Institute of Solid State Physics Technische Universität Graz 17. Phonons, Electrons May 24, 2018
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Institute of Solid State PhysicsTechnische Universität Graz
17. Phonons,Electrons
May 24, 2018
Waves and particles
The eigen function solutions of the wave equation are plane waves. The scattering time is one over the rate for scattering from a given plane wave solution to any other.
Phonons are particles. The scattering time is the time before the phonons scatter and randomly change energy and momentum.
The average time between scattering events is sc = 1/
p k
Phonon scattering
Scattering randomizes the momentum of the phonons.
Transition rates determined by Fermi's golden rule
Any process (3 phonon, 4 phonon, 5 phonon. ...) that conserves energy and momentum is allowed.
2
12
i f f i f iH E E
Results in attenuation of acoustic waves
1HOH H H
Umklapp Processes
Three phonon scattering
1 2 3k k k G
from: Hall, Solid State Physics
Heat transport (Kinetic theory)
Treat phonons as an ideal gas of particles that are confined to the volume of the solid.
Phonons move at the speed of sound. They scatter due to imperfections in the lattice and anharmonic terms in the Hamiltonian.
The average time between scattering events is sc
The average distance traveled between scattering events is the mean free path: l = vsc ~ 10 nm
Diffusion equation/ heat equation
2dn D ndt
j D n
dn jdt
21 exp44
rnDtDt
Continuityequation
Fick's law
Diffusion constant
Random walk
2exp x
Central limit theorem: A function convolved with itself many times forms a Gaussian
1 12ss s s
n n n nt
Thermal conductivity
vK Dc
Uj Ej
Average particle energy
u En
internal energy density
Uj ED n D u
U vduj D T Dc TdT
Uj K T
Thermal conductivity
0 as 0K T
Thermal conductivity Uj K T
Imperfections in the crystal or grain boundaries decrease the mean free path and the thermal conductivity.
At high temperatures, the mean free path is limited by Umklapp processes. At low temperatures the Umklapp processes freeze out and the mean free path is limited by imperfections.
~ T 3
~ 1/T
Material Thermal conductivity W/(m·K) Glass 1.1Concrete, stone 1.7Ice 2Sandstone 2.4Sapphire 35Stainless steel 12.11 ~ 45.0Lead 35.3Aluminum 237 Aluminum alloys 120—180 Gold 318Copper 401Silver 429Diamond 900 - 2320Graphene (4840±440) - (5300±480)
Uj K T
Thermal conductivity
Phonon student projects
Calculate a dispersion relation including next nearest neighbors.
Write a javascript program that plots the phonon dispersion relation in an arbitrary direction.
Calculate one column of the phonon table: hcp, NaCl, CsCl, ZnS, diamond, ...
Calculate the temperatures at which ZnO goes through a phase transition.
Institute of Solid State PhysicsTechnische Universität Graz
Free electron Fermi gas
Kittel, chapter 6
A simple model for a metal is electrons confined to box with periodic boundary conditions.
Like the problem of photons in a box except:Solve the Schrödinger equation instead of the wave equation.Electrons are fermions not bosons.
Free particles in 1-d
2 2
22d E
m dx
0V
2 2 2 2 2 2 2
2 28 2 2 2 2n h h p k mvEmL m m m
2Ln
Free particles in 1-d
i kx tk kA e
2 2
22di
t m dx
0V
Eigen function solutions:
Dispersion relation:2 2
212 2
kE mvm
k
E
Periodic boundary conditions
2( )D k
Density of states: Spin
2L
4L
i kx tkA e
L
( )LD k dkNumber of states between |k| and |k|+dk is
2 2( ) 2dkLD k dk
L
Free particles in 1-d
2( )D k
Density of states
Spin
k
E
1 2( ) ( ) dk mD E D kdE E
E
D(E)
Van Hove singularity
1 22
dk mdE E
2mEk
Fermi function
1( )1 exp
B
f EEk T
f(E) is the probability that a state at energy E is occupied.
= chemical potential
Chemical potential
( )( ) ( )1 exp
B
N D E dEn D E f E dEL E
k T
The chemical potential is implicitly defined as the energy that solves the following equation.
Here N is the total number of electrons.
E
D(E)f(E)
1( )1 exp
B
f EEk T
1 2( ) mD EE
E
D(E)f(E)
Chemical potential
is temperature dependent
Fermi energy
( )FE
n D E dE
EF = (T=0)
In one dimension,
0
1 2 2 2FE
Fmn dE mEE
2 2 2
8FnE
m
In semiconductor books, EF(T) = (T).
At T = 0
Free particles in 1-d
internal energy spectral density2 1( ) ( ) ( )
exp 1B
mEu E ED E f EEk T
( )
v
u u E dE
ducdT
analog to the Planck curve for electrons in 1-d
Not possible to do this integral analytically
Thermodynamic properties
1 2( ) mD EE
From the density of states, the thermodynamic properties can be calculated.
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