Transport properties  Boltzmann equation
goal: calculation of conductivity Boltzmann transport theory:
distribution function evolution from Boltzmann equation
number of particles in infinitesimal phase space volume around
collision integral for static po
Solid State Theory Solution Sheet 11
SS 11 Prof. M. Sigrist
Exercise 11.1
Relaxation time approximation
a) The transition rates are computed by use of Fermi's golden rule, relating the probability for a process to occur to the quantum mechanical matrix el
Solid State Theory Solution Sheet 10
FS 11 Prof. M. Sigrist
Exercise 10.1
Uniaxial Compressibility
We first write the deformation of the Fermi surface in terms of spherical harmonics:
0 0 0 kF (, ) = kF + kF [3 cos2  1] = kF + 4
0 k Y20 (, ). 5 F
(1)
a)
Solid State Theory Solution Sheet 9
FS 11 Prof. M. Sigrist
Exercise 9.1
Lowest Landau level in the Corbino geometry
a) Because Az = 0 and z A = 0 it follows that ( A)x = ( A)y = 0. Furthermore, for r = 0 we have ( A)z = 1 2 x B+ 0 r2 x+ 1 2 y B+ 0 r2 y =
Solid State Theory Solution Sheet 7
FS 11 Prof. M. Sigrist
Exercise 7.1
Phonons in One Dimension
Given the potential energy, v V = 2
N/2
(u2i  u2i+1 )2 + (u2i  u2i1 )2 ,
i=1
(1)
and the atomic masses M and m, we can immediately write down the classical
Solid State Theory Solution Sheet 6
FS 2011 Prof. M. Sigrist
Exercise 6.1
Lindhard function
At T = 0 the FermiDirac distribution function nF ( k ) reduces to ( F  k ). As usual, we go from the discrete summation to a ddimensional integral. Then, the st
Solid State Theory Solution Sheet 5
FS 11 Prof. M. Sigrist
Exercise 5.1
Graphene
a) We want to calculate the energy bands of graphene using the tightbinding method. Those bonds are due to electrons in the 2pz orbitals. Graphene has two inequivalent carb
Solid State Theory Solution Sheet 4
FS 11 Prof. M. Sigrist
Exercise 4.1
Onedimensional model of a semiconductor
The Hamilton operator is H1 = H0 + V where H0 = t
i
c ci+1 + c ci , i i+1 (1)i c ci . i
i
(1) (2)
V =v
[a] Let us consider the case v = 0. W
Solid State Theory Solution Sheet 3
FS 2011 Prof. M. Sigrist
Exercise 3.1
6Orbital tightbinding model
a) The Blochwaves constitute a basis of the (quasi1dimensional) Hilbert space of the atom chain. The structure of the functions in x and ydirectio
Solid State Theory Solution Sheet 2
FS 11 Prof. M. Sigrist
Exercise 2.1
Energy bands of almost free electrons on the fcc lattice
2
The Bloch equation is written in Fourier space as [see Eq. (1.21) of lecture notes] 2m (k + G)2  n,k cG +
G
VGG cG = 0.
(1
Solid State Theory Exercise 1
KronigPenney model
FS 11 Prof. M. Sigrist
We study a simple model for a onedimensional crystal lattice, which was introduced by Kronig and Penney in 1931. The atomic potentials are taken to be rectangular, where the minima
Quantum Hall effect
Hall bar geometry
integer
classical
quantum
classical Hall effect
flux quantum flux per electron
=
1
flux quanta per electron
Quantum Hall effect
integer
classical
quantum
conductivity / resistivity tensor
Quantum Hall effect
Landau le
pnjunction
depletion layer
equilibrium jdiff jdiff
n
dipole layer
p
+

jdrift jdrift
Solar cells
electric contacts
light
intensity
depletion layer
n

+
diffusion region
electronhole separation
p
used
J electric contact Jm Um JL UL
U
MOSFET
p
n source
Phonons in metals  Kohn anomaly
ion background oscillation relative to (fixed) electron analog to plasma oscillation
ion mass ion density ion charge
screening of Coulomb by (fast) electrons
Kohn anomaly
sound velocity singular at 2kF
Peierls instability
Solid State Theory
Spring Semester 2011
Manfred Sigrist Institut fr Theoretische Physik HIT K23.8 u Tel.: 0446332584 Email: [email protected] Website: http:/www.itp.phys.ethz.ch/research/condmat/strong/
Lecture Website: http:/www.itp.phys.ethz.ch
Metal  screening
ThomasFermi (static) screening
potential of point charge
Fourier
Coulomb
renormalize
Fourier
Yukawa
+


Metal  screening
static dielectric function
singularity at 2kF
dielectric function in 3D
1D 2D
singular at 2kF , unlike
smallq a
Group theory
Definition: group
is a set with a product
1
.
associative identity inverse Example:
d
with with
symmetry operation of square
h
nonabelian
Group theory
subgroup: group
subset of
2
examples:
number of elements:
devides
Group theory
transformat
Fermi liquid theory
Concept of Landau's Fermi liquid theory
elementary excitations of interacting Fermions are described by almost independent fermionic quasiparticles state of Fermi liquid described simply by quasiparticle distribution
Phenomenological T
Solid State Theory Exercise 13
SS 11 Prof. M. Sigrist
Exercise 13.1 = =
Critical temperature in the Stoner model
2 k2 2m
We consider three types of dispersion relations:
k k 0 0
(3D) and
+ k (1D).
Plot the critical temperature of the Stoner model for fix
Solid State Theory Exercise 12
SS 11 Prof. M. Sigrist
Exercise 12.1
Conductivity tensor
Calculate explicitly the conductivity tensor for a dispersion relation of the form k =
2 2 k
2m
(1)
in the static limit. Exercise 12.2 Residual Resistivity
Recapitula
Solid State Theory Exercise 11 Transport in metals
Exercise 11.1 Relaxation time approximation
FS 11 Prof. M. Sigrist
In this exercise we will show that the socalled singlerelaxationtime approximation, f (k) t =
coll
dd k W (k, k )[f (k)  f (k )] (2)
Solid State Theory Exercise 10
SS 11 Prof. M. Sigrist
Exercise 10.1
Uniaxial Compressibility
We consider a system of electrons upon which an uniaxial pressure in zdirection acts. 0 Assume that this pressure causes a deformation of the Fermi surface k kF
Solid State Theory Exercise 9
FS 11 Prof. M. Sigrist
Corbino disk and the integer quantum Hall effect
Exercise 9.1 The lowest Landau level in the Corbino geometry The Hamilton operator for an electron (e < 0) restricted to the plane z = 0 and exposed to a
Solid State Theory Exercise 8
FS 11 Prof. M. Sigrist
Exercise 8.1
BohrvanLeeuwenTheorem
Prove the BohrvanLeeuwentheorem, which states that there is no diamagnetism in classical physics. Hint: H(p1 , . . . , pN ; q1 , . . . , qN ) is the Hamiltonian
Solid State Theory Exercise 7
FS 11 Prof. M. Sigrist
Exercise 7.1
Phonons in One Dimension
In this exercise you will show that a chain of atoms that are harmonically coupled to each other (and may thus oscillate around their equilibrium positions) is equi
Solid State Theory Exercise 6
Exercise 6.1 Lindhard function
FS 2011 Prof. M. Sigrist
In the lecture it was shown how to derive the dynamical linear response function 0 (q, ) which is also known as the Lindhard function: 0 (q, ) = 1 nF ( k+q )  nF ( k )
Solid State Theory Exercise 5
FS 11 Prof. M. Sigrist
Exercise 5.1
TightBinding Model of Graphene
a) Compute the lowenergy band structure of graphene within a tightbinding description taking only nearestneighbor hopping into account! To get started, co
Solid State Theory Exercise 4 Excitons
Exercise 4.1 OneDimensional Model of a Semiconductor
SS 11 Prof. M. Sigrist
Let us consider electrons moving on a onedimensional chain. We use the socalled tightbinding approximation. Thus, we assume that each ato