## Factors Influencing the Modulation Bandwidth
### Crystal Orientation  
Another important factor that can influence the modulation bandwidth of LEDs is the crystal orientation. For c-plane LEDs, which are polar, the carrier lifetime can only be reduced at higher current densities, thereby achieving a high modulation bandwidth. Operating at high current density, however, is unfavorable due to the efficiency droop effect [22; 45; 46], so finding a way to reduce the current density for a given modulation bandwidth and also to achieve higher quantity efficiency is necessary. Considering the nonpolar and semipolar LEDs, a higher modulation bandwidth can be achieved at lower current density due to the larger overlap of the electron-hole wave function and shorter carrier lifetimes [25; 26; 47; 48; 49]. However, due to the higher ratio of radiative-to-Auger recombination coefficients, there is an efficiency droop problem for certain semipolar orientation, like (20-2-1) [50].  
[PERSON] et al. measured modulation bandwidths for different InGaN/GaN \(\upmu\)-LEDs with a QW thickness of 4 nm, 4 nm, and 6 nm for the polar, semipolar, and nonpolar structures in order to compare the modulation responses [51]. Figure 2a describes the 3 dB modulation bandwidth as a function of current density for the polar, semipolar, and nonpolar \(\upmu\)-LEDs. The nonpolar LED obviously has the highest modulation bandwidth of \(\sim\)1 GHz at a low current density of 500 A/cm\({}^{2}\) followed by the semipolar device and then by the polar device. The overlap of the electron-hole wave function will seriously affect the modulation bandwidth, as both radiative and non-radiative recombination rates depend on the overlap of the wave function that links to the speed. Since the average wave function overlap in the nonpolar QWs is around 3 to 8 times higher than that of polar QWs, the bandwidth is higher in the case of the nonpolar device. However, with an increase in the current density, the bandwidth of the polar LED increases at a faster rate compared to the semipolar and nonpolar LEDs due to the coulomb screening effect [19]. Figure 2b gives the differential carrier lifetimes (DLTs) for the three orientations as a function of current density using the relation. The DLTs can be calculated using the relation \(\tau_{\Delta n}=1/(2\pi f_{3\text{dB}})\), where \(\tau_{\Delta n}\) is the DLT. The carrier density can be calculated using \(n=\int_{0}^{G}\tau_{\Delta n}~{}dG\), where \(n\) is the carrier density and \(G\) is the generation rate, which can be calculated from the relation \(G=J/(e\times d)\), where \(J\) is the current density, \(e\) is the electron charge, and \(d\) is the active layer thickness [50; 52; 53]. Using the data from Figure 2a, the approximate  
Figure 1: **(a)** Time-resolved photoluminescence spectra (TRPL) for light-emitting diodes (LEDs) with different quantum well (QW) thickness; **(b)** external quantum efficiency (EQE) for LEDs with different QW widths; **(c)** 3-dB modulation bandwidths. [19] Figures reproduced with permission from American Institute of Physics Publishing.  
Shockley-Hall-Read (SHR) lifetimes have been extracted and found to be 0.4 ns for nonpolar, which is shorter than that of semipolar (2.0 ns) and polar (4.0 ns). At higher current densities, due to Coulomb screening of the polarization field, the DLT for the polar device decreases at a faster rate compared to the nonpolar and semipolar devices. The carrier density was calculated, and it was found that in the case of nonpolar, the current density is lower for a given current density followed by semipolar and then by polar, as shown in Figure 2c. This trend can be explained by the greater electron-hole wave function overlaps in nonpolar and semipolar, resulting in higher recombination rates [54]. Thus, the bandwidth of nonpolar and semipolar LEDs at low current density is higher than that of c-plane LEDs, and it can also operate more efficiently due to reduced drop in efficiency.