Characterization and Optimization of Skipper CCDs for the SOAR Integral Field Spectrograph

We fabricated eight backside illuminated AstroSkipper CCDs for the SIFS focal plane prototype. These detectors came from a fabrication run supported by the DOE Quantum Science Initiative, Early Career Award, and laboratory R&D funds. Figure 1 shows one of these wafers, which was fabricated at Teledyne DALSA. The wafers were processed to reach astronomy-grade qualifications following the same procedure as used for the Dark Energy Camera (DECam) and Dark Energy Spectroscopic Instrument (DESI; Holland et al. 2003; Bebek et al. 2015, 2017; Flaugher et al. 2015). Factors such as thickness and CCD surface coatings were developed to reach high QE from the near-infrared (NIR) to the near-ultra-violet, which are desirable for astronomical observations (Bebek et al. 2015). The wafers were thinned from a standard thickness of 650–675 μm to 250 μm at a commercial vendor and then backside processed at the LBNL Microsystems Laboratory. A thin (20–25 nm) in situ doped polysilicon layer was applied to form a backside n⁺ contact (Holland et al. 2007; Groom et al. 2017).

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The AstroSkipper CCDs are p-channel devices fabricated on high resistivity (>5 kΩ cm), n-type silicon. p-channel CCDs have demonstrated an improved hardness to radiation-induced CTI when compared to n-channel CCDs due to the dopants used to form the CCD channels (Gow et al. 2014, 2016; Wood et al. 2017). The p-channel nature of the Skipper CCD makes this technology attractive for space-based astronomical applications. Furthermore, high QE in the optical and near-infrared (O/NIR) makes these detectors candidates for ground- and spaced-based astronomical spectroscopy. To reach QE > 80% in the O/NIR, our eight AstroSkipper CCDs were treated with an anti-reflective (AR) coating at the LBNL Mycrosystems Laboratory. The AR coating was developed for the DESI detectors and consists of a 20 nm layer of indium tin oxide (ITO), 38 nm ZrO₂, and 106 nm of SiO₂. This three-layer AR coating resulted in substantial QE improvements at shorter wavelengths (<600 nm) than the two-layer ITO/SiO₂ coating that was used for the DECam detectors (Bebek et al. 2017; Groom et al. 2017). This result is reproduced in QE measurements of the AstroSkippers (Section 7.6).

Each silicon wafer contains 16 Skipper CCDs (Figure 1) with different readout and size configurations. The AstroSkipper detectors to be used for SIFS are standard wide-format Skipper CCDs (6k × 1k, 15 μm pixels) with four amplifiers (Figure 1). The choice of detector format and pixel size was informed by the current SIFS focal plane; a mosaic of four 6k × 1k Skipper CCD detectors will be used to cover the full ∼4k × 4k pixel area of the current SIFS detector in order to preserve the optical configuration of the instrument. More detailed plans for the construction of the Skipper CCD SIFS focal plane prototype can be found in Villalpando et al. (2022).

The AstroSkipper detector packaging was performed at Fermilab. We designed a four-side-buttable package (Villalpando et al. 2022) that is similar to DECam packages (Diehl et al. 2008). The AstroSkipper package has two main components: a flexible cable for carrying electrical signals to/from the CCD and a mechanical foot for mounting the CCD to the focal plane. The packaging process consists of attaching the flexible cable and CCD to a Si substrate with epoxy, wirebonding the CCD pads to the flexible cable, attaching the CCD and cable assembly to a gold-plated invar foot for focal plane mounting, and placing the packaged AstroSkipper within an aluminum carrier box for storage, transport, and laboratory testing (Figure 2). The carrier box is designed to mount directly to the cold-plate inside the testing vacuum dewar. A set of custom mechanical fixtures were developed to standardize and streamline the packaging process building upon experience from packaging DECam and DESI detectors (Flaugher et al. 2015; Villalpando et al. 2022).

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The AstroSkipper CCDs undergo testing employing the optical setup shown in Figure 3. Characterization of DECam and DESI detectors utilized a similar optical setup (Diehl et al. 2008; Bonati et al. 2020). This setup is located in a "dark room" in order to reduce external light entering the testing station. A single AstroSkipper CCD is housed in a thermally controlled vacuum dewar with a fused silica window for illumination purposes. The AstroSkipper carrier box (Figure 2) attaches to an aluminum plate that is screwed to a copper cold finger inside the vacuum dewar. The system is cooled by a closed-cycle cryocooler to an operating temperature of 140K, which is maintained by a LakeShore temperature controller. A standard set of optical devices consisting of a quartz tungsten halogen lamp, motorized filter wheel, monochromator, shutter, and integrating sphere are used to provide uniform illumination of the AstroSkipper surface in the targeted wavelength. Light intensity is measured independently by a National Institute of Standards and Technology (NIST)-traceable Oriel photodiode mounted on the integrating sphere. This first photodiode, in conjugation with a second Thorlabs NIST-traceable photodiode mounted at the position of the CCD, allows us to calibrate the photon flux for absolute QE measurements. The shutter, filter wheel, and monochromator are controlled using a serial-to-Ethernet interface.

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The readout chain consists of a second-stage flex cable, an output dewar board, which provides the pre-amplification stage, and a low-threshold acquisition (LTA) board. The flexible cable has two high performance LSJ689-SOT-23, p-channel junction-gate field-effect transistors, providing ultra-low noise (∼2.0 nV/$\sqrt{\mathrm{Hz}}$), four 20 kΩ resistors, and a 51-pin Omnetics connector. The LTA readout board was designed at Fermilab as an optimized readout system for p-channel, thick, high resistivity Skipper CCDs (Cancelo et al. 2021). The LTA's flexibility allows the operation of Skipper CCDs to be optimized for different applications (e.g., DM direct detection and astronomy). The LTA is a single-printed circuit board hosting four video amplifiers for readout, plus CCD biases and clock control. The LTA is controlled by a Xilinix Atrix XC7A200T FPGA, which sets programmable bias and clock voltages, video acquisition, telemetry, and data transfer from the board to the PC. The user can communicate with the LTA via terminal commands to perform board configuration, readout and telemetry requests, and sequencer uploading. The data acquisition comes in the form of images in FITS format for subsequent analysis.

Despite the timing optimization described in the previous section, it is clear from Figure 5 that it would still take several hours to achieve photon-counting resolution over the entire detector. These long readout times are unacceptable for astronomical applications. For astronomical observations, there is a minimum at which the signal-to-noise improvements due to reducing readout noise is overcome by the lost exposure time during long readout times, i.e., time used for readout could alternatively be used to collect more signal. Therefore, it is a priority to explore readout noise configurations for a particular application and optimize the AstroSkipper parameters (e.g., N_samp) to reach the desired noise performance in the least amount of time.

Drlica-Wagner et al. (2020) calculated the optimal readout noise for Lyα observations with the DESI multi-object spectrograph, considering two scenarios: maximize the signal-to-noise at a fixed observation time or alternatively minimize the observation time at a fixed signal-to-noise (see Figure 1 in Drlica-Wagner et al. 2020). In both instances, an optimum was found at σ_N ∼ 0.5 e⁻ rms pixel⁻¹, assuming that only 5% of the detector pixels need to be read with the improved signal-to-noise. It would take ∼40 minutes to achieve ∼0.5 e⁻ rms pixel⁻¹ in all detector pixels, and we attempt to shorten these long readout times by exploiting the region of interest (ROI) capability of the Skipper CCD (Drlica-Wagner et al. 2020; Chierchie et al. 2021). We can define a geometrical area on the Skipper CCD (e.g., 5% of the detector area) that is read with multiple samples. For our implementation, this ROI will correspond to a specific wavelength range produced by the SIFS spectrograph. The ROI can be read with reduced noise (e.g., 0.5 e⁻ rms pixel⁻¹) while the rest of the detector is read once with single-sample noise.

We use each integration window instance time and single-sample readout noise (top panel in Figure 5) to find the optimal configuration of pixel integration time and number of samples (N_samp) to reach 0.5 e⁻ rms pixel⁻¹ in 5% of the detector area and minimize the readout time per frame. From Equation (1), which describes the relationship between readout noise and the number of non-destructive read samples (N_samp), one sees that ${N}_{\mathrm{samp}}={\left(\tfrac{{\sigma }_{1}}{{\sigma }_{N}}\right)}^{2}$ where σ_N = 0.5 e⁻ rms pixel⁻¹ and σ₁ is the optimal single-sample readout noise in the top panel of Figure 5. For each N_samp and integration time (t_pixel), we use Equation (2) to calculate the total readout time. Figure 6 shows the optimal pixel integration time (∼20 μs) that minimizes the total readout time for a frame with σ_N = 0.5 e⁻ rms pixel⁻¹ in 5% of the pixels. We can then calculate the total readout time for a full frame where 5% of the detector is readout with 0.5 e⁻ rms pixel⁻¹ and the remaining 95% with single-sample readout noise, which comes out to t_readout ∼ 3.6 minutes.

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The readout noise of a Skipper CCD is tunable through multiple non-destructive measurements of the charge in each pixel. For uncorrelated Gaussian noise, the effective readout noise distribution from averaging N_samp non-destructive measurements will scale as ${\sigma }_{N}\propto 1/\sqrt{{N}_{\mathrm{samp}}}$ (Equation (1)). We note that the readout noise is a combination of intrinsic electronic noise and external noise sources, which are dependent on specific testing stations and factors such as electronic grounding. We measure the readout noise from processed images, i.e., overscan subtracted images. To account for non-uniformities in the overscan pedestal level, the overscan subtraction algorithm fits the pixel distribution of each row in the overscan region with a multi-Gaussian model and subtracts the mean of the 0 e⁻ peak to that row. Similarly to Drlica-Wagner et al. (2020) and Villalpando et al. (2022), the readout noise we measure is given by the standard deviation of the 0 e⁻ peak. Figure 7 shows a 600 × 3200 pixel image where the overscan region starts at pixel 3080. The middle panel plots the structure of the pedestal in the overscan and activated regions in a single-sample image. We observe a slowly decaying transient that is correlated with row number; this same behavior is present in both multi- and single-sample images. The pedestal subtraction algorithm is able to correct for this row-dependent variation as observed in the bottom panel (pedestal subtracted traces).

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We measure the readout noise for all 32 AstroSkipper amplifiers (eight detectors with four amplifiers per detector) and find values ranging from 3.5 e⁻ rms pixel⁻¹ to 5 e⁻ rms pixel⁻¹; the six astronomy-grade detectors maintain a readout noise < 4.3 e⁻ rms pixel⁻¹ for all 24 amplifiers. In Figure 8, we show an example of photon-counting (left) and noise reduction (right) achieved by one of the AstroSkippers with 400 samples per pixel. To simplify the calculation of the readout noise as a function of N_samp, we again implement the charge-less readout without integration charge to measure only the noise characteristics of the system. The measured noise varies from the expected Gaussian noise model (Equation (1)) by ≲20% for N_samp < 10 and 10% for N_samp > 150 (Figure 8, right). We note that the greatest contribution to variations in the noise model comes from instabilities in the first few samples. Similar behavior was seen in early testing of Skipper CCDs (Fernández Moroni et al. 2012), where it was postulated that variations in the readout noise at low N_samp are produced by residual low-frequency noise, and fluctuations at large samples might be due to external or unidentified sources of noise coupling to the readout electronics. At low sample numbers, fluctuations in the clock-induced transients can dominate the noise behavior; however these transients settle as N_samp increases. For instance, if the averaging starts from N_samp = 5, the variation in the noise measurements with respect to the Gaussian noise model reduces to <3% for all sample numbers. Figure 9 shows the single (σ₁) and multi-sample (σ₄₀₀) noise performance for each amplifier and detector.

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Since each peak in the pixel distribution histogram (Figure 8) differs from its neighbor by a single electron, we can obtain a direct measurement of the detector's gain by calculating the difference between adjacent electron peaks, i.e., the gain would be given by Δ_i,i+1 where Δ_i,i+1 = μ_i+1 − μ_i is the difference between the means of the Gaussian fits to the i and i + 1 adjacent electron peaks. Figure 10 shows gain measurements per amplifier from the six AstroSkippers; gain measurements depend on resolving the electron peaks in each amplifier, and applying the method described above. We use the 0 e⁻, 1 e⁻, and 2 e⁻ electron peaks (Δ_0,1, Δ_1,2) and take the average between both Δ_i,i+1 to calculate these gain values for all amplifiers. We measure gain values ranging from ∼123 ADU/e⁻ to ∼143 ADU/e⁻ (left) for amplifiers on all six of the astronomy-grade AstroSkippers with average variations of <6% between gain values from all amplifiers. We do not see any clear correlation between amplifier gain and amplifier noise.

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The AstroSkipper's large dynamic range and ability to count individual charge carriers enables a unique avenue to measure linearity at both low and high illumination levels. At high illumination levels, we follow a conventional approach to measure nonlinearity by increasing the exposure times. We study a wide range of images taken with high illumination levels from 1500 e⁻ pixel⁻¹ to values near saturation. The data taking procedure consists of taking ∼20 flat-fields with increasing illumination; we perform bias subtraction and sigma clipping to eliminate cosmic rays on each frame. We calculate nonlinearity following the procedure implemented by Bonati et al. (2020) to measure the nonlinearity of the DESI detectors. To compute the nonlinearity over our data set of 20 flat-field images, we implement a least-squares fit to minimize the sum of the squares of the error $E{(\alpha ,\beta )={\sum }_{n=1}^{20}({y}_{n}-(\alpha {x}_{n}+\beta ))}^{2}$. Here n indexes over the images, y_n is the mean of the pixel values for image n, x_n is the exposure time for image n, and α and β are the parameters of the linear fit. We find the values of α and β that minimize E(α, β), i.e., ∂E/∂α = 0, ∂E/∂β = 0. The nonlinearity factor is then given by the mean value of the square-root of the residuals E(α, β). We find nonlinearity values <0.05% for all of the amplifiers in the six AstroSkipper CCDs.

In conventional CCDs, low-signal-level nonlinearities are poorly understood since these CCDs lack the precision to quantize charge in the single-electron regime. In contrast, Skipper CCDs can quantify nonlinearities for all electron occupancies, i.e., they can resolve electron peaks for the full range. For instance, in Bernstein et al. (2017) nonlinearity measurements for a subset of DECam devices show poorly understood behavior at low illumination levels (few tens of electrons). The AstroSkipper allows us to precisely characterize nonlinearity in this regime of a few tens of electrons following a procedure similar to the one described in Rodrigues et al. (2021). In the photon-counting regime, one can probe nonlinearities by measuring variations in the relationship between the number of electrons in each pixel and the signal readout value in ADUs (i.e., the gain).

We take several flat-field images with 400 samples per pixel to reach single-electron resolution with σ₄₀₀ ∼ 0.18 e⁻ rms⁻¹ pixel⁻¹. Images are taken with increasing exposure time where the set of images produce different overlapping Poisson distributions with an increasing mean number of electrons (Rodrigues et al. 2021). We resolved up to 50 e⁻, i.e., it is possible to count individual peaks up to the 50th electron peak in the set of images. To perform the nonlinearity measurement, we fit each electron peak with a Gaussian and compute the gain from each peak by dividing the mean value of the difference between the electron peak and the 0 e⁻ peak in ADU by the peak's assigned electron number, (μ_i − μ₀)/i. For example the gain calculated from the 10th electron peak would be given by (μ₁₀ − μ₀)/10 e⁻ where (μ₁₀ − μ₀) is the difference (in ADU) between the means of Gaussian fits to the 0 e⁻ and 10 e⁻ peaks and 10 e⁻ is the assigned number of electrons for that peak. We note that this method for charge quantization results in increased variability when computing gain values at low electron counts and diminishes for higher electron counts since variations in the mean get divided by the amount of charge. If we instead calculate the gain from the peak-to-peak separation, i.e., μ_i+1 − μ_i , we find a consistent gain with a scatter of <4% between individual electron peaks (though we note that adjacent points are highly correlated).

Figure 11 (top) shows peak-to-peak gain calculations and as expected, we observe a greater spread between gain values. The bottom panel shows a low-signal nonlinearity measurement for one of the AstroSkippers where we represent the nonlinearity as the deviation from the unity of the ratio between the gain calculated as (μ_i − μ₀)/i and the fixed, conventional gain value measured from the slope of the variance versus the signal in the PTC (Section 7.3). We find nonlinearity values that are <1.5% at this low-signal regime of a few tens of electrons which agrees with values reported in Rodrigues et al. (2021) (<2.0%). Furthermore, this method for calculating nonlinearity also shows the agreement between PTC and charge quantization gain calculations. Similar to Drlica-Wagner et al. (2020), we find that the gain measurements from the two methods agree within ∼1%.

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CIC is generated during the clocking sequence when inverting clocks between the high and low-voltage states. When switching the clock to the non-inverted state, holes that become trapped at the Si–SiO₂ interface during clock inversion are accelerated with sufficient energy to create electron–hole pairs through impact ionization (Janesick 2001). Released electrons are then collected in the summing well and contribute to the overall readout signal. In conventional CCDs, where the noise floor can be >5 e⁻ rms pixel⁻¹, CIC is not apparent, i.e., the charge produced by CIC can be characterized as shot noise with a contribution of N_CIC = $\sqrt{{\mu }_{\mathrm{CIC}}}$ where μ_CIC is the average CIC in electrons (Janesick 2001) and for μ_CIC = 3 e⁻ pixel⁻¹ frame⁻¹ (typical value observed with an AstroSkipper at high operating voltages) the CIC noise contribution of ∼1.7 e⁻ rms pixel⁻¹ would be lower than the noise floor and thus undetectable. However, for ultra-low-noise detectors operating in the photon-counting regime, CIC is an important source of background for observations of faint sources (e.g., Kyne et al. 2016).

We focus on optimizing the CIC with respect to the horizontal clock swings since we find that CIC produced in the activated area is negligible compared to CIC generated in the serial register. First, we build a statistical model to predict the electron event rate from CIC as a function of the horizontal clock voltage swings. Since CIC can be characterized as shot noise, we assume it obeys Poisson statistics and therefore the expected CIC electron rate is given by the Poisson probability mass function (PMF)

$$\begin{eqnarray}&&P(X=k)=\displaystyle \frac{\exp (-\mu ){\mu }^{k}}{k!}\end{eqnarray} \tag{ 4 }$$

where P(X = k) gives the probability of observing k events (CIC electron rate) in a given interval, and μ is the average rate of CIC electron events for the full readout sequence. Furthermore, we assume that the average rate of CIC electron events (μ) can be modeled with an exponential, which describes the growth of CIC electron events with respect to the horizontal clock swings,

$$\begin{eqnarray}&&\mu =a\exp (b{\rm{\Delta }}H)+c.\end{eqnarray} \tag{ 5 }$$

Here ΔH is the horizontal clock voltage swing and a, b, and c are fit parameters. To calculate the best fit for μ, we take several dark frames with increasing ΔH and electron resolution (N_samp = 400, σ₄₀₀ ∼ 0.18 e⁻ rms pixel⁻¹) in order to resolve single electron rates from CIC. To get μ, we fit a single Gaussian model to the pixel distribution, containing CIC electron rate peaks, and subtract the background, which is calculated with the low-voltage configuration that generates minimum CIC (i.e., ≲1.45 × 10⁻³ e⁻ pixel⁻¹ frame⁻¹). We fit an exponential model to find the values of a, b, and c in Equation (5).

The number of transfers in the serial register is closely related to CIC (Janesick 2001); the probability of generating a CIC electron event increases as the pixel is clocked more times in the serial register. To investigate how the number of transfers (NT) affect our probabilistic model, we calculate CIC for one quadrant of the smaller format (1248 × 724, 15 μm × 15 μm pixels) Skipper CCD characterized in Drlica-Wagner et al. (2020). We use the voltage configurations tested in the AstroSkipper and repeat the same data taking procedure, i.e., dark frames (σ_N ∼ 0.18 e⁻ rms pixel⁻¹). We consider CIC_1i and CIC_2i : the average CIC electron rate per pixel per frame from the smaller Skipper CCD and the AstroSkipper, respectively. Assuming a linear relationship between both data sets (informed by CIC production in Janesick 2001), the linear regression model is

$$\begin{eqnarray}&&{\mathrm{CIC}}_{2i}={\beta }_{0}+{\beta }_{1}\cdot {\mathrm{CIC}}_{1i}+{\epsilon }_{i}\end{eqnarray} \tag{ 6 }$$

where β₀ is the rate of CIC generation, related to the relative NT between both detectors, β₁ is the intercept, and _i is the error term associated with the ith observation in CIC_2i . We perform a linear regression to find β₀ and β₁, minimizing the sum of squared residuals, $\sum {\epsilon }_{i}^{2}$. Figure 12 (left) shows the linear relationship between CIC_1i and CIC_2i with β₀ ∼ 7.1 (a factor of 7 increase in CIC for the AstroSkipper with 3200 transfers compared to 450 transfers for the smaller Skipper CCD).

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Figure 12 (right) shows the simulated CIC rate from the statistical model, i.e., random draws from the Poisson PMF, and the measured data; we see better than ∼10% agreement, for ΔH > 7.5 V, between the model and the measured data. We note that this model assumes a fixed horizontal clock filtering solution, and horizontal clock width. In future work, we will generalize the statistical model formalism to include the effect from varying clock pulse width, i.e., the time the clock spends in the non-inverted state immediately after inversion, and the CIC reduction from different clock shaping solutions (Janesick 2001; Daigle et al. 2009, 2010).

Because CIC is closely linked to horizontal clock voltage swing (ΔH) and the full-well capacity is also dependent on ΔH, we must optimize CIC and full-well capacity for the expected signal levels in the application. To mitigate CIC, we have implemented a simple filtering solution consisting of a first-order low-pass filter with a time constant τ = 5.1 μs placed between the pre-amplification stage and the LTA. This allows for a factor of ∼2 reduction in CIC electron events for ΔH > 9 V, which yields the highest full-well capacity.

Previous Skipper CCD operational parameters, such as clock voltage values, were primarily optimized for reducing CIC for DM direct detection and rare particle searches where operational processes that can produce a few electron events severely reduce sensitivity to rare events (Tiffenberg et al. 2017; Crisler et al. 2018; Barak et al. 2022). However, the small voltages used for these rare particle searches limit the dynamic range of the Skipper CCD, which can be problematic for most astronomical applications. We perform a voltage optimization for the AstroSkipper in order to increase the dynamic range while maintaining low CIC, stable readout noise, and photon-counting capabilities.

The full-well capacity is derived from the PTC. Due to the CIC and full-well dependence on the horizontal clocks swing voltage, we optimize ΔH for reducing CIC while maintaining a full-well capacity suitable for the expected signal levels from SIFS (≲1000 e⁻ for science images and >40,000 e⁻ for calibration products). Figure 13 shows the full-well capacity for increasing ΔH (top), which approaches levels comparable to other thick, fully depleted CCDs (Flaugher et al. 2015), and the CIC levels expected for each full-well (bottom). We measure an upper limit on the CIC using the low-voltage configuration from SENSEI (red arrow in Figure 13, bottom panel); however, the full-well capacity at these voltages (∼900 e⁻) is inadequate for the SIFS application. In our optimization tests, we find that a CIC rate of ∼1 e⁻ pixel⁻¹ frame⁻¹ is possible with a corresponding full-well of ∼10,000 e⁻, which is adequate for the expected signal levels in science data products. However, we expect signals of ∼40,000–50,000 e⁻ in calibration data products. Configuring voltages for these signal level results in a CIC rate of ∼3 e⁻ pixel⁻¹ frame⁻¹. The achieved CIC rate for the high-voltage configuration is an unacceptable source of noise if the expected signals are in the order of a few electrons. We note that it might be possible to maintain a large full-well capacity while retaining low CIC to levels (≲10⁻³ e⁻ pixel⁻¹ frame⁻¹) by employing waveform shaping (e.g., Barak et al. 2022). Shaping the clock pulse rise time and sharpness, which play a critical role in CIC generation, has been shown to reduce CIC in EMCCDs (Wilkins et al. 2014; Kyne et al. 2016).

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This is the first time that full-well capacities of ∼40,000–50,000 e⁻ have been demonstrated with a Skipper CCD. This was achieved in a configuration with a horizontal clock swing, ΔH = 9.5 V, a vertical clock voltage swing, ΔV = 5 V, and a transfer gate clock voltage swing, ΔT = 5 V. We discovered that the floating sense node reference voltage can be a limitation in increasing the dynamic range. Furthermore, it is important to optimize this reference voltage for both full-well and "skipping" functionality. In the Skipper CCD output stage, the charge packet is passed to the small capacitance, floating sense node where the charge packet is read out once. Then the summing well voltage is set to the low state, i.e., lower than the sense node fixed reference voltage, for the charge packet to move back to the summing well, repeating this "skipping" process N_samp times. We optimized the sense node reference voltage to achieve the targeted full-well while maintaining the ability for photon-counting.

To characterize CTI, we implement the extended pixel edge response method (EPER). EPER consists of measuring the amount of deferred charge found in the extended pixel region or overscan of a flat-field at a specific signal level. CTI is calculated from the EPER as

$$\begin{eqnarray}&&\mathrm{CTI}=\displaystyle \frac{{S}_{D}}{{S}_{\mathrm{LC}}{N}_{P}},\end{eqnarray} \tag{ 7 }$$

where S_D is the total deferred charge measured in the overscan in electrons, S_LC is the signal level (e⁻) of the last column in the detector's activated area and N_P is the number of pixel transfers in the serial register (Janesick 2001). For our CTI measurement, we take a number of flat-fields at increasing illumination levels (∼10,000 e⁻ to ∼50,000 e⁻); Figure 14 shows the average CTI for all of the amplifiers in one of the AstroSkippers versus signal level. We calculate an average CTI value of 3.44 × 10⁻⁷ from the 24 amplifiers on the six astronomy-grade AstroSkipper CCDs, which is about an order of magnitude lower compared to the one we reported previously in Drlica-Wagner et al. (2020). Studies of deferred charge at low signal levels in Skipper CCDs are ongoing and are left to future work.

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We use DC to refer to the electron events generated in the CCD during exposure and readout phases of data collection. These electron events are unrelated to the transfer of the charge between pixels (CIC). We note that the DC value reported for the AstroSkipper is an upper bound as it includes contributions from light leaks and other environmental sources.

The measurements of DC consist of acquiring 10 single-sample dark exposures with 400 s of exposure time. A combined dark, consisting of the median from 10 images, is calculated to remove cosmic rays and any transient effect; the combined dark is overscan subtracted and the signal mean is calculated over the activated area, divided by the exposure time, and normalized by the detector's gain. We measure DC values of ∼2 × 10⁻⁴ e⁻ pixel⁻¹ s⁻¹ for the six astronomy-grade AstroSkippers. Despite the fact that we tested the AstroSkippers in a dark room, light leaks dominate the electron event rates in our DC measurements and increase linearly with exposure and readout time. Previous DC measurements for astronomy performed with a similar setup in ambient lighting yielded DC values an order of magnitude higher (∼10⁻³ e⁻ pixel⁻¹ s⁻¹; Villalpando et al. 2022).

Barak et al. (2020) reports a significantly lower DC value of 6.82 × 10⁻⁹ e⁻ pixel⁻¹ s⁻¹ for a Skipper CCD operating underground (the lowest DC value measured for a CCD to date). In contrast to our measurements, the DC measurements reported in Barak et al. (2020) are performed underground with a Skipper CCD that is shielded from environmental radiation. For detectors with single-photon capabilities, external effects that increase the dark rate even by a small amount can be a problematic source of noise in the low-signal regime.

The PTC characterizes the response of a CCD to illumination and can be used to measure the detector's gain and dynamic range. A PTC is constructed by taking several flat-fields at increasing illumination levels, which can then be used to show how the variance in the signal changes with the mean flux level of uniformly illuminated images. To eliminate non-uniformities, e.g., variations in the illumination and CCD cosmetic defects, the PTC is calculated with the difference between pairs of flat-fields.

It is assumed that charge collection in pixels exactly follows Poisson statistics, and therefore, pixels are independent light collectors. In this idealized case, the variance versus the signal mean should be linear, once the readout noise is negligible, with a 1/gain slope until pixel saturation. However, at high signal levels this assumption breaks, causing a loss in variance as the PTC linear behavior flattens out. Furthermore, binning neighboring pixels improves the linearity of the PTC (Downing et al. 2006; Astier et al. 2019). This indicates that correlation arises between neighboring pixels as charges migrate from one pixel to another, producing transverse electric fields on incoming photocharges (Holland et al. 2014). The repulsion effect between photocharges in a pixel's potential well causes quasistatic changes in the effective pixel area, for astronomical observations, biasing the light profile from a bright source. This effect is known as the BFE, which can bias the point-spread function (PSF) from a source by ∼1% and the shear of faint galaxies, posing an unacceptable systematic for large imaging surveys if not corrected (Gruen et al. 2015; Lage et al. 2017; Coulton et al. 2018; Astier et al. 2019; Astier & Regnault 2023). At the detector level the BFE has been observed on DECam, Hyper Suprime-Cam, and LSSTCam fully depleted CCDs (Gruen et al. 2015; Astier et al. 2019; Astier & Regnault 2023). Astier et al. (2019) proposes an electrostatic model to characterize the time-dependent build-up of correlations in flat-fields. The model describes the resulting correlation between pixels that grow with increasing flux and decay rapidly as photocharges migrate to neighboring pixels, resulting in a loss of covariance. The covariance function, that describes the change in the effective area as a result of BFE, for a given signal level (μ), is given by

$$\begin{eqnarray}\begin{array}{rcl}{C}_{{ij}}(\mu ) & = & \displaystyle \frac{\mu }{g}\left[{\delta }_{i0}{\delta }_{j0}+{a}_{{ij}}\mu g+\displaystyle \frac{2}{3}{\left({\boldsymbol{a}}\otimes {\boldsymbol{a}}+{\boldsymbol{a}}{\boldsymbol{b}}\right)}_{{ij}}{\left(\mu g\right)}^{2}\right.\\ & & +\left.\displaystyle \frac{1}{6}{\left(2{\boldsymbol{a}}\otimes {\boldsymbol{a}}\otimes {\boldsymbol{a}}+5{\boldsymbol{a}}\otimes {\boldsymbol{a}}{\boldsymbol{b}}\right)}_{{ij}}{\left(\mu g\right)}^{3}+...\right]+\displaystyle \frac{{n}_{{ij}}}{g}\end{array}\end{eqnarray} \tag{ 8 }$$

where a_ij describes the strength of the changes in the pixel area due to the accumulated charge and has units of 1/e⁻, b_ij describes other contributions to the pixel area change, e.g., shortened drift time and asymmetries in how charges are stored in pixels, g is the detector's gain, n_ij is a matrix that contains noise components with n₀₀ being the traditional readout noise, and ⨂ refers to the discrete convolution. We follow the method in Astier et al. (2019) and perform the fit for our covariance function up to signal values close to saturation (∼8 × 10⁶ ADUs) up to O (a¹⁰) terms, resulting in a 11 × 11 covariance matrix as a function of mean signal from the difference of flat-field pairs, taken at an increasing illumination level. Figure 15 shows the recovered 11 × 11 a_ij and b_ij pixel matrices. We note that ∣a₀₀∣ > a_ij , therefore a₀₀ is the biggest contributor to pixel change, i.e., the quantity that describes the strength of the BFE (Astier et al. 2019; Astier & Regnault 2023). In Figure 16, we show the a and b matrices best-fit values averaged over all amplifiers from an AstroSkipper detector as a function of distance with error bars representing the uncertainty from all the averages. We see that a decays rapidly and becomes isotropic; similarly to Astier et al. (2019), we see that b is negative except for b₀₁, which might indicate a parallel distance increase in the charge cloud as charge accumulates.

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Astier et al. (2019) fit the electrostatic model (Equation (8)) for a LSST Teledyne e2V 250 device with a thickness of 100 μm, operated at a substrate voltage of 70 V. They find ∣a₀₀∣ = 2.377 × 10⁻⁶. Astier & Regnault (2023) perform a BFE analysis for the CCDs in the Hyper Supreme-Cam, which uses deep-depleted, 200 μm thick Hamamatsu CCDs, operated with a substrate voltage of <50 V (Miyazaki et al. 2017); they measure ∣a₀₀∣ = 1.24 × 10⁻⁶. We measure an average value of ∣a₀₀∣ = 6.153 × 10⁻⁶. We note that the AstroSkipper a₀₀ higher value might be due to the thickness (250 μm) and the operating substrate voltage (40 V) as explained by a physics-based model from Holland et al. (2014), which shows that the PSF size depends on detector thickness and substrate voltage.

For spectroscopy, especially applications where the line's structure profile is important, i.e., the radial velocity structure of radiative transfer effects in an object (Schmid 2012), the BFE can be an important systematic. Furthermore, as part of spectroscopic data reduction, sky subtraction and wavelength calibration depend on sky lines and calibration lamp data, which often have a signal level that is significantly higher than the science data, sometimes approaching the detector's full-well capacity. This situation could potentially bias science measurements due to the BFE. We note that further studies of the BFE in the context of spectroscopy are needed to fully determine how the BFE may bias various spectroscopic measurements, e.g., redshift recovery, equivalent widths, velocity dispersion, etc. For instance, assuming that DESI detectors will have similar BFE characteristics to those measured in the AstroSkipper, it would be possible to use DESI data to measure the impact of the BFE on DESI science and predict the potential impact of the BFE on future spectroscopic cosmology surveys (e.g., a Stage-V spectroscopic survey; Schlegel et al. 2022).

The shape of the PTC curve (variance versus signal mean) can be approximated by considering the first element (the variance) in the covariance matrix, i.e., C₀₀ and a₀₀ in the Taylor expansion of Equation (8). The PTC curve as a function of μ with g, n₀₀, and a₀₀ as fit parameters is given by

$$\begin{eqnarray}&&{C}_{00}(\mu )=\displaystyle \frac{1}{2{g}^{2}{a}_{00}}\left[\exp (2{a}_{00}\mu g)-1\right]+\displaystyle \frac{{n}_{00}}{{g}^{2}}.\end{eqnarray} \tag{ 9 }$$

Figure 17 shows a PTC curve for one of the AstroSkippers calculated with Equation (9). We implement public code from the LSST Science Pipelines for calculating and fitting the PTC (Bosch et al. 2018). We use 135 pairs of flat-fields, to compute the difference between them, taken at increasing signal rates from a few electrons to saturation (∼50,000 e⁻ for the AstroSkipper PTC shown in Figure 17). PTCs are constructed using ΔH = 9.5 V and the full-well capacity is determined by the last data point that is not cut by the outlier rejection algorithm; the algorithm assigns weights to data points based on residuals from deviations to the model. Figure 18 shows the full-well capacity averaged across amplifiers for the six astronomy-grade AstroSkipper detectors; we measure full-well values ranging from ∼40,000 e⁻ to 65,000 e⁻ which are suitable for the SIFS application.

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Cosmetic defect tests consist of characterizing pixels that are "hot" in dark exposure frames and "cold" in flat-fields at different illumination levels. We take 10 dark exposure frames with 400 s of exposure in the dark to measure bright pixels. Images are overscan-subtracted and sigma clipped to eliminate cosmic rays. We flagged bad pixels as those with mean values μ ± 3.5σ. The same statistical discrimination is applied for "cold" pixels in flat-fields, which also eliminates "hot" pixels that might be present (μ + 3.5σ). We use different illumination levels up to μ ∼ 30,000 e⁻ pixel⁻¹. We create a mask to include these pixels ("cold" and "hot") and apply it to the images for subsequent tests. Figure 19 shows cosmetic values (the fraction of "cold" and "hot" pixels with respect to the total number of pixels in the detector) for all amplifiers in the six AstroSkipper CCDs. We find cosmetic defects affect <0.45% of the pixels for all amplifiers.

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To characterize charge diffusion, we implement the method described in Lawrence et al. (2011) which is suitable for thick, backside illuminated, fully depleted CCDs. The method consists of exposing the CCD to low-energy X-rays from a ⁵⁵Fe source and statistically characterizing the charge clouds that result from the X-ray photon generating charge carriers in tight clusters. The charge diffuses laterally, producing a cloud with a Gaussian profile. The method uses the profile of the two-dimensional, Gaussian PSF to measure diffusion from these charge clouds. The pixel selection algorithm reconstructs events and selects those originating from conversions of 5.988 keV Mn Kα photons, producing 1590 electron–hole pairs. The algorithm (1) defines a "box" that is 2 × 2 pixels and calculates the charge in that region, (2) calculates local maxima by rejecting the box with minimum charge between two intersecting regions, (3) histograms remaining boxes, and (4) centers the window on the Mn Kα peak position with upper bound at the Kα and Kβ peaks. We take 10 images each with 5 minutes exposure to ⁵⁵Fe radiation, which are combined to measure the PSF of the charge clouds using the method described above. We test different bias substrate voltages ranging from 30 to 70 V and compute the PSF as a function of the substrate voltage. We decide to operate the AstroSkipper CCD at 40 V (similar to DECam; Diehl et al. 2008), since we find that cosmetic defects, e.g., hot columns, grow with increasing substrate voltage (>40 V). We measure PSF values for all amplifiers <6.75 μm for the six AstorSkippers, operating with a substrate voltage of 40 V. This is comparable to DECam charge diffusion requirements: PSF < 7.5 μm with a substrate voltage of 40 V (Diehl et al. 2008).

The LBNL Mycrosystems Laboratory CCD backside treatment and AR coating provides excellent (QE > 80%) long wavelength (NIR) and acceptable (QE > 60%) g-band response for 250 μm thick detectors (e.g., Diehl et al. 2008; Bebek et al. 2017). In Drlica-Wagner et al. (2020), we demonstrated that a 250 μm thick, backside illuminated Skipper CCD can achieve relative QE > 75% for wavelengths 450–900 nm. Here we report the first absolute QE measurements for astronomy-grade Skipper CCDs and demonstrate better QE than previous measurements.

We define the absolute QE as the ratio of the number of electrons generated and captured per incident photon at a given wavelength for a given unit area,

$$\begin{eqnarray}&&\mathrm{QE}(\lambda )=({N}_{\mathrm{ADU}}K)\displaystyle \frac{{hc}}{{{Pt}}_{\exp }\lambda },\end{eqnarray} \tag{ 10 }$$

where N_ADU is the signal from the detector in ADU, g is the detector's gain in ADU/e⁻, h is the Planck constant, c is the speed of light, P is the incident optical power at the CCD surface, ${t}_{\exp }$ is the exposure time used to take the flat-fields, and λ is the incident light wavelength. An accurate measurement of the absolute QE depends on an accurate determination of the incident optical power at the AstroSkipper, housed in the vacuum dewar (Figure 3). To measure the absolute incident power at the detector, we mount a Thorlabs NIST traceable calibrated Si photodiode, with a 10 mm × 10 mm activated area, on an AstroSkipper package (Figure 2). The photodiode plus AstroSkipper package is mounted inside the vacuum dewar at the same location that the AstroSkipper CCDs are mounted when testing.

Figure 20 shows the absolute QE for one quadrant of an AstroSkipper and the comparison with DECam and DESI detectors. We see good agreement with the QE of the DESI detectors, which is expected given that the AstroSkipper has a similar AR coating (Bebek et al. 2017). For all amplifiers in the six AstroSkippers, we see QE ≳ 80% between 450 and 980 nm, and QE > 90% for wavelengths from 600 to 900 nm.

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We note that the absolute calibration, i.e., the ratio of the incident optical power in the Thorlabs photodiode relative to the Oriel NIST traceable photodiode on the integrating sphere (Figure 3) is the greatest source of uncertainty; therefore, we take multiple absolute calibration measurements. We derive an uncertainty in the absolute QE at each wavelength; we find uncertainties <6% in the absolute QE values for all wavelengths. Error bars in Figure 20 represent the uncertainty at each wavelength calculated from the absolute calibration measurements.