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arxiv: 2606.12055 · v1 · pith:IRO6TJAJnew · submitted 2026-06-10 · ⚛️ physics.ins-det · physics.optics

Modeling of Dark Count Probability in Perimeter-Gated SPADs

Md Sakibur Sajal , Ziyad Alswaidan , Tathagata Srimani , Marc Dandin This is my paper

Pith reviewed 2026-06-27 07:56 UTC · model grok-4.3

classification ⚛️ physics.ins-det physics.optics
keywords dark count probabilityperimeter-gated SPADGompertz functionsingle-photon avalanche diodetemperature compensationCMOS sensor arrayactivation function
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The pith

Dark count probability in perimeter-gated SPADs follows a complementary Gompertz function from which pixel-specific midpoint voltages and temperature compensation rates are derived.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that dark count probability in perimeter-gated single-photon avalanche diodes follows a complementary Gompertz function. This functional form directly supplies a pixel-specific midpoint perimeter gate voltage that marks the equiprobable operating point for each device. From the same descriptor a compensation rate is obtained that adjusts the gate voltage to counteract temperature-driven shifts in the activation function. The approach is demonstrated on an array of 4096 devices fabricated in 0.35 micrometer CMOS and measured from -5 C to 55 C, producing consistent bias control across process and temperature spreads.

Core claim

The dark count probability PDC of perimeter-gated single-photon avalanche diodes follows a complementary Gompertz form. From this form a pixel-specific descriptor, the midpoint perimeter gate voltage, is derived that characterizes a pixel's equiprobable operating point. A perimeter gate voltage compensation rate obtained from this descriptor offsets temperature-induced changes in the pixel's activation function. The framework is validated on 4096 pg-SPADs in a 64 by 64 array in 0.35 micrometer CMOS across temperatures from -5 C to 55 C and perimeter gate voltages from 0 to 5 V.

What carries the argument

Complementary Gompertz function relating dark count probability to perimeter gate voltage, used to extract the midpoint voltage descriptor per pixel.

If this is right

  • Deterministic bias control of dark count probability becomes possible across process and temperature variations.
  • A compensation rate for perimeter gate voltage can be calculated from the midpoint descriptor to maintain stable activation.
  • Each pixel in an array receives an individual characterization by its midpoint perimeter gate voltage.
  • The model supports operation of large CMOS-fabricated SPAD arrays with reduced variation in dark counts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Gompertz-based descriptors could support per-pixel calibration routines in larger imaging arrays.
  • Testing the model on SPADs fabricated in other CMOS nodes would check whether the midpoint descriptor remains process-independent.
  • If the compensation rate proves stable, it could reduce the need for frequent temperature recalibration in fielded detectors.

Load-bearing premise

The complementary Gompertz function accurately describes dark count probability behavior across the full tested range of temperatures and voltages without needing extra device-specific adjustments.

What would settle it

Dark count probability measurements on pg-SPADs at temperatures or voltages outside the tested range that deviate systematically from the complementary Gompertz curve and cannot be fit without new parameters.

Figures

Figures reproduced from arXiv: 2606.12055 by Marc Dandin, Md Sakibur Sajal, Tathagata Srimani, Ziyad Alswaidan.

Figure 1
Figure 1. Figure 1: Spatial and one-dimensional characterization of a perimeter-gated [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Measured perimeter gating and dark count rate (DCR) dynamics of a [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental setup and simplified system-level block diagram of [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experimental characterization of model parameter dispersion due [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
read the original abstract

This Letter presents a novel analytical framework showing that the dark count probability (PDC) of perimeter-gated single-photon avalanche diodes (pg-SPADs) follows a complementary Gompertz function. Specifically, we show that PDC follows a complementary Gompertz form from which we derive a pixel-specific descriptor, the midpoint perimeter gate voltage, which characterizes a pixel's equiprobable operating point. We further show that a perimeter gate voltage compensation rate may be obtained from this descriptor to offset temperature-induced changes in the pixel's activation function. The proposed framework is experimentally validated using 4,096 pg-SPADs arranged in a 64 x 64 array and manufactured in a 0.35 $\mu$m CMOS process. The devices were characterized at temperatures ranging from -5 $^o$C to 55 $^o$C and perimeter gate voltage magnitudes of 0 to 5 V. The measured results demonstrate deterministic bias control of dark count probability across process and temperature variations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the dark count probability (PDC) of perimeter-gated SPADs follows a complementary Gompertz function. From this form the authors derive a pixel-specific midpoint perimeter gate voltage (characterizing the equiprobable operating point) and a perimeter-gate-voltage compensation rate to offset temperature-induced shifts in the activation function. The framework is validated on 4096 devices in a 64×64 array fabricated in 0.35 µm CMOS, characterized over −5 °C to 55 °C and Vpg = 0–5 V, with the results presented as demonstrating deterministic bias control across process and temperature variations.

Significance. If the complementary Gompertz description holds across the stated ranges with no additional device-specific parameters, the work supplies a compact analytical tool for characterizing and compensating dark-count behavior in pg-SPAD arrays. The scale of the experimental validation (4096 pixels) is a clear strength and would support claims of generality within the tested process and temperature window.

major comments (2)
  1. Abstract: the functional form is asserted to follow from an analytical framework, yet no derivation steps, fitting procedure, error analysis, or data-exclusion rules are supplied; these details are load-bearing for verifying that the measured data actually support the stated Gompertz claim rather than an empirical fit.
  2. Midpoint-voltage derivation: the midpoint perimeter-gate voltage is obtained directly from the Gompertz parameters; it is therefore necessary to show explicitly whether this quantity is independently measured or reduces to a fitted value by construction, as the latter would undermine the claim that it is a new pixel-specific descriptor.
minor comments (2)
  1. The temperature range is written as -5 $^o$C; adopt a consistent degree symbol and spacing throughout the manuscript.
  2. A table or figure summarizing the fitted Gompertz parameters (and their uncertainties) across the five temperature points would improve traceability of the compensation-rate extraction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the experimental scale. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: Abstract: the functional form is asserted to follow from an analytical framework, yet no derivation steps, fitting procedure, error analysis, or data-exclusion rules are supplied; these details are load-bearing for verifying that the measured data actually support the stated Gompertz claim rather than an empirical fit.

    Authors: The abstract summarizes the central result. The analytical framework deriving the complementary Gompertz form from the perimeter-gate physics, together with the fitting procedure, error metrics, and data-handling rules, appears in Sections II and III of the manuscript. To make these elements more immediately verifiable from the abstract onward, we will insert a concise methods paragraph in the revised version that explicitly lists the fitting algorithm, goodness-of-fit criteria, and any exclusion thresholds applied to the 4096-pixel dataset. revision: yes

  2. Referee: Midpoint-voltage derivation: the midpoint perimeter-gate voltage is obtained directly from the Gompertz parameters; it is therefore necessary to show explicitly whether this quantity is independently measured or reduces to a fitted value by construction, as the latter would undermine the claim that it is a new pixel-specific descriptor.

    Authors: The midpoint voltage is defined mathematically as the voltage at which the fitted complementary Gompertz function equals 0.5; it is therefore obtained from the model parameters by construction. Its utility as a pixel-specific descriptor lies in the fact that it collapses the entire activation curve into a single, temperature-compensable bias point that can be extracted uniformly across the array. We will add an explicit equation and a short paragraph in the revised text clarifying this definitional relationship while retaining the claim that the descriptor enables deterministic control. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained with external validation

full rationale

The abstract describes an analytical framework asserting that PDC follows a complementary Gompertz form, from which a midpoint perimeter gate voltage descriptor and compensation rate are derived. Experimental validation on 4096 pixels across -5°C to 55°C and 0-5 Vpg provides an independent test of the functional form. No quoted equations or self-citations in the available text reduce the midpoint or compensation rate to a fitted parameter by construction, nor does any load-bearing premise rely on prior author work as an unverified uniqueness theorem. The central modeling step is presented as empirically falsifiable rather than tautological, satisfying the criteria for a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are detailed beyond the central modeling assumption.

axioms (1)
  • domain assumption Dark count probability follows a complementary Gompertz function
    Stated as the basis of the novel analytical framework in the abstract.

pith-pipeline@v0.9.1-grok · 5709 in / 1235 out tokens · 21467 ms · 2026-06-27T07:56:44.194509+00:00 · methodology

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Reference graph

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