WeightField2 Simulation Program

• WeightField2 is a computational tool that simulates LGAD sensor behavior in high-radiation environments using detailed semiconductor transport and impact-ionization physics.
• It employs coupled Poisson, drift-diffusion, and signal induction frameworks with high-resolution finite-difference grids and adaptive time stepping to accurately model ultrafast current transients.
• Calibration against FBK LGAD data confirms that WF2 reliably predicts gain degradation and timing performance across materials like Si, 4H-SiC, and diamond under varied irradiation conditions.

WeightField2 (WF2) is a computational program developed for simulating the physical behavior and timing performance of Low Gain Avalanche Diodes (LGADs) in high-radiation environments, with particular attention to ultra-thin, n-in-p architectures. It integrates detailed semiconductor transport physics, radiation-induced damage mechanisms, and device geometry to model timing signals in planar or strip LGADs fabricated from silicon (Si), 4H-silicon carbide (4H-SiC), and diamond. The program’s predictive capabilities have been validated against measurements from FBK LGAD wafers, demonstrating quantitative agreement for gain, signal rise time, and collected charge across irradiation conditions (Kalani et al., 23 Jan 2026).

1. Physical and Mathematical Framework

WF2’s core simulation resolves the coupled, time-dependent drift-diffusion problem alongside electrostatic field evolution in semiconductor detectors. The primary equations implemented are:

• Poisson’s Equation for potential $\phi(x)$:

$$\nabla \cdot [\epsilon(x) \nabla \phi(x)] = -q [p(x) - n(x) + N_D(x) - N_A(x)]$$

with spatially dependent permittivity, carrier densities, and dopant concentrations.

• Drift–Diffusion Continuity Equations for electrons and holes:

$$\begin{align*} \frac{\partial n}{\partial t} &= \frac{1}{q} \nabla \cdot J_n + G - R, \qquad J_n = q \mu_n n E + q D_n \nabla n \ \frac{\partial p}{\partial t} &= -\frac{1}{q} \nabla \cdot J_p + G - R, \qquad J_p = q \mu_p p E - q D_p \nabla p \end{align*}$$

with $G$ generation and $R$ recombination terms.

• Impact-Ionization (Massey Model):
  • Coefficients $\alpha_{n,p}(T)$ parameterize temperature-dependent electron/hole ionization rates:

  $$\alpha_{n,p}(T) = A_{n,p}(T) \exp\left[ -\frac{B_{n,p}(T)}{E} \right],\quad B_{n,p}(T) = C_{n,p} + D_{n,p}\cdot T$$

  where $A$, $C$, $D$ are material- and carrier-specific constants.

• Signal Calculation via Ramo’s Theorem:

  • Induced current at time $t$, $i(t) = q\, v(x(t)) \cdot E_w(x(t))$, with $E_w(x)$ the “weighting field” (unit electrode bias).

This framework allows for the direct simulation of transient current signals following minimum ionizing particle (MIP) injection, including stochastic avalanche processes and the full device geometry.

2. Radiation Damage and Trapping Mechanisms

WF2 explicitly incorporates both bulk and surface radiation damage phenomena through established models:

• Acceptor Removal (Gain Layer Degradation):
  • Exponential depletion with fluence $\Phi$:

  $N_A(\Phi) = N_{A0} \exp(-c \Phi)$

  where $c$ is empirically determined: $c \approx 2.2 \times 10^{-16}\ \text{cm}^2$ for FBK W6, $c \approx 5.6 \times 10^{-16}\ \text{cm}^2$ for B-doped 4H-SiC.

• Carrier Trapping/Defect-induced Recombination:

  • Rate $$R_\text{trap} \simeq \beta_e\, n\, N_t + \beta_h\, p\, N_t$$
  • With capture coefficients $$\beta_e = 4.9 \times 10^{-16}\ \text{cm}^2/\text{ns}$$ and $$\beta_h = 6.2 \times 10^{-16}\ \text{cm}^2/\text{ns}$$, and defect density $N_t$ scaling with fluence.
  • Utilizes an effective lifetime model based on the standard SRH-recombination formalism.

A plausible implication is that WF2’s self-consistent integration of damage effects enables robust prediction of gain reduction and timing performance degradation under irradiation, as observed in High Luminosity LHC environments.

3. Numerical Implementation and Algorithmic Details

WF2 employs high-resolution finite-difference grids and explicit time-stepping algorithms:

• Spatial Discretization: 1D or 2D mesh; gain region resolved with submicron nodes ($\sim$0.5–1 μm).
• Time Stepping: Adaptive $\Delta t \leq 0.01\ \text{ps}$ for precise avalanche current tracking; output granularity is 0.01 ps.
• Field Solution Convergence: Iterative Poisson–drift–diffusion coupling at each time step.
• Signal Shaping: Final simulated signal passes through a Trans-Impedance Amplifier (TI-AMP; NA62 model) and a timing-binned (20 ps, $\sqrt{12}$ quantization) constant-fraction discriminator/TDC chain.

This methodology ensures accurate modeling of ultrafast current transients, critical for evaluating sub-50 ps resolution LGAD sensors.

4. Input Parameterization and Experimental Calibration

The input set for WF2 is designed to reflect both device-specific and operational configuration:

Parameter Class	Example Values	Calibration Context
Bulk Material	Si, 4H-SiC, diamond	Mobility, permittivity, ionization
Geometry	Thickness 20–120 μm; strip pitch 60 μm; width 45 μm	1–5 strips
Gain Layer	Depth 0.5–1 μm; $N_{A0}$ $\approx2.5$–$5.0\times10^{16}$/cm³	Doping removal for irradiation
Irradiation	$\Phi$ up to $5\times10^{15}$/cm²; beta’s, c	Dynamic coefficient computation
Operation	$T=243$–$293$ K; $V=115$–$385$ V	MIP injection: 57 e–h/μm (SiC), 75 (Si), 40 (diamond)

Experimental calibration against FBK W6 DC-LGAD (thickness $=55 \mu\text{m}$, $N_{A0}=4.71\times10^{16}$/cm³) confirms WF2’s accuracy for gain–bias and rise time relationships at fluences up to $3\times10^{15}$ n$_\text{eq}$/cm² (Kalani et al., 23 Jan 2026).

5. Validation and Performance Comparison

Quantitative validation of WF2 predictions is achieved through direct comparison with experimental data and parametric scans:

• FBK W6 (Si, 60 μm): Gain match within $5$–$10\%$ over $V_\text{bias}=100$–$300$ V; rise time (10–90%) within $<10\,$ps.
• Material Dependence (20 μm, $V_\text{bias}=150$ V, $T=243$ K):
  • $Q_\text{tot}$ at $\Phi=5\times10^{15}$: Si and SiC $\sim$2 fC, diamond $<$1 fC.
  • Timing resolution $\sigma_t$: Si $\approx23$ ps, SiC $<18$ ps, diamond $<18$ ps.
• Thickness Scan (SiC, unirradiated):
  • $\sigma_t$ improves by $\sim60\%$ when reduced from 100 μm→20 μm (35 ps→14 ps); slew-rate enhancement explained by induced-current scaling $\propto G/d$.
• Gain vs. Doping: Gain ($G$) vs. implant doping is quadratic; slope $\partial G/\partial N_A$ is $\sim2\times$ larger in SiC, indicating higher gain sensitivity compared to Si.
• SiC vs. Si at Fixed Gain (10), $T=243$ K, $V_\text{bias}=300$ V:
  • 20 μm SiC: $\sigma_t\approx14$ ps, Si: $\approx26$ ps ($\sim45\%$ better in SiC).
• Temperature Dependence: $\Delta G/G \approx -0.5\%$/K (Massey model); $\sigma_t$ increases $13$ ps @243 K to $22$ ps @293 K (SiC at 360 V).
  • SiC maintains $30$–$40\%$ better timing than Si across $243$–$293$ K.
• Radiation Tolerance (20 μm SiC):
  • $\sigma_t$ recovers by $23\%$ after raising $V_\text{bias}$ by $+65$ V for $\Phi=1\times10^{15}$ n$_\text{eq}$/cm².
  • At $\Phi=5\times10^{15}$ n$_\text{eq}$/cm², $\sigma_t$ can be restored below $20$ ps by $V_\text{bias}\approx385$ V (recovery $\sim65\%$).
  • SiC is $37$–$40\%$ better in $\sigma_t$ at $\Phi=1$–$5\times10^{15}$ n$_\text{eq}$/cm² due to higher carrier-saturation velocity and field endurance.

Validation outcomes confirm WF2’s capacity for predictive simulation across materials and irradiation regimes.

6. Significance for Ultra-Thin 4H-SiC LGADs

WF2 predicts that ultra-thin ($20 \mu\text{m}$) 4H-SiC AC-LGADs sustain high gain ($G\approx10$–$20$), large collected charge ($\sim2$ fC) and sub-$25$ ps timing resolution under fluences up to $5\times10^{15}$ n$_\text{eq}$/cm². Among Si, diamond, and 4H-SiC, the latter yields the highest gain for a fixed thickness and implant, and most effectively retains timing performance under irradiation. This suggests that 4H-SiC is a highly promising material for challenging timing applications in high-luminosity collider environments, particularly where radiation tolerance and timing precision are critical (Kalani et al., 23 Jan 2026).

A plausible implication is that self-consistent simulation platforms such as WF2, incorporating detailed radiation damage and advanced material models, are instrumental in the systematic optimization and benchmarking of next-generation ultra-fast, radiation-hard timing sensors.