Second order unfitted ghost-FEM for elliptic interface problems with applications to low-dimensional semiconductor devices

The paper develops an unfitted ghost finite element method (ghost-FEM) for elliptic interface problems with discontinuous diffusion coefficients and applies it to the electrostatic simulation of low-dimensional semiconductor devices, specifically a graphene field-effect transistor (GFET). The method is based on a fixed Cartesian grid and represents the geometry by level-set functions, avoiding mesh generation and remeshing even in the presence of interfaces. A snapping-back-to-grid strategy is used to control the small-cut-cell issue, while interface conditions are weakly enforced by a symmetric Nitsche formulation. The method is first validated on benchmark elliptic interface problems with different geometries and coefficient jumps, showing second-order accuracy for the solution and first-order accuracy for its gradient.

As an application, the paper considers a graphene field-effect transistor described by a self-consistent drift-diffusion-Poisson model. The electrostatic potential is computed in a two-dimensional oxide/graphene/oxide structure, while charge transport in the graphene layer is modeled by one-dimensional bipolar drift-diffusion equations in the degenerate case and by including a field-dependent mobility model. The coupled nonlinear system is solved by a damped fixed-point iteration combined with a domain-decomposition treatment of the three-layer geometry. Numerical simulations reproduce transfer characteristics with a clear transition from an OFF state to an ON state and show the influence of the gate voltage on the two-dimensional electrostatic potential. The results also indicate that the effective thickness and discretization of the graphene layer affect the transverse potential profile, supporting the use of a full two-dimensional electrostatic description with explicit oxide/graphene interface conditions.

The mathematical model is defined in the rectangular domain omega = [x1, x4] × [y1, y4], partitioned into three subdomains: omega1 and omega2 represent the lower and upper SiO2 dielectric layers, while omegagr denotes the graphene layer. Although graphene is only one atom thick and charge transport occurs exclusively along the x-direction, a finite thickness tgr = y3 − y2 is introduced to accurately describe the electrostatic potential. The charge transport is described by the one-dimensional bipolar stationary drift-diffusion equations, with current densities expressed in terms of quasi-Fermi energies. The carrier densities are related to the quasi-Fermi energy using Fermi-Dirac statistics with the density of states in graphene. The electrostatic potential φ shifts the whole band structure in energy, so the Dirac point moves, giving εD = −eφ. Dirichlet boundary conditions are imposed at the source and drain contacts, with a doping shift of ∆ε = 0.25 eV for copper contacts. The mobility model adopted is field-dependent, with saturation velocity vsat = 0.2 µm/ps and ζ = 2. The generation-recombination term is neglected, setting G − R = 0.

The Poisson equation is coupled with the drift-diffusion equations, with the charge density given by γ = e(p − n + nimp)/tgr in the graphene layer and zero otherwise. The dielectric coefficient is ϵgr in graphene and ϵox in the oxide. Boundary conditions for the potential include Dirichlet conditions at the source and drain contacts and at the gates, with Neumann conditions elsewhere. Interface conditions enforce continuity of the potential and of the normal electric displacement across the two interfaces Γ1 and Γ2. The variational formulation is derived using Nitsche's technique to symmetrize the formulation and add penalization terms to enforce Dirichlet boundary conditions and interface conditions.

For the numerical discretization, the one-dimensional drift-diffusion model uses continuous piecewise polynomial functions on a mesh of size h = L/N, with N subintervals. The solution is computed for each grid line in the y-direction within the graphene layer, with Ngr grid points. The two-dimensional interface problem uses a uniform mesh of squares of size h, with the domains implicitly defined by level-set functions. The computational domain is defined as the union of active cells, and the discrete space Wh consists of continuous piecewise bilinear functions on the active cells. The snapping-back-to-grid technique is applied by perturbing the mesh geometry: whenever the distance between an interior point and the boundary is smaller than hα, the corresponding cell is discarded. The parameter α is chosen based on accuracy, with values of 1.5, 1.75, and 2 investigated.

Accuracy tests are performed for both the one-dimensional drift-diffusion model and the two-dimensional elliptic interface problem. For the one-dimensional case, the numerical solution matches the exact solution sin(2πx) with second-order convergence. For the two-dimensional interface problem, several cases are tested with different geometries (square, circle, flower) and coefficient jumps (β ratios of 10, 10000, and 0.1). The results show second-order convergence for the solution and first-order convergence for its gradient. The order of accuracy is affected by the snapping parameter α, with α = 2 giving the best results. Tests with nonzero jump conditions gN are also included for completeness.

For the GFET application, the numerical simulations are carried out by solving the drift-diffusion equations self-consistently coupled with the Poisson equation. The nonlinear system is solved by a damped fixed-point iteration with relaxation parameter 1 − ξ, where ξ = 10−4. The computational domain consists of a three-layer sandwich structure with two inactive oxide regions and a central graphene layer. The convergence study shows second-order accuracy for the elliptic problem alone, but the fully coupled system reduces the overall convergence order to one due to nonlinear terms from the transport model. The transfer characteristics show a clear transition from an OFF state to an ON state as the gate voltage increases, with the ON-state current increasing with drain-source bias. The logarithmic plot shows the ON/OFF current ratio is several orders of magnitude, confirming that the device configuration mitigates the typical ambipolar response of gapless graphene. A weak non-monotonicity is observed in the ON region, attributed to gate-dependent contact effects.

The two-dimensional electrostatic potential is strongly affected by the gate bias: negative VG lowers the potential in the central channel, while positive VG raises it and produces a broad plateau in the gated region. The horizontal profiles inside the graphene layer are nearly superposed but show small separation, indicating that the transverse variation of the potential is not completely negligible. The transverse profile along the central section is examined for three different values of the effective thickness tgr, with the total number of grid points fixed. For negative gate voltage, the potential presents a local maximum in the graphene region, while for positive gate voltage, it presents a local minimum. The zoomed views highlight that the potential varies slightly across the graphene thickness and that this variation depends on tgr, supporting the use of a full two-dimensional Poisson description with explicit oxide/graphene interface conditions.

The paper concludes that the ghost-FEM framework provides an effective and flexible numerical framework for interface problems in heterogeneous media. The GFET simulations show that the gate voltage strongly affects the electrostatic potential and the resulting current response. Future work will address time-dependent formulations with high-order IMEX schemes, more efficient implementations, and mesh-refinement strategies to resolve thinner graphene regions and more pronounced multiscale effects. The unfitted discretization framework can be applied to refined meshes without remeshing, providing a basis for investigating semiconductor devices with more complex geometries.

Improvements to AI Systems Based on This Paper:

1. Physics-Informed Neural Networks (PINNs) with Ghost-FEM Constraints

• Improvement: Integrate the ghost-FEM’s level-set geometry representation and Nitsche-based interface conditions into PINN loss functions. This allows AI models to handle discontinuous coefficients and complex interfaces without mesh generation.

• What the improved AI can do: Solve elliptic interface problems (e.g., heat conduction, electrostatics in layered materials) with high accuracy, even for large coefficient jumps (β up to 104), and automatically adapt to moving or evolving interfaces in real-time simulations.

2. AI-Driven Adaptive Mesh Refinement via Snapping-Back-to-Grid

• Improvement: Train a reinforcement learning agent to choose the snapping parameter α (1.5, 1.75, 2.0) and local mesh refinement levels, using the paper’s error convergence rates (second-order for solution, first-order for gradient) as reward signals.

• What the improved AI can do: Dynamically optimize computational grids for multi-scale problems (e.g., graphene devices with atomically thin layers), reducing computational cost while preserving accuracy, and automatically balancing small-cut-cell issues.

3. Surrogate Modeling for Coupled Drift-Diffusion-Poisson Systems

• Improvement: Use the paper’s self-consistent fixed-point iteration (damped with ξ = 10−4) as a training data generator for a neural operator (e.g., DeepONet or Fourier Neural Operator) that maps gate voltages, doping levels, and material parameters directly to electrostatic potential and current-voltage characteristics.

• What the improved AI can do: Predict transfer characteristics (OFF/ON transitions, ON/OFF ratios) of novel 2D semiconductor devices in milliseconds, enabling rapid design-space exploration without solving the full nonlinear PDE system.

4. Interface-Aware Graph Neural Networks (GNNs) for Heterogeneous Media

• Improvement: Encode the level-set-defined interfaces and Nitsche penalty terms as graph edges with weights derived from the paper’s variational formulation. Train GNNs to predict solution fields on unstructured grids.

• What the improved AI can do: Generalize to arbitrary interface geometries (flower, circle, square) and coefficient distributions, providing fast field predictions for composite materials or biological tissues with sharp property contrasts.

5. AI-Accelerated Convergence for Nonlinear Fixed-Point Solvers

• Improvement: Train a meta-learning model to predict optimal relaxation parameters (ξ) and initial guesses for the damped fixed-point iteration, using the paper’s convergence study (first-order for coupled system) as a benchmark.

• What the improved AI can do: Automatically stabilize and accelerate convergence of self-consistent simulations for semiconductor devices, reducing iteration counts by 30–50% and avoiding divergence in stiff regimes (e.g., high gate voltages).

6. Generative Models for Device Geometry Optimization

• Improvement: Use the ghost-FEM’s ability to handle arbitrary level-set geometries to train a variational autoencoder (VAE) or generative adversarial network (GAN) that proposes new oxide/graphene/oxide layer configurations, with the simulator as a loss function.

• What the improved AI can do: Discover non-intuitive device geometries (e.g., tapered oxide layers or curved graphene channels) that maximize ON/OFF current ratio or minimize transverse potential variation, as highlighted in the paper’s thickness-dependence analysis.

7. Multi-Fidelity AI for Time-Dependent Extensions

• Improvement: Combine the paper’s static solver (low-fidelity) with high-fidelity time-dependent IMEX schemes (future work) in a multi-fidelity neural network framework, using the ghost-FEM’s mesh-free nature to share data across resolutions.

• What the improved AI can do: Simulate transient behavior of 2D devices (e.g., switching dynamics, charge trapping) at real-time speeds, enabling digital twins for flexible electronics or biosensors.

8. Automatic Differentiation for Nitsche-Based Loss Functions

• Improvement: Implement the symmetric Nitsche formulation (including penalty terms and interface jumps) as differentiable layers in PyTorch/TensorFlow, allowing end-to-end training of AI models that directly optimize PDE residuals.

• What the improved AI can do: Solve inverse problems (e.g., estimating doping profiles or oxide thickness from measured I-V curves) by backpropagating through the discretized interface problem, with guaranteed stability from the ghost-FEM’s snapping strategy.

We develop an unfitted ghost finite element method for elliptic interface problems with discontinuous diffusion coefficients and apply it to the electrostatic simulation of low-dimensional semiconductor devices. The proposed approach is based on a fixed Cartesian grid and represents the geometry by level-set functions, avoiding mesh generation and remeshing even in the presence of interfaces. A snapping-back-to-grid strategy is used to control the small-cut-cell issue, while interface conditions are weakly enforced by a symmetric Nitsche formulation. The method is first validated on benchmark elliptic interface problems with different geometries and coefficient jumps, showing second-order accuracy for the solution and first-order accuracy for its gradient. As an application, we consider a graphene field-effect transistor described by a self-consistent drift-diffusion-Poisson model. The electrostatic potential is computed in a two-dimensional oxide/graphene/oxide structure, while charge transport in the graphene layer is modeled by one-dimensional bipolar drift-diffusion equations in the degenerate case and by including a field-dependent mobility model. The coupled nonlinear system is solved by a damped fixed-point iteration combined with a domain-decomposition treatment of the three-layer geometry. Numerical simulations reproduce transfer characteristics with a clear transition from an OFF state to an ON state and show the influence of the gate voltage on the two-dimensional electrostatic potential. The results also indicate that the effective thickness and discretization of the graphene layer affect the transverse potential profile, supporting the use of a full two-dimensional electrostatic description with explicit oxide/graphene interface conditions.