Exploring Optimal Dark Current Design in HgCdTe Infrared Barrier Detectors: A TCAD and Semianalytic Investigation

The dark current is a fundamental figure of merit to characterize the performance of high-sensitivity, low-noise mid- and far-infrared barrier photodetectors. In the context of HgCdTe barrier photodetectors, the trend is to use very low doping concentrations, in an attempt to minimize recombination processes. In the present work, through TCAD simulations, we delve deeper into the design of low-dark-current <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula>B<inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> detectors, showing the possible existence of an optimum doping. This occurrence is investigated and interpreted also by means of closed-form expressions for the lifetimes, emphasizing the role of the interplay between Auger and Shockley-Read-Hall generation processes.

dark current in Hg 1−x Cd x Te IR detectors.In a few decades, the outstanding performance of this ternary alloy has enabled the development of four generations of large format IR detectors for imaging in space science, environmental monitoring, diagnostics, surveillance, security, defense, etc. [6], [8], [9], [10], [11], [12], [13], both for mid-wavelength (MWIR, λ ∈ [3,5] μm) and long-wavelength (LWIR, λ ∈ [8,12] μm) infrared bands.Rule 07 was originally intended for P -on-n photodiodes, but soon has been adopted as a reference by technologies other than the HgCdTe-based ones, such as type-II superlattice (T2SL) devices [14], [15], [16], quantum dot photodetectors [17], [18], [19], and devices based on two-dimensional materials [20]. 1  The dark current is largely determined by minority carrier lifetimes [8], [21], [22].After a summary of the concepts of dark current, generation rates, and lifetimes, Section II describes barrier detectors [23], [24], [25], [26], [27], [28], one of the popular solutions for fabricating photodetectors with low dark current and HOT performance.Focusing on one of their possible variants, Section III presents technology computer-aided design (TCAD) simulations of the dark current for a MWIR photodetector.Visualizing the dark current as a function of the absorber doping density reveals the existence of an optimal doping.This result indicates that the race to achieve extremely low residual doping in the absorber, required, for example, in fully-depleted detectors [29], [30], [31], is probably not an appropriate technological target for barrier photodetectors.This is an important point of interest to manufacturers, since the optimal doping can lower the dark current to values comparable to Rule 07.It is to be remarked that a reduction in dark current leads to a reduction of the NEDT (Noise Equivalent Difference Temperature) and increases the specific detectivity D * .The NEDT and D * definitions include the quantum efficiency, the dark current and its effect on thermal noise, and are the figures of merit most adopted in the infrared community to quantify the photodetector performance [1].
Aiming to provide a strong, yet simple physical interpretation of such optimal doping, Section IV presents three popular 1 Regarding P -on-n, P + nN + and similar strings, the conventional notation indicates with a capital P or N an acceptor-or donor-doped layer, respectively, with bandgap wider than the nor p-doped absorber; a "plus" superscript indicates a high doping level.approximations to obtain n and p as function of T , N D and N A (the absolute temperature, the donors and acceptors concentrations, respectively).These expressions can be plugged into the generation rates and diffusion currents discussed in Section IV to achieve estimates of the dark current.Although not universal, these approximations allow, as discussed in Section V, to interpret the optimal doping as a result of the interplay between the Auger and Shockley-Read-Hall (SRH) carrier generation processes, and to provide design guidelines for barrier detectors.Finally, Section VI summarizes the main results.

II. PHOTODETECTORS, LIFETIME AND DARK CURRENT:
GENERAL CONCEPTS Conventional photodetectors are conceptually not so different from a pn-diode operated in reverse bias, for which most of the detector volume consists of two quite thick, pand n-doped, quasi-neutral regions (QNRs), where the electric field appears negligible, whereas the space-charge region across the junction is normally much thinner (for precise definitions of QNRs and depleted regions see [32,).
This work is focused on a different kind of photodetectors, i.e., on barrier detectors [24], [25], [33], [34].Among the many variants of this class of devices, Fig. 1 represents a pBn detector: a thin layer of p-doped cap material, followed by a thin barrier of high bandgap material (represented by "B"), and finally the absorber, a layer of n-doped material whose thickness is in the order of the operating wavelength λ, therefore a few micrometers.Such detectors are also known as unipolar barrier detectors because, when a negative voltage is applied to the cap contact, the barrier blocks the electrons (in this case), but permits unimpeded flow of the absorber minority carriers (holes) towards the bias contact.
It is worth reminding that the dark current of a photodetector depends on the reverse bias voltage V bias .In general, J dark does not coincide with J diff , and should be evaluated using a numerical simulator [32,Ch. 1.8].In fact, the absorber can be partially depleted to an extent which depends on V bias , and consequently it behaves only partially as a QNR.Furthermore, other contributions to J dark come, e.g., from SRH generation in the barrier.Although J diff and J dark are different quantities, it will be interesting to compare their behavior against the absorber doping, when J dark is simulated at a moderate reverse bias of the order of a few hundred millivolts.
The dark current receives contributions mainly from Auger, SRH, and radiative generation processes.Auger and SRH generation rates are given by [1], [24], [31], [35], [36], [37], [38], [39] G Auger = (n + p) respectively, where τ i A1,A7 are the intrinsic A1, A7 Auger lifetimes [40], n i is the intrinsic density, and τ SRH is the SRH lifetime.G Auger can also be written in the alternative form which allows to single out the nand p-dependent expressions for the Auger lifetimes, The SRH lifetime τ SRH is only related to the material defect density and carrier trapping cross sections [41], and thus it may be considered a technology-dependent parameter, not associated with any fundamental process.We do not discuss here the contribution of radiative generation-recombination processes, since it depends on the detector geometry [42].However, when appropriate, the concepts treated in this work can be extended including radiative recombinations without changing the general message conveyed by this study.
When a QNR is part of a photodetector, carriers generated by Auger and SRH processes diffuse towards the junction, where they eventually drift and, ultimately, are collected by the contacts.Hence, they are said to form a diffusion current J diff originating from the QNR and concurring to build the dark current.In general, J dark does not coincide with J diff , and should be evaluated using a numerical simulator [32,Ch. 1.8].
It is worth reminding that the dark current of a photodetector depends on the reverse bias voltage V bias .In fact, the absorber can be partially depleted to an extent which depends on V bias , and consequently it behaves only partially as a QNR.Furthermore, other contributions to J dark come, e.g., from SRH generation in the barrier.Although J diff and J dark are different quantities, it will be interesting to compare their behavior against the absorber doping, when J dark is simulated at a moderate reverse bias of the order of a few hundred millivolts.J diff can be written as where q is the elementary charge and t the QNR thickness.Equation (4) shows that the Auger lifetime is inversely proportional to the square of the majority carriers density, hence Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.J diff, Aug increases proportionally to the latter, according to (1).Conversely, (2) shows that J diff, SRH decreases as the carrier density increases, and these two opposite behaviors have the interesting consequences discussed in the next sections.It can be kept in mind, however, that contributions to dark current other than Auger, SRH and possibly radiative can be at play (tunneling, surface recombinations, etc.).Their relative importance depends on the details of the detector composition and geometry, on the material quality, and on the technology adopted for its fabrication.For a discussion see, e.g., [43,Ch. 5.3.2].

III. DARK CURRENT IN BARRIER DETECTORS: LOOKING FOR OPTIMAL DOPING
We consider a simple one-dimensional model of a pBn photodetector (more precisely, a planar structure with a p + B p nN + doping scheme), whose doping and composition profiles are shown in Fig. 2. The layer stack is composed as follows: above a 0.5 μm-thick N + buffer layer (Hg 0.45 Cd 0.55 Te, N D = 2 × 10 17 cm −3 ) are a 5 μm-thick absorber (Hg 0.72 Cd 0.28 Te, donor-doped), followed by a 0.2 μm-thick barrier layer (Hg 0.38 Cd 0.62 Te) and a 0.3 μm-thick cap layer (Hg 0.7 Cd 0.3 Te), both acceptor-doped with N A = 1 × 10 17 cm −3 .Doping the barrier with acceptors helps reducing the barrier in the valence band (important to efficiently collect the photogenerated minority carriers in the absorber), increasing in turn the barrier in the conduction band [33], [44].The absorber doping varies in the interval N D ∈ [10 14 , 10 17 ] cm −3 , the operating temperature in T ∈ [120, 200] K, and the SRH lifetime in τ SRH ∈ [10 −1 , 10 2 ] μs.The absorber cutoff wavelength is around 4.8 μm at 160 K, and it is the same material considered in [24,Fig. 24], where the authors investigated its properties with approximated expressions similar to (5).Between the layers, composition and doping varies smoothly to increase realism.Possible effects of annealing, that, in nBn detectors, could contribute to smoothing the barrier composition profile to a larger extent, undermining the detector performance [45], have not been considered, since the adopted p + B p nN + profile is less affected by this detrimental effect [46].Finally, since at the upper limit of the MWIR band the absorption length is about 3 μm at 160 K and about 5.7 μm at 200 K, a 5 μm-thick absorber should provide efficient absorption, without overly increasing the thermal carriers generation, which is proportional to the absorber volume.
This device is simulated with a numerical TCAD analysis tool, based on solving the quasi-static Poisson's equation self-consistently with the continuity equations of electrons and holes, the latter being closed with the drift-diffusion constitutive relations.A commercial simulation suite is employed, which is based on discretizing the differential operators with a stabilized finite-box method, and allows for great flexibility in the grid definition, enabling local refinements for critical regions (such as junctions between different materials and/or doping concentrations) [47].Simulations take into account Fermi-Dirac statistics and incomplete dopant ionization, and employ temperature-and mole fraction-dependent material parameters according to [48] (Table I).The transport problem includes the SRH and Auger generation-recombination processes (modeled as in [41] and [49], respectively) and provides the steady-state solution at equilibrium and under reverse bias.A key difference vs. pnphotodetectors is that barrier detectors are quasi-unipolar and therefore do not require trap-assisted and band-to-band tunneling models [41], [50], [51], [52].In addition, the electron barrier is thick enough to prevent any contribution from intraband tunneling.
Fig. 3(a) shows the band diagrams at V bias = −0.2V, which indicates that only the electrons are blocked by the barrier, as required by a unipolar photodetector, while under illumination the photogenerated minority carriers -holes -are easily collected by the bias contact.
First, we simulated a set of JV characteristics for several values of N D in the absorber, varying the bias voltage from the equilibrium to −0.2 V. We obtained a quite standard behavior, since the JV curves reach a sort of plateau, typical of diffusive, bias quasi-independent regime.However, importantly and in some way unexpectedly, the dark current density depends non-monotonically on N D , as visible in Fig. 3(b).
Aiming at emphasizing this trend, Fig. 3(b) reports J dark versus N D at 160 K and fixed reverse voltage, for two values of τ SRH .As a benchmark, we also report the expected dark current density according to Rule 07 for this temperature.The non-monotonic behavior clearly indicates the presence of an optimal absorber doping.
With this encouraging result, we managed to find a useful figure of merit to be plotted in the 2D domains described at the beginning of this section, aiming at characterizing the location of the minimum of J dark .Since by changing the temperature and the SRH lifetime the current density varies of several orders of magnitude, the position of its minimum is most visible by plotting the logarithmic derivative of J dark instead of J dark itself, that is in the domain (T, N D ) for given τ SRH and in the domain (τ SRH , N D ) for given T .These results are shown in Fig. 4, where the large dark blue bands in the two panels identify the vanishing of the J dark derivative, i.e., the path followed by the Color maps such as those in Fig. 4 provide a vivid representation of the 2D parameter space and some qualitative clues, such as the "optimal path" indicated by the dark blue bands.However, much more quantitative information is provided by 1D cuts, such as those shown in Fig. 3(c), which provide direct indications of the sensitivity of the dark current to the device parameters.For example, Fig. 3(c) not only indicates that, e.g., for τ SRH = 10 μs, the optimal absorber doping is about 3 × 10 15 cm −3 , but also that a doping reduction to 10 14 cm −3 would make the dark current to increase by a factor of five, an effect that would thwart all the other efforts to obtaining high sensitivity photodetectors.
A further remark is in order, when discussing the available technology.The possibility to obtain minimal dark current for an intermediate doping, rather than for the lowest technologicallyachievable doping as in the fully-depleted detectors, is an interesting result whenever the employed technology does not allow to reduce the SRH-related defect density [32], [41] to values so low to provide τ SRH in the orders of milliseconds, as in the best fully-depleted detectors [31].In more common cases, where τ SRH is in the order of tens of microseconds as in Fig. 3(c), the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
dark current reaches its minimum value when the absorber doping is between 10 15 and 10 16 cm −3 , an undoubtedly interesting value that can be achieved and managed with ease [53].

IV. CHARGE CARRIER DENSITY IN QNR
When the absorber occupies the majority of the volume of the photodetector, it is often assumed in the literature [8], [21], [24] that the J diff originated in the absorber, treated as QNR, is a reliable estimate of the total detector J dark .Although this approach is not universally predictive, if the given constraint on the QNR thickness is verified, the general trend is sufficient to interpret the origin of the optimal doping.In this way, the simulation of the photodetector performance can be carried out by evaluating independently n and p, to be plugged in the generation rates described in Section II and then in the estimations of J diff .
In the literature, three approximated formulations of increasing complexity can be found: 1) simplest approximation (SIM), where it is assumed n = N D , p = 0 in donor-doped regions n = 0, p = N A in acceptor-doped regions in all generation-recombination rates and lifetimes expressions.It is an approximation adopted, e.g., in [8], [54], [55] 2) the full dopants ionization (FullDI), where n and p follow from the mass-action np = n 2 i and the charge neutrality n + N A = p + N D equations, which can be reformulated as and which include a dependence on temperature through n i 3) the incomplete dopants ionization (IncDI), where n and p are selfconsistently calculated from r the charge neutrality equation n whose expressions as function of N D , N A and T can be found in textbooks, (see, e.g., [32,Ch. 1.4]), assuming for dopants activation energies the values reported, e.g., in [1], [56] (≈ 1 meV for donors and ≈ 20 meV for acceptors).The accuracy of the estimates for n and p provided by SIM, FullDI, and IncDI varies and generally depends on temperature and doping.It must be noticed that the SIM approximation ignores the mass-action law.Consequently, in donor-doped semiconductors, it approximates p to zero instead of n 2 i /N D .This can be a sort of simplification, and it can be acceptable only when N D n i , which strongly depends on temperature (e.g., for MWIR when T is below ≈ 200 K).
Fig. 5 shows an example of calculation for a donor-doped Hg 1−x Cd x Te QNR with x = 0.294, for T = 240 K and N D ∈ [10 12 , 10 18 ] cm −3 .It is apparent that: r for N D >> n i , SIM and FullDI are quasi-equivalent: the calculated values of n are very similar, and FullDI estimates a value for p that is several orders of magnitude lower than n, hence negligible, at least with respect to it; r for high N D , e.g., N D ≈ 10 18 cm −3 , IncDI is the only acceptable approximation, as it provides an estimation for n which is one order of magnitude lower than that of the rougher FullDI and SIM, and the difference in some applications cannot be ignored; r for N D << n i , the SIM approximation is not appropriate and provides wrong results both for n and p, since in a QNR n can never be lower than n i .Instead, FullDI and IncDI are almost equivalent.

V. LITERATURE EXAMPLES: LIFETIMES AND DIFFUSION CURRENTS
Before investigating the origin of the optimal doping, it is important to validate the expressions in Section II by reproducing examples of calculation taken from the literature, comparing the results provided by the three approximations.

A. Example 1
As a first example, we have tried to reproduce the Auger lifetimes presented in [8, Fig. 3] calculating τ A1 and τ A7 at T = 77 K, respectively for n-doped and p-doped long-wavelength infrared (LWIR) Hg 1−x Cd x Te QNRs with x = 0.23, corresponding to a cutoff wavelength λ c = 9.5 μm at this temperature.Fig. 6(a) shows that the SIM and FullDI approximations are equivalent at this temperature (the curves overlap) and reproduce successfully the results from the reference, which are also included in the same figure.At high doping densities, the effects of incomplete ionization of the dopants cannot be neglected, and indeed the IncDI curves diverge progressively: for N D ≈ 10 18 cm −3 the deviation is of several orders of magnitude, and for this temperature the IncDI approximation would have been more appropriate.Fig. 6(b) shows that, conversely, as the temperature is increased toward room temperature, the effect of incomplete dopant ionization becomes increasingly less important and FullDI is a safe option.Instead, the SIM approach becomes completely inapplicable, as Fig. 5 had already indicated.

B. Example 2
As a second case study, we focus on the n-doped and pdoped mid-wave infrared (MWIR) photodetectors at T = 240 K from [8,Fig. 4], which are realized with Hg 1−x Cd x Te, x = 0.29, having, at this temperature, cutoff wavelength λ c = 4.6 μm.The results of our calculations are reported in Fig. 6(c), which also includes the Auger lifetimes estimated from the Auger diffusion currents reported in [8] (solid dots).It is apparent that the SIM approximation accurately reproduces the data taken from the cited reference.As for the effects of incomplete dopant ionization, we see that at this relatively high temperature the IncDI deviates only minimally from the FullDI, and only in the direction of the highest dopant value, so there is no obvious need for its use.
A further remark about the SIM approximation: Fig. 6(b) shows that for N D = 10 14 cm −3 and T > 200 K the Auger lifetime τ A1 calculated in the SIM approximation deviates by several orders of magnitude with respect to FullDI or IncDI, and the relative deviation increases rapidly with T .This because n i rapidly increases with T according to [32,Ch. 1.4] (E g is the material energy gap and k B is the Boltzmann constant), and in the SIM approximation it is τ A1 ∝ (n i /N D ) 2 , where N D is fixed.The obtained lifetime is unphysical: for comparison, when T tends to room temperature, the Auger lifetimes obtained by the FullDI and IncDI approximations correctly tend to the intrinsic lifetime τ i A1 .All this reflects on the calculation of the Auger diffusion current as shown in Fig. 6(c): the SIM approximation greatly underestimates J diff, Aug because it overestimates τ A1 .

C. Example 3: The Origin of the Optimal Doping
The third example is engineered in order to shed light on the physical origin of the optimal doping.Specifically, we attempted to reproduce the total diffusion current according to (5) for a 3 μm-thick QNR with composition, temperature and doping values as in [24,Fig. 24], i.e., n-doped MWIR Hg 1−x Cd x Te with x = 0.298, and T = 160 K (at this temperature λ c = 4.8 μm).This is the same material employed as absorber layer in the driftdiffusion simulations described in Section III (see in particular Fig. 3(c)).
We consider three values of τ SRH , representative of mediumquality material, i.e., 0.1, 1 and 10 μs, as in [24,Fig. 24].Fig. 7 was obtained using the IncDI approximation, achieving a good agreement with the behavior in the reference.Corresponding results obtained with FullDI and SIM, not reproduced here for brevity, show very similar behavior, including the SIM approximation.The reason is the following: around the lowest considered dopant concentration, it is τ SRH τ A1 , i.e., the material is SRH-limited.Consequently, J diff is dominated by J diff, SRH , which at this temperature depends on the adopted approximation to a very limited extent, in contrast to the scenario represented by Fig. 6(c).
It is important to remark that the optimal dopant concentration is approximately the same found when simulating the pBn photodetector with the TCAD numerical simulator.This is an important point, which was already remarked in [24], although without a detailed investigation.Fig. 7 clearly shows that the origin of the optimal doping is the interplay between the Auger and SRH generation, as anticipated when commenting (1)-( 2).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.It is important to note that, although for τ SRH = 10 μs the value of the optimal doping obtained by J dark (N D ) and J diff (N D ) is approximately the same, by contrast the minimum of J dark is an order of magnitude larger than the corresponding minimum of J diff .This confirms a partial depletion of the absorber -as also indicated by the band bending in Fig. 3(a) -which increases the contribution of SRH to dark current with respect of what calculated for an ideal QNR.

VI. CONCLUSION
This work presents an investigation about lifetime and dark current in HgCdTe IR photodetectors.In the first part we performed a set of numerical simulations with a semiclassical TCAD simulation suite for IR barrier detectors.The dark current presents an interesting non-monotonic behavior.In the parameter spaces (T, N D ) and (τ SRH , N D ) a region of optimal doping can be seen at a glance, allowing to minimize the dark current for a given temperature and material quality (the latter represented by τ SRH ).Figs. 3 and 4 show that, at least for the considered material, the optimal doping lies in the interval [10 15 , 10 16 ] cm −3 ; such values can be achieved by most manufacturers.Actually, considering that fully-depleted detectors require absorbers with ultra-low residual doping (N D ≈ 10 13 cm −3 ), this finding strengthens the idea that barrier photodetectors can represent a useful alternative whenever the available technology cannot provide materials with such record characteristics.
An important topic not addressed in the present work is represented by the effects of doping, thickness and molar composition of the barrier and cap layers.These points deserve a dedicated investigation, since these parameters could affect both dark current and quantum efficiency.An extension of the present investigation towards the exploration of their possible effects on the optimal doping in pB p n detectors is important and can be possibly the object of a future work.However, for a discussion concerning their effects on nBn detectors, see, e.g., [45].
In the second part of this work, aiming at achieving deeper insight into this property, we looked for a possible occurrence of an optimal doping targeting the minimum diffusion current generated in a QNR of doped HgCdTe.To this end, first we provided a critical review of three possible approximations employed in the literature to obtain the values of carrier density in HgCdTe based IR photodetectors.Each of them provides a different estimate of carrier lifetime and photodetector dark current, and the relative difference has been found to depend on temperature and doping concentration in a significant way.In some cases, such discrepancies cannot be neglected.Specifically, the SIM approximation tends to lose its validity when the doping is low and temperature is relatively high (see, e.g., Fig. 6(b) and (c) corresponding to 240 K), whereas it remains valid and in practice coincides with the FullDI when the temperature is particularly low (see, e.g., Fig. 6(a) corresponding to 77 K).Then, we calculated the diffusion current J diff generated in a donor-doped quasi-neutral (or field-free) region as function of doping, having considered the same HgCdTe composition adopted for the TCAD simulations.J diff has been found to have a non-monotonic behavior very similar to J dark , and this behavior is shown to be related with the interplay between the Auger and SRH generation rates.

Fig. 2 .
Fig. 2. Doping (a) and composition (b) profiles of the p + B p nN + barrier photodetector simulated in Section III, with a detail (c) of the computational grid across the barrier layer.

Fig. 3 .Fig. 4 .
Fig. 3. (a) Band diagram V bias = −0.2V, (b) J dark − V characteristics for τ SRH = 10 μs and two values of T (160 and 200 K).(c) J dark at fixed reverse bias as function of the absorber doping, for T = 160 K and two values of τ SRH (10 μs and 100 μs).A minimum for J dark at fixed reverse bias is clearly visible.

Fig. 5 .
Fig. 5. Carrier density evaluated in the SIM, FullDI and IncDI approximations, for donor-doped Hg 1−x Cd x Te with x = 0.294, for T = 240 K. Notice that in the SIM approximation a donor-doped material has always p = 0.