# Strain Relaxation Effect on the Peak Wavelength of Blue InGaN/GaN Multi-Quantum Well Micro-LEDs
## 3 Results and Discussion
### InGaN/GaN Quantum Well Finite Element Method (FEM) Simulation  
From the above section, the strain's relaxation was induced by the Raman shift of the GaN layer, we assume that the stress in the GaN layer is equal to that in the InGaN layer, and the direction is opposite. Then, we use the change of the GaN stress to characterize the stress relaxation in the InGaN quantum well but each InGaN quantum well strain cannot be described exactly. To support the edge strain relaxation and micro size effects described above, a solid mechanic's finite element model in COMSOL Multiphysics was built to simulate the strain distribution in a single micro-pillar. The isotropy of the in-plane strain in the [0001] growth direction and the cylindrical symmetry simplified the simulation to a two-dimensional case [20]. The simulation model was built according to the structure shown in Figure 1, with the sidewall edge of the pillar set as the start of the \(x\)-axis. The free boundary conditions were assigned to the micro-pillar side walls, and the up and bottom GaN surfaces. Meanwhile, we assumed that the quantum wells are pseudo-morphically grown. The initial strain \(\varepsilon_{0}\) between InGaN and GaN layers, calculated by the lattice mismatch Equation (3), was set as the initial strain boundary conditions [21]. The strain and stress were near the quantum well boundary and the carrier radiative recombination was mainly in the quantum wells so that compressive strain of the InGaN quantum wells was observed. The FEM calculated strain distribution of 12 In\({}_{0.25}\)Ga\({}_{0.75}\)N quantum wells is shown in Figure 6a. The height marks the vertical position of the quantum wells in the micro-LED pillar. At the transverse inside of the quantum well (distance from edge \(>\) \(\sim\)600 nm), the in-plane strain (\(\varepsilon_{\text{xx}}=\varepsilon_{\text{yy}}\)) of InGaN is about 2.67%, which is the complete strain of In\({}_{0.25}\)Ga\({}_{0.75}\)N on GaN. The strain decreases as the boundary approaches. The strain \(\varepsilon_{\text{xx}}\) is about 2.3% near the edge of pillar and the edge strain is too small to represent using a color scale. Therefore, the exact strain distribution is listed in Figure 6b. Figure 6b shows the calculated average strain of the 12 pairs of QW in a 19 \(\upmu\)m diameter pillar. The results show that in the micro-pillars, the strain distribution mainly can be divided into three regions. The first region is the complete strain region, which is located in the middle of the micro-pillars and the strain remains constant at 2.67%. The second region is the gradual strain release region, where the strain decreases from 2.67% to 2.36% within \(\sim\)600 nm to \(\sim\)10 nm of the pillar edge. The third area is the mutation region, where strain is released rapidly within 10 nm from the boundary and the strain change to 0.25% at the edge of the pillar. \begin{table}
\begin{tabular}{c c c c} \hline
**Material** & **InN** & **GaN** & **In\({}_{0.25}\)Ga\({}_{0.75}\)N** \\ \hline k (cm\({}^{-1}\)/GPa) & & 2.56 & \\ a (Å) & 3.545 & 3.189 & 3.278 \\ a\({}_{\text{c}x}\) (eV) & \(-3.5\) & \(-4.9\) & \(-4.55\) \\ a\({}_{\text{c}t}\) (eV) & \(-3.5\) & \(-11.3\) & \(-9.35\) \\ D\({}_{\text{D}}\) (eV) & \(-3.7\) & \(-3.7\) & \(-3.7\) \\ D\({}_{\text{2}}\) (eV) & 4.5 & 4.5 & 4.5 \\ D\({}_{\text{3}}\) (eV) & 8.2 & 8.2 & 8.2 \\ D\({}_{\text{4}}\) (eV) & \(-4.1\) & \(-4.1\) & \(-4.1\) \\ C\({}_{11}\) (Gpa) & 223 & 390 & 348.25 \\ C\({}_{12}\) (Gpa) & 115 & 145 & 137.5 \\ C\({}_{13}\) (Gpa) & 92 & 106 & 102.5 \\ C\({}_{33}\) (Gpa) & 224 & 398 & 354.5 \\ C\({}_{44}\) (Gpa) & 48 & 105 & 90.75 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters used for GaN and InGaN materials stress and strain analysis [18; 19]. Furthermore, based on Equations (3)-(8), the quantum well strain-related PL peak emission was estimated and shown in Figure 7a. In the complete strain region, the quantum well peak wavelength is 459 nm. In the strain mutation region, the strain relaxation makes the peak wavelength move to 416 nm.