Charge collection parameterization of MALTA2, a depleted monolithic active pixel sensor

This paper presents a fast, data-driven simulation method for the MALTA2 depleted monolithic active pixel sensor that accurately reproduces measured in-pixel efficiency without requiring proprietary process details, offering a computationally efficient alternative to TCAD simulations for optimizing future high-rate particle tracking and calorimetry designs.

The Gist

Imagine you are trying to build a super-sensitive camera that can see individual particles of light (or in this case, high-energy particles) flying through space. To do this, scientists use special chips called sensors. These sensors are like a giant grid of tiny buckets (pixels) designed to catch passing particles.

The problem is: How do you know if your bucket design is good before you actually build it?

Usually, scientists use complex computer programs (called TCAD) to simulate how electricity moves inside the chip. But there's a catch: the companies that make these chips keep their secret recipes (like the exact chemical mix inside the silicon) hidden. It's like trying to bake a perfect cake without knowing the exact amount of sugar or flour the baker used. You have to guess, and your simulation might be wrong.

This paper introduces a clever new way to solve that problem using the MALTA2 sensor. Here is the breakdown in simple terms:

1. The Problem: The "Black Box" Chip

The MALTA2 sensor is a high-tech chip made on a very thin layer of silicon (about 30 micrometers thick—thinner than a human hair). Inside, it has a complex layout of digital and analog circuits. Because the manufacturer won't share the blueprints, scientists couldn't easily simulate how the chip would react to particles using traditional methods.

2. The Solution: Learning from the "Real Thing"

Instead of guessing the internal chemistry, the team decided to measure the real sensor and use those measurements to build a simple simulation.

Think of it like this:

• Old Way (TCAD): Trying to predict how a car drives by studying the engine's blueprints (which you don't have).
• New Way (This Paper): Driving the car on a test track, recording exactly how it handles turns and bumps, and then writing a simple rulebook based on that driving data.

3. The Experiment: The Particle "Rain"

The team took the MALTA2 sensor to CERN (a giant particle lab in Europe) and shot a beam of high-energy particles at it.

• They mapped out exactly where the particles hit.
• They measured how much "charge" (electric signal) the sensor collected in the center of a pixel versus the edges.
• The Discovery: They found that when a particle hits the center of a pixel, the sensor catches almost all the signal. But when it hits the edge between two pixels, the signal gets "shared" or split between them, making the reading weaker.

4. The "Magic Formula"

The scientists took all that messy real-world data and distilled it into a simple mathematical formula (a "parameterization").

• Imagine the pixel grid as a checkerboard.
• They created a rule that says: "If a particle lands here, the signal is 100%. If it lands on the line between squares, the signal splits 50/50."
• They added a little bit of "fuzziness" to the formula to account for the fact that their measuring equipment isn't perfect (just like how a blurry photo makes it hard to tell exactly where a line is).

5. Why This Matters: The "Fast Forward" Button

This new method is a fast simulation.

• Traditional Simulation: Like running a full physics engine in a video game. It's accurate but takes hours or days to calculate one second of time.
• This New Method: Like using a pre-recorded video of the game. It's incredibly fast and lightweight.

Because the simulation is so fast and doesn't need secret factory data, scientists can now:

1. Design Better Chips: They can quickly test thousands of different digital designs to see which one works best for catching particles.
2. Optimize for the Future: They can prepare for future experiments (like the High-Luminosity LHC) where millions of particles will hit the detector every second.

The Bottom Line

The authors built a "shortcut" to understand how a complex sensor works. Instead of trying to reverse-engineer the factory's secret recipe, they just watched how the sensor actually behaved, wrote down the rules, and used those rules to build a super-fast computer model. This allows them to design the next generation of particle detectors much faster and more accurately than before.

Technical Summary

1. Problem Statement

The design and optimization of Depleted Monolithic Active Pixel Sensors (DMAPS) require accurate simulation of charge collection and signal formation.

• Limitations of TCAD: Traditional Technology Computer-Aided Design (TCAD) simulations are the standard for predicting sensor characteristics (capacitance, gain, timing, efficiency). However, they rely on proprietary process details (doping profiles, implant geometries, resistivity) that are often unavailable from commercial foundries.
• The Gap: When proprietary data is missing, TCAD models must rely on assumptions and systematic parameter variations. While these can offer qualitative insights, they lack the quantitative accuracy required for final sensor optimization without extensive experimental validation.
• Need: A fast, computationally efficient simulation method that derives inputs solely from experimental data, bypassing the need for proprietary process information, is needed to optimize digital sensor designs for high-rate tracking and calorimetry.

2. Methodology

The authors developed a measurement-driven parametric simulation for the MALTA2 sensor. The methodology consists of three main stages:

A. Experimental Data Acquisition

• Sensor: MALTA2, fabricated in a 180 nm CMOS imaging technology on a 30 µm thick epitaxial silicon layer.
• Setup: The sensor was tested at the CERN SPS using the MALTA beam telescope (six tracking planes, spatial resolution of $4.6 \pm 0.2$ µm) with a 180 GeV/c mixed hadron beam.
• Readout Strategy: Since the sensor has binary readout, charge reconstruction was performed by varying the in-pixel threshold settings.
• Analysis: Detection efficiency was mapped across the pixel area (specifically a $2 \times 2$ pixel matrix). The Most Probable Value (MPV) of the charge deposition was reconstructed for different pixel regions (center vs. corners) to quantify charge sharing effects.

B. Analytical Parameterization

The reconstructed charge distribution was modeled using an analytical function rather than complex physics simulations:

• 1D Model: A periodic, symmetric function based on the error function (erf) was defined to describe charge collection ($CC$) along the $x$-axis:
  $$CC(x) = \frac{C}{2} \left[ \text{erf}\left(\frac{x - \text{shift}}{\sigma_{\text{erf}}}\right) - \text{erf}\left(\frac{x - \text{shift} - \text{pitch}}{\sigma_{\text{erf}}}\right) \right]$$
  Where $C$ is the charge at the pixel center, and $\sigma_{\text{erf}}$ characterizes the transition width at pixel boundaries.
• Convolution: To account for the finite tracking resolution of the telescope, the model was convolved with a Gaussian distribution ($\sigma_{\text{gauss}}$).
• 2D Extension: A two-dimensional model ($CC_{2D}$) was constructed by multiplying the independent 1D models for $x$ and $y$ axes.
• Key Parameters: The fit yielded a central charge $C \approx 1718 e^-$, a charge sharing width $\sigma_{\text{erf}} = 4.3 \pm 0.3$ µm, and a tracking smearing $\sigma_{\text{gauss}} = 4.6 \pm 0.3$ µm.

C. Fast Simulation Implementation

• The simulation integrates with GEANT4. For any energy deposition at $(x, y)$, the charge is distributed among the seed pixel and the three adjacent pixels sharing the nearest corner.
• The charge fraction for each pixel is calculated using the $CC_{2D}$ function.
• The sum of fractions is normalized to unity, ensuring charge conservation.
• This approach restricts charge sharing to clusters of up to four pixels per single ionization event (larger clusters from delta-rays are handled naturally by multiple hits).

3. Key Contributions

• Proprietary-Data-Free Modeling: Demonstrated a robust simulation framework that does not require foundry-specific doping profiles or implant geometries.
• High Accuracy with Low Cost: The method reproduces measured in-pixel efficiency with high accuracy (residuals $< 2\%$) while being computationally lightweight compared to TCAD.
• Validation of Charge Sharing: Successfully quantified the charge sharing behavior of the MALTA2 sensor, identifying a specific width parameter ($\sigma_{\text{erf}}$) that accurately models the transition from pixel center to boundary.
• Layout Sensitivity Analysis: The residual maps revealed subtle asymmetries in charge collection caused by the analog circuit layout (e.g., deep p-well cut-outs and n-well shielding), providing insights into how physical layout affects signal formation.

4. Results

• Efficiency Matching: The simulation, when including the telescope's tracking uncertainty ($\sigma_{\text{gauss}} = 4.6$ µm), reproduced the experimental in-pixel efficiency data within a $1\%$ offset per bin.
• Threshold Range: The model accurately reproduced average efficiency and cluster size distributions across a wide threshold range (500 to 2500 $e^-$).
• Sensitivity: The study confirmed that the model is highly sensitive to the $\sigma_{\text{erf}}$ parameter. Simulations with incorrect $\sigma_{\text{erf}}$ values (e.g., 2.0 µm or 10.0 µm) failed to match the data, while the fitted value of 4.3 µm provided an excellent match.
• Charge Sharing Necessity: Simulations that neglected charge sharing significantly overestimated efficiency and underestimated cluster size, validating the necessity of the proposed parameterization.
• Residual Analysis: Residuals between the model and data remained below 2%, which is within the 3% uncertainty of the charge threshold calibration. Small deviations correlated with specific analog circuit features (e.g., n-wells reducing charge collection).

5. Significance

• Design Optimization: This method enables the rapid optimization of digital front-end designs and large-scale detector architectures for high-rate particle tracking and high-granularity calorimetry without waiting for TCAD iterations or foundry data.
• Scalability: The approach is applicable to other sensors where proprietary process details are unavailable, offering a general solution for data-driven sensor simulation.
• Future Applications: The framework is intended to be used for optimizing digital designs regarding hit-rates, multiple thresholds, and on-chip clustering, facilitating the development of next-generation detectors for future physics experiments.