a new expression for potential energy Vi (z). 5. If |φi (z) − φi−1 (z)| < α ∀z, terminate (where α is some error criteria). Else, return to step 1. Note that this procedure does not always converge. The numerical calculations are made using the finite difference method, in which the heterostructure is meshed into multiple small regions. If desired, the mesh size can be varied in different sections of the heterostructure, but the total number of elements can never exceed 1000 in 1DPoisson [61]. Boundary conditions must be applied at the surface of the heterostructure and and the interface with the substrate. For our purposes, a Schottky barrier of 0.6 eV is applied at the surface to account for the effect of surface states in GaAs [61, CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 76 64, 65]. At the substrate, the only boundary condition that is required is that the slope of the conduction band goes to zero, as the material remains the same and is far from the doped layers. The remaining inputs to the simulation are the layers (thickness and material), the doping concentration and type, the temperature and mesh size. The simulations were conducted at a temperature of 1 K, adjusting the doping concentration and geometry to produce a 2DEG with an electron density of ≈ 5 × 1011 cm−2 . Results At first, it was seen that a second 2DEG was forming at the interface of the sacrificial layer and the adjacent GaAs cap layer (at around 160 nm deep). This was not optimal since it meant having two layers of 2DEG outside regions where the wafer would be suspended; potentially having some transport effects between the two. Thus Zbig Wasilewski at NRC in Ottawa suggested that we make the lower GaAs cap layer a low-temperature grown layer. This made the layer non-conductive because of the many defects introduced in the crystal. This eliminates the lower 2DEG while maintaining a protective layer for the AlGaAs layer above. Figures 3.2 and 3.3 show plots of the conduction band in the as-is (not suspended) case and the suspended case, respectively. In the suspended heterostructure, an electron density of 4.750 × 1011 cm−2 is calculated for the 2DEG layer and 5.238 × 1011 cm−2 is calculated for the CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 77 non-suspended 2DEG. 3.2.2 Mechanics simulations Finite element models of a beam made out of the heterostructure were built and simulated using the commercial software package CoventorWare [67]. These simulations allowed us in first place to see the effect of gold electrodes would have on the vibrational properties of the beam, something not accounted for by Euler-Bernoulli theory. Secondly, we could verify that the actuation mechanism worked well by orienting the beam in the chosen crystallographic orientation and simulating a RF voltage signal applied on an actuation electrode with the 2DEG layer providing a ground plane. As the crystallographic orientations are fixed in CoventorWare, all of the material properties tensors had to be rotated in order to simulate a beam oriented in one of the h011i directions. With a target of designing a beam with a fundamental frequency of 1 GHz, the appropriate dimensions were calculated using the Euler-Bernoulli formalism giving l = 836 nm, w = 250 nm. The thickness of the heterostructure is h = 164 nm. The 230 nm-long gold electrodes were modeled to be 25 nm thick and were positioned at both beam ends, their length being determined by the optimal placement found in [10]. By a modal analysis, the fundamental mode is found to be out-of-plane flexural at CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 78 0.8 Conduction band energy (eV) 0.7 Conduction band energy (eV) 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0 100 200 300 400 500 Distance from surface (nm) 600 700 Figure 3.2: Plot of the conduction band plotted against depth in the heterostructure as simulated in 1DPoisson [61] in the nonsuspended case. The Fermi level is set at zero. Near the surface, the conduction band boundary condition was set to a 0.6 eV Schottky barrier to account for the effect of surface states according to the numbers found in literature for GaAs [61, 64, 65]. A second well is seen just above the sacrificial layer, however, this must be considered as non-conducting since a low-temperature GaAs was grown there. An electron density of 5.238 × 1011 cm−2 is obtained in the 2DEG layer. 79 CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 0.8 Conduction band energy (eV) 0.7 Conduction band energy (eV) 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0 20 40 60 80 100 Distance from surface (nm) 120 140 160 Figure 3.3: Plot of the conduction band plotted against depth in the heterostructure as simulated in 1DPoisson [61] for the suspended case. The Fermi level is set at zero. The conduction band boundary conditions is set to 0.6 eV Schottky barriers at the surface, to account for the exposed GaAs cap layer [61, 64, 65]. The bottom layer, however, is low-temperature GaAs and has was set to 0.47 eV as per the numbers found in reference [66]. The 2DEG layer has an electron density of 4.750 × 1011 cm−2 . CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 80 the frequency of 925.6 MHz, representing a slight deviation from the targeted 1 GHz. The second is found to be in-plane transverse and the third torsional, however these do not couple to the piezoelectric actuation scheme. A transient analysis with the beam actuated from rest by a 3 V amplitude RF voltage signal of frequency corresponding to the fundamental mode on the actuation electrode verifies that the actuation mechanism is working as predicted, as shown in figure 3.4. A steady-state, harmonic simulation for a Q ≈ 100 at resonance gives a maximum displacement of 0.26 nm at the beam’s midpoint. CHAPTER 3. DESIGN AND SIMULATION OF HETEROSTRUCTURE 81 Figure 3.4: Mosaic depicting the shape of a beam in time with the beam starting from rest and moving under the voltage applied to the actuation electrode on the right end. The time interval between each snapshot is ≈ 0.2 ns and the color bar gives the potential (blue is −3 V, red is +3 V). a) Positive voltage is applied to the beam at rest. The beam start to bend upwards. b) Beam has bent upwards while potential is reducing. c) Beam center reaches its apex; the voltage is near zero, becoming negative. d) Beam returns to its equilibrium position under the influence of a negative voltage. e) Voltage starts back once again towards zero; the beam moves down. f ) After the beam center reaches its minimum, the beam start moving up again under positive voltage. Chapter 4 Fabrication This chapter will present a survey of the fabrication process for making Hall bars and suspended beams using the designed heterostructure, which was discussed in chapter 3. Topics will range from the design of patterns to electron beam lithography, evaporation, etching and annealing of ohmic contacts to the embedded 2DEG. Appendix D examines safety issues encountered while performing fabrication. Several fabrication recipes were developed and tested over the duration of this work. Many of them did not produce the expected results. The most successful ones will be discussed here but also the biggest “pitfalls”. To start off, an explanation of the requirements for making a suspended structure with integrated 2DEG is made. In order to make useful suspended beams, several elements have to be in place. First a mesa has to be etched in the wafer, that is, 82 CHAPTER 4. FABRICATION 83 a relief created in the shape of the beam and the contacts by removing material all around. This is accomplished by depositing an etch mask over the desired region, preventing that surface from being washed away during an etching process. It is also desirable that this mask may be removed in the end leaving a clean surface over the device. Ohmic contacts must also be provided as a means of passing current though the 2DEG, but since the contacts’ material is not the same as the etch mask, these two patterns must be applied separately. Therefore use of a pattern alignment technique is made to align a pattern with its contacts, or vice-versa. Finally, one must remove the material under the beam in a step called sacrificial etch, detailed later on. 4.1 Wafer The wafer used was grown at the National Research Council in Ottawa by Zbig Wasilewski, to whom I am very grateful. The design of the crystal structure is discussed extensively in chapter 3. Recall that the wafer is constituted of a GaAs substrate covered by a Al0.7 Ga0.3 As sacrificial layer and topped with the Al0.3 Ga0.7 As/ GaAs heterostructure, where a 2DEG is formed. The surface of the wafer is GaAs and the wafer flats indicated a (100) GaAs wafer (see figure 2.9). For all practical purposes, the electron beam lithography instructions should apply CHAPTER 4. FABRICATION 84 to any patterns a user wishes to make on a GaAs wafer; however the numerous etch steps and annealing step remain relevant to the case of our particular AlGaAs/GaAs heterostructure. 4.2 Ohmic Contacts To connect a 2DEG to wiring and instruments, ohmic contacts must be made. These are metal layers that, once annealed, diffuse down into the heterostructure by forming spikes and thus provide electrical contact to the 2DEG from the surface of the wafer [68]. An often used alloy is that of nickel, germanium and gold. Successive layers of these elements can be heated in a rapid thermal annealer to make an ohmic electrical contact to a 2DEG, i.e. the current-voltage behavior is linear and passing through the origin. The recipe, suggested by Guy Austing, is to evaporate 25 nm of nickel, 55 nm of germanium and 80 nm of gold. Then, anneal in a rapid thermal annealer (RTA) at 415 ◦ C for 15 seconds in a N2 (95%) H2 (5%) atmosphere (a mixture called forming gas). It was found that a pure N2 atmosphere worked too, as long as sufficient time and flow was given to remove oxygen from the chamber. Through experimentation, it was determined that heating wafers up to 415 ◦ C in a convection oven (much slower heating rate than a RTA) while flowing forming gas does not produce ohmic contacts. CHAPTER 4. FABRICATION 4.3 85 Design of Patterns Design of electron beam lithography patterns is made using a commercial pattern generation software and hardware package called NPGS, or Nano Pattern Generation System [69]. Patterns first have to be drawn in a CAD program, organized by layers and entities ordered for patterning. The pattern file defines the geometry, the different layers, and the number of different dosages desired. A run file, created from the pattern file, will further specify the magnification and probe current used in each of the layers, the value of the dosages, the spacing between two exposure points, the spacing between two lines and other less important parameters. NPGS does not take the proximity effect into account (by which structures in close proximity effectively will have their dosages increased) so when designing patterns with large, prevalent and intricate geometry, the dosages will need to be modified from another “roomier” pattern. Figure 4.1 shows an example of how the proximity effect deforms the ends of a several beam patterns in close proximity. 4.4 Electron Beam Lithography Electron beam lithography (EBL) uses the focused electron beam of a scanning electron microscope to make patterns on top of a resist-coated wafer. This particular CHAPTER 4. FABRICATION 86 Figure 4.1: SEM picture of an array of four beam patterns in close proximity. The beam ends show a curvature that was not defined in the original pattern design and is caused by the proximity effect. CHAPTER 4. FABRICATION 87 technique is used because it is well-known and provides the means for making geometrical shapes with 10 nm resolution in a reliable manner. Line widths of 70 nm or less are not uncommon with our particular equipment. The working principle of EBL is based on polymers (EBL resists) that break down when sufficiently exposed to an electron beam, making them soluble in certain solvents. Because the solvent selectively dissolves the exposed regions of EBL resist layers, it is called a developer. This effectively cuts a window in the shape of the exposed region in the EBL resist layer. 4.4.1 Spin-coating of Resist Layers The process is as follows. Firstly, the wafer must be spin coated with EBL resist layers, namely PMGI and then PMMA [70, 71]. The reasons for using two layers of resist will become evident later on. To that end, the wafer is cleaned in deionized water, followed by a dip in isopropanol and baked using a hotplate at 180 ◦ C for five minutes in order to drive off moisture. The wafer is then held in a vacuum chuck in the spin-coater. To ensure that the whole surface gets coated in the most even fashion, liquid PMGI SF9 (9 % in cyclopentanone) is laid over all the surface of the wafer. Spinning occurs at 4000 RPM for 45 seconds, yielding a resist thickness of 500 nm (film thickness spin-curves are available from [71]). Hard baking for 7 minutes at CHAPTER 4. FABRICATION 88 180 ◦ C follows. Liquid PMMA (3% by weight in anisole) is placed over the surface of the wafer. The spin coating occurs using the exact same parameters as before, this time for a thickness of 120 nm [71]. Next, a hardbake for 10 minutes at 180 ◦ C in done. The wafer is then cleaved into small enough samples if necessary (usually, ≈ 7 mm by 5 mm) using a diamond-tipped scribe tool. Giving the chips a rectangular shape is desirable since it allows us to keep track of crystal orientations using the original wafer’s flats. If for some reason the coat obtained is not uniform, it may be necessary to remove the layer entirely. For example a dust particle may have been present, creating a streak in the resist layer. Uniform layers of resist are a requirement for precision electron beam lithography. PMMA can be removed using an acetone dip, while PMGI can be removed in Microchem’s proprietary solvent Nano Remover PG [71]. Also worth noting is the fact that because of its surface tension the resist forms bumps on the contour of the wafer, a so-called edge bead. As such the edges of a wafer may not be suitable for lithography. Thus, if possible, it is best to coat a whole wafer first, and then cleave it. If constrained to spin coating small samples, a good technique for reducing edge bead is to use a vacuum chuck with an off-center hole. This way the resist will be pushed mainly to one side of the sample, leaving most of its sides bead free. CHAPTER 4. FABRICATION 89 Table 4.1: Electron beam lithography settings. All patterning is done at an accelerating voltage of 40 kV. Magnification refers to the magnification factor. The feature size is usually the desired width of the smallest feature (e.g. the width of a beam). The center-to-center distance represents the spacing between the two centers of exposure points. The line spacing is the spacing between two lines of exposure. Offset is the pattern origin offset needed for good alignment between patterns at high magnification and patterns at low magnification. Magnification 1000 1000 1000 1000 200 110 Feature Size ≤ 200 nm ≈ 500 nm ≥ 1 µm ≥ 3 µm large large Dose (µC/cm2 ) ≥ 600 500 450 450 450 450 Current (pA) 10 10 10 30 500 ≥ 2000 Center-to-center (nm) 4.35 4.35 4.35 10.15 21.74 26.36 Line Spacing (nm) 4.35 4.35 4.35 10.15 21.74 26.36 Offset (µm) 0,0 0,0 0,0 0,0 -2,-2 -2,-2 4.4.2 Patterning The actual patterning is accomplished in a JEOL JSM-6400 SEM [72] retrofitted with an ion pump, LaB6 filament, computer-controlled beam blanker and NPGS. After attaching the sample to the sample mount a small scratch is made at one corner of the sample (by convention the lower left corner), as it will be used to obtain a good focus on the sample surface. The electron beam reacts with both the PMMA layer and the PMGI layer, therefore, to minimize unwanted exposure, one should sneak up on the scratch without passing over any regions where patterns are to be made. Once good focus is obtained, the beam should be blanked using the probe current detector (PCD) and the wafer moved so that the beam will be over an unexposed region. The PCD is a cup placed in the way of the beam that allows us to measure CHAPTER 4. FABRICATION 90 the beam current. With it, the current may be set to the desired value for the next lithographic layer. The magnification also needs to be manually set to the correct value, after which the beam blanker can be switched to external control, the PCD removed and the NPGS patterning routine launched. The specifics of the patterning settings are written in table 4.1, but further explanations are in order. A magnification factor of 1000 times is always used to pattern the smallest features, as the SEM activates fewer magnification circuits than at a magnification factor that is not a power of ten. This leads to reduced noise in the electron beam. Besides magnification, there are five other important parameters while patterning. These are dose, beam current, center-to-center distance and line spacing and origin offset. The dose specifies how much charge per area or per line the sample receives. Usually, the smaller the feature size is the higher the dose needs to be, because inside large patterns the proximity effect is more pronounced. For an area dose, it is measured in NPGS in µC/cm2 . The beam current will determine the amount of charge per unit of time hitting the sample. As such, for a given area and a given dose, a higher current will allow the patterning to finish sooner. In NPGS, the default unit for current is pA. The current can be measured by activating the PCD, which is connected to an ammeter. There are limitations, however, to using high current. Typically, a current no higher CHAPTER 4. FABRICATION 91 than 2000 pA is used, as greater currents can be difficult from time to time to obtain. Changing the current inside a patterning layer at a constant magnification will likely produce a small shift in the patterning. This brings us to origin offset. With a bit of testing, the offset needed to keep patterns aligned between the highest magnification layers and the lowest magnification layers can be found. The actual patterning proceeds in a point by point fashion. Lines are completed by a succession of exposure points and areas are swept by a succession of lines. Centerto-center distance allows the user to control the distance between two points, and line spacing the distance between two lines. This means that if sharp angles are desired in small structures, these values should be made small, on the order of 5 nm. For very large area however, they need not be so low and distances of 20 nm or more work well. 4.4.3 Pattern Developing Developing begins by dipping the sample in MIBK:isopropanol in a 1:3 ratio [71] by volume for one minute, then rinsed in isopropanol and dried with N2 . This first chemical removes the exposed PMMA. A color change can be seen under an optical microscope. Then, a 90 seconds dip in PG 101 or XP 101 [71], both PMGI developers, is done with the rinse being deionized water followed by isopropanol. The sample is CHAPTER 4. FABRICATION 92 dried with N2 as usual. This removes the exposed PMGI, leaving the wafer surface exposed where patterns were drawn. Next, a 5 seconds dip in MF319 [71] is made (rinse and dry procedure unchanged). This creates undercut, i.e. it removes some of the unexposed PMGI, “undercutting” the PMMA (see figure 4.2). Thus, when evaporating materials, the PMMA layer provides the mask while the PMGI will be used to lift off the excess material on top of the wafer. This is possible because of the undercut: it leaves a space for the Nano Remover PG to come in contact with the PMGI and lift off everything that was deposited on top of the remaining resist. The undercut of the PMGI is also essential to avoid tearing, ripping or shredding of the deposited film when lifting-off [71]. A note about the develop times. The times mentioned are appropriate for specific dosages and coats of resist (optimal dosage is particularly sensitive to bake time and spin speeds as well as feature size). They may need to be changed if some parameters were off, often indicated by abnormal shapes or colors when observing the sample under the microscope. One also has to be careful not to overdevelop. This is particularly true of the MF319 dip, as it removes unexposed PMGI and can cause a too large undercut leading to a collapse of the PMMA mask. CHAPTER 4. FABRICATION 4.4.4 93 Evaporation We used physical vapor deposition, a process by which metals are heated in high vacuum to evaporate and deposit on a relatively cooler wafer placed over the metal source. We are equipped with a dual source evaporator. The first is a standard, twomaterials, thermal evaporator in which the heat is provided resistively. It was mainly used to practice fabrication until the second source could be brought to operational status. This second source is a 3 kW Thermionics RC-series 3-crucibles linear electron beam evaporator [73]. Heating is caused by an electron beam directed at the materials. This source was the one primarily used for fabrication of our devices since the ohmic contacts to our 2DEG samples required three materials, namely nickel, germanium, and gold. The materials should be heated slowly and care should be taken that the electron beam does not veer off the materials to hit the crucibles nor the evaporator’s copper structure. When the desired evaporation speed is reached (say one to three Angstroms per second), the material is left to evaporate for approximately a minute to outgas any impurity and only then is the shutter opened. Once the target film thickness is reached the shutter is closed and the evaporator turned off and left to cool for at least ten minutes to ensure minimal oxidation once air is allowed back in to the bell jar, after venting with nitrogen. CHAPTER 4. FABRICATION 4.4.5 94 Lift-off When evaporation is done, the remaining resist on the wafer must be removed, leaving behind only metal in the shape of the desired pattern on the wafer’s surface. By dipping the wafer in an appropriate solvent, the PMGI is attacked and the PMMA and the metal adherent to it “lift-off”. The samples are put in a proprietary PMGI solvent, Nano Remover PG [71] heated at 78 ◦ C. They are left for several minutes with occasional stirring to ensure clean lift-off of the PMGI. In some cases, it may help to put the beaker in an ultrasound bath for a few seconds. This particularly helps in lifting-off nickel and aluminum films. Titanium, gold and germanium are relatively easy to lift-off and do not generally require this. To finish, the sample are rinsed in deionized water, followed by isopropanol and dried with nitrogen. This concludes the process of electron beam lithography (see figure 4.2 for a schematic overview). 4.4.6 Pattern Alignment Devices requiring layers of different metals to be evaporated will need multiple steps of EBL. To ensure that the subsequent patterns are aligned to the previous ones, one has to use the SEM to image patterns already on the wafer. In general, one should refer to the NPGS manual [69] for complete instructions, but a short explanation will 95 CHAPTER 4. FABRICATION e− 1111111111 0000000000 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 a) Spin-coating b) Exposure in SEM 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 c) PMMA developing 1111111111 0000000000 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 d) PMGI developing and undercut 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 e) Evaporation 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 f) Lift-off hν hν hν hν hν 1111111111 0000000000 0000000000 1111111111 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 1) Annealing T = 415 C, 15 s PMMA PMGI GaAs substrate 00 11 00 11 1111111111 0000000000 00 11 0000000000 1111111111 2) RIE or wet etch 11 00 00 11 00 11 3) Mask removal, HF suspension 00 11 00 Heterostructure 11 Exposed PMMA 00 11 00 11 Exposed PMGI 00 Sacrificial layer 11 Evaporated metal film (mask or contact) Figure 4.2: Diagram of a complete fabrication process. a) PMGI covered by PMMA are spin-coated on the wafer. b) Patterned is exposed in a SEM. c) Pattern is developed in a solution of MIBK:isopropanol in a 1:3 ratio. This develops the top layer of PMMA. d) The exposed PMGI on the bottom layer is removed under the opening made in the PMMA by use of a PMGI developer (XP101). Further ‘undercut’ is obtained by a dip in MF319. e) Metal film is evaporated (physical vapor deposition using an electron beam evaporator). f ) The remaining resist is lifted off by a proprietary solvent (Nano Remover PG). Two choices are available for the rest of the process. I. If RIE is used to create the mesa : 1) Evaporated Ni/Ge/Au ohmic contacts are first annealed at 415◦ C for 15 seconds. 2) A 60 nm thick Ni mask is applied and the mesa is created by RIE in BCl3 gas. 3) Finally the Ni mask is removed and the sacrificial layer removed by a solution of HF. II. If liquid etching is used to prepare the mesa : 2) The mesa is protected by a metal mask (Ti or Ni) and etched in a citric acid/hydrogen peroxide mixture. 1) The contacts are evaporated and annealed. 3) Mask is removed and removal of the sacrificial layer is made by dipping in HF. CHAPTER 4. FABRICATION 96 follow. For example, pattern A is first patterned, developed and a nickel mask is evaporated. Some processing is done on the wafer and the nickel mask is removed. The wafer receives a fresh coat of electron beam resist. Pattern B (ohmic contacts) must then be aligned to pattern A, for it consists a different material, namely an alloy of nickel, germanium and gold. The example of pattern A and pattern B will be kept in this section in order to explain the process of pattern alignment which is broken down into two steps. The first, coarse alignment, consists of positioning certain features of pattern A under alignment windows. These windows are regions of the field of view that will be scanned by the SEM. The coarse alignment is done by using the SEM’s mechanical stage (x-y translation and rotation). Additionally, the beam rotation knob on the SEM can be used which causes the image to be rotated (internally in the SEM’s circuits). It is possible to do coarse alignment without ever looking at pattern A. One only has to plan ahead when patterning A and pattern an alignment cross some distance from A. This cross is no more than a large “+” sign made to be easy to find when imaging. When doing coarse alignment to B, if the position of the cross relative to A was recorded, the user only has to creep over to the cross, using high current if necessary to see it, and center it on screen. The cross should also be kept level by rotating the stage (preferably) or the image. The electron beam can then be blanked CHAPTER 4. FABRICATION 97 and the necessary relative displacement entered in the mechanical stage’s computer interface. This procedure usually works very well, and will provide a coarse alignment to within ≈ 10 µm − 50 µm of the desired position, without even having to expose any part of pattern A. Once the coarse alignment is finished, one then proceeds with fine alignment. NPGS provides a graphical interface which allows the user to move virtual alignment marks over corresponding features in the image of pattern A using the computer’s mouse. NPGS then calculates a transformation matrix based of the displacement of these virtual alignment marks that will be applied to the new pattern (B). The suggested shape of these marks is that of an “L”. Figure 4.3 presents a schematic of the fine alignment step. The fine alignment step must be carried out at the same magnification and current that B is to be patterned with, because any change in these parameters could induce a shift in image position or rotation. However, the center-to-center distance and line spacing can be made much larger in order to minimize the exposure of the area under the alignment windows. The corrections made by NPGS at the fine alignment step have to stay within the field of view, or else and error will be produced. For this reason, it is best to use the mechanical stage as much as possible to align the features of A to the virtual alignment marks in order to minimize the NPGS’s corrections. CHAPTER 4. FABRICATION 98 Figure 4.3: Schematic view of the fine alignment process. In the drawing, coarse alignment is already done and four ohmic contacts with outgoing electric leads are visible in dark gray under the four alignment windows. The black represents the area of the field of view that is not scanned by the SEM and includes portions of pattern A which are not desirable to be exposed. Finally, the “L”-shaped polygons are positioned over their corresponding squares of pattern A with the computer mouse in NPGS [69]. CHAPTER 4. FABRICATION 99 To conclude the example of patterns A and B, the shapes of the ohmic contacts may once again by patterned over the corresponding squares of pattern A (which defined a ‘mesa’). The Ni, Ge and Au are evaporated and the wafer annealed, making ohmic contacts. 4.5 Wet Etching Wet etching is done by dipping the sample in a previously prepared etchant solution for a controlled amount of time. The sample is then dipped in an etch stop solution, often deionized water. This is followed by drying with N2 . In the case of a fragile suspended structure, a critical point dryer may be used, as the surface tension of the etchant or the rinsing liquid can break a suspended beam (or any other structure with a lot of exposed surface area). This allows the sample to be dried in supercritical carbon dioxide by avoiding the liquid to gas phase change. The chemicals used were almost exclusively obtained from Sigma-Aldrich, Transene and Acros Organics USA [74, 75, 76]. 4.5.1 Etching of the Wafer The original mesa etch was done in a citric acid and hydrogen peroxide mixture. This solution was chosen for two reasons : one is the availability of reliable information on this etchant, the other is that the etch rates of both GaAs and Al0.3 Ga0.7 As can be CHAPTER 4. FABRICATION 100 made close by controlling the volume ratio citric acid to hydrogen peroxide [77]. However, it was found that this etchant was not an optimal solution. Tested with both titanium and nickel masks, it attacked the mesa in several places, leaving holes in the mesa (see figure 4.4). These are likely caused by porous regions in the mask. Simply applying titanium over a whole wafer and stripping it off with HF did not produce these holes, therefore the likely culprit is the citric acid / H2 O2 based solution. It is also known that HF should not attack the GaAs surface, nor the heterostructure. This undesirable etching action occurred invariably with use of a titanium mask up to 50 nm thick, though increasing the mask thickness and applying a thin layer of gold (5 nm) on top reduced the occurrences of holes in the mesa. There is a limit to this however, as the Ti mask must be removed by a dip in HF. If the mask is too resilient, its removal in HF slows considerably and the sacrificial layer of the wafer is attacked to a too great extent, too soon in the fabrication process. Using a nickel mask has the advantage that a selective etchant can be used for its removal, so that the sacrificial layer is not attacked. However, even with a 60 nm thick Ni mask, holes were still observed in the mesa. It was thought that pores in metal masks may be caused by impurities left on the wafer’s surface prior to evaporation. However, improved cleaning procedures prior to evaporation did not eliminate the holes in the mesa. The procedures included CHAPTER 4. FABRICATION 101 Figure 4.4: SEM picture showing the holes etched unintentionally in the mesa of a large Hall bar structure that was protected by a titanium mask. cleaning the wafer in isopropanol and deionized water, dehydrating on a hotplate and blowing clean with nitrogen just prior to evaporating. A dry plasma cleaning process may improve the cleanliness of the wafer’s surface by removing any residual PMGI that was left after developing. Secondly, with the citric acid / H2 O2 mixture, the etch of the material was not uniform so that small structures were produced with uneven features, as seen in figure 4.5. This meant that, realistically, it could only be used to produce large structures with feature size of a micron or more. Finally, the etch was much too slow to produce structures of specific width reliably: the etch depth has to be greater or equal to 164 nm, meaning that a beam is etched sideways for ≈ 328 nm if the etch CHAPTER 4. FABRICATION 102 Figure 4.5: SEM picture showing the poor shape resulting from the definition of a mesa with the citric acid / hydrogen peroxide etchant for a ≈ 500 nm beam. The titanium mask was of the shape of a rectangular beam and was removed in HF. is completely isotropic. This is not always the case, as the different crystal planes of GaAs and AlGaAs present different etch rates. As such, it is even hard to control the width of the structure precisely using wet etch. Therefore it is advisable to use a dry etch technique for mesa definition. Nevertheless, this etchant mixture can still be successfully used to make mesa structures of large size. For example, sample A, discussed in chapter 5, was fabricated using citric acid and hydrogen peroxide. The recipe goes as follows and is adapted from reference [77]: 1. Prepare 50 % citric acid by mixing 1 g of citric acid to 1 mL of deionized water. CHAPTER 4. FABRICATION 103 The dissolution is an endothermic reaction and in consequence is quite slow. It is recommended that a magnetic stirrer be used and the solution be left stirring for at least 15 minutes after the citric acid has visibly dissolved. 2. The hydrogen peroxide solution used is of 30 % concentration and in our case was bought from Sigma-Aldrich [74]. H2 O2 degrades into water and dioxygen. To slow down this reaction, the solution contains a stabilizing agent and is kept in a refrigerator. 3. Mix the 50 % citric acid with the 30 % hydrogen peroxide solution in a Vcitric /VH2 O2 = 1/20 volume ratio (this makes the etch rates relatively the same between all materials of the heterostrucure). This should be done right before etching, so as to prepare a fresh mixture. 4. Etch wafer in mixture, typically from 3 1/2 minutes to 4 1/2 minutes. Successive color changes from green to red, then back to green should be visible. When the colors stop changing, it presents a good indication that the etchant has reached the sacrificial layer (which is quite thick compared to the heterostructure) and that the mesa is now formed. 5. Rinse in deionized water. Note that if wet etching is used to define the mesa, then ohmic contacts must not be patterned and annealed first. They must be deposited and annealed only once CHAPTER 4. FABRICATION 104 Figure 4.6: SEM picture showing the undercut made by etching the Al0.7 Ga0.3 As sacrificial layer for 1 minute in a 5% HF solution. The heterostructure being somewhat transparent at 20 kV accelerating voltage, we are able to see the shape of the sacrificial layer underneath. The sacrificial layer is slightly over-etched, as indicated by the two ends of the beam being just above and below the remaining Al0.7 Ga0.3 As support. the mesa is defined. If they are not, the liquid etchant tends to etch trenches in the mesa where it connects to the ohmic contact pads. This causes the pads to become completely disconnected from the mesa, making the contacts useless. The other wet etch step of the wafer is the HF dip for sacrificial layer etch. By submerging the samples in a 5% HF solution for 1 minute, a 1 µm wide beam can be suspended, as shown in figure 4.6. The etching of the sacrificial layer is a very time-dependent etch. If the sample is left too long under the etchant, more than the beam will be suspended, meaning CHAPTER 4. FABRICATION 105 that more than the beam will move under strain. One can expect that the mechanical response (hence the piezoelectric response as well) will be very different from that of a simple doubly-clamped beam. If the sample is not etched long enough, then of course the beam will not be suspended. It is also worth mentioning that, in truth, not only the sacrificial Al0.7 Ga0.3 As layer is attacked but also the intermediate Al0.3 Ga0.7 As layers, albeit much slower. For 10% HF at room temperature, the etch rate is of ≈ 500 nm/min for Al0.7 Ga0.3 As while it is of ≈ 0.2 nm/min for Al0.3 Ga0.7 As [60]. 4.5.2 Mask Removal A titanium mask can be removed by simply dipping in 5% HF for a few seconds or until the mask is gone by visual inspection. This will also attack the sacrificial layer slightly. A nickel mask may be removed in the commercially available nickel etchant TFG [75]. This etchant works best when heated to 50 ◦ C or else the etch is too slow. It is safe to use on GaAs and AlGaAs, however after prolonged use (e.g. over 45 minutes in the room-temperature solution) is is seen to begin attacking the GaAs/AlGaAs heterostructure. An etch time of seven minutes at 50 ◦ C effectively removes a nickel mask 60 nm thick without visibly attacking the wafer. CHAPTER 4. FABRICATION 106 Figure 4.7: SEM picture showing mesa as defined by RIE in a BCl3 gas using a 60 nm thick nickel mask. The beams of this pattern are, in ascending order, ≈ 300 nm, ≈ 400 nm, ≈ 500 nm, ≈ 650 nm and ≈ 1.1 µm wide. 4.6 Reactive Ion Etching Reactive ion etching (RIE) was done on the samples by Guy Austing of the National Research Council in Ottawa. This produced the mesa structure, leaving the wafers ready for etching of the sacrificial layer after nickel mask removal. BCl3 plasma was used with the etch depth set to 1050-1100 nm. Titanium is not a suitable mask for use with this gas, as it gets attacked. The samples were fixed in the plasma chamber with oil and so had to cleaned with acetone, isopropanol and deionized water before the next fabrication steps. Figure 4.7 shows the mesa defined for beams of several widths by RIE in BCl3 . CHAPTER 4. FABRICATION 107 The bottom part of figure 4.2 illustrates the various steps following electron beam lithography that were just discussed. 4.7 Observation of Patterns and Structures For quick observation, samples are put under an optical microscope. However, as many of the structures made are very small, this is not always the most useful observation tool. Further observation of the fabricated devices can be done either in our SEM or our atomic force microscope (AFM). The SEM is used mainly to measure lengths and widths of non-static sensitive samples with NPGS. It is also equipped with a tilting stage which can be useful in taking measurements and pictures of a suspended structure. On the other hand, the AFM has a greater resolution that can be used to observe very fine features. Mesa heights are better determined by imaging in the AFM than in the SEM. Chapter 5 Experiments The experiments conducted to characterize different parameters of the heterostructure will be described in this chapter. Specifically, resistance and I-V behavior of contacts to the 2DEG, electron density and mobility in the 2DEG, both in a large Hall bar and small suspended beams, were tested. The apparatus used in these experiments in discussed in greater details in appendix A, while the software programs are listed in appendix B. 5.1 Testing of Contacts The contacts deposited on the wafer had to be tested to ensure that, once annealed, they allowed ohmic contact to the 2DEG. For every wafer this was an essential step, since ohmic contact to the 2DEG is never guaranteed. In particular, too strong 108 109 CHAPTER 5. EXPERIMENTS V Contacts, 2DEG I Figure 5.1: Circuit used for testing contacts to 2DEG. variations of temperature, time interval or atmosphere at the annealing step may all result in failed ohmic contacts. Testing of the contacts is done with the use of a probe station. Sharp needles are gently lowered onto pairs of contacts. These needles are connected in series to a current source and in parallel to a voltmeter. The circuit is shown in figure 5.1. All instruments relay measurements to a computer using GPIB bus connections. Generally, the current is varied from −1 µA to 1 µA in steps of 10 nA. Then direction of sweep is reversed to see if any hysteresis is present. A LabVIEW [78] program was developed to accomplish this. It increments the applied current between a start current and a stop current (using a Keithley 220 current source [79]) while recording the voltage drop across the needles (using an HP multimeter [80]). A real-time I-V plot is displayed, giving the user an opportunity to see if the contacts are behaving in an ohmic fashion. Ohmic behavior exists if the 110 CHAPTER 5. EXPERIMENTS curve is linear and goes through the origin. When the sweep is completed, a simple linear regression is applied to give an estimate of the combined resistance of the two contacts and the 2DEG between them. A typical trace of the I-V data acquired for a good ohmic contact can be seen in figure 5.2. −6 1 Typical I−V curve of Ni/Ge/Au ohmic contacts x 10 0.8 I = 0.0014898 V − 1.9514e−9 0.6 0.4 Current (A) 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −5 −4 −3 −2 −1 0 Voltage (V) 1 2 3 4 5 −4 x 10 Figure 5.2: A typical I-V trace obtained for Ni/Ge/Au ohmic contacts with the microscope light turned off. The contacts were square-shaped, ≈ 200 µm of side and ≈ 400 µm apart, center-to-center. As can be seen, the curve is quite linear, indicating that the contacts are ohmic for this current range. A linear fit gives an intercept of −1.95 × 10−9 A and an overall resistance of ≈ 671 Ω (the inverse of the slope). Error bars are shown but difficult to see on this scale. CHAPTER 5. EXPERIMENTS 111 As a rule of thumb, it can be seen if there is indeed conduction between the two contacts by looking at the voltage reading on the voltmeter which stabilizes at values lower than 40 µV when it is the case. Then there is a good chance that the needles are making proper electrical contact with the pads. In order for this to happen, enough downwards pressure has to be applied to the needle so that it starts to move laterally. However, too much pressure will lightly scratch the contact, or the needle may even slip off the contact and damage the wafer’s surface. 5.2 Testing of 2DEG Measurements of the electron mobility and density in the 2DEG were done using the quantum Hall effect using the theory discussed in chapter 2. This meant that, for some source-drain excitation, the longitudinal voltage VL , hall voltage VH and current I had to be recorded as a function of the magnetic field B, all of which can be done using a Hall bar pattern. These measurements were conducted first in a nonsuspended, large Hall bar (sample A), then in a small suspended Hall bar (sample B). 5.2.1 Procedures for Quantum Hall Effect Measurements In the first experiment performed on sample A, the circuit was setup as in figure 5.3. A 0.5 V sinusoidal AC excitation at 17 Hz was sent through a large 10.093 MΩ 112 CHAPTER 5. EXPERIMENTS Input B on SR530 SR830 10.093 MOhms Hall bar Sine Out 0.500 V, 17 Hz TTL Out on SR830 Input A on SR530 Reference on SR530 17 Hz Input A on SR830 Input B on SR830 Figure 5.3: Circuit used for magnetoresistance measurements on sample A. SR830 and SR850 refer to Stanford Research Systems lock-in amplifiers [81]. resistor and the source and drain leads of the Hall bar. Stanford Research Systems [81] lock-in amplifiers were used to measure longitudinal voltage drop (model SR830) and Hall voltage (model SR530). The SR530 was frequency-locked to the TTL signal from the SR830. The SR830 provided as well the AC excitation through its signal generator. The magnetic field was changed slowly using the superconducting magnet’s power supply (model CS4-10V) fabricated by Cryomagnetics [82]. At the same time, continuous measurements were taken using the lock-in amplifiers. All instruments I/O was handled by programs written for Matlab [83] over a National Instruments [78] GPIB interface. The sweeps were performed at rates of ≈ 0.002 T/s or ≈ 0.01 A/s for our particular magnet power supply/superconducting magnet combination. Usually, 113 CHAPTER 5. EXPERIMENTS Input A on 5206 Input B on 5206 SR830 SR 5113 voltage pre−amplifier 10.093 MOhms Suspended Hall bar Input A on SR830 Sine Out 0.100 V, 17 Hz TTL Out on SR830 17 Hz Reference on SR850 Reference on 5206 SR 5113 voltage pre−amplifier Input A on SR830 Figure 5.4: Circuit used for magnetoresistance measurements on sample B. SR830 and SR850 refer to Stanford Research Systems lock-in amplifiers [81]. On the other hand, SR 5113 refers to the 5113 model pre-amplifier from Signal Recovery [84]. 5206 refers to the lock-in amplifier model by EG&G, which has since been bought by Signal Recovery. data was taken from zero field to ±9 T and then swept back to zero. In the second experiment, in which a suspended Hall bar was used (sample B), the circuit is shown in figure 5.4, where the main differences are that the voltage across the resistor in series with the excitation was monitored and that pre-amplifiers were used for measuring VH and VL . The Hall bar used is shown later on in figure 5.11. A different strategy was used for changing the magnetic field. In lieu of slowly sweeping the magnetic field from 0 T to 9 T and continuously taking measurements, the field was varied in small increments to specific values (for example in steps of CHAPTER 5. EXPERIMENTS 114 0.05 T). This is better in some ways, as the temperature of the sample goes up because of eddy currents induced in the metal structure of the cryostat whenever the field is changing. Changing the field in small steps means that time can be allowed for the temperature to return to normal at each field value. In addition, this allows for a longer measuring time at each field value (it is a good idea to wait 5-7 time constants when measuring with lock-in amplifiers). However, sweeps made with this method are coarser (less data) or simply much longer than sweeps like the ones used in the first experiment. 5.2.2 Procedures for Illumination of 2DEG A variable DC voltage supply was used to slowly increase the voltage until current flowed through a standard red LED mounted over the sample. A resistor limited current through the LED. In the case of sample A, only the minimal voltage for current to flow was used as indicated by the voltage drop across the resistor. With a resistor of 297 Ω in series with the LED, one-second pulses of 2.1 V giving a current of 0.1 mA were made until longitudinal resistance in the Hall bar did not drop significantly further, at zero magnetic field. The resistance of the wiring (without current-limiting resistor) was measured to be ≈ 200 Ω at room temperature. The circuit is depicted in figure 5.5. For sample B, a slightly different procedure was used. A Matlab program was 115 CHAPTER 5. EXPERIMENTS V 297 Ohms sample 293 K 300 mK Figure 5.5: Circuit used for illumination of samples. A simple standard red LED was used. written that allows the user to specify the desired current through the LED, the number of pulses of light, the duration of pulses and the relaxation time between pulses. The program then increases the voltage supplied to the LED until the desired current is reached. For the duration of the program, the resistance of the sample is monitored with a lock-in amplifier and recorded. The wiring being different (same circuit, but shorter wires), it took only 0.9 V to get 1 mA flowing through the LED for the second experiment. 5.2.3 Results for a Large Unsuspended Hall Bar The first Hall bar pattern used in characterizing the 2DEG is pictured in figure 5.6, while the actual Hall bar (not suspended) used is shown in figure 5.7 with the size CHAPTER 5. EXPERIMENTS 116 Figure 5.6: The first pattern used in characterizing the 2DEG. The scale bar at the bottom indicates a millimeter. The source and drain pads are left and right while the remaining pads connected to thin transverse leads are used to measurement the longitudinal and transverse voltages. of the bar being indicated. The transverse leads were connected to large Ni/Ge/Au ohmic contacts (visible in light gray on the images) that were be wire-bonded to a circuit board. As this is a large structure, concerns about the electron depletion width and occasional defect in the mesa can be put aside. A table summarizing the results of the 2DEG characterization of sample A is seen in table 5.1 (after illumination). The plot in figure 5.8 was used to determine the classical carrier density at zero-field. Figure 5.9 shows the Hall resistance and longitudinal resistance as a function of magnetic field. The longitudinal resistance clearly shows SdH oscillations, where the regions of zero resistance are aligned with the plateaus in the Hall voltage. The SdH CHAPTER 5. EXPERIMENTS 117 Figure 5.7: Sample A : the Hall bar (not suspended) used for taking magnetoresistance measurements. The dimensions of the bar are shown : the spacing between two longitudinal leads was 391.1 µm, the width of the bar was 105.1 µm and the total length of the bar was 1373.6 µm. Table 5.1: Results of 2DEG characterization and physical dimensions for sample A Characteristic Value na (zero field, classical) (7.04 ± 0.01) × 1011 cm−2 na (low field, SdH) (7.19 ± 0.03) × 1011 cm−2 na (high field, SdH) (6.9 ± 0.2) × 1011 cm−2 µ (1.72 ± 0.02) × 105 cm2/V·s l 782.2 µm w 105.1 µm 118 CHAPTER 5. EXPERIMENTS eVH vs IB −25 20 x 10 experimental linear eVH = 1.424e−016*IB − 1.151e−025 na = (7.04 ± 0.01) x 1011 cm−2 15 eVH (CV) 10 5 0 −5 0 0.2 0.4 0.6 0.8 IB (AT) 1 1.2 1.4 −8 x 10 Figure 5.8: eVH plotted against IB for the illuminated sample for 0.017 ≤ B ≤ 0.2 T. The inverse of the slope of the linear fit gives an electron sheet density of (7.04 ± 0.01) × 1011 cm−2 (see subsection 2.5.2 for the theory concerning this). The intercept is non-zero because of the uncertainty in the readings of our instruments for very small magnetic fields. CHAPTER 5. EXPERIMENTS 119 Table 5.2: Observable plateaus in the Hall resistance of sample A, where i = h/e2 RH and RH = VH /I. I was taken to be constant at 0.500 V/10.093 MΩ = 4.9539×10−8 A. Fitted by averaging VH over plateaus. i (theoretical) i (experimental) 3 3.019 ± 0.002 4 4.024 ± 0.002 5 5.030 ± 0.004 6 6.038 ± 0.005 7 7.045 ± 0.007 8 8.053 ± 0.007 10 10.07 ± 0.01 12 12.08 ± 0.01 14 14.10 ± 0.01 16 16.13 ± 0.02 oscillations start at B ≃ 0.25 T and the observable plateaus are listed in table 5.2. The plateaus are all measured higher than their theoretical value. The uncertainty included in the table is that of the standard deviation of linear fits over the plateaus. What is not included, however, is the uncertainty in the measured value for the large current-limiting resistor in series with the Hall bar. The current going through the resistor and Hall bar was considered constant, which is in itself an approximation. This experiment was never optimized for noise or sensitivity, as its purpose was only to characterize the 2DEG, so deviations are expected. Figure 5.10 shows the SdH oscillations plotted against inverse magnetic field. The graph can be used to calculate the electron density from the period of the oscillation both in the low-field case and in the high-field case. 120 CHAPTER 5. EXPERIMENTS Magnetoresistance measurements after illumination 10000 Hall resistance Longitudinal resistance Resistance (Ω) 8000 6000 4000 2000 0 0 1 2 3 4 5 Magnetic field (T) 6 7 8 Figure 5.9: Magnetoresistance measurements after illumination of sample A. 9 121 CHAPTER 5. EXPERIMENTS Longitudinal resistance vs inverse of magnetic field 6000 5000 Resistance (Ω) 4000 3000 2000 1000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1/B (1/T) Figure 5.10: Longitudinal resistance plotted against inverse magnetic field after illumination of sample A. Spin polarization is clearly visible at high magnetic fields. No evidence of a beat is present, which could indicate that two subbands are conducting in the 2DEG. Parallel channels as these would likely have different frequencies in 1/B and hence would produce a beat [47]. CHAPTER 5. EXPERIMENTS 122 Figure 5.11: Suspended Hall bar used for characterization of the 2DEG. The mesa definition etch (in this case RIE) went deeper than the bottom of the sacrificial layer and thus a similar shape to that of the suspended Hall bar can be seen under it. 5.2.4 Results for a Small Suspended Hall Bar and Beams Suspended Hall bar patterns (figure 5.11) and beam patterns were made in order to evaluate transport properties in small structures. The Hall bar pattern was to be used to measure electron density, mobility and depletion width. Several beams (such as those shown in figure 4.7) ranging from ≈ 300 nm to ≈ 1.1 µm would provide another way of evaluating the depletion width by seeing which of them had a conducting 2DEG. Upon trying out simple magnetoresistance measurements on the devices, however, it was found that they did not conduct unless they were exposed to a source of light, CHAPTER 5. EXPERIMENTS 123 such as that from a microscope light or the LED at the bottom of the cryostat being constantly lit. The suspension process had depleted the 2DEG and the latter would only conduct while submitted to some optical excitation. Even after the LED illuminated the samples and was turned off, the samples returned to high resistance. The effect of pulses of light from the LED is shown in figure 5.12 for the two-wire resistance of a micron-wide suspended beam at T ≃ 77 K and in figure 5.13 for the longitudinal resistance of a ≈ 500 nm-wide suspended Hall bar at T ≃ 300 mK (a four-wire measurement). In those two measurements, the voltage across the LED is increased in small increments until the desired current through the LED is reached, causing steps in the LED current before the first pulse. Concerning the quantum Hall effect, it could not be observed in suspended structures : some slight magnetoresistance effects were visible, but these were clearly not SdH oscillations. The Hall voltage remained near zero for every magnetic field value and no plateaus were observed. The measurements are displayed in figure 5.14. These figures (5.12, 5.13 and 5.14) were included for completeness, however the resistance values they display should not be considered to be the actual resistance of the samples. The absence of the quantum Hall effect suggests that no 2DEG is present in the samples and thus that their resistance is very high. Lower resistance values than in reality are measured in samples that do not conduct as artifacts of the measurement circuit. 124 CHAPTER 5. EXPERIMENTS 6 Longitudinal resistance (ΜΩ) Longitudinal resistance (MΩ) 5 4 3 2 1 0 0 100 200 300 400 500 600 700 Current through LED (mA) Time (s) 1.4 Current through LED (mA) 1.2 1 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 Time (s) Figure 5.12: Effect of pulses from the LED on the two-wire resistance of a suspended micron-wide beam such as the one in figure 5.17 at T ≃ 77 K. The pulse duration was one second and a relaxation time of one minute is allowed between each of the 10 pulses. The current through the LED at each of the pulses is approximately 1 mA. 125 CHAPTER 5. EXPERIMENTS 800 Longitudinal resistance (Ω) Longitudinal resistance (Ω) 700 600 500 400 300 200 100 0 0 100 200 300 400 500 600 700 Current through LED (mA) Time (s) 1.4 Current through LED (mA) 1.2 1 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 Time (s) Figure 5.13: Effect of pulses from the LED on the longitudinal resistance of a suspended Hall bar such as the one in figure 5.11 at T ≃ 300 mK (a four-wire measurement). The measured resistance is the longitudinal voltage difference across the Hall bar, divided by the measured source-drain current. The pulse duration was one second and a relaxation time of one minute is allowed between each of the 10 pulses. The current through the LED at each of the pulses is approximately 1 mA. 126 CHAPTER 5. EXPERIMENTS 390 50 Longitudinal resistance (Ω) Hall resistance (Ω) 380 370 30 360 20 Hall resistance (Ω) Longitudinal resistance (Ω) 40 350 10 340 330 0 0 1 2 3 4 5 Magnetic field (T) 6 7 8 9 Figure 5.14: Longitudinal resistance and Hall resistance as a function of magnetic field for a ≈ 500 nm-wide suspended Hall bar, such as the one in figure 5.11. CHAPTER 5. EXPERIMENTS 127 The same phenomenon was confirmed by suspending the leads of sample A (a large Hall bar) and testing the 2DEG using a probe station in the dark. For the 10 µm-wide suspended leads, resistance went from 66 kΩ under the microscope light to 226 kΩ by turning off the microscope light, room lights and covering the probe station with a dark cloth. Note that the two thin leads are connected through a large stretch that was not suspended, as its geometry was never designed for this (see figure 5.6). It was thought that there was five possible reasons for the 2DEG to deplete after going through HF suspension. 1. The low-temperature GaAs (LT-GaAs) layer at the bottom of the heterostructure is depleting the 2DEG. The presence of this layer is a major difference between our design and other designs that were successful in the past by Blick et al.[48, 49, 50]. 2. The HF suspension etch is attacking the heterostructure. 3. The HF suspension etch is attacking the contacts. Another difference is that Blick et al. deposited a gold metallization layer over the ohmic contacts, protecting them during the HF dip. 4. The edge depletion width in our samples was too large. 5. The reactive ion etching steps may damage the 2DEG by implanting ions in the CHAPTER 5. EXPERIMENTS 128 heterostructure. This was reported in reference [56] as a reduction in electron mobility. Even though these five hypotheses are not mutually exclusive, the first thing to do was to test them individually at least. LT-GaAs To test the first hypothesis, a relatively large (10 µm wide by ≈ 100 µm long) suspended Hall bar pattern was designed. It was made wide enough that edge depletion would not be a significant factor. The device was fabricated and, after the HF suspension etch, was put an additional time in citric acid to etch the LT-GaAs layer away (along with around 5 nm) on all sides. By controlling the volume ratio of 50% citric acid / 30 % H2 O2 to 1/2, the etch rate of GaAs can be slowed to only 6 nm per minute, while Al0.3 Ga0.7 As is etched at only 2.7 nm per minute [77]. This is slow enough that is it controllable, and by leaving the sample in the etchant for a minute, the LT-GaAs layer and the top GaAs were etched away. Even though Poisson-Schrödinger simulations (see figure 5.15) indicated that a 2DEG would still be present, the etch did not improve conduction on the 2DEG in any way. In fact, the device behaved as an open circuit after this step. This does not seem to support the hypothesis that the LT-GaAs is responsible for the depletion of the 2DEG, but neither does it constitute an absolute proof that the LT-GaAs isn’t. 129 CHAPTER 5. EXPERIMENTS 0.8 Unetched heterostructure Etched 5 nm Etched 10 nm 0.7 Conduction band energy (eV) 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0 20 40 60 80 100 Distance from original surface (nm) 120 140 160 Figure 5.15: Conduction band energy relative to the Fermi level (set at zero) for the heterostructure in the unetched case and for cases where it was etched 5 nm on all sides, and etched 10 nm on all sides. In all cases an electron density of ≈ 5×1011 cm−2 is predicted in the 2DEG (diminishing the more material is etched). CHAPTER 5. EXPERIMENTS 130 The etch rate of LT-GaAs in the etchant is unknown, so it remains unclear how much of it was etched. Because this layer is at the bottom of the suspended structures, it is very hard to discern in SEM images and impossible to scan using our AFM, so that we do not have the means to quantitatively evaluate the etching action. On the other hand, Zbig Wasilewski, at NRC in Ottawa, who suggested the use of LT-GaAs believes it to be safe. LT-GaAs has indeed been used with success a number of times before in FETs and in high-mobility modulation-doped 2DEGs [85, 86]. In none of these cases does the LT-GaAs layer come exposed to air or vacuum, as this is the case in our suspended structures. Still, it must be considered that because the mesa of this device was etched in citric acid, many holes are present in the suspended heterostructure. The presence of these holes allowed the final etch in citric acid to attack the heterostructure from within and the HF to attack the sacrificial layer in regions where it normally would not (like under large leads, or under the contacts). This is evident in SEM images where a ‘Swiss cheese’ motif in the sacrificial layer is visible under the heterostructure (see figure 5.16). Because we were not able to eliminate these defects caused by wet etching in the mesa, the test is not to be considered fully reliable. The unavailability of RIE at Queen’s University forces us to use wet etching when working on campus. The RIE process discussed in chapter 4 works well for obtaining mesas free of defects, however CHAPTER 5. EXPERIMENTS 131 Figure 5.16: ‘Swiss cheese’ motif found in the sacrificial layer of a suspended Hall bar. The circular gaps in the sacrificial layer are caused by the HF solution reaching the sacrificial layer through holes left in the mesa by the citric acid etch step that defined the mesa. CHAPTER 5. EXPERIMENTS 132 Figure 5.17: A suspended one-micron-wide beam. The sidewall of the heterostructure shows no sign of attack by the HF suspension. use of an outside facility is required. Heterostructure and HF dip Not by examining figure 5.11 nor figure 5.17 nor figure 5.18 can we detect evidence that the heterostructure is attacked at all by the HF suspension etch. It was remarked in chapter 4 that the etch speed of Al0.3 Ga0.7 As in 10% HF is only of 0.2 nm/min. For the one-minute, 5% HF dip needed to suspend 1 µm-wide beams, this represents an unsignificant amount. As for intrinsic GaAs, the other component of the heterostructure, it is known that HF does not attack it. This was tested in our lab by dipping a clean, unetched sample in HF and was not attacked. Therefore, it seems unlikely the CHAPTER 5. EXPERIMENTS 133 Figure 5.18: An undercut alignment mark with the sidewall of the heterostructure looking intact after an HF dip. HF attacking the heterostructure is the problem. Ohmic contacts and HF dip Another possibility is that the HF is attacking the contacts, rendering them useless unless exposed to light. Many experiments involving Ni/Ge/Au ohmic contacts to a 2DEG make use of small contacts covered by a metal overlay, often Ti/Pt/Au in successive layers [68]. Titanium, however, can not be used in our case since HF attacks it very fast, meaning that the HF suspension etch would remove the overlay. Blick et al. too used an overlay on top of their contacts [56], though it is unclear exactly what metals they chose. So far our single attempt using a metal overlay CHAPTER 5. EXPERIMENTS 134 (Ni/Au) did not produce an ohmic I-V behavior. Furthermore, the average resistance measured was ten times higher than with our traditional large ohmic contacts with no overlay. More experimentation is required. Edge Depletion If the edge depletion width was too great in each of our suspended samples, then this would explain the depletion of the 2DEG. The widest suspended beam without any defects in the mesa in which conduction was tested is just over one micron wide. If edge depletion is to blame, then the depletion width would be of ≈ 500 nm or more. This is more than twice what was observed (220 nm) for a similar heterostructure in reference [56]. The 2DEG was also found to be depleted in 10 µm-wide suspended Hall bars. These bars, on the other hand, were compromised by holes in the mesa, as they were fabricated using the citric acid/hydrogen peroxide etch. Their width is large enough that it is tempting to conclude that edge depletion is not a factor. However, as each hole in the mesa likely produces its own depleted region, a high enough density of them would probably deplete the 2DEG in its entirety. Therefore, fabrication of larger suspended structures free of defects using RIE is needed for a definitive answer. CHAPTER 5. EXPERIMENTS 135 Reactive Ion Etching Damage Blick et al. realized in [56] that their RIE step was reducing the electron mobility by a factor of 27, event through their nickel mask. In subsequent publications [49, 50], they mention using reducing the RF power of their RIE process to 10 W and using a bias of less than 40 V. Because an outside facility was used, we are unaware of the settings used for our single RIE process. Even though the reduction in mobility observed by Blick et al. is a benign problem in comparison with our observations, it would be advisable in the future to specify that a low power and bias are preferable. Chapter 6 Future Work 6.1 Heterostructure redesign In light of the depletion of the 2DEG in suspended structures a new design for the heterostructure is proposed. This new design does not contain a LT-GaAs or even a GaAs layer above the sacrificial layer. The sacrificial layer is made a full micron thick in order to facilitate taking pictures of suspended structures in a SEM. The absence of any GaAs cap layer at the bottom will prevent another 2DEG from forming there in unsuspended parts of the wafer. In suspended structures, its absence will allow the lower Al0.3 Ga0.7 As to oxidize. This should not affect the 2DEG more than the oxidation that occurs at the mesa’s sidewalls, so significant problems are not expected. 136 CHAPTER 6. FUTURE WORK 137 The GaAs central 2DEG layer is made thicker (from 10 nm to 20 nm) on recommendation from Zbig Wasilewski, which should increase the electron mobility. The 2DEG layer is also kept at the center (sharing the beam’s neutral axis) to take advantage of having the piezoelectrically active layer entirely off-axis, as explained in chapter 2. The resulting conduction band profile is shown in figure 6.1 for the unsuspended case and in figure 6.2 for the suspended case (the simulation input files are shown in appendix C). As before, the expected electron density in the 2DEG is ≈ 5 × 1011 cm−2 in both cases. A GaAs layer could still be included above the sacrificial layer, but this would sacrifice a bit the versatility of the wafer, as two separate 2DEGs may form in the unsuspended crystal (this was the approach used by Blick et al.). This would produce conduction band profiles similar to those shown in chapter 3 (see figures 3.2 and 3.3). 6.2 Improve Fabrication The definition of the mesa using wet etching has been problematic, as the masking is not completely effective and small holes are etched throughout the mesa. With the use of an oxygen plasma cleaning process before application of the mask and/or different masks materials, it is hoped that the occurrence of the defects will be reduced. On the other hand, Queen’s University will soon have its own RIE processing facility. The parts are currently disassembled, but once the space is found, plumbing 138 CHAPTER 6. FUTURE WORK 0.8 Conduction band energy (eV) 0.7 Conduction band energy (eV) 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0 50 100 150 200 250 Distance from surface (nm) 300 350 400 Figure 6.1: Conduction band energy relative to the Fermi level (set a zero) for the proposed new design of the heterostructure. The sacrificial layer is a full 1000 nm thick but is plotted only until a depth of 400 nm in the heterostructure in order to keep the figure clear. An electron density of 5.174 × 1011 cm−2 is predicted in the 2DEG. 139 CHAPTER 6. FUTURE WORK 0.8 Conduction band energy (eV) 0.7 Conduction band energy (eV) 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0 20 40 60 80 100 Distance from surface (nm) 120 140 160 Figure 6.2: Conduction band energy relative to the Fermi level (set a zero) for the proposed new design of the heterostructure when the sacrificial layer has been removed. The plot assumes a barrier of 0.72 eV for the AlGaAs/vacuum interface. An electron density of 5.508 × 1011 cm−2 is predicted in the 2DEG. CHAPTER 6. FUTURE WORK 140 done and machine re-assembled, RIE will be available and wet etching need not be used any longer for mesa definition. Mesas free of defects will then be easier to obtain. 6.3 Continue Characterization Once suspended structure with 2DEG are obtained, characterization should be continued by doing magnetoresistance measurements and evaluating the edge depletion width. 6.4 Experimental Test of Actuation It would be desirable to test the piezoelectric actuation mechanism experimentally. This would allow us to evaluate the damping and verify which resonant modes we are able to excite in our devices. For the first few tests, displacement would only have to be sensed using relatively simple methods, like optical interferometry or a solid-state FET-based electrometer. 6.5 Integration with Sensitive Amplifiers The ultimate goal of this project is, as mentioned in chapter 2, to construct an ultrasensitive displacement sensing device. Once the actuation scheme is tested and the mechanical response of beams characterized, sensitive amplifiers should be connected CHAPTER 6. FUTURE WORK 141 to the detection electrode. The design in [10] calls for a single electron transistor to be used, however, other amplifiers could also be used, such as a quantum point contact defined with electrodes since unsuspended parts of the wafer also contain a 2DEG. Note that since the frequency of resonance of the oscillators is expected to be high (≈ 1 GHz), the radio-frequency versions of these amplifiers will likely have to be implemented. If the frequency of the resonator used is high enough, and if a dilution refrigerator is used so that the temperature is low enough, we may then attempt to place the resonator in the “frozen-out” state of its fundamental mode. Some back-action cooling mechanism (see [19]) may be needed for this. If the detection scheme is sensitive enough, it might be then possible to observe displacements ever-approaching zeropoint motion. Chapter 7 Conclusions An AlGaAs/GaAs heterostructure was designed and modeled for use in the piezoelectric displacement sensing scheme described in [10]. The design incorporated several key elements : • Ability to make suspended structures. This was verified by making suspended beams using the selective wet etching of a sacrificial layer. • Presence of a 2DEG. A 2DEG of appropriate characteristics (µ = 1.72 × 105 cm2/V·s, na = 7.04 × 1011 cm−2 ) was found in unsuspended mesas. However the 2DEG was found to be depleted in suspended structures as no conduction can be observed unless the sample is exposed to light. • Use of piezoelectric materials. By using AlGaAs and GaAs to fulfill the previous 142 CHAPTER 7. CONCLUSIONS 143 two requirements, this requirement was already taken care of due to their piezoelectricity. The proper crystal orientation of the suspended beams and placement of an actuation electrode was determined. Piezoelectric actuation was simulated in FEM software and was found to be able to excite the fundamental, flexural out-of-plane mode of a 836 nm-long, 250 nm-wide and 164 nm-thick beam at 925.6 MHz. Fabrication recipes for making suspended structures out of this crystal were developed. Electron beam lithography settings were determined to be able to make beam patterns 300 nm-wide or less. Optimal etch times and etchant concentration for mesa definition, nickel mask removal and sacrificial layer removal were found. While no wet etching process was found that could define mesas free of defect holes, a dry etching recipe using RIE was developed and found to yield mesas without any holes. 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Fabrication and DC characterization of single electron transistors at low temperature. Master’s thesis, Queen’s University, 2007. Appendix A Apparatus A.1 Cryogenics The cryogenic equipment was supplied by Janis [87]. A model HE-3-SSV cryostat is fitted with a 9 T superconducting magnet. Its accompanying dewar is of insulated fiber glass and aluminum construction and holds 48 liters of liquid. The cryostat includes the refrigeration system and most of the wiring. The only wiring on the dewar are the superconducting magnet leads. Other than this, the dewar is simply a container vessel which allows the cryostat to bathe in liquid He-4 and allows us to recuperate the evaporated gas. 156 APPENDIX A. APPARATUS A.1.1 157 He-3 refrigerator The He-3 refrigerator has several key components. First, there is a closed He-3 system consisting of a He-3 reservoir and a He-3 pot, to which the sample mount is in good thermal contact, and a charcoal pump. It is by pumping on liquid He-3 in the He-3 pot with the charcoal pump that the sample mount can be cooled to ≈ 300 mK. The rate of pumping is controlled by varying the flow of liquid He-4 in a capillary wound around the charcoal pump. When pumping on the He-3 pot, the helium gathers in the charcoal. The helium can then be returned to the He-3 storage tank at the top of the cryostat by heating a wire wound heater mounted on the charcoal pump. Its temperature is measured by a silicon diode. Second, there are two separate system for handling the He-4. One is, as mentioned before, the capillary wound around the charcoal pump and has the purpose of cooling it down. The capillary has an intake directly in the liquid He-4 bath, and the flow rate of the helium inside it is simply controlled by a flow meter valve at the top of the cryostat. Pressure is simply supplied by the evaporating He-4 inside the dewar, which has been hermetically sealed. The other system is the 1 K pot system. A He-4 pot is situated around the tube connecting the He-3 pot and the charcoal pump and allows the He-3 to condense into the He-3 pot. By directly pumping on the 1 K pot with a mechanical pump, its temperature can be lowered to close to 1 K. A siphon feeds liquid He-4 from the bath APPENDIX A. APPARATUS 158 Figure A.1: Drawing of the HE-3-SSV He-3 refrigerator and cryostat [87]. in the dewar to the 1 K pot. The rate of pumping can be finely tuned with use a long needle valve. This rate must be kept just right, for pumping too much will result in an empty 1 K pot and thus a warm one. Not pumping enough will flood the 1 K pot with liquid He-4, which is at 4.2 K, much too high to condense He-3. Therefore, for continuous operation of the He-3 refrigerator at low temperatures, an equilibrium between the pumping rate and condensation rate of the He-3 must be established by tuning the various valves. APPENDIX A. APPARATUS 159 The cryostat is also equipped so that the mounted sample rests in vacuum with the addition of an inner vacuum can (IVC) covering all the elements mentioned above. This IVC can be evacuated using an access port on top of the cryostat. A drawing of the cryostat is seen in figure A.1. A.1.2 Cooling Procedures After the sample is mounted and all the circuitry connected and verified, the IVC must be mounted and sealed with indium gasket on the cryostat. It then must be leaked-checked with a leak detector. This is accomplished in three steps : first by back filling the 1 K pot with dry He-4 and checking the He-4 background in the IVC to check for a leak between the two and second by spraying He-4 on the outside seal of the IVC to verify the quality of this seal. The third check is made to check for leaks in the He-3 system. In this case, one only has to check the He-3 background in the IVC and record the pressure of the He-3 storage tank for future reference. Next, all leads, heaters and thermometers should be checked. If any problems are detected they can be more easily remedied at this stage than later on. This concludes the room temperature check. The liquid nitrogen cooldown follows. First the leak detector is disconnected from the cryostat, which is inserted into the dewar. ≈ 200 Torr of dry nitrogen gas is then introduced into the IVC through valve V3 for use as an exchange gas. Keeping APPENDIX A. APPARATUS 160 the 1 K pot needle valve closed (V2), the 1 K pot is pumped on. Dry He-4 gas is flowed through the charcoal pump cooling capillary to prevent air from blocking it. Next liquid nitrogen is inserted in the dewar until it is full. Pumping of the 1 K pot is made. Until the liquid nitrogen level reaches the 1 K pot needle valve or the 1 K pot temperature reaches below 100 K, the valve must be turned 1/4 of a turn and then closed approximately every 30 minutes to prevent it from freezing. After that, pumping of the 1 K pot should cease and ≈ 5 psi of dry He-4 gas is introduced into the 1 K pot to prevent liquid nitrogen from collecting inside. After the fill is finished, the He-4 flow through the charcoal pump capillary can stop. When the base of the cryostat reaches near liquid nitrogen temperatures, say ≈ 80 K, all heaters and thermometers as well as the wiring should be checked. This concludes the nitrogen cooldown. The last part of the process is the liquid helium cooldown which will lead to He-3 condensation in the He-3 pot. Now the IVC must be re-evacuated and can be checked again for He-3 background if desired. The liquid nitrogen is blown out of the dewar by dry He-4 gas. The He-4 background inside the IVC should be monitored to ensure that no leaks are present. After all the liquid nitrogen is blown out and to make sure that no liquid nitrogen is left in the charcoal pump cooling capillary, dry He-4 should be flowed in it. 1 K pot should be pumped from now on. Transfer of liquid He-4 begins while the He-4 background in the IVC is monitored. If it remains leak tight, APPENDIX A. APPARATUS 161 the leak detector can be disconnected and the IVC left under vacuum. The 1 K pot needle valve needs to be opened and closed at regular intervals just like in the case of the liquid nitrogen transfer. When the liquid He-4 transfer is finished, however, the needle valve can be left slightly open in order to cool the 1 K pot. Next helium gas must be allowed to escape the dewar through the charcoal pump capillary so as to cool the charcoal pump. One must take care that air is never sucked into that capillary, as it will freeze. Therefore, one must wait for positive pressure to build inside the dewar. After the 1 K pot has reached 1.5 K and the charcoal pump has reached below 5 K all of the He-3 should be gathered inside the charcoal pump and so pumping can cease by ceasing the flow of He-4 in the cooling capillary. By keeping the 1 K pot as cool as possible and heating the charcoal pump to 40 - 45 K, all of the He-3 can be condensed into the He-3 pot. After the He-3 pot temperature drops below 1.8 K, it is recommended to wait 1.5 hours while keeping the 1 K pot as cool as possible. After that period, the heater on the charcoal pump may be turned off and the flow inside its cooling capillary restored. The temperature of the charcoal pump has to remain below 5 K and can be controlled to a certain extent by adjusting the flow in the capillary. The sample mount is expected to cool down to base temperature within less than an hour. Warm-up of the cryostat is accomplished simply by closing the 1 K pot needle valve. This prevents liquid He-4 from reaching the 1 K pot, thus warming it up and APPENDIX A. APPARATUS 162 causing the He-3 to not condense but instead gather in the charcoal pump. As the temperature of the whole system increases, the He-3 will eventually be driven back to the He-3 storage tank. A.2 Wiring and Filters The wiring on the cryostat was installed by Mark Patterson and Greg Dubejsky. 19 DC wires and 4 coaxial RF lines go from the top of the cryostat down to the sample mount. All wires are heat-sunk at the 1 K plate, just below the 1K pot. Details of the wiring are available in Greg Dubejsky’s thesis [88]. The DC wires go through various filters. At room temperature, a Pi filter is placed, which attenuates signal with 10 MHz < f < 1 GHz with greater than 75 dB attenuation. At T = 1 K a metallic powder filter, fabricated by Greg Dubejsky, served to attenuate signals with frequency greater than 1 GHz [88]. Since signal used in lock-in detection were of low frequency (17 Hz), in the second experiment, band pass filters on the pre-amplifiers were used to eliminate the unwanted signals below the 10 MHz mark. To eliminate noise sources as much as possible, a signal analyzer was used by Kyle Kemp to identify sources and eliminate them by either physically moving the instruments or rewiring the power connections. APPENDIX A. APPARATUS 163 Figure A.2: Drawing of both sides on the mounting stage. On the left, the side facing down towards the superconducting magnet is where the sample is glued and wired bonded to the various DC and RF leads. On the right is the side facing up away from the magnet and towards the top of the cryostat where the wires exit. The black represents conductive metal. All of the backside of the board is a ground plane, as is the area under where the sample is glued. A.3 Mounting Stage The mounting stage with circuit board was designed by Greg Dubejsky, Mark Patterson, Jennifer Campbell and Rob Knobel. It consists of a circular board which can be screwed onto the very bottom of the cryostat. 19 pins are provided for DC wires, while two MMCX connectors are available for connecting radio-frequency (RF) lines. A drawing of both sides of the mounting stage is shown in figure A.2. APPENDIX A. APPARATUS A.4 164 Wire Bonding Very thin aluminum wires are bonded (soldered) using ultrasonic power to the leads of the circuit board and to the ohmic contacts on the wafer. On Dr. Lockwood’s Westbond wire bonder (at Queen’s University), the setting used were a power of 420, a time of 30 ms and a 45 degree feed angle for both the first bond and the second bond. A.5 Equipment This section lists in detail the equipment used in the course of the measurements. A.5.1 Lock-in Amplifiers Stanford Research Systems [81] lock-in amplifiers (models SR530, SR830 and SR850) were used in addition to an EG&G (now Signal Recovery) 5206 lock-in amplifier. A.5.2 Pre-Amplifiers Two Signal Recovery [84] pre-amplifiers (model 5113) were used. They were run on batteries whenever possible. These pre-amplifiers are equipped with variable bandpass filters, than were used to eliminate signals outside the frequency range of interest. APPENDIX A. APPARATUS A.5.3 165 Programmable Current Source A Keithley 220 [79] programmable current source was used in testing ohmic contacts and large devices. A.5.4 Current Source A “safe” current source, that is, a current source of low power and low current protecting devices under test from electrostatic discharges was used in testing small suspended beams that can not handle much power. The current source is equipped with a variable resistor parallel to the device under test, allowing the current to be slowly turned up in a safe manner. It was conceived and built by Mark Patterson in the summer of 2006. A current of I = 10 nA is flowed in the devices and the voltage V is measured using a voltmeter. The resistance of the device Rd can be calculated using the following formula (obtained from Greg Dubejsky) : V 106 Ω Rd = I 106 Ω − V /I A.5.5 (A.1) Multimeters A GPIB-enabled HP 34401A [80] multimeter was used whenever a multimeter was needed. APPENDIX A. APPARATUS A.5.6 166 Temperature Controller A LakeShore temperature controller was used to monitor the temperature of the 1 K pot, the charcoal pump, the sample mount and the bottom of the superconducting magnet. The controller can also be set to control the temperature of the charcoal pump with use of a PID controller and the heater mounted on the pump. Additionally, the various temperature readings can be monitored over GPIB. A.5.7 Superconducting Magnet and Power Supply A Cryomagnetics CS4-10V [82] magnet power supply was used, over a GPIB connection, to control the magnetic field produced by a 9T superconducting magnet placed at the bottom of the dewar. Appendix B List of Programs A non-negligible effort was put into writing data acquisition and control programs for our instruments. The programming was done using a combination of Matlab [83] and LabVIEW [78]. Use was made of the sample code written by Mark Patterson in the summer of 2006 as templates but was extended by the author. The programs themselves may be found on Dr. Knobel’s laboratory computers at Queen’s University. 1. ReadLockinStateSR830.m : Records all important settings of a SR850 or a SR850 lock-in amplifier. 2. ReadLockinStateSR530.m : Records all important settings of a SR530 lock-in amplifier. 3. ReadLockinState5206.m : Records all important settings of a EG&G 5206 lockin amplifier. 167 APPENDIX B. LIST OF PROGRAMS 168 4. ReadLockinSnapshotSR830.m : Reads an instantaneous snapshot of X and Y on a SR830 lock-in amplifier (this will not work for a SR850). 5. ReadLockin830.m : Reads the X value quickly followed by the Y value on a SR830 or SR850 lock-in amplifier. 6. ReadLockinSR530.m : Reads the X value quickly followed by the Y value on a SR530 lock-in amplifier. 7. ReadLockin5206.m : Reads the X value followed by the Y value on a EG&G 5206 lock-in amplifier. 8. GetTimeConstant830.m : Retrieves the time constant setting of a SR830 or SR850 lock-in amplifier. 9. GetTimeConstant530.m : Retrieves the time constant setting of a SR530 lock-in amplifier. 10. GetTimeConstant5206.m : Retrieves the time constant setting of a EG&G 5206 lock-in amplifier. 11. GetLPSlope5206.m : Retrieves the low-pass filer slope of a EG&G 5206 lock-in amplifier. 12. GetSensitivity5206.m : Retrieves the sensitivity of a EG&G 5206 lock-in amplifier. APPENDIX B. LIST OF PROGRAMS 169 13. GetReserveMode5206.m : Retrieves the reserve mode of a EG&G 5206 lock-in amplifier. 14. GetPhaseShift5206.m : Retrieves the phase shift of a EG&G 5206 lock-in amplifier. 15. SetSensitivity5206.m : Sets the sensitivity of a EG&G 5206 lock-in amplifier. 16. SetTimeConstant5206.m : Sets the time constant of a EG&G 5206 lock-in amplifier. 17. SetLockinAuxOut.m : Set the DC voltage on any of the aux out connectors on an SR830 or SR850 lock-in amplifier. 18. SetCurrentKeithley220.m : Sets the desired current on a Keithley 200 current source. 19. OperateKeithley220.m : Turns ON and OFF the current on a Keithley 220 current source. 20. SlowMagnetSweepLockin.m : Sweeps the magnet slowly while recording readings from two independent lock-in amplifiers. 21. SpecificFieldLockin2.m : Sweeps the magnetic field to specific values and records readings from three lock-in amplifiers at each value (designed for four-wires measurements). This program contains an option to wait at each value of APPENDIX B. LIST OF PROGRAMS 170 the magnetic field for the temperature to stabilize in a specified range so that measurements at an almost constant temperature may be taken. 22. SpecificFieldLockin 2WireMagnetoresistanceVersion.m : Similar to SpecificFieldLockin2.m except designed for two-wires measurements only. 23. StartMagnetSweep.m : Used to zero the magnet or sweep the magnetic field over a specified range at a specified rate using the Cryomagnetics CS4-10V [82] magnet power supply. 24. QuickMagnetSweep.m : Used to sweep the magnetic field to a new value using the Cryomagnetics CS4-10V [82] magnet power supply. 25. ReadMagnetField.m : Retrieves the magnetic field strength from Cryomagnetics CS4-10V [82] magnet power supply. 26. Illuminate.m : Increases the voltage applied to a LED circuit until current passes a certain threshold, and then makes several pulses of light of specified duration, waiting for the specified relaxation time in between pulses. The program reads from one lock-in amplifier to record longitudinal resistance in a Hall bar and one multimeter to record the current through the LED. 27. Current vs Voltage 2.vi : Sweeps the current of a Keithley current source from the specified range and records the voltage obtained from a HP multimeter. Used for testing ohmic contacts. APPENDIX B. LIST OF PROGRAMS 171 28. GetNoiseSpectrum.vi : Records the FFT spectrum from a SR725 signal analyzer and allows the user to save it to a file. Used for tracking down and eliminating noise sources. Appendix C 1DPoisson Input Files Heterostructure : surface GaAs AlGaAs AlGaAs AlGaAs GaAs AlGaAs AlGaAs AlGaAs GaAs AlGaAs substrate schottky=0.6 v1 t=5nm t=30nm Nd=0 t=2nm Nd=1.5e19 t=40nm Nd=0 t=10nm t=40nm Nd=0 t=2nm Nd=1.5e19 t=30nm Nd=0 t=5nm t=600nm Na=0 x=.3 x=.3 x=.3 x=.3 x=.3 x=.3 x=.7 v1 0.0 v2 0.0 schrodingerstart=0 schrodingerstop=164nm temp=1K dy=4 maxiterations=500 172 173 APPENDIX C. 1DPOISSON INPUT FILES Suspended heterostructure : surface GaAs AlGaAs AlGaAs AlGaAs GaAs AlGaAs AlGaAs AlGaAs GaAs substrate schottky=0.6 v1 t=5nm Nd=0 t=30nm Nd=0 t=2nm Nd=1.5e19 t=40nm Nd=0 t=10nm t=40nm Nd=0 t=2nm Nd=1.5e19 t=30nm Nd=0 t=5nm Nd=0 schottky=0.47 v2 x=.3 x=.3 x=.3 x=.3 x=.3 x=.3 v1 0.0 v2 0.0 schrodingerstart=0 schrodingerstop=164nm temp=1K dy=4 maxiterations=500 Redesigned heterostructure : surface GaAs AlGaAs AlGaAs AlGaAs GaAs AlGaAs AlGaAs AlGaAs AlGaAs substrate schottky=0.6 v1 t=5nm t=30nm Nd=0 t=2nm Nd=1e19 t=40nm Nd=0 t=20nm t=40nm Nd=0 t=2nm Nd=1e19 t=35nm Nd=0 t=1000nm Na=0 v1 0.0 v2 0.0 schrodingerstart=0 schrodingerstop=174nm temp=1K x=.3 x=.3 x=.3 x=.3 x=.3 x=.3 x=.7 APPENDIX C. 1DPOISSON INPUT FILES dy=4 maxiterations=500 Suspended redesigned heterostructure : surface GaAs AlGaAs AlGaAs AlGaAs GaAs AlGaAs AlGaAs AlGaAs substrate schottky=0.6 v1 t=5nm t=30nm Nd=0 t=2nm Nd=1e19 t=40nm Nd=0 t=20nm t=40nm Nd=0 t=2nm Nd=1e19 t=35nm Nd=0 schottky=0.72 v2 v1 0.0 v2 0.0 schrodingerstart=0 schrodingerstop=174nm temp=1K dy=4 maxiterations=1000 x=.3 x=.3 x=.3 x=.3 x=.3 x=.3 174 Appendix D Health and Safety Issues With the use of powerful chemicals comes a risk for injury or even death. Hence, when using the fabrication procedures just described there are several safety precautions one must take. Here are some general guidelines: • Read the MSDS for every chemical you are using. Pay attention specifically to the protective equipment needed, first aid and disposal sections. • Work in a well-ventilated area (we used a fume hood). • When working with developers, isopropanol, acetone and deionized water under a fume hood, the use of clean room gloves and safety glasses is sufficient. However when working with acids, bases, oxidizing agents or any other dangerous chemicals, full gowning with chemical-resistant clothing is required, as is the wearing of tri-polymer gloves. A full face shield can also be used as a better 175 APPENDIX D. HEALTH AND SAFETY ISSUES 176 choice than safety glasses. • HF-specific safety issues: 1. Have a tube of calcium gluconate ready for application on skin burns, which may not be felt immediately and yet can cause great harm. HF will react with the calcium in bones and bloodstream if splashed on skin, causing hypocalcemia and possibly cardiac arrest. The calcium gluconate minimizes this by reacting with most of the HF when applied after thorough rinsing of the exposed skin. 2. This is true for all acids, but relevant here also : when diluting, add acid to water to avoid an accidental acid splash. 3. Rinse all beakers, tweezers, etc. at least twice after contact with HF. • Always identify beakers, source containers and waste containers with their contents. • Don’t dispose of chemicals down the water drain. • Finally, use common sense, and think before you act.