In this application, a full 3D model of an AlGaN nanocolumn heterostructure with a GaN quantum disk has been designed and used to perform 3D quantum calculations and obtain eigenvalues and eigenfunctions of confined states in the quantum disk (QD).

We first perform a strain simulation, to get deformation potentials and piezopolarization, than we apply drift-diffusion model with an increasing bias to the contacts, until the nanocolumn diode is brought in conduction regime. Then quantum efa calculations are performed to get the electron and hole states in the 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

DEVICE STRUCTURE

Here is the structure of the AlGaN nanocolumn, with a GaN quantum disk region and two highly doped AlGaN contact regions (respectively p-type and n-type).

STRAIN AND DRIFT-DIFFUSION

First of all we define the calculations of strain and drift-diffusion transport.
In fact, we need 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 we apply driftdiffusion model for an increasing bias, until Vd = 4.5 V.

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.

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

QUANTIZED STATES OF ELECTRONS AND HOLES

We are going to study quantized states of electrons and holes in the quantum disk (QD).

Quantum calculation is restricted to a quantum region, which comprises the GaN QD and two AlGaN barrier regions in the nanocolumn at the sides of the QD.

• 30 eigenstates to be calculated for both electrons and holes
• electric potential and strain are applied

We define 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.

Above, 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, too.

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

QUANTUM DENSITY

We calculate the quantum density of electrons and holes, based on the eigenstates found by the EFA simulation.

Here are the quantum charge density respectively for electrons (left) and holes (right).

Below, these densities are shown again along a cutline in the z-axis 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. However, with tiberCAD it is also possible to perform this kind of more accurate calculation.

OPTICAL PROPERTIES

We calculate the optical matrix elements for a set of 10 initial and final states in the valence and the conduction band, resulting in the optical spontaneous emission spectrum for k = 0 which is shown below.

Here the result for a 3 nm-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.