Quantum Physics of Semiconductor Materials and Devices

Description

”Quantum Phenomena do not occur in a Hilbert space. They occur in a laboratory”. - Asher Peres Semiconductor physics is a laboratory to learn and discover the concepts of quantum mechanics and thermodynamics, condensed matter physics, and materials science, and the payoffs are almost immediate in the form of useful semiconductor devices. Debdeep Jena has had the opportunity to work on both sides of the fence - on the fundamental materials science and quantum physics of semiconductors, and in their applications in semiconductor electronic and photonic devices. In Quantum Physics of Semiconductors and Nanostructures, Jena uses this experience to make each topic as tangible and accessible as possible to students at all levels. Consider the simplest physical processes that occur in semiconductors: electron or hole transport in bands and over barriers, collision of electrons with the atoms in the crystal, or when electrons and holes annihilate each other to produce a photon. The correct explanation of these processes require a quantum mechanical treatment. Any shortcuts lead to misconceptions that can take years to dispel, and sometimes become roadblocks towards a deeper understanding and appreciation of the richness of the subject. A typical introductory course on semiconductor physics would then require prerequisites of quantum mechanics, statistical physics and thermodynamics, materials science, and electromagnetism. Rarely would a student have all this background when (s)he takes a course of this nature in most universities. Jena's work fills in these gaps and gives students the background and deeper understanding of the quantum physics of semiconductors and nanostructures. Cover Titlepage Copyright Dedication Preface Acknowledgement Content I Fundamentals And off We Go! Beyond belief A brief history of semiconductors Future These boots are made for walking Chapter summary section Further reading Exercises Quantum Mechanics in a Nutshell Planck's photon energy quanta Bohr's electron energy quanta Wave-particle duality The wavefunction Operators States of definite momentum and location States of definite energy: The Schrödinger equation Time-dependent Schrödinger equation Stationary states and time evolution Quantum current Fermions and bosons Fermion and boson statistics The Spin-statistics theorem The Dirac equation and the birth of particles Chapter summary section Further reading Exercises Damned Lies and Statistics Quantum statistics and entropy The physics of equilibrium Partition function for quantum systems The Fermi–Dirac distribution The Bose–Einstein distribution Properties of the distribution functions Quantum twist on thermodynamics Meaning of equilibrium in semiconductor devices Chapter summary section Further reading Exercises Electrons in the Quantum World In Schrödinger equation we trust The free electron Not so free: Particle on a ring The electron steps into a higher dimension: 2D Electrons in a 3D box The particle in a box The Dirac delta potential The harmonic oscillator The hydrogen atom Chapter summary section Further reading Exercises Red or Blue Pill: Befriending the Matrix The expansion principle Matrix mechanics Matrices and algebraic functions Properties of matrix eigenvlaues Looking ahead Chapter summary section Further reading Exercises Secrets of the Classical Electron Our ancestors knew metals Discovery of the electron and its aftermath Drude's model explains Ohm's law Metals are shiny Metals conduct heat Icing on the cake: The Wiedemann–Franz law All is not well Chapter summary section Further reading Exercises Quantum Mechanics in a Nutshell Planck’s photon energy quanta Bohr’s electron energy quanta Wave-particle duality The wavefunction Operators States of definite momentum and location States of definite energy: The Schr¨odingerequ tion Time-dependent Schr¨odinger equation Stationary states and time evolution Quantum current Fermions and bosons Fermion and boson statistics The Spin-statistics theorem The Dirac equation and the birth ofparticles Chapter summary section Further reading Exercises Damned Lies and Statistics Quantum statistics and entropy The physics of equilibrium Partition function for quantum systems The Fermi–Dirac distribution The Bose–Einstein distribution Properties of the distribution functions Quantum twist on thermodynamics Meaning of equilibrium in semiconductordevices Chapter summary section Further reading Exercises Electrons in the QuantumWorld In Schr¨odinger equation we trust The free electron Not so free: Particle on a ring The electron steps into a higherdimension: 2D Electrons in a 3D box The particle in a box The Dirac delta potential The harmonic oscillator The hydrogen atom Chapter summary section Further reading Exercises Red or Blue Pill:Befriending the Matrix The expansion principle Matrix mechanics Matrices and algebraic functions Properties of matrix eigenvlaues Looking ahead Chapter summary section Further reading Exercises Perturbations to the Electron's Freedom Degenerate perturbation theory Non-degenerate perturbation theory The Brillouin–Wigner perturbation results Rayleigh–Schrödinger perturbation results The Hellmann–Feynman theorem Perturbation theory example Chapter summary section Further reading Exercises II Bands, Doping, and Heterostructures Electrons in a Crystal Get Their Bands, Gaps, and Masses The free–electron Periodic perturbation Bands, gaps, and effective masses Non-degenerate perturbation theory Glimpses of the Bloch theorem Non-periodic potentials and scattering Chapter summary section Further reading Exercises Bloch Theorem, Bandstructure, and Quantum Currents The Bloch theorem Bloch theorem: aftermath Real and reciprocal lattice, Brillouin zones Velocity of Bloch states Dynamics of Bloch states Bloch wave velocity and ballistic current Transport by Bloch waves with scattering Energy (heat) current Any current Quantum Wiedemann–Franz law Metals, semiconductors, semimetals and insulators Chapter summary section Further reading Exercises Crystal Clear: Bandstructure of the Empty Lattice Diffraction as a sharp eye Bragg diffraction condition Broken symmetries and physical laws Bravais lattices Nearly free–electron bandstructure Chapter summary section Further reading Exercises Tight-Binding Bandstructure Atoms, bonds, and molecules Bandstructure of 1D, 2D, and 3D crystals 1D, 2D: nanotubes, graphene, BN, MX2 3D FCC: Si, GaAs 3D wurtzite: GaN, AlN, ZnO Tight-binding to design new properties Chapter summary section Further reading Exercises k p Bandstructure k p theory Symmetry Analytical model without spin Non-parbolicity and sum rules The Kane model with spin-orbit interaction Chapter summary section Further reading Exercises 1, 2, 3 ... : Pseudopotentials and Exact Bandstructure The empire strikes back Exact bandstructure of the Dirac comb potential Tight-binding models emerge from Kronig–Penney Point defects in Kronig–Penney models Green's functions from Kronig–Penney models Pseudopotentials: what they are and why they work Bandstructure of Si, Ge, and GaAs Bandstructure of AlN, GaN, and InN Pseudopotentials to DFT and beyond Chapter summary section Further reading Exercises Doping and Heterostructures: The Effective Mass Method Effective mass approximation, envelope functions 3D, 2D, 1D, 0D: heterostructures 3D bulk bandstructure Doped semiconductors 2D quantum wells 1D quantum wires 0D quantum dots Finite barrier heights Multilayers and superlattices Wannier functions Chapter summary section Further reading Exercises Carrier Statistics and Energy Band Diagrams Carrier statistics EF is constant at thermal equilibrium Metal-semiconductor Schottky junctions p-n homojunctions Heterojunctions Energy band diagrams: Poisson+Schrödinger Polarization-induced doping in heterostructures Chapter summary section Further reading Exercises Controlling Electron Traffic in the k-Space Electron energies in semiconductors Semiconductor statistics Ballistic transport in semiconductors Ballistic transport in non-uniform potentials/tunneling Scattering of electrons by phonons, defects and photons The Boltzmann transport equation Current flow with scattering: drift and diffusion Explicit calculations of scattering rates and mobility Semiconductor electron energies for photonics The optical joint density of states J() Occupation of electron states for photonics Absorption, and emission: spontaneous and stimulated Chapter summary section Further reading Exercises III Quantum Electronics with Semiconductors Game of Modes: Quantized R, L, and C Classical R, L, C circuits Quantized conductance Quantum capacitance Kinetic inductance Quantum R, L, C circuits Negative R, L, and C Chapter summary section Further reading Exercises Junction Magic: Schottky, pn and Bipolar Transistors Ballistic Schottky diodes pn diodes: discovery pn diodes: transport Bipolar junction transistors Deathniums! Chapter summary section Further reading Exercises Zeroes and Ones: The Ballistic Transistor The MOS capacitor The ballistic FET Ballistic I-V characteristics Quantum wire ballistic FET The drift-diffusion FET CMOS and HEMTs Source/drain ohmic contacts A brief history of FETs Chapter summary section Further reading Exercises Fermi's Golden Rule Fermi's golden rule Oscillating perturbations Transitions to continuum Kubo–Greenwood formula Decoherence in qubits Electron-electron scattering Dyson series and diagrams Zero-sum game: self energy Chapter summary section Further reading Exercises No Turning Back: The Boltzmann Transport Equation Micro vs. macro The Liouville theorem Boltzmann transport equation H-theorem and entropy Equilibrium distribution The RTA: time to relax! One formula to rule them all Electrical conductivity Thermoelectric properties Onsager relations Conservation laws Berry curvature correction Limitations of the BTE Chapter summary section Further reading Exercises Taking the Heat: Phonons and Electron-Phonon Interactions Phonon effects: A résumé Phonon dispersions and DOS Optical conductivity Lyddane–Sachs–Teller equation Acoustic wave devices Thermal conductivity Phonon number quantization Electron-phonon interaction Chapter summary section Further reading Exercises Scattering, Mobility, and Velocity Saturation Electron mobility: a résumé Scattering mechanisms Point defect scattering Coulomb impurity scattering Dipole scattering Dislocation scattering Alloy disorder scattering Interface scattering Phonon scattering Experimental mobilities High-field velocity saturation Chapter summary section Further reading Exercises Through the Barrier: Tunneling and Avalanches Tunneling: a résumé Single-barrier tunneling WKB tunneling theory WKB for semiconductors Nordheim supply function Fowler–Nordheim tunneling Interband Zener tunneling pn tunnel junctions in 1D, 2D, and 3D NDR, backward diodes Tunneling FETs Resonant tunneling Bardeen's tunneling theory Kubo formalism Landau–Zener theory Avalanche processes and impact ionization Tail of the dragon Chapter summary section Further reading Exercises Running Circles: Quantum Magnetotransport Magnetotransport: a résumé Hall effect Magnetoresistance Nernst and Ettingshausen effects Cyclotron resonance Faraday rotation Atoms: Bohr magneton and spin Landau levels in solids Shubnikov de Haas effect The quantum Hall effect Quantum Hall effect theories Hierarchy of Hall effects Chapter summary section Further reading Exercises IV Quantum Photonics with Semiconductors Let There Be Light: Maxwell Equations Maxwell equations in vacuum Light from Maxwell equations Maxwell equations in (k, ) space Maxwell equations in materials Classical light-matter interaction Kramers–Kronig relations Accelerating charges radiate Need for quantum theory of light Chapter summary section Further reading Exercises Light–Matter Interaction Photonic effects: a résumé Electron-photon matrix elements Absorption spectra of semiconductors Number of photons in light Photon absorption rate Equilibrium absorption coefficient Quantum wells, wires, and dots Critical points Forbidden and indirect absorption Exciton absorption Franz–Keldysh effect Intersubband absorption Free carrier, and impurity absorption Photoelectron spectroscopy Chapter summary section Further reading Exercises Heavenly Light: Solar Cells and Photodetectors Solar sells and photodetectors: a résumé Solar cells Shockley–Ramo theorem Photodetectors Avalanche photodiodes Quantum well infrared photodetectors Electro-absorption modulators Solar blind photodetectors Chapter summary section Further reading Exercises Reach for the Stars: Lasers and LEDs Lasers and LEDs: a résumé Einstein's A and B coefficients Semiconductor emission Entropy of optical transitions Gain and emission in bands Spontaneous emission: LEDs Stimulated emission: lasers Double heterostructure lasers Laser rate equations Case study: blue laser diode DFB, VCSELs and QCLs Towards field quantization Broadband field modes Quantization of fields Field quantization: aftermath Quantized light-matter interaction Fundamental optical processes Einstein's re-derivation of Planck's law: back to 12! Chapter summary section Further reading Exercises Every End is a New Beginning Smallest, fastest, brightest Looking back and forward Ferroelectric semiconductors Ferromagnetic semiconductors Multiferroic semiconductors Superconducting semiconductors Semiconductors for quantum communications Semiconductors for quantum computation Semiconductors for energy Semiconductors for healthcare and agriculture Semiconductors for space exploration Social impact of semiconductors Chapter summary section Further reading Exercises Appendix What is in the appendix? Semiconductor Formulae Physical properties of 3D and 2Dsemiconductors References for the appendix Physical constants References Index