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Abstract
A consistent trend in infrared imaging systems is a drive towards smaller pixel pitches in focal plane arrays. In this work, we present an extensive numerical study on how dark current, quantum efficiency, and modulation transfer function are affected when reducing the pixel pitch in SWIR InGaAs pixel arrays. From the results, we propose the introduction of diffusion control junctions into the pixel sub-architecture to lower dark current and improve modulation transfer function, with a minor decrease in specific detectivity.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Advances in the development of infrared (IR) imaging sensors across all IR spectral bands are driven by finding new ways to reduce the size, weight, power, and cost (SWaP-C) of the system while adding new functionalities without adversely affecting performance. One strategy that addresses this need is to move towards small-pitch focal plane arrays [1]. Considering an image sensor with fixed resolution, reducing the pixel size improves focal plane array (FPA) yield by enabling more sensors per wafer as the material footprint would decrease, assuming that any technological issues in the fabrication process are solved [2]. Moreover, a smaller die permits the use of smaller optical elements, effectively reducing the size of the whole imaging system [1,2]. To this end, it is critical to understand the implications on detector performance for reducing pixel size.
Another advantage of reducing pixel size is in terms of modulation transfer function (${\mathrm {MTF}}$). MTF is a common figure of merit that gives a quantitative description of the amount of aliasing an imaging system introduces when exposed to sinusoids of varying frquency [3]. The total MTF of the system is the product of the individual sub-components’ MTF’s; there are MTF components from the optics, electronics, and the sensor [3]. Assuming $f/\# = 1.4$ optics and considering a 1 ${\mathrm{\mu} \mathrm{m}}$ wavelength, a shortwave infrared (SWIR) system requires pixel pitches smaller than 3.4 ${\mathrm{\mu} \mathrm{m}}$ to transition from detector-limited to optics-limited [4]. In fact, modern systems are approaching this value, with a recent demonstration of a SWIR In$_{0.53}$Ga$_{0.47}$As FPA at 5 ${\mathrm{\mu} \mathrm{m}}$ pixel pitch [5].
Similar to the system, the MTF of the sensor is the product of several sub-components [6–8]:
The footprint of the detector is due to the discrete imaging elements that form the FPA, and sets a limit of the achievable MTF [3]. The footprint of a rectangular pixel within a rectangular pixel array has the analytic expression [3]
In this work, we focus on SWIR In$_{0.53}$Ga$_{0.47}$As FPAs utilizing the ubiquitous double layer planar heterojunction photodiode architecture [9]. This particular architecture offers benefits over others by offering an in situ passivation of the low bandgap In$_{0.53}$Ga$_{0.47}$As surface preventing extraneous surface leakage current. Additionally, using a planar architecture instead of mesas prevents the necessity of developing a process for passivating the etched III-V material; however, at a cost of a higher degree of crosstalk for small pixel pitches. We use numerical modeling to discuss the implication on dark current, quantum efficiency (QE), and MTF. From the results, we explore a potential modification to the pixel architecture aimed toward suppressing dark current and improving MTF.
This manuscript is organized as follows. Section 2 briefly introduces the simulation methodology used when calculating the results, Section 3 presents and discuss the results, and Section 4 concludes the article.
2. Methodology
The simulation methodology for modeling infrared photodetectors has been extensively discussed elsewhere, so only a brief introduction is given here [6,14,15]. The device architecture considered in this work is shown in Fig. 1. Unless otherwise stated, the epitaxial stack consisted of a 0.5 ${\mathrm{\mu} \mathrm{m}}$ InP substrate doped $N$-type $2\times 10^{18}$ cm$^{-3}$, a 3 ${\mathrm{\mu} \mathrm{m}}$ In$_{0.53}$Ga$_{0.47}$As absorber doped $N$-type $10^{16}$ cm$^{-3}$, and a 0.5 ${\mathrm{\mu} \mathrm{m}}$ InP cap doped $N$-type with the same doping level as the absorber. The $p$-well was modeled as an abrupt cylindrical region with a constant doping of $10^{18}$ cm$^{-3}$ that persisted 200 nm into the absorbing layer. Contacts were assumed to be ohmic, with the anodes placed on $P^{+}$ InP and the cathode as a ring centered around the InP substrate. The Shockley-Read-Hall lifetime was fixed at 107 ${\mathrm {\mu s}}$ [9,16]. Unless otherwise mentioned, the simulated temperature and bias were 300K and -200 mV respectively. The current-voltage characteristics, under dark and illuminated conditions, were calculated using the finite-element method to solve the drift-diffusion equations [17]. The optical generation throughout the device, included when solving the drift-diffusion equations as an additional generation term to determine the photocurrent, was computed using the finite-difference time-domain (FDTD) approach [18]. The quantum efficiency was determined by the computing the photocurrent while the device was uniformly illuminated by a planewave excitation incident on the back-side of the device, and is given by
where $\eta$, $J_\mathrm {ph}$, $\phi$, and $q$ denote the QE, photocurrent density, photon flux, and electron charge.Specific detectivity, $D^{*}$, is a useful figure of merit for comparing different optoelectronic devices, as it combines a detector’s spectral response with the noise characteristics. Assuming a shot-noise limited photovoltaic device, and a system noise bandwidth of 1 Hz, we estimated the specific detectivity from the computed dark current and QE using [19–21]
where $J_d$ is the dark current density and $\lambda$ is the incident wavelength in microns; $D^{*}$ is given with units cm Hz$^{1/2}$ W$^{-1}$.
Fig. 1. An example of an In$_{0.53}$Ga$_{0.47}$As FPA considered in this work. The inset highlights the refinement strategy used when creating the finite-element mesh for device simulations.
The model was validated against data available in literature as shown in Figs. 2 and 3 [10–13]. From Fig. 2, it is clear that a 10 ${\mathrm{\mu} \mathrm{m}}$ pitch detector array is diffusion-limited down to about 260K, when assuming a SRH lifetime of 107 ${\mathrm {\mu s}}$.
Fig. 2. An arrhenius plot comparing the simulated (lines) dark current density with data available in literature (markers) [10–13]. The applied voltage was -200 mV and the Shockley-Read-Hall lifetime was 107 ${\mathrm {\mu s}}$.
Fig. 3. A comparison of the simulated (line) QE with VISSWIR data in literature [12] for varied substrate thickness. The applied voltage, temperature, and photon flux were -200 mV, 300K, and $10^{14}$ ph cm$^{-2}$ s$^{-1}$ respectively.
The strategy we used to compute the modulation transfer function was adopted from previous works [6,8]. Generally, the method is analogous to the measurement of a sensor’s point spread function, but on the pixel scale [3]. Considering a 3x3 pixel array of photodiodes, the photocurrent observed in the central pixel is mapped as a function of the position of a Gaussian beam excitation, creating the spot-scan profile, $SS(x, y)$, of the detector. The 2D Fourier transform of the spot-scan, $\mathcal {F}(SS(x, y))$, yields the MTF in terms of both spatial frequency components, $\xi$ and $\eta$. The simulated MTF can be expressed as the product of the individual contributions that distort or degrade the system’s MTF:
An additional spot-scan profile, $SS_\mathrm {optgen}$, can be obtained from the FDTD simulations, by mapping the integrated optical generation rate within the central pixel as a function of the Gaussian beam position. The Fourier transform of $SS_\mathrm {optgen}$ yields an addition MTF that only contains the detector footprint, optical crosstalk, and the Gaussian beam excitation:
Using this relation, we can isolate the individual MTF components from the inter-pixel diffusion and optical scattering:
Note, the spot-scan profiles were constructed by simulating a 2x2 array and sweeping the Gaussian beam across one quadrant of the array. Then, using the symmetry of a rectangular array created the expected spot-scan for a 3x3 array.
3. Results
3.1 Dark current and QE in small-pitch In$_{0.53}$Ga$_{0.47}$As FPAs
The two dominant sources of dark current in SWIR In$_{0.53}$Ga$_{0.47}$As FPAs are diffusion and generation-recombination (GR) currents. Typically, at higher operating temperatures and with high-quality material, the detector will be diffusion-limited. Therefore, most of the following discussion on dark current is based on the diffusion characteristics of minority carriers. Other authors have noted that the diffusion current in planar photodiodes is largely determined by the device geometry; specifically, the ratio of the minority carrier diffusion length and quasineutral width of the semiconducting layer [15,22,23]. When considering a photodiode in three-dimensions, there are vertical and lateral components that contribute to the total dark current. The vertical component is due to the diffusion of minority carriers vertically beneath the $pn$-junction, and is controlled by the thickness of the absorbing layer and diffusion depth. The lateral component is controlled by the photodiode’s proximal environment; for example, an unguarded standalone diode will be limited by the diffusion length of the minority carriers. However, in small-pitch focal plane arrays, the pixel pitch can be reduced relative to the minority carrier diffusion length. For example, for an $n$-type doping density of $10^{1}6$ cm$^{-3}$, the minority carrier diffusion length for holes is estimated to be 25 ${\mathrm{\mu} \mathrm{m}}$ within the In$_{0.53}$Ga$_{0.47}$As absorber. Hence, the lateral component of the dark current will then be controlled by the pixel size. With reducing pixel size, we can reasonably expect the diffusion current due to the lateral diffusion of minority carriers to decrease. Shown in Fig. 4 is simulated dark current versus pixel pitch. The dark current is decomposed into diffusion and GR components by simulating the current-voltage characteristics with and without SRH enabled. It is observed that the total dark current, comprised of both diffusion and GR components, decreases with decreasing pixel pitch, due to the decreasing lateral diffusion current. For fixed junction radius, $r_j$, the GR current is constant as expected; the GR current is determined by the geometry of the depleted region. Last, an important result illustrated in Fig. 4 is that for small pixel pitches below 10 ${\mathrm{\mu} \mathrm{m}}$ and with larger junction radii may not be diffusion-limited as expected. For example, for an FPA with 7.5 ${\mathrm{\mu} \mathrm{m}}$ pixel pitch and a 2 ${\mathrm{\mu} \mathrm{m}}$ junction radius, the diffusion current is approximately equal to the GR current.
We also performed a study on how the QE and specific detectivity are affected by reducing pixel pitch, shown in Fig. 5. For fixed absorber thickness, $t_A$, the QE shown in Fig. 5(a) is positively affected by decreasing pixel pitch, increasing by about 10% for smaller pitches. The largest pixel pitch of 20 ${\mathrm{\mu} \mathrm{m}}$ approaches the minority carrier diffusion length of 25 ${\mathrm{\mu} \mathrm{m}}$. In this regime, the geometry of the diode is starting to transition from short-base to long-base laterally. When illuminated, carriers that are generated further from the junction will be lost to recombination, lowering the QE.
Fig. 4. Simulated total dark current versus pixel pitch for varied junction radii. Dashed and dotted lines denote the GR and diffusion contributions to the total current respectively. The applied voltage and temperature were -200 mV and 300K respectively.
Fig. 5. (a) Simulated (left) quantum efficiency and (right) specific detectivity for varied absorber thickness versus pixel pitch. The incident wavelength and flux were 1.5 ${\mathrm{\mu} \mathrm{m}}$ and $10^{14}$ ph cm$^{-2}$ s$^{-1}$ respectively. (b) Dark current (left) and dark current density (right) versus pixel pitch. For both figures, the applied voltage and temperature were -200 mV and 300K respectively.
However, note that while the QE rises with decreasing pitch, $D^{*}$ decreases slightly, by less than a factor of two. At first, this result seems counterintuitive as Fig. 4 indicates that the total dark current decreases, and Fig. 5(a) indicates that the QE should rise, meaning that it may be reasonable to expect $D^{*}$ to improve. The minor decrease in $D^{*}$ is explained in Fig. 5(b). While the total dark current decreases with pixel pitch, the dark current density increases. As previously mentioned, the total dark current becomes closer to dominated by GR current, which is invariant with pitch. Therefore once converted to a current density by dividing by the pixel pitch squared, manifests as a minor increased dark current density, lowering $D^{*}$.
3.2 MTF implications in small-pitch arrays
As mentioned previously, one motivation for decreasing pixel pitch is to improve the detector’s MTF. Shown in Fig. 6(a) is an example of a simulated spot-scan profile of a 10 ${\mathrm{\mu} \mathrm{m}}$ pitch FPA with $t_A=3\,{\mathrm{\mu} \mathrm{m}}$. Note, since the device is planar, there is a significant amount of photocurrent picked up by the central pixel when the Gaussian beam is incident on the adjacent pixels. The 2D Fourier transform of the profile yields the MTF in terms of both spatial frequency components, $\eta$ and $\xi$, and is presented in Fig. 6(b). The blue line indicates the footprint for a 10 ${\mathrm{\mu} \mathrm{m}}$ pitch. While useful to illustrate that we can compute the full MTF, it is helpful to limit the discussion to one spatial frequency component, while the other is held static. Shown in Fig. 7(a) is the MTF for $\xi =0$ for varied pixel pitch; the dashed lines indicate the corresponding footprint for each pitch. While the MTF indeed improves with decreasing pitch, we note that the deviation of the simulated MTF from the expected footprint increases, indicating diminishing returns in MTF improvement with smaller pitches. Last, the MTF at the Nyquist frequency, denoted by the star shaped markers, does increase slightly with decreasing pitch.
Fig. 6. (a) Two-dimensional spot-scan profile mapping the normalized photocurrent observed in the central pixel versus the position of a Gaussian beam excitation. Gray lines denote pixel borders and blue hatched circles denote the $p^{+}$ diffusions. (b) The two-dimensional MTF resulting from the Fourier transform of (a). The blue line denotes the expected detector footprint for a 10 ${\mathrm{\mu} \mathrm{m}}$ pitch.
Fig. 7. (a) One-dimensional MTF while $\xi =0$ for varied pixel pitch. Solid lines are the simulation results and dashed are the expected detector footprint. (b) Decomposed MTF for a 10 ${\mathrm{\mu} \mathrm{m}}$ pitch array. The MTF at Nyquist frequency is about 0.37.
One advantage of using numerical simulations to compute the MTF is that we can isolate the individual processes that lead to MTF degradation. Shown in Fig. 7(b) are the isolated MTF components for a 10 ${\mathrm{\mu} \mathrm{m}}$ pitch FPA. For a planar back-side illuminated device, the optical crosstalk component, ${\mathrm {MTF}_\mathrm {O}}$, as expected remains near unity with increasing spatial frequency; there are not any defining architectural features that would scatter light into the adjacent pixels. From Fig. 7(b), it is clear that the main source of MTF degradation is due to inter-pixel diffusion.
3.3 Diffusion control junctions
There were two important outcomes from the results presented in the previous sections. First, an important source of dark current in In$_{0.53}$Ga$_{0.47}$As FPAs is due to the lateral diffusion of minority carriers from the pixel boundaries. Second, the predominant source of MTF degradation in small-pitch planar In$_{0.53}$Ga$_{0.47}$As FPAs is due to inter-pixel diffusive crosstalk. To that end, we propose a modification to the pixel sub-architecture aimed to address both issues. Shown in Fig. 8, we have placed four additional $p^{+}$ diffused regions on the pixel corners. Under reverse bias, these junctions will suppress the minority carrier profile laterally through the diode, and reduce the lateral contribution to the diffusion current, and mitigate inter-pixel crosstalk from diffusing carriers [24]. Since these junctions are intended to modify the diffusion characteristics of the photodiode, we refer to them as diffusion control junctions (DCJs). More detail on their operation may be found elsewhere [24,25].
Here, we focused on the following device geometry and operation. The baseline photodiode is the same as previously discussed. The DCJs were assumed to be abrupt cylindrical junctions of 1 ${\mathrm{\mu} \mathrm{m}}$ radius, persisting to the same depth as the central sensing junction. While the DCJs are contacted separately, an equivalent reverse bias of -200 mV was applied.
Fig. 8. An example of an In$_{0.53}$Ga$_{0.47}$As photodiode with diffusion control junctions implemented on the pixel corners.
3.3.1 Dark current and quantum efficiency
Shown in Fig. 9(a) is a comparison of a photodiode with the DCJs implemented with a baseline device. As expected, the device with the DCJ presents with slightly lower dark current, as the lateral contribution is suppressed. Note, for the 10 ${\mathrm{\mu} \mathrm{m}}$ pixel pitch and 2 ${\mathrm{\mu} \mathrm{m}}$ central junction radius the lateral contribution to the dark current is not as large as it would be for a larger pixel pitch. As such, the dark current is only marginally affected, by less than a factor of two. At low temperatures, the affect is negligible as the DCJ only modifies the diffusion current characteristics of the photodiode.
The comparison of QE and $D^{*}$ for the two devices is shown in Fig. 9(b). While the DCJs lower the dark current by suppressing the minority carrier profile, it can be expected that the photocurrent will be similarly affected. In fact, the QE is also reduced by about 20% when compared to the baseline device. However, when comparing $D^{*}$, the DCJs only reduce the specific detectivity by less than a factor of two.
Fig. 9. A comparison of the simulated (a) dark current density and (b) quantum efficiency and specific detectivity for a baseline device and one with DCJs implemented.
3.3.2 Crosstalk mitigation
The secondary motivation for including the DCJs into the photodiode architecture was as a form of crosstalk mitigation. Shown in Fig. 10 are the 2D spot-scan profiles for the (a) baseline and (b) modified devices. In Fig. 10(b) a significant reduction in photocurrent measured in the central pixel is observed when the Gaussian beam is incident on adjacent diodes, especially when considering the diagonally adjacent pixels. The resulting decomposed MTF curves are shown in Fig. 11 for a single spatial frequency component ($\xi =0$). The baseline device has an MTF at Nyquist frequency of 0.37, and the device with DCJs is improved to 0.5. The isolated contributions to the MTF reveal that the improvement, as expected, stems from the suppression of inter-pixel diffusive crosstalk.
4. Conclusion
In this work, we presented a comprehensive analysis of how the relevant infrared focal plane array figures of merit are affected by reduction in pixel pitch. It was shown that from a 20 ${\mathrm{\mu} \mathrm{m}}$ to 5 ${\mathrm{\mu} \mathrm{m}}$ pitch the total dark current decreases from about 15 fA to 2.5 fA ($r_j=2\,{\mathrm{\mu} \mathrm{m}}$), the quantum efficiency rises from 80% to 90% ($t_A=3{\mathrm{\mu} \mathrm{m}}$), and the specific detectivity decreases from 2 to 1.4 cm Hz$^{1/2}$ W$^{-1}$ ($t_A=3\,{\mathrm{\mu} \mathrm{m}}$). The decrease in dark current is due to a suppression of the diffusion current by reducing the contribution from laterally diffusing holes from the pixel boundaries. The decrease in $D^{*}$ was shown to be due to a rise in the dark current density as the device becomes closer to GR-limited for smaller pixel sizes. It was also shown that while the MTF of the FPA improves with reduced pixel pitch, as expected, it further deviates from what would be expected by the detector footprint; this was shown to be due to an increase in inter-pixel diffuse crosstalk.
Fig. 10. A comparison of the simulated two-dimensional spot-scan profiles for (a) the baseline and (b) modified photodiode architectures.
Fig. 11. A comparison of the simulated MTF ($\xi =0$) for (a) the baseline and (b) modified photodiode architectures.
From the results, we proposed the addition of “diffusion control junctions” to the pixel sub-architecture aimed toward lowering dark current and mitigating the increase in pixel crosstalk in small-pitch planar FPAs. For a 10 ${\mathrm{\mu} \mathrm{m}}$ pitch pixel, the dark current was shown to decrease by a little less than a factor of two at room temperature. Alongside the decreased dark current is a decrease in photocurrent leading to a reduction of QE by about 20%. The specific detectivity, on the other hand, was shown to decrease marginally, by less than a factor of two. The MTF improves with the inclusion of the diffusion control junction, increasing the MTF at the Nyquist frequency from 0.37 to 0.5, indicating successful mitigation of inter-pixel crosstalk.
While this work focused on one specific implementation of the diffusion control junctions, previous work has shown that a higher degree of dark current suppression may be possible by exploiting radial symmetry with an annular structure [24]. However, the QE is of course also affected to a higher degree, requiring a separate mitigation strategy. If the reduction in QE can be addressed, an improvement in $D^{*}$ may be realized with the improved MTF performance.
Funding
Army Research Office (W911NF-19-1-0161); Army Research Laboratory (W911NF-18-2-0027); Defense Advanced Research Projects Agency (FA8650-16-C-7639).
Acknowledgments
The authors would like to thank their colleagues, Drs. P. Wijewarnasuriya, A. D’Souza, M. Reine, J. Schuster, and B. Pinkie for many useful discussions throughout this work. The authors also thank Mr. T. Hubbard for his help in the initial modeling effort. The authors credit Dr. A. Wichman for his discovery, and preliminary work, on the DCJ effect.
Disclosures
The authors declare no conflicts of interest. The views, opinions, and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. government.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
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