The Luttinger–Kohn model is a flavor of the k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors . The method is a generalization of the single band k· p theory.
In this model, the influence of all other bands is taken into account by using Löwdin 's perturbation method.[ 1]
All bands can be subdivided into two classes:
Class A : six valence bands (heavy hole, light hole, split off band and their spin counterparts) and two conduction bands.
Class B : all other bands.
The method concentrates on the bands in Class A , and takes into account Class B bands perturbatively.
We can write the perturbed solution,
ϕ
{\displaystyle \phi _{}^{}}
, as a linear combination of the unperturbed eigenstates
ϕ
i
(
0
)
{\displaystyle \phi _{i}^{(0)}}
:
ϕ
=
∑
n
A
,
B
a
n
ϕ
n
(
0
)
{\displaystyle \phi =\sum _{n}^{A,B}a_{n}\phi _{n}^{(0)}}
Assuming the unperturbed eigenstates are orthonormalized, the eigenequations are:
(
E
−
H
m
m
)
a
m
=
∑
n
≠
m
A
H
m
n
a
n
+
∑
α
≠
m
B
H
m
α
a
α
{\displaystyle (E-H_{mm})a_{m}=\sum _{n\neq m}^{A}H_{mn}a_{n}+\sum _{\alpha \neq m}^{B}H_{m\alpha }a_{\alpha }}
,
where
H
m
n
=
∫
ϕ
m
(
0
)
†
H
ϕ
n
(
0
)
d
3
r
=
E
n
(
0
)
δ
m
n
+
H
m
n
′
{\displaystyle H_{mn}=\int \phi _{m}^{(0)\dagger }H\phi _{n}^{(0)}d^{3}\mathbf {r} =E_{n}^{(0)}\delta _{mn}+H_{mn}^{'}}
.
From this expression, we can write:
a
m
=
∑
n
≠
m
A
H
m
n
E
−
H
m
m
a
n
+
∑
α
≠
m
B
H
m
α
E
−
H
m
m
a
α
{\displaystyle a_{m}=\sum _{n\neq m}^{A}{\frac {H_{mn}}{E-H_{mm}}}a_{n}+\sum _{\alpha \neq m}^{B}{\frac {H_{m\alpha }}{E-H_{mm}}}a_{\alpha }}
,
where the first sum on the right-hand side is over the states in class A only, while the second sum is over the states on class B. Since we are interested in the coefficients
a
m
{\displaystyle a_{m}}
for m in class A, we may eliminate those in class B by an iteration procedure to obtain:
a
m
=
∑
n
A
U
m
n
A
−
δ
m
n
H
m
n
E
−
H
m
m
a
n
{\displaystyle a_{m}=\sum _{n}^{A}{\frac {U_{mn}^{A}-\delta _{mn}H_{mn}}{E-H_{mm}}}a_{n}}
,
U
m
n
A
=
H
m
n
+
∑
α
≠
m
B
H
m
α
H
α
n
E
−
H
α
α
+
∑
α
,
β
≠
m
,
n
;
α
≠
β
H
m
α
H
α
β
H
β
n
(
E
−
H
α
α
)
(
E
−
H
β
β
)
+
…
{\displaystyle U_{mn}^{A}=H_{mn}+\sum _{\alpha \neq m}^{B}{\frac {H_{m\alpha }H_{\alpha n}}{E-H_{\alpha \alpha }}}+\sum _{\alpha ,\beta \neq m,n;\alpha \neq \beta }{\frac {H_{m\alpha }H_{\alpha \beta }H_{\beta n}}{(E-H_{\alpha \alpha })(E-H_{\beta \beta })}}+\ldots }
Equivalently, for
a
n
{\displaystyle a_{n}}
(
n
∈
A
{\displaystyle n\in A}
):
a
n
=
∑
n
A
(
U
m
n
A
−
E
δ
m
n
)
a
n
=
0
,
m
∈
A
{\displaystyle a_{n}=\sum _{n}^{A}(U_{mn}^{A}-E\delta _{mn})a_{n}=0,m\in A}
and
a
γ
=
∑
n
A
U
γ
n
A
−
H
γ
n
δ
γ
n
E
−
H
γ
γ
a
n
=
0
,
γ
∈
B
{\displaystyle a_{\gamma }=\sum _{n}^{A}{\frac {U_{\gamma n}^{A}-H_{\gamma n}\delta _{\gamma n}}{E-H_{\gamma \gamma }}}a_{n}=0,\gamma \in B}
.
When the coefficients
a
n
{\displaystyle a_{n}}
belonging to Class A are determined, so are
a
γ
{\displaystyle a_{\gamma }}
.
Schrödinger equation and basis functions
edit
The Hamiltonian including the spin-orbit interaction can be written as:
H
=
H
0
+
ℏ
4
m
0
2
c
2
σ
¯
⋅
∇
V
×
p
{\displaystyle H=H_{0}+{\frac {\hbar }{4m_{0}^{2}c^{2}}}{\bar {\sigma }}\cdot \nabla V\times \mathbf {p} }
,
where
σ
¯
{\displaystyle {\bar {\sigma }}}
is the Pauli spin matrix vector. Substituting into the Schrödinger equation in Bloch approximation we obtain
H
u
n
k
(
r
)
=
(
H
0
+
ℏ
m
0
k
⋅
Π
+
ℏ
2
k
2
4
m
0
2
c
2
∇
V
×
p
⋅
σ
¯
)
u
n
k
(
r
)
=
E
n
(
k
)
u
n
k
(
r
)
{\displaystyle Hu_{n\mathbf {k} }(\mathbf {r} )=\left(H_{0}+{\frac {\hbar }{m_{0}}}\mathbf {k} \cdot \mathbf {\Pi } +{\frac {\hbar ^{2}k^{2}}{4m_{0}^{2}c^{2}}}\nabla V\times \mathbf {p} \cdot {\bar {\sigma }}\right)u_{n\mathbf {k} }(\mathbf {r} )=E_{n}(\mathbf {k} )u_{n\mathbf {k} }(\mathbf {r} )}
,
where
Π
=
p
+
ℏ
4
m
0
2
c
2
σ
¯
×
∇
V
{\displaystyle \mathbf {\Pi } =\mathbf {p} +{\frac {\hbar }{4m_{0}^{2}c^{2}}}{\bar {\sigma }}\times \nabla V}
and the perturbation Hamiltonian can be defined as
H
′
=
ℏ
m
0
k
⋅
Π
.
{\displaystyle H'={\frac {\hbar }{m_{0}}}\mathbf {k} \cdot \mathbf {\Pi } .}
The unperturbed Hamiltonian refers to the band-edge spin-orbit system (for k =0). At the band edge, the conduction band Bloch waves exhibits s-like symmetry, while the valence band states are p-like (3-fold degenerate without spin). Let us denote these states as
|
S
⟩
{\displaystyle |S\rangle }
, and
|
X
⟩
{\displaystyle |X\rangle }
,
|
Y
⟩
{\displaystyle |Y\rangle }
and
|
Z
⟩
{\displaystyle |Z\rangle }
respectively. These Bloch functions can be pictured as periodic repetition of atomic orbitals, repeated at intervals corresponding to the lattice spacing. The Bloch function can be expanded in the following manner:
u
n
k
(
r
)
=
∑
j
′
A
a
j
′
(
k
)
u
j
′
0
(
r
)
+
∑
γ
B
a
γ
(
k
)
u
γ
0
(
r
)
{\displaystyle u_{n\mathbf {k} }(\mathbf {r} )=\sum _{j'}^{A}a_{j'}(\mathbf {k} )u_{j'0}(\mathbf {r} )+\sum _{\gamma }^{B}a_{\gamma }(\mathbf {k} )u_{\gamma 0}(\mathbf {r} )}
,
where j' is in Class A and
γ
{\displaystyle \gamma }
is in Class B. The basis functions can be chosen to be
u
10
(
r
)
=
u
e
l
(
r
)
=
|
S
1
2
,
1
2
⟩
=
|
S
↑
⟩
{\displaystyle u_{10}(\mathbf {r} )=u_{el}(\mathbf {r} )=\left|S{\frac {1}{2}},{\frac {1}{2}}\right\rangle =\left|S\uparrow \right\rangle }
u
20
(
r
)
=
u
S
O
(
r
)
=
|
1
2
,
1
2
⟩
=
1
3
|
(
X
+
i
Y
)
↓
⟩
+
1
3
|
Z
↑
⟩
{\displaystyle u_{20}(\mathbf {r} )=u_{SO}(\mathbf {r} )=\left|{\frac {1}{2}},{\frac {1}{2}}\right\rangle ={\frac {1}{\sqrt {3}}}|(X+iY)\downarrow \rangle +{\frac {1}{\sqrt {3}}}|Z\uparrow \rangle }
u
30
(
r
)
=
u
l
h
(
r
)
=
|
3
2
,
1
2
⟩
=
−
1
6
|
(
X
+
i
Y
)
↓
⟩
+
2
3
|
Z
↑
⟩
{\displaystyle u_{30}(\mathbf {r} )=u_{lh}(\mathbf {r} )=\left|{\frac {3}{2}},{\frac {1}{2}}\right\rangle =-{\frac {1}{\sqrt {6}}}|(X+iY)\downarrow \rangle +{\sqrt {\frac {2}{3}}}|Z\uparrow \rangle }
u
40
(
r
)
=
u
h
h
(
r
)
=
|
3
2
,
3
2
⟩
=
−
1
2
|
(
X
+
i
Y
)
↑
⟩
{\displaystyle u_{40}(\mathbf {r} )=u_{hh}(\mathbf {r} )=\left|{\frac {3}{2}},{\frac {3}{2}}\right\rangle =-{\frac {1}{\sqrt {2}}}|(X+iY)\uparrow \rangle }
u
50
(
r
)
=
u
¯
e
l
(
r
)
=
|
S
1
2
,
−
1
2
⟩
=
−
|
S
↓
⟩
{\displaystyle u_{50}(\mathbf {r} )={\bar {u}}_{el}(\mathbf {r} )=\left|S{\frac {1}{2}},-{\frac {1}{2}}\right\rangle =-|S\downarrow \rangle }
u
60
(
r
)
=
u
¯
S
O
(
r
)
=
|
1
2
,
−
1
2
⟩
=
1
3
|
(
X
−
i
Y
)
↑
⟩
−
1
3
|
Z
↓
⟩
{\displaystyle u_{60}(\mathbf {r} )={\bar {u}}_{SO}(\mathbf {r} )=\left|{\frac {1}{2}},-{\frac {1}{2}}\right\rangle ={\frac {1}{\sqrt {3}}}|(X-iY)\uparrow \rangle -{\frac {1}{\sqrt {3}}}|Z\downarrow \rangle }
u
70
(
r
)
=
u
¯
l
h
(
r
)
=
|
3
2
,
−
1
2
⟩
=
1
6
|
(
X
−
i
Y
)
↑
⟩
+
2
3
|
Z
↓
⟩
{\displaystyle u_{70}(\mathbf {r} )={\bar {u}}_{lh}(\mathbf {r} )=\left|{\frac {3}{2}},-{\frac {1}{2}}\right\rangle ={\frac {1}{\sqrt {6}}}|(X-iY)\uparrow \rangle +{\sqrt {\frac {2}{3}}}|Z\downarrow \rangle }
u
80
(
r
)
=
u
¯
h
h
(
r
)
=
|
3
2
,
−
3
2
⟩
=
−
1
2
|
(
X
−
i
Y
)
↓
⟩
{\displaystyle u_{80}(\mathbf {r} )={\bar {u}}_{hh}(\mathbf {r} )=\left|{\frac {3}{2}},-{\frac {3}{2}}\right\rangle =-{\frac {1}{\sqrt {2}}}|(X-iY)\downarrow \rangle }
.
Using Löwdin's method, only the following eigenvalue problem needs to be solved
∑
j
′
A
(
U
j
j
′
A
−
E
δ
j
j
′
)
a
j
′
(
k
)
=
0
,
{\displaystyle \sum _{j'}^{A}(U_{jj'}^{A}-E\delta _{jj'})a_{j'}(\mathbf {k} )=0,}
where
U
j
j
′
A
=
H
j
j
′
+
∑
γ
≠
j
,
j
′
B
H
j
γ
H
γ
j
′
E
0
−
E
γ
=
H
j
j
′
+
∑
γ
≠
j
,
j
′
B
H
j
γ
′
H
γ
j
′
′
E
0
−
E
γ
{\displaystyle U_{jj'}^{A}=H_{jj'}+\sum _{\gamma \neq j,j'}^{B}{\frac {H_{j\gamma }H_{\gamma j'}}{E_{0}-E_{\gamma }}}=H_{jj'}+\sum _{\gamma \neq j,j'}^{B}{\frac {H_{j\gamma }^{'}H_{\gamma j'}^{'}}{E_{0}-E_{\gamma }}}}
,
H
j
γ
′
=
⟨
u
j
0
|
ℏ
m
0
k
⋅
(
p
+
ℏ
4
m
0
c
2
σ
¯
×
∇
V
)
|
u
γ
0
⟩
≈
∑
α
ℏ
k
α
m
0
p
j
γ
α
.
{\displaystyle H_{j\gamma }^{'}=\left\langle u_{j0}\right|{\frac {\hbar }{m_{0}}}\mathbf {k} \cdot \left(\mathbf {p} +{\frac {\hbar }{4m_{0}c^{2}}}{\bar {\sigma }}\times \nabla V\right)\left|u_{\gamma 0}\right\rangle \approx \sum _{\alpha }{\frac {\hbar k_{\alpha }}{m_{0}}}p_{j\gamma }^{\alpha }.}
The second term of
Π
{\displaystyle \Pi }
can be neglected compared to the similar term with p instead of k . Similarly to the single band case, we can write for
U
j
j
′
A
{\displaystyle U_{jj'}^{A}}
D
j
j
′
≡
U
j
j
′
A
=
E
j
(
0
)
δ
j
j
′
+
∑
α
β
D
j
j
′
α
β
k
α
k
β
,
{\displaystyle D_{jj'}\equiv U_{jj'}^{A}=E_{j}(0)\delta _{jj'}+\sum _{\alpha \beta }D_{jj'}^{\alpha \beta }k_{\alpha }k_{\beta },}
D
j
j
′
α
β
=
ℏ
2
2
m
0
[
δ
j
j
′
δ
α
β
+
∑
γ
B
p
j
γ
α
p
γ
j
′
β
+
p
j
γ
β
p
γ
j
′
α
m
0
(
E
0
−
E
γ
)
]
.
{\displaystyle D_{jj'}^{\alpha \beta }={\frac {\hbar ^{2}}{2m_{0}}}\left[\delta _{jj'}\delta _{\alpha \beta }+\sum _{\gamma }^{B}{\frac {p_{j\gamma }^{\alpha }p_{\gamma j'}^{\beta }+p_{j\gamma }^{\beta }p_{\gamma j'}^{\alpha }}{m_{0}(E_{0}-E_{\gamma })}}\right].}
We now define the following parameters
A
0
=
ℏ
2
2
m
0
+
ℏ
2
m
0
2
∑
γ
B
p
x
γ
x
p
γ
x
x
E
0
−
E
γ
,
{\displaystyle A_{0}={\frac {\hbar ^{2}}{2m_{0}}}+{\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{\gamma }^{B}{\frac {p_{x\gamma }^{x}p_{\gamma x}^{x}}{E_{0}-E_{\gamma }}},}
B
0
=
ℏ
2
2
m
0
+
ℏ
2
m
0
2
∑
γ
B
p
x
γ
y
p
γ
x
y
E
0
−
E
γ
,
{\displaystyle B_{0}={\frac {\hbar ^{2}}{2m_{0}}}+{\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{\gamma }^{B}{\frac {p_{x\gamma }^{y}p_{\gamma x}^{y}}{E_{0}-E_{\gamma }}},}
C
0
=
ℏ
2
m
0
2
∑
γ
B
p
x
γ
x
p
γ
y
y
+
p
x
γ
y
p
γ
y
x
E
0
−
E
γ
,
{\displaystyle C_{0}={\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{\gamma }^{B}{\frac {p_{x\gamma }^{x}p_{\gamma y}^{y}+p_{x\gamma }^{y}p_{\gamma y}^{x}}{E_{0}-E_{\gamma }}},}
and the band structure parameters (or the Luttinger parameters ) can be defined to be
γ
1
=
−
1
3
2
m
0
ℏ
2
(
A
0
+
2
B
0
)
,
{\displaystyle \gamma _{1}=-{\frac {1}{3}}{\frac {2m_{0}}{\hbar ^{2}}}(A_{0}+2B_{0}),}
γ
2
=
−
1
6
2
m
0
ℏ
2
(
A
0
−
B
0
)
,
{\displaystyle \gamma _{2}=-{\frac {1}{6}}{\frac {2m_{0}}{\hbar ^{2}}}(A_{0}-B_{0}),}
γ
3
=
−
1
6
2
m
0
ℏ
2
C
0
,
{\displaystyle \gamma _{3}=-{\frac {1}{6}}{\frac {2m_{0}}{\hbar ^{2}}}C_{0},}
These parameters are very closely related to the effective masses of the holes in various valence bands.
γ
1
{\displaystyle \gamma _{1}}
and
γ
2
{\displaystyle \gamma _{2}}
describe the coupling of the
|
X
⟩
{\displaystyle |X\rangle }
,
|
Y
⟩
{\displaystyle |Y\rangle }
and
|
Z
⟩
{\displaystyle |Z\rangle }
states to the other states. The third parameter
γ
3
{\displaystyle \gamma _{3}}
relates to the anisotropy of the energy band structure around the
Γ
{\displaystyle \Gamma }
point when
γ
2
≠
γ
3
{\displaystyle \gamma _{2}\neq \gamma _{3}}
.
Explicit Hamiltonian matrix
edit
The Luttinger-Kohn Hamiltonian
D
j
j
′
{\displaystyle \mathbf {D_{jj'}} }
can be written explicitly as a 8X8 matrix (taking into account 8 bands - 2 conduction, 2 heavy-holes, 2 light-holes and 2 split-off)
H
=
(
E
e
l
P
z
2
P
z
−
3
P
+
0
2
P
−
P
−
0
P
z
†
P
+
Δ
2
Q
†
−
S
†
/
2
−
2
P
+
†
0
−
3
/
2
S
−
2
R
E
e
l
P
z
2
P
z
−
3
P
+
0
2
P
−
P
−
0
E
e
l
P
z
2
P
z
−
3
P
+
0
2
P
−
P
−
0
E
e
l
P
z
2
P
z
−
3
P
+
0
2
P
−
P
−
0
E
e
l
P
z
2
P
z
−
3
P
+
0
2
P
−
P
−
0
E
e
l
P
z
2
P
z
−
3
P
+
0
2
P
−
P
−
0
E
e
l
P
z
2
P
z
−
3
P
+
0
2
P
−
P
−
0
)
{\displaystyle \mathbf {H} =\left({\begin{array}{cccccccc}E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\P_{z}^{\dagger }&P+\Delta &{\sqrt {2}}Q^{\dagger }&-S^{\dagger }/{\sqrt {2}}&-{\sqrt {2}}P_{+}^{\dagger }&0&-{\sqrt {3/2}}S&-{\sqrt {2}}R\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\\end{array}}\right)}
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(July 2010 )