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We have investigated crosstalk in HgCdTe photovoltaic pixel arrays employing a photon trapping (PT) structure realized with a periodic array of pillars intended to provide broadband operation. We have found that, compared to non-PT pixel arrays with similar geometry, the array employing the PT structure has a slightly higher optical crosstalk. However, when the total crosstalk is evaluated, the presence of the PT region drastically reduces the total crosstalk; making the use of the PT structure not only useful to obtain broadband operation, but also desirable for reducing crosstalk in small pitch detector arrays.
© 2013 Optical Society of America
1. Introduction
Coupling light into semiconductor materials remains a challenging and active research topic. Towards this end micro- and nano-structured surfaces have become a widely used design tool to increase light absorption in solar cells [1] and enhance the performance of broadband detectors [2] without employing anti-reflection (AR) coatings. In this paper, we investigate the optical and electrical characteristics with an emphasis on crosstalk in mercury cadmium telluride (Hg1−xCdxTe) pixel arrays that incorporate a periodic micro-structured surface to enhance broadband absorption in the mid-wave infrared (MWIR) spectral region (0.5 – 5.0μm) without use of an AR coating. Although the focus of this work has been on the HgCdTe material system, the simulation model and design methodology presented here can be directly applied to other material systems, such as InGaAs or InAsSb. HgCdTe was chosen because it is widely used in high performance focal plane arrays (FPAs), it is technologically mature and very high quality epitaxial layers can be fabricated. Furthermore, it is a direct band-gap semiconductor with an absorption edge that is tunable from the near infrared (IR) to very-long wavelength IR spectral regions.
HgCdTe-based FPAs are normally backside illuminated where the incident light is absorbed through the transparent cadmium zinc telluride (CdZnTe) substrate on which the detector array is grown. Such conventional detectors suffer from significant reflection losses at the air/HgCdTe or air/CdZnTe interface. Consequently an 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. Additionally, an AR coating can introduce stress into a detector limiting its interoperability or lifespan. An alternative approach is to instead remove the CdZnTe substrate and etch a micro-structured photon trapping (PT) structure directly into the HgCdTe detector absorber layer. When properly engineered, the PT structure will significantly reduce reflection losses, negating the need for an AR coating [2]. Furthermore, by removing device volume, the detector dark current is potentially lowered, allowing higher operating temperatures to be achieved. In a previous work [3] we showed that PT structures offer superior device performance in terms of crosstalk compared to non-PT structures. In this work we extend that analysis to larger pixels and explore an alternative assessment of the crosstalk by performing a simulated spot scan. We then use the data from the spot scan to calculate the modulation transfer function (MTF) which is a measure of an array’s resolving ability. To test this, we consider a single-color detector array incorporating a PT structure that has been fabricated on the back surface of the array after the removal of the CdZnTe substrate.
To simulate HgCdTe pixel arrays we have developed a robust three-dimensional (3D) numerical simulation model that takes into account the composition, doping and temperature dependence of the HgCdTe alloy. Furthermore, the optical parameters depend on both alloy composition and temperature. The simulation procedure is a two step process. We first compute the optical response by performing an electromagnetic analysis of the entire structure using the finite-difference time-domain (FDTD) method [4, 5]. Subsequently, the electrical analysis is performed using the finite element method (FEM) [6] to solve the drift-diffusion equations.
The manuscript is organized as follows: Section 2 will describe the numerical model and geometry, section 3 will demonstrate the performance of these structures as an alternative to an AR coating, section 5 will present results on the J(V) characteristics and RDA to demonstrate the reduced dark current, section 4 will discuss the dependence of the optical generation rate under different types of illumination, section 6 will present results on the quantum efficiency (QE), section 7 will presents results on the crosstalk, specifically for optical crosstalk and total crosstalk in sections 7.1 and 7.2 respectively, and section 8 will present an alternative assessment of the crosstalk by performing a spot scan and calculating the MTF. Finally, section 9 will summarize the outcome of this work.
2. Numerical Model and Geometry
The procedure to simulate the arrays is a two step process. First, we compute the optical characteristics by performing an electromagnetic analysis of the structure using the FDTD method. In these simulations we implicitly assume that the entire 3 × 3 array is periodically replicated laterally in space. Specifically, a total field-scattered field approach is employed to compute the reflected and transmitted power. The formalism employs the use of absorbing boundary conditions in the active direction and periodic boundary conditions in the lateral direction which replicates an infinite array of pixels. Using the results of the FDTD simulation we calculate the optical generation profile in each pixel of the array, as well as the total reflectance and QE when the array is illuminated with planewaves. Alternatively, the array can instead be illuminated with a Gaussian beam to compute the crosstalk. Subsequently, the electrical analysis is performed using the FEM to solve the drift-diffusion equations employing ideal Neumann boundary conditions. The optical generation term in the electron and hole continuity equations is included by interpolating the optical generation rate evaluated using the FDTD calculation onto the finite element mesh. A detailed description of the FDTD modeling techniques used in this work is provided in [7] and the FEM modeling techniques in [8, 9].
A schematic representing a single pixel incorporating the PT geometry is shown in Fig. 1. We begin by considering a conventional non-PT MWIR single-color array with a 8.0μm thick narrow-gap Hg0.715Cd0.285Te absorbing layer, followed by a 1.5μm thick wide-gap Hg0.600Cd0.400Te cap layer. The narrow-gap absorbing region is lightly doped n-type at ND = 1.0 × 1015 cm−3 and the wide-gap non-absorbing region is heavily doped p-type at NA = 1.0 × 1017 cm−3. The pixel mesas are formed by etching through the cap layer and into the absorbing layer, thus delineating the mesa. To fabricate the micro-structured surface, the substrate is removed and pillars are etched into the absorbing layer, creating the PT structure.
Fig. 1 Schematic representing the geometry of a single pixel of an array with 8μm pixels incorporating a PT structure.
Depending on the specific etching process, different pillar profiles are obtained. In this work, we consider pillars having a nearly sinusoidal profile [10, 11]. The pillars are 5.0μm tall, 2.0μm in diameter at the base, 0.5μm in diameter just below their tip and are arranged in a square lattice with 2.0μm sides. Beneath the pillars there is another 3.0μm of absorbing layer (for a total of 8.0μm) followed by the cap layer. This work considers 6μm and 8μm pixel pitches. The 6μm {8μm} pixels have 9 {16} pillars per pixel for a total of 81 {144} pillars in the 3 × 3 array as is shown in Fig. 2 for an array with 8μm pixels. Additionally, the process of etching the pillars into the absorber layer can create defect surface states on the surface of the pillars that will lead to surface recombination and an increased dark current. To minimize surface effects the pillars are passivated with anodic oxide. However, the present work does not consider surface effects and instead assumes the device is dominated by bulk effects. Surface effects are beyond the scope of the present work and will instead be the focus of a future work. Furthermore, due to a lack of experimental data on the Shockley-Read-Hall (SRH) lifetime in these devices we assumed a lifetime of 10μs for electrons and holes.
Fig. 2 Three dimensional view of the geometry of the 3 × 3 pixel array with 8μm pixels incorporating a PT structure used in this work.
Furthermore, 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 0.2μm into the detector absorber layer. This region serves to repel minority carries generated in the pillars away from the cathode. In realized arrays the first few rows of pixels along the edge of the array are shorted out through deposition of a metal contact to form the cathode. 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, we have placed the cathode directly on the active pixels of interest, which is why the heavily doped n-type region is incorporated to repel carriers away from the cathode, mitigating effects related to placing the cathode on the active pixels. Additionally, a perfectly reflecting gold layer is placed at the top of the cap layer on each mesa during the FDTD simulations to account for the presence of the metal contacts in a physically realized device. Furthermore, in simulating these structures vacuum is inserted above and below the array to allow for proper simulation of the interfaces and calculation of the electric fields extending beyond the device boundaries.
To simulate the arrays it is first necessary to discretize the structures into a grid of points or mesh. It is important to point out the intricacies involved in generating the numeric meshes for the structures considered in this work. The FDTD structured tensor mesh used in the electromagnetic simulations is created based on the optical properties of the materials being used and is discretized using rectangular prisms. The mesh is produced such that there are a specified number of nodes per wavelength in any given direction in the material. Then, Maxwell’s equations are solved at every node in the mesh. In selecting the number of nodes per wavelength, there is a balance struck between numerical accuracy and the computational resources required. It has been found that optimal results are obtained using values between 15 and 20 nodes per wavelength. Values for typical tensor meshes incorporating the PT structure used in this work are given in Table 1 and a typical tensor mesh used in the electromagnetic simulation of a PT structure can be seen in Fig. 3 (left).
Table 1. Number of nodes in the FDTD Mesh in the PT array for 6μm and 8μm pixels.
Fig. 3 Left: Structured tensor mesh for the PT structure used in electromagnetic simulations (vacuum removed). Geometry is approximated by rectangular prisms with variable volume. Right: Finite element mesh for the PT structure used for the drift-diffusion simulations (vacuum removed). The mesh consists of triangular pyramids of varying dimensions. The approximation of the curved surfaces of the pillars requires a significant number of mesh points.
More attention is required in the generation of the non-structured FEM mesh. As pointed out previously, the computational burden increases vastly as the size of the simulation domain expands. In particular, three-dimensional simulations require highly optimized meshes to maintain a reasonable computation time especially when considering multiple pixels in a FPA. In general, it is essential that the structure be meshed more finely along the growth (z) axis than in the lateral directions in order to properly capture the physics of the device. Heterojunctions and doping boundaries should be further refined to account for the rapid changes in physical quantities that occur across these borders. Also, mesh refinements should be applied near contacts in the direction perpendicular to the face of the contact since the carrier dynamics are perturbed by the presence of the contact.
The most important consideration when deciding on a lateral mesh spacing is the optical generation profile to be used in the simulation. When a planar structure is uniformly illuminated, the carrier generation rate will not vary in the lateral directions and therefore a coarse lateral profile can be employed to maintain a small simulation domain. However, when simulating a device with a patterned surface or using an illumination source with a spatial variation, the lateral mesh spacing will need to be refined to account for the variation of the profile in those directions. This refinement can be eliminated far from the surface of illumination. In addition, it has been found that the borders of pixels should be laterally refined during crosstalk simulations in which the inter-pixel physics are being examined. Unlike the structured tensor mesh used in electromagnetic simulations, the finite element mesh consists of triangular prisms of varying dimensions. This allows curved surfaces to be approximated with a much higher degree of accuracy, although doing so can require a significant amount of mesh points. Table 2 shows the number of mesh points for the PT and non-PT meshes used in the FEM simulations in this work. Furthermore, a representative mixed-element mesh for the PT structure illustrating these principles can be seen in Fig. 3 (right).
Table 2. Number of mesh points in the FEM mesh in the PT and non-PT arrays for 6μm and 8μm pixels.
3. Reflectance
The micro-structured surface that is formed by etching into the detector absorber layer to create the PT structure has the benefit that it reduces the reflectance at the air/HgCdTe interface. To test this we perform FDTD simulations where we illuminate the PT structure using infinite planewaves and calculate the amount of light that is reflected off of the air/HgCdTe interface. To reduce complexity, in these simulations the input domain is not a 3 × 3 pixel array, but is instead a single pixel. The reflectance from a single pixel should be identical to that of a 3 × 3 array due to the use of infinite planewave illumination and periodic boundary conditions. The approach is identical to that described in section 2. Figure 4 shows the reflectance as a function of wavelength for the PT structure with pillars that are 2μm at their base and for the non-PT structure. We immediately observe that the planar surface on the standard non-PT structure suffers from ≈ 30% Fresnel reflection losses due to the refractive index mismatch between air (n = 1) and Hg0.715Cd0.285Te (n = 3.47 in the MWIR). However, properly designing the pillars can significantly reduce these losses to less than 1%. In fact a significant increase in the broadband light absorption is observed for the PT structure over the entire spectrum. A smaller pillar spacing makes it possible to extend the broadband light absorption to shorter wavelengths and in general offers the lowest reflectance, as shown in [2].
4. Optical Generation Profiles
It is very common in numerical simulations of pixel arrays to use the Beer-Lambert law (Beer’s law) to approximate the optical generation rate in the pixels. Using Beer’s law the optical generation rate is given by an exponential decay that depends on the incident photon flux, the absorber layer absorption coefficient, and the thickness of the absorber layer. While Beer’s law provides a satisfactory approximation when the device dimensions are much greater than the wavelength, it is incorrect when the device dimensions are comparable to or less than the wavelength. Furthermore, Beer’s law also does not offer a satisfactory approximation when the absorption coefficient is small and thus the absorption is weak. For these reasons, we cannot use Beer’s law to approximate the optical generation rate in the PT structure and instead we must use FDTD simulations to solve Maxwell’s equations which will yield solutions for the electric field from which the optical generation rate is calculated; for a complete description see [7].
Figure 5 presents the optical generation profile of the 3 × 3 PT array when it is back-illuminated with planewaves at a wavelength of 2.0 μm. In this example the wavelength is 2.0 μm and the pillars are 0.5 μm at their tip and 2.0 μm at their base, therefore the pillar dimensions are comparable to the wavelength of the light. Consequently, it is seen that there are diffraction and interference effects inside the pillars. Furthermore, it is observed that the optical generation rate does not resemble an exponential decay, further reinforcing the statement that Beer’s law is an incorrect approximation in this type of geometry. Finally, upon careful inspection of Fig. 5 it can be observed that the optical generation appears to be different along the x–axis and y–axis. This difference is attributed to the polarization of the incident light. The incident light is either TE or TM polarized, not a superposition of the two. Consequently, the optical generation profile is different along the x and y directions. If the polarization is switched then the optical generation profiles are reversed.
Fig. 5 Optical generation profile of the 3 × 3 array back-illuminated (along the +z–axis) with planewaves at a wavelength of 2.0 μm with a photon flux 1 × 1015 photons cm−2s−1.
Figure 6 presents the optical generation rate as a function of wavelength from 0.5 – 5.0μm. The optical generation rate is shown for a single pillar and the uppermost part of the absorber layer beneath the pillars. At 0.5μm it is observed that optical generation is concentrated at the edge of the pillars and as the wavelength is increased the majority of the optical generation gradually extends deeper into the pillars. In fact at 5.0μm there is very little optical generation at the top most tip of the pillars but instead there is significant optical generation further into the pillars and into the absorber layer.
Fig. 6 Optical generation profile of a single pillar back-illuminated with planewaves ranging from 0.5 – 5.0μm (figure courtesy of Dr. Craig Keasler from unpublished work).
Of great importance for pixel arrays is the inter-pixel crosstalk. To evaluate the crosstalk, it is necessary to use a localized optical excitation. A common experimental procedure for performing a crosstalk evaluation is to illuminate a small fraction of the array in question with a focused beam; which cannot be performed using the infinite planewaves that were used to generate Figs 5 and 6. A realistic simulation of a crosstalk experiment can be performed using a Gaussian beam (GB) to illuminate the center pixel of the virtual array. The beam should have a spot size small enough such that the majority of the beam’s intensity is delivered to the center pixel. This can be achieved by selecting an appropriate beam radius ω which defines the spatial coordinate where the beam intensity is equal to Iω = (1/e)Imax, where Imax = ϕhc/λ is the maximum intensity, ϕ is the photon flux, h is Planck’s constant and c is the speed of light in vacuum. However, the beam radius must be at least equal to or greater than the wavelength to minimize self-diffraction effects. Figure 7 presents the simulated optical generation profile for the PT array with 6μm pixels when the array is illuminated with a Gaussian beam having a wavelength of 2.0 μm and beam radius of 3.0 μm.
Fig. 7 Optical generation profile of the 3 × 3 array back-illuminated with a Gaussian beam with 2.0 μm wavelength. The beam radius was set to 3.0 μm and Imax was calculated using an incident photon flux of 1 × 1015 photons cm−2s−1.
5. J(V) Characteristics and RDA
Another significant advantage of the micro-structured surface approach is the possibility of reducing device noise and dark current as a consequence of removing device volume. To assess this possibility, in Fig. 8 (left) we have plotted the absolute value of the calculated dark current density as a function of the applied bias voltage at 140 K for the non-PT and PT arrays with 8 μm pixels. As expected, the dark current for the PT array is lower by ≈ 44% at the operating bias of −0.005V compared to the non-PT array due to the removal of device material. Furthermore Fig. 8 (right) presents the dynamic resistance multiplied by the pixel area (RDA). The dynamic resistance is computed according to RDA = (dIdark/dV)−1A where Idark is the dark current, V is the applied bias and A is the pixel area. From the RDA we observe that both devices are dominated by diffusion current under reverse bias, exhibiting near ideal behavior. This is to be expected since the SRH lifetime has been set at 10μs. It is desirable for devices to be dominated by diffusion current as this represents near ideal behavior and enables higher operating temperatures. The dynamic resistance saturates near −0.8 V due to the dependence of the diffusion current which leads to a marginal increase in RDA. If the device is biased more heavily the dark current will eventually begin to increase again as SRH recombination becomes more dominant. At very large reverse biases, dark current will further increase due to band-to-band tunneling. This will subsequently result in the dynamic resistance decreasing at large reverse bias.
Fig. 8 Left: Absolute value of the calculated dark current density as a function of the applied bias voltage at 140 K. Right: dynamic resistance multiplied by the area computed using the I(V) data.
6. Quantum Efficiency
The next quantity we will evaluate in our analysis is the quantum efficiency η of the detector as a function of the operating wavelength. To calculate the QE we illuminate the array with a planewave and compute the photocurrent. The QE for a uniform illumination source (planewave) is given by:
where Iph 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. 1 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. 1 is an expression for the total QE.The calculated QE as a function of the wavelength λ when the array is illuminated with planewaves is presented in Fig. 9 for both the PT and non-PT arrays. As explained before, such results automatically include the reflection losses at the surface. From Fig. 9, we can immediately notice that the maximum QE for the non-PT array is ≈ 70% which accounts for a 30% reflection loss at the back surface in the case when an AR coating in not used. This clearly contrasts with the result for the PT array for which the QE is always above 90 % even without an AR coating. It is also immediately possible to see that there is significant peaking in the photo-response near the cutoff. This is not a consequence of the PT structure since it is prevalent in both the non-PT and PT arrays. It is instead due to reflection from the contact, the mesa sidewalls, and the back of the array, which produces significant interference, enhancing the optical generation, for wavelengths near the cutoff wavelength. In particular, it has previously been shown that as the metal contact becomes larger, the peaking in the photo-response becomes more significant [7].
Fig. 9 Calculated QE of the center pixel of the 3 × 3 array with 6μm pixels versus wavelength when the entire array is uniformly (flood) back-illuminated with planewaves with an incident photon flux of 1 × 1015 photons cm−2 s−1, at 140 K.
7. Crosstalk
In this work we consider two types of crosstalk; optical crosstalk and total crosstalk. Optical crosstalk is due to the optical pulse being scattered into adjacent pixels and is calculated using solely the FDTD simulations. Due to the shape of the pillars it is expected that to some degree they might be scattering light out of the center pixel into adjacent pixels thus leading to an increased optical crosstalk with respect to non-PT structures. The second type of crosstalk is total crosstalk. Total crosstalk is due to both optical crosstalk as well as diffusion crosstalk (losses due to photo-generated carriers diffusing from the center pixel into neighboring pixels).
7.1. Optical Crosstalk
The optical crosstalk has been defined as the ratio of the integrated optical generation in a pixel neighboring where the beam is centered to the integrated optical generation in the entire array and is given by:
where the subscript np refers to either the nearest (edge) or next-nearest (corner) pixels, Nnp is the number of nearest or next-nearest pixels and Gopt is the optical carrier generation rate. Obviously, the value of Nnp depends on the cross-sectional geometry of the pixels in question. In this work, we consider square pixels and therefore set Nnp = 4.We begin by performing simulations of six separate cases. We explore 6μm pixels with the beam radius set to 3.0μm and 8μm pixels with the beam radii set to 3.0μm and 4.0μm for both the PT and non-PT arrays. For each case we first calculate the normalized optical generation of the center pixel (we divide the integrated optical generation in the center pixel by the integrated optical generation in the entire array). This provides a measure of the total amount of optical generation in the center pixel. Second, we calculate the optical crosstalk of the neighboring pixels using Eq. (2) to determine the amount of generation in the neighboring pixels.
Figure 10 presents the normalized optical generation of the center pixel with respect to the optical generation of the entire array (left) and optical crosstalk (right) of the nearest neighbors and next-nearest neighbors for the PT and non-PT arrays. The optical generation is plotted for 6μm pixels (beam radius set to 3.0 μm) and 8μm pixels (beam radii set to 3.0 μm and 4.0 μm) when the 3 × 3 array is back-illuminated with Gaussian beams using an incident photon flux of 1 × 1015 photons cm−2 s−1. Furthermore, the beam radius is set to a constant value to maintain a constant spot size at all wavelengths. However, even though ϕ is constant for all wavelengths Imax is still larger at shorter wavelengths due to its wavelength dependence.
Fig. 10 Normalized optical generation of the center pixel with respect to the optical generation of the entire array (left) and the optical crosstalk (right) of the nearest neighbors and next-nearest neighbors for the PT and non-PT arrays. The optical generation is plotted for 6μm pixels (beam radius set to 3.0 μm) and 8μm pixels (beam radii set to 3.0 μm and 4.0 μm) when the 3 × 3 array is back-illuminated with Gaussian beams using an incident photon flux of 1 × 1015 photons cm−2 s−1.
There are in fact two separate causes of optical crosstalk in Fig. 10. The first is the pillars reflecting the incident light out of the center pixel into neighboring pixels which is only prevalent in the PT array. The second (and by far the more dominant cause) is due to the fact that the Gaussian beam is not completely confined to the center pixel and that the decaying tail of the GB extends in neighboring pixels. This can be seen by analytically calculating the total amount of optical power in the center pixel of the array and the entire array respectively and taking their ratio should perfectly agree with Fig. 10 (left) for the non-PT array. Following the formalism outlined in [3] the optical power P is given by:
where Z0 is the vacuum impedance and the electric field profile of the Gaussian beam E is given by [5]: Substituting Eq. (4) into Eq. (3), evaluating the integral in Cartesian coordinates and truncating the limits of integration to the center pixel yields the optical power in the center pixel: where we have used the relation . Furthermore, the total optical power in the entire array is calculated by substituting Eq. (4) into Eq. (3), evaluating the integral in polar coordinates and letting r → ∞, yielding the optical power in the entire array: Finally, let Rpower = Pc/Parray. Rpower is the maximum analytical percentage of optical generation in the center pixel. The ratio Rpower is solely due to the effect of the Gaussian beam extending into neighboring pixels which is most relevant when the spot size (spot size= 2ω) approaches the pixel pitch. Rpower agrees with the normalized optical generation in the center pixel of the non-PT array shown in Fig. 10 (left) to within two decimal places. This result confirms that the only source of optical crosstalk in the non-PT array is the Gaussian beam extending into the neighboring pixels.From Fig. 10 (left) it is seen that when the beam radius is 3.0 μm for 6μm pixels roughly 91% of the optical generation is in the center pixel for the non-PT array and ≈ 89% for the PT array. Therefore, there is roughly 2–3% less optical generation in the center pixel of the PT array compared to the non-PT array, suggesting an increased optical crosstalk in the PT array due to the pillars reflecting light into adjacent pixels. Furthermore, when the pixel size is increased to 8μm for the same size beam radius roughly 98% of the optical generation is now in the center pixel (further confirming that the only source of optical crosstalk in the non-PT array is due to the Gaussian beam extending into neighboring pixels) with only a 1% difference between the PT and non-PT structures. This is to be expected since for a larger pixel and the same spot size more of the beam is confined to the center pixel. It is also possible to evince from Fig. 10 (right) that the optical crosstalk of the neighboring pixels is slightly higher in the PT array than the non-PT array. This furthermore shows that while the only source of optical crosstalk in the non-PT array is due to the Gaussian beam extending into neighboring pixels that that optical crosstalk in the PT array is also due to the pillars scattering the incident radiation into adjacent pixels. This effect is small, however, and should not present a major concern when considering the performance of these devices.
7.2. Total Crosstalk
In section 7.1 we showed that the PT arrays have a slightly higher optical crosstalk than the non-PT arrays. To determine if the optical crosstalk is sufficient to significantly impact the device performance it is necessary to calculate the total crosstalk, that is the sum of the diffusion and optical crosstalk. It is not practical to calculate solely the diffusion crosstalk since the optical generation term in the drift diffusion simulations depend on the FDTD results which automatically include the optical crosstalk. The standard definition for the total crosstalk is the ratio of the photocurrent in a peripheral pixel to the photocurrent in the center pixel when only the center pixel is illuminated:
where the subscript c denotes the center pixel.Figure 11 (left) presents the calculated total (optical + diffusion) crosstalk for 6μm and 8μm pixels with the beam radius set to 3.0μm and Fig. 11 (right) presents the calculated total crosstalk for 8μm pixels with the beam radius set to 4.0μm. The crosstalk is plotted for the nearest neighbors (NR) and next-nearest neighbors (NNR) for the PT and non-PT arrays when the 3 × 3 array is back-illuminated with Gaussian beams using an incident photon flux of 1 × 1015 photons cm−2 s−1. The first thing that can be noticed is the unusually high crosstalk in the non-PT array. The crosstalk is highest for 6μm pixels with the beam radius set to 3.0μm. The cause of the unusually high crosstalk can be traced back to the fact that the pixel size is 6μm and the detector absorber layer thickness is 8.0μm, while the minority carrier diffusion length is ≈ 30 μm. Therefore, with the pixel pitch being less than the absorber layer thickness and the absence of the electric field the carriers are very likely to diffuse to neighboring pixels rather than to the contact. This is compounded due to the absence of the graded alloy composition in the absorber layer that is typical of liquid phase epitaxy (LPE) grown material. The graded composition would have established a quasi-electric field causing minority carriers to drift towards the wide-gap region which would serve to decrease the crosstalk. The graded junction is not considered in this work in an effort to keep the simulation domain manageable. The graded junction requires a significant number of mesh points and the focus of this work is on effects due to the pillars not the graded junction. For a complete description of modeling graded junctions in LPE grown material see [9]. Obviously, if the pixel pitch is increased one expects that the total crosstalk contribution will decrease substantially. This can initially be seen by comparing the non-PT array with 6μm and 8μm pixels where the nearest neighbor crosstalk decreases by 21%. Furthermore, we expect the crosstalk to decrease substantially more as the pitch approaches the 12–15 μm range that is used in state of the art arrays. For the non-PT array the nearest neighbor crosstalk decreases from 0.67 to 0.24 when the pixel pitch is increased from 6μm to 12μm. We expect a similar trend for the PT array but 12–15 μm pixels were not used in this work because it is not practical using the computer hardware (a system with 512 GB of RAM) at our disposal.
Fig. 11 Left: calculated total (optical + diffusion) crosstalk for 6μm and 8μm pixels with the beam radius set to 3μm. Right: calculated total crosstalk for 8μm pixels with the beam radius set to 4μm. The crosstalk is plotted for the nearest neighbors (NR) and next-nearest neighbors (NNR) for the PT and non-PT arrays when the 3 × 3 array is back-illuminated with Gaussian beams using an incident photon flux of 1 × 1015 photons cm−2 s−1.
From Fig. 11 it is also important to notice that the PT array has a much lower total crosstalk than the non-PT array despite the fact that the non-PT array has a slightly lower optical crosstalk. For instance, considering 6μm pixels, there is 47% less crosstalk in the PT array compared to the non-PT array. From this observation we can distill two significant outcomes. First, the dominant crosstalk mechanism both in the non-PT and PT arrays is the diffusion crosstalk. Second, the PT structure not only dramatically increases the QE but it is also instrumental in reducing the total crosstalk. In fact, the total crosstalk in the PT array is mitigated by the presence of the pillars that prevent the minority carriers from diffusing to the adjacent pixels.
8. Simulated Spot Scan and MTF
The crosstalk results in Figs. 10 and 11 depend heavily on the beam radius. A more appropriate figure of merit that would be independent of beam radius is the MTF which is calculated by taking the Fourier transform of the line spread function (LSF). The MTF is a measurement that is used to determine the spatial resolution of the detector. The LSF is calculated by sweeping a beam with delta function extent across the array. Experimentally this is achieved by performing a bi-directional line scan using the image of a narrow slit that is scanned over the center pixel of the array to construct the pixel line spread function [12]. However, in our FDTD simulations it is impossible to produce a beam of near zero extent or use a slit without producing significant diffraction effects. An alternative approach is to use the narrowest beam possible. For these reasons, in these simulations ω = 2μm and λ = 2μm. We begin our analysis by first performing a spot scan. To perform the spot scan we sweep the beam across the x–axis of the array. While the beam is swept we record the photocurrent in the center pixel. Using this technique we plot the photocurrent in the center pixel as a function of beam position in Fig. 12.
Fig. 12 Spot scan (left) and normalized spot scan (right) of the 3 × 3 array with 8μm pixels back-illuminated with Gaussian beams at a wavelength of 2.0μm with the beam radius set to 2.0μm using an incident photon flux of 1 × 1015 photons cm−2 s−1. The vertical gray lines indicate the pixel boundaries with two edge pixels and center pixel visible in the plot.
Figure 12 (left) is a plot of the photocurrent density in the center pixel as a function of beam position. The light vertical gray lines correspond to the pixel boundaries with two edge pixels and the center pixel visible in the plot. A position at 0μm corresponds to the center of the center pixel while a position at 12μm corresponds to the edge of the array. We immediately observe that when the beam is centered over the center pixel that the photocurrent density for the PT array is 7.7μA higher than the photocurrent density for the non-PT array. This difference is due to both the higher reflectance losses of the non-PT array as well as losses due to diffusion of photo-generated carriers into adjacent pixels. Conversely when the beam is near the edge of the array the photocurrent density in the center pixel is higher for the non-PT array, further verifying that the crosstalk is higher for the non-PT array. Figure 12 (right) is identical to Fig. 12 (left) except that now the photocurrent has been normalized by the photocurrent in the center pixel when the Gaussian beam is centered over the center pixel, thus yielding a value of 1 for a beam position at 0μm. We immediately notice a significantly wider profile for the non-PT array due to the higher crosstalk. In fact the full-width at half-maximum (FWHM) is 14.45μm for the non-PT array but reduces to 8.93μm for the PT array. The significantly higher FWHM of the non-PT array is due the larger diffusion crosstalk compared to the PT array.
Using the data from the spot scan we now calculate the MTF by taking the Fourier transform of the spot scan profile. We have calculated the MTF following the formalism used in [13]. It is desirable to compute the MTF because it is a far more widely used parameter by system designers to asses an infrared detector array’s resolving ability than the crosstalk alone. Figure 13 plots the MTF for the PT array and non-PT array with 8μm pixels using the optical generation rate from the FDTD simulations (left) and using the photocurrent from the FEM simulations (right). Calculating the MTF from the optical generation neglects degradation of the MTF due to diffusion and only includes effects due to optical crosstalk, finite width of the Gaussian beam and spatial averaging (pixel footprint). The MTF contributions due to spatial averaging and the finite beam width have been calculated analytically in Fig. 13 (left) as detailed in [14]. The MTF due to spatial averaging represents the ideal limiting performance. We see that the finite extent of the Gaussian beam is not a limiting quantity. The MTF for the PT array is slightly below the non-PT due to the slightly higher optical crosstalk from the pillars scattering a small percentage of the incident light into adjacent pixels. While the optical crosstalk is a degrading effect it is only marginally limiting the performance.
Fig. 13 MTF calculated from taking the Fourier transform of the spot scan profile. Left: MTF calculated using the optical generation rate from the FDTD simulations. The spatial frequency has been normalized by the pixel pitch such that a detector limited by spatial averaging will intersect the horizontal axis at 1. Right: MTF calculated using the photocurrent from the FEM simulations. The 3 × 3 array with 8μm pixels was back-illuminated with Gaussian beams at a wavelength of 2.0μm with the beam radius set to 2.0μm using an incident photon flux of 1 × 1015 photons cm−2 s−1.
Conversely, Fig. 13 (right) plots the MTF calculated using the photocurrent from the FEM simulations. It is immediately seen that the total MTF for both geometries is significantly lower than the spatial averaging MTF. Furthermore, the MTF due to diffusion has been calculated by dividing the total MTF calculated from the photocurrent by the total MTF calculated by the optical generation rate. This removes all contributions from the finite beam width, spatial averaging and optical crosstalk. It is seen that the MTF for the non-PT array is severely limited by diffusion as one would expect due to the extremely high crosstalk seen in Fig. 11. Additionally, the MTF in the PT array is not nearly as limited by diffusion as the non-PT since the crosstalk is much lower. This further demonstrates the significance of using a PT structure to limit crosstalk and improve the spatial resolution for small pixel pitches.
9. Conclusions
In conclusion, we have developed a numerical model to analyze the optical and diffusion crosstalk in HgCdTe pixel arrays that incorporate a periodic micro-structured surface to reduce reflection. To simulate crosstalk we have incorporated a non-uniform illumination source into an FDTD simulation to replicated the behavior of a focused beam in an actual experiment. We have shown that compared to non-PT arrays that PT arrays have a slightly higher optical crosstalk, but significantly less diffusion crosstalk, thus indicating that PT arrays will have significantly better device performance than non-PT arrays in terms of crosstalk, especially for small pixel pitches. Furthermore, we have additionally characterized these structures by sweeping a Gaussian beam across the array to perform a spot scan. Using this technique we have calculated the FWHM to be significantly less for the PT array. Subsequently, we have calculated the MTF from the spot scan and shown that PT structures have superior resolving capability compared to non-PT structures. Consequently, as the detector array technology moves toward a reduction of the pixels size, the PT approach can also be an effective means to reduce the diffusion crosstalk in addition to increasing the QE without employing AR coatings
Acknowledgments
The work at Boston University has been supported by BAE Systems and the DARPA AWARE Program. The authors would like to acknowledge Mr. B. Pinkie for his assistance in computing the MTF and to thank Dr. C. Fulk, Dr. M. Reine, Dr. P. Zimmermann, Katherine Snell, and Dr. N. Dhar. Distribution Statement “A” (Approved for Public Release, Distribution Unlimited). The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government.
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