Band_structure
Band structure
Dung Nguyen
Content
1.
Lattice structure
Lattice symmetry
2.
3.
Reciprocal lattice
Brillouin zone
Schrodinger equation
Bloch theorem
Tight-binding method
Lattice structure
Solid state has lattice structure.
Lattice structure has translation symmetry, so that atomic structure (lattice structure) remains invariant under translation through combination of basic vectors:
Bravais lattice: all the points are constructed by basic translation:
Where are integers.
Unit cell
The whole crystal is made of repetition of object called unit cell.
As we see the definition of unit cell is not unique.
Wigner-Seitz cell
If there is some center symmetry about a lattice point ( and hence all equivalent points) . We chose it as center of unit cell.
Draw the perpendicular bisector plane of the translation vector from center point to nearest equivalent lattice sites.
The volume inside the bisector planes is a unit cell called
Wigner-Seitz cell
Reciprocal lattice
Consider periodic function in rectangular lattice:
Then we can Fourier transform function f(r) as:
In which the reciprocal vector:
We have:
With non-rectangular lattice:
Where are integers.
Then we have:
Brillouin zone
Wigner-Seitz cell of the reciprocal lattice is called Brillouin zone.
Schrodinger equation
The Hamiltonian of perfect crystal can be written as:
Obviously, the Hamiltonian is unsolvable, we need simplifications.
Firstly, we separate valence electrons and core electrons. Core electron are those in filled orbital. They are mostly localized around atoms and combine to ion core.
Now
The second approximation is Born-Oppenheimer or adiabatic approximation. -The ion cores are much heavier than valence electrons, so they move slowly. As a result, compared to electron the ions are essentially stationary. The Hamiltonian can be rewritten:
In calculation for band structure, we mainly interested in electron
Hamiltonian:
The last approximation is mean-field approximation. We assume that electrons feel the same potential V(r):
Each eigenstate can have up to 2 electrons due to Pauli principle.
Bloch theorem
Consider periodic potential:
Schrodinger equation:
Hamiltonian under translation:
Consider eigen-state under translation by a lattice vector:
Translation invariant of Hamiltonian:
Eigen-state equation:
The equation is identical with:
So we have:
If eigen-state is non-degenerate, we have:
By normalized condition:
So that:
Similarly:
In general:
Define a vector in the form:
Then we have:
So for each eigen-wave function, there exist a vector such that:
For the case of degenerate eigen-state, we use the fact that:
In which
Thus can be diagonalized together.
Reduction to first BZ
By Bloch theorem, we can label every eigen-wave function by its wave vector
We know that free electron wave satises this form:
So we want to make eigen-wave functions as much as possible like free electron wave:
Such that
Then we have:
Then the wave-vector is not uniquely defined, it can only defined up to differences by reciprocal lattice vectors.
Example:
Consider one-dimension lattice with lattice size a, the reciprocal lattice vector:
we may assign any wave number in the set:
Thus is only defined by modulo
So how we define unique wave-vector? We always choose for a value wave-number in first BZ in reciprocal lattice space.Any state may be characterized by its reduced wave vector.
There will be multiple states with the same reduced wave vector in first BZ with different energies.
Ex: Free electron energy:
In reduced zone:
If we turn on small potential, consider free electron eigen-state:
The perturbative energy at first order:
With periodic potential:
With is Fourier transform of potential. We only sum over reciprocal lattice vectors.
With degeneracy of energies:
We have energy gap:
Because:
Tight binding calculation:
There are some different methods to calculate band structure, of course with different order of uncertainty.
I just want to introduce a simple method is Tight binding or
LCAO method. In which, we use atomic's valence electron orbitals as basic wave functions.
Toy example I
Consider one-dim solid composed by N atoms, each atom has one valence electron orbital.
Hamiltonian in valence electron orbitals basic:
In which we consider overlap of electron orbitals wave function of nearest atoms.
In order to determine band structure (eigen-value problem), we have to diagonalize NxN matrix to fine eigen-energies at each wave vector k.
We have eigen-value equation:
Or:
In which we have N-component vector:
By consequence of Bloch theorem ( translation invariant), we consider ansatz:
Thus we have:
Therefore:
Toy example II
Consider one-dimension solid whose unit cell consists of two orbitals
We have Hamiltonian:
Eigen-value equation:
We rewrite eiegen-value equation:
And Hamiltonian:
We again solve the eigen-value by ansatz:
Thus we have:
We used:
General case
Consider particular unit cell n that has non-zero Hamiltonian coupling to its neighboring unit cells m by a matrix [Hnm] of size (bxb), where b being the number of basic functions per unit cell.
We can write the overall matrix equation:
We solve it by ansatz:
Band structure can be derived by diagonalizing bxb matrix:
Band structure of Graphene
Carbon Sp
2
orbital
2s, 2p x and 2p y electrons hybridize to sp 2 orbital:
Graphene lattice
sp 2 electrons combine together by sigma-bond to make hexagonal graphene lattice on x-y plane:
At low energy, we only consider the electron in p z orbitals.
We use nearest-neighbor tight binding:
We found egien-energy:
We have band structure and first Brillouin zone:
Dirac Cone
Brillouin Zone
Dirac Point
We have 6 Dirac points but only 2 independent points K and K’(-K), other can be derived with translation by reciprocal lattice vectors.
Band structure of GaAs
• Lattice structure and BZ:
We have each unit cell includes 2 atoms. Each atom has 5 valence electron orbitals basic per unit cell.
, thus we have 10 orbitals
Tight binding Hamiltonian:
In which we have parameters:
We found band structure:
In order to modify tight binding method for counting spin orbit and also other magnetic effect, we have to write down Hamiltonian for spin up and spin down electrons in each orbital ( double number of basic wave function)
In the case of GaAs, we have to diagonalize 20x20 Hamiltonian. matrix.
Reference:
J.M. Ziman, Principles of the theory of solids.
Peter Y. Yu, Fundamental of semiconductor.
S. Datta, Quantum transport: Atom to Transitor.
