In this tutorial we will see how to calculate quantum and optical properties of a AlGaN/GaN LED diode with three GaN quantum wells.

First of all, strain and polarization fields are calculated for this wurtzite heterostructure, then an increasing bias is applied to the diode until conduction regime is reached. Schroedinger equation is solved, with a single-band effective mass model for conduction band and with a 6-band k.p model for valence band. Eigenvalues and eigenfunctions are calculated to get energy levels and wavefunctions in the three quantum wells; then k.p optical matrix elements are calculated and used to find the emission spectrum, which finally is integrated in k-space.

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

 

 

Let's see the input file:

Device structure

After a buffer n-doped AlGaN layer (Region n-AlGaN ), a Al_0.78InN barrier layer is present (Region AlInN), then a series of three AlGaN/GaN quantum wells, each 2 nm-wide.
The AlGaN barriers are described with several Regions, in order to distinguish regions where quantum calculations will be applied.
For example, the Region AlGaN_barrier_quantum comprises all the GMSH mesh regions corresponding to barrier layers and which will be associated to quantum calculations:

• The Region GaN_QWell , instead, comprises in a unique TiberCAD region all the GMSH mesh regions corresponding to GaN well layers:

 

• Finally, the p-doped AlGaN layer is described in this way by Region p_AlGaN:

• For convenience, we define 3 Clusters, to collect the mesh regions (with different materials) which will be associated to each quantum calculation:

Quantum_1 collects the regions related to the first quantum well.

Quantum_2 and Quantum_3 collect respectively the mesh regions related to the second and to the third quantum well.

 

Simulation Models

 

1. Drift-diffusion

A simulation model for drift-diffusion calculation is defined.
Physical models for srh recombination and direct (radiative) recombination are declared inside it.

 

2. Strain

A macrostrain simulation (simulation_name = strain) is defined with a boundary condition located in the AlGaN substrate (BC_Region n_contact ).

 

3. Quantum

For the quantum calculations, we define several efaschroedinger model simulations, two (one for electrons and one for holes) for each quantum well.

To each simulation, one Cluster is associated, for example
physical_regions = Quantum_1 for simulations QW1_electrons and QW1_holes.

In Solver section, we set Dirichlet b.c. and number_of_eigenstates = 20:

These settings are valid for all the efaschroedinger simulations; each simulation, instead, needs the proper particle specification (el or hl)

In $Physics, we choose to use single-band model for conduction band:

and 6x6 kp model for valence bands

 

4. Optical properties

For the optical properties, a simulation of the model opticskp is to be defined:
simulation_name = optics,
for one of the quantum clusters (in this case physical_regions = Quantum_3).

We want to calculate the (k-space integrated) optical spectrum, so we need an opticalspectrum simulation :

In Solver section, we specify, in optics block:

• the quantum simulations related to the initial and final state (initial_state_model = QW3_electrons, final_state_model = QW3_holes), which give the electron and hole states used in optical calculations;
• which of the eigenstates we want to take in account for the transitions : initial_eigenstates = (0, 19), final_eigenstates = (0, 19)

 

 

As for optical spectrum calculation, we need to specify several information for spectrum simulation:

• dimension of k-space ( =2 for 1D simulation, k_space_dimension = 2) and corresponding k vectors
• parameters for the adaptive refimement of the k-space mesh (refine_fraction, number_of_nodes , etc.)
• name of the simulation from which optical matrix elements are taken (optical_matr_elem_model = optics)
• light polarization and energy range (Emin, Emax, dE) for the spectrum calculation

 

 

Run simulations

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

solve = (strain, sweep, QW3_electrons,QW3_holes, spectrum)

First we run strain simulation, to get deformation potentials and piezopolarization, than a sweep on drift-diffusion, until the diode is in conduction, than quantum efa calculations are performed (QW3_electrons, QW3_holes), to get the states in the third quantum well, which are finally used in optical calculations (spectrum, which by itself runs the opticskp model simulation as defined in Solver section).

tibercad led_spectrum.tib

 

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Output

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

• driftdiffusion_elemental.dat: current vectors, etc.
• driftdiffusion_nodal.dat: bands profiles, densities, etc.
• sweep_driftdiffusion_Vb.dat: integrated currents at contacts (IV characteristic)

and for strain:

• strain_elemental.dat: strain tensor components

Conduction and valence band profile for an applied bias Vb=4 V; quasi-fermi levels for electrons and holes are indicated, showing that the three GaN quantum wells levels are populated both in conduction and in valence band.

for quantum:

• QW3_electrons.dat, QW3_electrons_nodal.dat
• QW3_holes.dat, QW3_holes_nodal.dat

 

 

First conduction band state (left) and valence band state (right) in the third quantum well.

 

Finally, since the plot variables optical_spectrum and k-space are defined, we get the output files

• spectrum_spectrum.dat: emission spectrum in the defined energy range
• spectrum_k_space.gmv: k-space mesh

Attachment	Size
led_spectrum.geo	2.34 KB
led_spectrum.tib	8.83 KB
led_spectrum.msh	113.14 KB