Lattice Vibrations
Classical theory of lattice vibrations
Adiabatic Approximation
Average velocity of valence electrons (106M/sec)
>
Average velocity of atomic cores (103M/sec)
Valence electrons are always in the ground state of the instantaneous potentia
1
Notes on Group theory
G-group
1. Closure: ab G
2. Associativity: a(bc) = a(bc)
3. Identity: exists e: ae = a
4. Inverse element: for every a exists a1 so a1 a = e
One can show that (ab)1 = b1 a1 :
b1 a1 ab = e
In general elements do not commute: ab = ba
1
k p method, general formulation
The idea of the method is to work in the basis of periodic functions unk (r) instead of the
Bloch functions, nk = exp(ikr)unk (r).
For this consider the action of
/i on nk :
( /i)nk = ( /i) exp(ikr)unk (r) =
exp(ikr)( /i)
Band Structure of Graphene
Graphene-a monolayer of carbon atoms packed into a honeycomb lattice, is one of the most intriguing systems in
solid state physics today. The most popular description of graphene band structure is the tight binding one, rst
done
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EXCITON
The coupled hydrogen-like electron-hole states are called excitons. We shall consider
weak coupling between electron and hole, as occur in high low eective mass materials.
High radius excitons in such materials are called Wannier-Mott excitons i
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Dispersion law for electrons in conduction band in eective mass approximation:
k2
c (k) =
2mc
We assume for simplicity spherical conduction band in, e.g., point.
In the valence band, the dispersion relationship for e.g. light holes looks like:
e (k) = v
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INTERBAND ABSORPTION
We choose vector potential in the Lorentz gauge:
E=
1 A
c t
B=
A = 0 and choose = 0.
A
and
The term in Hamiltonian which contains the electromagnetic eld is:
P
eA
c
2
e
e2
= P 2 2 AP + 2 A2
c
c
One can safely neglect the quadratic
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DISPERSION IN THE ZINC-BLEND LATTICE
We are going to discuss the dispersion in the point. It is generally believed, that for
the spinless particles the wave functions at the the top of the valence band transform like
4 : cfw_x, y, z
(notations of Lage a
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DISPERSION IN THE DIAMOND LATTICE
The most important dierence between semiconductors with diamond and the zinc-blend
lattices is following: the two FCC lattices, shifted by 1/4(1,1,1), in diamond lattice are
equivalent and in the zinc-blend lattice are
Crystal Field
24.10.2002
1
Crystal field, quenching of orbital moment, Kramers
doublets, van Vleck susceptibility
Introduction
We want to consider ferromagnets such that to a fair approximation, we can consider
composed of ions that retain at least some o
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EFFECTIVE MASS APPROXIMATION (WANNIER EQUATION)
Consider motion of an electron in perturbed lattice in presence of perturbation potential
U (r) we will require the U (r) to be smooth on the scale of the lattice constant. Consider
nondegenerate (expect K
1
Kramers theorem
Let us take complex conjugation of the Shrdinger equation and the same time change t
o
by t. this will lead to the Shrdinger equation, but for complex conjugated wave function
o
and with the complex conjugated hamiltonian:
i
h
= H
t
I
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EFFECTIVE MASS AND G-FACTOR
Let us consider electrons in the conduction band in the center of the Brullien zone. In
the absence of magnetic eld the energy to the second order in k is
(k) = D k k ,
where
D =
1
1
+ 2
2m
m
n| P |n n | P |n
n n
The tensor
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NEARLY FREE ELECTRONS
Hamiltonian for the electron motion in a periodic crystal can be written as:
2
= P + V (x) ,
H
2m
where V (x) is periodic function: V(x)= V(x + a),
Vg eigx ,
V (x) =
g
where
g=
2
j, and j = 0, 1, .
a
The V (x) is real, therefore V
Crystal Lattices
Bravais Lattice
(a) infinite array of points, in which the arrangement of points around any point is exactly the same.
r
(b) Position vector R of any point in a Bravais lattice can be expressed as
r
r
r
r
R = n1a1 + n2 a2 + n3 a3
r
a1, 2,
Electrons in a weak periodic potential
General properties of electrons in a periodic potential
2 2
(
k
K
)
ck K U K ' K ck K ' 0
K'
2m
For a fixed k , the above set of equations has N number of equations with
each equation having N number of coefficien
Semiclassical theory of electron dynamics
Semiclassical theory of electron dynamics
Free electron gas: the equations governing the dynamics
hk
m
r
r 1
hk& = e E + v H
c
v = r& =
Quantum mechanics: Wave packet
hk 2
(r, t ) = a (k') exp i k' r
t
2
m
Electrons in a periodic potential
General Properties
Independent electron approximation: effective potential U (r )
U (r R) U (r )
Character of the problem: Solve one electron Schrdinger equation
2 2
H
U ( r )
2m
Electrons in a periodic potential
Bill of Resources and Priority-Capacity Balancing
5
BILL OF RESOURCES AND
PRIORITY-CAPACITY BALANCING
MGT2405, University of Toronto, Denny Hong-Mo Yeh
Load and capacity are terms that are often used interchangeably. The two terms, however,
have very diff
1
SPIN-ORBIT INTERACTION
Hand waving derivation. Energy of a magnetic moment in a eld B: = B When
an electron moves in an electric eld E, the magnetic eld in its rest frame is:
1
1
B= Ev=
U/e v
c
c
where U is the potential energy of the electron in the el
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Velocity in a crystal
As an example for application of the k P method we shall obtain an expression for the
velocity of the electron in a non degenerate state (besides may be Kramers degeneracy, see
later) with a wave vector k:
vk =
1 (k)
h k
For this w
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SYMMETRY IN CRYSTALS
Let R be some rotational transformation of the coordinate. P 2 is invariant under rotation:
RP 2 R1 = P 2 . If R belongs to the point group of the crystal, it will keep the V(r) unchanged.
So, the hamiltonian is invariant under R: R
Geometric Phase and Matrix Examples; Kramers
Doublets
Frank Wilczek
Center for Theoretical Physics, MIT, Cambridge MA 02139 USA
September 24, 2009
Abstract
We illustrate use of geometric phases and matrices in several simple but meaningful physical exampl
6.012 - Microelectronic Devices and Circuits - Spring 2003
Lecture 5-1
Lecture 5 - PN Junction and MOS
Electrostatics (II)
pn Junction in Thermal Equilibrium
February 20, 2003
Contents:
1. Introduction to pn junction
2. Electrostatics of pn junction in th
Effect of Spin-Orbit Coupling in
Weak-Localization in Graphene
par Arnaud Raoux
January, 7th 2011
Work supervisor: Christophe Texier
Examiner:
Christophe Voisin
1
matre de confrences luniversit Paris-Sud
matre de confrences au LPA
Contents
1 Weak-Localiza
About the course
Sunday, January 03, 2010
3:17 PM
Instructor:
Teaching Assistant/Grader:
H R Krishnamurthy ([email protected])
Subhro Bhattacharjee ([email protected])
Course Objective :
To provide a thorough grounding in the basic,