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Institute of Solid State Physics
Technische Universität Graz
Advanced Solid State Physics
Solid state physics is the study of how atoms arrange themselves into solids and
what properties these solids have.
Calculate the macroscopic properties from the microscopic structure.
Peter Hadley
Institute of Solid State Physics
Technische Universität Graz
Advanced Solid State Physics
Quantization
Review: Photons (noninteracting bosons), photonic crystals
Review: Free electrons (noninteracting fermions), electrons in crystals
Electrons in a magnetic field
Fermi surfaces
Quantum Hall effect
Linear response theory
Dielectric function / optical properties
Transport properties
Quasiparticles (phonons, magnons, plasmons, exitons, polaritons)
Mott transition, Fermi Liquid Theory
Ferroelectricity, pyroelectricity, piezoelectricity
Landau theory of phase transitions
Superconductivity
Institute of Solid State Physics
Technische Universität Graz
Books
e-Books
http://lamp.tu-graz.ac.at/~hadley/ss2/
TUG -> Institute of Solid State Physics -> Courses
Institute of Solid State Physics
Technische Universität Graz
Student projects
Something that will help other students pass this course
2VO + 1UE
Improve the script (1 student needed to compile)
Solutions to exam questions
Example calculations (phonon dispersion relation for GaAs)
Javascript calculations
Lecture videos
You can decide not to put your project on the webpage.
Institute of Solid State Physics
Technische Universität Graz
Examination
1 hour written exam
half of the questions will be from the website
Oral exam
Student project
Mistakes on written exam
General questions about the course
Institute of Solid State Physics
Technische Universität Graz
Quantization
Start with the classical equations of motion
Find the normal modes
Construct the Lagrangian
From the Lagrangian determine the conjugate variables
Perform a Legendre transformation to the Hamiltonian
Quantize the Hamiltonian
Harmonic oscillator
Newton's law:
Euler - Lagrange equations:
Lagrangian (constructed
by inspection)
Conjugate variable:
Legendre transformation:
Quantize:

p  i
x
ma  Kx
d L L

0
dt  qi  qi
mx 2 Kx 2
L  x,x  

2
2
p
L
 mx
x
p 2 Kx 2
H  px  L 

2m
2
 2 d 2 Kx 2
H 


2
2m dx
2
LC circuit
Classical equations
V L
dI
dt
I  C
dV
dt
Q  CV
Q
d 2Q
 L 2
C
dt
Euler - Lagrange equation:
Lagrangian (constructed
by inspection)
Conjugate variable:
Legendre transformation:
d  L L

0
dt  Q  Q


LQ 2 Q 2

L Q,Q 
2
2C
p
L
 LQ
Q
LQ 2 Q 2
H  QQ L 

2
2C
LC circuit
Conjugate variable:
Hamiltonian:
Quantize:
L
p
 LQ
Q
LQ 2 Q 2
H

2
2C

p  i
Q
 2 d 2 Q2
H 
   E
2
2L dQ 2C
Transmission line
iCZ
L inductance/m
C capacitance/m
2
dV
dI
L
dx
dt
  2 LCZ  i L   0
dI
dV
C
dx
dt
d 2V
d 2V
 LC 2
2
dx
dt
normal mode solution:
Vk  V0 exp  i  kx  t  
Each normal mode moves independently from the other normal modes
Transmission line
iCZ
Substituting the normal mode solution
into the wave equation
C
V
L
  2 LCZ  i L   0
V  V0 exp  i  kx  t  
d 2V
d 2V
 LC 2
2
dx
dt
yields the dispersion relation
I
2

k
 ck
LC
Z
V
L

I
C
An infinite transmission line is resistive, typically ~ 50 W.
Transmission line
iCZ
2
  2 LCZ  i L   0
Wave equation
d 2V d 2V
c
 2
2
dx
dt
2
c is the speed of waves
Not clear what mass we should use in the Schrödinger equation
The Schrödinger equation is for amateurs
Lagrangian (constructed L  x,x 
by inspection)
Euler - Lagrange equations:
d L L

0
dt  Q  Q
Conjugate variable:
L
p
x
Legendre transformation:
Quantize:

p  i
x
H  px L
Classical equations of motion
(Newton's law)
Lagrangian
Construct the Lagrangian 'by inspection'. The Euler-Lagrange equation and
the classical equation of motion for a normal mode are,
  L

t  Vk
 L
 0.

 Vk
2

Vk
2 2
c k Vk 
.
2
t
classical equation for the mode k
The Lagrangian is,
L
Vk 2 c 2 k 2 2


Vk
2
2
Hamiltonian
L
Vk 2 c 2 k 2 2


Vk
2
2
The conjugate variable to V is,
L
 Vk
Vk
The Hamiltonian is constructed by performing a Legendre transformation,
H  VkVk  L
Vk 2 c 2 k 2 2


Vk
2
2
To quantize we replace the conjugate variable by

i
Vk
 2 d 2 c 2 k 2 2

Vk   E
2
2 dVk
2
Quantum solutions
 2 d 2 c 2 k 2 2

Vk   E
2
2 dVk
2
This equation is mathematically equivalent to the harmonic oscillator.
E    j  12 
spring constant

K
m
j  0,1, 2,
  c2 k 2
wave mode
mass - spring
 c k
j is the number of photons.
Dissipation in Quantum mechanics
Transmission line
=
C
I
V
L
V
L
Z 
I
C
An infinite transmission line is resistive
Dissipation in quantum mechanics
Quantum coherence is maintained until the decoherence time. This depends on
the strength of the coupling of the quantum system to other degrees of freedom.
Decay is the decoherence time.
Dissipation in solids
At zero electric field, the electron eigen states are Bloch states. Each Bloch state
has a k vector. The average value of k = 0 (no current).
At finite electric field, the Bloch states are no longer eigen states but we can
calculate the transitions between Bloch states using Fermi's golden rule. The
final state may include an electron state plus a phonon. The average value of k is
not zero (finite current).
The phonons carry the energy away like a transmission line.
Institute of Solid State Physics
Technische Universität Graz
The quantization of the electromagnetic field
Wave nature and the particle nature of light
Unification of the laws for electricity and magnetism (described by
Maxwell's equations) and light
Quantization of fields
Planck's radiation law
Serves as a template for the quantization of noninteracting bosons:
phonons, magnons, plasmons, electrons, spinons, holons and other
quantum particles that inhabit solids.
http://lamp.tu-graz.ac.at/~hadley/ss2/emfield/quantization_em.php
Maxwell's equations

E 
0
B  0
B
 E  
t
E
  B  0 J  0 0
t
The vector potential
B   A
A
E  V 
t
Maxwell's equations in terms of A
A
0
t
   A  0


A 
  A
t t



2 A
    A    0 0 2
t
Coulomb gauge
 A  0

 A  0
t
Vector identity


 A   A
t
t
The wave equation
2 A
   A   0 0 2
t
    A  (  A)   2 A,
Using the identity
wave equation
2 A
c  A 2 .
t
2
2
normal mode solutions have the form:
A(r , t )  A exp(i (k  r  t ))
Substituting the normal mode solution in the wave equation results in the dispersion relation
 c k
EM waves propagating in the x direction
A  A0 cos(kx x  t ) zˆ
The electric and magnetic fields are
A
E
  A0 sin(k x x  t ) zˆ,
t
B    A  k x A0 sin(k x x  t ) yˆ.
Lagrangian
To quantize the wave equation we first construct the Lagrangian 'by
inspection'. The Euler-Lagrange equation and the classical equation of
motion are,
  L

t  As
 L
 0.

 As
The Lagrangian is,
2

As
2 2
c k As 
.
2
t
classical equation for the
normal mode k
As2 c 2 k 2 2
L

As
2
2
Hamiltonian
As2 c 2 k 2 2
L

As
2
2
The conjugate variable to As is,
L
 As
As
The Hamiltonian is constructed by performing a Legendre transformation,
As2 c 2 k 2 2
H  As As  L 

As
2
2
To quantize we replace the conjugate variable by

i
As
 2 d 2 c 2 k 2 2

As   E
2
2 dAs
2
Quantum solutions
 2 d 2 c 2 k 2 2

As   E
2
2 dAs
2
This equation is mathematically equivalent to the harmonic oscillator.
Es  s  js  12 
js  0,1, 2,
s  c k s
js is the number of photons in mode s.
Partition function
At constant temperature the relevant partition function is the canonical
partition function
 E 
Z (T ,V )   exp   q 
q
 kBT 
The sum over all microstates is a sum over each mode s (polarization and k)
containing js photons.:
 1
exp


js max
 kBT
Z (T ,V )  
j1
smax

s 1

s  js  12  

The sum in the exponent can be converted to a product ea+b+c+... = eaebec...
Z (T ,V )  
j1
 s  js  12  
exp  



k
T
js max s 1
B


smax