Demonstrating a broadband Photon Detection Efficiency model on VUV sensitive Silicon Photomultipliers

Imagine you are trying to catch raindrops with a bucket, but the rain is actually made of tiny, invisible particles of light called photons. In the world of physics, scientists use special devices called Silicon Photomultipliers (SiPMs) to catch these light particles. These devices are like high-tech buckets made of silicon, covered in millions of tiny "traps" (pixels) that scream when a photon hits them.

However, catching light isn't perfect. Some photons bounce off the surface, some get lost inside the bucket, and some hit the wrong spot. The efficiency of this catching process is called Photon Detection Efficiency (PDE).

This paper is about building a super-smart calculator (a mathematical model) that predicts exactly how good these light-catching buckets are, even in conditions we haven't tested yet.

Here is the breakdown of what the authors did, using some everyday analogies:

1. The Problem: The "Black Box" Mystery

Scientists use these light detectors to study things like dark matter and quantum computers. They need to know exactly how many photons the detector catches.

The Challenge: It's incredibly hard to test the detector under every possible condition. You can't easily test it in liquid xenon (which is super cold and dense) or at every single color of light (wavelength) and every angle.
The Analogy: Imagine you have a new fishing net. You want to know how many fish it catches in the deep ocean, in a storm, at night, and with different types of bait. You can't go to the deep ocean every day to test it. You need a simulation that tells you, "Based on how it works in your backyard pond, here is exactly how it will perform in the deep ocean."

2. The Solution: The "Recipe" Model

The authors created a mathematical recipe (a model) that breaks down the detector's performance into three main ingredients:

Transmission (The Glass Window): How much light gets through the surface of the detector without bouncing off?
Absorption (The Sponge): Once the light is inside, how much of it gets soaked up by the silicon?
Triggering (The Alarm): Once the light is absorbed, does it actually set off the alarm (create an electrical signal)?

They took real-world measurements of two specific detectors (one made by Hamamatsu and one by FBK) and "tuned" their recipe until the math matched the real data perfectly.

3. The "Secret Sauce": The Oxide Layer

One of the most important discoveries in the paper is about the oxide layer (a thin coating of glass-like material on the silicon).

The Analogy: Think of the detector surface like a stage. The oxide layer is like a specific thickness of fog on the stage.
- If the fog is too thick, the light gets scattered and lost.
- If the fog is too thin, the light might bounce right off.
- There is a "Goldilocks" thickness where the light waves bounce around inside the fog and interfere with each other in a way that pushes more light into the detector.
The Finding: The Hamamatsu detector has a very thin layer of fog (about 17 nanometers), while the FBK detector has a very thick layer (about 1,358 nanometers). The model figured out exactly how thick these layers are just by looking at how the light bounced off them.

4. The "Shadow" Effect

The paper also looked at what happens when light hits the detector from an angle (not straight on).

The Analogy: Imagine the detector pixels are like little wells in a field. Between the wells are small walls (resistors). If the sun is high noon (straight down), the light goes straight into the wells. But if the sun is low in the sky (an angle), the walls cast shadows over the wells, blocking the light.
The Discovery: The model successfully predicted that for one of the detectors, these "shadows" significantly reduce the number of photons caught when the light comes in at a steep angle. This helps scientists understand why the detector behaves differently depending on where the light comes from.

5. Why Does This Matter? (The "Crystal Ball" Effect)

Once the model was tuned, the authors used it as a crystal ball:

Predicting the Future: They used the model to predict how these detectors would work in Liquid Xenon and Liquid Argon (super-cold liquids used in giant physics experiments). They didn't need to build a new machine to test this; the math told them the answer.
Optimizing Design: They used the model to design better detectors. They calculated that if you change the thickness of the "fog" (oxide layer) and the depth of the "wells" (junctions), you could theoretically catch 80% to 90% of the light. This is huge for quantum computing and finding dark matter.
Stopping "Ghost" Signals: When a detector catches a photon, it sometimes accidentally creates a second, fake signal (called "crosstalk"). The model helps predict how many of these ghosts will appear in different liquids, helping scientists clean up their data.

Summary

In simple terms, this paper is about teaching a computer to understand the physics of light-catching buckets.

Instead of guessing how a detector will work in a new environment, the authors built a detailed map. They measured the map's landmarks (the oxide thickness, the shadowing, the absorption) and then used that map to navigate to places they haven't been yet (like deep inside liquid xenon). This allows scientists to design better experiments, save money on testing, and understand the universe a little bit better.

Here is a detailed technical summary of the paper "Demonstrating a broadband Photon Detection Efficiency model on VUV sensitive Silicon Photomultipliers" by Austin de St Croix et al.

1. Problem Statement

Silicon Photomultipliers (SiPMs) have largely replaced Photomultiplier Tubes (PMTs) in modern particle physics experiments (e.g., DUNE, nEXO, Darkside-20K) due to their compact size, radiopurity, and low operating voltage. However, accurately characterizing their Photon Detection Efficiency (PDE) across the full operational parameter space is experimentally challenging.

Key challenges addressed in this work include:

Multi-variable Complexity: PDE depends on wavelength ( $\lambda$ ), angle of incidence ( $\theta$ ), overvoltage ( $V_{ov}$ ), operating medium (vacuum vs. liquid noble gases), and temperature ( $T$ ). Measuring all combinations is impractical.
VUV Sensitivity: Experiments using Liquid Argon (LAr, 127 nm) and Liquid Xenon (LXe, 175 nm) require specialized VUV-sensitive SiPMs. Predicting their performance in these dense media based on limited vacuum data is difficult.
External Crosstalk (ExCT): SiPM avalanches emit secondary long-wavelength photons that can trigger neighboring sensors. Accurate broadband PDE modeling is essential to predict ExCT rates in large-scale detectors.
Design Optimization: There is a need for a predictive tool to optimize SiPM geometry (e.g., oxide thickness, junction depth) for specific applications like Quantum Key Distribution (QKD) or astroparticle physics.

2. Methodology

The authors developed a versatile analytic model that factors PDE into transmission and internal efficiency components, using device-specific parameters as inputs.

A. The Physical Model

The PDE is modeled as:
$PDE(\lambda, \theta, V) = FF \cdot T_{Si}(\lambda, \theta) \cdot iPDE(V, \lambda)$
Where:

Fill Factor ( $FF$ ): The fraction of the surface sensitive to light. The model includes a "shadowing" effect where surface structures (quenching resistors) block light at oblique angles.
Transmission ( $T_{Si}$ ): Calculated using wave optics (Fresnel equations) for a vacuum/SiO $_2$ /Silicon stack. It accounts for interference effects in the oxide layer and absorption in the VUV.
Internal Efficiency ( $iPDE$ ): The probability that an absorbed photon triggers an avalanche. This is the sum of electron-driven and hole-driven avalanches:
$iPDE = P_e(V) \cdot W_p(\lambda) + P_h(V) \cdot W_n(\lambda)$
- $W_p, W_n$ : Absorption probabilities in the p-doped and n-doped regions, determined by junction depths ( $d_p^*, d_w^*$ ) and the junction center ( $X_{PN}$ ).
- $P_e, P_h$ : Avalanche triggering probabilities, parameterized empirically as a function of overvoltage ( $V$ ).

B. Experimental Setup

Devices Tested: Hamamatsu VUV4 (S13371-6050CQ-02) and FBK VUV-HD (VUV-HD3).
Facility: TRIUMF's VERA (Vacuum Efficiency Reflectivity and Absorption) setup.
Conditions: Measurements taken at 163 K (LXe temperature) and room temperature.
Data Collected:
- Absolute PDE: 350–830 nm at various overvoltages.
- Relative Angular PDE: 160–850 nm at angles from -15° to 60°.
- Reflectivity: Used to constrain oxide thickness for thin-oxide devices.
- Avalanche Probability: Measured via pulse counting rates to isolate electron vs. hole triggering.

C. Fitting Strategy

The authors employed a sequential fitting method followed by a global fit:

Optics: Constrained oxide thickness ( $t_{ox}$ ) using angular PDE oscillations (for thick oxides) or VUV reflectivity (for thin oxides).
UV Region: Constrained p-region depth ( $d_p^*$ ) and electron probability ( $P_e$ ) using UV data where hole absorption is negligible.
Vis-NIR Region: Constrained junction center ( $X_{PN}$ ), n-region depth ( $d_w^*$ ), and hole probability ( $P_h$ ) using visible/IR data.
Global Fit: All parameters were floated simultaneously to refine the model.

3. Key Results

A. Device Characterization

Hamamatsu VUV4: Features a thin oxide layer ( $\sim$ 17 nm) and a larger junction area. The model successfully fit the data, revealing that the fill factor decreases at high angles of incidence due to resistor shadowing (confirmed by AFM measurements of resistor height $\approx$ 2.6 $\mu$ m).
FBK VUV-HD: Features a very thick oxide layer ( $\sim$ 1359 nm) and a smaller, asymmetric junction. The thick oxide creates strong interference patterns in the angular PDE data, which were used to precisely determine the oxide thickness.
Avalanche Probabilities: Both devices showed similar electron triggering probabilities ( $P_e$ ) that saturate near unity at high overvoltage. Hole triggering probabilities ( $P_h$ ) showed more variance between devices and fitting methods.

B. Model Validation and Extrapolation

VUV Prediction: The model successfully extrapolated PDE from the measured 350–830 nm range down to the VUV (127 nm and 175 nm). The predictions aligned well with literature data for LAr and LXe, validating the use of UV/IR data to predict VUV performance.
Temperature Dependence: The model incorporated temperature-dependent photoabsorption corrections. It showed that increasing temperature from 160 K to 273 K improves Vis-NIR PDE predictions, matching datasheets.
Extrapolation to ExCT: The model was used to predict PDE up to 1050 nm, which is critical for estimating the yield of secondary photons causing external crosstalk in liquid noble detectors.

C. Optimization Insights

The model was used to simulate theoretical limits for specific applications:

Liquid Noble Gases: For LAr, PDE could approach 35% with optimized thin oxides. For LXe, current devices achieve $\sim$ 90% internal efficiency.
Quantum Sensing: By optimizing oxide thickness and junction geometry for specific wavelengths (e.g., 532 nm, 780 nm), the model suggests PDEs exceeding 80% are theoretically achievable, provided fill factors can be maximized (e.g., via backside illumination).

4. Significance and Impact

Predictive Power: The model provides a robust framework to predict SiPM performance in unmeasured conditions (e.g., different liquids, angles, or temperatures) using limited vacuum data. This is crucial for designing future multi-tonne detectors where full characterization is impossible.
ExCT Mitigation: By accurately modeling broadband emission and detection, the model helps quantify external crosstalk, a major source of background noise in dark matter and neutrino experiments.
Design Tool: The analytic approach allows for "what-if" scenarios, guiding the fabrication of next-generation SiPMs with optimized oxide layers and junction depths for specific physics goals (e.g., maximizing LAr scintillation detection or QKD efficiency).
Open Science: The authors plan to release the model and lookup tables online, enabling the broader community to simulate and characterize SiPMs without needing extensive experimental campaigns.

Conclusion

This work establishes a comprehensive, physics-based analytic model for SiPM PDE that bridges the gap between limited experimental data and the complex operational environments of next-generation particle physics and quantum experiments. By successfully fitting two distinct VUV-sensitive devices and validating extrapolations against literature, the authors demonstrate that SiPM performance in liquid noble gases can be reliably predicted and optimized, significantly aiding the design of future large-scale detectors.