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Band Structure Calculation – using a plane wave basis set
^
^
2
2
U
m
k^
k^
k^
k
r^
r^
r^
r
Fourier transform:
i
U
U e
K r K K
r
^
^
^
^
^
^
^
i^
i^
i^
i^
i
e^
u^
e^
c^
e^
e^
c^
e
k r
k r
K r
k r
K r
k^
k^
k-K
k+K
K^
K
r^
r
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
2
i^
i^
i
i
c^
e^
U
e
c^
e^
c^
e
m
k^ K^
r^
k^ K^
r^
k^ K^
r
K^
r
k-K
K^
k-K
k^
k-K
K^
K^
K^
K
k^
K
nd 2 term
^
^
^
^
^ ^
i
c^
U
e
k K
K
r
k-K
K
K K
Let
^
^
K
K
K
nd term
^
^
^
i
c^
U
e^
k^ K^
r
k-K
K^
K
K,K
^
^
^
^
^
^
^
^
^
i^
i^
i
e^
e^
e
k^ K^
r^
k r
K r
K^
K
Each Fourier coefficient must be separately zero.
^
^
^
^
^
^
^
^
^
^
^
2
c^
U
c
m
k^
k-K
K^
K^
k^ K
K
k^
K
^
An infinite set of linear homogeneous equations ^
Nontrivial
c
’s
determinant = 0
secular equation of
order.
^
Often use a truncated set
diagolization
approx.
^
n
k
and
c k
K
^
Accuracy depends on truncation. Nearly free electron approx:
U
small, using perturbation theory
Unperturbed states:
^
i e V
k r
k^
;^
^
^
^
2
0
k 2 m
k^
in extended zone.
Perturbation theory:
^
^
^
^
^
^
^
^
^
^
^
^
^
^
2
0
3
0
0 U
U
O U
k
k^
k
k^
k^
k^
k^
k^
k^
assuming it is convergent.
k^
k^
K
k^
k^
Laue condition
Bragg diffraction (just like x-rays)
k
and
k strongly coupled.
Assume just two states
k
&^
k^
K
are strongly coupled:
^
^
^
^
^
^
^
U
K
k^
k^
Consider the subspace spanned by
k
&^
k^
K
. Ignore coupling to all other
states,
^
O U
. Set
0
U
From
^
^
^
^
^
^
^
^
^
^
^
2
c^
U
c
m
k^
k-K
K^
K^
k^ K
K
k^
K
^
^
^
^
^
^
^
^
^
^
^
2
c^
U
c
m
k^ K^
Κ^
k
k^
K
k
^
^
^
^
^
^
^
2
k^
c^
U
c
m
k^
K^
k^ K
k
^
^
U
U
K^
K^
because
U
is real. With
^
^
^
2
0
k 2 m
k^
^
^
^
^
^
0
*^
c^
U c
k^ K^
k^
k-K
K^
k
^
^
^
^
^
0
c^
U c
k^
k^
k^
K^
k^ K
Determinant = 0
^
^
^
^
^
^
^
^
^
^
^
^
^
2
2
0
0
0
0
U
k^
k^
k^ K^
k^
k^ K^
K
i e^ k r
,^
^
^
i k e K
r^
, and
^
0
0 k^
k-K
^
^
^
^
^
^
^
i
i c e
c^
e^
k^ K^
r
k r k^
k^ K^
, and
k
^
^
c^
U
c
k^
K
k^ K
k^
k
. This and normalization determine the coefficients.
Without coupling, exact degeneracy at
^
k^
k^
, or
^
^
0
0 k^
k^
. K
With coupling,
^
^
^
0
U
k^
k^
K^
. Degeneracy removed. Gap =
^
U
O U
K^
, a 1
st
order effect.
Band Structure Calculations
Plane wave basisAtomic or molecular orbital basis; tight-bindingGaussian basisWigner-Seitz-cell or cellular method (somewhat like the KP model)Augmented plane waves (APW)Green’s function (KKR)Orthorgonalized plane waves (OPW) Self consistency
: approximate potential
U
^
n
(electron density)
^
^
Columb
exchange
correlation
U
U
U
U
repeat until convergent.
Density functional theory
: a systematic (but approx) way of finding
U
from
n
Psudopotential
:^ U
can be approximated by a weaker and smoother potential.
Often used for core electrons to reduce the work. For valence electrons,
U
~ a K
few eV and converges fairly quickly.GW, Quantum Monte Carlo, Bethe-Saltpeter, dynamic mean field theory, …
Si band structure, empirical pseudo potential method with three formfactors: V
= V 3
111
= -2.85726 eV; V
= V 8
220
= 0.54424 eV; V
11 = V
311
=
1.08848 eV (from Cohen)Plane wave basis includes: G = (000), (111), (200), (220), and (311) inunits of 2
/a. In all, 51 plane waves.
Results are valid in an irreducible wedge in the first zone, first quadrant,and kx <ky < kz.Left: empty lattice (all form factors set to zero; same as a free electronmodel).Middle: all form factors set to 30%.Right: full potentials applied.
Energy (eV)^
^
X Z W Q
L
^
^
X
(^5051015)
(^5051015)
(^5051015)
Tight Binding (TB) Method
Starting with individual atoms
^
^
2
atomic
a^
a
p
H
U
m ^
^
;^
a^
n^
n^
n
H
E
^
Assume
e
^ are tightly bound, mostly confined to one atom.
^
^
^
^
^
^
^
^
^
^
^
2
2
a^
a^
a
p^
p
H
U
U
U
U
H
U
m
m
U
small within atom, treated as perturbation; big elsewhere, but OK, because
^
is fairly localized. Construct TB Bloch function using
n
^
^
i
n^
n
e^ k R
k
R r^
r^
R
as trial wave function.
Bloch condition is satisfied.
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
i
i^
i
n^
n^
n^
n
e^
e^
e
k^
R^
R
k R
k R
k^
k
R^
R
r^
R
r^
R
R
r^
R
r
Good near the atom, and therefore good for the whole solid.
^
^
^
^
n^
n
n
n^
n H k
k
k^
k
k^
to first order.
Numerator =
^
^
^
^
^
*^
3
,
i^
i
n^
n
e^
H
e
d r
k R
k R
R R
r^
R
r^
R
=^
^
^
^
^
^
^
^
^
^
*^
3
,
i
i
n^
n
e^
He
d r
k^
R^
R
k R
R R
r^
R
r^
R
R
=^
^
^
^
^
^
^
*^
3
,
i
n^
n
H e
d r
k R
R R
r^
R
r^
R
R
H
is invariant under translation; shift origin from
r to
r
R
Numerator =
^
^
^
^
^
^
^
^
^
*^
3
*^
3
i^
i
n^
n^
n^
n^
n
N
H
e^
d r
N
E^
U
e^
d r
k R
k R
R^
R
r^
r^
R
r^
r^
R
Example, fcc crystal, s-band (atomic wave function is spherical):Nearest neighbors
^
^
^
^
^
^
^
^
^
^
^
1,^
1,^
1,^
a^
a^
a
R
overlap integral =
^
^
^
^
^
*^
3
n^
n U
d r
r^
R
same for all = -
^
^
^
^
^
..^0
const.
n n
i
n^
n E^
e^ k R R
k
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
.^
cos
cos
n^
x^
y
E^
const
k a
k a
cycl
Band width
^
^
^
^
^
^
^
max
min
const.
overlap integral
k^
k
TB is excellent for narrow bands (e.g., d and f bands). Works OK for wider bandswith enough parameters (longer range overlaps).Program available at the NRL website
Band widths get wider as the interatomic spacing gets smaller.Use LCAO (linear combination of atomic orbitals) where bands overlap or wherethere is degeneracy to begin with.
^
^
i
m^
mn
n
n
e^
b
k R
k
R
r^
R
a
En
