Study on the Theoretical Limitation of the Mid-Infrared PbSe N^{+}-P Junction Detectors at High Operating Temperature ()

Xinghua Shi^{1*}, Quang Phan^{1}, Binbin Weng^{1}, Lance L. McDowell^{1}, Jijun Qiu^{1,2}, Zhihua Cai^{1,2}, Zhisheng Shi^{1*}

^{1}The School of Electrical and Computer Engineering, University of Oklahoma, Norman, USA.

^{2}Nanolight, Inc., Norman, USA.

**DOI: **10.4236/detection.2018.61001
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This paper provides a theoretical study and calculation of the specific detectivity-*D** limit of photovoltaic (PV) mid-wave infrared (MWIR) PbSe n^{+}-p junction detectors operating at both room temperature and TE-cooled temperature. For a typical PbSe p-type doping concentration of 2 × 10^{17} cm^{-3} and with high quantum efficiency, the *D** limits of a photovoltaic PbSe n^{+}-p junction detector are shown to be 2.8 × 10^{10} HZ^{1/2}/W and 3.7 × 1010 HZ1/2/W at 300 K and 240 K, with cut-off wavelength of 4.5 μm and 5.0 μm, respectively. It is almost one magnitude higher than the current practical MWIR PV detector. Above 244 K, the detector is Johnson noise limited, and below 191 K the detector reaches background limited infrared photodetector (BLIP) *D**. With optimization of carrier concentration, *D** and BLIP temperature could be further increased.

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Shi, X. , Phan, Q. , Weng, B. , McDowell, L. , Qiu, J. , Cai, Z. and Shi, Z. (2018) Study on the Theoretical Limitation of the Mid-Infrared PbSe N^{+}-P Junction Detectors at High Operating Temperature. *Detection*, **6**, 1-16. doi: 10.4236/detection.2018.61001.

1. Introduction

The MWIR light detection has widespread applications in the fields of health monitoring, environmental protection, defense and national security, as well as space exploration and other fields. Existing technologies with high sensitivity are mainly based on semiconductor photo-detectors. In the past half-century, many semiconductor material systems have been intensively studied, and significant progress has been made [1] [2] . Currently, the leading photodetector approaches are mainly based on HgCdTe (MCT) [3] [4] III-V type-II superlattices, and InSb-based material systems. Among them, MCT is the premier material of interest for MWIR Focal Plane Arrays (FPA) applications. The challenges have been reducing the cost, size, weight, and power requirements, while improving the detection range and resolution. So far, these MWIR FPAs still require cryogenic cooling to achieve high detectivity, which is bulky and expensive. It is known that the major fundamental hurdle of the MWIR photodetectors at high operation temperature is the high Auger recombination rate. Among these detectors, the MCT technology has demonstrated its state-of-the-art performance, which is closely approaching the Auger theoretical limit.

It is well known that Auger coefficient in IV-VI semiconductors [5] [6] [7] is about an order of magnitude lower than those in Sb-based type-II QWs, [8] [9] [10] which are in turn significantly suppressed relative to other III-V and II-VI semiconductors such as MCT [11] [12] for the same waveleng. Such low Auger recombination should result in superior device performance such as high detectivity for detectors at a high operating temperature. High specific detectivity (D*) of PbSe MWIR photoconductive (PC) detector at room temperature was reported, 2.8 × 10^{10} cm・Hz^{1/2}/W and 4.2 × 10^{10} cm・Hz^{1/2}/W at ~3.8 µm, with and without antireflective coating [13] [14] [15] . Northrop Grumman has demonstrated monolithic integrated PbSe PC detector FPA with Si read-out circuitry (ROIC) with thermoelectric (TE) cooling [16] . These exciting results have stimulated renewed interests in using Pb-salt cameras for TE-cooled or uncooled applications [17] .

It is worth nothing that photovoltaic (PV) detectors offer advantages over their PC detector counterparts, such as low power consumption, high pixel density, and small pixel size. In addition, a PV detector FPA operating at low bias with no or low flicker noise eases the design of ROIC which further reduces cost. Although the D* of the IV-VI Pb-salt semiconductor junction at a low temperature has been investigated via experimental and theoretical research, the performance limit of PbSe at high temperature has not been well investigated. In this paper, we investigate the performance limit of a PbSe n^{+}-p junction detector operating at room temperature and TE-cooled temperature.

2. The Specific Detectivity D^{*}

Specific detectivity, or D*, for a photodetector is an important figure of merit and is determined by [18] :

${D}^{*}=\frac{\sqrt{A\Delta f}}{NEP}$ (1)

where A is the active area of the detector and NEP is the noise equivalent power (the power needed to generate the signal current which is equal to the noise current)

$NEP=\frac{{\Phi}_{e}}{{i}_{sig}/{i}_{n}}$ (2)

Here ${\Phi}_{e}$ [19] is the radiant power incident on a detector.

In a junction device, the fluctuation of diffusion rates in the neutral region and the generation-recombination (g-r) fluctuation in both depletion region and quasi-neutral region are indistinguishable. All of them give rise to the shot noise which has the form of [20]

${i}_{n}^{2}=2q\left({I}_{D}+2{I}_{S}\right)\Delta f$ (3)

where ${I}_{D}$ is the diode current, ${I}_{S}$ is the dark reverse-bias saturation current of the device and $\text{\Delta}f$ is the electrical bandwidth.

${I}_{D}={I}_{S}\times \left({\text{e}}^{\frac{qV}{\beta kT}}-1\right)$ (4)

β is the ideality factor which determines how the actual diode deviates from the ideal diode. In most of our later calculations, we assume the ideal diode case where $\beta =1$ .

For the diode under the illumination of photon density flux ${\Phi}_{b}$ , which generates the photon current ${I}_{ph}=q\eta A{\Phi}_{b}$ , the total noise contribution is:

${I}_{n}^{2}=2q\left(q\eta A{\Phi}_{b}+\frac{kT}{q{R}_{0}}{\text{e}}^{\frac{qV}{kT}}+\frac{kT}{q{R}_{0}}\right)\Delta f$ (5)

where R_{0} is the zero-voltage resistance which is defined by

${R}_{0}={\left(\frac{\partial {I}_{D}}{\partial V}\right)}_{V=0}^{-1}=\frac{kT}{q{I}_{S}}$ (6)

Under the zero-voltage case, the expression of shot noise in (3) has the same form as the Johnson-noise associated with the R_{0} of the junction, thus it is often called Johnson noise.

Substituting (1) and (2) to (5), the expression for the detectivity of the device is

${D}^{*}=\frac{q\eta \lambda}{hc}{\left[\frac{4kT}{{R}_{0}A}+2{q}^{2}\eta {\Phi}_{b}\right]}^{-1/2}$ (7)

In Equation (7), η is the quantum efficiency, λ is the wavelength of the incident radiation, q is the charge of the carrier, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, R_{0} is the diode incremental resistance at 0 V, A is the detector sensitive area, T is the temperature of the detector in Kelvin, and
${\Phi}_{b}$ is the photon flux incident on the detector from its surroundings. It is clear that both R_{0}A and η are key factors on the specific detectivity D*. Therefore, the resistance-area (R_{0}A) product at zero bias needs to be calculated to determine D* based on Equation (6).

3. R_{0}A Analysis and Discussions

The total dark current density flowing through the junction can be found with the following equation:

${J}_{s}=\frac{{I}_{s}}{A}={J}_{DF}+{J}_{GR}+{J}_{T}+{J}_{L}$ (8)

In Equation (7), J_{DF} is the diffusion current density, J_{GR} is generation-recombination current density that is often dominated in the depletion layer, J_{T} is tunneling current density and J_{L} is leakage current density. J_{L} may be due to bulk as well as surface defects of the material. When suitable diode technology and construction are used, the contribution of the J_{L} component is negligible.

The maximum R_{0}A is obtained from the one-sided abrupt junction [21] , giving the limitation on the specific detectivity D*. The minority carriers on the lightly doped side of the n^{+}-p junction will dominant the diffusion current. The R_{0}A product determined by the diffusion current in the case of radiative recombination and Auger recombination is

${\left({R}_{0}A\right)}_{DF}=\frac{{\left(kT\right)}^{1/2}}{{q}^{3/2}{n}_{i}^{2}}{N}_{A}{\left(\frac{{\tau}_{e}}{\mu}\right)}^{1/2}$ (9)

where n_{i} is the intrinsic carrier concentration, N_{A}(N_{D}) is the concentration of donors (acceptors), τ_{h}(τ_{e}) is the minority carrier lifetime of holes (electrons) and μ_{h}(μ_{e}) is the minority carrier mobility.

The resistance-area product (R_{0}A) of PbSe photo-voltaic detectors under conditions of zero bias is calculated for one-side abrupt junction. Highly doped n-side homojunction (n^{+}-p) is discussed in this paper. At room temperature, the effective R_{0}A product on the PbSe n^{+}-p junction can be expressed by:

$\begin{array}{c}{\left({R}_{0}A\right)}^{-1}={\left({R}_{0}A\right)}_{DF}^{-1}+{\left({R}_{0}A\right)}_{T}^{-1}+{\left({R}_{0}A\right)}_{SRH}^{-1}\\ ={\left({R}_{0}A\right)}_{RA}^{-1}+{\left({R}_{0}A\right)}_{Auger}^{-1}+{\left({R}_{0}A\right)}_{T}^{-1}+{\left({R}_{0}A\right)}_{SRH}^{-1}\end{array}$ (10)

where
${\left({R}_{0}A\right)}_{RA}$ is the R_{0}A product due to radiative recombination,
${\left({R}_{0}A\right)}_{Auger}$ is the R_{0}A product due to Auger recombination, and
${\left({R}_{0}A\right)}_{SRH}$ is the R_{0}A product due to Shockley-Read-Hall recombination. Different R_{0}A products due to each recombination mechanism will be discussed in the following section.

3.1. Radiative Recombination Contribution

The R_{0}A product determined by the diffusion current in the case where radiative recombination has the relationship on the lifetime
${\tau}_{r}^{-1}$ . The radiative lifetime
${\tau}_{r}^{-1}$ for PbSe follows as [22] :

${G}_{R}={10}^{-15}{n}_{r}{\left(kT\right)}^{-\frac{3}{2}}{\left(2+\frac{1}{K}\right)}^{-\frac{3}{2}}{\left(\frac{{m}^{*}}{m}\right)}^{-\frac{5}{2}}{\left(K\right)}^{-\frac{1}{2}}{E}_{g}^{2}{\text{cm}}^{\text{3}}/\text{sec},\text{\hspace{0.17em}}\text{\hspace{0.17em}}K={m}_{l}^{*}/{m}_{t}^{*}$ (11)

${\tau}_{r}^{-1}={G}_{R}\left({n}_{0}+{p}_{0}\right)$ (12)

${n}_{0}+{p}_{0}=2{\left(\frac{{N}_{D}^{2}}{4}+{n}_{i}^{2}\right)}^{1/2}$ (13)

The radiative lifetime is inversely proportional to the total concentration of free carriers under all circumstances. The radiative lifetime decreases when the doping concentration increases.

G_{R} is the capture probability for radiative recombination. The band gap of a semiconductor E_{g} is expressed in electron-volts. n_{r} is the index of refraction, and
${m}^{*}={\left[1/3\left(2/{m}_{t}+1/{m}_{l}\right)\right]}^{-1}$ is the density-of-states electron and hole effective masses. n_{i} is intrinsic carrier concentration. n_{0} and p_{0} represent the equilibrium carrier concentrations. According to Equation (14), the (R_{0}A)_{RA} product can be found after the radiative recombination lifetime is calculated.

${\left({R}_{0}A\right)}_{RA}=\frac{{\left(kT\right)}^{1/2}}{{q}^{3/2}{n}_{i}^{2}}{N}_{A}{\left(\frac{{\tau}_{r}}{\mu}\right)}^{1/2}$ (14)

Figure 1 describes the radiative life time on the dopant concentrations for the PbSe junction at 300 K. When carrier concentration is between 10^{17} and 10^{18}, the radiative recombination life time of PbSe is between 10^{−}^{7} s and 10^{−}^{8} s at 300 K.

3.2. Auger Recombination Contribution

The R_{0}A product determined by the diffusion current in the case of Auger recombination has the relationship on the lifetime
${\tau}_{A}^{-1}$ . The PbSe Auger’s recombination coefficient C_{A} is given by [7] :

${C}_{A}=\frac{3{q}^{4}{\left(2\text{\pi}\right)}^{5/2}{\left({k}_{B}T\right)}^{1/2}E{g}^{-7/2}{\u045b}^{3}}{{\left(16\text{\pi}{\epsilon}_{0}{\epsilon}_{\infty}\right)}^{2}{m}_{l}^{\ast 1/2}{m}_{t}^{\ast 3/2}}\times \mathrm{exp}\left[-\frac{Eg}{2{k}_{B}T}{\left(\frac{{m}_{l}^{*}}{{m}_{t}^{*}}\right)}^{-1}\right]$ (15)

where ${m}_{l}^{*}$ and ${m}_{t}^{*}$ are the longitudinal and transverse effective mass, ${\epsilon}_{\infty}$ is high frequency dielectric constant, and ${\epsilon}_{0}$ is vacuum permittivity. During the Auger recombination process, the carrier lifetime is defined as:

${\tau}_{A}={\left[{C}_{A}\left({N}_{A}^{2}+2{n}_{i}^{2}\right)\right]}^{-1}$ (16)

Figure 2 shows the Auger life time of different dopant concentrations for the PbSe junction at room temperature. When carrier concentration is between 10^{17}^{ }

Figure 1. The dependence of the radiative life time on the dopant concentrations for the PbSe junction at 300 K.

Figure 2. The dependence of the Auger life time on the dopant concentrations for the PbSe junction at 300 K.

and 10^{18}, the Auger recombination life time of PbSe is between 10^{−}^{7} s and 10^{−}^{8} s at 300 K. According to Equation (17), the (R_{0}A)_{Auger} product can be derived when the Auger recombination lifetime is calculated.

${\left({R}_{0}A\right)}_{Auger}=\frac{{\left(kT\right)}^{1/2}}{{q}^{3/2}{n}_{i}^{2}}{N}_{A}{\left(\frac{{\tau}_{A}}{\mu}\right)}^{1/2}$ (17)

3.3. Tunneling Current Contribution

The (R_{0}A)_{T} determined by tunneling is given by

${\left({R}_{0}A\right)}_{T}=4\ast {\text{\pi}}^{3}{\u045b}^{2}{\left({\epsilon}_{s}{\epsilon}_{0}\right)}^{\frac{1}{2}}\mathrm{exp}\left[\frac{\text{\pi}{\left({m}_{x}{\epsilon}_{s}{\epsilon}_{0}\right)}^{\frac{1}{2}}Eg}{{2}^{\frac{3}{2}}q\u045b{N}_{D}^{\frac{1}{2}}}\right]/\left({q}^{3}\sqrt{\frac{2{N}_{A}{m}_{y}{m}_{z}}{{m}_{x}}}\right)$ (18)

Tunneling simulation of PbSe depends on the crystal orientation due to the difference in effective masses. The four effective masses to consider are: ${m}_{el}$ (conduction band, longitudinal), ${m}_{et}$ (conduction band, transverse), ${m}_{hl}$ (transverse band, longitudinal) and ${m}_{ht}$ (transverse band, transverse). ${m}_{0}$ is the electron resting mass [23] .

${m}_{el}={\left[11.4\times \frac{0.145}{Eg}+2.9\right]}^{-1}\times {m}_{0}$

${m}_{et}={\left[20.7\times \frac{0.145}{Eg}+4.3\right]}^{-1}\times {m}_{0}$

${m}_{hl}={\left[11.4\times \frac{0.145}{Eg}+3.3\right]}^{-1}\times {m}_{0}$

${m}_{ht}={\left[20.7\times \frac{0.145}{Eg}+8.7\right]}^{-1}\times {m}_{0}$

For PbSe [100] directions, the effective mass on different orientations can be determined by the Table 1.

In order to compare Auger, radiative and tunneling recombination, R_{0}A production on those three recombination mechanisms versus different temperatures is analyzed. In the following calculation
${N}_{A}=2\times {10}^{17}{\text{cm}}^{-3}$ , a typical PbSe hole concentration is used. To satisfy n^{+}-p junction, N_{D}_{ }is chosen to be 1 × 10^{18} cm^{−}^{3}, which is five times higher than N_{A}. Throughout this paper, we use N_{D} ≥ 5N_{A} for our n^{+}-p junction calculation. Looking at Figure 3(a) for the given carrier concentration
${N}_{A}=2\times {10}^{17}$ , at a higher temperature, (R_{0}A)_{T} increases significantly for the PbSe [100] junction so that the tunneling (R_{0}A)_{T} can be ignored. This is mainly because the energy bang gap of PbSe increases with temperature. When

Table 1. PbSe [100] effective mass in different directions.

Figure 3. The dependence of the tunneling R_{0}A product on the dopant concentrations for the PbSe [100] junction at 160 K and 240 K. For Figure 3(a) the n side carrier concentration is 1 × 10^{18} cm^{−}^{3}, and in Figure 3(b) n side carrier concentration is 5 × 10^{18} cm^{−}^{3}.

the temperature is around 160 K, (R_{0}A)_{T} is comparable to the Auger R_{0}A and radiative R_{0}A. For even lower temperatures, (R_{0}A)_{T} becomes dominant. At temperatures above 180K K (R_{0}A)_{T} is at least five times higher than the other mechanisms and thus (R_{0}A)_{T} can be neglected relative to the total R_{0}A product.

To keep the same N_{A} and increase N_{D} doping concentrations, Figure 3(b) shows that (R_{0}A)_{T} cannot be neglected even at room temperature. At about 275 K, (R_{0}A)_{T} is five time lower than the other mechanisms and at higher than 275 K tunneling needs to be considered in the total R_{0}A product. Comparing Figure 3(a) and Figure 3(b), it is apparent that carrier concentration has a large influence on R_{0}A, and therefore, the total device performance.

3.4. SRH Recombination Contribution

The carrier generation-recombination mechanisms in detector devices are distinguished as Auger, radiative, tunneling, and Shockley-Read-Hall’s (SRH) generation-recombination mechanisms. Auger, radiative, and tunneling mechanisms are determined by energy band structures. However, SRH’s mechanism is determined by the material quality. In Equation (4), the depletion generation and recombination R_{0}A term is largely dominated by the lifetime associated with SRH centers. However, the value of this lifetime is extremely uncertain as has been noted in previous studies [7] [24] . These studies often assume longer SRH lifetimes in their investigations, typically on the order of 10^{−8}s^{25}. Thus, using this assumed SRH lifetime, higher temperature operations produce high values for (R_{0}A)_{GR} which may be neglected in our case. However, SRH recombination may need to be considered with a shorter SRH lifetime.

3.5. Discussion of the Overall R_{0}A Limitation

Utilizing the R_{0}A expressions shown in the above sections, calculations of R_{0}A for Auger, radiative, and tunneling were described in Figure 4. For a fixed operating temperature, R_{0}A dependence on carrier concentration reveals the contribution of each mechanism, revealing an optimized condition for the maximum R_{0}A limit. Three temperatures of interest are used. 300 K, 240 K, and 160 K represent temperatures of an uncooled, cut-off wavelength of 5 μm, and that the tunneling effect needs to be considered.

Table 2 shows simulation parameters at 300 K. Between 50 K and 300 K, the mobility varies as [25]

$\mu ={\mu}_{0}{T}^{-2.65}$ (19)

And the expression of energy gap is given as [26]

${E}_{g}\left(T\right)=125+\sqrt{400+0.256\times {T}^{2}}$ (20)

At 300 K, the total effective R_{0}A product is around 1.3 × 10^{−}^{4} ohm・m^{2}. At 240 K, the total effective R_{0}A product is around 1.6 × 10^{−}^{4} ohm・m^{2}. When temperature is higher, the total effective R_{0}A product becomes smaller. According to Equation (10), Figure 4 gives the results of the total effective R_{0}A product on the

Figure 4. The PbSe R_{0}A product determined by the diffusion current in the case of Auger recombination and radiative recombination at different high temperature. The black curve is the total effective PbSe R_{0}A product on the diffusion current in the case of Auger recombination and radiative recombination.

Table 2. The parameters are applied in the simulations for 300 K [27] .

diffusion current in the case of Auger recombination and radiative recombination at a different temperature. At 160 K, tunneling R_{0}A value is comparable to the Auger and radiative R_{0}A value.

Figure 4(a) shows the R_{0}A limitation at 160 K when the n^{+} doping concentration is
${N}_{D}=1\times {10}^{18}{\text{cm}}^{-3}$ . In this case,
${\left({R}_{0}A\right)}_{T}$ is comparable with
${\left({R}_{0}A\right)}_{RA}^{-1}$ and
${\left({R}_{0}A\right)}_{Auger}^{-1}$ . When the tunneling effect on the total R_{0}A cannot be ignored, there exists an optimized carrier concentration for the largest R_{0}A. In this case, the largest R_{0}A occurs at
${N}_{A}=1.8\times {10}^{17}{\text{cm}}^{-3}$ . When the n side carrier concentration was increased from
$1\times {10}^{18}{\text{cm}}^{-3}$ to
$5\times {10}^{18}{\text{cm}}^{-3}$ , tunneling could not be ignored at 300 K, as shown in Figure 4(b). The contribution from tunneling recombination results in a lower total R_{0}A in Figure 4(b). In Figure 4(c) and Figure 4(d), where tunneling can be ignored, it seems that higher PbSe doping concentration (N_{A}) gives a larger total R_{0}A. This is because a larger N_{A} leads to a smaller minority carrier concentration, which in turn reduces the saturation current. However, for a n^{+}-p junction n-doping concentration has to be increased, as p-doping concentration increases, which will increase the tunneling effect. In our simulations, the maximum R_{0}A limit at 240 K (Figure 2(c)) and room temperature (Figure 2(d)) occurs when carrier concentration on the p side is at a maximum, while satisfying the condition for the n^{+}-p junction where
${N}_{D}=1\times {10}^{18}{\text{cm}}^{-3}$ and
${N}_{A}=2\times {10}^{17}{\text{cm}}^{-3}$ . The recombination mechanisms in n^{+} side are not taken into consideration in this paper. Recombination in n^{+} side poses a further limitation of n^{+} concentration and thus p-doping concentration for optimized D*. Assuming a wide bandgap N^{+}-doping layer with negligible Auger and Radiative recombination rate, and negligible tunneling effect, then higher p-type doping concentration could be used to further increase the R_{0}A.

4. Thermal Noise and Background Photon Shot Noise

As can be seen from Equation (6), the two main mechanisms contributing to the homojunction device are the thermal noise and the background photon shot noise. By using calculated R_{0}A when
${N}_{D}=1\times {10}^{18}{\text{cm}}^{-3}$ and
${N}_{A}=2\times {10}^{17}{\text{cm}}^{-3}$ , the temperature dependence current noise of these two sources is described in Figure 5.

From Figure 5, we can easily see that at room temperature the dominant source of noise in the device is the Johnson noise. For temperatures around 191 K to 244 K, the contribution of both noise sources is in the same order of magnitude. Below 200 K, photon noise becomes the dominating mechanism, with the background photon noise current five times greater than the Johnson noise. Therefore, if we ignore one noise source when it is five times lower than the other source, the detectivity expressed in (7) can be simplified to high temperature Johnson noise limited D* and background limited infrared photodetector (BLIP) D*.

Figure 5. The comparison between the thermal noise and photon background noise on PbSe PV detectors related to temperature.

${D}^{*}=\frac{q\eta \lambda}{hc}{\left[\frac{4kT}{{R}_{o}A}\right]}^{-1/2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}T>244\text{\hspace{0.17em}}\text{K}$ (21)

${D}^{*}\equiv {D}_{BLIP}^{\ast}=\frac{q\eta \lambda}{hc}{\left[2{q}^{2}\eta {\Phi}_{b}\right]}^{-1/2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}T<191\text{\hspace{0.17em}}\text{K}$ (22)

5. Theoretical Limitation on the Specific Detectivity D^{*} at High Temperature^{ }

To find the theoretical limit for PbSe photovoltaic detector D* at 300 K, we apply the R_{0}A product from the above conditions with Equation (23) because photon shot noise can be negligible at room temperature.

${D}^{*}=\frac{q\eta \lambda}{hc}{\left[\frac{4kT}{{R}_{o}A}\right]}^{-1/2}$ (23)

At 240 K, to calculate the theoretical limit for PbSe photovoltaic detector D* following by:

${D}^{*}=\frac{q\eta \lambda}{hc}{\left[\frac{4kT}{{R}_{o}A}+2{q}^{2}\eta {\Phi}_{b}\right]}^{-1/2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Phi}_{b}=1.23\times {10}^{16}\text{\hspace{0.17em}}{\text{cm}}^{-2}\cdot {\mathrm{sec}}^{-1}$ (24)

Theoretically, the quantum efficiency should be calculated by the number of electron-hole pairs generated per incident photon. In order to get the D* limitation, the detector quantum efficiency is simplified to a constant for a different cutoff wavelength. The D* of the PbSe junction is simulated for a quantum efficiency of 75% and 100%, λ is the wavelength of the incident radiation, q is the charge of the carrier, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, R_{0} is the diode incremental resistance at 0 V, A is the detector sensitive area, T is the temperature of the detector in Kelvin. With cutoff wavelength λ = 5 μm at T = 240 K, the photon flux
${\Phi}_{b}$ equals to 1.23 × 10^{16} cm^{−2} sec^{−1}. Figure 6 shows D* for the PbSe junction as it depends on the absorbing light wavelength at 240 K and 300 K. At 300 K with 100% quantum efficiency, the theoretical limit of D* for the photovoltaic PbSe n^{+}-p junction detector is 2.8 × 10^{10} HZ^{1/2}/W at 4.5 μm. At 240 K, the D* theoretical limit is 3.7 × 10^{10} HZ^{1/2}/W at the cut-off wavelength of 5 μm.

Due to reabsorption in the crystal, the effect of radiative recombination may be reduced and carrier radiative life time becomes larger [28] . Photon reabsorption is an effect by which a photon created by the recombination of an electron-hole pair is reabsorbed to produce a new pair and the process is repeated over and over again until a photon can either escape from the sample, or is absorbed by a process which does not create electron-hole pairs, such as free carrier absorption [29] . Considering this, Auger recombination becomes the dominating mechanism at high temperature. The D* of the PbSe junction in this case is shown in Figure 7.

At 300 K, the limitation of D* now increases from 2.8 × 10^{10} HZ^{1/2}/W to 3.9 × 10^{10} HZ^{1/2}/W at 4.5 μm when radiative recombination is ignored.

(a)(b)

Figure 6. The specific detectivity D* on the PbSe junction versus the absorbing light wavelength at 240 K and 300 K.

In this n^{+}-p homojunction junction model, the n^{+} PbSe was not considered for its impact on diffusion R_{0}A product. As discussed in previous sections, high carrier concentration will lead to high Auger and radiative recombination and thus reduced R_{0}A product. To achieve the calculated D* in this paper, a heterojunction structure with a wide bandgap N^{+} doped layer with much reduced Auger and radiative recombination is needed.

6. Optimization on Carrier Concentration

According to Figure 4(a), when tunneling can be ignored, R_{0}A value increases with increasing N_{A}. However, to keep the device n^{+}-p junction N_{D} needs to be increased with N_{A}, which increases the tunneling effect. Therefore, there exists an optimized N_{A} which provides the maximum R_{0}A at a given temperature. In the following calculation, N_{D}/N_{A} is set to be 5 to satisfy the condition for a n^{+}-p junction. N_{A} is then optimized at 300 K and 240 K, as show in Figure 8. The optimized R_{0}A for PbSe homojunction at 300 K and 240 K are 2.5 × 10^{18} cm^{−3} and 1.5 × 10^{18}, respectively. For optimized carrier concentration the detector reaches BLIP limited D* at 210 K, 10 degrees higher than the “typical” carrier concentration.

At 240 K, the optimized
${N}_{D}=1.5\times {10}^{18}{\text{cm}}^{-3}$ ^{ }and
${N}_{A}=3\times {10}^{18}{\text{cm}}^{-3}$ are used to calculate maximum R_{0}A product in Figure 9(a). By using the maximum R_{0}A value, the limitation D* is increased to 5.6 × 10^{10} HZ^{1/2}/W. At 300 K, the optimized
${N}_{D}=2.5\times {10}^{18}$ and
${N}_{A}=5\times {10}^{17}{\text{cm}}^{-3}$ are applied to get the limitation D* which can be increased to 3 × 10^{10} HZ^{1/2}/W in Figure 9(b). Therefore, at different temperatures the carrier concentration can be optimized to achieve the highest D*.

Figure 7. The specific detectivity D* on the PbSe junction with Auger limitation versus the absorbing light wavelength at 300 K.

Figure 8. The n side doping concentration dependent PbSe R_{0}A products at 240 K and 300 K.

(a)(b)

Figure 9. D* versus wavelength at 240 K and 300 K after carrier concentration optimization.

7. Conclusion

Performance limitation of the PbSe homojunction at high temperatures is theoretically studied in this paper. For such PV detectors thermal noise is dominating at temperatures higher than 240 K. BLIP limit could be achieved at temperatures around 210 K. The calculated peak D* are 3 × 10^{10} HZ^{1/2}/W and 5.6 × 10^{10} HZ^{1/2}/W at 300 K and 240 K respectively. Achieving D* of more than 10^{10} HZ^{1/2}/W in a MWIR detector at uncooled temperatures allows for its use in practical applications of high sensitivity.

Acknowledgements

The authors thank helpful discussions with Dr. Sung-Shik Yoo from Northrop Grumman. This work is partially supported by the DARPA WIRED program through Northrop Grumman, US army research office (ARO) under Grant No. W911NF-14-1-0312. The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

Conflicts of Interest

The authors declare no conflicts of interest.

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