# Reduced Dimensionality Multiphysics Model for Efficient VCSEL Optimization
## 2 3D-1D Model Alignment
### Thermal Problem  
Heat is the main performance killer in VCSELs, the responsibility of power roll-over and of degradation at high ambient temperatures [21; 47; 48; 49; 50; 51]. This can be grasped in Figure 2, where we mimic the effects of a pulsed operation regime by varying the duty cycle of the injected current. The heating not only shifts and quenches the optical gain spectrum (see Figure 3), but it also impacts on the current leakage, as shown in Figure 2b. As discussed in [10; 11], while optical and electrical domains can be much reduced compared to actual device size (typically 150-250 \(\upmu\)m lateral size and 100-300 \(\upmu\)m height), the solution of the steady-state heat equation  
\[\
abla\cdot(\kappa\
abla T)=-Q_{tot} \tag{1}\]  
must include the full actual domain. In Equation (1), \(\kappa\) is the spatially-varying thermal conductivity and \(Q_{tot}\) is the sum of all the heat sources, which include Joule heating, free-carrier absorption, non-radiative recombinations, and QW capture heating. Therefore, to stretch the actual thermal problem to a 1D longitudinal equation is the most challenging and demanding task in our investigation. To help the discussion, in Figure 4a we show the actual 3D thermal profile from VENUS. The heat sources (apart from Joule heating, which is not dominating) are focused in the MQW (multi-quantum well) region, i.e., a tiny part of the whole device, about 50 nm in longitudinal direction and 4 \(\upmu\)m in the radial direction, accounting for carrier lateral diffusion compared to the 2.35 \(\upmu\)m oxide aperture radius. Figure 3: Gain spectra at different ambient temperatures, with electron and hole sheet densities set at \(4\times 10^{12}\) cm\({}^{-2}\). The concept of cavity detuning is sketched at the top of the figure. \(\lambda_{nom}\) refers to actual VCSEL design (\(\Delta\lambda=3\,\)nm), \(\lambda_{opt}\) to larger detuning values. Figure 2: Effect of the heating in VCSEL operation (CW—continuous wave; DC—duty cycle) at 20 \({}^{\circ}\)C. The impact of reduced internal heating is parametrically investigated for the output power (**a**) and leakage current (**b**) on our test device. This has to be compared with the 12 times larger transverse size, which, in terms of area, means a factor of 144. This could be interpreted as a heat source filling factor of the thermal transverse domain. In a 1D perspective, Equation (1) loses azimuthal and radial dependence (shown as an inset of Figure 4b), and retains just the longitudinal variable \(z\); the gradient operator reduces to a derivative along \(z\). Due to the much focused heat source, the temperature radial decay is strong at the MQW section, as shown by the different longitudinal cuts in Figure 4b (solid lines) and also by the black arrows in the inset of Figure 4a, representing the heat flow. Therefore we must carefully investigate and discuss how to treat such a major difference. As stated, the optical and electrical environments are mainly developing at the structure axis. The oxide aperture serves as both current and optical confinement. Therefore, carriers also nearly follow the oxide aperture size, allowing of course for some lateral spreading due to diffusion. Tightly connected to carriers is the gain profile, which is therefore also defined by the oxide aperture. The interaction with the optical gain of the guided modes and, in particular, of the fundamental mode (which is crucial, as most of the relevant sensing and datacom applications are aiming at single-mode operation) also completely occurs within the oxide aperture. Therefore, the temperature values relevant in VCSEL operation are those close to the axial region and we decided to make a surface average of the transverse temperature profile over the oxide area, which is shown in Figure 4b by the dashed black line. In Figure 5, such average is reported for two currents, together with the achieved 1D fits.