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These are the Lecture Slides of Advanced Device Simulation which includes Analytic Expressions, Band Dispersion, Optical Matrix Elements, Effective Masses, Zone Center Energy Gaps, Bardeen and Seitz, Shockley, Dresselhaus etc. Key important points are: Band Structure Calculations, Hamiltonian of System, Born-Oppenheimer, Adiabatic Approximation, Equilibrium Positions, Hamiltonian for Electrons, Schrodinger Equation, Energy of Electron
Typology: Slides
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The total Hamiltonian of the system is of the following
general form:
eraction
electron electron
ii (^) i i
eraction
electron ion
j i i j
j
ions
potential
jj (^) j j
j j
ions
KET
j (^) j
j
electrons
KET
i (^) i
i
r r
e
r R
Z e
Z Z e M
m
p H
int
' (^0) '^2
2 2
1
int
, 0 2
2
' (^0) '^2
2 ' 2
1
2 2
^
The following are approximations made in solving this large scale problem:
The electronic part to the Hamiltonian is of the following form
Since there are on the order of 10^23 electrons in the solid, one uses the so-called mean-field approximation according to which the Schrödinger equation is a one-electron approximation of the form:
Hamiltonian, the wavefunction and the energy of an electron in an eigenstate labeled by n.
(^2) ' (^) '
1 (^2 )
2
'
(^22)
ii r R
Z e
ii r r
e i m
p e i j
j i i i
H i
2 1 V r r E r m
p H (^) e n r n nn
One electron Hamiltonian
Typical approximations Hartree, Hartree-Fock, Local Density approximation (LDA) ….
These approximations reduce the many body system to the problem of one electron moving in an effective field
2 V r m
p H
To find the one electron energy En, one has to solve the Schroedinger (or Kohn-Sham) equation
H (^) n k r En k n k r
V has the crystal symmetry V( r + R )=V( r )
The atomic orbital basis
We attempt to solve the one electron Hamiltonian in terms of a Linear Combination of Atomic Orbitals (LCAO)
i
Ci i i
site,
atomic orbitals,
Ci = coefficients, i = atomic orbitals (s,p,d)
Lowdin theorem: One can perform the orthogonalization procedure in such a manner as to preserve the symmetry properties of the original atomic basis
Y
Ci
orbital
The Schroedinger equation in LCAO basis
matrix notation:
H C=EC
(^0)
site, j
atomic orbitals,
,
Hi j En Cj
where the Hamiltonian matrix element
H (^) i , j i (^) H j i H j d r *r RjH rRi
The LCAO expansion of the one electron wave function is only an approximate variational ansatz unless one uses a complete set of basis functions
The approach can be implemented “ab-initio” where the orbitals are the basis functions and Hia ,jb is evaluated numerically
i
Ci i i site,
atomic orbitals,
r r R H r E r
The assumption is that the wavefunction is written as a Bloch function of the form:
The energy expectation value is then given by the following matrix element:
We now define the on-sight and the nearest- neighbor matrix elements to get:
j j
j
ik r k kj j N ( r ) c (r r )^1 e j^ (r r )
j
j
ik
j m
j m
ik r r k k N
e H d
H e r r H r r d
j
m j
(^1) *( ) ( ) r
n.n.
ik k k H e j
where:
Example:
We now consider simple cubic lattice with atoms at: (±a,0,0), (0,±a,0) and (0,0,±a)
The dispersion relation in the vicinity of the band extrema is:
The effective electron mass is then given by:
H d
H d *( j ) ( )
2
2 2
2 2
2 (^2 )
1 1 2 a
a dk
d E m m^
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, FIREBALL, 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.
The sp^3 s* 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.)
Tight-Binding sp^3 d^5 s*^ model for nitrides
Ab-Inito Plane Wave DFT-LDA Band Structure for GaN Wurtzite TB Wurtzite GaN Band Structure
Nearest-neighbours sp^3 d^5 s*^ tight- binding parametrization for wurtzite GaN, AlN and InN compare well with Ab-Initio results.
Boundary conditions
Finite chain
Periodic
Open boundary conditions
After P planes the structure repeats itself. Suitable for superlattices
After P planes the structure end. Suitable for quantum wells
After P planes there is a semiinfinite crystal Suitable for current calculations
BULK P BULK
P
P
∞ (^) ∞
Example: Strain and Pseudomorphic
growth An epitaxial layer is grown, on a substrate with different lattice constant. The epilayer deforms (strain)
0 11
12
as
as
a 0
a 0
as
as
as
a
as as
as
R '( 1 ) R Strain tensor Docsity.com
Calculation of strain tensor
Elastic energy
GaN
SiN
