## Abstract

We have numerically analyzed the electromagnetic and electrical characteristics of InAsSb nBn infrared detectors employing a photon-trapping (PT) structure realized with a periodic array of pyramids intended to provide broadband operation. The three-dimensional numerical simulation model was verified by comparing the simulated dark current and quantum efficiency to experimental data. Then, the power and flexibility of the nBn PT design was used to engineer spectrally filtering PT structures. That is, detectors that have a predetermined spectral response to be more sensitive in certain spectral ranges and less sensitive in others.

© 2014 Optical Society of America

## 1. Introduction

There is a constant pursuit in the infrared detector industry to develop detectors with lower dark current, higher operating temperature and broadband (0.5 – 5.0*μ*m) quantum efficiency (QE). One such technique to tackle these requirements is to incorporate a micro-structured photon-trapping (PT) structure into the detector absorber layer (AL). This was performed on two separate detector array architectures: HgCdTe *P*-on-*n* photodiodes [1–3] and InAsSb nBn detectors [4, 5]. These focal plane arrays are normally backside illuminated where the incident light is absorbed through the transparent substrate on which the detector array is grown. Such conventional detectors suffer from significant reflection losses at the air/semiconductor interfaces. Consequently, an anti-reflection (AR) coating is incorporated to reduce these losses. Unfortunately, it is very difficult to engineer an AR coating that works effectively over a wide spectral range, that is one that offers nearly zero reflectance and consequently 100% external QE, assuming unity internal QE. Additionally, an AR coating can introduce stress into a detector limiting its interoperability or lifespan. An alternate approach is to instead remove the substrate and etch the PT structure directly into the detector AL. When properly engineered, the PT structure will significantly reduce reflection losses, negating the need for an AR coating [2, 6, 7]. Furthermore, by removing device volume, the detector dark current is potentially lowered, allowing higher operating temperatures to be achieved. However, this technique exposes the narrow gap AL mandating that the surface be properly passivated to reduce surface recombination current and maintain high device performance.

The nBn detector is a new type of quantum semiconductor detector that was introduced relatively recently [8, 9]. This new detector consists of a narrow-gap *n*-type AL, a wide-gap depleted barrier layer (BL), and a narrow-gap *n*-type contact layer (CL). The BL presents a large barrier to electron flow, and the nBn detector operates as a unipolar unity-gain minority carrier device. A key benefit of the nBn architecture is that, for a wide range of design parameters, there is no depletion region in the narrow-gap layers, thereby eliminating the space-charge generation-recombination (G-R) dark currents that have plagued conventional InAs and InAsSb junction photodiodes and severely limited their applicability for high sensitivity requirements to lower temperatures. Another key benefit of the nBn architecture is self-passivation. The narrow-gap AL is buried beneath the wide-gap BL effectively eliminating surface effects, thereby negating any additional dark current caused by the exposing the AL. The focus of the current work is now to leverage the power and flexibility of the nBn PT design to engineer spectrally filtering PT structures. That is, detectors that have a predetermined spectral response to be more sensitive in certain spectral ranges and less sensitive in others. The target performance will of course depend on the specific application and will be system dependent. The aim of this paper is to establish general techniques and methodologies that can be used in any region of interest and then tailored to specific applications.

To simulate the InAsSb nBn detectors a robust three-dimensional (3D) numerical simulation model has been developed that takes into account the composition, doping and temperature dependence of the InAsSb alloy. Furthermore, the optical parameters depend on both alloy composition and temperature. The simulation procedure is a two step process. First, the optical response is computed by performing an electromagnetic analysis of the entire structure using the finite-difference time-domain (FDTD) method [10,11]. Subsequently, the electrical analysis is performed using the finite element method (FEM) [12] to solve the drift-diffusion equations.

The manuscript is organized as follows: Section 2 describes the numerical model and geometry, Section 3 presents the experimental verification of the numerical model, Section 4 presents the electromagnetic and electrical response of the spectrally filtering detectors and finally, Section 5 summarizes the outcome of this work. Appendix A analyzes the dependence of the PT structure on non-normal angles of incidence; and appendices B and C describe the InAsSb and AlAsSb material models used in this work.

## 2. Numerical model and geometry

The procedure to simulate the devices is outlined in Fig. 1. The first step is to discretize the device into a grid of points or mesh. The FDTD and FEM simulations utilize two completely distinct meshes. The FDTD structured tensor mesh, shown in Fig. 1(a), is created based on the optical properties of the materials being used and is discretized using rectangular prisms. Unlike the structured tensor mesh, the finite element mesh, shown in Fig. 1(c), consists of triangular prisms of varying dimensions. Next, the optical characteristics are computed by performing an electromagnetic analysis of the device using the FDTD method. The simulations employ the use of absorbing boundary conditions in the active direction and periodic boundary conditions in the lateral direction which periodically replicates the device to form an infinite array of pixels. Solving Maxwell’s equations at every node in the mesh at each point in time yields the absorbed power density from which the optical generation rate is calculated. Specifically, a total-field scattered-field approach is employed to compute the reflected and transmitted power. Using the results of the FDTD simulation the optical generation rate is calculated in the device, as well as the total reflectance when the device is illuminated with planewaves. Alternatively, an array can instead be illuminated with a Gaussian beam to compute the optical crosstalk by integrating the optical generation rate in each pixel of the array. The output from the FDTD simulation is shown in Fig. 1(b) and Fig. 1 box I.

Subsequently, the electrical analysis is performed using the FEM to solve the drift-diffusion equations employing ideal Neumann boundary conditions. This requires solving self-consistently the electron and hole continuity equations coupled with Poisson’s equation. The optical generation terms in the electron and hole continuity equations are included by interpolating the optical generation rate evaluated from the FDTD simulations onto the finite element mesh, shown in Fig. 1(d). The FEM simulations yield quantities such as the hole current density, shown in Fig. 1(e), and currents at the contacts from which the QE, crosstalk and modulation transfer function (MTF) can be calculated, shown in Fig. 1 box II. Note that calculation of the crosstalk and the MTF requires at least a 3 × 3 pixel array. Such an array was not considered here since the current work is focused on studying the underlying physics and spectral sensitivity and as such a single pixel is sufficient. A detailed description of the FDTD method is provided in [10]. Utilizing the FEM method to simulate infrared detectors is discussed in [13,14] and [3,15,16] extensively cover the coupled usage of the FDTD and FEM methods to simulate infrared detectors.

The nBn device modeled in this work was designed to closely resemble those reported [4,5] which are grown on GaAs substrates. On the substrate is grown a thick narrow-gap InAs_{1−x}Sb* _{x}* AL. On the AL a thin wide-gap BL made of AlAs

*Sb*

_{x}_{1−x}, which is nearly lattice matched to InAs

_{1−x}Sb

*, is deposited. Then, a thin narrow-gap InAs*

_{x}_{1−x}Sb

*CL is grown. Finally, a 100 nm thick gold layer terminates the device. The specifications for the AL, BL and CL are given in Table 1. A complete description of the InAs*

_{x}_{1−x}Sb

*and AlAs*

_{x}*Sb*

_{x}_{1−x}material models is given in Appendices B and C respectively. Using the formulas in Appendix B,

*x*= 0.210 and

*N*= 1.2 × 10

_{D}^{16}cm

^{−3}, InAsSb has a total lifetime of 304 ns (

*L*= 21

_{p}*μ*m) at 150 K and 408 ns (

*L*= 24

_{p}*μ*m) at 200 K and the dominant mechanism effecting the lifetime is Auger recombination. The total lifetimes were calculated assuming a Shockley-Read-Hall (SRH) lifetime of 10

*μ*s. To create the PT structure the GaAs substrate is removed and pyramids are etched into the AL [4, 5]. Several techniques exist to etch the pyramids including inductively coupled plasma dry etching [17] and interference lithography [6, 7]. The pyramids are initially assumed to be 3

*μ*m at the base (

*w*

_{PT}= 3

*μ*m) and 4

*μ*m in height (

*t*

_{PT}= 4

*μ*m), leaving another 1

*μ*m of AL beneath the pyramids (

*t*

_{AL}= 1

*μ*m). A schematic representing a single pixel incorporating the PT structure is shown in Fig. 2. There are several advantages to removing the substrate and etching the pyramids directly into the detector AL. First, is that etching the PT structure directly into the detector AL, and not the substrate, should decrease the dark current due to the removal of device volume. Second, by removing the substrate the detector spectral response can be extended down to

*λ*= 0.5

*μ*m. At this wavelength the GaAs substrate is absorbing and must be removed. Finally, etching the pyramids directly into the detector AL should greatly decrease the inter-pixel crosstalk in pixel arrays [3], especially when reducing the pixel pitch.

The model assumes ideal contacts with a common cathode implemented as an external ring. Beneath the cathode contact is a heavily doped *n*-type region that extends 0.2*μ*m into the detector AL. This region serves to repel minority carries away from the cathode. In realized arrays several rows of outer pixels along the edge of the array are shorted out through deposition of a metal contact to form the cathode [5]. This metal contact extends along the outside boundary of the array. The PT structure is not etched into these edge pixels to allow for the placement of the cathode. In the simulations it is not realistic to simulate these extra pixels solely to form the cathode. Consequently, the cathode was placed directly on the sole active pixel of interest, necessitating the need for the heavily doped *n*-type region to repel carriers away from the cathode, mitigating effects related to placing the cathode on the active pixel. Additionally, a perfectly reflecting gold layer is placed at the top of the CL on the mesa during the FDTD simulations to account for the presence of the metal contact in a physically realized device. This step is crucial for longer wavelengths where the detector operates as a multi-pass device. To allow for proper simulation of the interfaces vacuum is inserted above and below the device which allows for calculation of the electric fields extending beyond the device boundaries.

## 3. Experimental verification

Before simulating the electromagnetic and electrical response presented in Section 4 the numerical and material models must first be verified. This is accomplished by computing the dark current density and QE and comparing the numerical results to experimental data reported in [18]. The experimental structure employed a MWIR InAsSb alloy compound barrier (CB) detector similar to that described previously in [4, 5] with the exception that the detectors fabricated, characterized and reported in [4, 5] incorporated a PT structure, while the data in [18] is of a non-PT structure. It should be noted that in all cases, both non-PT and PT, that the devices are diffusion limited at the operating bias, with very similar activation energies, calculated from the slope of an Arrhenius plot of dark current *versus* temperature, see ([5], Fig. 11). This is due to the unique properties of dark current suppression in nBn detectors. Therefore, there is no significant contribution to the dark current due to defects attributed to etching the PT structure directly into the detector AL. Consequently, it is acceptable to validate the numerical simulations using non-PT nBn device data.

The devices were grown with an alloy composition of *x* = 0.195 for the InAs_{1−x}Sb* _{x}* AL providing a cutoff wavelength of 4.87

*μ*m at 150 K. The InAsSb alloy material was processed into detector structures in a 1024 × 1024 format on an 18

*μ*m pitch. The 1024 × 1024 detector structure was hybridized to a fanout chip that had metalization of different areas on it to permit the shorting of multiple 18 × 18

*μ*m

^{2}detectors, thereby forming large area detectors that could be characterized through a fanout chip. Several detectors of different areas were characterized. An example of the detectors that were characterized is a 64 × 64 substructure (i.e. 4096 18 × 18

*μ*m

^{2}detectors) of detectors that were connected together at the fanout metal level to form a large 1152 × 1152

*μ*m

^{2}detector structure. The measured dark current density for two such substructures, a 64 × 64 substructure and a 20 × 20 substructure are plotted in Fig. 3(a) at two different temperatures: 150 K and 200 K. In Fig. 3(a) is also plotted the simulated dark current density at the same two temperatures. The simulations in Fig. 3(a) are of a very large single non-PT pixel in two dimensions and the CB has been modeled as a single AlAsSb layer. It is observed that there is excellent agreement between the measurement and the simulations, especially where the reverse bias current saturates at −1.0 V. For low reverse bias the current is due to diffusion current and at larger reverse bias it is believed that the increase in current is due to tunneling as the dynamic impedance decreases as the magnitude of the reverse bias is increased beyond −1.0 V.

The data also indicates that the experimented devices have a large barrier in the valence band. Figure 3(b) plots the QE *versus* voltage for a 3D device with *x* = 0.195 at 150 K. The measured devices do not begin to turn-on until *V* ≈ −0.5*V* and in fact the QE does not reach 66 % until *V* = −1.0 V and eventually the QE increases to 70 % at *V* = −2.0 V. Consequently, there is a substantial valence band offset (VBO) that is blocking the flow of photo-generated holes from the AL to the CL. Currently, no data in the literature exists to calculate the VBO for InAs_{1−x}Sb* _{x}* at these compositions. Therefore, the VBO as been empirically determined by matching the turn-on voltage from the J(V) characteristics and the QE. Excellent agreement was achieved between the dark current density for the data and simulations in Fig. 3(a) using a VBO of 0.18 eV at 150 K and 0.20 eV at 200 K. The VBO offset is the only fitting parameter used in the simulations.

The InAsSb detectors are back-illuminated through the GaAs substrate which has a reflectance of 29 % due to the absence of an AR coating. Consequently, the QE can be at most 71 %. Therefore, a ≈ 0.70 % QE implies an internal QE that is close to 100 %. At *V* = −1.0 V and a wavelength (*λ*) of 2*μ*m the simulations yielded a QE of of 65.6%, which is in good agreement with analytical calculations at 65.3% and the data at 66.1%. Calculating the QE analytically [19] yields an internal QE of 96.2%, which reduces to 65.3% after factoring in 32.1% reflection losses (reflection at air/InAsSb interface at *λ* = 2.0*μ*m). Therefore, at *V* = −1.0 V the simulations agree to within 1% with the experimental data and the analytical calculation. Note that the simulations in Fig. 3(a) were 2D simulations and used a molar fraction of *x* = 0.195. All subsequent simulations in this paper are 3D simulations with molar fraction of *x* = 0.210.

## 4. Electromagnetic and electrical response

In this section the electromagnetic and electrical characteristics of the InAs_{0.79}Sb_{0.21} nBn PT device are evaluated. For this analysis the pixel was illuminated with an infinite planewave and the reflectance (*R*) from the structure was calculated according to

*I*

_{0}=

*ϕhc/λ*where

*ϕ*is the incident photon flux,

*h*is Planck’s constant,

*c*is the speed of light in vacuum and

*λ*is the wavelength. The reflected intensity

*I*is computed from the simulations. Then, the QE for a uniform (un) illumination source (planewave)

_{R}*η*

_{un}is given by [20] where

*I*

_{ph}is the photocurrent,

*q*is the elemental charge,

*ϕ*is the photon flux and

*A*is the illumination area. One should notice that although Eq. 2 does not formally include reflection losses, such losses are automatically incorporated when using the optical generation rate from the FDTD simulation to compute the photocurrent. Therefore, Eq. 2 is an expression for the total QE. All subsequent simulations are of a single pixel at a 12

*μ*m pitch,

*T*= 150 K,

*V*= −1.0 V and

*ϕ*= 1 × 10

^{17}photonscm

^{−2}s

^{−1}at normal incidence. Appendix A discusses the dependence of the PT structure on non-normal angles of incidence.

#### 4.1. Uniform square lattice of pyramids

We begin by considering a pixel with a uniform lattice of pyramids. Figure 4(a) shows the simulated reflectance *versus* wavelength for pyramids whose base (*w*_{PT}) was varied from 1 – 4*μ*m, calculated according to Eq. (1). In all cases the pyramid height was held constant (*t*_{PT} = 4*μ*m). For reference the reflectance for the non-PT structure has also been plotted in Fig. 4(a) as well as in all subsequent reflectance plots. To maintain a fair comparison, the total thickness of the non-PT pixel is always identical to the total thickness of the PT pixel. The reflectance is always calculated in *λ* = 0.05*μ*m increments with finer resolution occasionally used in specific areas of rapid change. It is seen that the reflectance is significantly lower for the four PT structures compared to the non-PT structure. The reflectance for the non-PT structure is ≈ 30%, but the reflectance for the PT structure can be tailored to be below 1 % by adjusting *w*_{PT}. Note that as *w*_{PT} is increased the sides of the pyramids become less steep. In fact as *w*_{PT} is increased from 1*μ*m to 4*μ*m the angle of the sides decreases from 82.0° to 63.4° (as measured from the pyramid base). These results are in agreement with [4] and demonstrate that the PT structure successfully acts as an AR coating. Similar results are obtained using different geometries such as pillars (see [2]). The key to the PT structures success is the following: at shorter wavelengths the PT structure scatters the incident radiation between the pyramids causing multiple reflections between pyramids which increases the absorption; and at longer wavelengths the PT structure acts as a continuously graded effective medium.

Figure 4(b) presents the calculated QE *versus* wavelength for both the non-PT and PT structures, calculated according to Eq. (2). Only the PT structure with *w*_{PT} = 2*μ*m was simulated. As explained before, such results automatically include the reflection losses at the surface. From Fig. 4(b), it is immediately noticed that the maximum QE for the non-PT structure is ≈65% which is limited by reflection losses at the backside at the air/InAs_{0.79}Sb_{0.21} interface (*R* ≥ 31.5%) in the case when an AR coating in not used. This clearly contrasts with the result for the PT structure for which the QE is in general above 90 % even without an AR coating. The PT QE decreases steadily past *λ* = 3.0*μ*m, in accordance with the reflectance increasing. An effect that may be mitigated by utilizing taller pyramids. It is also immediately possible to see that there is significant peaking in the photo-response near the cutoff (especially for the non-PT structure). This is due to reflection from the contact, the mesa sidewalls, and the back of the structure, which produces significant interference for wavelengths near the cutoff wavelength [15].

#### 4.2. Uniform square lattice of pyramids with pseudo-cavity

The goal of the current work is to engineer the PT structure to yield a specific predetermined spectral response. This can be accomplished by removing pyramids from the center of the pixel as shown in Fig. 5. It should be understood that the structure in Fig. 5(b,c) does not strictly constitute a cavity in the conventional sense and as such it is referred to as a pseudo-cavity. The pseudo-cavity area *A*_{cav} can be varied be removing subsequent rows of pyramids. Such a structure can be physically realized by not etching pyramids into the center of the pixel and instead etching away nearly all of the AL. The reflectance for pixels incorporating a pseudo-cavity is plotted in Fig. 6(a) for pixels with pseudo-cavity areas ranging from 4 × 4*μ*m^{2} to 10 × 10*μ*m^{2}. It is observed that the presence of the pseudo-cavity results in the reflectance assuming a periodic behavior. The strength (magnitude) of the periodicity significantly depends on the pseudo-cavity area. This can be explained in the following way. The presence of the pseudo-cavity perturbs the structure, where the strength of the perturbation is proportional to the pseudo-cavity area. The larger the pseudo-cavity the larger the perturbation and the more dominant is the periodicity. As pyramids are removed the amount of absorbing material is decreased which increases the reflectance. Furthermore, when the pseudo-cavity is large almost all of the pyramids are removed and the majority of the absorbing material that remains is the 1*μ*m of AL beneath the pyramids. The periodicity is caused by incident light being absorbed into the AL and being internally reflected between the AL/air interface and the gold contact on the front-side of the pixel. In effect, the AL is acting as a Fabry-Pérot cavity.

The reflectance for pyramids ranging from 2 – 4*μ*m in height for a constant pseudo-cavity area of 10 × 10*μ*m^{2} is plotted in Fig. 6(b). It is observed that the pyramid height does not significantly affect the reflectance. This is because for such a large pseudo-cavity area the perturbation due to the pseudo-cavity is the dominant mechanism effecting the reflectance which screens out the dependence of the reflectance on the pyramid height.

The QE is plotted in Fig. 7 for the non-PT structure, the PT structure with a uniform lattice of pyramids and the PT structure with a cavity, The total AL thickness in Fig. 7 is 3*μ*m compared to the 5*μ*m in Fig. 4(b) and consequently, the devices have a softer cutoff. It is seen that the periodicity in the reflectance is translated into the QE resulting in a 20% contrast at longer wavelengths. The period of the periodicity increases at shorter wavelengths but the magnitude of the difference of the QE is much less than that of the reflectance. Therefore, it is necessary for a very large contrast in the reflectance to be achieved for a strong periodicity to be evident in the QE. While this approach is successful, the magnitude of the QE is much lower than without a cavity. It is expected that if a smaller pseudo-cavity is used that the overall magnitude of the QE will increase, but that the magnitude of the periodicity will decrease and eventually diminish to that of the PT structure without a cavity.

The dependence of the reflectance on the thickness of the AL beneath the pseudo-cavity is shown in Fig. 8(a) as the AL thickness is reduced from 1.0*μ*m to 0.1*μ*m. It is seen that the location of the inflection points significantly depends on the AL thickness. In fact for strong periodicity at shorter wavelengths a thinner AL is required and for a stronger periodicity at longer wavelengths a thicker AL is required. To obtain strong periodicity at shorter wavelengths, the AL thickness must be decreased to 0.5*μ*m or even less. It is difficult to go thinner since there is now nowhere to place the cathode, without simulating extra edge pixels without a cavity. While this is possible it will greatly increase the size of the problem with no real advantages. The magnitude of the periodicity of the QE is much stronger at shorter wavelengths with a 0.5*μ*m thick AL, but the overall QE is much lower, as is shown in Fig. 8(b).

#### 4.3. Uniform hexagonal lattice of holes

As an alternative to pyramids, holes are now used to tune the spectral sensitivity. Initially, consider only holes that are perfectly cylindrical. It was found that for holes to be most effective they must be very densely packed. This is best accomplished by using a hexagonal lattice, whereas the pyramids in the previous sections were arranged in a square lattice (see [2] for a discussion on the differences between arranging PT structures in square *versus* hexagonal lattices). Figure 9(a) shows a single pixel with a hexagonal lattice of 128 holes that are 1.0*μ*m in diameter (*D*_{base} = *D*_{top} = 1.0*μ*m). The total thickness of the AL is 3.0*μ*m and the holes are 2.0*μ*m deep (*d* = 2.0*μ*m). Figure 9(b) depicts a cross section of the holes taken along the dashed black line in Fig. 9(a) and Fig. 9(c) shows the hole dimensions *D*_{base}, *D*_{top} and *d*. The reflectance *versus* wavelength for holes of different diameters is shown in Fig. 10(a). It is observed that the reflectance is on average 10 %. This is still 20 % lower than the non-PT structure, but it is clear that the holes are not nearly as effective at reducing the broadband reflectance as pyramids. It is observed that there is a sharp dip (≈ 10%) in the reflectance at wavelengths immediately longer than *D*_{top}. This feature is prevalent only in a densely packed lattice and will diminish if the number of holes is reduced.

Figure 10(b) plots the QE for the non-PT structure and the PT structure with 1.0*μ*m diameter perfectly cylindrical holes. It is seen that the holes do offer a significantly higher QE than the non-PT structure. For short wavelengths the QE for the hole structure is ≈ 15% higher than the non-PT structure and is consistently higher than the non-PT structure until *λ* ≈ 3.5*μ*m. It is seen that the sharp dip in the reflectance from *λ* = 1.05 – 1.45*μ*m translates into an increase in the QE over that spectral range. However, the two very sharp dips in the reflectance at *λ* = 1.05*μ*m and *λ* = 1.40*μ*m have no effect on the QE. That is because these dips decrease the reflectance from 1% to 0.1% which is negligible in terms of the QE.

In reality, it may not be possible to etch holes that are perfect cylinders. For this reason holes that are in the shape of cones are now considered (see Fig. 9(b)). Figure 11(a) shows the reflectance for three structures: holes in the shape of perfect cylinders (*D*_{base} = 1.0*μ*m, *D*_{top} = 1.0*μ*m), holes in the shape of perfect cones (*D*_{base} = 1.0*μ*m, *D*_{top} = 0.0*μ*m) and holes in the shape of truncated cones (*D*_{base} = 1.0*μ*m, *D*_{top} = 0.5*μ*m), with truncated cones being a truly realistic geometry. *D*_{base} is the diameter at the device edge where the light is incident (*z* = 0*μ*m) and *D*_{top} is the diameter at the other end of the hole (*z* = 2*μ*m). Then, there is another 1.0*μ*m of AL beneath the holes, followed by the BL, CL and finally the gold contact. The holes in the shape of perfect cones now have a single pronounced sharp dip near *λ* = 1.3*μ*m, whereas the perfect cylinder had two. The truncated cones do not have any pronounced sharp dips, but still have the 8 % drop in reflectance from *λ* = 1.05–1.45*μ*m indicating that even with sloped sides the holes still reduce the reflectance and can offer greater sensitivity over a short range. There are also slight differences in the periodic behavior at longer wavelengths, especially the height and location of the maximum near *λ* = 4*μ*m.

Figure 11(b) plots the QE *versus* wavelength for the hole structure with holes in the shape of perfect cylinders (*D*_{top} = 1.0*μ*m) and truncated cones (*D*_{top} = 0.5*μ*m). The two structures have relatively similar QEs with four exceptions. The first is that the truncated cones have a more “square”-like profile from *λ* = 1.05 – 1.45*μ*m than the perfect cylinders. The second is that at wavelengths below *λ* = 1.00*μ*m the QE increases for the truncated cones, but decreases for the perfect cylinders. Third, is that in the vicinity of *λ* = 4.00*μ*m the truncated cones have a slight peak whereas the perfect cylinders has a substantial dip and lastly the perfect cylinders have a slightly softer cutoff. These attributes make the hole structure with truncated cones more desirable than its counterparts and it has the added benefit that it should be slightly easier and more realistic to fabricate.

## 5. Conclusions

The authors have developed a three-dimensional numerical simulation model and material model to realistically predict the performance of InAsSb nBn detectors incorporating a photon-trapping structure. The model was validated by comparing the numerical results to data provided by DRS with excellent agreement in the dark current density *versus* voltage and the peak quantum efficiency. The material model is fully temperature and molar fraction dependent and utilizes minimal fitting parameters. The authors have simulated the performance of currently realized photon-trapping structures and have shown that photon-trapping structures can be engineered to yield a predetermined spectral response. Various types of photon-trapping structures have been explored including pyramids and holes in the shape of cylinders and cones. The ideal PT structure is application dependent. If the absolute minimum reflectance is desired over a broadband spectrum then pyramids and pillars [2] are the best structures. Holes are not as effective over a broadband range as pyramids or pillars, but are effective over narrower ranges. Other structures exist that have not been treated in this paper, including random texture AR micro-structures [7] or asymmetric (sawtooth) and general Fourier series micro-structures [21]. Using the techniques developed in this paper it will be possible to engineer individual detector pixels to be particularly and uniquely sensitive (compared to their neighbors) over any spectral range that is desired.

## A. Angular dependence of PT structure

Infrared detectors are seldom illuminated at normal incidence. For example, an optical system with *f*/# = 3 would have an angle of incidence of 9.46°, *f*/# = 2 would have an angle of incidence of 14.04°, and the angle of incidence increases substantially as *f*/# is decreased [22]. Therefore, it is desirable to asses the angular dependence of the PT structure which is plotted in Fig. 12(a) for the non-PT structure in linear scale and Fig. 12(b) for the non-PT and PT structures in logarithmic scale. Also plotted in Fig. 12(a) is an analytical expression for the reflectance from a solid slab which will serve as a reference case. The reflection coefficients are given analytically in [23, Eqs. (6.2-6) and (6.2-7)] by

*μ*

_{1}=

*μ*

_{2}=

*μ*

_{0}(non-magnetic material),

*μ*

_{0}is the free space permeability in units of [H/m],

*ε*

_{1}=

*ε*

_{0}(vacuum),

*ε*

_{2}= 13.4492

*ε*

_{0}(InAs

_{0.805}Sb

_{0.195},

*λ*= 3

*μ*m),

*ε*

_{0}is the free space permittivity in units of [F/m], ${\eta}_{1}=\sqrt{{\mu}_{1}/{\epsilon}_{1}}$, ${\eta}_{2}=\sqrt{{\mu}_{2}/{\epsilon}_{2}}$ and ${\theta}_{2}=\text{arccos}\sqrt{1-{({n}_{1}/{n}_{2})}^{2}{(\text{sin}{\theta}_{1})}^{2}}$ where

*θ*

_{1}is the initial angle of incidence of the incoming radiation. The reflected power (

*R*= |

*r*

_{x(TE),y(TM)}|

^{2}) is plotted in Fig. 12(a).

It is seen that the reflectance for the non-PT structure increases for TE polarized radiation and decreases for TM polarized radiation. This is in agreement with the analytical expressions. The reflectance is initially 30% at normal incidence and does not significantly change until the angle of incidence is beyond 20°. Consider now the reflectance for the PT structure, shown in Fig. 12(b). At normal incidence the reflectance for the PT structure is 0.1%, it increases to 1.0% at 30° and 3.3% at 50°. Therefore, even at very large angles of incidence the PT structure still maintains a very low reflectance.

## B. InAsSb material model

To simulate the nBn device structures in this work a detailed model of the InAs_{1−x}Sb* _{x}* material characteristics are required. A comprehensive set of all of these parameters is not available in the open literature and as such the authors had to gather the material characteristics from multiple sources and in some instances interpolate quantities from the binary compounds. For convenience the comprehensive set of parameters is presented here.

The electron affinity has been taken from [24] where a value of 4.90 eV ([24], Chap. 17, pp. 601) is given for InAs and 4.59 eV ([24], Chap. 18, pp. 642) is given for InSb. The value for InAsSb is obtained by linearly interpolating between the values for the binaries. The static dielectric constant is given by *ε* = 15.15 + 1.65*x* and the high frequency dielectric constant is given by *ε*_{∞} = 12.3 + 3.4*x* [25] where *x* is the molar fraction which is equal to zero for InAs and one for InSb. The effective masses are given by [25]

The model that has been implemented uses the widely accepted equations for the InAs_{1−x}Sb* _{x}* energy gap [26]

*T*is the temperature in degrees kelvin and the energy gap is given in eV. The InAs

_{1−x}Sb

*intrinsic carrier concentration is given by [26]*

_{x}*n*(

_{i}*x*,

*T*) is given in cm

^{−3}and

*k*is the Boltzmann constant.

_{B}The material model uses the mobility values of InAs and InSb from [27] as an initial input. For convenience these values are recorded in Table 2. Note that [27] only contains values for the binary constitutes at 77 K and 300 K. To obtain values for all intermediate temperatures, namely 150 K and 200 K, the values from [27] are fitted to

*ζ*from fitting Eq. (7) to the data are also provided in Table 2. Note that Eq. (7), using the values tabulated in Table 2, is also in acceptable agreement with the range of data provided in [28]. To obtain the mobility for InAs

_{e,h}_{1−x}Sb

*at an arbitrary temperature*

_{x}*T*, the mobility values for InAs and InSb at that temperature are determined and then a linear interpolation is performed between the two binaries to obtain the value for InAs

_{1−x}Sb

*.*

_{x}The device model also takes into consideration both radiative and Auger-1 recombination mechanisms [13]. The band-to-band radiative recombination rate is given by [13]

in units of [cm^{−3}s

^{−1}]. In the limit of low level injection the lifetime can be given by [13] where

*n*

_{0}and

*p*

_{0}are the equilibrium electron and hole concentrations respectively. The radiative coefficient

*G*[cm

_{R}^{3}s

^{−1}] is given by [13]

*ε*

_{∞}is the high frequency dielectric constant and ${m}_{e}^{*}$ and ${m}_{h}^{*}$ are the effective masses for electrons and holes respectively.

The Auger recombination rate is given by [13]

In the limit of low level injection the Auger lifetime is given by [13] and the Auger coefficients*C*and

_{n}*C*[cm

_{p}^{6}s

^{−1}] are given by [13]

*F*

_{1}

*F*

_{2}| range from 0.1 to 0.3 [26]. In this work a value of 0.1 has been chosen as this agrees best with the data and other independent modeling of this particular alloy [29].

The model also takes into consideration SRH recombination. The SRH lifetime is very dependent on the material quality and the substrate used. For these reasons the SRH lifetime in the simulations was set at 10*μ*s for both electrons and holes. This ensured that in the simulations SRH recombination was not dominant in comparison to radiative and Auger recombination.

Additionally, the model of [30] has been adopted to account for the narrowing of the bandgap in InAs due to doping and is given by

*N*is taken to be the electron dopant concentration. At doping levels of 10

_{d}^{15}– 10

^{16}cm

^{−3}the bandgap narrowing should be negligible. However, as the doping approaches 10

^{17}– 10

^{18}cm

^{−3}, the bandgap narrowing becomes quite considerable. Consequently, unless otherwise stated, we only activated bandgap narrowing in the heavily doped CL and not the lighter doped AL; as it is believed that Eq. (15) significantly overestimates the bandgap narrowing in lightly doped material. Equation (15) is valid only for InAs. However, due to the very small variation in bandgap narrowing between InAs and InSb and the lack of experimental data, Eq. (15) has also been used for the InAs

_{0.805}Sb

_{0.195}and InAs

_{0.790}Sb

_{0.210}alloys.

The wavelength dependent refractive index was obtained by fitting a polynomial curve to the data in [31] for InAs_{1−x}Sb* _{x}* with

*x*= 0.20. The data and the fitted curve are plotted in Fig. 13(a). The wavelength dependent optical absorption coefficient

*α*has been calculated according to

*α*= 4

*πk/λ*using the values of the extinction coefficient

*k*that have been extracted from the review book by Palik and Holm ([32], pp. 485, Table VI). The absorption data was fitted to the following expressions [5]

*E*

_{0}=

*E*+ 0.001,

_{g}*K*= 10000 + 20000

*E*,

_{g}*E*=

*hc/λ*is the photon energy and

*λ*is the wavelength. The absorption coefficient

*versus*wavelength is plotted in Fig. 13(b) for the molar fractions and temperatures used in this work.

## C. AlAsSb material model

The static dielectric constant has been taken from [24] where a value of 11.21 is given for AlSb and 10.06 is given for AlAs. The value for AlAsSb is obtained by linearly interpolating between the values for the binaries. In order to calculate the energy gap for the ternary alloy AlAs* _{x}*Sb

_{1−x}the energy gap of its binary compounds AlAs and AlSb are first calculated according to the Varshni form for the energy gap of a binary semiconductor [33]

*α*and

*β*are the Varshni parameters for AlAs and AlSb, extracted from [33] and provided in Table 3 for conviencence, and

*E*(

_{g}*T*= 0) is the energy gap at zero kelvin.

In order to now determine the energy gap of AlAs* _{x}*Sb

_{1−x}it is assumed that the alloy energy gap depends on its constituent binary compounds according to

*C*is the bowing parameter that accounts for the deviation from a linear interpolation between the two constituent compounds. For AlAs

*Sb*

_{x}_{1−x}, AlSb corresponds to compound

*A*in Eq. (18), AlAs corresponds to compound

*B*and the bowing parameter is

*C*= 0.28 eV. All recombination lifetimes are taken to be very long. Such an assumption is valid since AlAsSb is a wide-gap material at cryogenic temperatures. Additionally, the AlAsSb alloy is assumed to be non-absorbing in the spectral range of interest.

_{AlAsSb}The VBO at the InAsSb/AsAsSb interface is a very important parameter in devices and can be immediately determined for the InAs/AlAs_{0.18}Sb_{0.82} interface. Consider the VBO values for InAs (−0.59 eV), AlAs (−1.33 eV), and AlSb (−0.41 eV) (all relative to that for InSb) taken from the review by [33]. Linear interpolation between the VBO for AlAs and AlSb gives the value of −0.5756 eV for the VBO for AlAs_{0.18}Sb_{0.82}. These values place the valence band maximum of AlAs_{0.18}Sb_{0.82} at an energy 14.4 meV above the valence band maximum of InAs.

## Acknowledgments

The authors would like to acknowledge DRS Sensors & Targeting Systems for funding this work. We would also like to thank Mr. Benjamin Pinkie for his assistance with establishing the model for the optical constants and Dr. Chris Grein for many insightful discussions on InAsSb recombination parameters.

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