Multiscale Device Simulator

In this tutorial, we will see how to perform 3D calculations of strain and quantum properties of a wurtzite GaN conic quantum dot embedded in an AlGaN nanocolumn heterostructure.

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

Device structure

The structure of the nanocolumn is shown here, together with the 3D mesh obtained with GMSH.

Fig.1. Nanocolumn structure	Fig.2. 3D mesh (quantum dot region)

The conical wutzite GaN quantum dot (blue in Fig. 1 and 2) is modelled by the Physical Volume qdot in quantum_dot.geo and associated to Region qdot in TiberCAD.

Then, an AlGaN buffer region surrounding the quantum dot is created (Physical Volume("barrier"))

Finally, the two AlGaN side regions of the nanocolumn ( Physical Volume buffer1 and buffer2) :

The Cluster Quantum_dot is defined to comprises the mesh regions qdot and barrier (see fig.1) in the quantum calculations:

Strain calculations

To calculate strain in the nanocolumn, we define a macrostrain simulation (str) with a substrate b.c. referring to AlGaN material layer.

Strain Output

From the file str_elemental.plt, we plot the strain tensor component and the sum of diagonal components, which shows the relative volume deformation in the nanocolumn.

Fig.3. component of the strain tensor	Fig.4. Relative volume change

Poisson calculation

We want to solve Poisson to find band profiles; to do this, as usual, we create a driftdiffusion simulation (dd), with no explicit boundary conditions to the two side surfaces of the nanocolumn. This means that flat band b.c. are implemented for top and bottom surfaces.
This time, however, we include a new feature, that is we apply to the lateral surface a surface charge b.c. (type = interface).

This option has two effects:

• a surface charge with the value Ns = -1e10 is imposed to the lateral surface of the nanocolumn
• the piezo and pyro polarization at the (lateral) surface is included in calculations.

N.B. Usually, by default, piezo and pyro polarization at the boundaries are NOT included in the model, since it is assumed that either they are screened by free carriers and electric field, or the boundaries are very far from the active region of the device, so that polarization at boundaries are negligible.
Only in the case a b.c. with type = interface is defined, polarization at boundaries are taken into account; if Ns =0, only charge due to polarization is present, otherwise the surface charge is the sum of polarization charge and of Ns.

In Solver section, we specify Poisson calculation in this way:

Poisson Output

In Fig. 5 the total polarization is shown , together with the electric potential (from the output file dd_nodal.plt). One can see how the potential distribution is related to polarization, which generates a polarization charge.

Fig.5. a) z component of total (piezoelectric and pyroelectric) polarization; b) electric potential

Quantum calculations

For the quantum calculations, we define two efaschroedinger simulations: quantum_el for electrons and quantum_hl for holes

In Physics section, we associate quantum_el simulation to a single-band calculation of conduction band states

and quantum_hl simulation to a k.p 6x6 calculation of valence states

In Solver we select the krylovshur solver, the poisson model (dd) and the strain model (macrostrain)

In Simulation section :

solve = (str, dd, quantum_el, quantum_hl )

defines the following flow of simulations:

1. Strain calculation (str)
2. Poisson calculation (dd)
3. Quantum calculations for electrons (quantum_el) and
4. for holes (quantum_hl)

Quantum Output

Eigenvalues and wavefunctions can be found in the output files quantum_el_nodal.plt and quantum_hl_nodal.plt.

Here are the first four quantum states in the conduction band, which are confined inside the conical quantum dot.

And here is the first quantum state in the valence band.