# Trends in Performance Limits of the HOT Infrared Photodetectors
## 3 The Ultimate HgCdTe Photodiode Performance
### Dark Current Density  
In general, for the fully depleted P-i-N photodiode, the current is built by diffusion from N and P regions (that depends on SRH and Auger generations) and depletion current only determined by SRH generation in the space charge region. Influence of radiative recombination is still debatable but is not considered as a limiting factor of the small pixel HgCdTe photodiodes. Moreover, due to the photon recycling effect, the radiative recombination contribution can be significantly reduced [27]. By that reason in our discussion the radiative recombination is omitted. The diffusion current of P-i-N HgCdTe photodiode structure arises from the thermal generation of carriers in thick, non-depleted absorber and is dependent on the Auger and SRH generation in n-type semiconductor [20]  
\[J_{dif}=\frac{qn_{i}^{2}t_{dif}}{n}\bigg{(}\frac{1}{\tau_{A1}}+\frac{1}{\tau_ {SRH}}\bigg{)}, \tag{4}\]  
where \(q\)--Electron charge, \(n_{i}\)--Intrinsic carrier concentration, \(n\)--Electron concentration, \(t_{dif}\)--Diffusion region thickness, \(\tau_{A1}\)--Auger 1 lifetime, and \(\tau_{SRH}\)--SRH lifetime. Auger 1 lifetime relates to the hole, electron, and intrinsic carrier concentrations, and \(\tau_{A1}\) is given by equation  
\[\tau_{A1}=\frac{2\tau_{A1}^{i}n_{i}^{2}}{n(n+p)}, \tag{5}\]  
where \(p\)--Hole concentration and \(\tau_{A1}\)--Intrinsic Auger 1 lifetime. For a low temperature operation or a non-equilibrium active volume, when the majority carrier concentration is held equal to the majority carrier doping level [and intrinsically generated majority carriers are excluded (p << n \(\approx\) N\({}_{\text{dop}}\))], Equation (5) becomes  
\[\tau_{A1}=\frac{2\tau_{A1}^{i}n_{i}^{2}}{n^{2}}. \tag{6}\]  
Figure 2: P-i-N detector: (**a**) energy band profile under reverse bias, (**b**) heterojunction architecture (adapted after reference [26]). The shortest SRH lifetime occurs through centers located approximately at the intrinsic energy level in the semiconductor bandgap. Then, for the field-free region in an n volume (\(n>>p\)), \(\tau_{SRH}\) is given by  
\[\tau_{SRH}=\frac{\tau_{no}n_{i}+\tau_{po}(n+n_{i})}{n}, \tag{7}\]  
where \(\tau_{no}\) and \(\tau_{po}\)--specific SRH lifetimes. At low temperatures, where n \(>n_{i}\), we have \(\tau_{SRH}\approx\tau_{po}\). At high temperatures where n \(\approx\) n\({}_{i}\), we have \(\tau_{SRH}\approx\tau_{no}+\tau_{po}\). For a non-equilibrium active volume, \(\tau_{SRH}\approx(\tau_{no}+\tau_{po})n_{i}/n\) exhibits a temperature dependence given by n\({}_{i}\). The second component is the depletion current arising from the portion of the absorber that becomes depleted. The depletion current density can be assessed by the relation  
\[J_{dep}=\frac{qn_{i}t_{dep}}{\tau_{no}+\tau_{po}}, \tag{8}\]  
where \(t_{dep}\)--Depletion region thickness. The P-i-N HOT detector is characterized by useful properties at reverse voltage. Figure 3 shows the calculated reverse voltage required to completely deplete a 5-\(\upmu\)m-thick absorber for selected doping level. For the Rule 7 with doping range about 10\({}^{15}\) cm\({}^{-3}\), a 5-\(\upmu\)m-thick absorber can be fully depleted by applying a relatively high reverse bias between 10 V and 30 V. On the other hand, for the doping level reached presently at Teledyne Technologies (-10\({}^{13}\) cm\({}^{-3}\)), the 5-\(\upmu\)m-thick active layer can be fully depleted for reverse bias from zero up to 0.4 V.  
If P-i-N photodiode operates under reverse bias, Auger suppression effect should be considered. This effect is important in HOT condition when n\({}_{i}>>\) N\({}_{\rm dop}\). In non  
Figure 3: Calculated reverse voltage versus doping concentration required to deplete a 5-\(\upmu\)m-thick MWIR HgCdTe absorber. Inset: absorber depletion thickness versus reverse bias and selected doping concentration. equilibrium, large number of intrinsic carriers can be swept-out of the absorber region. It is expected that this impact is larger for lower n-doping levels since n\({}_{\rm i}\) is proportionately higher.