WF2: Silicon Detector Signal Simulation

• WeightField2 is a 1D transient drift–diffusion simulation tool that models signal formation in advanced silicon detectors, including LGADs.
• It numerically solves Poisson’s, drift–diffusion, and Ramo’s equations along with radiation damage models to achieve high physics fidelity.
• The simulation framework supports modular workflows and extensive parameter sweeps, validated against experimental data for optimized sensor performance.

WeightField2 (WF2) is a one-dimensional, transient drift–diffusion simulation package widely employed for modeling signal formation in advanced silicon detectors, particularly Low-Gain Avalanche Detectors (LGADs) and related ultra-fast sensors. It numerically solves the time-resolved transport, avalanche multiplication, and signal induction of charge carriers in silicon, including radiation damage effects, with an emphasis on high timing precision and physics fidelity. WF2 has been validated against experimental data and is a common reference for device optimization, sensor characterization, and frontend-electronics design in 4D-tracking and radiation-hard detector R&D.

1. Core Physical Models in WF2

WF2 numerically implements the microscopic physics of signal generation in silicon detectors. The primary equations solved include:

• Poisson’s equation for electrostatic potential:

$$\frac{\partial^2 \varphi(x,t)}{\partial x^2} = -\frac{q}{\varepsilon} \left[ p(x,t) - n(x,t) + N_D^+(x,\phi) - N_A^-(x,\phi) \right]$$

with $\varphi(x,t)$ the potential, $n$ and $p$ the carrier densities, $N_D^+$ and $N_A^-$ the (fluency-dependent) donor and acceptor profiles.

• Carrier drift–diffusion and continuity equations:

$$\frac{\partial n}{\partial t} = -\frac{\partial J_n}{\partial x} + G_n - R_n,\quad J_n = q\, \mu_n(E,T)\, n\, E + q\, D_n\, \frac{\partial n}{\partial x}$$

with analogous form for holes.

• Impact ionization (internal gain) in the gain layer using Chynoweth’s law:

$\alpha_n(E) = A_n \exp(-B_n/E)$

and similar for holes, with empirical coefficients $A_n$, $B_n$.

• Radiation damage parameterizations:
  • Acceptor removal: $N_{GL}(\phi) = N_{GL}(0)\, \exp(-c\,\phi)$, $c \sim 8 \times 10^{-16}$ cm$^2$.
  • Acceptor creation (traps): $N_{bulk}(\phi) = N_{bulk}(0) + g_A \phi$.
  • Trapping lifetimes: $1/\tau_n(\phi) = \beta_n \phi$, $1/\tau_p(\phi) = \beta_p \phi$.
• Signal formation (Ramo’s theorem): For a parallel plate geometry, the induced current is

$$i(t) = -q \left[ v_n(t)\, n(t) + v_p(t)\, p(t) \right] E_w(x), \quad E_w(x) = 1/d$$

where $d$ is the detector thickness.

Carrier dynamics are evolved in time using explicit Runge–Kutta schemes with self-consistent updates to space charge and electric fields. The mesh and time-step are chosen for numerical stability (e.g., $\Delta x \sim 0.1\,\mu$m, $\Delta t\sim0.5$ ps) (Galloway et al., 2017).

2. Simulation Workflow and Implementation

WF2 executes a modular, stepwise workflow for the simulation of signal generation:

1. Device Geometry and Meshing: Define a 1D slab of thickness $d$ (e.g., $50\,\mu$m), discretize spatially.
2. Doping Profiles: Specify gain layer (e.g., $N_{GL}(0)\sim1$–$2\times10^{17}$ cm$^{-3}$) and bulk for a given irradiation fluence.
3. Electric Field Solution: Numerically integrate Poisson’s equation for $E(x)$.
4. Weighting Field Calculation: Typically $E_w(x)=1/d$ for slab devices.
5. Primary Ionization Sampling: Model a minimum-ionizing-particle (MIP) via Landau-distributed e–h clusters along $x$.
6. Carrier Transport and Gain: Time-step carrier densities with drift–diffusion, field-dependent mobility, multiplication, and trapping.
7. Signal Readout: Calculate time-dependent induced current $i(t)$.
8. Electronics Response: Convolve $i(t)$ with front-end impulse response and noise models (e.g., 4.7 k$\Omega$ transimpedance, GHz bandwidth).
9. Analysis: Extract pulse amplitude, timing, rise time, and other observables, repeating over $\sim\!10^3$–$10^4$ simulated events for statistical robustness (Galloway et al., 2017).

WF2 is typically coded in C++ with modular routines for each physical process. No modification to the physical models is reported in its applications to DC-coupled resistive silicon detectors (Menzio et al., 2022).

3. Application to Detector Design and Performance

WF2 underpins the optimization and analysis of advanced silicon detector concepts, particularly in the context of timing and spatial resolution for high-radiation environments:

• In DC-Coupled Resistive Silicon Detectors (RSD), the signal formation is simulated in WF2 with the output current $I(t)$ serving as the input to a macroscopic LTspice network model representing the resistive sheet, capacitance, and readout electronics. This split allows rapid scanning of sensor parameters in WF2 and decoupled study of electrode topology in circuit simulation (Menzio et al., 2022).
• Key simulated metrics:
  • Timing performance: WF2 predicts pulse shapes with rise times $\sim$300–400 ps and amplitudes 1–2 mA, corresponding to per-hit timing of 30–40 ps when combined with typical frontend noise.
  • Spatial resolution: Using the WF2-generated pulse in a resistive network yields reconstructed position error $20$–$30\,\mu$m, well below typical strip pitch.
• Modeling radiation damage: WF2 quantitatively reproduces gain loss, pulse rise time reduction, and amplitude suppression in heavily irradiated LGADs. Experimental pulse shapes, gain–voltage curves, and timing resolution are found to match WF2 predictions within 10% across neutron fluences up to $6 \times 10^{15}$ n/cm$^2$ (Galloway et al., 2017).
• Workflow best practices: Sensor physics (gain layer, bias, etc.) are isolated in WF2, with network and charge-sharing analyses offloaded to circuit simulators. This modular approach accelerates optimization and supports rapid parameter sweeps.

4. Numerical and Algorithmic Details

WF2 emphasizes computational efficiency and adaptability:

• Spatial mesh: Uniform $\Delta x \sim 0.1\,\mu$m for standard LGADs.
• Time integration: Explicit second-order (RK2) or higher Runge–Kutta method, with time steps $\Delta t$ small enough for carrier motion per step $<\Delta x$ (CFL condition).
• Self-consistency: Updates to carrier-induced fields and densities are synchronized at each step, enforcing convergence (field tolerance $10^{-4}$ V).
• Electronics simulation: Electronics convolution includes transimpedance amplifiers (e.g., 4700 Ω), GHz bandwidth shaping, and white-noise sources for SNR determination.

Randomized sampling of energy loss and trapping events is essential to build realistic statistical output distributions for key observables such as time resolution.

5. Validation and Accuracy

WF2 is benchmarked in multiple studies:

• Pre- and post-irradiation pulse shapes: Agreement with measured rise times (20–50 ps), pulse height changes, and normalized trailing edge is within 10%.
• dV/dt and gain curves: Simulated slopes and gain–voltage dependence accurately reproduce experimental trends before and after fluence.
• Radiation-damage effects: WF2's inclusion of acceptor removal, acceptor creation, and field-dependent trap capture is found to sufficiently model the performance loss of UFSDs under neutron damage (Galloway et al., 2017).

No major systematic discrepancies are observed, with minor residuals attributed to uncertainties in input dopant profiles or trapping coefficients.

6. Impact and Best Practices in Detector R&D

WF2 enables a rapid, physics-driven workflow for the design and optimization of ultra-fast silicon sensors:

• Decoupled simulation pipeline: Separates microscopic (WF2) from macroscopic (LTspice, network) physics, maximizing computational efficiency (Menzio et al., 2022).
• Parameter sweeps: Supports large-scale scans of gain-layer doping, bias voltage, and thickness, suitable for multi-geometry optimization.
• Empirical validation: WF2 results are typically cross-checked against full TCAD simulation once, after which wide sweeps can proceed using WF2 for efficiency.
• Automation: Simple scripts are recommended for WF2–LTspice interface, automating the import of WF2-generated current–time data as current sources in larger network simulations.

This approach allows systematic exploration of sensor physics and readout designs, forming the basis for iterative improvements in timing and spatial resolution in radiation environments (Menzio et al., 2022, Galloway et al., 2017).

7. Summary Table: WF2 Simulation Inputs and Outputs

Parameter	Typical Value / Use	Origin in Workflow
Sensor thickness	50 μm (e.g., LGAD)	Geometry, drift distance
Gain-layer doping	W13 UFSD3.2 or HPK profiles	Sets gain, pulse rise time
Bias voltage	200–600 V	Controls $E(x)$, gain
Carrier mobility	Field- and temperature-dependent	Drift velocity calc.
Radiation fluence $\phi$	$0$ to $6\times10^{15}$ n/cm$^2$	Doping/trapping profile
Electronics response	4.7 k$\Omega$ TIA, 2.5 GHz BW	S/N, pulse shaping
Output metric	$I(t)$ waveform, rise time, timing resolution	Pulse analysis, R&D

WF2 is established as a validated, high-fidelity tool for LGAD and UFSD simulation, widely adopted in detector R&D for both fundamental performance understanding and device-to-readout co-design (Galloway et al., 2017, Menzio et al., 2022).