### Band_structure

```Band structure
Dung Nguyen
Content
 Lattice structure
Lattice symmetry
2. Reciprocal lattice
3. Brillouin zone
 Schrodinger equation

Bloch theorem
 Tight-binding method
1.
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
Then we have:
are integers.
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 Sp2 orbital
 2s, 2px and 2py electrons hybridize to sp2 orbital:
Graphene lattice
 sp2 electrons combine together by sigma-bond to make
hexagonal graphene lattice on x-y plane:
 At low energy, we only consider the electron in pz orbitals.
 We use nearest-neighbor tight binding:
 We found egien-energy:
 We have band structure and first Brillouin zone:
Brillouin Zone
Dirac Cone
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.
 Tight binding Hamiltonian:
, thus we have 10 orbitals
 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.
```