Nanoelectronic Device and Circuit Modeling

Introduction and Context

Nanoelectronic modeling connects physical phenomena at atomic and nanometre scales with the electrical behavior required for circuit and system design. At conventional dimensions, drift–diffusion equations, empirical compact models, and classical circuit abstractions often provide adequate accuracy. As device dimensions approach the de Broglie wavelength, carrier mean free path, screening length, or phase-coherence length, these assumptions become incomplete. Quantum confinement, tunneling, ballistic transport, discrete dopants, contact resistance, self-heating, variability, and nonlocal electrostatics must then be treated explicitly or represented through carefully calibrated compact models.

A useful modeling hierarchy contains three levels. Atomistic and quantum models resolve electronic states, charge density, and quantum transport. Device-level multiphysics models describe electrostatics, carrier transport, heat flow, and sometimes mechanical or electrochemical effects. Compact models reduce this information to computationally efficient equations suitable for circuit simulation. No single level is universally superior: the appropriate model depends on the question, available parameters, required accuracy, and computational budget.

This chapter develops three central approaches: self-consistent Poisson–Schrödinger modeling, nonequilibrium Green’s function (NEGF) transport, and semiclassical Monte Carlo simulation. It then addresses parameter extraction, compact-model construction, circuit implementation, validation, uncertainty, and current research practice.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain when classical, quantum, ballistic, and atomistic descriptions are appropriate.

2. Formulate and interpret coupled Poisson–Schrödinger equations.

3. Describe the NEGF method using Hamiltonians, Green’s functions, self-energies, and contact distributions.

4. Explain ensemble and ensemble Monte Carlo methods for nanoscale carrier transport.

5. Extract model parameters from experimental or numerical data.

6. Construct and validate compact device models for circuit simulation.

7. Diagnose convergence, calibration, and model-transferability problems.

8. Evaluate contemporary multiphysics and data-assisted modeling methods critically.

1. Modeling Scales and Physical Regimes

The first modeling decision is to identify the dominant length, time, and energy scales. Important quantities include the device length, carrier mean free path, phase-coherence length, thermal de Broglie wavelength, screening length, and energy level spacing.

A useful estimate of quantum confinement is the energy spacing in an ideal one-dimensional well of width L:

E_n = (n^2 pi^2 hbar^2)/(2 m* L^2),

where n is a positive integer, hbar is the reduced Planck constant, and m* is the effective mass. This expression assumes an infinite square well, a parabolic band, and a constant effective mass. It is a qualitative estimate rather than a direct model of a real nanowire or quantum well. If the spacing is comparable with kBT, where kB is Boltzmann’s constant and T is temperature, subband quantization affects carrier occupation and transport.

The electrostatic potential is governed by Poisson’s equation,

nabla · (epsilon nabla phi) = -rho,

where phi is electrostatic potential, epsilon is permittivity, and rho is charge density. With electrons, holes, ionized donors, and ionized acceptors,

rho = q(p - n + N_D^+ - N_A^-),

where q is the elementary charge and the concentrations are expressed per unit volume. The sign convention assumes phi is the electrostatic potential and electron potential energy is -q phi.

Classical drift–diffusion is generally suitable when carriers are near local equilibrium, scattering is frequent, and device dimensions exceed relevant nonlocal scales. Hydrodynamic and energy-transport models retain some nonequilibrium effects by adding carrier temperature or energy equations. Boltzmann transport and Monte Carlo methods resolve the distribution function more directly. NEGF is appropriate when phase coherence, tunneling, contact broadening, and quantum interference are important.

2. Poisson–Schrödinger Modeling

2.1 Coupled equations

In a confined direction, the envelope-function Schrödinger equation is commonly written as

[-hbar^2/2 nabla · (1/m*(r) nabla) + V(r)] psi_i(r) = E_i psi_i(r).

Here psi_i is the normalized envelope wavefunction, Ei is the eigenenergy, m*(r) is position-dependent effective mass, and V(r) is the total potential energy. The divergence form is important at material interfaces because it supports the appropriate effective-mass boundary conditions.

The potential energy may be expressed as

V(r) = V_band(r) - q phi(r) + V_xc(r),

where Vband represents band offsets, phi is electrostatic potential, and Vxc is an optional exchange-correlation correction. In an effective-mass Poisson–Schrödinger model, Vxc is often omitted unless a density-functional or quantum-correction treatment is specifically intended.

The electron density reconstructed from occupied states is, in a simple discrete-state representation,

n(r) = sum_i N_i |psi_i(r)|^2,

where Ni is the occupation of state i per appropriate in-plane area, length, or state normalization. For a two-dimensional electron gas with parabolic in-plane dispersion, the sheet occupation of subband i at temperature T can be calculated using Fermi–Dirac statistics. The exact prefactor depends on degeneracy and normalization, so it must be specified consistently with the density units.

Poisson and Schrödinger equations are nonlinear because the charge determines phi, while phi determines the wavefunctions and occupations. A self-consistent loop is therefore required:

1. Initialize the potential using a classical solution or a previous bias point.

2. Solve the Schrödinger eigenproblem.

3. Compute state occupations and quantum charge.

4. Solve Poisson’s equation with the updated charge.

5. Mix old and new potentials or densities.

6. Repeat until energy, potential, and charge residuals satisfy specified tolerances.

Simple linear mixing may use

phi_new = alpha phi_calculated + (1-alpha) phi_old,

with 0 < alpha <= 1. Small alpha improves stability but slows convergence. Newton, quasi-Newton, Anderson, or predictor–corrector methods can be more efficient near difficult bias points.

2.2 Boundary conditions and interfaces

Dirichlet conditions prescribe potential, usually at ohmic contacts or specified gates. Neumann conditions prescribe normal electric displacement:

n_hat · epsilon nabla phi = -sigma_s,

where sigma_s is surface charge density. At an ideal interface without an explicit sheet charge, electrostatic potential and normal displacement are continuous. Material band offsets and effective-mass discontinuities must be treated separately from electrostatic boundary conditions.

The model must also specify dielectric environment, fixed charge, interface traps, dopant ionization, and carrier statistics. Omitting interface charge can produce an apparently accurate fit at one bias but incorrect threshold voltage, subband occupancy, and transconductance elsewhere.

2.3 Example: confinement-energy estimate

Consider an idealized 5 nm-wide infinite well with electron effective mass m* = 0.20m0. The ground-state energy is

E1 = pi^2 hbar^2/(2m*L^2).

Using hbar = 1.055 x 10^-34 J s, m0 = 9.11 x 10^-31 kg, and L = 5 x 10^-9 m gives approximately E1 = 0.075 eV. This value is much larger than kBT, approximately 0.026 eV at 300 K, indicating strong confinement in this idealized structure. Finite barriers, nonparabolicity, valley degeneracy, strain, and dielectric effects would modify the actual result.

3. Nonequilibrium Green’s Function Modeling

3.1 Physical basis

NEGF provides a quantum-statistical description of open systems. The device is represented by a Hamiltonian H coupled to source and drain reservoirs. In a localized basis, the retarded Green’s function is

G^r(E) = [(E + i eta)I - H - Sigma_S^r(E) - Sigma_D^r(E)]^-1,

where E is energy, eta is a positive infinitesimal or numerical broadening, I is the identity matrix, and SigmaS,D are retarded contact self-energies. The self-energies incorporate both energy shifts and broadening caused by the semi-infinite contacts.

The spectral function is

A(E) = i[G^r(E) - G^a(E)],

where Ga is the advanced Green’s function, equal to the Hermitian adjoint of Gr for a suitable stationary formulation. The broadening matrix of contact alpha is

Gamma_alpha(E) = i[Sigma_alpha^r(E) - Sigma_alpha^a(E)].

For coherent two-terminal transport, the transmission is

T(E) = Tr[Gamma_S G^r Gamma_D G^a].

The Landauer current is

I = (q/h) integral T(E)[f_S(E)-f_D(E)] dE,

where h is Planck’s constant and fS and fD are Fermi functions of the source and drain reservoirs. This expression assumes elastic coherent transport and ideal reservoir descriptions. Spin or valley degeneracy must be included explicitly if not already contained in the trace or Hamiltonian.

For inelastic scattering, self-energies representing phonons, impurities, or other interactions are introduced. Their inclusion produces a self-consistent Born approximation or related approximation, but increases computational cost and requires physically justified scattering parameters.

3.2 Charge and self-consistency

The nonequilibrium density matrix can be obtained from the lesser Green’s function:

G^<(E) = G^r(E) Sigma^<(E) G^a(E).

For contacts in a coherent model,

Sigma^<(E) = i[Gamma_S(E)f_S(E) + Gamma_D(E)f_D(E)].

The real-space density is obtained by energy integration of the diagonal elements of G<, with prefactors determined by the chosen basis and spin convention. This charge is inserted into Poisson’s equation, producing a Poisson–NEGF self-consistent calculation.

Numerical difficulties include energy-grid resolution, treatment of evanescent modes, contact matching, charge neutrality, open-boundary conditions, and nonlinear oscillations. A converged current alone is not sufficient evidence of correctness; charge residuals, current conservation, and stability under grid refinement must also be checked.

3.3 When NEGF is preferable

NEGF is particularly valuable for ultrashort transistors, resonant tunneling diodes, molecular junctions, quantum dots, tunnel field-effect transistors, nanowires, and devices with strong source-to-drain tunneling. It can reveal mode filtering, resonance broadening, contact-induced states, and quantum interference that a local mobility model cannot represent.

Its limitations include high memory and computational requirements, sensitivity to Hamiltonian quality, difficulty of modeling long devices with many inelastic events, and the possibility of obtaining numerically precise results from physically inaccurate material parameters.

4. Monte Carlo Transport

4.1 Boltzmann transport foundation

The semiclassical carrier distribution f(r,k,t) obeys the Boltzmann transport equation,

partial f/partial t + v(k) · nabla_r f - (q/hbar)E · nabla_k f = (partial f/partial t)_coll,

where v(k) is group velocity, E is electric field, and the collision term represents scattering. The equation assumes well-defined semiclassical wave packets and is not intended to describe coherent tunneling through barriers.

Monte Carlo methods solve this equation statistically by tracking representative particles. Between scattering events, particles evolve under the applied electric field according to the band structure. A scattering mechanism is then selected probabilistically from rates for acoustic phonons, polar optical phonons, intervalley transitions, ionized impurities, surface roughness, or other processes.

In an ensemble Monte Carlo algorithm, the total scattering rate is

Gamma_total = sum_j Gamma_j.

If rates are locally constant during a free-flight interval, the time to the next event is sampled as

tau = -ln(r)/Gamma_total,

where r is a uniformly distributed random number in (0,1). The probability of selecting mechanism j is Gamma_j/Gamma_total. In realistic nonparabolic bands or spatially varying fields, self-scattering or null-collision techniques are often used to maintain efficient sampling.

4.2 Outputs and interpretation

Monte Carlo simulation provides velocity distributions, energy distributions, occupation of valleys, transient response, velocity overshoot, mobility, energy relaxation, and hot-carrier behavior. Statistical uncertainty decreases approximately with the inverse square root of the number of independent samples, so reducing noise by a factor of two requires roughly four times as many samples. Correlated samples, weighting schemes, and transient initialization can alter this simple estimate.

Monte Carlo is powerful for semiclassical nonequilibrium transport but has limitations. It does not naturally capture phase-coherent interference or tunneling unless supplemented by specialized quantum algorithms. Scattering rates and band parameters may be uncertain, and nanoscale boundaries require physically defensible roughness and surface-scattering models.

5. From Physics-Based Simulation to Compact Models

A compact model is a computational representation of a device terminal behavior, normally expressed as algebraic and differential equations for use in SPICE-like circuit simulators. It must balance physical meaning, numerical robustness, parameter identifiability, and speed.

A compact transistor model may define terminal currents as functions of terminal voltages, internal states, temperature, and model parameters:

I_k = F_k(V_1,...,V_m, T, x, theta),

where x represents internal dynamic variables and theta denotes parameters. Charge-based models are preferable for transient simulation because terminal charges determine capacitive currents:

I_k = I_k^cond + dQ_k/dt.

The terminal charges should satisfy charge conservation, such as the sum of terminal charges equaling the relevant stored charge under the adopted partitioning convention. Reciprocity and passivity are also important for stable circuit operation.

5.1 Parameter extraction

Parameter extraction maps measured or simulated observables to model parameters. A practical sequence is:

1. Define the intended operating domain, including voltage, temperature, frequency, and geometry.

2. Identify directly measurable or independently known parameters, such as dimensions, dielectric thickness, or contact area.

3. Extract low-coupling parameters first, for example series resistance from appropriate linear-region data.

4. Extract electrostatic parameters from threshold, subthreshold, capacitance, and charge data.

5. Extract transport parameters from transconductance, output conductance, mobility, or ballistic-current data.

6. Fit dynamic parameters using capacitance or S-parameter measurements.

7. Validate all parameters using datasets not used for fitting.

A weighted least-squares objective can be written as

J(theta) = sum_i w_i [y_i^meas - y_i^model(theta)]^2,

where yi are observations and wi are weights. If measurement uncertainty standard deviation is sigmai, a statistically motivated choice is wi = 1/sigmai^2, assuming independent Gaussian errors. Correlated errors require a covariance-weighted objective rather than independent scalar weights.

Parameter identifiability must be examined. Two parameters may produce nearly identical changes in a limited dataset, making their individual values unreliable even when the overall fit is good. Sensitivity matrices, profile likelihoods, Bayesian posterior distributions, or singular-value analysis can reveal this problem.

5.2 Validation metrics

Validation should include more than visual agreement. Useful metrics include absolute error, normalized root-mean-square error, maximum error in critical regions, threshold-voltage error, subthreshold-slope error, current-ratio error, capacitance error, and transient timing error. The metric must reflect the intended application: a small absolute current error may be unacceptable in leakage prediction, while a small relative error can be misleading near zero current.

Validation should test interpolation and extrapolation separately. A model calibrated only at room temperature should not be assumed valid at cryogenic temperature. Similarly, a model fit to one device length may fail when short-channel electrostatics or contact resistance changes.

6. Multiphysics and Circuit-Level Considerations

Nanoelectronic devices frequently exhibit electrothermal feedback. Joule heating raises temperature, temperature changes mobility, bandgap, contact resistance, and phonon populations, and these changes alter current. A basic steady-state thermal equation is

P = (T_device - T_ambient)/R_th,

where P is dissipated power and Rth is thermal resistance. This lumped relation assumes linear heat flow and a temperature-independent thermal resistance over the operating range. More detailed models solve the heat equation with spatially varying thermal conductivity and boundary conditions.

Other coupled effects include:

• Electromechanical coupling: strain changes band edges, mobility, and tunneling barriers.
• Electrochemical effects: ionic motion and redox reactions alter local charge and contact potentials.
• Radiation and reliability: energetic carriers and defects modify trap populations and leakage.
• Statistical variability: random dopants, line-edge roughness, metal-grain work-function variation, and interface traps produce device-to-device variation.

Circuit simulation introduces additional requirements. The model must provide continuous and differentiable terminal equations where possible, consistent derivatives for Newton iteration, physically bounded behavior outside nominal data ranges, and robust initialization. A model that is accurate in isolated device simulation may fail in a circuit because of discontinuous derivatives, nonconserved charge, or negative differential resistance interacting with parasitic elements.

7. Illustrative Case: Modeling a 5 nm-Scale Gate-All-Around Nanowire FET

Background

Consider an illustrative, fictional case involving a silicon gate-all-around nanowire transistor with a nominal 5 nm square cross-section. The objective is to predict subthreshold leakage, on-current, quantum capacitance, and temperature sensitivity for circuit design.

Problem

A classical drift–diffusion model overestimates inversion charge because it treats the carrier distribution as continuous and does not adequately represent source-to-drain tunneling. A fully atomistic simulation would be too expensive for repeated circuit-level evaluation.

Analysis

First, a two-dimensional Poisson–Schrödinger calculation is performed in the nanowire cross-section. It estimates subband energies and quantum charge as functions of gate voltage. The electrostatic solution includes gate work function, oxide permittivity, interface charge, and source/drain boundary conditions.

Second, a reduced mode-space NEGF model is used in the transport direction. The lowest relevant subbands form the basis, while source and drain self-energies represent open contacts. The model calculates transmission and current under several drain and gate biases. Poisson–NEGF self-consistency is used in the subthreshold and near-threshold regimes, where tunneling and quantum charge are important.

Third, a compact model is constructed. The extracted parameters include effective subband thresholds, contact resistance, a gate-dependent quantum capacitance, and a tunneling correction calibrated against NEGF. Temperature-dependent parameters are extracted from simulations at 250, 300, and 400 K. The compact model uses charge-based terminal equations and includes a thermal node represented by Rth.

Validation and outcome

The model is validated against NEGF results not used during extraction. It reproduces transfer characteristics, subthreshold slope, and gate capacitance over the selected voltage range. It is less accurate at very high drain bias because the reduced mode basis omits higher subbands and the contact model becomes inadequate. Rather than silently extrapolating, the model reports a validity range and flags the need for a larger basis.

Learning points

The case demonstrates a hierarchical workflow: quantum confinement is obtained from Poisson–Schrödinger analysis, open-system transport from NEGF, and circuit efficiency from a calibrated compact model. It also shows why validation must include bias, temperature, and physical residuals—not merely a single current curve.

8. Challenges, Governance, and Best Practices

Major challenges include uncertain material parameters, incomplete knowledge of interface states, computational cost, numerical stiffness, and model non-identifiability. Atomistic calculations may depend strongly on the selected exchange-correlation functional or tight-binding parameter set. Continuum models may conceal discrete defects that dominate variability. Statistical calibration should therefore distinguish numerical error, measurement noise, parameter uncertainty, and model-form error.

Reproducibility requires recording geometry, mesh, basis, material parameters, boundary conditions, solver tolerances, random seeds, and software versions. Published current values without these details are difficult to assess. Data-driven surrogate models should document training distributions, feature definitions, error metrics, and out-of-domain behavior. AI can accelerate parameter fitting, surrogate construction, and anomaly detection, but it should not replace physical constraints. A neural surrogate that violates charge conservation or predicts negative dissipation can produce unsafe circuit conclusions.

Best practice is to use a hierarchy of verification:

1. Code verification: confirm that equations are implemented correctly using analytical or manufactured solutions.

2. Numerical verification: test mesh, energy-grid, time-step, and particle-number convergence.

3. Physical validation: compare with trusted experiments or higher-fidelity calculations.

4. Circuit validation: test convergence, transient stability, noise behavior, and parameter variation.

5. Uncertainty assessment: report confidence intervals or prediction bands where possible.

Contemporary developments include atomistic tight-binding and density-functional workflows, adaptive mode-space NEGF, quantum-corrected hydrodynamic models, electrothermal NEGF, differentiable simulators, Bayesian calibration, and physics-informed machine learning. These developments improve efficiency but introduce additional assumptions that must be evaluated rather than accepted automatically.

Chapter Summary

Nanoelectronic modeling requires a deliberate match between physical regime and model hierarchy. Poisson–Schrödinger methods describe self-consistent electrostatics and quantum confinement. NEGF models open-system quantum transport through Green’s functions, contact self-energies, spectral functions, and transmission. Monte Carlo methods solve semiclassical nonequilibrium transport statistically and are especially useful for hot-carrier and scattering analysis. Compact models translate high-fidelity results into efficient, charge-consistent circuit equations. Reliable parameter extraction requires identifiable parameters, weighted objectives, independent validation, and explicit uncertainty analysis. Multiphysics effects, numerical convergence, reproducibility, and validity limits are as important as the governing equations themselves.

Key Terms

Quantum confinement: Modification of energy states caused by spatial restriction of carriers.

Poisson–Schrödinger model: Self-consistent solution of electrostatic and quantum mechanical equations.

Effective mass: Band-structure parameter used to approximate carrier dynamics near a band extremum.

Subband: Quantized energy branch arising from confinement in one or more directions.

NEGF: Quantum transport formalism for open, nonequilibrium systems.

Self-energy: Mathematical representation of contact or interaction effects on a device region.

Broadening matrix: Matrix describing reservoir-induced coupling and level broadening.

Transmission function: Energy-dependent probability for carriers to pass between contacts.

Lesser Green’s function: Green’s function used to calculate nonequilibrium occupations and charge.

Monte Carlo transport: Statistical simulation of carrier free flight and scattering events.

Self-scattering: Null event used to maintain a convenient total scattering rate.

Compact model: Efficient device equation set intended for circuit simulation.

Parameter extraction: Estimation of model parameters from data.

Identifiability: Ability to determine individual parameters uniquely from available data.

Charge-based modeling: Formulation using terminal charges to obtain consistent dynamic currents.

Electrothermal feedback: Mutual dependence of electrical behavior and device temperature.

Mode-space method: Reduced quantum basis using selected transverse modes.

Model-form error: Error caused by an inadequate model structure rather than uncertain parameters.

Physics-informed machine learning: Data-driven modeling constrained by physical laws or invariants.

Review Questions

Short Answer Questions

1. Why does quantum confinement become important when the energy spacing is comparable with kBT?

2. What physical effects are represented by contact self-energies in NEGF?

3. State the main assumptions of the Landauer current expression used for coherent transport.

4. Why is charge conservation important in a compact model?

5. Distinguish code verification from physical validation.

Long Answer Questions

1. Explain the self-consistent Poisson–Schrödinger workflow, including boundary conditions, occupations, mixing, and convergence.

2. Derive the role of retarded Green’s functions, broadening matrices, and transmission in a two-terminal NEGF calculation.

3. Compare Monte Carlo, NEGF, and drift–diffusion methods in terms of physical assumptions, outputs, computational cost, and applications.

4. Develop a parameter-extraction strategy for a nanoscale transistor compact model and discuss identifiability.

5. Explain why a device model that fits one bias and temperature condition may fail in circuit simulation.

Analytical / Application Questions

1. Estimate the ground-state confinement energy of an electron in a 4 nm infinite well with effective mass 0.15m0. Compare it with kBT at 300 K and discuss the limitations of the estimate.

2. Design a validation plan for a Poisson–NEGF nanowire model that includes grid refinement, energy-grid refinement, current conservation, temperature variation, and comparison with measurement.

3. A compact model matches current accurately but produces unstable transient circuit simulations. Identify likely causes and propose diagnostic tests and corrective actions.

Multiple Choice Questions with Answers

Question 1

Which equation directly couples electrostatic potential to charge density in a Poisson–Schrödinger calculation?

A. The Landauer equation

B. Poisson’s equation

C. The Monte Carlo free-flight equation

D. The least-squares objective

Correct Answer: B. Poisson’s equation

Question 2

In the coherent two-terminal NEGF formalism, what does Gamma_alpha represent?

A. The contact-induced broadening matrix

B. The electron effective mass

C. The lattice temperature

D. The dielectric permittivity

Correct Answer: A. The contact-induced broadening matrix

Question 3

Which expression gives the coherent transmission in the standard NEGF formulation?

A. Tr[H G^r]

B. Tr[Gamma_S G^r Gamma_D G^a]

C. q n mu E

D. epsilon nabla phi

Correct Answer: B. Tr[Gamma_S G^r Gamma_D G^a]

Question 4

The Monte Carlo free-flight time tau = -ln(r)/Gamma_total assumes that r is what type of quantity?

A. A uniformly distributed random number between zero and one

B. The device resistance

C. The carrier density

D. The thermal conductivity

Correct Answer: A. A uniformly distributed random number between zero and one

Question 5

Why are charge-based compact models particularly useful for transient circuit simulation?

A. They eliminate all parasitic resistance

B. They obtain dynamic currents consistently from terminal charges

C. They require no parameter extraction

D. They apply only to ballistic devices

Correct Answer: B. They obtain dynamic currents consistently from terminal charges

Question 6

If independent measurement errors have standard deviations sigma_i, a statistically motivated least-squares weight is

A. sigma_i^2

B. 1/sigma_i^2

C. 1/sigma_i

D. sigma_i + 1

Correct Answer: B. 1/sigma_i^2

Question 7

Which method is most directly suited to modeling coherent tunneling and quantum interference?

A. Classical drift–diffusion

B. Equilibrium mobility extraction

C. NEGF

D. Lumped thermal resistance alone

Correct Answer: C. NEGF

Question 8

What is the primary purpose of a contact self-energy in an open-device NEGF calculation?

A. To represent the effect of semi-infinite reservoirs

B. To define the lattice crystal structure only

C. To calculate the dielectric constant

D. To remove all scattering from the device

Correct Answer: A. To represent the effect of semi-infinite reservoirs

Question 9

Reducing independent Monte Carlo statistical noise by approximately a factor of two generally requires approximately

A. Half as many samples

B. Twice as many samples

C. Four times as many samples

D. Ten times fewer samples

Correct Answer: C. Four times as many samples

Question 10

Which is an example of numerical verification rather than physical validation?

A. Comparison with measured drain current

B. Comparison with a trusted experimental capacitance

C. Testing convergence under mesh refinement

D. Comparison with a fabricated device at several temperatures

Correct Answer: C. Testing convergence under mesh refinement