## 2 Theoretical Background of Micro-LEDs for VLC Applications  
Frequency response is one of the most significant parameters to be addressed in developing LEDs for use in visible light communication systems, and the cut-off or -3 dB frequency is found to be correlated to certain diode parameters. The modulated photocurrent signal of LEDs can be detected using a high-speed photodiode amplifier to evaluate their frequency response by applying a fixed bias current, which will eventually determine the speed of the LED. The cut-off frequency of the LEDs is constrained by the lifetime of the carrier and a time constant consisting of the capacitance of a depletion layer and the junction differential resistance. The net recombination rate \(R\) is usually defined in terms of the equilibrium hole concentration \(p_{o}\), the equilibrium electron concentration \(n_{o}\), the excess carrier concentration \(\Delta n\), and the recombination constant \(B\), which is given by the following relation  
\[R=B(n_{o}+\Delta n)(p_{o}+\Delta n)-Bn_{o}p_{o}=B(n_{o}+p_{o}+\Delta n)\Delta n \tag{1}\]  
The carrier lifetime of the injected carrier is related to recombination rate by the following equation  
\[\tau=\frac{\Delta n}{R}=\frac{1}{B(n_{o}+p_{o}+\Delta n)} \tag{2}\]  
However, when we consider an n-type semiconductor, \(p_{o}\) can be neglected as majority carrier, and \(n_{o}\) is much greater than minority carrier, \(p_{o}\); hence Equation (2) becomes  
\[\tau=\frac{1}{B(n_{o}+\Delta n)} \tag{3}\]  
When carriers are injected into the LEDs, if the injected carrier uniformly recombines, then the injected carrier concentration \(\Delta n\) can be attributed to the injected current density \(J\) and the active layer thickness \(d\) by the following equation  
\[\Delta n=\frac{J\tau}{qd} \tag{4}\]  
where \(q\) is the elementary charge. However, when \(d\) is greater than carrier diffusion length, the injected carrier concentration is insignificantly relative to the majority carrier concentration, and the carrier lifetime equation can be written as  
\[\tau=\frac{1}{Bn_{o}} \tag{5}\]  
From this equation, we observed that the carrier lifetime is related to the recombination coefficient, which is related to the modulation frequency. Hence, since the modulation bandwidth can be attributed to the speed for VLC application, the -3 dB bandwidth of the LEDs can be related to the minority carrier lifetime \(\tau\) and Resistance-Capacitance (RC) time constant, \(\tau_{RC}\), which consist of differential resistance and parasitic geometrical capacitance of the LED (i.e., \(\tau_{RC}=R\cdot\)C), as shown in the following Equation [28]  
\[f_{\
eg 3 dB}=\frac{\sqrt{3}}{2\pi\tau}=\frac{\sqrt{3}}{2\pi}\bigg{(}\frac{1}{ \tau_{r}}+\frac{1}{\tau_{nr}}+\frac{1}{\tau_{RC}}\bigg{)} \tag{6}\]  
where \(\tau_{r}\) and \(\tau_{nr}\) represent the radiative and nonradiative carrier lifetime, respectively. Hence, it is possible to monitor the modulation bandwidth of LEDs by optimizing the active area thickness that will evidently control the lifetime of the carrier. By utilizing Equation (5), the modulation bandwidth expression can take the form of, after assuming a bimolecular recombination mechanism,  
\[f_{\
eg 3 dB}=\frac{\sqrt{3}}{2\pi}\times\sqrt{\frac{BJ}{qd}} \tag{7}\]Hence, LEDs of micro-size can have a high modulation bandwidth by achieving a small carrier lifetime, as they can sustain higher injection current densities. In addition, having a small active region will lead to a decrease in geometric capacitance, thus reducing the RC time constant, which is a limiting factor of modulation bandwidth. Furthermore, the modulation bandwidth can be enhanced by growing the LEDs on semipolar or nonpolar substrates, as they corresponds to a larger radiative recombination rate which will be further discussed in the following section.