Charge-Carrier transport simulations in diamond detectors with electric-field-dependent mobility and charge-collection-distance-based trapping

This paper extends the \allpix{} simulation framework with diamond-specific transport models featuring electric-field-dependent mobility and charge-collection-distance-based trapping, enabling realistic detector-level simulations that accurately reproduce experimental drift velocities and transient current signals for both single-crystalline and polycrystalline CVD diamond sensors.

The Gist

Imagine you are trying to catch raindrops in a bucket while running through a storm. The size of the bucket, how fast you run, and how many raindrops get lost along the way all determine how much water you collect at the end.

This paper is about building a super-accurate computer simulation to predict exactly how a special kind of "bucket" (a diamond sensor) catches "raindrops" (electric charges) when it's hit by radiation.

Here is the breakdown of what the scientists did, using simple analogies:

1. The Problem: Why Diamond?

Diamonds are tough. They can survive in "harsh radiation environments" (like inside a nuclear reactor or a particle collider) where normal silicon sensors would melt or break.

• The Goal: Scientists want to use these diamond sensors to detect particles.
• The Challenge: To design good sensors, they need to know exactly how the electric charges move inside the diamond. But diamonds are tricky. Sometimes the charges get stuck (trapped) on the way, or they move at different speeds depending on how hard you push them (the electric field).

2. The Solution: A New "Traffic Simulator"

The team used a famous software called Allpix Squared. Think of this software as a video game engine for particle detectors. It usually simulates silicon chips. The authors added a new "mod" (an update) to make it understand diamond.

They added two main rules to the game:

• Rule A: The Speed Limit (Mobility): In normal materials, cars (charges) drive at a steady speed. In diamond, the speed changes based on how hard you press the gas pedal (electric field). The team programmed the simulation to know that electrons and holes (the two types of charges) speed up and then hit a "speed limit" (saturation) just like a car on a highway.
• Rule B: The Potholes (Trapping): In a perfect diamond, charges run a straight line. In real diamonds (especially the cheaper, multi-crystal kind), there are "potholes" (defects) where charges get stuck and disappear. The team created a way to simulate this using a concept called CCD (Charge Collection Distance).
  • Analogy: Imagine the sensor is 500 meters long. If the "CCD" is 500 meters, every runner makes it to the finish line. If the CCD is only 200 meters, it means the average runner gets stuck after 200 meters. The simulation uses this number to figure out how many runners get lost before finishing.

3. The Test Drive: Did it Work?

The team didn't just guess; they tested their simulation against real-world experiments.

• Test 1: The Perfect Diamond (Single-Crystal)

  • They used a high-quality, flawless diamond.
  • Result: The simulation predicted exactly how fast the charges moved and what the electrical signal looked like. It matched the real data perfectly. It's like their video game physics engine perfectly replicated real-world driving on a smooth highway.

• Test 2: The "Real World" Diamond (Polycrystalline)

  • They used a diamond made of many tiny crystals stuck together (like a mosaic). This has more "potholes."
  • Result: They fed the simulation the "CCD" number they measured in the lab. The simulation then successfully predicted that the signal would be weaker and "messier" (slower to rise and fall) than the perfect diamond.
  • The "Pumping" Trick: They found that if they "primed" the sensor with radiation first (like warming up a car engine), the charges moved better. The simulation could handle this by adjusting the "CCD" number, showing it's a flexible tool.

4. Why Does This Matter?

Before this, if you wanted to design a diamond detector, you had to guess a lot or build expensive prototypes to test.

Now, engineers can:

1. Plug in numbers: Take a measurement of a diamond's quality (how many "potholes" it has).
2. Run the simulation: Let the computer predict exactly how the detector will behave.
3. Optimize: Figure out the best voltage to use or how thick the sensor should be before they ever build it.

The Bottom Line

The authors have built a digital twin for diamond detectors. They taught the computer how charges move and get stuck in diamond. This allows scientists to design better, faster, and more radiation-hard detectors for future space missions, medical imaging, and particle physics experiments, all by running a simulation on a laptop instead of building a lab full of expensive hardware.

Technical Summary

1. Problem Statement

Diamond detectors are highly valued for particle physics applications in harsh radiation environments due to their radiation tolerance, fast signal formation, and low leakage current. However, accurately simulating their response is challenging because:

• Material Complexity: Charge carrier transport in diamond is governed by field-dependent mobility and trapping mechanisms that vary significantly between single-crystalline (scCVD) and polycrystalline (pcCVD) materials.
• Simulation Gaps: Existing simulation frameworks, such as Allpix Squared, lacked specific, validated models for diamond. Standard silicon models do not account for the unique mobility parameterizations or the specific trapping behaviors (often characterized by Charge Collection Distance, CCD) found in diamond, particularly in polycrystalline structures where grain boundaries and radiation damage play a major role.
• Need for Realism: Without accurate transport models, it is difficult to translate material-level properties into detector-level observables (signal amplitude, timing, pulse shape) required for optimizing detector design and studying radiation damage.

2. Methodology

The authors extended the Allpix Squared modular simulation framework to include diamond-specific transport physics. The methodology involved three main components:

A. Mobility Models

The authors implemented field-dependent mobility parameterizations for electrons and holes, replacing the constant mobility assumptions often used in simpler models.

• Electrons: Modeled using the Piecewise (PW) parametrization.
• Holes: Modeled using the Caughey–Thomas (CT) parametrization.
• Implementation: The simulation calculates drift velocity ($v_d = \mu(E)E$) and diffusion at every propagation step based on the instantaneous local electric field, rather than using a fixed global velocity.

B. Trapping Model (CCD-based)

To handle trapping without requiring a microscopic simulation of every defect, the authors implemented an effective trapping model based on the Charge Collection Distance (CCD).

• Hecht Relation: The model maps the experimentally measured CCD to an effective mean free path ($\lambda$) for electrons and holes using the Hecht relation.
• Probabilistic Trapping: Trapping is treated as an exponential decay probability over the simulation time-step/distance-step: $P_{trap} = 1 - \exp(-\Delta x / \lambda)$.
• Input Flexibility: The model accepts CCD as a primary input, allowing users to simulate radiation damage by parameterizing CCD as a function of fluence ($\Phi$). A configurable ratio ($r = \lambda_h / \lambda_e$) allows tuning between electron and hole trapping lengths.

C. Experimental Validation

The simulations were validated against two distinct experimental scenarios:

1. scCVD (Single-Crystalline): Validated in the "negligible-trapping" limit. The authors compared simulated drift velocities and transient current waveforms against published literature data.
2. pcCVD (Polycrystalline): Validated against new experimental data from a 500 µm thick Element Six detector.
   • Electrical Characterization: IV/CV measurements to ensure stable biasing.
   • CCD/CCE Measurements: Using a $^{90}$Sr $\beta$-source to measure charge collection efficiency as a function of electric field and pumping state.
   • Transient Current Technique (TCT): Using an $^{241}$Am $\alpha$-source to measure time-domain response (pulse shapes, rise/fall times) for both electron and hole drift.

3. Key Contributions

• Allpix Squared Extension: Successfully integrated diamond-specific mobility and trapping models into a standard end-to-end detector simulation framework.
• CCD Interface: Developed a practical interface that links macroscopic experimental measurements (CCD) directly to the microscopic transport simulation, bridging the gap between material characterization and detector performance simulation.
• Polarity-Aware Modeling: The framework accounts for asymmetric charge transport in pcCVD, where the growth side (grain boundaries) affects electron and hole collection differently depending on bias polarity.
• Comprehensive Validation: Provided a rigorous validation suite covering both drift velocity (time-of-arrival) and transient signal shapes (TCT) for both carrier types in different diamond qualities.

4. Results

• scCVD Mobility Validation:
  • The simulated drift velocities for electrons (PW model) and holes (CT model) showed excellent agreement with published literature data across a wide range of electric fields.
  • Simulated TCT waveforms reproduced the main features of experimental signals (steep rise, plateau, fall-off), though experimental waveforms showed slightly broader edges due to the finite bandwidth of the readout electronics not fully modeled in the simulation.
• pcCVD Trapping Validation:
  • Using experimentally measured CCD values as inputs, the simulation successfully reproduced the reduced charge collection and degraded transient response observed in polycrystalline diamond.
  • Polarity Dependence: The model correctly captured the asymmetry where negative bias (electrons drifting to the growth side) yielded higher CCD than positive bias (holes drifting to the growth side), consistent with higher hole trapping at grain boundaries.
  • Waveform Comparison: At $E = 1$ V/µm, the simulated TCT waveforms matched the experimental pulse amplitude, timing of the maximum, and overall duration.
  • Discrepancies: Minor differences remained in the leading edge (simulated rise was sharper) and late-time baseline recovery. The authors attribute this to the effective nature of the model, which captures integral charge loss but does not resolve microscopic field distortions or specific grain-boundary transport paths.

5. Significance and Outlook

• Practical Framework: This work provides a robust, experimentally accessible tool for detector developers to simulate charge collection, pulse formation, and timing performance in diamond sensors without needing to simulate every atomic defect.
• Radiation Damage Studies: By parameterizing CCD as a function of fluence, the model enables predictive studies of how diamond detectors degrade under radiation, crucial for future high-luminosity collider experiments.
• Future Work: The authors plan to extend the model to include polarization effects, non-uniform internal electric fields, and spatially varying trapping to further improve the realism of pcCVD simulations.

In conclusion, this paper establishes a validated, modular simulation capability for diamond detectors within Allpix Squared, significantly enhancing the ability to design and optimize diamond sensors for extreme radiation environments.