## Abstract

We report on the modulation transfer function (MTF) in short-wave infrared indium gallium arsenide (InGaAs) on indium phosphide (InP) planar photodetector arrays. Our two-dimensional numerical method consists of optical simulations using the finite-difference time domain method and drift-diffusion simulations using the finite-element method. This parametric study investigates MTF dependence on pitch, the addition of refractive microlenses, the thickness of the InGaAs absorber, and the doping concentration of the InGaAs absorber. A focus is placed on the connection between the lateral diffusion of photogenerated holes in InGaAs and the MTF. It is found that the MTF of small-pitch arrays exhibit sub-ideal behavior due to pixel cross-talk resulting from a long minority carrier diffusion length. By incorporating monolithic microlenses with the InP substrate, the MTF response is improved for all pitches investigated, particularly for spatial frequencies near the respective cutoff frequencies. We also find a strong dependence of the MTF on the thickness and doping concentration in the absorbing region. Trends in dark current and quantum efficiency are reported.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The reduction of pixel pitch in detector arrays is an important technological trend for infrared imaging systems. If the outer dimensions of an array are held constant while pixel pitch is reduced, the pixel density and therefore resolution is increased. If instead the number of pixels are held constant, a reduction in pitch leads to a smaller array and thus lower fabrication costs.

From a system perspective, pitch reduction is beneficial only to a point. If the diffraction-limited blur spot of a system’s optics (the Airy disk of diameter 2.44*λF*, where *λ* is the wavelength and *F* is the f-number) is much larger than the pixel pitch, a further reduction in pixel pitch yields a negligible improvement in system resolution. On the other hand, if a system’s optics produce a blur spot that is smaller than the pixel pitch, reducing the pitch can provide better resolution. In this respect, one may delineate two regimes of system operation: *detector-limited* or *optics-limited*. Consideration of this balance [1–3] has led to the recommendation of *F λ*/*d*, where *d* is the detector pitch, as a figure of merit for predicting the regime of operation; the transition from detector-limited operation to optics-limited operation occurs between *F λ*/*d* = 1 2.44 to *F λ*/*d* = 2. For *F* = 1.2 optics and short wavelength infrared (SWIR) illumination of *λ*/= 1 *µ*m, this figure of merit suggests that a pixel pitch of 3 *µ*m or smaller is needed to ensure optics-limited operation. Indeed, state-of-the-art SWIR detector pitch is approaching this value [4].

From a device perspective, pitch reduction leads to the so-called dense array condition which is characterized by a minority carrier diffusion length in the absorbing material that is longer than the pixel pitch and the thickness of the absorbing region. New behavior that arises in dense planar arrays such as suppression of dark current [4,5] can be beneficial; at the same time, a long minority carrier diffusion length can introduce severe cross-talk effects between pixels.

The planar, lattice-matched In_{.53}Ga_{.47}As on InP photodiode structure is an ideal material system for the study of cross-talk effects in the dense array regime. Its absorbing range of 0.9 *µ*m–1.7 *µ*m places it in the SWIR region of the IR spectrum. Using the *F λ*/*d* figure of merit, the useful minimum pitch of this technology is 3 *µ*m or smaller. However, high-quality InGaAs material is known to have a long minority carrier diffusion length–oftentimes tens of microns at room temperature [6,7]. Therefore, if a single pixel of the detector array is illuminated, photogenerated carriers are very likely to diffuse to adjacent pixels and be collected there. The delineation of pixels with mesa structures is commonly used to reduce diffusive cross-talk between pixels. However, the use of mesas reduces fill factor and requires effective passivation of the sidewalls of the active device regions. In this study, we focus solely on planar arrays.

To reduce pixel cross-talk, we investigate the addition of spherical refractive microlenses to the array. Microlenses focus the incident illumination into the center of each pixel region and effectively increase the distance that photogenerated carriers must diffuse to reach an adjacent pixel. Additionally, we evaluate the diffusion of photogenerated carriers in the InGaAs absorbing region as a function of its thickness and its doping concentration. Both of these parameters are expected to influence the lateral diffusion of minority carriers. Modifying absorber thickness changes the area of the cross-sectional surface through which lateral diffusion may occur. Modifying absorber doping changes carrier lifetime and mobility, and therefore, diffusion length.

This paper is organized as follows. Section 2 presents the device structure and details on microlens design. Section 3 explains our numerical method and how the MTF can be used to quantify optical and diffusive cross-talk. Section 4 presents simulation results. This section begins by examining how pitch reduction affects the MTF. We then show how the MTF may be improved by microlenses and the design of the absorber. Finally, Section 5 summarizes the results of this work.

## 2. Device structure

A schematic of a 3-pixel array structure is shown in Fig. 1. The labeled dimensions and doping concentrations correspond to the baseline case in the parametric study, except for the microlenses, which are not part of the baseline case. The pixel width (pitch) is denoted as *d*. The array is shown with its illuminated side (backside of the substrate) at the top. The detector structure is that of a double layer planar heterostructure and consists of a heavily-doped InP substrate (*N _{D}* = 2 × 10

^{18}cm

^{−3}) of thickness

*f*, an In

_{.53}Ga

_{.47}As absorber (

*N*= 1 × 10

_{D}^{16}−3 cm

^{−3}) of thickness

*t*, and finally an InP cap (

*N*= 1 × 10

_{D}^{16}cm

^{−3}) of thickness 1

*µ*m. The pixel boundaries are indicated in the absorber by vertical dashed lines; these are illustrative and not physical boundaries. As will be discussed in Section 3, the number of pixels in the array are chosen to ensure valid optical and electrical results. Centered in each pixel region is an abrupt rectangular

*p*

^{+}diffusion (

*N*= 1 × 10

_{A}^{18}cm

^{−3}) of width

*d*/2 that forms the sensing junction for the pixel. The depth of the

*p*

^{+}diffusion extends 200 nm into the absorber region. Ideal Ohmic contacts are placed on the surface of each

*p*

^{+}diffusion to sense photocurrent as well as on each side of the InP substrate to serve as common cathode contacts. For all simulations, a reverse bias of 100 mV is applied to each diode. The parametric study varies

*d*,

*t*, the absorber doping concentration, and the inclusion of microlenses.

In Fig. 1, a circular monolithic microlens is centered on each pixel. The focal length of the microlens is always chosen so that incident light is focused on the interface between the InP substrate and the InGaAs absorber. In other words, the focal length of the lens is precisely the thickness of the substrate, *f*. To determine the radius of curvature, *R*, of the lens, we use the conventional formula for refraction at a spherical surface under the paraxial approximation [8]

*n*

_{0}is the index of refraction outside the lens,

*S*is the distance from a point source of illumination to the lens,

_{o}*n*is the refractive index of the lens, and

_{L}*S*is the distance from the apex of the lens to its focal point. Setting

_{i}*S*= ∞,

_{o}*n*

_{0}= 1, and requiring that

*d*is the length of the chord formed by the intersection of the circular lens and the substrate gives The substrate thickness is set to 2.5

*d*so that the f-number of the lens remains constant as pitch is varied. The parametric study varies pitch, the inclusion of microlenses, the absorber thickness, and the absorber doping. All other parameters remain fixed.

## 3. Methodology

#### 3.1. Numerical method

Our two-dimensional numerical simulations consist of two stages. The first is an optical simulation using the finite difference time domain (FDTD) method to solve Maxwell’s equations. The second is an electrical simulation using the finite element method (FEM) to solve a drift-diffusion transport model. The FDTD method is chosen for its robust ability to capture diffractive effects which are particularly important in simulations of micro-optics. All optical simulations are conducted using a wavelength of 1 *µ*m which is near the lower absorption cutoff for the InP/InGaAs structure. The selection of a short wavelength ensures that diffractive effects for small microlenses will be minimized. The optical simulation provides electric field and optical intensity distributions throughout the array in response to plane wave or Gaussian beam illumination. The FDTD-calculated optical intensity profile is used to determine the carrier generation profile in the absorber which is provided as an input to the FEM solver. Shockley-Read-Hall (SRH), radiative, and Auger generation-recombination physics are included. All simulations assume a temperature of 300 K. A detailed overview of the FEM drift-diffusion model and the InGaAs/InP material parameters used are provided elsewhere [9]. These methods are used to determine the dark current, external quantum efficiency, and modulation transfer function (MTF) for a given set of array parameters in the parametric study.

#### 3.2. Modulation transfer function

A straightforward method for estimating pixel cross-talk is to illuminate a single pixel and compare the illuminated pixel’s photocurrent to the photocurrent sensed in adjacent pixels [10,11]. Because the adjacent pixels are nominally dark, cross-talk is responsible for any photocurrent that is sensed in these pixels. A more robust but also more computationally expensive method for analyzing cross-talk is the calculation of the MTF [12, 13]. The advantages of the MTF method are its ability to characterize cross-talk as a function of spatial frequency and to correct for the spatial distribution of the illumination source. The MTF for a detector array also provides relevant information for designers of optical systems.

The optical transfer function (OTF) of a system describes its magnitude and phase response to sinusoidal intensity inputs as a function of their spatial frequencies. For a detector array, the OTF is in general a function of two spatial frequency coordinates, *ξ* and *η*, defined in the same plane as the surface of the array

*H*(

*ξ, η*)| and the phase transfer function is defined as

*θ*(

*ξ, η*) [14]. Because this work utilizes two-dimensional simulations, the proceeding MTF analysis is restricted to a single spatial frequency variable,

*ξ*. Due to symmetry, it is reasonable to expect that the spatial response with respect to

*η*(defined in the third dimension) is the same.

Ideally, the MTF of an optical system is constant and does not vary as a function of spatial frequency. However, such a system would have infinite spatial resolution. In a detector array, the MTF is ultimately limited by the finite size, or footprint, of each pixel. The MTF for a pixel with a finite width *d* is obtained by calculating the Fourier transform of the box function with width *d* [14]. The result is normalized to a maximum value of one to give a sinc function

_{ideal}occurs at

*ξ*= 0 and it decreases monotonically to its first zero at

*ξ*= 1/

*d*. The location of this first zero is commonly used to define the cutoff frequency of the detector. In real detectors, the MTF can be different from the ideal MTF because of optical and diffusive cross-talk factors.

We will now explain how we use numerical simulations to calculate the MTF for a given detector design. This is done by simulating a spot scan profile in which a Gaussian beam is swept laterally across the surface of the detector array. A Gaussian beam has at its waist an intensity distribution of the form

where*I*

_{0}is the peak intensity,

*r*is the radial distance from the center axis of the beam, and

*w*

_{0}is the waist radius of the beam. The waist radius of the beam is made much smaller than the length of the array so that the optical generation and the resulting photocurrent in each pixel is a strong function of the beam’s lateral position. For each position of the beam, an FDTD simulation is done to compute the profile of optical generation rate in the absorber. The optical generation profile is then used in the drift-diffusion FEM simulation to determine the photocurrent extracted from each pixel. As the beam is swept across the array, two important quantities are calculated for a single pixel as a function of the lateral position of the beam center. The first quantity is the optical generation rate integrated over the pixel’s absorbing region (regions

*r*in Fig. 1) which we call SS

_{j}_{generation}(

*x*). The variable

*x*is the lateral distance between the beam center and the pixel center. The second quantity is the photocurrent extracted from the pixel, SS

_{photocurrent}(

*x*).

The two spot scan profiles, SS_{generation}(*x*) and SS_{photocurrent} (*x*), are modeled as convolution operations between the intensity distribution of the Gaussian beam and the box function that represents the pixel footprint. By the convolution theorem, the Fourier transform of the spot scan is given by the product of the Fourier transforms of *g*(*x*) and the pixel footprint. Thus, we may write

_{optical},

_{optical}is introduced to model any deviation from the ideal MTF due to optical cross-talk. If MTF

_{optical}is equal to one up to the cutoff frequency, it can be said that there is no significant optical cross-talk in the system. On the other hand, if MTF

_{optical}is different from one, it must be the case that the footprint of the pixel is not well-described by the box function. Physically, MTF

_{optical}represents pixel cross-talk that occurs as the electric field (originating from some illumination source) propagates from the surface of the array to the absorber region, where the field decays as semiconductor carriers are generated. For example, any diffraction caused by microlenses that leads to cross-talk is captured by MTF

_{optical}. Because SS

_{generation}(

*x*) is obtained purely from the calculation of optical generation rates, MTF

_{optical}does not capture cross-talk associated with the lateral diffusion of photogenerated carriers.

We now turn to the other spot scan quantity, SS_{photocurrent}, which gives the photocurrent of a single pixel as a function of the position of the Gaussian beam. To calculate this quantity, drift-diffusion FEM simulations are performed that include the optical generation profiles in the absorber as an input. Because photocurrent is a response to both optical generation and charge carrier movement in the absorber, the photocurrent spot scan is sensitive both to optical and diffusive cross-talk. To model diffusive cross-talk, we introduce another cross-talk factor, MTFdiffusion, in order to write

_{diffusion}represents any cross-talk that occurs between the time a minority carrier is generated and it is collected by a sensing junction. This factor is named MTF

_{diffusion}because diffusion is the physical process by which photogenerated carriers can move between pixels and contribute to cross-talk.

Equation (8) is the overall MTF for the detector. If both MTF_{optical} and MTF_{diffusion} are equal to one up to the cutoff frequency, the detector has an ideal response. With Eqs. (4) and (7), the only unknown in Eq. (8) is MTF_{diffusion}. Therefore, one may separately examine how the optical design of the detector array and the diffusive behavior of carriers in the absorber influence the overall cross-talk.

Finally, we note a few practical considerations. The first regards sampling frequency. To resolve the MTF to the cutoff frequency 1/*d*, by the Nyquist criterion the sampling interval must be no larger than *d*/2. We use a value of *d*/6. Second, we assume that the spot scan profile for a given pixel is symmetric about its center and collect only a single-sided profile. To collect a spot scan, we begin with the lateral position of the Gaussian beam at a pixel center and sweep it toward the right or left boundary of the array. We sweep the beam sufficiently far from the center of the selected pixel so that its total current falls to within 1% of its known dark current. The number of pixels in the array are set so that this is possible. Third, we select a beam waist radius based on two key facts. The first is that the paraxial solution for a Gaussian beam breaks down when *w*_{0} ≈ *λ* or smaller [15]. The second is that, for a Gaussian beam of waist *w*_{0} ≈ *λ* focused through a lens, the position of the tightest focus does not necessarily lie at the focal point of the lens [16]. Both of these non-idealities are resolved when *w*_{0} is sufficiently larger than *λ*. As a confirmation of our methodology, we have verified that our MTF results converge to a single solution as *w*_{0} is increased from *λ* to 5*λ*. We have chosen *w*_{0} = 3*λ* because it provides an ideal balance between solution accuracy and computational cost. From these requirements, the collection of a single spot scan profile can sometimes require many pairs of FDTD and FEM simulations, particularly when pitch is small.

For the optical simulation of MTF, perfectly absorbing boundary conditions are used along all four sides of the simulation domain. However, for quantum efficiency calculations, separate simulations are performed using uniform plane wave illumination with absorbing boundary conditions at both limits of the optical axis and periodic boundary conditions at the sides of the array. The application of periodic boundary conditions at the sides allows the pixels to behave as if they were part of an array with infinitely many pixels.

#### 3.3. Parametric study

An overview of the parametric study is shown in Table 1. We begin by calculating the MTF for a baseline set of array parameters at pitches of 4, 6, and 12 *µ*m (Section 4.1). The MTF calculations show significant diffusive cross-talk for all pitches. The cross-talk worsens as pitch is reduced. In an effort to improve the MTF, we then evaluate the addition of microlenses (Section 4.2). We find that microlenses improve the MTF for all pitches. Notably, the 12 *µ*m pitch variant attains a near-ideal MTF but the 4 and 6 *µ*m pitch variants remain significantly sub-ideal. We then focus on these two small pitches and evaluate how the MTF changes with the reduction of absorber thickness (Section 4.3) and the variation of absorber doping (Section 4.4). In these last two sections we also report dark current and quantum efficiency with respect to the parameter that is varied.

## 4. Simulation results

#### 4.1. Baseline performance as a function of pitch

We first evaluate the MTF of an array design without microlenses and with typical parameters for SWIR operation. This baseline array structure has no microlenses, an absorber of thickness *t* = 3 *µ*m, and an absorber doping concentration *N _{D}* = 1 × 10

^{16}cm

^{−3}. At this doping concentration, the minority carrier diffusion length in the absorber is 27

*µ*m. All other details of the baseline array structure were presented in Section 2. The MTF is calculated for pixel pitches of 4, 6, and 12

*µ*m in order to examine how the detector MTF changes as pitch is scaled deep below the minority carrier diffusion length. The results of the MTF simulations are shown in Fig. 2. The left plot shows SS

_{photocurrent}for each pitch. For comparison, ideal spot scans (computed as the inverse Fourier transform of the ideal MTF) are shown for the 4 and 12

*µ*m pitch arrays. The right plot shows the corresponding detector MTFs for each pitch calculated using Eq. (8). In the MTF plot, the spatial frequency axis is normalized by the nominal cutoff frequency so that the 1/

*d*frequency lies at a value of one. The ideal MTF response given by Eq. (4) is plotted for comparison.

From the spot scan results, it is clear that even at a 12 *µ*m pitch, the simulated spot scan response is broader than the ideal response. The spot scan simulations for the 6 *µ*m and 4 *µ*m yielded comparable spot scan curves, indicating that such a pitch reduction in this baseline case provides only marginally improved spatial resolution. The corresponding MTF calculation echoes the sub-ideal behavior seen in the spot scans. The detector MTF moves farther from the ideal response as pitch is reduced. The most severe deviation from the ideal response occurs for spatial frequencies near the cutoff. This is an undesirable feature as fine details in images correspond to higher spatial frequencies.

The detector MTFs shown in Fig. 2 include both MTF_{optical} and MTF_{diffusion} factors, to which the departure from the ideal response must be attributed. To quantify the degree of optical cross-talk, we compare the simulated MTF_{ideal}MTF_{optical} product to the analytical MTF_{ideal} given by Eq. (4). If these two quantities are well correlated, optical cross-talk is negligible, and diffusive cross-talk dominates. For each of the three pitches simulated, the coefficient of determination (*R*^{2}) between these two quantities is better than 0.99 up to the cutoff frequency. Therefore, because the simulated MTF_{ideal}MTF_{optical} product is very nearly MTF_{ideal}, MTF_{diffusion} is the dominant contributor to the sub-ideal MTF response. In fact, for the remainder of the parametric study, MTF_{optical} is never found to be responsible for a sub-ideal response; there is always very good agreement between the MTF_{ideal}MTF_{optical} product and the analytical expression for MTF_{ideal}. Achieving an ideal response reduces to the problem of controlling diffusive cross-talk.

#### 4.2. Effect of microlenses on the modulation transfer function

The addition of circular refractive microlenses lead to a concentration of optical generation near the pixel centers. In Fig. 3, this effect is shown for a 6 *µ*m pitch array with and without microlenses under uniform plane wave illumination. In the substrate region (*y* > 0), the squared magnitude of the electric field is shown. In the absorber region (*y* < 0), the optical generation rate is shown. In this example, adding lenses increases the peak optical generation rate in the absorber by a factor of eight; foci with this peak rate lie at each pixel center. For Gaussian beam illumination there is a similar focusing effect. Figure 4 shows the FDTD results in the absorber and substrate regions during spot scan simulation for the 6 *µ*m pitch array. The beam begins at a pixel center and continues laterally in 1 *µ*m steps. As the beam is swept, the lenses split the beam and direct illumination toward pixel centers.

The MTF was evaluated for all three baseline array configurations with microlenses added. The MTF results are shown in Fig. 5. Substantial improvement in MTF response is observed for all three pitches with respect to the baseline results presented in Fig. 2. With microlenses, the 12 *µ*m pitch array exhibits a nearly-ideal MTF. The responses for the 6 and 4 *µ*m arrays are also improved, especially for spatial frequencies near the cutoff. From these results, it is evident that adding microlenses can reduce diffusive cross-talk. This happens because the optical generation is now concentrated at pixel centers and it is less likely for carriers to diffuse to neighboring pixels. The reduction in pixel cross-talk resulting from the use of microlenses is particularly valuable because the response for high spatial frequencies (which was a problem with the baseline cases) is greatly improved.

Although the addition of microlenses lends a substantial reduction in diffusive cross-talk in small-pitch arrays, the results show that their MTF responses remain sub-ideal. It is of interest for these small-pitch arrays to investigate other device parameters that may improve diffusive cross-talk. To this end, the variation of the absorber thickness and its doping concentration are investigated for 4 and 6 *µ*m pitch arrays with lenses in the following two subsections. We report on dark current and quantum efficiency for all variants as these parameters are varied.

#### 4.3. Effect of absorber region thickness on the modulation transfer function

For this analysis, the structures of the arrays are identical to those in the previous section, except for the variation of the absorber region thickness. The MTF results for 4 and 6 *µ*m arrays with microlenses are shown in Fig. 6. There is a clear trend of improving MTF response as the thickness of the absorber is decreased. For an absorber thickness of 1.5 *µ*m, which is half of the baseline value, both arrays attain near-ideal MTFs. The trend observed is consistent with the long diffusion length in InGaAs being the primary culprit for diffusive cross-talk. As absorber thickness is decreased, the length of the vertical cut in the *x*-direction through which lateral diffusion can occur is decreased.

To illustrate how absorber thickness influences the lateral diffusion of minority carriers, we plot in Fig. 7 minority hole concentration in the absorber for a 12 *µ*m pitch array without lenses for absorber thicknesses of 3 and 1.5 *µ*m. In this plot, the array is illuminated by a Gaussian beam with *w*_{0} = 3 *µ*m centered on the middle pixel so that the optical generation rates in the neighboring (left and right) pixels is negligible. The hole concentration shown is the *difference* between illuminated and dark hole populations; i.e., we show *p*′(*x, y*) = *p* (*x, y*) − *p*_{0}(*x, y*) where *p* is the hole concentration in the illuminated case and *p*_{0} is the hole concentration in the dark case. In this manner, we may visualize the diffusion of photogenerated holes. Qualitatively, it can be seen that the lateral spread of photogenerated holes is suppressed in case of the thinner absorber.

Because a reduction in absorber thickness can have an impact on the dark current and quantum efficiency of the array, these figures have been calculated and are shown in Fig. 8. Results for the 12 *µ*m array are included for completeness. The dark current curves show a linear dependence on absorber thickness. In fact, we find that the slopes of the dark current curves are approximately proportional to the area of the absorber regions *r _{j}*, which is simply

*td*. Linear least-squares fits for the dark current as a function of absorber thickness reveal that the fit for 6

*µ*m array has a slope that is 1.50 times that of the 4

*µ*m array, which reflects the ratio between the two pitches. Similarly, the fit for the 12

*µ*m array has a slope that is 3.05 times that of the 4

*µ*m array. This relationship of dark current to absorber area is explained by a nearly uniform carrier generation rate throughout the absorber regardless of its thickness. This phenomenon is due to a dense array condition that prevents the absorber from behaving as a quasi-neutral region. The depletion region edge of the sensing junction extends approximately 400 nm into the absorber; there, holes are extracted under reverse bias. However, because the absorber is much thinner than the minority carrier diffusion length, the hole population outside the depletion region does not return to its equilibrium value anywhere within the absorber. In response, there is substantial radiative generation throughout the absorber. Its dark current contribution scales with absorber area.

Quantum efficiency decreases as the absorber thickness is decreased, but the QE loss is confined to approximately 4% provided that the absorber thickness is 1.5 *µ*m or greater. From Beer’s law, an absorption coefficient of 1.88 *µ*m^{−1} at *λ* = 1 *µ*m and the reflectance at the vacuum-substrate interface taken to be 31%, the 1.5 *µ*m absorber should absorb 3.9% less incident radiation than the 3 *µ*m one. This is in line with the simulated reduction in quantum efficiency.

#### 4.4. Effect of absorber region doping concentration on the modulation transfer function

To evaluate the effect of absorber doping concentration on lateral diffusion, we concentrate again on the 4 and 6 *µ*m arrays with lenses. The thickness of the absorber is restored to the baseline value of 3 *µ*m. The MTF results are shown in Fig. 9 for *N _{D}* between 1 × 10

^{15}and 1 × 10

^{18}cm

^{−3}. The ideality of the MTF response is not a monotonic function of doping. For both pitches, the MTFs for

*N*= 1 × 10

_{D}^{15}and 1 × 10

^{18}cm

^{−3}are comparable and provide the best response, while the MTFs for

*N*= 1 × 10

_{D}^{16}and 1 × 10

^{17}cm

^{−3}are comparable and provide the worst.

The change in MTF as a function of doping concentration is not monotonic because both the minority carrier diffusion length and the electric field in the absorber vary with doping. A smaller doping concentration provides a longer diffusion length, which would be expected to worsen cross-talk. The influence of an electric field in the absorber on MTF has been explicitly studied by Gravrand et al., who found that the presence of a field that sweeps carriers to the sensing junction does improve the MTF [12]. To illustrate how the electric field varies as a function of doping concentration, Fig. 10 provides a plot of band profile in the absorber for the 6 *µ*m pitch array. At high doping concentrations, the electric fields (as indicated by band bending) developed at the *N*-InP–*n*-InGaAs heterointerface and the *p*^{+}-*n* junction are of small spatial extent in the *y*-direction. Transport inside the absorber is dominated by diffusion. Conversely, at low doping concentrations the spatial extent of the electric fields is larger, and drift plays a greater role in transport. The MTF for the case of *N _{D}* = 1 × 10

^{15}cm

^{−3}is better than the baseline case, despite its long diffusion length, because the electric field in the absorber sweeps photogenerated carriers toward the sensing junction. In such a case of very low absorber doping, the drift component of hole transport is more significant and the MTF is actually improved even though the diffusion length increases. On the other hand, at very high absorber doping concentrations, the electric field inside the absorber is negligible but the diffusion length is substantially decreased by enhanced radiative and Auger recombination. The MTF for the case of

*N*= 1 × 10

_{D}^{18}cm

^{−3}is better than the baseline case because photogenerated carriers tend to recombine before they can contribute to cross-talk. This recombination will obviously have severe implications for quantum efficiency.

Plots of the photogenerated hole population in the absorber are shown for absorber doping concentrations of 1 × 10^{15}, 1 × 10^{16}, and 1 × 10^{18} cm^{−3} in Fig. 11. It is evident from this comparison that lateral diffusion is worse in the 1 × 10^{16} cm^{−3} case than the others. In the 1 × 10^{15} cm^{−3} case, the lobe of peak hole concentration is noticeably removed from the absorber–substrate interface (at *y* = 0) when compared to the other cases. The hole density decays rapidly inside the wide depletion region. These two features are attributed to the enhanced built-in fields. In the 1 × 10^{18} cm^{−3} case, drift plays a smaller role in transport. Lateral diffusion is limited by enhanced recombination.

Dark current and QE are compared as a function of absorber doping in Fig. 12. For completeness, the 12 *µ*m pitch and no-lens variants are included. Relative to the 1 × 10^{16} cm^{−3} doping concentration, the case of 1 × 10^{15} cm^{−3} leads to approximately twice as much dark current and comparable QE for all pitches. The case of 1 × 10^{18} cm^{−3} shows reduced dark current but a drastic reduction in QE. The trend of decreasing dark current with increasing doping concentration is due to a suppression of generation in the depletion region at the sensing junction. Figure 13 shows the generation profile along a slice in the absorber for a 6 *µ*m array for all four absorber doping concentrations. The position of this slice in the *x*-direction is identical to that shown in Fig. 10, but total generation rate is plotted instead of band profile. For each doping level, peak generation occurs inside the sensing junction depletion region due to the SRH mechanism. This peak value is comparable for all doping concentrations. The generation in the absorber outside of the depletion region is dominated by radiative and Auger generation.

Manipulation of the doping concentration to improve the MTF clearly carries some caveats. Though a very high doping concentration offers an improved MTF, it is not viable due to the QE impact. On the other hand, using a very low doping concentration to improve the MTF preserves the QE but increases dark current.

## 5. Conclusions

We have reported on the MTF for small-pitch SWIR detector arrays. For the baseline array configurations, which had no microlenses and an absorber with thickness 3 *µ*m and doping concentration 1 × 10^{16} cm^{−3}, it was shown that pitch reduction degrades the MTF response because of diffusive cross-talk. Adding refractive microlenses improved the MTFs substantially. Microlenses permitted the 12 *µ*m pitch array to operate with a nearly-ideal MTF. The MTFs for the 4 and 6 *µ*m pitch arrays were also improved, yet their responses remained sub-ideal. In the interest of making further improvements to the 4 and 6 *µ*m cases, the dependence of the MTF on the absorber thickness and its doping concentration were explored. It was shown that while a thinner absorber improves the MTF, a price is paid in quantum efficiency. With respect to doping concentration, it was found that a reduction from 1 × 10^{16} cm^{−3} to 1 × 10^{15} cm^{−3} improved the MTF but increased the dark current by a factor of two.

In summary, it is evident that all parameters investigated are closely linked to the MTF, dark current, and quantum efficiency of InGaAs-InP planar photodetector arrays. Optimization of these parameters is crucial to array performance, especially for small-pitch arrays. Our simulation results indicate that reasonable MTF performance is possible for ultra-small-pitch planar InGaAs-InP arrays.

Further investigation might explore how the MTF and QE change with wavelength. For longer wavelengths, diffraction introduced by the microlenses may be more significant and contribute to optical cross-talk. The absorption coefficient of InGaAs decreases with longer wavelengths and a thicker absorber may be needed to retain high quantum efficiency. With respect to these two factors, our selection of a 1 *µ*m wavelength for simulations has demonstrated the best-case MTF performance. Also of interest for the MTF may be the introduction of an electric field in the absorber by grading the InGaAs composition. Other nonidealities such as interface behavior and non-rectangular doping profiles also may play an important role in MTF. Additionally, the optical and electrical simulations might be extended to three dimensions in order to capture the full geometry of the diode junctions and microlenses.

## Acknowledgments

Boston University Ignition Program; Raytheon Advanced Studies Program.

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