††thanks: corresponding author<EMAIL_ADDRESS>
# Ultrahigh-performance superlattice mid-infrared nBn photodetectors at high
operating temperatures

Rohit Kumar1 Bhaskaran Muralidharan1 1Department of Electrical Engineering,
Indian Institute of Technology Bombay, Powai, Mumbai-400076, India

(August 28, 2024)

###### Abstract

While advancing a physics-based comprehensive photodetector-simulation model,
we propose a novel device design of the mid-wavelength infrared nBn
photodetectors by exploiting the inherit flexibility of the InAs1-xSbx ternary
alloy material system. To further explicate the physics of such
photodetectors, we calculate several crucial transport and optoelectronic
parameters, including the dark current density, absorption coefficient,
responsivity, and the quantum efficiency of nBn photodetectors. A remarkable
maximum efficiency of 57.39% is achieved at room temperature at a bias of
-0.25 V, coupled with a radiation power density of 50 mW/$cm^{2}$. The
proposed structure features a maximum quantum efficiency of 44.18% and 37.87%
at 60% and 70% of the $\lambda_{c}$, respectively. Furthermore, a maximum
responsivity of 0.9257 A/W is shown within the mid-wavelength infrared
spectrum. Through our comprehensive analysis, we also demonstrate that our
proposed device design effectively reduces the dark current density by
confining the electric field inside the barrier while preserving a superior
level of quantum efficiency, and the current in such detectors is diffusion-
limited. Insights uncovered here could be of broad interest to critically
evaluate the potential of the nBn structures for mid-wavelength infrared
photodetectors.

Mid-wavelength infrared (MWIR) photodetectors Soibel et al. (2014); Kumar et
al. (2023); Soibel et al. (2019); Gautam et al. (2012); Wu et al. (2020) are
in high demand for a wide range of civilian, military, and space applications,
including environmental monitoring, chemical sensing, medical diagnostics, and
infrared (IR) imaging. This is because they meet high operating temperature
requirements, exhibit better performance, and have numerous advantages over
other photodetector candidates Martyniuk and Rogalski (2013); Martyniuk et al.
(2014); Klipstein (2008); Rogalski et al. (2020). The design of such a high
performance MWIR photodetector relies on achieving a balance between low dark
current and high quantum efficiency. A majority of photodiodes in the market
today are p-n junction photodiodes made from conventional materials like
HgCdTe (MCT) Itsuno et al. (2012), InGaAs Martinelli et al. (1988) and InSb
Shimatani et al. (2020), etc., which are plagued by space-charge generation-
recombination (G-R) dark currents that significantly restrict their efficacy
for applications demanding high sensitivity at lower temperatures Reine et al.
(2014). In order to prevent excessive dark currents, these devices typically
need to be cooled down to cryogenic temperatures.
The nBn photodetector Rodriguez et al. (2007); Chen et al. (2021); Maimon and
Wicks (2006); Kim et al. (2012); Haddadi et al. (2017); Baril et al. (2016);
Craig et al. (2013), in which the barrier is sandwiched between two n-type
regions, features a distinct design that is less susceptible to crystalline
defects and effectively reduces the dark current and noise brought on by the
Shockley-Read-Hall (SRH) generation, surface states, and various other
processes. Due to the high cost of fabrication and the complexity of such
structures, an accurate theoretical modeling is essential for developing the
physics of such IR photodetectors Kumar et al. (2023). In this work, we
develop an accurate physics-based theoretical reliable simulation model and
propose a novel device design for the nBn photodetector Reine et al. (2014);
Maimon and Wicks (2006) that offers relatively high performance and excellent
quantum efficiency within the MWIR spectrum D’souza et al. (2012). We then use
our in-depth computational analysis to elucidate intriguing physics and
predict the various performance limiting factors for an InAs1-xSbx (IAS) Krier
et al. (2007); Lackner et al. (2009); Rogalski et al. (2020); Martyniuk and
Rogalski (2013); Rogalski (1989); Shaveisi and Aliparast (2023) based nBn MWIR
photodetector, where the barrier is designed with a large band gap
Al0.7In0.3As0.3Sb0.7 (AIAS) Bank et al. (2017); Maddox et al. (2016); Ren et
al. (2016) material.
The device structure, depicted in Fig. 1, consists of three principal layers:
a thick absorber layer (AL) of n-type narrow gap IAS material with a thickness
of 2.7 $\mu$m, a barrier layer (BL) of n-type lattice-matched AIAS material
with a thickness of 0.25 $\mu$m, and a contact layer (CL) of n-type IAS with a
thickness of 0.27 $\mu$m. The thickness of the BL (tBL) is considered to be
sufficiently large to inhibit electron tunneling between the CL and the AL
layers. As a result, the majority current is impeded by the barrier material
when a proper bias is applied. The absence of a significant electric field in
the narrow gap material prevents the SRH generation and the Band-to-band (BTB)
tunneling thereby, the nBn photodetectors operating in the MWIR region exhibit
lower levels of dark currents and noise Martyniuk and Rogalski (2013). This
characteristic enables a reduction in the cooling demands associated with
these devices.

FIG. 1: Preliminaries for the proposed MWIR nBn photodetector (a) schematic
illustration of the carrier transport in the nBn photodetector (b) layout of
the considered nBn photodetector with IAS as the AL and the CL, and AIAS as
the BL that hinders the movement of electrons.

FIG. 2: Energy band profiles for the nBn photodetector under (a) positive
bias, (b) equilibrium, i.e., zero bias, and (c) negative applied bias voltage.
The high potential barrier in the VB that appears within the BL at zero bias
prevents holes from moving to the CL on the left. Under equilibrium, Efh is
superimposed to Efe due to their identical spatial alignment. At negative
bias, the holes will be able to traverse the potential barrier and reach the
CL. The presence of a strong electric field within the barrier region
facilitates the movement of carriers, thus contributing to the overall drift
current. The electrons within the CL are unable to traverse to the AL due to
the large CBO.

FIG. 3: Electrostatic potential and the absorption coefficient of the MWIR nBn
photodetector under consideration (a) variation in the electrostatic potential
across the device structure as a function of temperature with an applied bias
of -0.5 V (b) behavior of the absorption coefficient, $\alpha$, in the
absorber region as a function of the incoming radiation wavelength at various
temperatures.

Our methodology comprises the finite difference method in conjunction with the
linear interpolation technique to solve the Poisson and the continuity
equations for the carriers while taking into account the temperature, doping,
and structural parameters of the ternary alloy material system. We obtain the
electrostatic potential of the heterojunction, hole quasi-Fermi-level outside
the thermal equilibrium to build the band structure of the considered device
design. The BTB tunneling, the trap-assisted tunneling (TAT), the SRH G-R, and
the Auger G-R processes are some of the primary sources of the dark current,
as shown in Fig. 1 (a).
Incorporating a BL into the design of IR detectors has the potential to reduce
the unfavorable extrinsic SRH G-R contribution substantially. In this study,
we thereby focus on the radiative recombination, the Auger G-R, and other
thermally generated processes that exhibit dominance in nBn photodetectors.
The device layout using GaSb as the substrate is shown in Fig. 1 (b). The n+
CL (i.e., bottom contact) made up of IAS material is the collector layer for
the photogenerated electrons, which also serves as the buffer layer to reduce
epitaxial strain between the GaSb and the AL for the device. The nBn structure
is shown on top of this buffer layer. We use an iterative approach to solve
the specific equations and relations used in the device’s modeling. The entire
sequence of the numerical simulation is outlined in Fig. A1. The performance
of the proposed device is analyzed based on the various calculated transport
and optoelectronic parameters such as the carrier density, the electric field,
the electrostatic potential, the absorption coefficient for the absorber
region, the dark current density, the responsivity, and the quantum
efficiency, etc., as a function of the applied bias voltage, operating
temperatures, and the structural parameters. The specifications for the CL,
BL, and AL to design the considered nBn MWIR photodetectors are given in Tab.
1.

FIG. 4: Bias dependence of the diffusion-limited dark current density for the
nBn photodetectors at various operating temperatures for (a)
$t_{{}_{AL}}=2.7\leavevmode\nobreak\ \mu m$ and
$N_{{}_{D}}\leavevmode\nobreak\ (BL)=1\times 10^{16}\leavevmode\nobreak\
cm^{-3}$, (b) $t_{{}_{AL}}=3.0\leavevmode\nobreak\ \mu m$ and
$N_{{}_{D}}\leavevmode\nobreak\ (BL)=1\times 10^{16}\leavevmode\nobreak\
cm^{-3}$, and (c) $t_{{}_{AL}}=2.7\leavevmode\nobreak\ \mu m$ and
$N_{{}_{D}}\leavevmode\nobreak\ (BL)=2\times 10^{16}\leavevmode\nobreak\
cm^{-3}$. The dark current density, as shown in (a) and (b), exhibits an
exponential relationship at low bias until it reaches the saturation voltage.
Beyond this voltage, the dark current plateaus at a constant value. In
contrast, the saturation level shifts to higher voltages when the doping of
the BL is increased, as shown in (c).

Figures 2 (a), (b), and (c) depict the potential energy profile of the nBn
structure at three distinct applied voltages: positive bias, equilibrium, and
negative bias. It is assumed that the electron quasi-Fermi levels in the BL
and AL are in equilibrium with each other. As shown in Fig. 2 (c), by applying
a voltage of V = - 0.23 V to the structure, the energy barrier for holes is
lowered by nearly two orders of magnitude (from 163 to 80 meV) compared to an
equilibrium condition. It is evident that the applied voltage primarily
affects $\Delta E_{v}$, and when the nBn detector is reverse-biased, it
indicates that a positive voltage is applied to the absorber contact.
The nBn band structure, as illustrated in Fig. 2 (c), hinders the flow of the
majority carriers (electrons) through a large conduction band offset (CBO) but
enables the flow of the minority carriers (holes) through a near-zero valence
band (VB) offset. Therefore, when a relatively low operating voltage is
applied, it falls almost entirely across the barrier, thereby separating the
photogenerated carriers. For $\Delta E_{v}$ $>$ 3$k_{B}$T, the minority
carriers in the nBn architecture are effectively blocked therefore, the
condition for unhindered minority carrier transport to the CL ($\Delta E_{v}$
$<$ 3$k_{B}$T) is met when applied bias exceeds –0.23 V. The band bending
phenomena is also observed at the boundaries of the BL, which signifies the
accumulation of the majority charge carriers in the vicinity of the space
charge region, which spans the depth of the BL. The mobile charges have
migrated to the neighboring IAS-based AL and CL, depleting the entire BL.

TABLE 1:  Parameters used to design the device structures.  $x_{{}_{Sb}}$ | 0.09 |
---|---|---
T | 300 K |
Cut-off wavelength ($\lambda_{c}$) | 4.33 $\mu$m |
Layers |  Thickness / Depth | Doping
CL | 0.27 $\mu$m | $5\times 10^{15}$ $cm^{-3}$
BL | 0.25 $\mu$m | $8\times 10^{15}$ $cm^{-3}$
AL | 2.7 $\mu$m | $1\times 10^{16}$ $cm^{-3}$

In Appendix A, an in-depth investigation of the electric field and the carrier
density is provided. There, we demonstrate how the electric field and the
carrier density in the nBn photodetector are affected by the temperature and
applied bias.

In Fig. 3 (a), we show the distribution of the electrostatic potential across
the nBn detector as a function of the temperature and the layer thicknesses,
which has a direct impact on the carrier concentration. An inference can be
made that as the temperature increases, the electrostatic potential becomes
more concentrated within the width of the BL. The potential drop in the AL
varies hardly with temperature therefore, the complete depth of the AL is not
shown, whereas a noticeable change can be seen in the CL.

FIG. 5:  Room temperature responsivity and the quantum efficiency for the MWIR
nBn photodetector (a) responsivity as a function of the incoming radiation
wavelength (b) wavelength dependence of the quantum efficiency at various AL
thicknesses (c) bias dependence of the quantum efficiency at various
percentages of cut-off wavelength.

In Fig. 3 (b), we show the absorption coefficient of the IAS absorber in
relation to the incoming radiation wavelength and temperature. The energy
threshold of absorption is expected to shift as the band gap of AL material
varies with temperature. The absorption coefficient exhibits maximum values
within the wavelength range of 3.1 to 4.3 $\mu$m for all simulated
temperatures. The corresponding $\alpha$ values fall within the range of
1557-1644 $cm^{-1}$. The absorption coefficient exhibits a decreasing trend in
its maximum value as temperature increases. This behavior appears to be
affected by the band gap of the absorber material.

In Figs. 4 (a), (b), and (c), we demonstrate the dark current density of the
nBn MWIR photodetector. It is important to note that the nBn structure
operates in a minority carrier manner, hence the hole transport from AL to CL
is the primary cause of the dark current. The “turn-on voltage” is assumed to
be V $\approx$ –(0.23 - 0.25) V. This is the voltage after which the dark
current saturates and $\Delta E_{v}$ falls below 3$k_{B}$T. The dark current
density is shown in Fig. 4 (b) when the $t_{AL}$ is slightly increased and
held constant at 3 $\mu$m. Figure 4 (c) demonstrates the dark current density
when the BL doping is increased from $1\times 10^{16}\leavevmode\nobreak\
cm^{-3}$ to $2\times 10^{16}\leavevmode\nobreak\ cm^{-3}$. Changing the
thickness of the AL has a low impact, while changing the doping of the BL has
a noticeable effect, as shown in Figs. 4 (b) and (c), respectively. The dark
current density is shown to be highly voltage-dependent between 0 to –0.23 V.
Alternatively, it is less sensitive to voltage changes between -0.23 V and
-0.7 V, as shown in Figs. 4 (a) and (b). As previously mentioned, the $\Delta
E_{v}$ value for V = -0.23 V is comparable with 3$k_{B}$T, indicating that the
holes are almost freely transported to the CL and hence contribute to the net
dark current. It is observed that for reverse voltages V $<$ –0.23 V, the dark
current density rises sharply due to a rapid rise in the hole concentration,
whereas above V $>$ –0.23 V, the dark current approaches saturation. But when
BL doping is increased, the saturation level shifts to the higher voltages, as
shown in Fig. 4 (c). Moreover, the diffusion-limited barrier structure
prevents tunneling in the simulated voltage range.

The spectral response of the considered nBn photodetector as a function of the
wavelength at a temperature of 300 K for a constant bias of -0.25 V with a
radiation power density of 50 mW/$cm^{2}$ is presented in Fig. 5(a). The
maximum responsivity obtained at lower and higher wavelengths within the MWIR
spectrum is recorded as 0.9257 A/W. The same responsivity value is also
recorded at $\lambda_{c}$ = 4.33 $\mu$m. The nBn device here consist $t_{AL}$
of 2.7 $\mu$m. The VB barrier reduces carrier collection, resulting in
slightly lower responsivity values close to the cut-off wavelength. The
effects of photogenerated carriers altering the nBn barrier height at higher
temperatures could be responsible for the slight increase in the responsivity
observed at shorter wavelengths. Due to its large responsivity and low dark
current density, the proposed nBn design has a very high external quantum
efficiency.

The quantum efficiency, which in turn affects the flux of the photogenerated
carriers transported to the CL, is directly affected by the $\Delta E_{v}$.
The dependence of the quantum efficiency on the wavelength and applied bias
voltage is depicted in Figs. 5 (b) and (c). In Fig. 5 (b), we present the
dependence of the quantum efficiency on the wavelength at a temperature of 300
K for two different AL thicknesses. The photodetector exhibits a maximum
efficiency of 57.39% and 63.74% at a bias voltage of -0.25 V for 2.7 $\mu$m
and 3.0 $\mu$m AL thicknesses, respectively, with a radiation power density of
50 mW/$cm^{2}$. In Fig. 5 (c), we show the quantum efficiency at two different
wavelengths, 60%, and 70% of the $\lambda_{c}$ and it rises sharply to a
maximum value of 44.18% and 37.87%, respectively, at voltage nearly -0.25 V as
the reverse bias increases. Beyond this threshold of applied reverse bias, the
quantum efficiency remains unaffected by the VB offset. Since there is no
tunneling contribution and the dark current density slightly saturates,
showing the photoconductive effect, while quantum efficiency reaches its
maximum value, it can be stressed that such nBn structure may be operated
above -0.25 V.
In conclusion, a physics-based theoretical simulation model has been developed
via an iterative approach for the Poisson and the continuity solver to develop
a framework for the nBn MWIR photodetectors. A remarkable maximum efficiency
of 57.39% was achieved at room temperature when applying a bias of -0.25 V,
coupled with a radiation power density of 50 mW/$cm^{2}$. We have recorded the
maximum quantum efficiency of 44.18% and 37.87% at 60% and 70% of the
$\lambda_{c}$, respectively. Furthermore, a maximum responsivity of 0.9257 A/W
has been recorded within the MWIR spectrum. Through a comprehensive analysis,
we have demonstrated that our proposed device design effectively reduces the
dark current density by confining the electric field inside the barrier while
preserving a superior level of quantum efficiency, and the current in such
detectors is diffusion-limited. Hence, the G-R and tunneling currents do not
limit the high performance of the nBn architecture. Insights uncovered here
could be of broad interest to critically evaluate the potential of the nBn
structures for MWIR IR photodetectors.

## Acknowledgments

The authors acknowledge funding from ISRO, under the ISRO-IIT Bombay Space
Technology Cell. B.M. also acknowledges funding from the Science and
Engineering Research Board (SERB), Government of India, under Grant No.
CRG/2021/003102.

## Author declarations

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Rohit Kumar: Investigation (equal); Conceptualization (equal); Visualization
(lead); Data curation (lead); Methodology (lead); Formal analysis (lead);
Validation (lead); Writing-original draft (lead); Writing-review & editing
(lead). Bhaskaran Muralidharan: Investigation (equal); Conceptualization
(equal); Funding acquisition (lead); Project administration (lead); Resources
(lead); Supervision (lead); Writing- review & editing (equal).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Methodology

We solve the Poisson and the continuity equations for the carriers using the
finite difference method and obtain the electrostatic potential of the
heterojunction, hole quasi Fermi-level outside thermal equilibrium to build
the band structure of the considered device design. We assume that no G-R
processes are taking place within the BL, and the absorber region is uniformly
illuminated. The entire sequence of the numerical simulation is outlined in
Fig. A1. The subsequent equations presented herein are relevant for the
calculation of various transport and optoelectronics parameters associated
with the MWIR nBn photodetectors.

FIG. A1: Flowchart to calculate the transport and optoelectronics parameters
using the Poisson and continuity solver.

By using the linear interpolation technique, the lattice constant or the
mobility of IAS can be expressed as Adachi (2017)

$(a/\mu)_{{}_{IAS}}=x_{{}_{Sb}}\times(a/\mu)_{{}_{InSb}}+(1-x_{{}_{Sb}})\times(a/\mu)_{{}_{InAs}}\>,$
(1)

where $a$ denotes the lattice constant, $\mu$ represents the mobility of the
carriers and $x_{{}_{Sb}}$ is the molar fraction. In order to determine the
mobility at any given temperature, the mathematical expression is employed as
Schuster et al. (2014)

$\mu|_{{}_{T}}=\mu|_{{}_{(300\leavevmode\nobreak\
K)}}\times\Big{[}\dfrac{T}{300}\Big{]}^{-\zeta}\>,$ (2)

where $\mu|_{{}_{(300\leavevmode\nobreak\ K)}}$ is the mobility of the
carriers at room temperature. The value of $\zeta$ can be extracted from TABLE
S1. The diffusion current in the nBn photodetector is limited by the intrinsic
carrier concentration $n_{i}$ of the IAS material, which can be calculated
asRogalski and Jóźwikowski (1989); Rogalski (1989)

$\displaystyle(n_{i})_{{}_{IAS}}=(8.50\leavevmode\nobreak\
x_{{}_{Sb}}-6.73\leavevmode\nobreak\ x_{{}_{Sb}}^{2}-1.53\times
10^{-3}x_{{}_{Sb}}\leavevmode\nobreak\ T$ (3) $\displaystyle+4.22\times
10^{-3}\leavevmode\nobreak\ T+1.35)E_{g}^{0.75}\leavevmode\nobreak\
T^{1.5}\exp\Big(\dfrac{-E_{g}}{2k_{B}T}\Big{missing})\times 10^{14}\>,$

where $E_{g}$ is the bandgap of the material. The band gap of the IAS-based CL
and the AL as a function of molar composition and temperature has been
calculated using the following relation Wieder and Clawson (1973); Shaveisi
and Aliparast (2023); Schuster et al. (2014); Rogalski (1989); D’souza et al.
(2012)

TABLE A1:  Parameters for the binary materials at T = 300 K Schuster et al. (2014); Rogalski (1989); Adachi (2017); InS ; InA ; Vurgaftman et al. (2001); Sai-Halasz et al. (1977); Rogalski et al. (2020); Martyniuk et al. (2014). Material |  $a$ (Å) |  $\mu_{e}\leavevmode\nobreak\ (cm^{2}\leavevmode\nobreak\ V^{-1}\leavevmode\nobreak\ s^{-1}$) |  $\mu_{h}\leavevmode\nobreak\ (cm^{2}\leavevmode\nobreak\ V^{-1}\leavevmode\nobreak\ s^{-1}$) | $\zeta_{e}$ | $\zeta_{h}$
---|---|---|---|---|---
InSb | 6.4794 | $8\times 10^{4}$ | 800 | 1.8572 | 1.8572
InAs | 6.0583 | $3\times 10^{4}$ | 500 | 0.7212 | 0.5097
TABLE A2:  Parameters used for the device simulations Lackner et al. (2009); IAS ; Ren et al. (2016); Maddox et al. (2016); Krier et al. (2007); Bank et al. (2017).  Parameters | Unit | CL | BL | AL
---|---|---|---|---
$E_{g}|\leavevmode\nobreak\ _{(300\leavevmode\nobreak\ K)}$ | meV | 286 | 1160 | 286
$\epsilon_{r}$ | - | 15.298 | 15.5 | 15.298
$m^{*}_{e}$ | $m_{0}$ | 0.0197 | 0.071 | 0.0197
$m^{*}_{h}$ | $m_{0}$ | 0.416 | 0.35 | 0.416
$\mu_{h}|\leavevmode\nobreak\ _{(300\leavevmode\nobreak\ K)}$ | $cm^{2}\leavevmode\nobreak\ V^{-1}\leavevmode\nobreak\ s^{-1}$ | $5.27\times 10^{2}$ | $3\times 10^{3}$ | $5.27\times 10^{2}$
$D_{h}|\leavevmode\nobreak\ _{(300\leavevmode\nobreak\ K)}$ | $cm^{2}\leavevmode\nobreak\ s^{-1}$ | 13.636 | 77.625 | 13.636
$P_{d}$ | $mW/cm^{2}$ | 50 |  |
$|F_{1}F_{2}|$ | - | 0.2 |  |
$m_{0}$ | Kg | $9.109\times 10^{-31}$ |  |

$\begin{split}E_{g}(x_{{}_{Sb}},T)=0.411-\bigg{[}\frac{(3.4\times
10^{-4})\leavevmode\nobreak\ T^{2}}{(T+210)}\bigg{]}-0.876\>\\\
x_{{}_{Sb}}+\>0.70\>x^{2}_{{}_{Sb}}+\>(3.4\times
10^{-4})\>x_{{}_{Sb}}\leavevmode\nobreak\ T\leavevmode\nobreak\
(1-x_{{}_{Sb}})\>,\end{split}$ (4)

where $x_{{}_{Sb}}$ is the molar fraction, $E_{g}$ is the bandgap, and T is
the operating temperature. The absorption coefficient $\alpha$ for the IAS-
based AL depends on the incoming radiation wavelength (or photon energy) and
temperature and can be calculated using the following equations D’souza et al.
(2012); Shaveisi and Aliparast (2023); Schuster et al. (2014). When ($E_{g}$
$\geq$ $h\nu$), the absorption coefficient can be expressed using the Urbach
tail model as

$\alpha=948.23\leavevmode\nobreak\ \times\leavevmode\nobreak\
\exp\>\>[170\leavevmode\nobreak\ (h\nu-E_{g}-0.001)]\>,$ (5)

where $h\nu$ is the photon energy. Similarly, when ($E_{g}$ ¡ $h\nu$), the
absorption coefficient can be expressed as D’souza et al. (2012); Shaveisi and
Aliparast (2023); Schuster et al. (2014)

$\begin{split}\alpha=800\leavevmode\nobreak\ +\leavevmode\nobreak\
&\frac{K(h\nu-E_{g}-\Xi)\sqrt{(h\nu-E_{g}-\Xi)^{2}-\Xi^{2}}}{h\nu}\>,\\\
&\Xi=\frac{E_{g}}{2}+0.1\>,\\\ &K=10^{4}\leavevmode\nobreak\
[1+2\>E_{g}]\>.\end{split}$ (6)

We demonstrate the temperature and bias dependence of the carrier density of
the nBn photodetector in Fig. A2. The exponential relationship between the
carrier concentration and applied bias voltage results in a significant change
of the carrier density when there is a relatively small change in the
electrostatic potential distribution, as depicted in Fig. A2. As a result of
the reverse bias applied over the heterojunction, changes are observed in the
BL and the accumulation regions. The potential drop is anticipated to occur in
the BL because it is the constituent layer with more dopant atoms. An increase
in the reverse bias voltage results in a notable increase in the charge
density within the accumulation region at the contact-barrier junction. In
contrast, a substantial drop in the charge density within the accumulation
region at the barrier-absorber junction is observed, resulting in a decrease
in the carrier density at this junction. A positive carrier density indicates
the presence of ionized donors or holes in that region, whereas a negative
carrier density indicates that the majority charge carriers occupy that
region. Up to V $\approx$ -0.5 V, the carrier density in the BL is constant,
implying that only ionized donors are present. This behavior persists up to a
voltage of -0.5 V when the flat band condition is met, indicating that there
is no charge density at the barrier absorber edge. Figure A2 (b) provides a
clear illustration of the carrier density both before and after it meets the
flat band condition.

We demonstrate the dependence of the electric field on temperature and bias in
Fig. A3. The analysis of the nBn photodetector involves considering the
electric field, which provides valuable insights into the electrostatic
properties. In addition, it is an important consideration in the carrier
collection, as a high electric field in the BL may improve the photodetector’s
response time. The occurrence of undesired phenomena, such as BTB tunneling,
can be attributed to the presence of a high electric field. However, this
limitation in the nBn photodetector is effectively reduced through the
incorporation of a substantial CBO. The magnitude of the electric field in the
AL decreases as the applied bias becomes more negative, as illustrated in Fig.
A3. At approximately -0.5 V, the flat band condition takes place, and the
electric field is close to zero. As the AL is depleted in response to an
increasing negative applied bias, a negative electric field is produced.
However, the magnitude of the electric field at the contact barrier junction
grows as the applied bias is made more negative.

FIG. A2: Variations in the carrier density across the heterostructure in
relation to the applied bias voltages at (a) T = 192 K, (b) T = 250 K, and (c)
T = 300 K. The full extent of the AL region is not shown. Carrier density with
a negative value signifies the allocation of electrons in that region, whereas
a positive value indicates the presence of holes and positively ionized
donors.

FIG. A3: Bias and position dependence of the electric field across the
heterostructure at (a) T = 192 K, (b) T = 250 K, and (c) T = 300 K. The
complete width of the AL region is not depicted in the illustration as the
electric field remains constant beyond the depletion region.

##  Appendix B Recombination rate calculations

The radiative recombination mechanism is characterized by the recombination of
an electron in the conduction band with a hole in the valence band, resulting
in the emission of a photon due to an excess of energy. Therefore, the BTB
recombination coefficient, B included in our model, can be written as Bellotti
and D’Orsogna (2006); Pierret and Neudeck (1987); Rogalski (1989); Hall (1959)

$\begin{split}B=\Bigg{[}(5.8&\times
10^{-13})\>\epsilon^{0.5}_{\infty}\Big{(}\dfrac{m_{0}}{m^{*}_{e}+m^{*}_{h}}\Big{)}^{1.5}\Big{(}1+\dfrac{m_{0}}{m^{*}_{e}}+\dfrac{m_{0}}{m^{*}_{h}}\Big{)}\\\
\times&\Big{(}\dfrac{300}{T}\Big{)}^{1.5}(E^{2}_{g}+3k_{B}TE_{g}+3.75\leavevmode\nobreak\
k^{2}_{B}T^{2})\Bigg{]}\>,\\\ &R_{rad}=B\>(np-n^{2}_{i})\>,\end{split}$ (7)

where $R_{rad}$ is the BTB recombination rate, $k_{B}$ is the Boltzmann
constant, $\epsilon_{\infty}$ is the high-frequency dielectric constant, and
$m^{*}_{e}$ and $m^{*}_{h}$ are the effective masses of the electrons and
holes, respectively. Depending on the shape of the bands involved, there are
various types of Auger recombination processes. In the context of n-type
material, such as the IAS layers studied in the present work, the predominant
recombination mechanism is Auger 1. The Auger 1 process, characterized by its
non-radiative nature, exhibits dominance at higher temperatures. The Auger
carrier coefficients can be defined as Schuster et al. (2014)

$\begin{split}C_{n}=&\dfrac{\Big{(}\dfrac{m^{*}_{e}}{m_{0}}\Big{)}\leavevmode\nobreak\
|F_{1}F_{2}|^{2}}{2(n_{i}\leavevmode\nobreak\
\epsilon_{\infty})^{2}\leavevmode\nobreak\ (3.8\times
10^{-18})\Big{(}1+\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}^{0.5}\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}}\times\\\
&\Big{(}\dfrac{E_{g}}{k_{B}T}\Big{)}^{-1.5}\exp[-\dfrac{\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}E_{g}}{\Big{(}1+\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}k_{B}T}\Bigg{]}\>,\end{split}$ (8)

$C_{p}=C_{n}\left[\dfrac{1-\dfrac{3E_{g}}{k_{B}T}}{6\Bigg{(}1-\dfrac{5E_{g}}{4k_{B}T}\Bigg{)}}\right]\>,\\\
$ (9)

$R_{A1}=\Big{[}p\leavevmode\nobreak\ C_{p}+n\leavevmode\nobreak\
C_{n}\Big{]}\>(np-n^{2}_{i})\>,\\\ $ (10)

where $R_{A1}$ is the Auger 1 recombination rate. The overlap integral
$|F_{1}F_{2}|$ values range from 0.1 to 0.3. The key factor constraining the
carrier lifetime in the IAS layers is the Auger 1 recombination. Given that
holes are the sole carrier type capable of moving around the heterostructure,
the intrinsic Auger 1 carrier lifetime $\tau^{A1}_{i}$ can be expressed as
Bellotti and D’Orsogna (2006); Rogalski and Orman (1985)

$\begin{split}\tau^{A1}_{i}=&3.8\times 10^{-18}\leavevmode\nobreak\
\dfrac{\epsilon^{2}_{\infty}\Big{(}1+\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}^{0.5}\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}}{\Big{(}\dfrac{m^{*}_{e}}{m_{0}}\Big{)}\leavevmode\nobreak\
|F_{1}F_{2}|^{2}}\times\\\
&\Big{(}\dfrac{E_{g}}{k_{B}T}\Big{)}^{1.5}\exp[\dfrac{\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}E_{g}}{\Big{(}1+\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}k_{B}T}\Bigg{]}\>,\\\ \end{split}$ (11)

$\tau^{A1}_{h}=\dfrac{2\tau^{A1}_{i}}{1+\Big{(}\dfrac{n_{0}}{n_{i}}\Big{)}^{2}}\>,$
(12)

where $\tau^{A1}_{h}$ is the hole carrier lifetime due to the Auger 1
recombination process, and $n_{0}$ is the equilibrium electron concentration.

##  Appendix C Analysis of the dark current, optical responsivity, and the
quantum efficiency

The nBn architecture effectively mitigates the dark current contribution
resulting from the SRH recombination. Consequently, the dark current in this
architecture is mainly diffusion limited and can be calculated as Martyniuk et
al. (2013); Klipstein (2008); Savich et al. (2015); Kwan et al. (2021);
Martyniuk et al. (2014); Alchaar et al. (2019)

$J_{D}=q\dfrac{n^{2}_{i}}{N_{D}}\dfrac{L_{D}}{\tau}\dfrac{\tanh\Big(\dfrac{t_{{}_{AL}}}{L_{D}}\Big{missing})+\beta}{1+\beta\leavevmode\nobreak\
\tanh\Big(\dfrac{t_{{}_{AL}}}{L_{D}}\Big{missing})}\>,$ (13)

where $L_{D}$ is the diffusion length, $N_{D}$ is the donor density in the
absorber region, $\tau$ is the minority carrier lifetime, and $t_{{}_{AL}}$ is
the thickness of the AL. The surface recombination velocity, denoted as
$\beta$, can be neglected due to the boundary conditions that enforce the
absence of hole current at the interface between the AL and the CL. The
valence band potential barrier resulting from the electrostatics of the
junction has been taken into account. Only the thermally generated holes in
the AL that possess sufficient kinetic energy to overcome the barrier will
contribute to the dark current. Therefore, the dark current equation can be
modified as

$J_{D}=q\dfrac{n^{2}_{i}}{N_{D}}\dfrac{L_{D}}{\tau}\tanh\Big(\dfrac{t_{{}_{AL}}}{L_{D}}\Big{missing})\exp[\dfrac{E_{a}-3k_{B}T}{k_{B}T}\Big{]}\>.$
(14)

The photocurrent density $J_{photo}$ caused by the incident power density
$P_{d}$ was used to calculate the optical responsivity R as Itsuno et al.
(2011); Martyniuk et al. (2013); Shaveisi and Aliparast (2023); Schuster et
al. (2014); Kwan et al. (2021)

$R=\dfrac{J_{photo}}{P_{d}}\>.$ (15)

The quantum efficiency is a crucial parameter for assessing the optical
response of the nBn detector and can be calculated as Shaveisi and Aliparast
(2023); Martyniuk et al. (2013); Itsuno et al. (2011); Schuster et al. (2014);
Kwan et al. (2021); Akhavan et al. (2022); Martyniuk and Rogalski (2013)

$\eta=R\times\dfrac{h\nu}{q}\>.$ (16)

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