Order 1252715: Condensed Matter physics
tutorthammy
Lect-10-Nearly-Free-Electron-Model.pdf
13/10/2017
1
1
4-50 keV electrons
“grazing incidence”,
angle 1-4o
Reflection High Energy Electron Diffraction
(RHEED)
Used to characterize the surface of
a crystalline material, e.g., growing GaAs –
can monitor growth in situ and measure
how many layers you have (thickness) and
quality of surface (how rough it is),
orientation of crystal, lattice constant
Low Energy Electron Diffraction (LEED)
2
Si(111)-7x7
Need Multiple
Scattering Theory
13/10/2017
2
Bloch’s Theorem
3
The coefficients, Ck only involve k’s:
For each q, there is a wavefunction
where we have put in
Bloch’s Theorem: An electron in a periodic potential has eigenstates of the form
where has the periodicity of the lattice and q (the crystal momentum) can be
chosen within the first Brillouin zone.
Felix Bloch,
Nobel 1952
4
Example
Representation of the real part of an eigenstate in 1D
(black solid line), a Bloch function (green solid line)
and plane wave (blue dashed line).
13/10/2017
3
Example
5
Another
example
• A Bloch wave (bottom) can be broken up into the product of a periodic function (top) and a plane-wave (center).
• The left and right sides represent the same Bloch wave split in two different ways, one involving wave vector k1 (left) the other k2
(right). The difference (k1−k2) is a reciprocal lattice vector. Blue is
real part and red is imaginary part. 6
13/10/2017
4
Conclusions that follow from Bloch’s Theorem
• Bloch’s theorem introduces a wave vector k which plays the same fundamental role in the general problem of motion in
a periodic potential that the free electron wave vector k
plays in free-electron theory.
• In the Bloch case, k is not proportional to the electronic momentum. It is known as the crystal momentum or quasi-
momentum of the electron.
• The wave vector k can always be confined to the first Brillouin zone, since any k’ not in the first Brillouin zone can
be written as k’=k+G where G is a reciprocal lattice vector
and k is in the first zone.
7
• The appearance of energy bands,
where
• We can regard this equation as an eigenvalue problem restricted to a single primitive cell of the crystal.
• The wave functions are therefore denoted as indicating that the value of the band index n and the vector k specifies an
electron state.
8
Conclusions that follow from Bloch’s Theorem
R
13/10/2017
5
9
Conclusions that follow from Bloch’s Theorem Each energy level for a given k varies as k varies, -
obtain a description of the energy levels of an
electron in a periodic potential in terms of a family of
continuous functions
The information contained in these
functions for different n and k is called
the band structure of a solid.
Group Velocity
An electron in a level specified by band index n and
wave vector k has a mean velocity given by
10
Conclusions that follow from Bloch’s Theorem
Extended zone
scheme
Reduced zone
scheme
Repeated zone
scheme
13/10/2017
6
Nearly free electron model
11
• Going to consider two limits, very weak periodic potential
(treated by perturbation theory) and a very strong periodic
potential (treated by tight-binding theory).
• Both extreme limits give rise to bands, with band gaps between
them. In both extremes the bands are qualitatively very similar,
so real potentials, must lie somewhere between
Firstly, recall, for the free electron
And the solutions to the SE are plane waves
Each band (1,2,3 ..) corresponds to a different value of G in the extended scheme
Very weak potential
12
The matrix elements of this potential are the Fourier components
which is zero unless k’-k is a reciprocal lattice vector.
Now turn on weak potential (see Handout 5)
Ch 9 Ashcroft & Mermin
13/10/2017
7
13
Very weak potential
Apply perturbation theory
At first order in the perturbation V, we have:
just a constant energy shift, set to zero
At second order we have:
Possible that is close to or equal to
– will be degenerate when
dash means G not equal to 0
• In one-dimension, this will occur when
• This means that k must lie on the Brillouin zone boundary (i.e. on a Bragg plane determined by reciprocal lattice G)
14
Very weak potential
The states at the two
zone boundaries are
separated by a reciprocal
lattice vector G and have the
same energy.
13/10/2017
8
15
Very weak potential Degenerate perturbation theory
Consider two plane wave states k and k’=k+G that are approximately of the same energy and close to the zone boundaries,
We have from above,
Can write any wave function as:
Solving the effective SE:
since potential real
16
Very weak potential Degenerate perturbation theory
Secular equation determining E is:
And if exactly on the zone boundary where
It reduces to:
That is, a gap opens up
at the zone boundary!
13/10/2017
9
17
Very weak potential
group velocity = 0
Very weak potential
18
One dimension with
At zone boundary, we have:
leads to:
13/10/2017
10
Very weak potential
19
But how do we understand
the parabolic nature at the zone
boundaries?
Consider not quite on zone boundary
It can be shown that
Adv – go through working in Handout and show this is the case
Nearly free electron model
20
The dispersion is quadratic in near the band gap
Dispersion of a Nearly
Free Electron Model
plotted in repeated
zone scheme
13/10/2017
11
Nearly free electron model
21
Can rewrite below,
as:
both exhibit
“band gaps”
22
Band structures of real materials: Al
13/10/2017
12
23
Band structures of real materials: Si and GaAs
