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| A primer on electron transport | |
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| Nanoscale systems | |
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| Generating currents | |
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| Finite versus infinite systems | |
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| Electron sources | |
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| Intrinsic nature of the transport problem | |
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| Measuring currents | |
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| Microscopic states | |
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| The current operator | |
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| The measurement process | |
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| Complete measurement and pure states | |
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| The statistical operator and macro-states | |
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| Pure and mixed states | |
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| Quantum correlations | |
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| Time evolution of the statistical operator | |
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| Random or partially specified Hamiltonians | |
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| Open quantum systems | |
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| Equilibrium statistical operators | |
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| Current measurement and statistical operator truncation | |
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| One current, different viewpoints | |
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| Summary and open questions | |
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| Exercises | |
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| Drude model, Kubo formalism and Boltzmann equation | |
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| Drude model | |
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| Resistance, coherent and incoherent transport | |
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| Relaxation vs. dephasing | |
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| Mean-free path | |
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| The meaning of momentum relaxation time | |
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| Kubo formalism | |
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| The current-current response function | |
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| The use of Density-Functional Theory in the Kubo approach | |
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| The fluctuation-dissipation theorem | |
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| Ohmic vs. ballistic regimes | |
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| Chemical, electrochemical and electrostatic potentials | |
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| Drift-diffusion equations | |
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| Diffusion coefficient of an ideal electron gas in the non-degenerate limit | |
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| Generalization to spin-dependent transport | |
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| Distribution functions | |
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| Boltzmann equation | |
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| Approach to local equilibrium | |
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| Entropy, loss of information, and macroscopic irreversibility | |
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| The classical statistical entropy | |
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| Quantum statistical entropy | |
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| Information content of the N- and one-particle statistical operators | |
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| Entropy of open quantum systems | |
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| Loss of information in the Kubo formalism | |
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| Loss of information with stochastic Hamiltonians | |
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| Entropy associated with the measurement of currents | |
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| Summary and open questions | |
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| Exercises | |
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| Landauer approach | |
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| Formulation of the problem | |
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| Local resistivity dipoles and the "field response" | |
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| Conduction from transmission | |
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| Scattering boundary conditions | |
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| Transmission and reflection probabilities | |
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| Total current | |
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| Two-probe conductance | |
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| The Lippmann-Schwinger equation | |
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| Time-dependent Lippmann-Schwinger equation | |
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| Time-independent Lippmann-Schwinger equation | |
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| Green's functions and self-energy | |
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| Relation to scattering theory | |
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| The S matrix | |
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| Relation between the total Green's function and the S matrix | |
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| The transfer matrix | |
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| Coherent scattering of two resistors in series | |
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| Incoherent scattering of two resistors in series | |
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| Relation between the conductance and the transfer matrix | |
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| Localization, ohmic and ballistic regimes | |
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| Four-probe conductance in the non-invasive limit | |
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| Single-channel case | |
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| Geometrical "dilution" | |
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| Multi-channel case | |
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| Multi-probe conductance in the invasive limit | |
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| Floating probes and dephasing | |
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| Generalization to spin-dependent transport | |
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| Spin-dependent transmission functions | |
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| Multi-probe conductance in the presence of a magnetic field | |
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| Local resistivity spin dipoles and dynamical effects | |
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| The use of Density-Functional Theory in the Landauer approach | |
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| Summary and open questions | |
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| Exercises | |
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| Non-equilibrium Green's function formalism | |
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| Formulation of the problem | |
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| Contour ordering | |
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| Equilibrium Green's functions | |
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| Time-ordered Green's functions | |
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| Dyson's equation for interacting particles | |
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| More Green's functions | |
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| The spectral function | |
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| Contour-ordered Green's functions | |
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| Equations of motion for non-equilibrium Green's functions | |
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| Application to steady-state transport | |
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| Coulomb blockade | |
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| Quantum kinetic equations | |
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| Summary and open questions | |
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| Exercises | |
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| Noise | |
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| The moments of the current | |
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| Shot noise | |
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| The classical (Poisson) limit | |
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| Quantum theory of shot noise | |
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| Counting statistics | |
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| Thermal noise | |
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| Summary and open questions | |
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| Exercises | |
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| Electron-ion interaction | |
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| The many-body electron-ion Hamiltonian | |
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| The adiabatic approximation for a current-carrying system | |
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| The phonon subsystem | |
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| Electron-phonon coupling in the presence of current | |
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| Inelastic current | |
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| Inelastic current from standard perturbation theory | |
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| Inelastic current from the NEGF | |
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| Local ionic heating | |
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| Lattice heat conduction | |
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| Thermopower | |
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| Current-induced forces | |
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| Elastic vs. inelastic contribution to electro-migration | |
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| One force, different definitions | |
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| Local resistivity dipoles and the force sign | |
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| Forces at equilibrium | |
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| Forces out of equilibrium | |
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| Are current-induced forces conservative? | |
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| Local ionic heating vs. current-induced forces | |
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| Summary and open questions | |
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| Exercises | |
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| The micro-canonical picture of transport | |
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| Formulation of the problem | |
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| Transport from a finite-system point of view | |
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| Initial conditions and dynamics | |
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| Electrical current theorems within dynamical DFTs | |
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| Closed and finite quantum systems in a pure state | |
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| Closed quantum systems in a pure state with arbitrary boundary conditions | |
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| Current in open quantum systems | |
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| Closure of the BBGKY hierarchy | |
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| Functional approximations and loss of information | |
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| Transient dynamics | |
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| Properties of quasi-steady states | |
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| Variational definition of quasi-steady states | |
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| Dependence of quasi-steady states on initial conditions | |
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| A non-equilibrium entropy principle | |
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| Approach to steady state in nanoscale systems | |
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| Definition of conductance in the micro-canonical picture | |
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| Summary and open questions | |
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| Hydrodynamics of the electron liquid | |
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| The Madelung equations for a single particle | |
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| Hydrodynamic form of the Schrodinger equation | |
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| Quantum Navier-Stokes equations | |
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| Conductance quantization from hydrodynamics | |
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| Viscosity from Time-Dependent Current Density-Functional Theory | |
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| Functional approximation, loss of information, and dissipative dynamics | |
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| Effect of viscosity on resistance | |
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| Turbulent transport | |
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| Local electron heating | |
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| Electron heat conduction | |
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| Hydrodynamics of heat transfer | |
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| Effect of local electron heating on ionic heating | |
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| Summary and open questions | |
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| Exercises | |
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| Appendices | |
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| A primer on second quantization | |
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| The quantum BBGKY hierarchy | |
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| The Lindblad equation | |
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| The Lindblad theorem | |
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| Derivation of the Lindblad equation | |
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| Steady-state solutions | |
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| Ground-state Density-Functional Theory | |
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| The Hohenberg-Kohn theorem | |
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| The Kohn-Sham equations | |
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| Generalization to grand-canonical equilibrium | |
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| The local density approximation and beyond | |
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| Time-Dependent DFT | |
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| The Runge-Gross theorem | |
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| The time-dependent Kohn-Sham equations | |
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| The adiabatic local density approximation | |
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| Time-Dependent Current DFT | |
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| The current density as the main variable | |
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| The exchange-correlation electric field | |
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| Approximate formulas for the viscosity | |
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| Stochastic Time-Dependent Current DFT | |
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| The stochastic Schrodinger equation | |
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| Derivation of the quantum master equation | |
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| The theorem of Stochastic TD-CDFT | |
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| Inelastic corrections to current and shot noise | |
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| Hydrodynamic form of the Schrodinger equation | |
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| Equation of motion for the stress tensor | |
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| Cut-off of the viscosity divergence | |
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| Bernoulli's equation | |
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| References | |
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| Index | |