## 2 DBQW RTD Photo-Detectors
### DC Photo-Detection Characteristics
Therefore, the RTD photo-generated current \(I_{ph}\) in response to an incident optical signal \(P(\lambda)\) can be expressed by:  
\[I_{ph}=\eta_{ph}\frac{q\lambda}{hc}\,P\left(\lambda\right) \tag{2}\]  
If the surface reflections and the finite dimensions of the RTD waveguide photo-detector are considered, then the quantum efficiency, \(\eta_{ph}\) is given by light coupling factor \(\kappa\) multiplied by the absorbance [28]:  
\[\eta_{ph}=\kappa(1-R_{ref})\left(1-e^{-\alpha(\lambda,V)_{ph}\lambda}\right) \tag{3}\]  
with \(R_{ref}\) is the waveguide facet reflectivity (sidewall imperfections and refractive index nonuniformity contribute to this kind of losses), \(\alpha(\lambda,V)\) is the waveguide core absorption coefficient at wavelength \(\lambda\) and bias \(V\), \(\gamma_{ph}\) is the overlap integral of the electric field and the optical field, and \(\Lambda\) is the waveguide photo-detector absorbing active length. This expression does not take into account with propagation losses. In Table 1 are presented the typical physical parameters of the RTD photo-detectors. For the waveguide structure described in Figure 2a, we estimate \(\gamma_{ph}\sim\)0.25 and \(\kappa\sim\)0.35 at 1.55 \(\,\mathrm{\SIUnitSymbolMicro m}\), and an absorption coefficient \(\alpha(\lambda,V)\), in the valley region, around 400 \(\mathrm{cm}^{-1}\), estimated from absorption spectra characterization reported in [11], which for devices with absorbing active length \(\Lambda\geq 150\)\(\,\mathrm{\SIUnitSymbolMicro m}\) yields a quantum efficiency \(\eta_{ph}\sim\)0.2. Finally, the external responsivity is expressed by:  
\[R_{\lambda}=\frac{\lambda}{1.24}\eta_{ph} \tag{4}\]  
that gives a responsivity of 0.25 A/W in the valley region, which compares well with the typical values measured between 0.25 A/W and 0.3 A/W for RTD-PD devices biased in the valley region (see measured responsivity in the valley region shown in Figure 3a). According to the calculations, most of the light is not coupled in the waveguide (25% coupling efficiency which gives a coupling loss of 6 dB), which seems to be the major factor limiting the responsivity. Although the responsivity of 0.25 A/W at these wavelengths (\(\sim\)1,550 nm) is low and below available commercial devices, as shown in the next section, the values reported here are sufficient to achieve an efficient optical control of the RTD microwave signals when the RTD is operating in the oscillating regime. Moreover, considering the estimated losses, we expect RTD-PD devices can provide PDs with responsivities up to 1 A/W which will further improve the efficiency of the optical control. The losses can be reduced by increasing the overlap integral of the input beam profile with the waveguide cross-section, reducing facet reflections, and design an optical waveguide with the RTD-PD active region closer to the input facet. In addition, calculations show that there would be no significant increase in the responsivity even with a much higher overlap, indicating that \(\sim\)150 \(\upmu\)m is long enough to absorb almost all light. In fact, calculations suggest there is not a substantial increase in responsivity for waveguides longer than \(\sim\)100 \(\upmu\)m. Therefore, the waveguide structure will occupy much less area, and the RTD could operate at higher frequencies. Currently, we are also investigating RTD-based surface illuminated photo-detectors which can provide advantages in terms of overlap integral of the input beam profile and coupling efficiency.