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.


  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:





fn(y) = En fn(y),
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+1yi−1]/2 , i ∈ {2, …, N−1 }, we obtain [1]:

fn (yi+1)

m (yi+1)
fn (yi−1)

m (yi−1)


2 ∆y
= 2

[V0(yi) − En ] fn(yi).
Inserting fn(yi) = [fn(yi+1) − fn(yi−1)]/(2 ∆y) into (1.2) we get:
fn (yi+2)

m (yi+1)

2 (2 ∆y )2

[V0(yi) − En ]+ 1

m (yi+1)
+ 1

m (yi−1)

fn(yi) − fn (yi−2)

m (yi−1)
The next step involves the following transformation: ∆y ⇒ ∆y/2, leading to:
fn (yi+1) =

2 (∆y )2

[ V0(yi) − En ] m (yi+1/2) + 1+ m (yi+1/2)

m (yi−1/2)

                        − m (yi+1/2)

m (yi−1/2)
fn (yi−1),
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 ≤ im ) and from the right (m < iN) 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)
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)
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 (−ρ3D) 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) = esum3D |fn(yi)|2 − ρ3D(yi)] ∆y    where   ρsum3D = N

The definition of the 2-D charge density σ2D ensures charge neutrality: ∑i=1N σ2D(yi) ≡ 0. The electric field due to the sheet charge density σ2D(yk) is given by:
Eel(yi) = [σ2D(yk)/2 ϵ] sign(yiyk).
Due to the superposition principle, the total electric field can be calculated by:
Eel(yi) = N


2 ϵ
sign(yiykwhere sign(yiyk) =

1     for     yi > yk
0     for     yi = yk
−1     for     yi < yk
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 ∆y Eel(yi) + Vrel(yi+1),
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. Integration Scheme

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 In0.2Ga0.8As/GaAs QW. The QW includes a thin layer of In0.65Ga0.35As that simulates the wetting layer.
We have assumed that the QW and the WL are p-doped with C at a concentration of 3×1018 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.


Paul Harrison. Quantum Wells, Wires and Dots. John Wiley & Sons, Chichester, 2001.

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