Drift-Diffusion Model Analysis

• Drift-diffusion models are mathematical frameworks using coupled PDEs to simulate deterministic drift and random diffusion of charge carriers in semiconductors and insulators.
• They incorporate self-consistent boundary conditions and dynamic trap-assisted SRH kinetics to prevent non-physical behavior and ensure accurate simulations.
• The approach is applied to electron-beam diagnostics and material design, with systematic parameter calibration yielding realistic current and emission predictions.

A drift-diffusion model (DDM) quantitatively describes stochastic evolution under the concurrent influence of deterministic drift and stochastic diffusion, typically realized through partial differential equations (PDEs) or stochastic processes. Drift-diffusion models are foundational in fields ranging from semiconductor device physics and materials science to neuroscience and statistical decision modeling. Drift-diffusion model analysis encompasses the mathematical, physical, and computational paper of DDMs: formulating PDE systems, incorporating appropriate physical or chemical processes, assigning boundary and interface conditions, designing numerical schemes, performing sensitivity and calibration analyses, and interpreting model results in relation to underlying mechanisms and experimental data.

1. Model Structure and Governing Equations

A typical drift-diffusion(-reaction) model describes the time evolution of multiple interacting species—such as electrons, holes, and traps in semiconductors or insulators—subject to applied or internally generated fields. The mathematical structure is represented by coupled PDEs for densities $n$, $p$ (electrons, holes), trapped states $n_T$, and potential $V$: $$\begin{align*} -\nabla \cdot (\varepsilon \nabla V) &= \frac{q}{\varepsilon_0} (C + p - n - n_T) \ \frac{\partial n}{\partial t} + \nabla \cdot \mathbf{J}_n &= S_n - (R_n - G_n) \ \frac{\partial p}{\partial t} + \nabla \cdot \mathbf{J}_p &= S_p - (R_p - G_p) \ \frac{\partial n_T}{\partial t} &= (R_n - G_n) - (R_p - G_p) \end{align*}$$ with current densities

$$\mathbf{J}_n = -D_n \nabla n + \mu_n n \nabla V \qquad \mathbf{J}_p = -D_p \nabla p - \mu_p p \nabla V$$

and source terms $S_n$, $S_p$ for carrier generation (e.g., by irradiation), with trapping/detrapping and recombination modeled via rates $R_n$, $G_n$, $R_p$, $G_p$. The dimensionless or SI units, scaling, and nonlinearities (e.g., field or density dependence) are specified by the physical system under paper.

The paper by Cazaux and Dapor (Raftari et al., 2016) exemplifies a detailed modeling context: time-domain analysis of charging in electron-beam-irradiated insulators, integrating self-consistent field effects, carrier transport, interfacial emission and return, and dynamic trapping phenomena.

2. Boundary Conditions and Physical Constraints

Boundary conditions greatly impact drift-diffusion analysis. For semiconductor and insulator models, bulk boundaries and especially vacuum or interface boundaries must encode relevant physical processes:

• Classical Emission: Surface emission is described by a velocity term, $v_e (n - n_i)$, for secondary electron (SE) emission.
• Tertiary Electron Return: At finite surface potential, emitted SEs can be recaptured as tertiary electrons (TE). The model (Raftari et al., 2016) introduces a novel Robin-type (mixed) boundary:

$$J_n \cdot \nu = \begin{cases} v_e(n - n_i) - \alpha \left(\frac{\partial V}{\partial \nu}\right)^-, & n > n_i \ 0, & \text{otherwise} \end{cases}$$

where $(\cdot)^-$ denotes the negative part (inward field), and $\alpha$ dynamically encodes the TE return, ensuring total charge conservation and preventing unphysical surface potential divergence.

• Spatial Resolution: The boundary condition is applied locally, distributing currents according to the actual electrostatic field at each point, enabling self-limiting charge buildup.

This advanced boundary representation reconciles simulation with the self-limited charging, as higher surface potentials draw back all emitted SEs as TE, enforcing physically realistic yield limits.

3. Dynamic Trap-Assisted Generation-Recombination Schemes

Accurate modeling of long-time, transient, and non-steady-state response requires dynamic treatment of carrier trapping, detrapping, and recombination. Instead of ad hoc or steady-state kinetics, a fully dynamic trap-assisted Shockley-Read-Hall (SRH) formalism is employed: $$\frac{\partial n_T}{\partial t} = (R_n - G_n) - (R_p - G_p)$$ with

$$\begin{aligned} R_n &= \sigma_n v_{th} n (N_T - n_T) \ R_p &= \sigma_p v_{th} p n_T \ G_p &= \sigma_p v_{th} n_i (N_T - n_T) \ G_n &= \sigma_n v_{th} n_i n_T \end{aligned}$$

where trapping/detrapping rates depend on carrier densities, trap occupation, thermal velocity $v_{th}$, capture cross-sections $\sigma_{n,p}$, and intrinsic carrier density $n_i$. This approach allows a single variable for the trap population while capturing the dynamic population exchange, suitable for both short- and long-timescale analysis.

The integrated treatment of traps and a dynamic population model is crucial for reproducing observed physical behaviors, especially under sustained irradiation or strong field conditions.

4. Correction of Non-Physical Behavior and Model Calibration

A central aim of refined drift-diffusion analysis is preventing non-physical predictions (e.g., runaway surface charging or yields exceeding unity). The dual modifications—implementing a self-consistent TE boundary and dynamic SRH trapping—correct such artifacts:

• Charge Neutralization: The tertiary electron return boundary automatically balances strong positive potential, capping further charging.
• Time- and Space-Resolved Dynamics: Removing steady-state approximations and using a dynamic trap model suppresses non-physical accumulation of free carriers or unphysical recombination events.

Systematic parameter calibration is necessary for physically predictive simulation:

• Parameter Fitting: Empirical yield-energy curves under conditions minimizing charging (low-dose, short-pulse, or defocused irradiation) are matched by carefully adjusting emission velocities, trap densities, cross-sections, and penetration depths.
• Calibration is informed by physical experiments, semi-empirical formulas, and, when necessary (e.g., at low energies), direct fitting to experimental data.
• Model validation is performed by comparison of simulated emitted current yields to measured experimental data over broad energy ranges.

Comprehensive sensitivity analysis demonstrates which parameters have the largest effect on model outputs, notably emission velocity (affects yield height), trap density/cross-section (affect curve shape), and penetration depth scaling (affects shift).

5. Analysis under Experimental and Realistic Beam Conditions

Simulations are performed with spatially and temporally resolved drift-diffusion models under various irradiation regimes:

• Stationary, Defocused Beams: Broad irradiation reduces localized charging, and model predictions reproduce transient and equilibrium behavior consistent with early analytical models, but capture the full geometry and electrode effects in 3D.
• Focused, Stationary Beams: Localized charge accumulation is pronounced, but tertiary electron return rapidly drives yields to unity, with return current balancing the SE emission exactly as required for steady-state (a mechanism distinguishing these simulations from explanations based solely on energy shift arguments).
• Moving/Scanning Beams: Drift-diffusion simulations show that non-static beams prevent persistent local accumulation, enabling yields above or below unity depending on scan speed and current. In materials with spatial inhomogeneity, scan history and direction produce observable "memory" and local potential effects in yield maps.

These results connect directly to real-world concerns in scanning electron microscopy (SEM), insulator imaging, charge artifact mitigation, and device design, demonstrating that careful DDM analysis provides predictive, physically grounded answers.

6. Sensitivity to Material Parameters and Practical Implications

Key parameters subject to experimental or intrinsic uncertainty include:

• Electron penetration depth scaling (as a function of PE energy): dominant at low energies; sometimes best fit empirically.
• Trap density and capture cross-section: crucial for the steepness of the yield-vs-energy curve.
• SE emission velocity (surface property): primarily determines overall yield normalization.

Despite intrinsic materials variability and parameter uncertainty, systematic calibration to yield curves renders the advanced DDM physically robust and predictive, as the key non-idealities are absorbed into observable macroscopic behaviors. The model demonstrates notable resilience to uncertainties when physical constraints (charge conservation, trapping dynamics) are correctly imposed.

7. Broader Impact of Drift-Diffusion Model Analysis

The improved DDM framework enables, for the first time, quantitatively accurate three-dimensional, time-dependent predictions of charging and electron emission in insulators exposed to electron irradiation, with direct application to SEM diagnostics and design, material selection for charge mitigation, and a wide range of dielectric measurement scenarios. Physically motivated dynamic boundary and recombination schemes furnish realism without over-fitting, and robust convergence is seen under experimental parameter ranges.

The results generalize to other systems where boundary-driven drift, nonlinear reaction kinetics, and dynamic trapping are essential; the analysis and model structuring in (Raftari et al., 2016) stands as a reference for such extensions. The methodology provides a paradigm for correcting, calibrating, and validating DDMs in complex material and device scenarios.