Dark Current Density
In general, for fully depleted P-i-N photodiodes, the limiting dark currents are diffusion currents in the N and P regions (depending on SRH and Auger generations) and depletion current ruled only by SRH generation in the space charge region. The influence of radiative recombination is still debatable but is not considered to be a limiting factor of small-pixel HgCdTe photodiodes. Moreover, due to the photon recycling effect, the influence of radiative recombination can be significantly reduced . For 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 a thick, undepleted absorber and is dependent on the Auger and SRH generation in n-type semiconductors 
where q stands for the electron charge, n is the electron concentration, tdij- is the diffusion region thickness, и, is the intrinsic carrier concentration, rAl is the Auger 1 lifetime, and rSRH is the SRH lifetime. Auger 1 lifetime is related to the hole, electron, and intrinsic carrier concentrations, and xM is given by the equation:
where p is the hole concentration and тд, is the intrinsic Auger 1 lifetime.
For a low-temperature operation or a non-equilibrium active volume, when the majority carrier concentration is held to be equal to the majority carrier doping level [and intrinsically generated majority carriers are excluded (p«nxNdop)], Eq. (3.28) becomes:
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), rSRH is given by
where тпо and tpo are the specific SRH lifetimes. At low temperatures, where n > n(, we have tsrh я тро. At high temperatures where n « и,, we have
TSRH~'rno + Tpo■ F°r a non-equilibrium active volume, rSRH~(tll0 + r^njn exhibits a temperature dependence given by
A comparison of the Auger 1 and SRH lifetimes in equilibrium and non-equilibrium modes is shown in Fig. 3.7 for MWIR (At. = 5 pm) and LWIR (Ac= 10 pm) HgCdTe with n-type doping concentration of 5 X 1013 cm-3. In calculations, the Hansen and Schmit analytical expression for the intrinsic carrier concentration was used . The strong increase in SRH lifetimes for the non-equilibrium mode of operation at high temperatures is caused by the decreasing electron population of the SRH level, which, in consequence, results in the decrease of the minority carrier capture rates .
The second component is the depletion current arising from the portion of the absorber that becomes depleted. The depletion current density can be estimated by the following expression:
where tdep is the width of the depletion region.
The P-i-N HOT photodiode is characterized by useful properties at a reverse-biased operation. Figure 3.8 shows the calculated reverse-biased voltage which is required to completely deplete a 5-pm thick absorber doped at different doping levels. For the Rule 7 doping range of about 1015 cm'3,
FIGURE 3.7 Equilibrium and non-equilibrium Auger 1 and SRH minority carrier lifetimes versus inverse temperature for MWIR and LWIR HgCdTe with n-type doping concentration of 5 X 1013 cm J.
FIGURE 3.8 Calculated reverse-biased voltage versus doping concentration required to deplete a 5-pm thick MWIR HgCdTe absorber. Inset: Width of absorber depletion versus reverse-biased voltage and doping concentration.
a 5-pm 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 range of doping reached presently at Teledyne (about 1013 cm 3), the 5-pm thick absorber can be fully depleted for reverse bias from zero up to 0.4 V.
If P-i-N photodiode operates under reverse bias, the Auger suppression effect should be taken into account. This effect is important under HOT conditions, when n,» Ndop. At non-equilibrium, large numbers of intrinsic carriers can be swept-out the absorber region. It is expected that this impact is larger for lower n-doping levels, since я, will be proportionately higher under these conditions. At very low levels of n-type doping (about 1013 cm'3), the ultimate performance of P-i-N photodiode is limited by SRH recombination and neither Auger recombination nor Auger suppression.
As is shown in Fig. 3.9, for sufficiently long SRH-carrier lifetime in HgCdTe, the internal photodiode current is suppressed, and the performance is limited by the background radiation. The current density is shown at four background temperatures: 300, 200, 100, and 50 K. In , it is proposed to replace “Rule 07” with “Law 19”. Law 19 corresponds exactly to the background-limited curve at room temperature. The internal
FIGURE 3.9 Current density of p-on-n HgCdTe photodiodes versus 1/(Л,.Т) product (adapted after Ref. ). Experimental data are gathered from Teledyne and alternative technologies.
photodiode current can be several orders of magnitude below Rule 07 in dependence on a specific cutoff wavelength and operating temperature. It can also be seen that Rule 07 coincides well with a theoretically predicted curve for Auger-suppressed p-on-n photodiode with doping concentration in an active region equal to Nd = 1015 cmr3.
Figure 3.9 collates published experimental data for p-on-n HgCdTe photodiodes (Teledyne)  and for alternatives to HgCdTe material systems like III-V barrier detectors (Raytheon  and SCD ), operated at about 80 K, and room temperature inter-band quantum cascade infrared photodetectors IB QCIP . It is easy to notice that experimental values for III-V barrier detectors are slightly poorer in comparison with those from p-on-n HgCdTe photodiodes, but III-V IB QCIPs operated at 300 К perform even better in the long wavelength spectral region. Figure 3.9 also shows representative data for both InSb (Ac=53 pm, T= 78 K) and InGaAs ((2f = 1.7 and 3.6 pm, T=300 K) photodiodes. The InSb detector is characterized by several orders of magnitude higher dark current density than the HgCdTe one, although, for optimal InGaAs photodiodes, the dark current density is close to HgCdTe data .
Theoretical simulations presented in Fig. 3.9 show that the background limitation has the greatest impact on photodiode current density for small 1/(ДCT) products, in other words, for photodiodes operated at the long wavelength region and under high operating-temperature conditions. HgCdTe photodiodes operated at low temperatures become generation- recombination-limited due to the influence of SRH centers having lifetimes in the millisecond range.
Figure 3.10 shows the current density calculated using Rule 07 (determined for diffusion limited P-on-n photodiodes) or Law 19 (which is exactly equal to the background radiation current density) as a function of temperature for the SWIR (3 pm), MWIR (5 pm), and LWIR (10 pm) absorbers.
If the fully depleted P-i-N photodiode is to be limited by the background radiation current, a certain minimal value of SRH lifetime is required. The calculations of SRH lifetime were made for which the depletion dark current is equal to the background radiation current:
FIGURE 3.10 Calculated current density versus temperature relationship, using Law 19 and Rule 07 for SWIR (3 pm), MWIR (5 pm), and LWIR (10 pm) HgCdTe absorber.
It was assumed that the 5-pm thick absorber is fully depleted.
The SRH lifetime, at which the fully depleted P-i-N photodiode reaches the BLIP limit, is presented in Fig. 3.11. As we can see, the SRH lifetime required to reach the BLIP limit decreases with increasing temperature, although fully depleted P-i-N photodiodes are particularly interesting under HOT conditions. Furthermore, for LWIR detectors, achieving BLIP performances is possible for shorter carrier lifetimes. At 300 K, these carrier lifetimes are 15 ms for SWIR (3 pm), 150 ps for MWIR (5 pm), and 28 ps for LWIR (10 pm) 5-pm thick fully depleted absorbers.
The Teledyne experimentally measured SRH lifetimes, extracted at 30 К for 10-pm cutoff HgCdTe, are reproducibly greater than 100 ms . Despite the fact that the lifetimes at 300 К are likely to be a minimum of
FIGURE 3.11 The SRH lifetime versus temperature relationship, for which the depletion dark current of the fully depleted P-i-N HgCdTe photodiode is equal to the background radiation current. The calculations are carried out for SWIR (3 pm), MWIR (5 pm), and LWIR (10 pm) absorbers.
10 times lower (due to the increase in thermal velocity, which increases the probability of carrier capture by a recombination center), there are still values of SRH lifetimes which enable it to reach the BLIP limit. This prediction is supported by theoretical simulation presented in .
The photodiode D' is specified on the basis of current responsivity, R(, and noise current, i„, and can be written as:
For the non-equilibrium devices, the i„ value can be calculated including thermal Johnson-Nyquist noise and shot noise, using the following expression
where к is the Boltzmann constant, RdA is the dynamic resistance area product, and Jdark is the dark current density.
The performance of P-i-N MWIR HgCdTe photodiode (Ac = 5 pm) is presented in Fig. 3.12. Figure 3.12(a) shows the diffusion and depletion dark current components versus temperature relationship, assuming the value of the SRF1 carrier lifetime to be 1 ms, with an absorber thickness of 5 pm and doping of 5 X 1013 cm'3. For this doping level, a 5-pm thick absorber can be fully depleted at the reverse bias of 0.4 V (see Fig. 3.8). The diffusion component associated with the Auger 1 mechanism is eliminated because of the absence of majority carriers due to exclusion and extraction effects [23,24]. The background radiation calculated from //3 optics has a decisive influence on the dark current. It should be mentioned here that the background flux current is defined by the total flux through the optics (limited by//#), plus the flux from the cold shield. This effect is shown by increasing influence of the background-limited performance (BLIP) (//3) on dark current at temperatures above 220 K.
As is shown in Fig. 3.12(a), the Teledyne Judson experimentally measured current densities, at the bias of -0.3 V, are close to the BLIP (fl3) curve, being located less than one order of magnitude above this limit.
FIGURE 3.12 Performance of MWIR P-i-N HgCdTe photodiode with the value of rSRH= 1 ms and absorber doping level of 5 X 1013 cm 3: (a) diffusion and depletion current components versus temperature, (b) detectivity versus temperature. The thickness of active region is t = 5 pm and consists of tdij= 2 pm and tdep = 3 pm. The experimental data are taken from different sources as marked. PV - photodiode; CQD - colloidal quantum dot (CQD); IB QCIP - inter-band quantum cascade infrared photodetector.
The current density at room temperature is even lower than that predicted by Rule 07. The measured current densities presented by VIGO are about one order of magnitude higher, although, in this case, they were measured at lower reverse bias, -0.1 V, with a less effective contribution from Auger suppression. It is interesting to note that the performance of IB QCIPs, based on type-II superlattices (T2SLs) fabricated with InAs/GaSb, coincide well with the upper experimental data for HgCdTe photodiodes at room temperature .
Figure 3.12(b) shows calculated detectivity versus temperature relationship for a MWIR P-i-N HgCdTe photodiode, assuming identical parameters taken in calculation, as presented in Fig. 3.12(a) (Ac = 5 pm, tsrh= 1 ms, t = 5 pm, Ndop = 5 X 1013 cm 3). The current responsivity was calculated, assuming quantum efficiency (QE) = 1 (although typical QE reaches a reasonable value of about 0.7). As is shown, for a MWIR photodiode with a 5-pm cutoff wavelength and low doping in the active region, D' is limited by background and is about one order of magnitude higher than predicted by Rule 07. The experimental data given for HgCdTe photodiodes in Teledyne Judson and VIGO catalogues are more than one order of magnitude below the background flux limitation for the//3 optics.