### VCSEL

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#### Introduction

In VCSEL structures the light is emitted perpendicular direction to the top surface, while in edge emitting lasers this direction is parallel to the top surface. The fabrication of such devices requires more precision, and theoretical modelling. In this example we investigate the active region of such devices.

#### Structure

A simple VCSEL structure is plotted in figure 1., which is built from an active region and two DBRs (Distributed Bragg Reflector).

The active region is built from an p-n junction and in the depletion region InGaAs multiple quantum well structure was placed. In the simulation we set ohmic contacts to the both sides of the active region.

#### Band-structure

The band-structure of the device without external bias is plotted in figure 2., where we considered the built-up strain in the structure which splits the light-hole and heavy-hole bandedges.

#### Voltage characteristics

In the presence of external bias in the forward direction in the diode the quantum wells start to fill-up with charge carriers. The carriers can recombine in the multiple quantum well structure (MQW) in a radiative process, which makes the laser work.

##### Band-structure on applied bias

The band-structure of the device is bias voltage $U_{bias}= 1.2V$ is applied on the device is plotted in figure 3. It shows that the bandgap is the lowest in the MQW structure.

##### Eigenfunctions in the MQW

The quantum well structure creates a quantum confinement for the electrons, and for the holes. In order to calculate correctly the carrier densities we need to use quantum mechanical density calculation.

The eigenfunctions are the mixture of the independent single quantum well eigenfunctions. They are delocalized over the full MQW structure for electron states, but for hole states the they are more localized to single quantum wells as it is depicted in figures 4 and 5 . The hole states are calculated with 6 band k.p method, which calculates with the LH and HH characteristics of the hole states.

##### Density Calculation

It is important to calculate the carrier densities according to the wave-function calculation.

We compared the classical density calculation for the electron, and hole states with the quantum mechanical density calculation in figures 6, 7.

It shows that due confinement effects the carrier densities become lower.

##### Gain spectrum

It is also important to know on which bias the population-inversion occurs. If we plot the gain spectrum in the function of the applied bias, the threshold bias can be estimated.

It shows in figure 8 that around $U_{bias}= 1.0V$ the device is capable to amplify light.