The Physics of Nanoelectronics

Transcription

1 The Physics of Nanoelectronics Transport and Fluctuation Phenomena at Low Temperatures Tero T. Heikkilä Low Temperature Laboratory, Aalto University, Finland OXFORD UNIVERSITY PRESS

2 Contents List of symbols xviii 1 Introduction Studied systems Metallic wires and metal-to-metal contacts Semiconductor systems Carbon nanotubes and molecules Graphene Classical vs. quantum transport Drude formula Quantum effects 10 Further reading 13 Exercises 14 2 Semiclassical theory Semiclassical Boltzmann equation Observables Relaxation time approximation Elastic scattering and diffusive limit Currents in the diffusive limit Inelastic scattering Electron electron scattering Electron phonon scattering Junctions Magnetic heterostructures Thermoelectric effects 34 Further reading 35 Exercises 36 3 Scattering approach to quantum transport Scattering region, leads and reservoirs Transverse modes in semi-infinite leads Current carried by a transverse mode Wire between two reservoirs Quantum point contacts Scattering matrix Some properties of the scattering matrix Combining scattering matrices: Feynman paths Conductance from scattering 48

3 xii Contents Diffusive wire and Drude formula Resonant tunnelling Models for inelastic scattering and dephasing Further developments Time-dependent transport Non-linear transport Application to magnetic systems 56 Further reading 58 Exercises 58 4 Quantum interference effects Aharonov Bohm effect Localization Weak localization Localization length Weak localization from enhanced backscattering Dephasing Magnetic field effect an weak localization Universal conductance fluctuations Effect of dephasing Persistent currents 72 Further reading 76 Exercises 76 5 Introduction to superconductivity Cooper pairing Main physical properties Current without dissipation Meissner effect BCS theory briefly Energy gap and BCS divergente Coherence length Josephson effect Main phenomena characteristic for mesoscopic systems Andreev refiection Andreev bound states Proximity effect 91 Further reading 92 Exercises 92 6 Fluctuations and correlations Definition and main characteristics of noise Motivations for the study of noise Fluctuation dissipation theorem Thermal and vacuum fluctuations Shot noise Scattering approach to noise Two-terminal noise 99

4 Contents xiii 6.3 Langevin approach to noise in electric circuits Boltzmann Langevin approach Cross-correlations Equilibrium correlations Finite-voltage cross-correlations Effect of noise an quantum dynamics Relaxation Dephasing Full counting statistics Basic statistics Full counting statistics of charge transfer Heat current noise 117 Further reading 118 Exercises Single-electron effects Charging energy Single-electron box Single-electron transistor (SET) Tunnel Hamiltonian and tunnelling rates Master equation Cotunnelling Dynamical Coulomb blockade Phase fluctuations Single-electron devices Coulomb blockade thermometer Radio frequency SET Single-electron pump 137 Further reading 138 Exercises Quantum dots Electronic states in quantum dots Spectral function Weakly interacting limit Weakly transmitting limit Coulomb blockade Metallic limit Two-state limit Addition spectrum Charge sensing with quantum point contacts Kondo effect Double quantum dots Artificial molecules Spin states in double quantum dots Pauli spin blockade Spin qubits in quantum dots 156 Further reading 157 Exercises 157

5 xiv Contents 9 Tunnel junctions with superconductors Tunnel contacts without Josephson coupling NIS contact SIS contact Superconducting SET SINIS heat transport and pumping Thermometry with (SI)NIS junctions Electron cooling and refrigeration Josephson junctions SQUIDs Resistively and capacitively shunted junction model Overdamped regime Underdamped regime Escape process Quantum effects in small Josephson junctions `Tight-binding limit' `Nearly free-electron limit' Superconducting qubits 177 Further reading 181 Exercises Graphene Electron dispersion relation in monolayer graphene Massless Dirac fermions in graphene Eigensolutions in monolayer graphene Bilayer and more Multilayer graphene Ray optics with electrons: np and npn junctions Graphene np junction Klein tunnelling Pseudodiffusion Graphene nanoribbons Zigzag ribbons Armchair ribbons 200 Further reading 201 Exercises Nanoelectromechanical systems Nanomechanical systems Basic elastic theory Flexular eigenmodes of a doubly clamped beam without tension Effect of tension an the vibration modes Driving and dissipation Coupling to nanoelectronics Magnet omot ive actuation and detection Capacitive actuation and detection 212

6 Contents xv SQUID detection Detection through single-electron effects 11.3 Coupling to microwave resonant circuits 11.4 Quantum effects Creating a quantum superposition of vibration states in an oscillator qubit system Describing dissipation 226 Further reading 227 Exercises 228 A Important technical tools 229 A.1 Second quantization: a short introduction 229 A.1.1 Bosons 229 A.1.2 Fermions 231 A.2 Heisenberg and Schrödinger pictures 232 A.2.1 Int eraction picture 233 A.3 Fermi golden rule 234 A.3.1 Higher order: generalized Fermi golden rule 235 A.4 Describing magnetic field in quantum mechanics 237 A.5 Chemical potential and Fermi energy 237 A.6 Pauli spin matrices 240 A.7 Useful integrals 241 Exercises 241 B Current operator for the scattering theory 243 C Fluctuation dissipation theorem 245 C.1 Linear response theory and susceptibility 245 C.2 Derivation of the fluctuation dissipation theorem 247 D Derivation of the Boltzmann Langevin noise formula 249 E Reflection coefficient in electronic circuits 253 References 255 Index 275