We consider a GaN quantum well embedded into a Al_0.3Ga_0.7N barrier. The structure is grown on a Al_0.3Ga_0.7N substrate.

The structure geometry file is quantum_well.geo.

There are three physical regions (the well and two barriers, defined by means of Physical Lines 1, 2 and 3) and two physical points, at the simulation domain boundary, which will be associated to the boundary conditions (see Tutorial 0 and Tutorial 1). We'll study the following properties:

• Strain
• Band profile
• Quantized states of electrons and holes

The $Device section of the input file reads:

Calculation of strain

The strain is calculated over all the structure. Since the structure is grown over a substrate, we define an appropriate boundary condition. The substrate boundary condition is associated with Physical Point 1. The $Models section reads:

The $Solver section defines that the structure has a substrate:

In the output section we specify the variable name strain.

The results look as follows:

The well is strained, but the barriers are not, because the substrate material and the barrier material coincide.

Calculation of the band profile

The band profile calculation needs the Poisson equation to be solved. Therefore, we have to define a drift-diffusion model and solve it without any voltage applied. The $Models section reads:

This section does not contain any boundary conditions. It means that we rely on the default boundary condition, that require zero gradient of electric potential near the boundary. This boundary condition, known as the flat band boundary condition, is usually applied if the simulation domain represents a small part of a real heterostructure. It assumes that the free charge accumulated somewhere outside of the simulation domain screens the electric field.

The $Solver section reads:

The most important parameter is coupling = poisson that defines that no transport equation is solved. The rest of the parameters define properties of the numerical solver. The important thing is to define a physical model in the model $Physics, that reads as follows:

Here we impose that the drift diffusion model takes into account both the deformation potential effect and the piezoelectric effect. The latter is very important for wurtzite materials. In order to plot the band diagram we ask for two variables to be written out, namely Ec, Ev. The band diagram calculated with and without taking into account the effect of strain is shown:

Quantized states of electrons and holes

We are going to study quantized states of electrons and holes in the quantum well. Since the structure is 1D, the eigenstate is characterized by the energy level number n and the k_|| vector that is perpendicular to the growth direction. First, we define two simulations that solve Schrödinger for a single k-vector, for electrones and holes. In order to simplify the tutorial we solve the Schrödinger equation over all the structure. The $Models section reads:

The $Solver section reads as follows:

Here we defined two different option sets for electrons and holes. We specified that

• Dirichlet boundary condition is imposed at the simulation domain boundary
• Electric potential and strain is applied

The $Physics section defines the 8x8 k.p model for electrons and holes:

We choose to plot the following variables: EigenEnergy, EnergyLevels, EigenFunctions.

The results (levels and |ψ|² ) for electrons and holes are shown:

Now we would like to study the dispersion of the two first electron and hole states. In this tutorial we show how to define different k-spaces and calculate the particle dispersion over them.

First, we would like to calculate the dispersion along a line in k-space. We define two simulations belonging to the model quantumdispersion: dispersion1D_el and dispersion1D_hl.

We have to connect the quantumdispersion simulations to the related quantum (efaschroedinger) ones.
Thus, in $Solver section: