Eigenfunctions in quantum well

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#### Introduction

In that example we investigate a AlGaAs/GaAs quantum well in quantum mechanical point of view. The simulation involves band profile calculation in the hetero-structure, and compare its influence with different QM solvers.

#### Band-structure calculation

The growth direction of the sample is in the **001** direction on $Al_{0.4}Ga_{0.6}As$. The lattice miss-match of the GaAs and AlAs is very low, which leaves the structure un-strained. The calculated profile including bowing parameters is depicted in figure 1.

#### Eigenfunction calculation

In this example we are going to consider just the electron wave-functions, but the same ideas could be used for the hole functions.

##### Single-band method

In figure 2. the single-band wave-functions are calculated, which means the effective mass of the electron is a constant energy-independent value in the Scrödinger-equation:

\begin{equation} - \nabla \frac{\hbar^2}{2m_e(x)} \nabla \Phi + V(x) \Phi = E\phi. \end{equation}

The $m_e(x)$ is the effective mass of the electron, while the $V(x)$ is the conduction band edge - both of them are position dependent.

##### 8 band k.p method

We are able to calculate the wave functions of the quasi-particles in the sample with the coupling of the hole and electron bands \citep{pryor1998eight}. Which can be used for more realistic calculations, due to the fact the effective mass of the electron is energy dependent.

The first confined state in the Quantum well is plotted in figure 3.

It has a bit different energy, which reflects in the different electron density calculated in the next section.

##### Lateral dispersion

In the section before we calculated the wavefunctions if the electron has zero in-plane momentum($k_{||}= 0$). But if this lateral momentum is not zero it changes the eigenenergy of the electrons. It can be described according to an E(k) dispersion relation in figure 4.

As it shows it can be treated a constant mass problem for some energy range. In our approach we mix out a constant mass calculated from the 8 band k.p wavefunction at $k_{||} = 0$.

\section{Charge density calculation}

For the constant mass approach we can calculate the density in the sample according the equation: \begin{equation} n(x) = \sum_{i = 0}^{N} |\Phi_i(x)|^2 k_bT \frac{1}{4 \pi} \frac{2 m_{DOS}(x)}{\hbar^2} ln(1+exp((E_i-E_f)/k_bT)) \end{equation}

in one dimension, with two dimensional k-space. Where $E_i, \Phi_i$ is the i-th eigenenergy and eigenfunction of the sample.

In figure 5. we compared 3 different density calculations with effective mass. It shows that due the eigenenergy of the electron function is higher than the conduction band-edge, which results less electrons in the band.

We can calculate the charge carrier density without assuming constant mass in the lateral dimension, but for this we have to calculate the wave-function for each $k_{||}$ parallel point in the sample. It ends up nearly at the same result as in figure 5.