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

TL;DR

The paper develops and validates a broadband analytic PDE model for P-on-N SiPMs, decomposing PDE into transmission and internal efficiency terms and fitting device-specific parameters to UV–NIR data for HPK VUV4 and FBK VUV-HD. By incorporating thin-film optics, temperature-corrected photoabsorption, and avalanche probabilities $P_e(V)$ and $P_h(V)$, the authors demonstrate robust fits across wavelengths (350–830 nm), angles, and cryogenic conditions, and extend predictions into liquid noble media (LXe/LAr) and the VUV. Key results include oxide thickness estimates (roughly ~17 nm for HPK and ~1.36 μm for FBK) and inferred junction geometries, with strong predictive power for extrapolations to unmeasured regimes, enabling design optimization for ExCT suppression and high PDE in quantum sensing and astroparticle experiments. The framework provides practical guidance for maximizing PDE (e.g., maximizing $FF$, optimizing $T$, and tailoring $W_p$) and offers a route to compute PDE under dense media and cross-talk scenarios, supported by publicly available PDE tables for simulation.

Abstract

We present a versatile analytic model describing Photon Detection Efficiency (PDE) for P-on-N silicon photomultipliers, with possible applications in device characterization, PDE extrapolation from limited data, simulation and design optimization. Using device specific parameters, SiPM PDE is modeled as a function of wavelength, angle of incidence, voltage, and a range of temperatures. By factoring the PDE into transmission and internal efficiency, the efficiency in liquid nobles or other dense media can be predicted. We present the measurement of the absolute PDE from 350 to 830~nm at 163~K for two VUV sensitive SiPMs: a Hamamatsu VUV4 and Fondazione Bruno Kessler VUV-HD Technology. Additional measurements of relative PDE versus angle are also included. We successfully fit the model to the data, compare with literature and show the model's predictive power by extrapolating PDE to new wavelengths and operation in liquid xenon and argon, which is useful for estimating the impact of external cross-talk in future large-scale experiments. Lastly we use the model investigate optimizing efficiency for specific applications in astroparticle physics and quantum computing.

Figures (11)

• Figure 1: (Left) Diagram of the modeled SiPM structure, with illustration of the minority carriers drifting into the high field region and initiating avalanches. (Right) Ray diagram of photon paths in SiPM (for real part of internal angle $\theta_3$). Thick lines in silicon correspond to that wavelength's attenuation length $1/\mu$, with fainter lines drawn at length $4/\mu$. $n^-$ denotes the insensitive bulk silicon. The refracted angles are drawn accurately, highlighting the similarity in path length within silicon for different angles of incidence.
• Figure 2: The input optical data used in this work. (Main figure) solid and dashed lines represent $n$ and $k$, respectively. All silicon $n$ values asymptote to $\sim$3.55 at 1000 nm. The inset figure shows Vis-IR $k$ values on a log scale; the differences between these curves drive variation in fit results due to different photoabsorption. The thicker gray 'default, 150 K' $k$ curve under the purple line is the default data scaled by the photoabsorption model. Data references are given in the text. The liquid nobles $n$ values are drawn above where $k = 0$.
• Figure 3: Vacuum transmission for the vacuum-quartz-silicon thin film interface used in the PDE model. The dashed pink line is the transmission for ray optics; the thin-film transmission (using \ref{['eq:transmission']}) oscillates about this mean value. The 20 nm (gold) transmission is interference-enhanced below 300 nm while the thicker 90 nm curve (purple) oscillates in the VUV and is constructive above 380 nm. The 1$\mu\text{m}$ oxide oscillates quickly about the median (truncated in VUV for clarity) for monochromatic light. Inset graph shows the region of interest for liquid noble applications.
• Figure 4: Absorption probability vs wavelength for the electron collection region $W_p$ (solid lines) and hole collection region $W_n$ (dashed). The different curves highlight the following influence of PN geometry: proximity to surface ($dp^*$) improves absorption in the UV, a deeper high-field region $X_{PN}$ shifts the tradeoff between electron or hole driven avalanches, and deeper junction ($dw^*$) increases absorption for NIR wavelengths.
• Figure 5: Three measurements of FBK device's relative PDE versus angle, fit to extract the oxide thickness. Monochromator FWHM (MC width) are noted in the legends, and fitted oxide thicknesses displayed in the lower left. The UV data (red) are shifted by an additional wavelength offset of -10 nm (upper) and -15 nm (middle), which is attributed to the spectral non-uniformity of the light source.