The chemical bonds between atoms are not rigid : Act like spring Vibration Lattice vibrations are responsible for transport of energy in many solids Quanta of lattice vibration are called phonons. The 1-D Diatomic Chain Phonon Dispersion and Scattering

harmonic oscillator Equation of motion : Solution : Natural frequency :  Spring-mass model

Equilibrium Position : Deformed position : Assumption All the atoms and the springs between them are the same The force on the n-th atom is only by its neighboring atoms  Lattice vibrations with monatomic basis

Solution Substitute into equation

Frequency,  Wavevector, k 0 First Brillouin zone Slope( ) = Group velocity 1st Brillouin zone : : Dispersion relation

transmission velocity of a wave packet - Wave packet or Wave group - Beats The velocity of energy propagation in the medium This is zero at the edge of the zone Standing wave  Group velocity

- : Long wavelength or continuum limit : Velocity of sound in a crystal Speed of sound Frequency,  Wavevector, k 0 1 Edge of First Brillouin zone - : Edge of first Brillouin zone Standing wave Acoustic branch  Acoustic branch

 Lattice vibrations with Two atoms per primitive basis two atoms per unit cell

Solution Substitute into equation The determinant must be zero for nontrivial solutions

Two roots for Optical branch Acoustic branch

Acoustic branch : The atoms move together, as in long wavelength acoustical vibration. Optical branch : If the two atoms carry opposite charges, we may excite a motion of this type with the electric field of a light wave. Influences optical properties of a crystal Optical branch  Optical branch and Acoustic branch

First Brillouin zone

High frequency First Brillouin zone  Characteristics of optical branch : Group velocity is negligible Between and :no solution

ss+1s+2s+3s-1 u s-1 usus u s+1 u s+2 u s+3 u s-1 usus u s+1 u s+2 Three modes of wave vectors for one atom per unit cell One longitudinal mode Two transverse modes  Transverse vs. Longitudinal polarization

If there are q atoms in the primitive cell, there are 3q branches to the dispersion relation - Number of branches 3 acoustic branches : 1 longitudinal acoustic (LA) 2 transverse acoustic (TA) 3q - 3 optical branches : q - 1 longitudinal optical (LO) 2q - 2 transverse optical (TO) Dispersion Relation for Real Crystal

- Si[100] direction - SiC The group velocity of phonons in the optical branches is small contribute little to the thermal conduction At low temperatures : TA are dominant contributors to the heat conduction At high temperatures : LA are dominant contributors to the heat conduction Frequency gap

Governs the thermal transport properties of dielectric and semiconductor Inelastic scattering : the phonon frequency before the scattering event is different from that after the event Normal (or N) – process : Inside the 1 st brillouin zone Phonon scattering  Phonon-phonon scattering : energy conservation : crystal momentum conservation

Umklapp(or U) process : Outside the 1 st brillouin zone N processU process

N - process U - process Energy Conservation Momentum Conservation Thermal conductivity Conserved Net momentum not conserved Dominant Not dominant Act as a direct resistance to heat flow Distributing the phonon energy Thermophysical role N-process vs. U-process

Scattering rate of the U-process Positive constants Above room temperature Below room temperature - Fig Thermal conductivity of silicon Phonon scattering: Temperature dependence At high temperature specific heat does not change significantly

Four – phonon scattering (Temperature range : 300 K ~ 1000 K) : Negligible Phonon – defect scattering Elastic scattering Independent of temperature Defendant on the phonon wavelength  Phonon scattering – 4 phonon & defect

Dominant at high temperature Scattering by acoustic phonon is essentially elastic. Scattering by optical phonon is inelastic : Polar scattering negligible Facilitates heat transfer between optical phonon and electron (Joule heating) ( i : initial state f : final state)  Phonon scattering- phonon-electron : energy conservation : momentum conservation + : phonon absorption - : phonon emission

Phonon and photon inelastic scattering called Raman scattering, X-ray scattering, neutronscattering, and Brillouin scattering i : incident photon s : scattered photon ph : phonon  Phonon scattering- Raman scattering Stokes shift (phonon emission) anti-Stokes shift (phonon absorption)

Dependence on temperature used for surface temperature measurements

Photoelectric Effect: electromagnetic wave metal plate Heinrich Hertz observed the photoemission in 1887 J. J. Thomson discovered electron as a subatomic particle Albert Einstein explained the photoelectric effect in 1905 (Nobel Prize in 1921) Photoemission Electron Emission and Tunneling

 Measuring the Ejected Photoelectrons electrode A load vacuum incident photon Frequency, of incident radiation is not high enough no electric current threshold frequency for photoemission in given material

 Work Function  work function: energy needed to remove electron from metal Ag, Al, Au, Cu, Fe: 4 ~ 5 eV (ultraviolet region) Na, K, Cs, Ca: 2 ~ 3 eV (visible region) maximum kinetic energy of ejected electron A photon can interact only one electron at a time. electron right at the Fermi level

 Application of Photoemission XPS (x-ray photoelectron spectroscopy) measurement of the electron binding energy, E bd sample chemical composition of the substance near the surface electron energy analyzer

Thermionic Emission hot cold A J Load vacuum Similarity to photoemission → Work function current density

EFEF Fermi-Dirac distribution at T = 0, E E F all states are empty when T > 0, Some electron having more than E F + , Small fraction of electron must occupy energy levels exceeding E F + .

 Current Density Particle flux Current density velocity space: : fraction of electron reflected

number of electrons per unit volume Recall current density in the x direction ejected velocity ( v x ) of electron > binding E bd

when less than 2% error

Let

Richardson-Dushman eq. Richardson constant A RD = 1.202×10 6 A/(m 2 K 2 )

heat transfer associated with electron flow  Heat Flux Flux of energy → similarity to current density : average energy of the “hot electron”

 Derivation of Heat Flux Let

heat transfer associated with electron flow

Field Emission and Electron Tunneling Tunnel Electron wave Electron energy Potential U Potential barrier (hill),  When the field strength is very high, electrons at lower energy levels than the height of the barrier can tunnel through the potential hill. Thermionic emission may be enhanced or even reversed by an applied electric field  Current density Fowler-Nordheim equation  Field emission  (x) : width of potential at E

 Current Density Electron motion: governed by Schrödinger’s wave equation Wavefunction form Time dependent Schrödinger equation when E > U when E < U

 Tunneling current density kinetic energy in the x direction, E energy at the top of potential barrier, E max reference energy, E min number of available electrons, n(E)  Transmission probability or transmission coefficient

Energy barrier with two electrodes  Fowler-Nordheim tunneling chemical potential current density by various approximations electric field, Potential U Tunnel

 Current density positive constant, C positive constant, 

Electrical Transport in Semiconductor Devices Number Density, Mobility, and the Hall Effect Number density of electrons and holes determines the electrical, optical, thermal properties of semiconductor materials.  Number Density electron and hole → Fermi-Dirac distribution function number density of electrons and holes E C : minimum of conduction band E V : maximum of valence band

densities of states in the conduction and valance bands M C : number of equivalent minima in conduction band effective mass for density of states geometric average of 3 masses = longitudinal mass + 2 transverse mass

M C : number of equivalent minima in the conduction band Most semiconductor can be described one band minimum at k = 0 as well as several equivalent anisotropic band minima k ≠ 0 Simplified E-k diagram of silicon within the 1 st Brillouin zone along the (100) direction The energy is chosen to be to zero at the edge of the valence band. Lowest band minimum at k = 0 is not the lowest minimum above the valence band at ( In here x = 5 nm -1 ) There are 6 equivalent minima, these are minimum energy. On the other hand, maxima of valence band only has one k state. So in calculation D e, we must multiply 6 (M C )

we have to consider the effective mass The effective mass of electrons is a geometric average over the 3 major axes because the effective mass of silicon depends on the crystal direction. The effective mass of holes is an average of heavy holes and light holes because there exist different sub-bands. The effective mass calculation for density of states = The geometric average of the 3 masses: (one longitudinal mass m l, two transverse mass m t ) must include the fact that several equivalent minima exist : M C in the conduction band

At moderate temperatures, approximation with M-B distribution N C, N V : effective density of states in conduction band and valance band

number density for intrinsic and doped semiconductors thermally excited electron-hole pairs per unit volume number density of intrinsic carriers Fermi energy for an intrinsic semiconductor when n e = n h Fermi energy for an intrinsic semiconductor in the middle of the forbidden band or the bandgap

 n -type  p -type  Fully ionized impurities, charge neutrality requirement N A, N D : number densities of donors and acceptors impurities of donors ( P, As ) involved → ionization of donors increases the number of free electrons Impurities of acceptors ( B, Ga ) involved → Ionization of acceptors increases the number of holes

Electric Field = 0 so net motion =0 Electric Field ≠ 0 so net motion ≠ 0 Electric Field Electrons and holes are accelerated by electric field, but lose momentum due to scattering processes.  Mobility mobility: ratio of the drift speed to applied electric field mean time between collision, applied electric field to electron, drift velocity,

mobilityconductivity Electrical conductivity of a semiconductor Average energy of semiconductors thermal velocity at equilibrium T

At sufficiently high T, contribution of carrier-phonon scattering Impurities scattering N d : concentration of the ionized impurities Matthiessen’s rule Overall mobility

 Hall Effect useful in measuring the mobility of semiconductors → van der Pauw method (4 probe technique) Net current flow Lorentz force in the y direction

Hall coefficient

Generation and Recombination Photoconductivity: Excitation of electrons from valence band to conduction band by the absorption of radiation increases the conductivity of the semiconductor  Generation Conductivity at thermal equilibrium before incident radiation Relative change in electrical conductance after incident radiation  rc : recombination lifetime or recombination time

 Recombination Relaxation process, related to electron scattering, lattice scattering, defect scattering because the excess charge is not at thermal equilibrium Non-radiative: Auger effect, multiphonon emission Radiative: using in luminescence application (LED) Net rate of change = generation rate – recombination rate

Under steady-state incident radiation  : absorptance I : spectral irradiance of incoming photon ( W/m 2 Hz ) Sensitivity of a photoconductive detector

The p - n Junction Diffusion of electrons and holes → Fick’s law Diffusion coefficients Assume Diffusion coefficient, Einstein relation

pn x N nh(x)nh(x)ne(x)ne(x) p -region n -region ECpECp EVpEVp EFEF ECnECn EVnEVn Concentration gradient holes will diffuse right and electrons will diffuse left. 2. As they leave the host material, ions of opposite charges are left behind. 3. This results in a charge accumulation and consequently it leads to a built-in potential in the depletion region. 4. The energy in the p-doped region rise relatively Therefore, forward bias removes the barrier for elements to diffuse, on the other hand, a reverse bias creates an even stronger barrier → the junction has characteristic of rectification. 1 : electron drift2 : electron diffusion 3 : hole diffusion4 : hole drift depletion region

Current density in semiconductor for charge transfer drift termdiffuse term Under equilibrium condition, J = 0 V bi : built-in potential

 Derivation of Current Density Applied voltage → Non-equilibrium occurs → Elements move J s : Saturation current density

Optoelectronic Applications  Photovoltaic effect Incident upon a p - n junction generates electron-hole pairs → Built-in electric field in the p - n junction → Solar cell and photovoltatic detector  TPV (thermophotovoltatic) devices Incident radiation with a photon energy greater than the bandgap energy strikes the p - n junction Drift current: Electron-hole pair is generated → swept by the built-in electric field → collected by electrodes at ends of cell Diffusion current: For radiation absorbed near the depletion region → minority carriers diffuse toward the depletion region