Post on 09-Feb-2022
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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 !