Biomedical Engineering Reference
In-Depth Information
to ab initio band structures from the literature to reproduce key properties such as
the band gaps. The TB parameters for In
x
Ga
1
−
x
N are obtained from a modified
virtual crystal approximation [
57
], which allows us to take the band gap bowing
into account. The band gap bowing parameter used for InGaN has been taken from
[
44
]. Such an approach has been often used to calculate the electronic structure of
alloyed semiconductor materials [
57
-
59
].
Starting from the bulk TB parameters, nanostructures can then be modeled at
an atomistic level. To this end one chooses the matrix elements at each lattice site
according to the occupying atom. In general, the resulting
i
th TB wave function of
the nanostructure
|
ψ
i
is expressed in terms of the localized orbitals
|
α
,
ω
,
σ
,
R
:
∑
α
,
ω
,
σ
,
R
c
i
α
,
ω
,
σ
,
|
ψ
i
=
|
α
,
ω
,
σ
,
R
.
(6.6)
R
Here,
R
denotes the unit cell,
α
the orbital type,
σ
the spin and
ω
labels the anions
E
i
and cations in the given unit cell. The Schr odinger equation
H
|
ψ
i
=
|
ψ
i
can then
be expressed as the following finite matrix eigenvalue problem:
∑
α
,
ω
,
σ
,
R
α
,
ω
,
σ
,
R
|
c
i
α
,
ω
,
σ
,
E
i
c
i
H
|
α
,
ω
,
σ
,
R
−
α
,
ω
,
σ
,
R
=
0
,
(6.7)
R
where
E
i
is the corresponding eigenvalue. In the following we use the abbreviation
α
,
ω
,
σ
,
R
|
=
α
,
ω
,
σ
|
α
,
ω
,
σ
,
=
H
R
H
l
R
,
m
R
for the matrix elements with
l
=
α
,
ω
,
σ
and
m
.
In setting up the Hamiltonian, one has to include the local strain
(
)
ε
r
and
ij
=
φ
+
φ
the total built-in potential
sp
to ensure an accurate description of the
electronic properties of QDs and QDMs. Several authors have shown that this can
be done by introducing on-site corrections (
R
φ
tot
pz
R
,
ω
=
ω
,
σ
=
σ
)totheTB
matrix elements
H
l
R
,
m
R
[
60
,
61
]. Therefore, we proceed here in the following way.
The strain dependence of the TB matrix elements is included via the Pikus-Bir
Hamiltonian [
47
,
62
] as a site-diagonal correction:
=
⎛
⎞
S
s
000
0
S
x
S
xy
S
xz
0
S
xy
S
y
S
yz
0
S
xz
S
yz
S
z
⎝
⎠
,
H
str
l
R
=
(6.8)
,
m
R
with
S
s
=
a
ct
(
ε
11
+
ε
22
)+
a
cp
ε
zz
,
S
x
=(
D
2
+
D
4
)(
ε
11
+
ε
22
)+
D
5
(
ε
11
−
ε
22
)+(
D
1
+
D
3
)
ε
33
,
S
y
=(
D
2
+
D
4
)(
ε
11
+
ε
22
)
−
D
5
(
ε
11
−
ε
22
)+(
D
1
+
D
3
)
ε
33
,
S
z
=
D
2
(
ε
11
+
ε
11
)
,
