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QUANTUM ELECTRONICS IN SEMICONDUCTORS C. H. W. Barnes Cavendish Laboratory, University of Cambridge Contents 1 The Free Electron Gas page 3 1.1 Reference material 3 1.2 Introduction 3 1.3 Si, GaAs and AlxGa1¡xAs properties 4 1.3.1 Real space lattice 4 1.3.2 Reciprocal space lattice 4 1.4 E®ective mass theory 6 1.5 Doping semiconductors 8 1.6 Band engineering 8 1.6.1 Modi¯cation of chemical potential and carrier densities 8 1.6.2 Band bending 11 1.7 The Si metal-oxide-semiconductor junction 13 1.8 The GaAs-AlxGa1¡xAs heterostructure 17 1.9 Capacitor model 17 1.10 Bi-layer heterostructures: Weak coupling 19 1.10.1Magnetotunnelling spectroscopy [Eisenstein] 22 1.11 Bi-layer heterostructures: Strong coupling [Davies] 25 1.12 Exercises 27 2 Semi-classical electron transport. 28 2.1 Reference material 28 2.2 Introduction 28 2.3 The non-linear Boltzmann equation 28 2.4 The linear Boltzmann equation 31 2.5 Semi-classical conductivity 33 2.6 Drude conductivity 34 2.7 Impurity scattering 36 2.8 Exercises 38 iii iv Contents 3 Particle-like motion of electrons 39 3.1 Sources 39 3.2 Introduction 39 3.3 The Ohmic contact 39 3.4 The electron aperture 40 3.4.1 The e®ective potential 41 3.5 Cyclotron motion 41 3.6 Experiments 44 3.7 Detection of ballistic motion L. W. Molenkamp [1] 44 3.8 Collimation 45 3.9 Skipping orbits J. Spector [2] 45 3.10 The cross junction C. J. B. Ford [3] 46 3.11 Chaotic Motion C. M. Marcus [4] 47 3.12 Classical refraction 49 3.13 Refraction J. Spector [5,6] 52 3.14 Exercises 53 4 The quantum Hall and Shubnikov de Haas e®ects 56 4.1 Sources 56 4.2 Introduction 56 4.3 Boltzmann prediction 56 4.4 Conductivity 58 4.5 Experiment 58 4.6 Eigenstates in a magnetic Field 60 4.7 Density of electrons in a Landau level 62 4.8 Disorder broadening of Landau levels 63 4.9 Oscillation of the Fermi energy 65 4.10 Oscillation of the capacitance 68 4.11 Conductivity and resistivity at high magnetic ¯eld 71 4.12 A simple model 75 4.13 Appendix I 78 4.14 Exercises 79 5 Quantum transport in one dimension. 80 5.1 Sources 80 5.2 Introduction 80 5.3 Experimental realization of the quasi-one-dimensional system [1] 81 5.4 Eigenstates of an in¯nitely long quasi-one-dimensional system 81 5.5 Kardynal et al [2] 83 5.6 Density of States in a quasi-one-dimensional system 92 5.7 Oscillation of the Fermi energy and Capacitance 94 5.7.1 Macks et al [4] 94 Contents v 5.7.2 Drexler et al [5] 96 5.8 The Conductance of a quasi-one-dimensional system 97 5.9 Thomas et al [9] 98 5.10 The Landauer Formalism 100 5.11 The saddle point potential 103 5.12 Determination of saddle point potential shape 106 5.13 Exercises 111 6 General quantum transport theory 113 6.1 Sources 113 6.2 Introduction 113 6.3 Eigenstates of an in¯nite quasi-one-dimensional system 113 6.4 Group Velocity 118 6.5 Density of states 119 6.6 E®ective potential at high magnetic ¯eld [1] 119 6.7 Two-terminal conductance in a magnetic ¯eld 120 6.8 The saddle point in a ¯nite magnetic ¯eld [2] 122 6.9 Magnetic Depopulation [3,4] 126 6.10 Multi-probe Landauer-BÄuttiker formalism [5] 126 6.11 Edge states and obstacles 129 6.12 The Quantum Hall E®ect 130 6.13 Exercises 137 7 Quasi-particles in two dimensions 138 7.1 Sources 138 7.2 Introduction 138 7.3 Electrical forces 139 7.4 Landau's theory of quasi-particles [2-4] 140 7.5 Quasi-particles 143 7.6 Quasi-particle decay 144 7.7 Resistivity 145 7.8 Temperature dependence of quasi-particle decay [5] 146 7.9 Density of states dependence of quasi-particle decay 149 7.10 Formation of new quasi-particles in a quantising magnetic ¯eld 149 7.11 Willett, Goldman and Du [8,11-12] 151 7.12 Composite fermions [16-17] 155 7.13 Composite fermion e®ective magnetic ¯eld 158 7.14 Composite fermion e®ective electric ¯eld 159 7.15 Composite fermion Hamiltonian 160 7.16 Fractional quantum Hall e®ect 160 7.17 Fractional edge-state model [18] 161 7.17.1Fractional quantum Hall e®ect 161 Contents 1 7.17.2Reflection of composite fermion edge states 162 7.18 Exercises 164 8 Quantum dots 165 8.1 Sources 165 8.2 Introduction 165 8.3 Quasi-zero dimensional systems [1,2] 165 8.4 The single-particle eigen-spectrum of a quantum dot [3] 167 8.4.1 Zero ¯eld limit 167 8.4.2 High ¯eld limit 168 8.5 Conductance of a quantum dot 169 8.6 McEuen et al [2] 171 8.6.1 Magnetic ¯eld dependence of resonant peaks [1,2] 171 8.7 Classical Coulomb blockade [4]. 172 8.8 Quantum Coulomb blockade 176 8.9 Arti¯cial atoms 178 8.10 The Aharonov-Bohm e®ect [5] 179 8.11 Webb et al [6] 181 8.12 High¯eld Aharonov-Bohm e®ect. 183 8.13 Edge-state networks, dots and anti-dots. 184 8.14 Mace et al [7] 186 8.15 Exercises 188 9 Quantum computation 190 9.1 Sources 190 9.2 Introduction 190 9.3 Classical vs Quantum computation 190 9.4 The DiVincenzo rules 190 9.5 The charge qubit 190 9.6 Fujisawa 190 9.7 g-factor in a semiconductor 190 9.8 Hanson 190 9.9 Single spin detection: ... 190 9.10 Interaction in the Hubbard model 190 9.11 Single-spin rotation 190 9.12 Surface acoustic wave current quantisation 190 9.13 The surface acoustic wave quantum processor 190 9.14 Exercises 190 Appendix 1 Occupation probabilities. 191 Appendix 2 Density of states in three dimensions. 193 Appendix 3 Band bending from a delta layer 194 2 Contents Appendix 4 Density of states in two dimensions. 196 1 The Free Electron Gas 1.1 Reference material [A M] Solid State Physics, N. W. Ashcroft and N. D. Mermin. [Ziman] Principles of the Theory of Solids, J. M. Ziman. [Kittel]Introduction to Solid State Physics, C. Kittel. [Sze] Semiconductor Devices: Physics and Technology, S. M. Sze. [Kelly] Low-Dimensional Semiconductors : Materials, Physics, Technology, Devices, M. J. Kelly. [Eisenstein] J. P. Eisentein et al Phys. Rev. B 44 6511 (1991) [Davies] G. Davies, et al Phys. Rev. B 54 R17331-17334 (1996). 1.2 Introduction It is a remarkable fact that a free-electron-like gas can be made to form in a semiconductor crystal. As an interacting Fermi gas, it has many complex properties and behaviours. In particular, it has been shown that by ma- nipulating these gases with electric and magnetic ¯elds, they can be made to exhibit all of the familiar quantum e®ects of undergraduate and post- graduate quantum courses. This having been said though, many of the experimental indicators of these quantum e®ects can only be fully under- stood once the basic electrostatic building blocks of the host semiconductor devices have been understood. In particular, since quantum e®ect are more easy to see in lower-dimensional systems, we concentrate here on the essen- tial physics needed to understand semiconductor devices containing single, or many parallel two-dimensional electron or hole gases. This section of the notes covers: the basic properties of Si, GaAs and AlxGa1¡xAs; e®ective mass theory; semiconductor doping; band engineer- ing; the Si MOSFET; and the GaAs-AlGas heterostructure. 3 4 The Free Electron Gas a Fig. 1.1. (a) Diamond lattice structure of Si a = 5:4Aº (b) Face centred cubic space lattice. 1.3 Si, GaAs and AlxGa1¡xAs properties 1.3.1 Real space lattice Intrinsic crystalline Silicon has a diamond lattice structure Fig. 1.1a. Its underlying space lattice is face-centred cubic (Fig. 1.1b) and its primitive basis has two identical atoms, one at co-ordinate (0; 0; 0) and the other at co- ordinate (1=4; 1=4; 1=4) measured relative to each lattice point. Each atom has four nearest neighbours that form a tetrahedron and the structure is bound by directional covalent bonds. Intrinsic crystalline GaAs has a zincblende crystal structure (Fig. 1.2). This structure also has a space lattice that is face-centered cubic but the primitive basis has two di®erent atoms, one at co-ordinate (0; 0; 0) and the other at co-ordinate (1=4; 1=4; 1=4) measured relative to each lattice point. Each atom has four nearest neighbours of the opposite type that form a tetrahedron. AlxGa1¡xAs has the same lattice structure and approximately the same lattice constant as GaAs but with occasional Al atoms substituting for Ga atoms. 1.3.2 Reciprocal space lattice The reciprocal lattice of a face-centered cubic lattice is a body-centred cubic lattice (Fig 1.3a). The principal symmetry points in the body-centred cubic reciprocal-space lattice are: ¡ = (0; 0; 0), X = (1; 0; 0) + 6 equivalent points, L = (1; 1; 1) + 8 equivalent points (Fig. 1.3b). 1.3 Si, GaAs and AlxGa1¡xAs properties 5 Fig. 1.2. Zincblende structure of GaAs a = 5:6Aº. X L L X X L Fig.
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