Semiclassical Electron Transport
|
|
- Basil Todd
- 6 years ago
- Views:
Transcription
1 Semiclassical Electron Transport Branislav K. Niolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 64: Introduction to Solid State Physics
2 Quasiparticle Propagation: Bloch Wavepacets Wavepacets represent quasiparticle localized in space: In the wea external field we can neglect the transitions between different bands fix the zone index n=const. Ψ ( r ) = C ( t) Φ ( r )
3 Semiclassical Dynamics of Bloch Wavepacets Wavepacet in external field described by potential ħ Ψ + U ( r ) + Wext ( r ) Ψ ( r ) = i m ħ ε t Introduce new operator ε( i ) Electron in a crystal represented as wavepacet of Bloch states propagates as a free quasiparticle with charge e, energy ( ), and Hamiltonian ε( i ) : P ˆ, ε ( i ) = 0 W ( r ) Ψ C ( t) ( ) + Wext ( r) Φ ( r) = iħ t ε( ) εe R R R ir ε( i ) Φ ( r) = εe Φ ( r) = εφ ( r + R) = e εφ ( r) = ε( ) Φ( r) Band energy is periodic function in the reciprocal space ext = ir R R R R R R Ψ Ψ( r) C ( t) [ ε( i ) + W ( )] ( ) [ ( ) ( )] ext r Φ r = iħ ε i + Wext r Ψ ( r) = iħ t t ε
4 Hamiltonian, Velocity, and Effective Mass of Bloch Electron ħ * m = d ε( ) d v Ehrenfest theorem: Schrödinger evolution of the center of wavepacet can be obtained from the trajectory of the corresponding classical particle: p correspondence principle: i ħ p H ( r, p ) = E + W ext ( r ) ħ H ( r, p ) 1 ε ( ) = = p ħ tensor of inverse effective mass: d 1 ε ( ) 1 ε( ) ħ a = = dt ħ ħ t ħ H = = W ext ( r ) t r 1 1 ε ( ) = m αβ α ħ β
5 Limitations of Semiclassical Dynamics The spatial scale of all external potentials must be much larger than interatomic spacing, maing it possible to construct wave pacets spanning many unit cells, but seeing the external potential as very slowly varying The magnitude of the electric field cannot be too large, or else they induce Zenner tunneling between bands (electrons are quite effective in screening external fields so this strengths are hard to achieve in metals): ee F ε gap ε ε gap F The magnitude of magnetic fields cannot be too large: ħeb ε m ħω = ε gap The fields must be slowly varying to avoid excitations across the gap. gap ε ε gap F
6 Currents in Bands: Insulators Add Pauli principle to Bloch electron propagation under the influence of an applied electric field Full band of states is insulating! e ( ) j = ε d 8 πħ ( π ) First BZ time reversal invariance: ε ( ) = ε ( ) ε ( ) = ε ( ) 1 v = ε ( ) = ε ( ) = v j 0 for insulator ħ 1
7 Currents in a Band: Metals Fermi sea of a partially filled band will shift under the influence of an applied electric field this destroys inversion symmetry of the Fermi sea therefore causing a net current: e ε j = ( ) d 8 πħ ( π) Occupied 1 e ε( ) j = d 8 πħ ( π) Empty e 1 e 1 j = ε( ) d ε( ) d 8 π ( π) 8 π ( π) ħ ħ First BZ Empty 1
8 Properties of Holes A nearly full simple band has states near the Fermi surface that can be thermally or electrically excited of a negative mass Density of states with holes at the top of the band which have positive charge and positive mass 1 d ε * d ε ħ m * top ħ ħ = ± π / a 1 1 ( ) ( ) = = = < 0 m d d ta ħ 1 d ee ε top = ε 0 t + ta = ε 0 t + vɺ = ( ) * ε = * m ħ dt m
9 Currents in Many Bands: Insulators at Finite Temperature An insulator forms when Fermi energy falls within the gap of the density of states as the temperature is raised, electrons are - Eg / BT promoted over the gap n e and both electrons and holes contribute to the conductivity which increases with temperature.
10 Sources of Electric Resistivity Bloch states are stationary states that describe unperturbed propagation of electrons perfect lattice yields no resistivity! Resistivity is dominated by the scattering off the deviations from a perfect lattice: defects and lattice vibrations=phonons Resistivity arises also due to electron-electron interactions (from simple order of magnitude arguments based on relative strength of the interactions, their contribution should dominate but due to the Pauli principle it does not).
11 Distribution Function In and Out of Equilibrium equilibrium: f ( ) = f ( r,, t) = eq E = 0 ( ( ) F ) e β ε ε no scattering: (,, ) (, ee f r t = f r vdt + dt, t dt) ħ ee f f ( r,, t) = f ( r vdt, + dt, t dt) + dt ħ t scattering e n( r ) = d f ( r,, t) 8π e j( r ) = d v( ) f ( r,, t) 8π
12 Boltzmann Equation ee f f ( r,, t ) = f ( r,, t ) v + f f + r ħ t t ee f f + v f + f = t r ħ t scattering scattering If the phonon and defect perturbations are small, time-independent, and described by Hamiltonian Ĥ, then the scattering rate from a Bloch state to (occupied to unoccupied) is w = π Hˆ ħ f V = d (1 f ( )) w f ( ) (1 f ( )) w f ( ) t ( π ) scattering The Boltzmann equation is valid under assumptions of semi-classical transport: Effective mass approximation (which incorporates the quantum effects due to periodicity of the crystal); Born approximation for the collisions, in the limit of small perturbation for the electron-phonon interaction and instantaneous collisions; no memory effects, i.e. no dependence on initial condition terms. The phonons are usually treated as in equilibrium, although the condition of non-equilibrium phonons may be included through an additional equation.
13 Relaxation Time Approximation Ansatz: The rate at which a system returns to equilibrium is proportional to its deviation from equilibrium (i.e., we mae the assumption that scattering merely acts to drive a non-equilibrium system bac to equilibrium): f t = f ( ) feq( ) τ( ) scattering If E 0, t < 0 and then at t 0, E 0 the external electric field is switched off, then for a homogeneous system we find: f f f feq = = f feq = f ( t = 0) f eq e t t scattering τ f f e f f f ( ) f ( ) = 0, = 0 E = = t r ħ t τ( ) In the steady state transport regime induced by a time-independent external electric field: e f ( ) = f eq ( ) + τ ( ) E ħ f ( ) scattering t / τ
14 Linear Semiclassical Response For small electric field (Ohmic regime), the relaxation time approximation solution can be linearized: e f ( ) feq ( ) + τ ( ) E feq ( ) ħ e E = E ˆ xx f ( ) feq + τ ( ) Ex ħ According to the linear Boltzmann equation, the effect of the electric field E x is to shift the Fermi surface by δ x = eτ E / ħ x Note that elastic scattering cannot restore equilibrium; rather they would cause Fermi surface to expand inelastic scattering (i.e., from phonons) is needed to explain relaxation.
15 Drude Conductivity: Naïve Derivation Drude (1900) assumptions: all electrons participate and electron-lattice -1 scattering yields a scattering rate τ which introduces the mean free time (or relaxation time or collision time which electrons travel, on average, between collisions): τ m mvɺ + ( v vtherm ) = ee τ drift velocity: v v = v friction: eτ m v τ D therm ne τ steady state: vd = E j = envd = E m m ne τ Ohmic conductivity: j = σ E σ = m eτ Mobility: σ = µ ne µ = m Drift
16 Drude Conductivity: Bloch-Boltzmann Derivation Quantum Mechanics (197) maes Drude reasoning problematic not all electrons can participate in the conduction due to the Pauli principle! e e eτ ( ) eq j = d v( ) f ( ) d v( ) f ( ) eq + Ex 8π 8π ħ x isotropic material: jy = jz = 0 ( ) v = v v f 0 eq d = feq ( ) f E feq = = ħvx δ ( E EF ) ħvx E E x x e f j E d v E 8π E eq x ( ) x xτ = σ x f
17 Drude-Boltzmann Conductivity as a Fermi Surface Property d = ds d = ds E E de ħv( ) j e v ( ) e v ( ) = = ds de ( ) ( E E ) = ds ( ) E v v ħ spherical Fermi surface: ds ( ) ( E ) v( E ) ( E ) m x x x E F E x 8π ħ ( ) 8π ħ E= E ( ) F vx ( ) 4π 4π E τ = Fτ F F = Fτ F ( ) E= E v σ τ δ τ F 4 BT E e π ħ F F e τ E F ( ) Fτ F * * π F = π n ( ) σ = E = n 8 ħ m m F *
18 Temperature Dependence of Drude-Boltzmann Conductivity Mattheiesen Rule: the phonon and defect scattering mechanism are independent = + ρ = ρ ρ phonon + τ τ τ phonon defect defect T T ρ = at + Debye ρ defect
19 Drift-Diffusion approximation to Boltzmann Equation for Semiconductors * The drift-diffusion equations are derived introducing the mobility µ = eτ / m and * replacing v = BT / m with its average equilibrium value, therefore neglecting thermal effects. The diffusion coefficient D = µ (Einstein's relation) is BT / e also introduced, and the resulting drift-diffusion current is: dn( x) n 1 jn = eµ nn( x) E( x) + ed = jn + U n n dx t e C ontin uity: dp( x) p 1 j p = eµ p p( x) E( x) ed = j p + U p dx t e + q( n p + N A N D ) Poisson: V = ε The choice of equilibrium (thermal) velocity means that the drift-diffusion equations are only valid for very small perturbations of the equilibrium state (low fields). The validity of the drift-diffusion equations is empirically extended by introducing field-dependent mobility µ ( E) and diffusion coefficient D( E), obtained from empirical models or detailed calculations. 0 p
Unit III Free Electron Theory Engineering Physics
. Introduction The electron theory of metals aims to explain the structure and properties of solids through their electronic structure. The electron theory is applicable to all solids i.e., both metals
More informationMinimal Update of Solid State Physics
Minimal Update of Solid State Physics It is expected that participants are acquainted with basics of solid state physics. Therefore here we will refresh only those aspects, which are absolutely necessary
More informationElectrical Transport. Ref. Ihn Ch. 10 YC, Ch 5; BW, Chs 4 & 8
Electrical Transport Ref. Ihn Ch. 10 YC, Ch 5; BW, Chs 4 & 8 Electrical Transport The study of the transport of electrons & holes (in semiconductors) under various conditions. A broad & somewhat specialized
More informationCondensed matter theory Lecture notes and problem sets 2012/2013
Condensed matter theory Lecture notes and problem sets 2012/2013 Dmitri Ivanov Recommended books and lecture notes: [AM] N. W. Ashcroft and N. D. Mermin, Solid State Physics. [Mar] M. P. Marder, Condensed
More informationCondensed Matter Physics 2016 Lecture 13/12: Charge and heat transport.
Condensed Matter Physics 2016 Lecture 13/12: Charge and heat transport. 1. Theoretical tool: Boltzmann equation (review). 2. Electrical and thermal conductivity in metals. 3. Ballistic transport and conductance
More informationChapter 2. Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene. Winter 05/06
Winter 05/06 : Grundlagen und Anwendung spinabhängiger Transportphänomene Chapter 2 : Grundlagen und Anwendung spinabhängiger Transportphänomene 1 Winter 05/06 2.0 Scattering of charges (electrons) In
More information7. FREE ELECTRON THEORY.
7. FREE ELECTRON THEORY. Aim: To introduce the free electron model for the physical properties of metals. It is the simplest theory for these materials, but still gives a very good description of many
More informationChapter 6 Free Electron Fermi Gas
Chapter 6 Free Electron Fermi Gas Free electron model: The valence electrons of the constituent atoms become conduction electrons and move about freely through the volume of the metal. The simplest metals
More information7.4. Why we have two different types of materials: conductors and insulators?
Phys463.nb 55 7.3.5. Folding, Reduced Brillouin zone and extended Brillouin zone for free particles without lattices In the presence of a lattice, we can also unfold the extended Brillouin zone to get
More informationElectrons in a periodic potential: Free electron approximation
Dr. A. Sapelin, Jan 01 Electrons in a periodic potential: ree electron approximation ree electron ermi gas - gas of non-interacting electrons subject to Pauli principle Wealy bound electrons move freely
More informationElectrons & Phonons. Thermal Resistance, Electrical Resistance P = I 2 R T = P R TH V = I R. R = f( T)
lectrons & Phonons Ohm s & Fourier s Laws Mobility & Thermal Conductivity Heat Capacity Wiedemann-Franz Relationship Size ffects and Breadown of Classical Laws 1 Thermal Resistance, lectrical Resistance
More informationSemiconductor Physics and Devices Chapter 3.
Introduction to the Quantum Theory of Solids We applied quantum mechanics and Schrödinger s equation to determine the behavior of electrons in a potential. Important findings Semiconductor Physics and
More informationChapter 5. Carrier Transport Phenomena
Chapter 5 Carrier Transport Phenomena 1 We now study the effect of external fields (electric field, magnetic field) on semiconducting material 2 Objective Discuss drift and diffusion current densities
More informationJournal of Atoms and Molecules
Research article Journal of Atoms and Molecules An International Online Journal ISSN 77 147 Hot Electron Transport in Polar Semiconductor at Low Lattice Temperature A. K. Ghorai Physics Department, Kalimpong
More informationOhm s Law. R = L ρ, (2)
Ohm s Law Ohm s Law which is perhaps the best known law in all of Physics applies to most conducting bodies regardless if they conduct electricity well or poorly, or even so poorly they are called insulators.
More informationCarrier Action: Motion, Recombination and Generation. What happens after we figure out how many electrons and holes are in the semiconductor?
Carrier Action: Motion, Recombination and Generation. What happens after we figure out how many electrons and holes are in the semiconductor? 1 Carrier Motion I Described by 2 concepts: Conductivity: σ
More informationIntroduction to Engineering Materials ENGR2000. Dr.Coates
Introduction to Engineering Materials ENGR2000 Chapter 18: Electrical Properties Dr.Coates 18.2 Ohm s Law V = IR where R is the resistance of the material, V is the voltage and I is the current. l R A
More informationFermi surfaces which produce large transverse magnetoresistance. Abstract
Fermi surfaces which produce large transverse magnetoresistance Stephen Hicks University of Florida, Department of Physics (Dated: August 1, ) Abstract The Boltzmann equation is used with elastic s-wave
More informationChapter 12: Semiconductors
Chapter 12: Semiconductors Bardeen & Shottky January 30, 2017 Contents 1 Band Structure 4 2 Charge Carrier Density in Intrinsic Semiconductors. 6 3 Doping of Semiconductors 12 4 Carrier Densities in Doped
More informationChapter 3 Properties of Nanostructures
Chapter 3 Properties of Nanostructures In Chapter 2, the reduction of the extent of a solid in one or more dimensions was shown to lead to a dramatic alteration of the overall behavior of the solids. Generally,
More information3. LATTICE VIBRATIONS. 3.1 Sound Waves
3. LATTIC VIBRATIONS Atoms in lattice are not stationary even at T 0K. They vibrate about particular equilibrium positions at T 0K ( zero-point energy). For T > 0K, vibration amplitude increases as atoms
More informationCarriers Concentration and Current in Semiconductors
Carriers Concentration and Current in Semiconductors Carrier Transport Two driving forces for carrier transport: electric field and spatial variation of the carrier concentration. Both driving forces lead
More informationSemiclassical formulation
The story so far: Transport coefficients relate current densities and electric fields (currents and voltages). Can define differential transport coefficients + mobility. Drude picture: treat electrons
More informationMotion and Recombination of Electrons and Holes
Chater Motion and Recombination of Electrons and Holes OBJECTIVES. Understand how the electrons and holes resond to an electric field (drift).. Understand how the electrons and holes resond to a gradient
More informationPhysics 127a: Class Notes
Physics 127a: Class Notes Lecture 15: Statistical Mechanics of Superfluidity Elementary excitations/quasiparticles In general, it is hard to list the energy eigenstates, needed to calculate the statistical
More informationChapter 3: Introduction to the Quantum Theory of Solids
Chapter 3: Introduction to the Quantum Theory of Solids Determine the properties of electrons in a crystal lattice. Determine the statistical characteristics of the very large number of electrons in a
More informationElectrical conduction in solids
Equations of motion Electrical conduction in solids Electrical conduction is the movement of electrically charged particles through a conductor or semiconductor, which constitutes an electric current.
More informationLecture 15: Optoelectronic devices: Introduction
Lecture 15: Optoelectronic devices: Introduction Contents 1 Optical absorption 1 1.1 Absorption coefficient....................... 2 2 Optical recombination 5 3 Recombination and carrier lifetime 6 3.1
More informationSemiconductor Physics. Lecture 3
Semiconductor Physics Lecture 3 Intrinsic carrier density Intrinsic carrier density Law of mass action Valid also if we add an impurity which either donates extra electrons or holes the number of carriers
More informationCalculating Band Structure
Calculating Band Structure Nearly free electron Assume plane wave solution for electrons Weak potential V(x) Brillouin zone edge Tight binding method Electrons in local atomic states (bound states) Interatomic
More informationNasser S. Alzayed.
Lecture #4 Nasser S. Alzayed nalzayed@ksu.edu.sa ELECTRICAL CONDUCTIVITY AND OHM'S LAW The momentum of a free electron is related to the wavevector by mv = ћk. In an electric field E and magnetic field
More informationCondensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras
Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras Lecture - 10 The Free Electron Theory of Metals - Electrical Conductivity (Refer Slide Time: 00:20)
More informationSolid State Physics. Lecture 10 Band Theory. Professor Stephen Sweeney
Solid State Physics Lecture 10 Band Theory Professor Stephen Sweeney Advanced Technology Institute and Department of Physics University of Surrey, Guildford, GU2 7XH, UK s.sweeney@surrey.ac.uk Recap from
More informationSpins and spin-orbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spin-orbit coupling in semiconductors, metals, and nanostructures Behavior of non-equilibrium spin populations. Spin relaxation and spin transport. How does one produce
More informationsin[( t 2 Home Problem Set #1 Due : September 10 (Wed), 2008
Home Problem Set #1 Due : September 10 (Wed), 008 1. Answer the following questions related to the wave-particle duality. (a) When an electron (mass m) is moving with the velocity of υ, what is the wave
More informationPHYSICAL SCIENCES PART A
PHYSICAL SCIENCES PART A 1. The calculation of the probability of excitation of an atom originally in the ground state to an excited state, involves the contour integral iωt τ e dt ( t τ ) + Evaluate the
More informationChapter 5 Phonons II Thermal Properties
Chapter 5 Phonons II Thermal Properties Phonon Heat Capacity < n k,p > is the thermal equilibrium occupancy of phonon wavevector K and polarization p, Total energy at k B T, U = Σ Σ < n k,p > ħ k, p Plank
More informationLecture 4: Basic elements of band theory
Phys 769 Selected Topics in Condensed Matter Physics Summer 010 Lecture 4: Basic elements of band theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 Introduction Most matter, in particular most insulating
More informationPHYS208 p-n junction. January 15, 2010
1 PHYS208 p-n junction January 15, 2010 List of topics (1) Density of states Fermi-Dirac distribution Law of mass action Doped semiconductors Dopinglevel p-n-junctions 1 Intrinsic semiconductors List of
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More informationSEMICONDUCTOR PHYSICS REVIEW BONDS,
SEMICONDUCTOR PHYSICS REVIEW BONDS, BANDS, EFFECTIVE MASS, DRIFT, DIFFUSION, GENERATION, RECOMBINATION February 3, 2011 The University of Toledo, Department of Physics and Astronomy SSARE, PVIC Principles
More informationMetals: the Drude and Sommerfeld models p. 1 Introduction p. 1 What do we know about metals? p. 1 The Drude model p. 2 Assumptions p.
Metals: the Drude and Sommerfeld models p. 1 Introduction p. 1 What do we know about metals? p. 1 The Drude model p. 2 Assumptions p. 2 The relaxation-time approximation p. 3 The failure of the Drude model
More informationMicroscopic Ohm s Law
Microscopic Ohm s Law Outline Semiconductor Review Electron Scattering and Effective Mass Microscopic Derivation of Ohm s Law 1 TRUE / FALSE 1. Judging from the filled bands, material A is an insulator.
More informationParallel Ensemble Monte Carlo for Device Simulation
Workshop on High Performance Computing Activities in Singapore Dr Zhou Xing School of Electrical and Electronic Engineering Nanyang Technological University September 29, 1995 Outline Electronic transport
More informationNon-Continuum Energy Transfer: Phonons
Non-Continuum Energy Transfer: Phonons D. B. Go Slide 1 The Crystal Lattice The crystal lattice is the organization of atoms and/or molecules in a solid simple cubic body-centered cubic hexagonal a NaCl
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationN independent electrons in a volume V (assuming periodic boundary conditions) I] The system 3 V = ( ) k ( ) i k k k 1/2
Lecture #6. Understanding the properties of metals: the free electron model and the role of Pauli s exclusion principle.. Counting the states in the E model.. ermi energy, and momentum. 4. DOS 5. ermi-dirac
More informationRelativistic magnetotransport in graphene
Relativistic magnetotransport in graphene Markus Müller in collaboration with Lars Fritz (Harvard) Subir Sachdev (Harvard) Jörg Schmalian (Iowa) Landau Memorial Conference June 6, 008 Outline Relativistic
More information1. Introduction of solid state 1.1. Elements of solid state physics:
1. Introduction of solid state 1.1. Elements of solid state physics: To understand the operation of many of the semiconductor devices we need, at least, an appreciation of the solid state physics of homogeneous
More informationLecture 3: Optical Properties of Insulators, Semiconductors, and Metals. 5 nm
Metals Lecture 3: Optical Properties of Insulators, Semiconductors, and Metals 5 nm Course Info Next Week (Sept. 5 and 7) no classes First H/W is due Sept. 1 The Previous Lecture Origin frequency dependence
More informationNiS - An unusual self-doped, nearly compensated antiferromagnetic metal [Supplemental Material]
NiS - An unusual self-doped, nearly compensated antiferromagnetic metal [Supplemental Material] S. K. Panda, I. dasgupta, E. Şaşıoğlu, S. Blügel, and D. D. Sarma Partial DOS, Orbital projected band structure
More informationMat E 272 Lecture 25: Electrical properties of materials
Mat E 272 Lecture 25: Electrical properties of materials December 6, 2001 Introduction: Calcium and copper are both metals; Ca has a valence of +2 (2 electrons per atom) while Cu has a valence of +1 (1
More informationTransport Properties of Semiconductors
SVNY85-Sheng S. Li October 2, 25 15:4 7 Transport Properties of Semiconductors 7.1. Introduction In this chapter the carrier transport phenomena in a semiconductor under the influence of applied external
More informationAssumptions of classical free electron model
Module 2 Electrical Conductivity in metals & Semiconductor 1) Drift Velocity :- The Velocity attain by an Electron in the Presence of applied electronic filed is Known as drift Velocity. 2) Mean free Path:-
More information8.1 Drift diffusion model
8.1 Drift diffusion model Advanced theory 1 Basic Semiconductor Equations The fundamentals of semiconductor physic are well described by tools of quantum mechanic. This point of view gives us a model of
More informationThree Most Important Topics (MIT) Today
Three Most Important Topics (MIT) Today Electrons in periodic potential Energy gap nearly free electron Bloch Theorem Energy gap tight binding Chapter 1 1 Electrons in Periodic Potential We now know the
More informationCarrier Mobility and Hall Effect. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India
Carrier Mobility and Hall Effect 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013 calculation Calculate the hole and electron densities
More informationCorrelations between spin accumulation and degree of time-inverse breaking for electron gas in solid
Correlations between spin accumulation and degree of time-inverse breaking for electron gas in solid V.Zayets * Spintronic Research Center, National Institute of Advanced Industrial Science and Technology
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2014 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 POP QUIZ Phonon dispersion relation:
More informationMaterials & Properties II: Thermal & Electrical Characteristics. Sergio Calatroni - CERN
Materials & Properties II: Thermal & Electrical Characteristics Sergio Calatroni - CERN Outline (we will discuss mostly metals) Electrical properties - Electrical conductivity o Temperature dependence
More informationReview of Semiconductor Physics
Solid-state physics Review of Semiconductor Physics The daunting task of solid state physics Quantum mechanics gives us the fundamental equation The equation is only analytically solvable for a handful
More informationCHARGE TRANSPORT (Katharina Broch, )
CHARGE TRANSPORT (Katharina Broch, 27.04.2017) References The following is based on these references: K. Seeger Semiconductor Physics, Springer Verlag, 9 th edition 2004 D. Jena Charge Transport in Semiconductors
More informationPhysics of Semiconductors
Physics of Semiconductors 13 th 2016.7.11 Shingo Katsumoto Department of Physics and Institute for Solid State Physics University of Tokyo Outline today Laughlin s justification Spintronics Two current
More informationConducting surface - equipotential. Potential varies across the conducting surface. Lecture 9: Electrical Resistance.
Lecture 9: Electrical Resistance Electrostatics (time-independent E, I = 0) Stationary Currents (time-independent E and I 0) E inside = 0 Conducting surface - equipotential E inside 0 Potential varies
More informationBonding in solids The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together.
Bonding in solids The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together. For example Nacl In the Nacl lattice, each Na atom is
More informationPHYS208 P-N Junction. Olav Torheim. May 30, 2007
1 PHYS208 P-N Junction Olav Torheim May 30, 2007 1 Intrinsic semiconductors The lower end of the conduction band is a parabola, just like in the quadratic free electron case (E = h2 k 2 2m ). The density
More informationCurrent mechanisms Exam January 27, 2012
Current mechanisms Exam January 27, 2012 There are four mechanisms that typically cause currents to flow: thermionic emission, diffusion, drift, and tunneling. Explain briefly which kind of current mechanisms
More informationMTLE-6120: Advanced Electronic Properties of Materials. Intrinsic and extrinsic semiconductors. Reading: Kasap:
MTLE-6120: Advanced Electronic Properties of Materials 1 Intrinsic and extrinsic semiconductors Reading: Kasap: 5.1-5.6 Band structure and conduction 2 Metals: partially filled band(s) i.e. bands cross
More informationNonlinear resistance of two-dimensional electrons in crossed electric and magnetic fields
Nonlinear resistance of two-dimensional electrons in crossed electric and magnetic fields Jing Qiao Zhang and Sergey Vitkalov* Department of Physics, City College of the City University of New York, New
More informationNote that it is traditional to draw the diagram for semiconductors rotated 90 degrees, i.e. the version on the right above.
5 Semiconductors The nearly free electron model applies equally in the case where the Fermi level lies within a small band gap (semiconductors), as it does when the Fermi level lies within a band (metal)
More informationIn an electric field R and magnetic field B, the force on an electron (charge e) is given by:
Lecture 17 Electric conduction Electrons motion in magnetic field Electrons thermal conductivity Brief review In solid state physics, we do not think about electrons zipping around randomly in real space.
More informationFYS Vår 2017 (Kondenserte fasers fysikk)
FYS3410 - Vår 2017 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/fys3410/v16/index.html Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9, 11, 17, 18,
More informationElectronic and Optoelectronic Properties of Semiconductor Structures
Electronic and Optoelectronic Properties of Semiconductor Structures Jasprit Singh University of Michigan, Ann Arbor CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE INTRODUCTION xiii xiv 1.1 SURVEY OF ADVANCES
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 7: Magnetic excitations - Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations. External parameter, as for
More informationSemiconductor Device Physics
1 Semiconductor Device Physics Lecture 3 http://zitompul.wordpress.com 2 0 1 3 Semiconductor Device Physics 2 Three primary types of carrier action occur inside a semiconductor: Drift: charged particle
More informationLecture. Ref. Ihn Ch. 3, Yu&Cardona Ch. 2
Lecture Review of quantum mechanics, statistical physics, and solid state Band structure of materials Semiconductor band structure Semiconductor nanostructures Ref. Ihn Ch. 3, Yu&Cardona Ch. 2 Reminder
More informationLecture 9 - Carrier Drift and Diffusion (cont.), Carrier Flow. September 24, 2001
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-1 Lecture 9 - Carrier Drift and Diffusion (cont.), Carrier Flow September 24, 2001 Contents: 1. Quasi-Fermi levels 2. Continuity
More informationThe Oxford Solid State Basics
The Oxford Solid State Basics Steven H. Simon University of Oxford OXFORD UNIVERSITY PRESS Contents 1 About Condensed Matter Physics 1 1.1 What Is Condensed Matter Physics 1 1.2 Why Do We Study Condensed
More informationADVANCED UNDERGRADUATE LABORATORY EXPERIMENT 20. Semiconductor Resistance, Band Gap, and Hall Effect
ADVANCED UNDERGRADUATE LABORATORY EXPERIMENT 20 Semiconductor Resistance, Band Gap, and Hall Effect Revised: November 1996 by David Bailey March 1990 by John Pitre & Taek-Soon Yoon Introduction Solid materials
More informationReview of Optical Properties of Materials
Review of Optical Properties of Materials Review of optics Absorption in semiconductors: qualitative discussion Derivation of Optical Absorption Coefficient in Direct Semiconductors Photons When dealing
More informationList of Comprehensive Exams Topics
List of Comprehensive Exams Topics Mechanics 1. Basic Mechanics Newton s laws and conservation laws, the virial theorem 2. The Lagrangian and Hamiltonian Formalism The Lagrange formalism and the principle
More information5. Superconductivity. R(T) = 0 for T < T c, R(T) = R 0 +at 2 +bt 5, B = H+4πM = 0,
5. Superconductivity In this chapter we shall introduce the fundamental experimental facts about superconductors and present a summary of the derivation of the BSC theory (Bardeen Cooper and Schrieffer).
More informationThe Electromagnetic Properties of Materials
The lectromagnetic Properties of Materials lectrical conduction Metals Semiconductors Insulators (dielectrics) Superconductors Magnetic materials Ferromagnetic materials Others Photonic Materials (optical)
More informationCourse overview. Me: Dr Luke Wilson. The course: Physics and applications of semiconductors. Office: E17 open door policy
Course overview Me: Dr Luke Wilson Office: E17 open door policy email: luke.wilson@sheffield.ac.uk The course: Physics and applications of semiconductors 10 lectures aim is to allow time for at least one
More informationUnified theory of quantum transport and quantum diffusion in semiconductors
Paul-Drude-Institute for Solid State Electronics p. 1/? Unified theory of quantum transport and quantum diffusion in semiconductors together with Prof. Dr. V.V. Bryksin (1940-2008) A.F. Ioffe Physical
More informationChap 7 Non-interacting electrons in a periodic potential
Chap 7 Non-interacting electrons in a periodic potential Bloch theorem The central equation Brillouin zone Rotational symmetry Dept of Phys M.C. Chang Bloch recalled, The main problem was to explain how
More informationLecture 8 - Carrier Drift and Diffusion (cont.), Carrier Flow. February 21, 2007
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 8-1 Lecture 8 - Carrier Drift and Diffusion (cont.), Carrier Flow February 21, 2007 Contents: 1. Quasi-Fermi levels 2. Continuity
More informationAnderson Localization Looking Forward
Anderson Localization Looking Forward Boris Altshuler Physics Department, Columbia University Collaborations: Also Igor Aleiner Denis Basko, Gora Shlyapnikov, Vincent Michal, Vladimir Kravtsov, Lecture2
More informationElectron levels in a periodic potential: general remarks. Daniele Toffoli January 11, / 46
Electron levels in a periodic potential: general remarks Daniele Toffoli January 11, 2017 1 / 46 Outline 1 Mathematical tools 2 The periodic potential 3 Bloch s theorem and Born-von Karman boundary conditions
More informationSection 10 Metals: Electron Dynamics and Fermi Surfaces
Electron dynamics Section 10 Metals: Electron Dynamics and Fermi Surfaces The next important subject we address is electron dynamics in metals. Our consideration will be based on a semiclassical model.
More informationQuantum Condensed Matter Physics Lecture 9
Quantum Condensed Matter Physics Lecture 9 David Ritchie QCMP Lent/Easter 2018 http://www.sp.phy.cam.ac.uk/drp2/home 9.1 Quantum Condensed Matter Physics 1. Classical and Semi-classical models for electrons
More informationElectrical Transport in Nanoscale Systems
Electrical Transport in Nanoscale Systems Description This book provides an in-depth description of transport phenomena relevant to systems of nanoscale dimensions. The different viewpoints and theoretical
More informationMP464: Solid State Physics Problem Sheet
MP464: Solid State Physics Problem Sheet 1 Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred rectangular
More information8.512 Theory of Solids II Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 8.5 Theory of Solids II Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture : The Kondo Problem:
More informationPhonons Thermal energy Heat capacity Einstein model Density of states Debye model Anharmonic effects Thermal expansion Thermal conduction by phonons
3b. Lattice Dynamics Phonons Thermal energy Heat capacity Einstein model Density of states Debye model Anharmonic effects Thermal expansion Thermal conduction by phonons Neutron scattering G. Bracco-Material
More informationQuantum Molecular Dynamics Basics
Quantum Molecular Dynamics Basics Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Depts. of Computer Science, Physics & Astronomy, Chemical Engineering & Materials Science, and Biological
More informationThermoelectrics: A theoretical approach to the search for better materials
Thermoelectrics: A theoretical approach to the search for better materials Jorge O. Sofo Department of Physics, Department of Materials Science and Engineering, and Materials Research Institute Penn State
More informationSession 5: Solid State Physics. Charge Mobility Drift Diffusion Recombination-Generation
Session 5: Solid State Physics Charge Mobility Drift Diffusion Recombination-Generation 1 Outline A B C D E F G H I J 2 Mobile Charge Carriers in Semiconductors Three primary types of carrier action occur
More informationIntroduction to Quantum Theory of Solids
Lecture 5 Semiconductor physics III Introduction to Quantum Theory of Solids 1 Goals To determine the properties of electrons in a crystal lattice To determine the statistical characteristics of the very
More informationSemiconductor Physics
Semiconductor Physics Motivation Is it possible that there might be current flowing in a conductor (or a semiconductor) even when there is no potential difference supplied across its ends? Look at the
More information