QUANTUM ELECTRONICS IN SEMICONDUCTORS C. H. W. Barnes Cavendish Laboratory, University of Cambridge Contents 1 The Free Electron Gas page 1 1.1 Sources 1 1.2 Introduction 1 1.3 Si and GaAs properties 2 1.3.1 Real space lattice 2 1.3.2 Reciprocal space lattice 2 1.3.3 E®ective mass theory 3 1.4 Doping 5 1.5 Band engineering 7 1.5.1 Modi¯cation of chemical potential and carrier densities 7 1.5.2 Band bending 8 1.6 The Si metal-oxide-semiconductor junction 11 1.7 The GaAs-AlxGa1¡xAs heterostructure 15 1.8 Capacitor model 15 1.9 Bi-layer heterostructures: Weak coupling 16 1.10 Magnetotunnelling spectroscopy J. P. Eisentein et al [Eisenstein] 19 1.11 Bi-layer heterostructures: Strong coupling 23 1.12 Exercises 25 2 Semi-classical electron transport. 26 2.1 Sources 26 2.2 Introduction 26 2.3 The non-linear Boltzmann equation 26 2.4 The linear Boltzmann equation 29 2.5 Semi-classical conductivity 31 2.5.1 Drude conductivity 32 2.6 Impurity scattering 34 2.7 Exercises 36 iii iv Contents 3 Particle-like motion of electrons 37 3.1 Sources 37 3.2 Introduction 37 3.3 The Ohmic contact 37 3.4 The electron aperture 38 3.4.1 The e®ective potential 39 3.5 Cyclotron motion 39 3.6 Experiments 42 3.7 Detection of ballistic motion L. W. Molenkamp [1] 42 3.8 Collimation 43 3.9 Skipping orbits J. Spector [2] 43 3.10 The cross junction C. J. B. Ford [3] 44 3.11 Chaotic Motion C. M. Marcus [4] 45 3.12 Classical refraction 47 3.13 Refraction J. Spector [5,6] 50 3.14 Exercises 51 4 The quantum Hall and Shubnikov de Haas e®ects 54 4.1 Sources 54 4.2 Introduction 54 4.3 Boltzmann prediction 54 4.4 Conductivity 56 4.5 Experiment 56 4.6 Eigenstates in a magnetic Field 58 4.7 Density of electrons in a Landau level 60 4.8 Disorder broadening of Landau levels 61 4.9 Oscillation of the Fermi energy 63 4.10 Oscillation of the capacitance 66 4.11 Conductivity and resistivity at high magnetic ¯eld 69 4.12 A simple model 73 4.13 Appendix I 76 4.14 Exercises 77 5 Quantum transport in one dimension. 78 5.1 Sources 78 5.2 Introduction 78 5.3 Experimental realization of the quasi-one-dimensional system [1] 79 5.4 Eigenstates of an in¯nitely long quasi-one-dimensional system 79 5.5 Kardynal et al [2] 81 5.6 Density of States in a quasi-one-dimensional system 90 5.7 Oscillation of the Fermi energy and Capacitance 92 5.7.1 Macks et al [4] 92 Contents v 5.7.2 Drexler et al [5] 93 5.8 The Conductance of a quasi-one-dimensional system 95 5.9 Thomas et al [9] 96 5.10 The Landauer Formalism 98 5.11 The saddle point potential 101 5.12 Determination of saddle point potential shape 104 5.13 Exercises 109 6 General quantum transport theory 111 6.1 Sources 111 6.2 Introduction 111 6.3 Eigenstates of an in¯nite quasi-one-dimensional system 111 6.4 Group Velocity 116 6.5 Density of states 117 6.6 E®ective potential at high magnetic ¯eld [1] 117 6.7 Two-terminal conductance in a magnetic ¯eld 118 6.8 The saddle point in a ¯nite magnetic ¯eld [2] 120 6.9 Magnetic Depopulation [3,4] 124 6.10 Multi-probe Landauer-BÄuttiker formalism [5] 124 6.11 Edge states and obstacles 127 6.12 The Quantum Hall E®ect 128 6.13 Exercises 135 7 Quasi-particles in two dimensions 136 7.1 Sources 136 7.2 Introduction 136 7.3 Electrical forces 137 7.4 Landau's theory of quasi-particles [2-4] 138 7.5 Quasi-particles 141 7.6 Quasi-particle decay 142 7.7 Resistivity 143 7.8 Temperature dependence of quasi-particle decay [5] 144 7.9 Density of states dependence of quasi-particle decay 147 7.10 Formation of new quasi-particles in a quantising magnetic ¯eld 147 7.11 Willett, Goldman and Du [8,11-12] 149 7.12 Composite fermions [16-17] 153 7.13 Composite fermion e®ective magnetic ¯eld 156 7.14 Composite fermion e®ective electric ¯eld 157 7.15 Composite fermion Hamiltonian 158 7.16 Fractional quantum Hall e®ect 158 7.17 Fractional edge-state model [18] 159 7.17.1Fractional quantum Hall e®ect 159 vi Contents 7.17.2Reflection of composite fermion edge states 160 7.18 Exercises 162 8 Quantum dots 163 8.1 Sources 163 8.2 Introduction 163 8.3 Quasi-zero dimensional systems [1,2] 163 8.4 The single-particle eigen-spectrum of a quantum dot [3] 165 8.4.1 Zero ¯eld limit 165 8.4.2 High ¯eld limit 166 8.5 Conductance of a quantum dot 167 8.6 McEuen et al [2] 169 8.6.1 Magnetic ¯eld dependence of resonant peaks [1,2] 169 8.7 Classical Coulomb blockade [4]. 170 8.8 Quantum Coulomb blockade 174 8.9 Arti¯cial atoms 176 8.10 The Aharonov-Bohm e®ect [5] 177 8.11 Webb et al [6] 179 8.12 High¯eld Aharonov-Bohm e®ect. 181 8.13 Edge-state networks, dots and anti-dots. 182 8.14 Mace et al [7] 184 8.15 Buks et al 186 8.16 Exercises 186 1 The Free Electron Gas 1.1 Sources [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] \Probing a 2D Fermi Surface by tunneling" J. P. Eisentein et al Phys. Rev. B 44 6511 (1991) 1.2 Introduction It is a remarkable fact that a free-electron 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. This section of the notes provides an understanding of the basic building blocks of semiconductor device structures. In particular, since quantum ef- fect are more easy to see in lower-dimensional systems, it concentrates on the essential physics necessary to understand semiconductor devices containing single, or many parallel two-dimensional electron or hole gases. 1 2 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 and GaAs 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 at co-ordinates (0; 0; 0) and (1=4; 1=4; 1=4) centred at each point of the lattice. 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), centred at each point of the lattice. Each atom has four nearest neighbours of the opposite type that form a tetrahedron. 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). Figs. 1.4a,b show the band structure of Si and GaAs along the directions de¯ned by X and L starting at ¡. These directions are the most important 1.3 Si and GaAs properties 3 Fig. 1.2. Zincblende structure of GaAs a = 5:6Aº. X L L X X L Fig. 1.3. (a) Body-centred cubic lattice. (b) Brillouin zone boundaries for a face- centered cubic lattice. for both band structures because their principal band minima, which de¯ne their band gaps, lie along them. 1.3.3 E®ective mass theory At zero temperature the valence-band states in intrinsic semiconductors are full and the conduction-band states are empty. As a result intrinsic semicon- ductors are insulating at zero temperature because there are no free carriers. If by some means electrons are introduced to the conduction band then their Fermi surface will be de¯ned by a constant energy surface. Figs 1.5a,b show constant energy surfaces close to the conduction band edges in Si and GaAs. 4 The Free Electron Gas 1 1 Fig. 1.4. (a) The band structure of Si: indirect band gap Eg=1.17 eV.