# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 3 Microcavity Lasers for Energy-Efficient Communications
Here, we have assumed a logarithmic dependence of the material gain on carrier density [23]. Furthermore, defect-assisted recombination, radiative recombination and Auger recombination have been modeled, respectively, as linear, quadratic and cubic with carrier density [23]. The simulation parameters closely reflect those of [67] and can be found therein. In particular, we have assumed \(N_{\rm well}=3\), \(d=6\,\)nm, \(w=300\,\)nm and \(\alpha_{i}=15\,\)cm\({}^{-1}\). Here, \(N_{\rm well}\) is the number of active layers, \(d\) is the thickness of each active layer, \(w\) is the active region width and \(\alpha_{i}\) is the total internal loss. Figure 5a,b emphasizes that increasing the Q-factor or optical confinement factor marginally improves the threshold current if the Q-factor is already high enough. On the contrary, the threshold current is significantly reduced by shortening the cavity length. Overall, PhC lasers have a small threshold current [54; 59; 60] because they allow one to scale down the active volume, while keeping high the Q-factor and optical confinement factor. It should be noted that heterogeneous [60; 63] and epitaxial [6] integration of PhC lasers on silicon have also been demonstrated. The energy cost, EC, is a common figure of merit encompassing the static and dynamic characteristics of semiconductor lasers in terms of energy efficiency. It is defined as \(\mathrm{EC}=P_{\rm in}/B=\left(IV_{b_{0}}+R_{s}I^{2}\right)/B\)[67]. Here, \(B=1.3f_{\rm 3 dB}\) is the data rate, which is assumed to be achievable under direct non-return-to-zero modulation [68]. \(f_{\rm 3 dB}\) is the 3 dB modulation frequency and \(P_{\rm in}=IV_{b_{0}}+R_{s}I^{2}\) is the electrical power. \(V_{b_{0}}\) is the built-in bias voltage and \(R_{s}\) is the series electrical resistance. This is related to the electrical resistivity, \(\rho_{s}\), by \(R_{s}=\rho_{s}/L\). Figure 5c illustrates the energy cost as a function of the modulation frequency for \(Q=10^{3}\) and \(\Gamma_{\rm well}=3\%\). Each color denotes a different value of the cavity  
Figure 5: Threshold current versus Q-factor for different values of (**a**) the cavity length, \(L\), and (**b**) confinement factor of a single active layer, \(\Gamma_{\rm well}\). In (**a**), \(\Gamma_{\rm well}\) is 3%, and in (**b**) \(L\) is 2.5 μm. (**c**) Energy cost versus 3 dB direct modulation bandwidth, with line colors corresponding to different cavity lengths. The electrical resistivity, \(\rho_{s}\), is \(10^{4}\,\Omega\times\
u\)m (solid) and 0 (dotted). The Q-factor is \(Q=10^{3}\) and the confinement factor of a single active layer \(\Gamma_{\rm well}=3\%\). length, with the solid (dotted) line corresponding to \(\rho_{s}=10^{4}\,\Omega\times\upmu\)[67] (\(\rho_{s}=0\)). For simplicity, we have neglected the spontaneous emission coupled to the lasing mode, similarly to [67]. The figure highlights that an optimum modulation bandwidth exists which minimizes the energy cost, as also noted in [62]. An optimum occurs irrespective of \(R_{s}\), although a smaller electrical resistance obviously reduces the energy cost. The trend can be easily understood by assuming a low damping factor, leading to \(f_{\rm 3 dB}\approx\frac{\omega_{R}}{2\pi}\sqrt{1+\sqrt{2}}\)[23], with \(\omega_{R}\) being the relaxation resonance frequency. In this case, by neglecting for simplicity the electrical resistance, one may express the energy cost as  
\[\text{EC}\approx\bigg{(}\frac{I_{\text{th}}}{B}+\frac{B}{\sigma}\bigg{)}V_{b_{ 0}} \tag{6}\]  
with \(\sigma=\frac{1.69}{4\pi^{2}}\Big{(}1+\sqrt{2}\Big{)}\frac{\eta\cdot\upmu\upmu N _{\text{sub}}}{qV_{\rm act}}v_{g}g_{N}\) being a constant independent of the bias current. This equation reveals that at low data rates the power consumption is dominated by the threshold current, with the energy cost being inversely proportional to the data rate. However, as larger data rates are demanded, the current required in excess to threshold increases and gradually dominates the power consumption.