In this tutorial, we come back to the nanostructure studied in Tutorial 05 (AlGaN nanocolumn diode with a GaN quantum disk)

This time we show how to perform 3D quantum calculations with efaschroedinger model, to obtain eigenvalues and eigenfunctions of confined states in the quantum disk (QD). Then, from these states in conduction and valence band , the optical emission spectrum is calculated. Finally, the quantum density for electrons and holes in the QD is calculated and compared to classical densities.

• Quantized states of electrons and holes

• Quantum density

• Optical properties

• Output

In order to execute correctly this example you should have the following files in the same working directory:
qdisk.tib: input file for TiberCAD
qdisk.msh : mesh file produced by the GMSH script qdisk.geo

Device structure

In the Device section, the structure of the nanocolumn is described: the GaN quantum disk region , the AlGaN barriers and
two highly doped AlGaN contact regions (respectively p-type and n-type).
For the details you can refer to Tutorial 05.

Simulation models

1. Strain and Drift-diffusion

First of all we define the calculations of strain and drift-diffusion transport ( analogously to Tutorial 05).
In fact, first we want to use strain and piezoelectric polarization results for the drift-diffusion current calculation.
Then, the information about band structure and Fermi level position is used in the quantum calculation to calculate the correct occupation of quantum states.
We choose a bias of 4.5 V to get a reasonable population of the QD states, so a sweep of driftdiffusion model until Vd = 4.5 V is performed.
Be sure that you add the keyword strain_simulation = strain for driftdiffusion model In Physics section. In this way we specify that strain effects will be taken in account correctly in the calculation of current. In particular, the piezoelectric polarization arising from strain in the wurtzite nitrides materials of this nanocolumn will enter in Poisson equation and heavily modify the band profiles.

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2. Quantized states of electrons and holes

We are going to study quantized states of electrons and holes in the quantum disk (QD). Thus, we define two simulations that solve Schrödinger equation for electrons and holes. Quantum calculation is restricted to the "quantum" cluster defined in in Device section, which comprises the GaN QD and two AlGaN barrier regions in the nanocolumn at the sides of the QD.

The $Solver section reads as follows:

Here we define two different option sets for electrons and holes. In particular we specify:

• 30 eigenstates to be calculated for both electrons and holes
• Electric potential and strain is applied (strain_model_name and poisson_model_name keywords)

The $Physics section defines the 6x6 k.p model for holes and the single band (conduction band) model for electrons :

This choice is justified by the use of large-gap materials like nitrides, where the interaction between conduction and valence band can be neglected in a first approximation.

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3. Quantum density

For the calculation of quantum density of electrons and holes, we define two simulations belonging to the model quantumdensity:
simulation_name = dens_el,
simulation_name = dens_holes,
to be solved in the quantum cluster (in this case physical_regions = quantum).

This model calculates the quantum density of the given particle, based on the eigenstates found by an efaschroedinger simulation

In Solver section, we specify (e.g. for electron case):

• which efaschroedinger simulation calculates eigenstates: quantum_simulation = quantum_el
• calculation with analytical formula (faster) : analytic = true
  
• the actual number of eigenstates we use in the quantum density calculation : initial_eigenstates_number (10 for electrons and 80 for holes)

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4. Optical properties

For the optical properties, we define a simulation of the model opticskp:
simulation_name = optics,
to be solved in the quantum cluster (in this case physical_regions = quantum).

This model calculates the optical emission spectrum for k=0 (no k-space-integration)

In Solver section, we specify, in optics block:

• the quantum simulations related to the initial and final state (initial_state_model = quantum_el, final_state_model = quantum_hl), which give the electron and hole states used in optical calculations;
• which eigenstates we want to take in account for the transitions : initial_eigenstates = (0, 9), final_eigenstates = (0, 9)
• the light polarization and the spectrum energy range.

Finally, in the plot statement, we use the variable optical_spectrum_k_0 to get a plot of the optical spectrum in k=0

Run simulations

In Simulation section we define the flow scheme of our calculations:

solve = (strain,sweep ,quantum_el,  quantum_hl, dens_el,  dens_holes,  optics   )

Thus, we first run strain simulation, to get deformation potentials and piezopolarization, than a sweep on drift-diffusion, until the nanocolumn diode is brought in conduction regime, than quantum efa calculations are performed (quantum_el, quantum_hl), to get the electron and hole states in the QD. Quantum and drift-diffusion (for the occupation function) results are used to get quantum densities for electrons (dens_el) and holes (dens_holes) . Finally, optical spectrum is calculated (optics).

tibercad qdisk.tib

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Output

As usual, we get the output files for drift-diffusion and for strain;

Here you can see, along a cut in the z-axis of the nanocolumn, the band profiles for conduction and valence bands and respectively the electron (black) and hole (red) charge density as obtained from the classical drift-diffusion calculations.

then for quantum calculations, the output files quantum_el_nodal.vtk and quantum_hl_nodal.vtk contain the wavefunctions which are plotted below:

Here, in blue, the first conduction band state and in red the first valence band state in the GaN quantum disk , obtained from the k.p calculation, without self-consistence with drift-diffusion. The electric potential distribution in the nanocolumn is shown.

It is clearly visible the quantum confinement of eigenstates in the QD; note also the clear spatial separation between hole and electron states, due to the strong band-bending arising from the strain-induced piezoelectric polarization.

The output of the quantumdensity simulation is the files dens_el_elemental.vtk and dens_holes_elemental.vtk which contain respectively the quantum charge density for electrons and holes.

Below, these densities are shown again along a cutline in the z-axiz direction in the centre of the nanocolumn.
Note that these results, both for quantum states and densities, and for band profiles, still do not take in account a self-consistent calculation of Poisson and Schroedinger equations. We will see in a next Tutorial how to perform this kind of more accurate calculation, and its effects on the results.

Finally , the optics simulation calculates the optical matrix elements for a set of initial and final states in valence and conduction band, resulting in the optical emission spectrum below ( optics_spectrum_k_0.dat file ):

Here the result of the present Tutorial structure, with a 3nm-thick QD, is compared to that of the same structure with a 2 nm QD, showing a blue-shift of emission peak with a reduced quantum disk thickness.

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