Quantum Well States: Numerical Integration
Overview: In the following, we present a numerical algorithm for the calculation
of 2-D wavefunctions describing carriers in a quantum well with finite potential and arbitrary
potential shape. The alogrithm is applied to a
self-consistent Schroedinger-Poisson integration scheme, used to include the effect of doping
on the band-profile of a 2-D semiconductor structure.
Contents
1 Numerical Integration via a Shooting Method
In order to determine
fn(
y) , the part of
the wave-function characterising the confined 2-D states with sub-band index n,
the following equation has to be solved:
|
| ⎡ ⎣
|
− |
ħ2
2
|
|
d
dy
|
|
1
m∗(y)
|
|
d
dy
|
+V0(y) | ⎤ ⎦
|
fn(y) = En fn(y), |
| | (1.1) |
where
V0 , the confinement potential, is given by the band offset at the
semiconductor interfaces (see Fig.
1.1),
m∗(
y)
is the effective mass in the different semiconductor regions and
En is
the sub-band energy.
Figure 1.1: Typical band diagram of an In0.2Ga0.8As QD surrounded by an In0.35Ga0.65As QWell.
The surrounding material is GaAs. Note: The diagram is not to scale!
Discretizing (
1.1) using the differential operator:
∆
y = [
yi+1 −
yi−1]/2 ,
i ∈ {2, …,
N−1 }, we obtain [
1]:
|
| ⎡ ⎣
|
f′n (yi+1)
m∗ (yi+1)
|
− |
f′n (yi−1)
m∗ (yi−1)
| ⎤ ⎦
|
|
1
2 ∆y
|
= |
2
ħ2
|
[V0(yi) − En ] fn(yi). |
| | (1.2) |
Inserting
f′n(
yi)
= [
fn(
yi+1) −
fn(
yi−1)]/(2 ∆
y) into
(
1.2) we get:
| |
fn (yi+2)
m∗ (yi+1)
|
= | ⎧ ⎨
⎩
|
2 (2 ∆y )2
ħ2
|
[V0(yi) − En ]+ |
1
m∗ (yi+1)
|
+ |
1
m∗ (yi−1)
| ⎫ ⎬
⎭
|
fn(yi) − |
fn (yi−2)
m∗ (yi−1)
|
. |
| | (1.3) |
|
The next step involves the following transformation: ∆
y ⇒ ∆
y/2, leading to:
|
fn (yi+1) = | ⎧ ⎨
⎩
|
2 (∆y )2
ħ2
|
[ V0(yi) − En ] m∗ (yi+1/2) + 1+ |
m∗ (yi+1/2)
m∗ (yi−1/2)
| ⎫ ⎬
⎭
|
fn(yi) |
| |
| − |
m∗ (yi+1/2)
m∗ (yi−1/2)
|
fn (yi−1), |
| | (1.4) |
|
where the effective mass at intermediate grid positions is given by the mean of the
effective masses at the neighbouring points:
m∗ (
yi+1/2) = [
m∗ (
yi+1) +
m∗ (
yi)]/2.
We are interested in the bound eigenstates of the 2-D quantum well.
Due to the confinement potential we expect exponentially decaying solutions for the
wave-functions
fn(
y) for large values of
y:
fn(
y) → 0 for
y → ±∞.
As boundary conditions we choose:
fn(
y1) = 0.0 and
fn(
yN) = 0.0.
To start the integration from the left, we set
fn(
y2) = 1.0 . This choice is
arbitrary since the wave-functions obtained in this way are not normalised.
A shooting method is used to generate
the numerical solution. We first guess two different values for the eigenstate:
En(0) and
En(1). Then we perform the integration using (
1.4) to obtain the
corresponding wave-functions
fn(0) and
fn(1). The error values of the wave-functions with respect to
the boundary condition:
fn(
yN) = 0.0 can be used to get an improved guess of the eigenvalue
En(2).
The numerical procedure presented above closely follows the approach of Harrisson [
1].
In order to improve the stability of the method we integrate from the left ( 1 ≤
i ≤
m )
and from the right (
m <
i ≤
N) and match the first derivative of the
solutions at the joining point (e.g. the middle point). We set:
fn(
yN) = 1.0 as an additional initial condition to start the integration from the right.
The improved
guess of the eigenvalue is calculated using:
| En(k+1) = En(k)−χ(k) |
En(k)−En(k−1)
χ(k) − χ(k−1)
|
, |
| | (1.5) |
|
where χ is a measure of the difference between
the derivative of the left and right solution of (
1.4) at the joining point
m.
The value of χ is given by:
|
χ(k+1) = [ f(k−1)n(ym+1) − f(k−1)n(ym)] f(k)n(ym) −[ f(k)n(ym+1) − f(k)n(ym)] f(k−1)n(ym) | | | (1.6) |
|
In each iteration step an improved guess of the energy eigenvalue
En is obtained.
The iteration cycle is stopped once the value of χ is lower than the targeted
precision of the numerical solution. The wave-function generated in this way
has to be normalised:
fn(
y) ⇒
fn(
y)/∫
∞−∞|
fn(
y)|
2 dy.
A further complication arises if the quantum well or the surrounding semiconductor
layers are doped. In this case the potential
V0(
y) depends on the
wave-functions
fn(
y). To obtain a self-consistent numerical solution we
use a Schrödinger-Poisson integration scheme.
2 Self-consistent Schrödinger-Poisson Integration
As numerical integration method, we will use the procedure described above.
In each iteration step, we use the
wave-function to calculate the electro-static potential due to the ion cores of
ionized acceptors/acceptors and the mobile charge carriers. The electro-static
potential is then added to the confinement potential
V0(
y) and an improved
energy eigenvalue and wave-function are calculated. We follow the numerical
procedure presented by Harrison [
1] p. 105.
In this work, we are focusing on p-doped semiconductor structures.
Let (−ρ
3D) be the background density of ionized acceptors
in the semiconductor structure and
e the
elementary charge unit. Neglecting the influence of minority charge carriers, the
2-D charge density in a sheet of width ∆
y is given by:
| σ2D(yi) = e [ρsum3D |fn(yi)|2 − ρ3D(yi)] ∆y where ρsum3D = |
N ∑
i=1
|
ρ3D(yi). |
| | (2.1) |
|
The definition of the 2-D charge density
σ
2D ensures charge neutrality: ∑
i=1N σ
2D(
yi) ≡ 0.
The electric field due to the sheet charge density σ
2D(
yk)
is given by:
| Eel(yi) = [σ2D(yk)/2 ϵ] sign(yi − yk). |
| | (2.2) |
|
Due to the superposition principle, the total electric field can be calculated by:
| Eel(yi) = |
N ∑
k=1
|
|
σ2D(yk)
2 ϵ
|
sign(yi − yk) where sign(yi − yk) = | ⎧ ⎪ ⎨
⎪ ⎩
|
|
|
| | (2.3) |
|
The electro-static potential related to the electric field defined in (
2.1)
can be calculated using:
| |
|
−2 ∆y Eel(yi) + Vlel(yi−1), |
| | (2.4) |
| |
|
2 ∆y Eel(yi) + Vrel(yi+1), |
| | (2.5) |
|
where the superscript
l and
r indicate integration from the left or right.
To start the integration using (
9) and (
10)
we set the potential to zero at the boundary.
We calculate the `symmetrized' electro-static potential using:
Vel = [
Vlel +
Vrel ]/2. The potential defined in this way has the same
functional form as
Vlel and
Vrel but is shifted with respect to the
energy scale such that: |
Vel(
y1)| = |
Vel(
yN)|. The self-consistent
Schrödinger-Poisson integration
scheme is presented in Fig.
2.1.
Figure 2.1:
Self-consistent Schrödinger-Poisson integration method. As an intermediate step the electro-static potential V0 is calculated.
As an application of the self-consistent Schrödinger-Poisson integration scheme,
we present Fig.
2.2, illustrating the effect of p-doping on the
band profile of an In
0.2Ga
0.8As/GaAs QW.
The QW includes a thin layer of In
0.65Ga
0.35As
that simulates the wetting layer.
We have assumed that
the QW and the WL are p-doped with C at a concentration
of 3×10
18 cm
−3. The doping leads to a bending of the band diagram.
The asymmetry of the semiconductor structure is reflected in the shape of the 2-D wave-functions describing the
confinement of charge carriers in growth direction. The overall tilt of the band profile
arises due to the applied bias voltage. Note: The two figures representing the band diagram of the conduction band (CB) and
the valence band (VB), respectively, have been stacked in order to resemble the band diagram shown in Fig.
1.1
Figure 2.2:
Band diagram of a p-doped In0.2Ga0.8As/GaAs
QW. A sheet of In0.65Ga0.35As is positioned asymmetrically within the QW and
simulates the wetting layer.
The red curves represent the wave-functions, the broken green lines indicate the energy level of
the 2-D sub-bands.
Having determined the energy levels of the quantum well sub-bands, we are now in a position to calculate the
chemical potential of the doped semiconductor structure.
Bibliography
- [1]
-
Paul Harrison.
Quantum Wells, Wires and Dots.
John Wiley & Sons, Chichester, 2001.
File translated from TEX by TTHgold,version 4.00.