Tight Binding Method(Linear Combination of

Atomic Orbitals)

Dragica VasileskaDragica Vasileskaandand

Gerhard Gerhard KlimeckKlimeck

Almostfree

particles

Tightlybound

particles

Perturbationapproach

Scalability of TB approaches

DFT local basis approaches provide transferable and accurate interaction potentials. The numerical efficiency of the method allows for molecular dynamics simulations in large super cells, containing several hundreds of atoms.

Density Functional based Tight-Binding (DFTB, FIREBAL, SIESTA)

Empirical Tight-Binding

Semi-Empirical Hartree-Fock

Hamiltonian matrix elements are obtained by comparison of calculated quantities with experiments or ab-initio results. Very efficient, Poor transferability.

Methods used in the chemistry context (INDO, PM3 etc.). Medium transferability.

Why Tight-Binding ?

Allows us to describe the band structure over the entire Brillouin zone

Relaxes all the approximations of Envelope Function approaches

Allows us to describe thin layer perturbation (few Å)

Describes correctly band mixing

Gives atomic details

The computational cost is low

It is a real space approach

Molecular dynamics

Scalability (from empirical to ab-initio)

Bulk Hamiltonian

Step 1: Bloch sum (discrete Fourier

Transform) of each localized orbital.

Step 2: Write wavefunction as linear

combination of Bloch sums.

Step 3: Expand the Hamiltonian in

terms of the Bloch sums.

a

(001)

(100)

(010)

(111)

(110)

inbR

iR

i iik R v

ii

nbk e nbR

,n b

k c nbk

11 12

21 22

H HH k

H H

Type 1 Type 2Size of each block is NbXNb

11 11 12 12

21 21 22 22

B

E V k V k V kH k

V k V k E V k

3NN

21V k

11 11 12

21 22 22

B

E V k V kH k

V k E V k

2NN

22V k

Nearest neighbors only

Nearest + Distant neighbors

Tight-binding Models

Models: Interaction Range

11E22E

11 12

21 22

B

E V kH k

V k E

NN

21V k

Interaction

sub-matrices

Models: Atomic Basis Set

CB from NN-sp3s*

Bulk germanium

Bulk silicon

NN-sp3 vs. NN-sp3s*

NN-sp3 model captures key features of

valence band (VB), but fails for

conduction band (CB) in indirect

bandgap materials.

NN-sp3s* reproduces indirect conduction

bandgap but with wrong effective

masses.

s xp *szpyp

sp3

sp3s*

NB=4

NB=5

The sp3s* Hamiltonian [Vogl et al. J. Phys. Chem Sol. 44, 365 (1983)]

In order to reproduce both valence and conduction band of covalently bounded semiconductors a s* orbital is introduced to account for high energy orbitals (d, f etc.)

Models: Atomic Basis SetCB from 2NN-sp3s*

CB from NN-sp3s*d5

sp3s* Nb=5

sp3s*d5

Nb=102 2x y

dzxdyzdxyd

s xp *szpyp

2 2 / 3z rd

2NN-sp3s* vs. NN-sp3s*d5

� Both models offer correct effective

mass.

� 2NN-sp3s* has smaller basis size

compared to NN-sp3s*d5.

� Accurate modeling strain is difficult

for distant neighbor interactions.

NN-sp3s*d5 is the appropriate model for device simulation

The sp3d5s* Hamiltonian[Jancu et al. PRB 57 (1998)]

Many parameters, but works quite well !