In this Tutorial we will see how to perform the simulation of a GaN/AlN Quantum Dot (QD) structure by employing the Empirical Tight Binding (ETB) Module.

We first define an atomistic structure corresponding to the quantum cluster where we want to apply ETB calculations, than we apply continuous simulations on the whole device structure, in order to calculate strain map and equilibrium solution for potentials in device.

Then quantum calculations are performed with empirical_tb Module to get the electron and hole states in the QD.

The device structure is defined in the geometry .geo file and is the following:
a spherical GaN QD with radius 1 nm inside a 5X5 nm AlN cubic region

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

In the following, some features of the input file will be described. For further details you can refer to the program reference manual.

DEVICE STRUCTURE

In the Device section, the QD heterostructure is described: the GaN quantum dot region (ball), the AlN qbox, an intrinsic buffer AlN region and two n and p-doped AlN regions.
The crystal directions are defined for this wurtzite structure:

x-growth-direction = (-1,2,-1,0)

y-growth-direction = (1,0,-1,0)

z-growth-direction = (0,0,0,-1)

The regions are defined in the usual way:

Region ball

{

 material = GaN

  Doping 

  {

  Nd = 1e15

  type = donor
     
  Ed = 0.025

  }

}

................

A Cluster named atomistic is declared, including the ball QD region and the qbox AlN barrier region

Cluster atomistic

{

regions = (ball, qbox)

}

ATOMISTIC STRUCTURE

An atomistic representation of the above defined atomistic cluster is generated by means of the Atomistic block

Atomistic tb1

{

reference_region = nside

regions = atomistic

passivation = yes

print = ( xyz, gen, tgn)

translation = (0.0, 0.8983559, -4.39363)

}

The reference region is chosen to provide the lattice parameters with which the crystalline structure is built. In this case the lattice is that of AlN, the material composing the nside region.

reference_region = nside

In the QD region, the GaN atoms are then substituted in the lattice basis.
We will see in the following how Elasticity Module is then used to apply strain induced deformation to the mesh which is then projected to the atomic structure of GaN qdot.

Finally, passivation is performed at the boundaries of the heterostructure

passivation = yes

A print instruction gives in output the atomic structure for a visualization.

SIMULATION MODULES

1. Elasticity

Elasticity is used here to calculate lattice mismatch induced strain tensor in the whole structure. Moreover, in this case a non-linear calculation is performed, to feed back obtained displacements to the mesh elements. The mesh is thus deformed and a new strain calculation is performed with the new mesh.

Through these keywords:

mesh_deformation = true

shape_iterations = 1

the strain is computed iteratively until the convergence on the structure deformation is reached.

NOTE: This step is required to obtain a correct atomistic representation of  a heterostructure.
See also in the following, in  ETB section.

2. Drift-diffusion

As for drift-diffusion, as usual we define a simulation

name = dd

belonging to the model driftdiffusion and associated to the whole device (default choice).
Polarization is included through:

polarization (piezo, pyro) {}

The Boundary Regions for drift-diffusion are the two contact regions, defined by the two boundary surfaces anode and cathode.

3. Empirical Tight-Binding

For the atomistic quantum calculations, we define two empirical_tb simulations, named tb1 and tb2. In the first we will calculate the ground states, in the second one these calculate states will be loaded from output file, to complete the calculation of tb states.

Module empirical_tb
{

 regions = atomistic 

 name = tb1 

 atomistic_structure = tb1

 potential_simulation = dd

 plot = (tbstates, MeshStatesNodes )

 Solver

 {

 num_valence_eigenvalues = 1 

 num_conduction_eigenvalues = 1

 long_tolerance = 1e-4  

 }

}

In both cases the associated region is the Cluster atomistic and the associated potential simulation is the drift-diffusion dd simulation. Note that with this link the potential profile including built-in and polarization fields are correctly included into the TB Hamiltonian, through a correction on the on-site elements.

Another critical point is that Harrison scaling of ETB parameters is here applied
(it's a deafult whenever a strain simulation is performed on the system, as in this case).
Scaling is required in presence of material deformation which causes atom displacement from
equilibrium position.

In tb1, only one state is calculated for holes and electrons:

num_valence_eigenvalues = 1

num_conduction_eigenvalues = 1

In tb2:

Module empirical_tb

{

 regions = atomistic 

 name = tb2

 atomistic_structure = tb1

 potential_simulation = dd

 plot = (tbstates, MeshStatesNodes)


Solver

 {

 load_path = output_tb1

 num_valence_eigenvalues = 4 # 2

 num_conduction_eigenvalues = 4 # 2

 long_tolerance = 1e-4 

 }

}

calculated states are loaded from output file by defining in the Solver block:

load_path = output_tb1

and the calculation of 4 states is completed, starting from those alreay available.

num_valence_eigenvalues = 4 # 2

num_conduction_eigenvalues = 4 # 2

This option may be useful to continue a long calculation of many eigenstates after that it has been stopped by the user for any reason.

4. TB Optics

With the Module opticstb it is possible to perform the calculation of the optical matrix from the TB Hamiltonian.

Module opticstb

{

 name = opt

 regions = atomistic 

 initial_state_model = tb2

 final_state_model = tb2


 compute_strengths = true

 plot = (matrix_elements, optical_spectrum_k_0)

 output_format = grace

 Emin = 4.25

 Emax = 4.75

 dE = 0.001

}

Note that, differently from the analogous Optics Module for EFA, here the initial state and the final state source models must be the same. In this case, we choose tb2 simulation, which compute four eigenstates.
The other keywords are similar to those of EFA case.
Note that we choose

compute_strengths = true

to calculate optical strength output.

Run simulations

We may now run tiberCAD to calculate strain with Elasticity (str) and driftdiffusion (dd) for an equilibrium solution.
Then, for tight-binding, we first run tb1 for calculation of the ground state

solve = (str, dd, tb1)

then we run tb2 with

load_states = true

to load existing ground state solution and calculate further states and then finally the optical properties (opt)

solve = (str, dd, tb2, opt)

Now we can execute tiberCAD:

tibercad quantum_dot_GaN_TB.tib

Output

After the execution, the output directory contains, as usual, the results for the simulations performed. As for tight-binding simulation, the file tb2.dat contains a table with the calculated eigenvalues for electrons and holes, together with their occupation index.
The files .cube have been also generated, which contain the information on the electron and hole wavefunction (square module). The format of these data files is supported by the visualization software jmol, an open-source Java viewer for chemical structures in 3D http://jmol.sourceforge.net.
By using jmol, we can load the atomic structure generated by tiberCAD Atomistic Generator, contained in the file tb1.xyz. Here Ga atom are shown in red.

Then we can visualize the isosurface of e.g. the electron ground state, with this command:

isosurface mo1 color green cutoff 0.00005 "mo_cb_01.cube"

Analogously, we can visualize the hole state.

The figure below shows the confinement of the two states inside the spherical GaN quantum dot: in green the conduction state and in yellow the valence state.

ATTACHMENTS

• quantum_dot_GaN_TB.geo
  
• quantum_dot_GaN_TB.msh
  
• quantum_dot_GaN_TB.tib