Quantum Well States: Numerical Integration
Overview: In the following, we present a numerical algorithm for the calculation
of 2D wavefunctions describing carriers in a quantum well with finite potential and arbitrary
potential shape. The alogrithm is applied to a
selfconsistent SchroedingerPoisson integration scheme, used to include the effect of doping
on the bandprofile of a 2D semiconductor structure.
Contents
1 Numerical Integration via a Shooting Method
In order to determine
f_{n}(
y) , the part of
the wavefunction characterising the confined 2D states with subband index n,
the following equation has to be solved:

 ⎡ ⎣

− 
ħ^{2}
2


d
dy


1
m^{∗}(y)


d
dy

+V_{0}(y)  ⎤ ⎦

f_{n}(y) = E_{n} f_{n}(y), 
  (1.1) 
where
V_{0} , 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
E_{n} is
the subband energy.
Figure 1.1: Typical band diagram of an In_{0.2}Ga_{0.8}As QD surrounded by an In_{0.35}Ga_{0.65}As QWell.
The surrounding material is GaAs. Note: The diagram is not to scale!
Discretizing (
1.1) using the differential operator:
∆
y = [
y_{i+1} −
y_{i−1}]/2 ,
i ∈ {2, …,
N−1 }, we obtain [
1]:

 ⎡ ⎣

f^{′}_{n} (y_{i+1})
m^{∗} (y_{i+1})

− 
f^{′}_{n} (y_{i−1})
m^{∗} (y_{i−1})
 ⎤ ⎦


1
2 ∆y

= 
2
ħ^{2}

[V_{0}(y_{i}) − E_{n} ] f_{n}(y_{i}). 
  (1.2) 
Inserting
f^{′}_{n}(
y_{i})
= [
f_{n}(
y_{i+1}) −
f_{n}(
y_{i−1})]/(2 ∆
y) into
(
1.2) we get:
 
f_{n} (y_{i+2})
m^{∗} (y_{i+1})

=  ⎧ ⎨
⎩

2 (2 ∆y )^{2}
ħ^{2}

[V_{0}(y_{i}) − E_{n} ]+ 
1
m^{∗} (y_{i+1})

+ 
1
m^{∗} (y_{i−1})
 ⎫ ⎬
⎭

f_{n}(y_{i}) − 
f_{n} (y_{i−2})
m^{∗} (y_{i−1})

. 
  (1.3) 

The next step involves the following transformation: ∆
y ⇒ ∆
y/2, leading to:

f_{n} (y_{i+1}) =  ⎧ ⎨
⎩

2 (∆y )^{2}
ħ^{2}

[ V_{0}(y_{i}) − E_{n} ] m^{∗} (y_{i+1/2}) + 1+ 
m^{∗} (y_{i+1/2})
m^{∗} (y_{i−1/2})
 ⎫ ⎬
⎭

f_{n}(y_{i}) 
 
 − 
m^{∗} (y_{i+1/2})
m^{∗} (y_{i−1/2})

f_{n} (y_{i−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^{∗} (
y_{i+1/2}) = [
m^{∗} (
y_{i+1}) +
m^{∗} (
y_{i})]/2.
We are interested in the bound eigenstates of the 2D quantum well.
Due to the confinement potential we expect exponentially decaying solutions for the
wavefunctions
f_{n}(
y) for large values of
y:
f_{n}(
y) → 0 for
y → ±∞.
As boundary conditions we choose:
f_{n}(
y_{1}) = 0.0 and
f_{n}(
y_{N}) = 0.0.
To start the integration from the left, we set
f_{n}(
y_{2}) = 1.0 . This choice is
arbitrary since the wavefunctions 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:
E_{n}^{(0)} and
E_{n}^{(1)}. Then we perform the integration using (
1.4) to obtain the
corresponding wavefunctions
f_{n}^{(0)} and
f_{n}^{(1)}. The error values of the wavefunctions with respect to
the boundary condition:
f_{n}(
y_{N}) = 0.0 can be used to get an improved guess of the eigenvalue
E_{n}^{(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:
f_{n}(
y_{N}) = 1.0 as an additional initial condition to start the integration from the right.
The improved
guess of the eigenvalue is calculated using:
 E_{n}^{(k+1)} = E_{n}^{(k)}−χ^{(k)} 
E_{n}^{(k)}−E_{n}^{(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}(y_{m+1}) − f^{(k−1)}_{n}(y_{m})] f^{(k)}_{n}(y_{m}) −[ f^{(k)}_{n}(y_{m+1}) − f^{(k)}_{n}(y_{m})] f^{(k−1)}_{n}(y_{m})    (1.6) 

In each iteration step an improved guess of the energy eigenvalue
E_{n} is obtained.
The iteration cycle is stopped once the value of χ is lower than the targeted
precision of the numerical solution. The wavefunction generated in this way
has to be normalised:
f_{n}(
y) ⇒
f_{n}(
y)/∫
^{∞}_{−∞}
f_{n}(
y)
^{2} dy.
A further complication arises if the quantum well or the surrounding semiconductor
layers are doped. In this case the potential
V_{0}(
y) depends on the
wavefunctions
f_{n}(
y). To obtain a selfconsistent numerical solution we
use a SchrödingerPoisson integration scheme.
2 Selfconsistent SchrödingerPoisson Integration
As numerical integration method, we will use the procedure described above.
In each iteration step, we use the
wavefunction to calculate the electrostatic potential due to the ion cores of
ionized acceptors/acceptors and the mobile charge carriers. The electrostatic
potential is then added to the confinement potential
V_{0}(
y) and an improved
energy eigenvalue and wavefunction are calculated. We follow the numerical
procedure presented by Harrison [
1] p. 105.
In this work, we are focusing on pdoped 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
2D charge density in a sheet of width ∆
y is given by:
 σ_{2D}(y_{i}) = e [ρ^{sum}_{3D} f_{n}(y_{i})^{2} − ρ_{3D}(y_{i})] ∆y where ρ^{sum}_{3D} = 
N ∑
i=1

ρ_{3D}(y_{i}). 
  (2.1) 

The definition of the 2D charge density
σ
_{2D} ensures charge neutrality: ∑
_{i=1}^{N} σ
_{2D}(
y_{i}) ≡ 0.
The electric field due to the sheet charge density σ
_{2D}(
y_{k})
is given by:
 E_{el}(y_{i}) = [σ_{2D}(y_{k})/2 ϵ] sign(y_{i} − y_{k}). 
  (2.2) 

Due to the superposition principle, the total electric field can be calculated by:
 E_{el}(y_{i}) = 
N ∑
k=1


σ_{2D}(y_{k})
2 ϵ

sign(y_{i} − y_{k}) where sign(y_{i} − y_{k}) =  ⎧ ⎪ ⎨
⎪ ⎩



  (2.3) 

The electrostatic potential related to the electric field defined in (
2.1)
can be calculated using:
 

−2 ∆y E_{el}(y_{i}) + V^{l}_{el}(y_{i−1}), 
  (2.4) 
 

2 ∆y E_{el}(y_{i}) + V^{r}_{el}(y_{i+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' electrostatic potential using:
V_{el} = [
V^{l}_{el} +
V^{r}_{el} ]/2. The potential defined in this way has the same
functional form as
V^{l}_{el} and
V^{r}_{el} but is shifted with respect to the
energy scale such that: 
V_{el}(
y_{1}) = 
V_{el}(
y_{N}). The selfconsistent
SchrödingerPoisson integration
scheme is presented in Fig.
2.1.
Figure 2.1:
Selfconsistent SchrödingerPoisson integration method. As an intermediate step the electrostatic potential V_{0} is calculated.
As an application of the selfconsistent SchrödingerPoisson integration scheme,
we present Fig.
2.2, illustrating the effect of pdoping on the
band profile of an In
_{0.2}Ga
_{0.8}As/GaAs QW.
The QW includes a thin layer of In
_{0.65}Ga
_{0.35}As
that simulates the wetting layer.
We have assumed that
the QW and the WL are pdoped 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 2D wavefunctions 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 pdoped In_{0.2}Ga_{0.8}As/GaAs
QW. A sheet of In_{0.65}Ga_{0.35}As is positioned asymmetrically within the QW and
simulates the wetting layer.
The red curves represent the wavefunctions, the broken green lines indicate the energy level of
the 2D subbands.
Having determined the energy levels of the quantum well subbands, 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 T_{E}X by T_{T}Hgold,version 4.00.