# Modulation Characteristics of High-Speed Transistor Lasers
## 2 Inherent Advantages of TL
### Drastically Reduced Effective Carrier Recombination Lifetime
In Figure 4b, \(\tau_{\text{sch2}}\) is the lifetime of carriers passing through the base region from the left SCH layer to the right SCH layer, \(\tau_{\text{sch1}}\) is the diffusion time of carriers from the left SCH layer, \(\tau_{\text{cap1}}\) is the QW\({}_{1}\) trapping lifetime, \(\tau_{\text{bar}}\) is the diffusion time between QWs, \(\tau_{\text{qw1}}\) is the effective radiative recombination of carriers in the QW\({}_{1}\) region lifetime, including spontaneous emission and stimulated emission within the QW, \(\tau_{\text{sch}}\) is the spontaneous emission recombination lifetime of carriers in the SCH layer, \(\tau_{\text{tun}}\) is the tunneling carrier time between wells, \(\tau_{\text{therm}}\) is the hot-electron escape time, \(\text{W}_{\text{B}}\) is the base width, \(\text{L}_{\text{b}}\) is the barrier width, \(\text{W}_{\text{QW}}\) is the quantum well thickness, and \(\text{W}_{\text{SCH1}}\) and \(\text{W}_{\text{SCH2}}\) are the upper and lower base, respectively, of the SCH layer thicknesses. The green line in Figure 4b is the forward route of the carriers in the base region; most of the carriers participate in the transition from the base region, and the rest enter the QW to participate in the recombination. The carriers entering the QW generate photons through radiative transition, and some of them escape through hot electrons or tunnel electrons into other QWs. The frequency response \(\mathbf{H}(\mathbf{\omega})\) of TL is expressed as  
\[\mathbf{H}(\mathbf{\omega})=\frac{\mathbf{A_{0}}}{\mathbf{1}-\frac{\mathbf{\omega}^{ 2}}{\omega^{2}_{\text{B}}}+\mathbf{j}\mathbf{2}(\omega/\omega_{\text{n}}) \mathbf{\xi}}, \tag{1}\]  
Figure 4: (**a**) Carrier density model of the base region; the triangular distribution of Q1 and Q2 corresponds to the capture of carriers by QW and the transition of the base region [24]. (**b**) Simplified TL carrier transport model for any number of QWs. where \(\omega_{n}\), the relaxation oscillation damping factor \(\xi\), relaxation frequency \(f_{R}\), and the relaxation peak amplitude H(\(\omega_{R}\)) are defined as follows:  
\[\omega_{n}^{2}=\frac{\Gamma\mathrm{vg}^{\prime}N_{\mathrm{ph}}}{\tau_{\mathrm{ph }}}=\frac{\eta(I_{\mathrm{B}}/I_{\mathrm{TH}}-1)}{\tau_{\mathrm{ph}}\tau_{ \mathrm{B,spon}}}, \tag{2}\]  
\[\xi=\frac{1}{2\omega_{n}\tau_{\mathrm{B,spon}}}+\frac{\tau_{\mathrm{ph}}\omega _{n}}{2}, \tag{3}\]  
\[f_{R}=\frac{\omega_{n}}{2\pi\sqrt{1-2\xi^{2}}}, \tag{4}\]  
\[|H(\omega_{R})|^{2}=\frac{A_{0}^{2}}{4(1-\xi^{2})\xi^{2}}. \tag{5}\]  
The modulation bandwidth \(\omega_{\mathrm{3 dB}}\) denotes the response frequency when H(\(\omega_{R}\)) = \(-3\) dB. The parameters involved in the formulas include normalization factor A\({}_{0}\), optical confinement factor \(\Gamma\), group velocity \(\
u\), differential gain g\({}^{\prime}\), spontaneous emission recombination lifetime \(\tau_{\mathrm{B,spon}}\), photon lifetime \(\tau_{\mathrm{ph}}\), photon number N\({}_{\mathrm{ph}}\), and slope efficiency \(\eta\). Although DML is the most commonly used light source for optical communication, its modulation capability is inherently limited by relaxation oscillation [27]. The relaxation oscillation deteriorates the modulation characteristics mainly by reducing the bandwidth \(\omega_{\mathrm{3 dB}}\) and increasing the relaxation peak H(\(\omega_{R}\)). The shorter carrier lifetime of TL [13] can improve the relaxation oscillation damping factor \(\xi\), thus increasing \(\omega_{n}\) and modulation bandwidth \(\omega_{\mathrm{3 dB}}\) while effectively reducing the relaxation oscillation resonance peak H(\(\omega_{R}\)) [23]. The increase in H(\(\omega_{R}\)) is the main reason for distortion during optical signal transmission. [PERSON] et al. used a 40 Gb/s NRZ pseudorandom bit sequence with a word length of 27-1 to simulate eye diagrams of DML and TL.