麻省理工大学固体物理课件

1分子-简单固体(PDF)

Lecture 1: Molecules–the Simple Solid (PDF)

2氢分子中的振动和旋转态(PDF)

Lecture 2: Vibrational and Rotational States in Hydrogen (PDF)

3金属中的自由电子气(PDF)

Lecture 3: Metal as a Free Electron Gas (PDF)

4固体中的振动(PDF)

Lecture 4: Vibrations in Solids (PDF)

5格波的比热(PDF - 1.4 MB)

Lecture 5: Specific Heat of Lattice Waves (PDF - 1.4 MB)

8一维单原子晶体中的格波(PDF)

Lecture 8: Lattice Waves in 1D Monatomic Crystals (PDF)

9一维双原子链中的格波(PDF)

Lecture 9: Lattice Waves in 1D with Diatmomic Basis (PDF)

10 &11离散晶格的比热(PDF - 1.1 MB)

Lecture 10 & 11: Specific Heat of Discrete Lattice (PDF - 1.1 MB)

12周期结构的固体中的电子(PDF)

Lecture 12: Electrons in a Periodic Solid (PDF)

13周期结构的固体中的电子(PDF)

Lecture 13: Electrons in a Periodic Solid (PDF)

14周期结构的固体中的电子(PDF - 3.2 MB)

Lecture 14: Electrons in a Periodic Solid (PDF - 3.2 MB)

15周期结构的固体中的电子(PDF - 3.1 MB)

Lecture 15: Electrons in a Periodic Solid (PDF - 3.1 MB)

16近自由电子能带(PDF - 1.2 MB)

Lecture 16: Nearly Free Electron Bands (PDF - 1.2 MB)

17近自由电子能带（part III）(PDF)

Lecture 17: Nearly Free Electron Bands (Part III) (PDF)

18 Bloch函数的属性(PDF)

Lecture 18: Properties of Bloch Functions (PDF)

19电子波包的运动(PDF)

Lecture 19: Motion of Electronic Wavepackets (PDF)

20杂质态(PDF)

Lecture 20: Impurity States (PDF)

21 & 22电子及空穴运动的半经典方程(PDF - 1.1 MB)

Lecture 21 & 22: Semi Classical Equations of Motions & Electrons and Holes I (PDF - 1.1 MB)

23 有效质量(PDF - 1.1 MB)

Lecture 23: Effective Mass (PDF - 1.1 MB)

24 化学势和平衡态(PDF)

Lecture 24: Chemical Potential and Equilibrium (PDF)

25 化学势和非平衡态(PDF)

Lecture 25: Chemical Potential and Non-equilibrium (PDF)

26 多相固体(PDF)

Lecture 26: Inhomogeneous Solids (PDF)

27 Bloch函数的散射(PDF)

Lecture 27: Scattering of Bloch Functions (PDF)

28 电子-声子散射(PDF)

Lecture 28: Electron-phonon Scattering (PDF)

6.730PSSA

6.730 Physics for Solid State Applications

Lecture 1: Molecules – the Simple Solid

Rajeev J. Ram

6.730PSSA

• Molecules• Approximate models for molecular bonding• Vibrational and rotational modes

• Continuum Models of Solids• Elasticity Phonons• Free electron gas Electron

• Lattice (‘Atomic’) Models of Solids• Crystal structure• Phonons (lattice waves) on discrete lattice• Bloch electrons

• Single Electron Transport• ‘Electrons’ and ‘holes’• Chemical potential and band bending

• Electron Scattering• Electron-phonon scattering; mobility• Electron-photon scattering; absorption and gain

• Statistical Theory of Electron Transport• Boltzmann transport• Drift-diffusion; hot electron effects

Syllabus

6.730PSSA

General Course InformationGeneral Course InformationRequired Text :Fundamentals of Carrier Transport, Second Edition, by Mark Lundstrom, Cambridge University Press, 2000 Notes will be handed out to cover the first two-thirds of the class. This required text be used for the last third of the class.

Suggested Text :Solid State Physics , N.W. Aschroft and M.D. Mermin, Saunders College Publishing, 1976. This text is not required, but if you wish to purchase a textbook that covers some of the material in the first two-thirds of the class I would suggest this text because it is the one used in 6.732.

Problem Sets :All homework sets are due at the beginining of class on the assigned due date. You may work together on the problem sets but you are required to write up your own solution and code. Students will make oral presentations on the homework.

FINAL :There will be a final examination for the class.

GRADES:40% Group Project (30% written report and 10% presentation) 40% Problem Sets (25% written and 15% presentation) 20% Final Examination

6.730PSSA

Band Formation in 1Band Formation in 1--D SolidD SolidSimple model for a solid: the one-dimensional solid, which consists of a single, infinitely long

line of atoms, each one having one s orbital available for forming molecular orbitals (MOs).

When the chain is extended:

The range of energies covered by the MOs is spread

This range of energies is filled in with more and more orbitals

The width of the range of energies of the MOs is finite, while the number of molecular orbitals is infinite: This is called a band .

6.730PSSA

Band Formation in 1Band Formation in 1--D Solid with s & p D Solid with s & p orbitalsorbitals

Before we can build models for the solid, we need to understand a simple diatomic molecule.

6.730PSSA

The Simplest Molecule: HThe Simplest Molecule: H22

, the wavefunction for the entire system of nuclei and electrons

M1 M2

me

r

R2

R1

6.730PSSA

Approximate Models: Simplifying HApproximate Models: Simplifying H22

Born-Oppenheimer ApproximationThe electrons are much lighter than the nuclei (me/mH≅1/1836), their

motion is much faster than the vibrational and rotational motions of the nuclei within the molecule.

A good approximation is to neglect the coupling terms between the motion of the electrons and the nuclei: this is the Born-Oppenheimerapproximation. The Schrödinger equation can then be divided into two equations:

Linear Combination of Atomic Orbitals (LCAO)Even the electron part of the problem is too hard to solve exactly

6.730PSSA

BornBorn--Oppenheimer Oppenheimer ApproximationApproximation

M1 M2

me

r

R2R1

Electronic Part:

Nuclear Part:

where E is the energy of the entire molecule

6.730PSSA

Electronic Part: LCAOElectronic Part: LCAO

For example, if we consider 1s orbitals only…

6.730PSSA

Normalization of Electronic Part: LCAO cont.Normalization of Electronic Part: LCAO cont.

Normalize to guarantee the probability of finding an electron anywhere is still 1

where

6.730PSSA

Approximate Electronic EnergyApproximate Electronic Energy

6.730PSSA

Nuclear and Electronic Energy TogetherNuclear and Electronic Energy Together(E

nerg

y -I

H) /

I H

6.730PSSA

First Excited State Energy: First Excited State Energy: AntibondingAntibonding

6.730PSSA

First Excited State Energy: LCAOFirst Excited State Energy: LCAO

(Ene

rgy

-IH) /

I H


Lecture 2: Vibrational and Rotational States in Hydrogen

Rajeev J. Ram

Review Lecture 1: HReview Lecture 1: H22

, the wavefunction for the entire system of nuclei and electrons

M1 M2

me

r

R2R1

Approximate Models: Simplifying HApproximate Models: Simplifying H22

Born-Oppenheimer ApproximationThe electrons are much lighter than the nuclei (me/mH≅1/1836), their

motion is much faster than the vibrational and rotational motions of the nuclei within the molecule.

Works since vibrational and rotational energy of molecule is typically much less than the binding energy

Linear Combination of Atomic Orbitals (LCAO)Even the electron part of the problem is too hard to solve exactly

BornBorn--Oppenheimer Oppenheimer ApproximationApproximation

M1 M2

me

r

R2R1

Electronic Part:

Nuclear Part:

where E is the energy of the entire molecule

Approximate Electronic EnergyApproximate Electronic Energy

Nuclear and Electronic Energy TogetherNuclear and Electronic Energy Together(E

nerg

y -I

H) /

I H

First Excited State Energy: First Excited State Energy: AntibondingAntibonding

First Excited State Energy: LCAOFirst Excited State Energy: LCAO

(Ene

rgy

-IH) /

I H

Decrease of electron density

Increase of electron density

A Closer Look at Nuclear MotionA Closer Look at Nuclear MotionMolecular Vibration and RotationMolecular Vibration and Rotation

M1 M2

me

r

R2R1

LCAO for electronin Veff

Divide and ConquerDivide and Conquer

M1 M2

me

r

R2R1 Born-Oppenheimer

Center-of-mass and Relative nuclear motion

LCAO for electronin Veff

Vibrational and rotational motion

CenterCenter--ofof--Mass and Relative Nuclear MotionMass and Relative Nuclear Motion

Relative Center-of-mass

Note that R is the C-of-M coordinate now

CenterCenter--ofof--Mass and Relative Nuclear MotionMass and Relative Nuclear Motion

Total energy is the sum of CM motion and relative:

If this is the total energy, where is the electron energy ?

CenterCenter--ofof--Mass Nuclear MotionMass Nuclear Motion

Schrodinger equation for center-of-mass is same as free particle:

Eigenstate: Eigenenergy:

center-of-mass

M1 M2

meM1 M2

me

Relative Nuclear MotionRelative Nuclear Motion

Schrodinger equation for relative motion is a central potential problem just like the hydrogen atom:

Radial kinetic energy Angular kinetic energy

is the angular momentum operator

Relative Nuclear MotionRelative Nuclear MotionSeparation of Radial and Angular ComponentsSeparation of Radial and Angular Components

is the spherical Bessel function

Vibrational Vibrational Motion of NucleiMotion of NucleiHarmonic OscillatorHarmonic Oscillator

For no rotation, this simplifies to…(E

nerg

y -I

H) /

I H

Vibrational Vibrational Motion of NucleiMotion of NucleiHarmonic OscillatorHarmonic Oscillator

Approximation: Born-Oppenheimer, parabolic effective potential

Vibrational Vibrational Motion of NucleiMotion of NucleiRigid RotorRigid Rotor

Assuming that the vibrational motion produces only small displacements…

Divide and ConquerDivide and Conquer

M1 M2

me

r

R2R1

Approximations• Born-Oppenheimer

Nuclei inside electron cloud act as if they are embedded in an elastic medium (Veff)

• Effective potential (Veff) is parabolicVibrations of simple harmonic oscillator

• Rigid rotorVibrations only displace nuclei slightly from equilibrium bond length

Total Energy of the HTotal Energy of the H22 MoleculeMolecule

0.3-3 THz3-30 THz30 THz

30 l (l +1) meV0.1 (n + ½) eV1.0 eVO2

7.5 l (l+1) meV0.5 (n + ½) eV1.4 eVH2

TranslationalRotationalVibrationalBinding

Generalizations from Molecules to SolidsGeneralizations from Molecules to Solids

• The source of the binding energy is primarily the electrostatic potential between the nuclei and the electrons. The localization energy can also play a role (metal).

• Nuclear motions of the ions contribute a very small part to the binding energy.

• Sharing electrons between nuclei lowers the energy of the solid.

• The potential between the nuclei is of the same form as the molecule.

• Exicted states exists.

Assumptions for Electronic StatesAssumptions for Electronic States

• One electron energy levels

• No spin or exchange energies

• LCAO a good approximation

• Ignore motion of the nuclei to first order


Lecture 3: Metal as a Free Electron Gas

Rajeev J. Ram

Generalizations from Molecules to SolidsGeneralizations from Molecules to Solids

• The source of the binding energy is primarily the electrostatic potential between the nuclei and the electrons. The localization energy can also play a role (metal).

• Nuclear motions of the ions contribute a very small part to the binding energy.

• Sharing electrons between nuclei lowers the energy of the solid.

• The potential between the nuclei is of the same form as the molecule.

• Exicted states exists.

Assumptions for Electronic StatesAssumptions for Electronic States

• One electron energy levels

• No spin or exchange energies

• LCAO a good approximation

• Ignore motion of the nuclei to first order

Overview of Electron TransportOverview of Electron Transport

Goal: Calculate electrical properties (eg. resistance) for solidsApproach:

• Macroscopic theory: V, I, R

• Microscopic theory: J, E, σ

• Phenomenological model of transport: n, τ, m

• Relate parameters in phenomenological theory to electronic energy levels and wavefunction

Overview of Electron TransportOverview of Electron Transport

Goal: Calculate electrical properties (eg. resistance) for solids

Approach:In the end calculating resistance boils down to calculating the electronic energy levels and wavefunctions; to knowing the bandstructure

You will be able to relate a bandstructure to macroscopic parameters for the solid

Why this approach ?:This first principles approach will make assumptions and approximations explicit. The phenomenological theory fails for modern devices – the channel in the MOSFET on the Pentium chip.

Microscopic Variables for Electrical TransportMicroscopic Variables for Electrical TransportDrude Drude TheoryTheory

Balance equation for forces on electrons:

In steady-state when B=0:

Microscopic Variables for Electrical TransportMicroscopic Variables for Electrical Transport

Recovering macroscopic variables:


Balance equation for energy of electrons:

In steady-state:

In the continuum models, we assume that electron scattering is sufficiently fast that all the energy pumped into the electrons is randomized; all additional energy heats the electrons

How do we relate ∆E and T ?

Equipartition Equipartition TheoremTheorem


The theorem of equipartition of energy states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion

So in this simple theory, ∆E and T are proportional to each other…

Specific Heat and Heat CapacitySpecific Heat and Heat Capacity

Again assume that the heat and change in internal energy are the same:

(heat capacity)

Take constant volume since this ensures none of the extra energy is going into work(think ideal gas)

(specific heat)

Specific heat is independent of temperature…Law of Dulong and Petit

Specific Heat MeasurementsSpecific Heat Measurements

(hyperphysics.phy-astr.gsu.edu)

Specific heat is independent of temperature…NOT TRUE !To get this correct we will need to (a) quantize electron energy levels, (b) introduce discreteness of lattice and (c) the heat capacity of lattice

Quantum Free Electron GasQuantum Free Electron GasCrystal as Infinite Well PotentialCrystal as Infinite Well Potential

Electron confined in crystal of size L on a sideno interaction with nucleisingle particle approximationperiodic boundary conditions

not for periodic b.c.


Quantum Free Electron GasQuantum Free Electron GasPeriodic Boundary ConditionsPeriodic Boundary Conditions

Estimating Electron NumberEstimating Electron Number

Probability of a particular energy level being occupied by an electron:

Total number of electrons:

spin

Limit for Large CrystalsLimit for Large Crystals

ZeroZero--Temperature LimitTemperature Limit

ZeroZero--Temperature LimitTemperature LimitFermi Fermi Energy and TemperatureEnergy and Temperature

ZeroZero--Temperature LimitTemperature LimitElectronic EnergyElectronic Energy

Average energy per electron:

Finite TemperaturesFinite Temperatures

Ensemble Averages at Finite TemperaturesEnsemble Averages at Finite Temperatures

Where Fk is any property of the electron

where g(E) is number of states at E per unit volume

By comparing the above two expressions…

Density of States in Large 3D SolidDensity of States in Large 3D Solid

Density of States in Different SolidsDensity of States in Different Solids

Low Temperature Specific Heat of the Free Electron GasLow Temperature Specific Heat of the Free Electron GasSommerfeldSommerfeld ApproximationApproximation



To get this correct we will need to (a) quantize electron energy levels, (b) introduce discreteness of lattice and (c) the heat capacity of lattice

Conductivity of the Free Electron GasConductivity of the Free Electron GasSommerfeldSommerfeld ApproximationApproximation

Only electrons near EF contribute to current !


Sommerfeld recovers the phenomenological results !

SommerfeldSommerfeld ExpansionExpansion

SommerfeldSommerfeld Expansion for Electron DensityExpansion for Electron Density

SommerfeldSommerfeld Expansion for Electron EnergyExpansion for Electron Energy

Density of States is the Central Character in this StoryDensity of States is the Central Character in this Story





Lecture 4: Vibrations in Solids

Outline

• Review Lecture 3

• Sommerfeld Theory of Metals

• 1-D Elastic Continuum

• 1-D Lattice Waves



Microscopic Variables for Electrical TransportMicroscopic Variables for Electrical TransportDrude Drude TheoryTheory

Balance equation for forces on electrons:

In steady-state when B=0:

Density of StatesDensity of States



In steady-state:

In the continuum models, we assume that electron scattering is sufficiently fast that all the energy pumped into the electrons is randomized; all additional energy heats the electrons

How do we relate ∆E and T ?

Specific Heat and Heat CapacitySpecific Heat and Heat Capacity

Again assume that the heat and change in internal energy are the same:

(heat capacity)

Take constant volume since this ensures none of the extra energy is going into work(think ideal gas)

(specific heat)

Specific heat is independent of temperature…Law of Dulong and Petit



Specific heat is independent of temperature…NOT TRUE !To get this correct we will need to (a) quantize electron energy levels, (b) introduce discreteness of lattice and (c) the heat capacity of lattice

Outline







Low Temperature Specific Heat of the Free Electron GasLow Temperature Specific Heat of the Free Electron GasSommerfeldSommerfeld ApproximationApproximation


Only electrons near EF contribute to current !


Sommerfeld recovers the phenomenological results !

SommerfeldSommerfeld ExpansionExpansion

SommerfeldSommerfeld Expansion for Electron DensityExpansion for Electron Density

SommerfeldSommerfeld Expansion for Electron EnergyExpansion for Electron Energy



To get this correct we will need to (a) quantize electron energy levels, (b) introduce discreteness of lattice and (c) the heat capacity of lattice

Density of States is the Central Character in this StoryDensity of States is the Central Character in this Story




Outline







11--D Elastic ContinuumD Elastic ContinuumStress and StrainStress and Strain

uniaxial loading

Lo

LStress:

Normal strain:Strain:

If ux is uniform there is no strain, just rigid body motion.

11--D Elastic ContinuumD Elastic ContinuumYoung’s Modulus METALS :

Tungsten (W) 406Chromium (Cr) 289Berylium (Be) 200 - 289Nickel (Ni) 214Iron (Fe) 196Low Alloy Steels 200 - 207Stainless Steels 190 - 200Cast Irons 170 - 190Copper (Cu) 124Titanium (Ti) 116Brasses and Bronzes 103 - 124Aluminum (Al) 69

PINE WOOD (along grain): 10

POLYMERS :Polyimides 3 - 5Polyesters 1 - 5Nylon 2 - 4Polystryene 3 - 3.4Polyethylene 0.2 -0.7Rubbers / Biological Tissues 0.01-0.1

Young’s Modulus

Young’s Modulus For Various Materials (GPa)from Christina Ortiz

CERAMICS GLASSES AND SEMICONDUCTORSDiamond (C) 1000Tungsten Carbide (WC) 450 -650Silicon Carbide (SiC) 450Aluminum Oxide (Al2O3) 390Berylium Oxide (BeO) 380Magnesium Oxide (MgO) 250Zirconium Oxide (ZrO) 160 - 241Mullite (Al6Si2O13) 145Silicon (Si) 107Silica glass (SiO2) 94Soda-lime glass (Na2O - SiO2) 69

Dynamics of 1Dynamics of 1--D ContinuumD Continuum11--D Wave EquationD Wave Equation

Net force on incremental volume element:


Velocity of sound, c, is proportional to stiffness and inverse prop. to inertia

Dynamics of 1Dynamics of 1--D ContinuumD Continuum11--D Wave Equation SolutionsD Wave Equation Solutions

Clamped Bar: Standing Waves

Dynamics of 1Dynamics of 1--D ContinuumD Continuum11--D Wave Equation SolutionsD Wave Equation Solutions

Periodic Boundary Conditions: Traveling Waves

33--D Elastic ContinuumD Elastic ContinuumVolume DilatationVolume Dilatation

Lo Lapply load

Volume change is sum of all three normal strains

33--D Elastic ContinuumD Elastic ContinuumPoisson’s RatioPoisson’s Ratio

ν is Poisson’s Ratio – ratio of lateral strain to axial strain

Poisson’s ratio can not exceed 0.5, typically 0.3

33--D Elastic ContinuumD Elastic ContinuumPoisson’s Ratio ExamplePoisson’s Ratio Example

Aluminum: EY=68.9 GPa, ν = 0.35

20mm75mm5kN

5kN

33--D Elastic ContinuumD Elastic ContinuumShear StrainShear Strain

Shear plus rotationφ

φ

Pure shearShear loading

2φ

Pure shear strain

Shear stress

G is shear modulus

33--D Elastic ContinuumD Elastic ContinuumStress and Strain TensorsStress and Strain Tensors

For most general isotropic medium,

Initially we had three elastic constants: EY, G, e

Now reduced to only two: λ, µ


If we look at just the diagonal elements

Inversion of stress/strain relation:

33--D Elastic ContinuumD Elastic ContinuumExample of Example of Uniaxial Uniaxial StressStress

Lo

L



Total force is the sum of the forces on all the surfaces


Net force in the x-direction:


Finally, 3-D wave equation….

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumFourier Transform of 3Fourier Transform of 3--D Wave EquationD Wave Equation

Anticipating plane wave solutions, we Fourier Transform the equation….

Three coupled equations for Ux, Uy, and Uz….

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumDynamical MatrixDynamical Matrix

Express the system of equations as a matrix….

Turns the problem into an eigenvalue problem for the polarizations of the modes (eigenvectors) andwavevectors q (eigenvalues)….

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumSolutions to 3Solutions to 3--D Wave EquationD Wave Equation

Transverse polarization waves:

Longitudinal polarization waves:

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumSummarySummary

1. Dynamical Equation can be solved by inspection

2. There are 2 transverse and 1 longitudinal polarizations for each q

3. The dispersion relations are linear

4. The longitudinal sound velocity is always greater than the transverse sound velocity


Lecture 5: Specific Heat of Lattice Waves

Outline




• Lattice Density of Modes

• Specific Heat of Lattice

33--D Elastic ContinuumD Elastic ContinuumPoisson’s Ratio ExamplePoisson’s Ratio Example

A prismatic bar with length L = 200 mm and a circular cross section with a diameter D = 10 mm is subjected to a tensile load P = 16 kN. The length and diameter of the deformed bar are measured and determined to be L’ = 200.60 mm and D’ = 9.99 mm. What are the modulus of elasticity and the Poisson’s ratio for the bar?

33--D Elastic ContinuumD Elastic ContinuumShear StrainShear Strain

Shear plus rotationφ

φ

Pure shearShear loading

2φ

Pure shear strain

Shear stress

G is shear modulus


For most general isotropic medium,

Initially we had three elastic constants: EY, G, e

Now reduced to only two: λ, µ


If we look at just the diagonal elements

Inversion of stress/strain relation:

33--D Elastic ContinuumD Elastic ContinuumExample of Example of Uniaxial Uniaxial StressStress

Lo

L



Total force is the sum of the forces on all the surfaces


Net force in the x-direction:


Finally, 3-D wave equation….

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumFourier Transform of 3Fourier Transform of 3--D Wave EquationD Wave Equation

Anticipating plane wave solutions, we Fourier Transform the equation….

Three coupled equations for Ux, Uy, and Uz….

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumDynamical MatrixDynamical Matrix

Express the system of equations as a matrix….

Turns the problem into an eigenvalue problem for the polarizations of the modes (eigenvectors) andwavevectors q (eigenvalues)….

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumSolutions to 3Solutions to 3--D Wave EquationD Wave Equation

Transverse polarization waves:

Longitudinal polarization waves:

Direct Measurements of Sound VelocityDirect Measurements of Sound Velocity

Bolo

met

er s

igna

l Ge at 1.9 K

Time (microseconds)

LA phonons are faster, since real solids are not isotropic the TA phonons travel at different velocity

Dynamics of 3Dynamics of 3--D ContinuumD ContinuumSummarySummary

1. Dynamical Equation can be solved by inspection

2. There are 2 transverse and 1 longitudinal polarizations for each q

3. The dispersion relations are linear

4. The longitudinal sound velocity is always greater than the transverse sound velocity

Counting Counting Vibrational Vibrational ModesModesSolid as an Acoustic CavitySolid as an Acoustic Cavity

For each of three polarizations:

If the plane waves are constrained to the solid with dimension Land we use periodic boundary conditions:

number of states in dω :

Specific Heat of SolidSpecific Heat of SolidHow much energy is in each mode ?How much energy is in each mode ?

Need to approximate the amount of energy in each mode at a given temperature…

If we assume equipartition, we will againDulong-Petit which is not consistent with experiment for solids…

Approach:

• Quantize the amplitude of vibration for each mode

• Treat each quanta of vibrational excitation as a bosonic particle, the phonon

• Use Bose-Einstein statistics to determine the number of phonons in each mode

Lattice Waves as Harmonic OscillatorLattice Waves as Harmonic Oscillator

Treat each mode and each polarization as an independent harmonic oscillator:

is the quantum number associated with harmonic

Now, we think of each quantum of excitation as a particle…

lattice waves electromagnetic wavesacoustic cavity (solid) optical cavity (metal box)quanta observed quanta observed

by light scattering by photoelectric effectbosons ? bosons (eg. laser)

Lattice Waves in Thermal EquilibriumLattice Waves in Thermal Equilibrium

Lattice waves in thermal equilibrium don’t have a single well define amplitude of vibration…

For each mode, there is a distribution of amplitudes…

Bose-Einstein distribution

Total Energy of a Lattice in Thermal EquilibriumTotal Energy of a Lattice in Thermal Equilibrium

number of states in dω :

Specific Heat of a Crystal LatticeSpecific Heat of a Crystal Lattice

Aside: Thermal Energy of PhotonsAside: Thermal Energy of Photons

Energy density of blackbody:

Specific heat :


Lecture 8: Lattice Waves in 1D Monatomic Crystals

Outline

• Overview of Lattice Vibrations so far

• Models for Vibrations in Discrete 1-D Lattice

• Example of Nearest Neighbor Coupling Only

• Relating Microscopic and Macroscopic Quantities

Continuum ModelsContinuum Models11--D Wave EquationD Wave Equation

ω

κ

Velocity of sound, c, is proportional to stiffness and inverse prop. to inertia

Periodic Boundary Conditions: Traveling Waves

Continuum ModelsContinuum ModelsTT33 Specific HeatSpecific Heat


The Atomistic PerspectiveThe Atomistic PerspectiveArrangement of Atoms and Bond OrientationsArrangement of Atoms and Bond Orientations

The Atomistic PerspectiveThe Atomistic PerspectiveArrangement of Atoms and Bond OrientationsArrangement of Atoms and Bond Orientations

Diamond Crystal Structure:Silicon

Bond angle = 109.5º

• Add 4 atoms to a FCC

• Tetrahedral bond arrangement

• Each atom has 4 nearest neighbors and

12 next nearest neighbors

The Atomistic PerspectiveThe Atomistic PerspectiveVibrational Vibrational Motion of NucleiMotion of Nuclei

(Ene

rgy

-IH) /

I H

Strain in a Discrete LatticeStrain in a Discrete LatticeExample of Nearest Neighbor InteractionsExample of Nearest Neighbor Interactions

n n+1 n+2 n+3n-3 n-2 n-1

a

un-3 un-2 un-1 un un+1 un+2 un+3

strained

equilibrium

is the discrete displacement of an atom from its equilibrium position

Strain in a Discrete LatticeStrain in a Discrete LatticeGeneral ExpansionGeneral Expansion

The potential energy associated with the strain is a complex function ofthe displacements.

where

and the force on each lattice atom

Harmonic MatrixHarmonic MatrixSpring Constants Between Lattice AtomsSpring Constants Between Lattice Atoms

Harmonic Matrix:

Dynamics of Lattice AtomsDynamics of Lattice Atoms

Force on the jth atom (away from equilibrium)…

Solutions of Equations of MotionSolutions of Equations of MotionConvert to Difference EquationConvert to Difference Equation

Time harmonic solutions…

Plugging in, converts equation of motion into coupled difference equations:

Solutions of Equations of MotionSolutions of Equations of Motion

We can guess solution of the form:

This is equivalent to taking the z-transform…

Solutions of Equations of MotionSolutions of Equations of MotionConsider Consider Undamped Undamped Lattice VibrationsLattice Vibrations

We are going to consider the undamped vibrations of the lattice:

Solutions of Equations of MotionSolutions of Equations of MotionDynamical MatrixDynamical Matrix


n n+1 n+2 n+3n-3 n-2 n-1

a


strained

equilibrium


Harmonic matrix:

Dynamical matrix:


ω

kk=π/ak= - π/a


ω

kk=π/ak=-π/ak=-2π/a k=2π/a

1st Brillouin zone 2nd Brillouin zone2nd Brillouin zone

A B

From what we know about Brillouin zones the points A and B (related by a reciprocal lattice vector) must be identical

This implies that the wave form of the vibrating atoms must also be identical.


n n+1 n+2 n+3 n+4 n+5n-1n-2n-3n-4n-5

κ=-2π/a κ=2π/a

ω

κκ=π/aκ=-π/a A B

A: k=-0.7π/a

B: k=1.3π/a

But: note that point B represents a wave travelling right, and point A one travelling left


ω

κκ=π/aκ=-π/aκ=-2π/a κ=2π/a

c Consider point C at the zone boundary

When k=π/a, λ=2a, and motion becomes that of a standing wave (the atoms are bouncing backward and forward against each other

n n+1 n+2 n+3 n+4 n+5n-1n-2n-3n-4n-5

λ=2a


In the limit of long-wavelength, we should recover the continuum model…

Linear dispersion, just like the sound waves for the continuum solid


Lecture 9: Lattice Waves in 1D with Diatmomic Basis

Outline


• 1-D Lattice with Basis

• Example of Nearest Neighbor Coupling

• Optical and Acoustic Phonon Branches

Strain in a Discrete 1Strain in a Discrete 1--D Monatomic LatticeD Monatomic LatticeGeneral ExpansionGeneral Expansion

n n+1 n+2 n+3n-3 n-2 n-1

a


strained

equilibrium

Equations of Motion for Lattice AtomsEquations of Motion for Lattice Atoms

Harmonic Matrix:

Force on the jth atom (away from equilibrium)…

Solutions of Equations of MotionSolutions of Equations of Motion

Assuming time-harmonic solutions, converts into coupled difference equations:


n n+1 n+2 n+3n-3 n-2 n-1

a


strained

equilibrium


ω



A B


Summary of Phonon Dispersion CalculationSummary of Phonon Dispersion Calculation

• Taylor series expansion for total potential stored in all bonds• Neglect first order since in equilibrium F=0

• Truncate expansion at second order, assume small amplitudes

• Determine harmonic matrix from potential energy

• Represents bond stiffness

• Assume time harmonic and discrete ‘plane wave’ solutions

• Determine dynamical matrix from harmonic matrix plus phase progression

• Determine dispersion relation

Strain in a Discrete Lattice with BasisStrain in a Discrete Lattice with BasisExample of Nearest Neighbor InteractionsExample of Nearest Neighbor Interactions

n n+1n-1

a

uj[n-2] ui[n-1] uj[n-1] ui[n] uj[n] ui[n+1] uj[n+1]strained

α1 α2equilibrium

Harmonic Matrix for 1Harmonic Matrix for 1--D Lattice with BasisD Lattice with Basis

Equations of MotionEquations of Motion

The force on the l th basis atom in the nth unit cell…

Matrix Representation of Equations of MotionMatrix Representation of Equations of Motion

Can collect system of equations for each atom in the basis as a matrix…

Plane Wave Solutions & the Dynamical MatrixPlane Wave Solutions & the Dynamical Matrix


equilibrium

n n+1n-1

a

u2[n-2] u1[n-1] u2[n-1] u1[n] u2[n] u1[n+1] u2[n+1]strained

α1 α2

Dynamical Matrix for 1Dynamical Matrix for 1--D Lattice with BasisD Lattice with BasisExample of Nearest Neighbor CouplingExample of Nearest Neighbor Coupling

Dispersion Relation for 1Dispersion Relation for 1--D Lattice with BasisD Lattice with BasisExample of Nearest Neighbor CouplingExample of Nearest Neighbor Coupling

Dispersion Relation for 1Dispersion Relation for 1--D Lattice with BasisD Lattice with BasisExample of Nearest Neighbor CouplingExample of Nearest Neighbor Coupling

k

optical branch

acoustic branch

Lattice Waves at k=0Lattice Waves at k=0Example of Nearest Neighbor CouplingExample of Nearest Neighbor Coupling

Lattice Waves at Small kLattice Waves at Small kExample of Nearest Neighbor CouplingExample of Nearest Neighbor Coupling

Lattice Waves Near Zone BoundaryLattice Waves Near Zone BoundaryExample of Nearest Neighbor CouplingExample of Nearest Neighbor Coupling

Dispersion Relation for 3Dispersion Relation for 3--D LatticesD Lattices


Lecture 10, 11: Specific Heat of Discrete Lattice

Outline

• 2-D Lattice Waves Solutions

• Review Continuum Specific Heat Calculation

• Density of Modes

• Quantum Theory of Lattice Vibrations

• Specific Heat for Lattice

• Approximate Models

Lattice Waves in 3Lattice Waves in 3--D CrystalsD Crystals

Second order Taylor series expansion for total potential energy:

Harmonic Matrix:

Equation of motion for lattice atoms assuming ‘plane wave’ solutions:

Dynamical Matrix:

Lattice Waves in 3Lattice Waves in 3--D CrystalsD Crystals

Dimension of system is given by (number of basis atoms) x (dimension of lattice)

Bond Stretching and BendingBond Stretching and Bending

Example: 1Example: 1--D Diatomic Lattice with Bond Stretching and BendingD Diatomic Lattice with Bond Stretching and BendingPotential EnergyPotential Energy

αsA αsB

M2M1y

x

Example: ‘1Example: ‘1--D’ Diatomic Lattice with Bond Stretching and BendingD’ Diatomic Lattice with Bond Stretching and BendingPotential EnergyPotential Energy

αsA αsB

M2M1y

x

Example: 2Example: 2--D Lattice with Bond StretchingD Lattice with Bond StretchingPotential EnergyPotential Energy

α2

α1a1a2

Example: 2Example: 2--D Lattice with Bond StretchingD Lattice with Bond StretchingElements of the Dynamical MatrixElements of the Dynamical Matrix

Example: 2Example: 2--D Lattice with Bond StretchingD Lattice with Bond StretchingDynamical MatrixDynamical Matrix

Example: 2Example: 2--D Lattice with Bond StretchingD Lattice with Bond StretchingDispersion RelationDispersion Relation

Longitudinal Waves:

Transverse Waves:

Example: 2Example: 2--D Lattice with Bond StretchingD Lattice with Bond StretchingDispersion RelationsDispersion Relations


Approach:




Simple Harmonic OscillatorSimple Harmonic Oscillator

10 2E ω= h

31 2E ω= h

52 2E ω= h

73 2E ω= h

0n =

1n =

2n =

3n =

( ) 2 212 ω=U x m x( ) 2xψ

x

E

Hamiltonian for Discrete LatticeHamiltonian for Discrete Lattice

Potential energy of bonds in 3-D lattice with basis:

For single atom basis in 3-D, µ & ν denote x,y, or z direction:

Hamiltonian for Discrete LatticeHamiltonian for Discrete LatticePlane Wave ExpansionPlane Wave Expansion

The lattice wave can be represented as a superposition of plane waves (eigenmodes) with a known dispersion relation (eigenvalues)….

σ denotes polarization

Commutation Relation for Plane Wave DisplacementCommutation Relation for Plane Wave Displacement

…commute unless we have same polarization and k-vector

Creation and Creation and Annhilation Annhilation Operators for Lattice WavesOperators for Lattice Waves

Operators for the Lattice DisplacementOperators for the Lattice Displacement

We will use this for electron-phonon scattering…

Specific Heat with Continuum Model for SolidSpecific Heat with Continuum Model for Solid

3-D continuum density of modes in dω :

Specific Heat with Discrete LatticeSpecific Heat with Discrete LatticeDensity of Modes from DispersionDensity of Modes from Dispersion


ωωm

k

ωωm

Specific Heat with Discrete LatticeSpecific Heat with Discrete LatticeDensity of Modes from DispersionDensity of Modes from Dispersion


Cu


Approach:





And we are done…


Lecture 12: Electrons in a Periodic Solid

Outline

• Review Lattice Waves

• Brillouin-Zone and Dispersion Relations

• Introduce Electronic Bandstructure Calculations

• Example: Tight-Binding Method for 1-D Crystals

Solutions of Lattice Equations of MotionSolutions of Lattice Equations of MotionConvert to Difference EquationConvert to Difference Equation

Time harmonic solutions…

Plugging in, converts equation of motion into coupled difference equations:

Solutions of Lattice Equations of MotionSolutions of Lattice Equations of Motion

We can guess solution of the form:

This is equivalent to taking the z-transform…

Solutions of Lattice Equations of MotionSolutions of Lattice Equations of MotionConsider Consider Undamped Undamped Lattice VibrationsLattice Vibrations

We are going to consider the undamped vibrations of the lattice:

Solutions of Lattice Equations of MotionSolutions of Lattice Equations of MotionDynamical MatrixDynamical Matrix

Solution of 1Solution of 1--D Lattice Equation of MotionD Lattice Equation of MotionExample of Nearest Neighbor InteractionsExample of Nearest Neighbor Interactions

ω



A B


This implies that the wave form of the vibrating atoms must also be identical.

Solution of 3Solution of 3--D Lattice Equation of MotionD Lattice Equation of Motion

Phonon Dispersion in FCC with 2 Atom BasisPhonon Dispersion in FCC with 2 Atom Basis

http://debian.mps.krakow.pl/phonon/Public/phrefer.html

Approaches to Calculating Electronic Approaches to Calculating Electronic BandstructureBandstructure

Nearly Free Electron Approximation:

Cellular Methods (Augmented Plane Wave):

• Plane wave between outside rs• Atomic orbital inside rs (core)

• Superposition of a few plane waves

Pseudopotential Approximation:

• Superposition of plane wavescoupled by pseudopotential

k.p:• Superposition of bandedge (k=0) wavefunctions

Tight-binding Approximation (LCAO):

• Superposition of atomic orbitals

Band Formation in 1Band Formation in 1--D SolidD SolidSimple model for a solid: the one-dimensional solid, which consists of a single, infinitely long

line of atoms, each one having one s orbital available for forming molecular orbitals (MOs).

“s” band

4β

When the chain is extended:

The range of energies covered by the MOs is spread

This range of energies is filled in with more and more orbitals

The width of the range of energies of the MOs is finite, while the number of molecular orbitals is infinite: This is called a band .

TightTight--binding (LCAO) Band Theorybinding (LCAO) Band Theory

LCAO LCAO WavefunctionWavefunction

Energy for LCAO BandsEnergy for LCAO Bands


Reduced Overlap Matrix:Reduced Hamiltonian Matrix:

Reduced Overlap Matrix for 1Reduced Overlap Matrix for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basis

Reduced Hamiltonian Matrix for 1Reduced Hamiltonian Matrix for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basis

Energy Band for 1Energy Band for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basis

LCAOLCAO WavefunctionWavefunction for 1for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basis


kk = 0= 0

kk ≠≠ 00

kk = = ππ / / aa

)/(2 Napk π=


remember H2 ?lowest energy (fewest nodes)

H2

highest energy (most nodes)

Bloch’s TheoremBloch’s Theorem

Translation of wavefunction by a lattice constant…

…yields the original wavefunction multiplied by a phase factor

Consistent that the probability density is equal at each lattice site

Wavefunction Wavefunction NormalizationNormalization

Using periodic boundary conditione for a crystal with N lattice sites between boundaries…

Counting Number of States in a BandCounting Number of States in a Band

Combining periodic boundary condition…

…with Bloch’s theorem…

…yields a discrete set of k-vectors

Within the 1st Brillouin Zone there are N states or 2N electrons

TightTight--binding and Lattice Wave Formalismbinding and Lattice Wave Formalism

Electrons (LCAO) Lattice Waves



Outline

• Review Electronic Bandstructure Calculations

• Example: 1-D Crystals with Two Atomic Functions

• Example: 1-D Crystals with Two Atom Basis

OverviewOverview

2N electrons each for px,py,pz

2N electrons

TightTight--binding and Lattice Wave Formalismbinding and Lattice Wave Formalism

Electrons (LCAO) Lattice Waves


Reduced Hamiltonian Matrix: Reduced Overlap Matrix:

Reduced Overlap Matrix for 1Reduced Overlap Matrix for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basis

Reduced Hamiltonian Matrix for 1Reduced Hamiltonian Matrix for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basis


kk = 0= 0

kk ≠≠ 00

kk = = ππ / / aa

)/(2 Napk π=

Energy Band for 1Energy Band for 1--D LatticeD LatticeTwo orbital, single atom basisTwo orbital, single atom basis

2N electrons each for px,py,pz

2N electrons


Reduced Hamiltonian and Overlap Matrices:


Hamiltonian MatrixHamiltonian Matrix


Overlap MatrixOverlap Matrix


SolutionsSolutions

At k=0:

pure s

pure p


SolutionsSolutions

At k=π/a:

pure s

pure p

For k away from zone center and zone boundary, bands are mixture of s and pbut will have a dominant s-like or p-like character….

At high symmetry points tight-binding returns pure orbitals…


SolutionsSolutions

Energy Band for 1Energy Band for 1--D LatticeD LatticeSingle orbital, two atom basisSingle orbital, two atom basis


At k=0:


At k=π/a:



Outline

• Review LCAO for 1-D Crystals

• Preview Problem for 2-D Crystal

• 2-D and 3-D Tight-binding

• Example: 2-D Crystal, single atom basis, 4 orbitals


lowest energy (fewest nodes)

highest energy (most nodes)


Es= - 12 eV, Ep= - 6 eV, Vssσ= - 1 eV, Vppσ = Vppπ = + 1 eV


Es= - 0.9 eV, Vs,a= - 0.4 eV, Vs,a-d = - 0.2 eV

Preview Problem: 2D Monatomic Square Crystals Preview Problem: 2D Monatomic Square Crystals

LCAO Basis for FCC CrystalsLCAO Basis for FCC Crystals

Ga: [Ar]3d10 4s2 4p1

As: [Ar]3d10 4s2 4p3

TightTight--binding for 3binding for 3--D Crystals D Crystals

Best estimate for energy with LCAO basis….

Hamiltonian matrix….

Overlap matrix….


Since the probability of finding electrons at each lattice site is equal…

Consequently…

Orbital Overlaps for 3Orbital Overlaps for 3--D Crystals D Crystals

distance from positive to negative lobe of p-orbital

Orbital Overlaps for 3Orbital Overlaps for 3--D CrystalsD CrystalsDiamond and Diamond and ZincblendeZincblende

2D Monatomic Square Crystals 2D Monatomic Square Crystals

2D Monatomic Square Crystals2D Monatomic Square CrystalsDispersion RelationsDispersion Relations

Es= - 10.11 eV

Ep= - 4.86 eV

a = 5.5 A


WΓX

X

W

Γ

2D Monatomic Square Crystals2D Monatomic Square CrystalsDispersion Relations at Dispersion Relations at ΓΓ=0=0

2D Monatomic Square Crystals2D Monatomic Square CrystalsVariations with Lattice ConstantVariations with Lattice Constant

X WΓ

X WΓX

W

Γ

a = 8.3 A


X WΓ

X WΓ

a = 5.5 A

X

W

Γ

a = 2.8 A

2D Monatomic Square Crystals2D Monatomic Square CrystalsFermi Fermi EnergyEnergy

How many states per band ?

where n is the areal density of atoms

To estimate Fermi energy we need to know the number of outermost valence electrons each atom has…

11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar

I II III IV V VI VII VIII


11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar


conductor

insulator

conductor

a = 5.5 A

Al

Mg

X

W

Γ Na

Reducing a, makes Mg a conductor !

Name: __________________

Matrix element (s-px) _________________________

Matrix element (s-py) _________________________

Matrix element (px -px) _________________________

Matrix element (px - py) _________________________



Outline

• Review 2-D Tight-binding

• 3-D Tight-binding

• Semiconductor Fermi Energy

• Silicon Bandstructure

2D Monatomic Square Crystals 2D Monatomic Square Crystals


Es= - 10.11 eV

Ep= - 4.86 eV

a = 5.5 A


WΓX

X

W

Γ


X WΓ

X WΓX

W

Γ

a = 8.3 A


X WΓ

X WΓ

a = 5.5 A

X

W

Γ

a = 2.8 A

2D Monatomic Square Crystals2D Monatomic Square CrystalsFermi Fermi EnergyEnergy

How many states per band ?

where n is the areal density of atoms

To estimate Fermi energy we need to know the number of outermost valence electrons each atom has…

11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar



11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar


conductor

insulator

conductor

a = 5.5 A

Al

Mg

X

W

Γ Na

Reducing a, makes Mg a conductor (semimetal) !

LCAO Basis for FCC CrystalsLCAO Basis for FCC Crystals

Ga: [Ar]3d10 4s2 4p1

As: [Ar]3d10 4s2 4p3


Best estimate for energy with LCAO basis….

Hamiltonian matrix….

Overlap matrix….


Since the probability of finding electrons at each lattice site is equal…

Consequently…


+-+-

+-

= +

+=


+-

+-

+-

+-+-

+- = +

+=


109o

Zincblende Zincblende LCAO BandsLCAO BandsReduced Hamiltonian MatrixReduced Hamiltonian Matrix

Zincblende Zincblende LCAO BandsLCAO BandsNearest NeighborsNearest Neighbors

Zincblende Zincblende LCAO BandsLCAO BandsReduced Hamiltonian MatrixReduced Hamiltonian Matrix

Silicon Silicon Bandstructure Bandstructure

Si: [Ne] 3s2 3p2

4 e- per silicon atom2 silicon atoms per lattice site

total: 8 electrons at each site

Silicon and GermaniumSilicon and Germanium Bandstructure Bandstructure

Si: [Ne] 3s2 3p2 Ge: [Ar]3d10 4s2 4p2

LCAO and Nearly Free Electron LCAO and Nearly Free Electron Bandstructure Bandstructure


Lecture 16: Nearly Free Electron Bands

Outline

• Fun: Application of 1-D Tight Binding

• Free Electron in Reduced Zone Representation

• Nearly Free Electron Bands

• Labeling Eigenvectors

B. Ethene and frontier orbitals Ethene: CH2=CH2Within the Hückel approximation, the secular determinant becomes:

022EE

E

E- = - energy of the Lowest Unoccupied Molecular Orbital (LUMO)

E+ = + energy of the Highest Occupied Molecular Orbital (HOMO)

LUMO= 2 *

HOMO= 1

HOMO and LUMO are the frontier orbitals of a molecule.

those are important orbitals because they are largely responsible for many chemical andoptical properties of the molecule.

Note: The orbitals together give rise to an cylindricaldistribution of charge. Electrons can circulate around this toruscan create magnetic effect detected in NMR

2| |+ +

- -

+ -

+-

Sketch Calculated (Hartree-Fock level)

Courtesy of

LCAO and Nearly Free Electron LCAO and Nearly Free Electron BandstructureBandstructure

Free Electron Dispersion RelationFree Electron Dispersion Relation

E

/a/a /a/ak

Nearly Free Electron Dispersion RelationNearly Free Electron Dispersion RelationFor weak lattice potentials, E vs k is still approximately correct…

Dispersion relation must be periodic….

E

/a/a /a/ak

Nearly Free Electron Dispersion RelationNearly Free Electron Dispersion Relation


Expect all solutions to be represented within the Brillouin Zone (reduced zone)

E

/a/a /a/ak




/a/a /a/a

E

k




/a/ak

/a/a

E

Nearly Free Electron Dispersion RelationNearly Free Electron Dispersion RelationExtension to 3-D requires, translation by reciprocal lattice vectors

in all directions…

Ge

/a/ak

/a/a

E



/a/ak

/a/a

EGe

LCAO and Nearly Free Electron LCAO and Nearly Free Electron BandstructureBandstructure

Finite Basis Set Expansion with Plane Waves Finite Basis Set Expansion with Plane Waves

Fourier series expansion of Bloch function

Basis functions in expansion are…

Finite Basis Set Expansion with Plane Waves Finite Basis Set Expansion with Plane Waves Hamiltonian MatrixHamiltonian Matrix

Basis functions are exactly orthogonal…overlaps are all zero.


Fourier Series coefficients for the lattice potential…

InfiniteInfinite Basis Set Expansion with Plane Waves Basis Set Expansion with Plane Waves Hamiltonian MatrixHamiltonian Matrix

Eigenvectors for Nearly Free Electron BandsEigenvectors for Nearly Free Electron Bands

Fourier transform

Sample eigenvector…

Eigenvectors for Nearly Free Electron BandsEigenvectors for Nearly Free Electron Bands


Lecture 17: Nearly Free Electron Bands (Part III)

Outline

• Free Electron Bands

• Nearly Free Electron Bands

• Approximate Solution of Nearly Free Electron Bands

• Bloch’s Theorem

• Properties of Bloch Functions

Free Electron Dispersion RelationFree Electron Dispersion Relation

E

π/a−π/a 3π/a−3π/ak

Nearly Free Electron Dispersion RelationNearly Free Electron Dispersion RelationFor weak lattice potentials, E vs k is still approximately correct…


E





E





3π/a−3π/a π/a−π/a

E

k




3π/a−3π/ak

π/a−π/a

E



Ge

3π/a−3π/ak

π/a−π/a

E



3π/a−3π/ak

π/a−π/a

EGe

Finite Basis Set Expansion with Plane Waves Finite Basis Set Expansion with Plane Waves

Fourier series expansion of Bloch function

Basis functions in expansion are…


Fourier Series coefficients for the lattice potential…

InfiniteInfinite Basis Set Expansion with Plane Waves Basis Set Expansion with Plane Waves Hamiltonian MatrixHamiltonian Matrix

LCAO and Nearly Free Electron LCAO and Nearly Free Electron Bandstructure Bandstructure

Why Is Lattice Potential Important Near Crossing Points ? Why Is Lattice Potential Important Near Crossing Points ? Let’s consider lattice potential to be a perturbation on free electrons….

Periodic Perturbation of Free Electron Bands Periodic Perturbation of Free Electron Bands

Energy up to second-order in perturbation expansion….

Matrix elements for periodic potential…


If the potential is sufficiently weak, this is a small perturbation on the

free electron bands, unless

Since these are free electron energies, we can relate this easily to the wave vectors…

, when k is at edge of B-Z


If only two bands cross…

Periodic Perturbation of Free Electron BandsPeriodic Perturbation of Free Electron BandsSolutionsSolutions

Eigen-values…

Eigen-vectors…

Periodic Perturbation of Free Electron BandsPeriodic Perturbation of Free Electron BandsSolutionsSolutions

high energy solutionlow energy solution

Plots are for a potential of the form…


‘When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal….By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation’

F. BLOCH

For wavefunctions that are eigenenergy states in a periodic potential…

or

Proof of Bloch’s TheoremProof of Bloch’s Theorem

Step 1: Translation operator commutes with Hamiltonain…so they share the same eigenstates.

Translation and periodic Hamiltonian commute…

Therefore,

Step 2: Translations along different vectors add…so the eigenvalues of translation operator are exponentials

Normalization of Bloch FunctionsNormalization of Bloch Functions

Conventional (A&M) choice of Bloch amplitude…

6.730 choice of Bloch amplitude…

Normalization of Bloch amplitude…

Momentum and Crystal MomentumMomentum and Crystal Momentum

where the Bloch amplitude is normalized…

Physical momentum is not equal to crystal momentum

So how do we figure out the velocity and trajectory in real space of electrons ?


Lecture 18: Properties of Bloch Functions

Outline

• Momentum and Crystal Momentum

• k.p Hamiltonian

• Velocity of Electrons in Bloch States


‘When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal….By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation’

F. BLOCH

For wavefunctions that are eigenenergy states in a periodic potential…

or




Therefore,


Normalization of Bloch FunctionsNormalization of Bloch Functions

Conventional (A&M) choice of Bloch amplitude…

6.730 choice of Bloch amplitude…

Normalization of Bloch amplitude…


where the Bloch amplitude is normalized…

Physical momentum is not equal to crystal momentum

So how do we figure out the velocity and trajectory in real space of electrons ?


canceling exponentials from both sides

A useful identity, for the action of the momentum operator on the Bloch amplitude….

Leads us to, the action of the Hamiltonian on the Bloch amplitude….

k.p Hamiltoniank.p Hamiltonian(in our case q.p)(in our case q.p)

If we know energies as k we can extend this to calculate energies at k+qfor small q…

k.p Hamiltoniank.p Hamiltonian

Taylor Series expansion of energies…

Matching terms to first order in q…

Velocity of an Electron in a Bloch Velocity of an Electron in a Bloch EigenstateEigenstate

Electron Electron Wavepacket Wavepacket in Periodic Potentialin Periodic Potential

Wavepacket in a dispersive media…

So long as the wavefunction has the same short range periodicity as the underlying potential, the electron can experience smooth uniform motion at a constant velocity.

Energy Surface for 2Energy Surface for 2--D Crystal D Crystal

In 2-D, circular energy contours result in parallel to


In general, for higher lying energies is not parallel to


4 valence bands4 conduction bands

Semiclassical Semiclassical Equation of MotionEquation of Motion

Ehrenfest’s Theorem:

Consider some external force that perturbs the electron in the lattice…

An elegant derivation can be made if we consider the equation of motion for the lattice translation operator

Since the lattice translation and Hamiltonian commute with each other…


Lets consider a specific external force…an external uniform electric field…

Equation of motion for translation operator becomes…

Can evaluate the commutation relation in the position basis…


Plugging in this commutation relation into the equation of motion…

Solving the simple differential equation…

From Bloch’s Thm. We know the eigenvalues of TR…

Electron Motion in a Uniform Electric FieldElectron Motion in a Uniform Electric Field22--D CrystalD Crystal

http://www.physics.cornell.edu/sss/ziman/ziman.html


Lecture 19: Motion of Electronic Wavepackets

Outline

• Review of Last Time

• Detailed Look at the Translation Operator

• Electronic Wavepackets

• Effective Mass Theorem




Therefore,



In 2-D, circular energy contours result in parallel to


In general, for higher lying energies is not parallel to

Properties of the Translation OperatorProperties of the Translation Operator

Definition of the translation operator…

Bloch functions are eigenfunctions of the lattice translation operator…

Lattice translation operator commutes with the lattice Hamiltonian (Vext=0)

The translation operator commutes with other translation operators…


Lets see what the action of the following operator is…

This is just the translation operator…

Another Look at Electronic Another Look at Electronic BandstructureBandstructure


E

As we will see, it is often convenient to represent the bandstructure by its inverse Fourier series expansion…

Translation Operator and Lattice HamiltonianTranslation Operator and Lattice Hamiltonian

From before, the eigenvalue equation for the translation operator is….

If we multiply this by the Fourier coefficients of the bandstructure…

…and sum over all possible lattice translations…

…we see that the eigenvalue on the left is just the bandstructure (energy)

This suggests the operator on the left is just the crystal Hamiltonian !

No wonder

Electron Electron Wavepacket Wavepacket in Periodic Potentialin Periodic Potential

Wavepacket in a dispersive media…

So long as the wavefunction has the same short range periodicity as the underlying potential, the electron can experience smooth uniform motion at a constant velocity.

Wavefunction Wavefunction of Electronic of Electronic WavepacketWavepacketThe eigenfunction for k~k0 are approximately…

A wavepacket can therefore be constructed from Bloch states as follows…

G is a slowly varying function…

Wavefunction Wavefunction of Electronic of Electronic WavepacketWavepacket

Since we construct wavepacket from a small set of k’s…

…the envelope function must vary slowly…wavepacket must be large…

Action of Crystal Hamiltonian on Action of Crystal Hamiltonian on WavepacketWavepacket

It appears that the Hamiltonian only acts on the slowly varying amplitude…

Effective Mass TheoremEffective Mass TheoremIf we can consider an external potential (eg. electric field) on the crystal…

The influence of the external field on the wavepacket…

We can solve Schrodinger’s equation just for the envelope functions…

Normalization of the Envelope FunctionNormalization of the Envelope Function

Since the envelope is slowly varying…it is nearly constant over the volume of one primitive cell…

What is the Position of What is the Position of Wavepacket Wavepacket ??

Proof that…

What is the Momentum of What is the Momentum of WavepacketWavepacket

SummarySummary

Without explicitly knowing the Bloch functions, we can solve for the envelope functions…

Bandstructure shows up in here…

The envelope functions are sufficient to determine the expectation of position and crystal momentum for the system…


Lecture 20: Impurity States

Outline

• Semiclassical Equations of Motion

• Review of Last Time: Effective Mass Hamiltonian

• Example: Impurity States


Definition of the translation operator…

Bloch functions are eigenfunctions of the lattice translation operator…

Lattice translation operator commutes with the lattice Hamiltonian (Vext=0)

The translation operator commutes with other translation operators…


Lets see what the action of the following operator is…

This is just the translation operator…

Another Look at Electronic Another Look at Electronic BandstructureBandstructure


E

As we will see, it is often convenient to represent the bandstructure by its inverse Fourier series expansion…

Wavefunction Wavefunction of Electronic of Electronic WavepacketWavepacketThe eigenfunction for k~k0 are approximately…

A wavepacket can therefore be constructed from Bloch states as follows…

Since we construct wavepacket from a small set of k’s…

…the envelope function must vary slowly…wavepacket must be large…

Summary of Last TimeSummary of Last Time

Without explicitly knowing the Bloch functions, we can solve for the envelope functions…


The envelope functions are sufficient to determine the expectation of position and crystal momentum for the system…

Using Using Bandstructure Bandstructure in Effective Mass Hamiltonianin Effective Mass Hamiltonian

where


For example…

Donor Impurity States Donor Impurity States Example of Effective Mass ApproximationExample of Effective Mass Approximation

Replace silicon (IV) with group V atom…


This is a central potential problem, like the hydrogen atom…


Hydrogenic wavefunction with an equivalent Bohr radius..

EC

EV

ED

Egap~ 1 eV

n-type SiDonor ionization energy…


When there are Nd donor impurities…

E

Acceptor Impurity States Acceptor Impurity States Example of Effective Mass ApproximationExample of Effective Mass Approximation

Replace silicon (IV) with group III atom…


Another central potential problem…


Hydrogenic wavefunction with an equivalent Bohr radius..

EA

EC

EVp-type Si

Acceptor ionization energy…


Lecture 21:

Outline

• Dynamical Effective Mass

• Fermi Surfaces

• Electrons and Holes

Semiclassical Semiclassical Equations of MotionEquations of Motion



Lets try to put these equations together….

Looks like Newton’s Law if we define the mass as follows…

dynamical effective mass

mass changes with k…so it changes with time according to k

Dynamical Effective Mass (3D)Dynamical Effective Mass (3D)

Extension to 3-D requires some care, F and a don’t necessarily point in the same direction

where

Dynamical Effective Mass (3D)Dynamical Effective Mass (3D)Ellipsoidal Energy SurfacesEllipsoidal Energy Surfaces

Fortunately, energy surfaces can often be approximate as…


11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar


conductor

insulator

conductor

a = 5.5 A

Al

‘Mg’

X

W

Γ Na



Si: [Ne] 3s2 3p2

4 e- per silicon atom2 silicon atoms per lattice site

total: 8 electrons at each site

Free Electron Free Electron Fermi Fermi Surfaces (2D)Surfaces (2D)T=0T=0

For free electrons energy surfaces are simple spheres (circles)…Valence (# of electrons) determines radius of energy surface… 1st zone

2nd zone

3rd zone

FermiFermi Surfaces (3D)Surfaces (3D)

When k near to BZ boundary:When k near to BZ boundary:E contours become distortedE contours become distorted

2D2D

FermiFermi Surfaces (3D)Surfaces (3D)

NNee = 1= 1 monovalentmonovalent metals, e.g. Na, Cu, with values ~ f.e. theorymetals, e.g. Na, Cu, with values ~ f.e. theory

other cases, e.g. Be (other cases, e.g. Be (NNee=2), Al (=2), Al (NNee=3), there are serious differences=3), there are serious differences

Overview of Electron DistributionsOverview of Electron Distributions

Metal Insulatoror

Semiconductor T=0

n-DopedSemi-

Conductor

Semi-Conductor

T=0

Electron Distributions in Doped SemiconductorsElectron Distributions in Doped Semiconductors

+1 e -

n-DopedSemi-

Conductor

Semi-Conductor

T=0

EC

EV

ED

Egap~ 1 eV

n-type Si

Electron and HolesElectron and Holes

Semi-Conductor

T=0

Electrons in conduction bandElectrons in conduction band

Holes in valence bandHoles in valence band

Motion of Valence ElectronsMotion of Valence Electrons

electrons have negative chargeelectrons have negative charge

kk--spacespace

Valence electrons (and vacancy) all move in the positive Valence electrons (and vacancy) all move in the positive kkxx direction…direction…



valence electrons have negative mass !valence electrons have negative mass !

Real spaceReal space

Vacancy ends up moving in the direction of the Vacancy ends up moving in the direction of the electric field as if it had a positive chargeelectric field as if it had a positive charge

Hole is a quasiHole is a quasi--particle with positive charge and positive mass…particle with positive charge and positive mass…


Hole is a quasiHole is a quasi--particle with particle with positive charge and positive positive charge and positive mass…mass…

kk--spacespace

hole dispersionhole dispersion


Lecture 23: Effective Mass

Outline

• Review of Last Time

• A Closer Look at Valence Bands

• k.p and Effective Mass


Lets try to put these equations together….

Looks like Newton’s Law if we define the mass as follows…

dynamical effective mass

mass changes with k…so it changes with time according to k

Dynamical Effective Mass (3D)Dynamical Effective Mass (3D)

Extension to 3-D requires some care, F and a don’t necessarily point in the same direction

where

Dynamical Effective Mass (3D)Dynamical Effective Mass (3D)Ellipsoidal Energy SurfacesEllipsoidal Energy Surfaces

Fortunately, energy surfaces can often be approximate as…

Motion of Valence Electrons (and Holes)Motion of Valence Electrons (and Holes)


valence electrons have negative mass !valence electrons have negative mass !

Real spaceReal space

Vacancy ends up moving in the direction of the Vacancy ends up moving in the direction of the electric field as if it had a positive chargeelectric field as if it had a positive charge

Hole is a quasiHole is a quasi--particle with positive charge and positive mass…particle with positive charge and positive mass…


Increasing the orbital overlap, reduces the effective mass…


X WΓ

X WΓX

W

Γ

a = 8.3 A



X WΓ

X WΓ

a = 5.5 A

X

W

Γ

a = 2.8 A


3D Band Structures3D Band StructuresDispersion RelationsDispersion Relations

Lighter effective mass Larger overlap between orbitals

basisorbital

lightmass

heavymass

heavymass

a a a

Bandstructure Bandstructure of of GaAsGaAs

(k)

s like -orbital

p-like orbital

What is this split-off band ?

SpinSpin--orbit Coupling orbit Coupling WavefunctionsWavefunctions

heavy hole charge distribution light hole charge distribution

Orbital Angular MomentumOrbital Angular Momentum

Angular momentum for quantum state with l = 2:

z

24ºm = 1

m = 0

m = −1

55ºm = 2

m = −2

l = 2

SpinSpin--Orbit CouplingOrbit Coupling

-q +Zq

L, Bl

-q

+Zq

The effective current from the motion of a nucleus in a circular orbit…

…generates an effective magnetic field…

SpinSpin--Orbit Splitting

-q +ZqS

Spin up:High Energy

µs

L, Bl

-q +ZqS

Spin down:Low Energy

µs L, Bl

2P3/2L S2P

1S

2P1/2

B

Orbit Splitting

J = L + S = 3/2

J = L + S = 1/2L SB

SpinSpin--Orbit Splitting in HydrogenOrbit Splitting in Hydrogen

-q +ZqS

Spin up:High Energy

µs

L, Bl

-q +ZqS

Spin down:Low Energy

µs L, Bl

Angular Momentum Addition RulesAngular Momentum Addition Rules

Quantum NumbersVectors

j = 3/2 j = 1/2

Example: l = 1, s = ½

SpinSpin--orbit Coupling orbit Coupling WavefunctionsWavefunctions

heavy hole charge distribution light hole charge distribution

heavy mass (along kz) light mass (along kz)

Bandstructure Bandstructure of of GaAsGaAs

Spin-orbit splitting

Another Approach to Another Approach to BandstructureBandstructure: k.p: k.p

Often it is easier to know the energies at a particular point (ex. Bandgap) than it is to measure the effecitve mass

k.p is a way to relate your knowledge of energy levels at k to the effective mass…using perturbation theory

k.p Effective Massk.p Effective Mass

Second-order perturbation theory…


k.p Effective Massk.p Effective Mass

k.p Effective Massk.p Effective MassExampleExample

Lets only consider two bands (valence and conduction) and assumethey are spherical…

k.p Effective Massk.p Effective MassExampleExample

Level repulsion causes bands to curve as bandgap is reduceed…

Effective Mass and Effective Mass and BandgapBandgap

Experimental Data

Courtesy of Jasprit Singh; Used with Permission http://www.eecs.umich.edu/~singh/semi.html

farnaz

(see "Semiconductor Bandstructure")


Lecture 24: Chemical Potential and Equilibrium

Outline

• Microstates and Counting

• System and Reservoir Microstates

• Constants in Equilibrium

Temperature & Chemical Potential

• Fermi Integrals and Approximations

Microstates and CountingMicrostates and CountingEnsemble of 3 ‘2Ensemble of 3 ‘2--level’ Systemslevel’ Systems

Total Energy # of Microstates

E=0 g=1E=1 g=3E=2 g=3E=3 g=1

As we shall see, g is related to the entropy of the system…

Microstates and CountingMicrostates and CountingEnsemble of 4 ‘2Ensemble of 4 ‘2--level’ Systems

E=2level’ Systems

Total Energy # of Microstates

E=0 g=1E=1 g=4E=2 g=6E=3 g=4E=4 g=1

E=2

Microstates and CountingMicrostates and Counting

The larger the systems, the stronger the dependence on E

For most mesoscopic and macroscopic systems, g is a monotonically increasing function of E

System + Reservoir MicrostatesSystem + Reservoir Microstates

Gibb’s Postulate = all microstates are equally likely

reservoirsystem

Example

Consider a system of 3 ‘2-levels’ + a reservoir of 10 ‘2-levels’

Probability of finding: Es = 0 45/78Es = 1 30/78Es = 2 3/78

Most electrons are in the ground state so reservoir entropy is maximized !

System + Reservoir MicrostatesSystem + Reservoir Microstates

reservoirsystem

For sufficiently large reservoirs….

…we only care about the most likely microstate for S+R

Now we have a tool to look at equilibrium…

System + Reservoir in EquilibriumSystem + Reservoir in Equilibrium

reservoirsystem

Equilibrium is when we are sitting in this max entropy (g) state…

is the same for two systems in equilibrium


reservoirsystem

We observe that two systems in equilibrium have the same temperature, so we hypothesize that…

This microscopic definition of temperature is a central result of stat. mech.

Boltzmann Boltzmann DistributionsDistributions

S is the thermodynamic entropy of a system

Boltzmann observed that…

and

…so he hypothesized that

Boltzmann Boltzmann DistributionsDistributions

reservoir controls system distribution


Now we allow system and reservoir to exchange particles as well as energy… reservoir

system


reservoir

system

Entropy of reservoir can be expanded for each case…

Difference in entropy of the two configurations is…

..where µ is the electrochemical potential


Chemical potential is change in energy of system if one particle is added without changing entropy

System + Reservoir in EquilibriumSystem + Reservoir in EquilibriumExample: Example: FermiFermi--Dirac Dirac StatisticsStatistics

Consider that the system is a single energy level which can either be…occupied:unoccupied:

Normalized probability…

Two Systems in EquilibriumTwo Systems in Equilibrium

reservoir

system 1 system 2

Particles flow from 1 to 2…

Particles flow from 2 to 1…

In equilibrium…

Counting and Counting and Fermi Fermi IntegralsIntegrals33--D Conduction Electron DensityD Conduction Electron Density

Counting and Counting and Fermi Fermi IntegralsIntegrals33--D Hole DensityD Hole Density


Exact solution !


Lecture 25: Chemical Potential and Non-equilibrium

Outline

• Fermi Integrals and Approximations

• Rate Equations for Non-equilibrium Electrons

• Quasi-Fermi Levels

Counting and Counting and Fermi Fermi IntegralsIntegrals33--D Hole DensityD Hole Density

Boltzmann Boltzmann ApproximationApproximation

Boltzmann Approximation:

Approximations for Approximations for Fermi Fermi IntegralsIntegrals33--D Carrier DensitiesD Carrier Densities

Sommerfeld Approximation:

Unger Approximation:

where

Approximations for Approximations for Fermi Fermi IntegralsIntegrals33--D Carrier DensitiesD Carrier Densities

Approximations for InverseApproximations for Inverse FermiFermi IntegralsIntegrals

Inverse First-order Sommerfeld Approximation:

for 0.04 error

Inverse Second-order Unger Approximation:

for 0.04 error

Near Equilibrium Electron DistributionsNear Equilibrium Electron DistributionsOptical ExcitationOptical Excitation

E3

E2E1

E3

E2E1

Intraband scattering: electron-electronelectron-acoustic phonon

Interband scattering: electron-holeelectron-phonon with defects

What are f1, f2, & f3 under illumination (non-equilibrium) ?

Rate Equation FormalismRate Equation Formalismnumber of electrons = number of states x probability of occupancy

assume total number of electrons in N1, N2, & N3 is contant

Rate Constants in EquilibriumRate Constants in EquilibriumDetailed BalanceDetailed Balance

In equilibrium:

Detailed balance:In equilibrium, each scattering process balances with its inverse

Rate EquationsRate Equations

Assume the rate constants don’t change out of equilibrium…

SteadySteady--State SolutionsState SolutionsNonNon--equilibriumequilibrium

For example when intraband scattering is much faster than interbandscattering…


Non-equilibrium Quasi-Fermi-Diracdistribution:

Equilibrium Fermi-Dirac distribution:

Intraband states have same chemical potentialin ‘equilibrium’ with each other because of fast intraband scattering


Interband states have different chemical potentialsunless

Counting in NonCounting in Non--equilibrium Semiconductorsequilibrium Semiconductors

Equilibrium Quasi-equilibrium


Lecture 26: Inhomogeneous Solids

Outline

• Last Time: Quasi-Fermi Levels

• Inhomogenous Solids in Equilibrium

• Quasi-equilibrium Transport

• Heterostructures

Near Equilibrium Electron DistributionsNear Equilibrium Electron DistributionsOptical ExcitationOptical Excitation

E3

E2E1

E3

E2E1

Intraband scattering: electron-electronelectron-acoustic phonon

Interband scattering: electron-holeelectron-phonon with defects

What are f1, f2, & f3 under illumination (non-equilibrium) ?


For example when intraband scattering is much faster than interbandscattering…


Non-equilibrium Quasi-Fermi-Diracdistribution:

Equilibrium Fermi-Dirac distribution:

Intraband states have same chemical potentialin ‘equilibrium’ with each other because of fast intraband scattering


Interband states have different chemical potentialsunless

Counting in NonCounting in Non--equilibrium Semiconductorsequilibrium Semiconductors

Equilibrium Quasi-equilibrium

Inhomogeneous Semiconductors in EquilibriumInhomogeneous Semiconductors in Equilibrium

Consider a solid with a spatially varying impurity concentration…

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2El

ectro

stat

ic P

oten

tial (

V)Microns

0

2 1016

4 1016

6 1016

8 1016

1 1017

0 0.5 1 1.5 2

Elec

tron

Con

cent

ratio

n (c

m-3

)

Microns

In equilibrium, the carrier concentration is balanced by an internal electrostatic potential…

Inhomogeneous Semiconductors in EquilibriumInhomogeneous Semiconductors in EquilibriumIf electrostatic potential varies slowly compared to wavepacket…

Dividing solid into slices where φi is uniform…

…the envelope function has solutions of the form…

…therefore the eigenenergies are…


Given the modified energy levels, the 3-D DOS becomes….

…in equilibrium the carrier concentration is…

Boltzmann approx.


The slowly varying electrostatic potential can be incorporated in

QuasiQuasi--equilibrium Transportequilibrium Transport

Species of Species of HeterjunctionsHeterjunctions

Type IType II

Type III

Type I

Type III

http://www.utdallas.edu/~frensley/technical/hetphys

TightTight--binding Calculation of Band Alignmentsbinding Calculation of Band Alignments

LCAO internally references bandstructures to each other…

InAsGaAs

Unfortunately, this doesn’t take into account the details of thecharges and bonding at the interface…

…need a self-consistent LCAO theory…still a research topic !

TightTight--binding Calculation of Band Alignmentsbinding Calculation of Band Alignments

Example GaAs/InAs

Ga: As: In:

InAs:GaAs:

InAsGaAsGaAs/InAs: (LCAO)

(experiment)

Experimentally Determined Band AlignmentExperimentally Determined Band Alignment

Ener

gy (e

V)

Valence Band Alignment

Courtesy of Sandip Tiwari, Cornell University; Used with Permission


Ener

gy (e

V)

InAs

Conduction Band Alignment



Ener

gy (e

V)


SimWindows SimWindows SoftwareSoftware

Self-consistent solution of modified drift-diffusion & Poisson’s Equation…

http://www-ocs.colorado.edu/SimWindows/simwin.html


Lecture 27: Scattering of Bloch Functions

Outline

• Review of Quasi-equilibrium

• Occupancy Functions

• Fermi’s Golden Rule

• Bloch electron scattering

Occupancy Functions and QuasiOccupancy Functions and Quasi--Fermi Fermi FunctionsFunctions

E2E1

E3

Equilibrium occupancy function…

Quasi-equilibrium occupancy function…

Properties of the Occupancy FunctionProperties of the Occupancy FunctionMoments of Moments of ff ((r,k,tr,k,t))

Carrier density…

Current density…

Energy density…

All the classical information about the carriers is contained in f (r,k,t)

Rate Equations for Occupancy FunctionRate Equations for Occupancy Function

Previously we developed rate equation for model 3-level system…

E2E1

E3

Now, generalize for the whole occupancy function…

Rate Equations for Occupancy FunctionRate Equations for Occupancy Function

rate of scattering from k’ to k

rate of scattering from k to k’

Perturbations that cause scattering….• Impurities or defects• Electron-phonon scattering• Electron-photon scattering

Use Fermi’s Golden Rule to calculate scattering between Bloch functions…

Fermi’s Fermi’s Golden RuleGolden Rule

Scattering rate from k to k ’

• For weak collisions to continuum of nearby states…

where…

• Energy conservation holds for infrequent collisions …

General Scattering PotentialGeneral Scattering Potential

We will only consider scattering potentials of the form…

We can consider each potential term separately…

…Fermi…


final state energy is greater than initial absorption

final state energy is less than initial emission

Initial and Final States for ScatteringInitial and Final States for Scattering

Envelope (effective mass) approximation…

∆ is volume of primitive cellN is numer of primitive cells in solid

are slowly varying over ∆

Normalization of Envelope FunctionsNormalization of Envelope Functions

Since envelope functions are slowly varying…

Normalization of envelope functions…

Matrix Elements for Bloch StatesMatrix Elements for Bloch States

Approximation for periodic scattering potential…

Approximation for slowly varying scattering potential…

Scattering from a Slowly Varying PotentialScattering from a Slowly Varying Potential

Matrix element is just the Fourier component of the scattering

potential at

Scattering Rate CalculationsScattering Rate CalculationsExample: 1Example: 1--D Scattering from DefectD Scattering from Defect

• Sharply peaked potential scatters isotropically

• Static potential scatters elastically

Scattering Rate CalculationsScattering Rate CalculationsExample: 1Example: 1--D Scattering from Traveling WaveD Scattering from Traveling Wave

• Periodic potentials conserve total momentum..

Scattering TimesScattering Times

Scattering time out of state k …

…at low densities…

…relaxation time is a function of state k

We usually measure some ensemble averaged relaxation time…

…which means we have to know

Scattering TimesScattering Times

Relaxation time for z-directed momentum…

Relaxation time for energy…


Lecture 28: Electron-phonon Scattering

Outline

• Bloch Electron Scattering

• Deformation Potential Scattering

• LCAO Estimation of Deformation Potential

• Matrix Element for Electron-Phonon Scattering

• Energy and Momentum Conservation


final state energy is greater than initial absorption

final state energy is less than initial emission

Scattering from a Slowly Varying PotentialScattering from a Slowly Varying Potential

Matrix element is just the Fourier component of the scattering

potential at

Scattering Rate CalculationsScattering Rate CalculationsExample: 1Example: 1--D Scattering from Traveling WaveD Scattering from Traveling Wave

• Periodic potentials conserve total momentum..

Scattering Rate CalculationsScattering Rate CalculationsOverviewOverview

Step 1: Determine Scattering Potential

Step 2: Calculate Matrix Elements

Step 3: Calculate State-State Transition Rates

Step 4: Calculate State Lifetime

Step 5: Calculate Ensemble Lifetime

ElectronElectron--Phonon Scattering PotentialPhonon Scattering PotentialBeyond the Born-Oppenheimer Approximation…

• Phonons change the electron energies by changing the bond displacement• Both shear strain and local volume changes alter the electron energy

…change in the bandstructure due to a dilatation of solid by sound wave…

Relate the phonons to local changes in the volume (lattice constant)….

ElectronElectron--Phonon Scattering PotentialPhonon Scattering Potential

Only LA phonons cause local changes in the volume (lattice constant)….


Conduction band (diamond):

Valence band (diamond):

ElectronElectron--Phonon Scattering PotentialPhonon Scattering PotentialSilicon ExampleSilicon Example

Phonon Displacement OperatorPhonon Displacement Operator

See Lecture 11…phonon displacement operator

relating mass for continuum solid and discrete lattice

ElectronElectron--Phonon Matrix ElementPhonon Matrix Element

Phonon absorption…

ElectronElectron--Phonon Matrix ElementPhonon Matrix Element

For long wavelength phonons, can make slowly-varying approx…

ElectronElectron--Phonon Scattering RatePhonon Scattering Rate

Energy and Momentum ConservationEnergy and Momentum Conservation


For acoustic phonons…


Typical acoustic phonon velocity…

Velocity of typical electron (300 K)…

final inital


Maximum momentum exchange…

Maximum energy exchange…

Acoustic phonon scattering is essentially elastic for 300K electrons…

ElectronElectron--Phonon Scattering TimePhonon Scattering TimePreviewPreview

麻省理工大学固体物理课件

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