Dynamic Capacitance Models: Theory & Design

• Dynamic capacitance models are frameworks that capture time, bias, and state dependencies using detailed representations of quantum, geometric, and material effects.
• They are applied in nanoscale systems, ferroelectric devices, and RC networks, revealing nonlinear phenomena like negative capacitance and fringing field corrections.
• Advanced extraction techniques, including integral solvers and machine learning surrogates, enable efficient prediction, design, and optimization of complex electronic systems.

A dynamic capacitance model characterizes the time-dependent, bias-dependent, or configuration-dependent nature of capacitance in physical and engineered systems. Such models capture phenomena ranging from distributed charge evolution and nonlinear switching in ferroelectric and graphene-based devices to geometric, material, and quantum effects in miniaturized and nanoscale structures. The following sections synthesize advanced research developments underlying a detailed understanding and practical implementation of dynamic capacitance models across diverse material platforms and devices.

1. Theoretical Foundations and General Frameworks

Dynamic capacitance is fundamentally rooted in the interplay between electrostatics, nonequilibrium charge transport, and, in the case of nanoscale or quantum-confined systems, quantum mechanical effects. In classical systems, the capacitance $C$ relates accumulated charge $Q$ and the resulting voltage $V$, with $C = dQ/dV$. In realistic devices, especially with multiple terminals, the full picture is captured by a capacitance matrix $\mathbf{C}$, reflecting both self- and mutual capacitances between terminals (Abt et al., 2022). This matrix is, in general, bias- and state-dependent, and needs to be extracted or constructed self-consistently as the device evolves dynamically.

Field-effect transistors (FETs), quantum dots, nanogaps, and hybrid circuits with dissipative and nonlinear elements (such as ferroelectric domains) all introduce time-dependent, configuration-dependent, and nonlinear corrections to the capacitance function, necessitating models far beyond the static parallel plate approximation (Jiménez, 2011, Thorbeck et al., 2012, Lu et al., 2013, Sandu et al., 2014, Chang et al., 2017, Hoffmann et al., 2017, Koduru et al., 20 Jul 2024).

2. Dynamic Capacitance in Nanoelectronic and Quantum Devices

Quantum Dots and Nanoscale Effects

In silicon nanowire quantum dots, standard parallel-plate models break down because fringing fields, edge effects, and quantum confinement dominate the field distribution. Experimentally, capacitance can be underpredicted by factors of five or more in small QDs unless fringing corrections are included, which are parametrized by the lateral and vertical device geometries (Thorbeck et al., 2012). The corrected expression,

$$C_{LGC} = \epsilon (2 t_{Si} + w_{Si}) (L_{LG} + t_{ox,1}/2),$$

includes width/height and oxide-related fringing lengths, and suggests the dynamic capacitance model must account for both geometric and voltage-dependent effects (since the dielectric constant $\epsilon$ can itself depend on carrier density and field).

For graphene-based field-effect devices, dynamic capacitance is dictated by the interplay of classical geometrical capacitance (top/bottom gate oxides), quantum capacitance (from the density of states in the channel), and field-induced carrier re-partitioning. This results in explicit closed-form expressions for self- and transcapacitances as functions of bias, covering the full range from unipolar to ambipolar conduction (Jiménez, 2011, Zebrev et al., 2011).

Nanoscale Charge-Dipole and Electrode Contributions

At the atomic scale, especially for nano-gap structures (e.g., carbon nanotube electrodes or molecular junctions), capacitance is ill-described by macroscopic models because the electrode response (redistribution of charge and induced dipoles) determines the dynamic screening. The dynamic capacitance $C$ for a nanogap system is defined as $C = Q / \Delta V$ (with $Q$ the applied electrode charge and $\Delta V$ the resulting potential difference). When an active device (such as C$_{60}$) is inserted, the effective capacitance is $C_a = C - C'$, where $C'$ is the capacitance without the device (Lu et al., 2013). Notably, as electrode length increases, the electrode contribution to $C$ keeps growing, while $C_a$ converges—demonstrating that electrodes are the key player in dynamic nanoscale capacitance, and must be treated dynamically (not as static boundaries).

Quantum and Cross Quantum Capacitance

In atomically-thin or strongly correlated systems, the concept of quantum capacitance is paramount: it describes the additional series capacitance arising from the finite density of states of the conducting plates. More recently, the role of cross quantum capacitance—stemming from interlayer electronic polarizability—has been rigorously articulated. The total capacitance is given by

$$\frac{1}{C} = \frac{1}{C_G} + \frac{1}{e^2} \frac{(\Pi_{11} + \Pi_{22} - \Pi_{12} - \Pi_{21})}{\Pi_{11}\Pi_{22} - \Pi_{12}\Pi_{21}},$$

where $\Pi_{ij}$ are intra- and interlayer polarizabilities. Notably, interlayer correlations can either enhance or suppress the total capacitance, and may result in nonmonotonic dependence on layer separation—a feature with direct experimental ramifications for van der Waals heterostructures and energy-storage devices (Berthod et al., 2021).

3. Distributed and Nonlinear Capacitance in Complex Systems

Ferroelectric and Negative Capacitance Effects

Dynamic capacitance in ferroelectric capacitors is fundamentally nonlinear and time-dependent due to polarization switching and domain wall motion. The key phenomena—transient negative capacitance (NC)—are observed during polarization reversal, where $dQ/dV$ can become negative (Chang et al., 2017, Hoffmann et al., 2017, Orlova et al., 2022, Koduru et al., 20 Jul 2024). This effect is rooted in a mismatch between the rate of free charge supplied to the interface and the bound (polarization) charge switching within the ferroelectric, often formalized using coupled Landau-Khalatnikov or time-dependent Ginzburg-Landau dynamics: $$\gamma \frac{dP}{dt} = -\left[\mathrm{linear}\, +\, \mathrm{nonlinear}\, +\, \mathrm{depolarization} + E_{FE}\right],$$ where $\gamma$ is the viscosity coefficient, and $E_{FE}$ is the internal field. The transient drop in $V_{FE}$ (and thus negative $C$) is pronounced for fast, large domain wall motion and can be tuned by resistance, viscosity, and geometric parameters. Furthermore, the memory window and nonmonotonic C–V dependence in HZO and GeTe-based ferroelectrics critically depend on the thickness-dependent domain density and the concurrent evolution of dielectric and polarization switching components of capacitance (Koduru et al., 20 Jul 2024, Orlova et al., 2022).

Signal-Line and Interconnect RC Networks

In complex RC-line or VLSI interconnects, dynamic capacitance arises from distributed loading effects subject to time-varying input signals. The dynamic capacitance matching (DCM) algorithm divides the driver–load network into small segments and matches the instantaneous effective capacitance using pre-characterized driver data and frequency-domain (y-parameter) RC network models, producing highly accurate current waveforms and capturing subtle over/undershoot effects without the need for full SPICE simulation (Wu et al., 16 Jan 2024). Symbolic expressions for current response are constructed via Laplace-domain algebraic expansions.

In static timing analysis, the effective capacitance ($C_{eff}$), crucial for delay computation, is now determined using graph neural networks (GNNs). The GNN-Ceff model learns to map RC network topology and features to $C_{eff}/C_{total}$, leveraging massive parallel GPU evaluation and outperforming traditional iterative heuristics (e.g., O’Brien/Dartu) in both accuracy and speed (Dogan et al., 4 Jul 2025).

4. Numerical and Machine-Learning Enhanced Extraction

Field-Solver and Integral Equation Approaches

For extracting capacitance in arbitrarily shaped conductors, advanced integral equation formulations—such as the second-kind boundary integral equation with Neumann–Poincaré operators—are employed (Sandu et al., 2014). The resulting eigenfunction expansions enable both analytical and spectral–numerical evaluation of capacitance, and can easily incorporate corrections for geometry (curvature effects in thin films, finite rods, spheroids) and even nontrivial boundary conditions relevant to supercapacitors or biological membranes.

For large-scale voxelized structures, FFT-enhanced solvers (e.g., VoxCap) utilize the block Toeplitz structure of the system’s Green’s function matrices. Efficient, memory-reduced solutions are enabled via tensor decompositions (Tucker) and hybrid block-diagonal preconditioners, facilitating rapid, accurate extraction across millions of voxels (Wang et al., 2020).

Deep Neural Networks and Surrogate Modeling

Recent trends integrate data-driven inference with physical modeling. Machine-learned surrogates—deep neural networks trained on GPU-accelerated field-solver outputs—enable full Bayesian inference of material parameters (e.g., impurity density in Ge detectors) in high-dimensional parameter spaces, using capacitance–voltage (C–V) data as the observational input (Abt et al., 2022). Neural architecture search (NAS) for 3D CNN models (NAS-Cap) and domain-inspired data augmentation (rotational and reflection invariance) have established new standards for accurate, transferable 3D capacitance extraction in IC designs—surpassing both pattern-matching and traditional field solvers in both scalability and accuracy (Li et al., 23 Aug 2024).

5. Special Topics: Biological and Neural Systems

Membrane capacitance in neurons is typically considered fixed, but recent studies introduce time-dependent models motivated by observations of capacitance changes under physical (e.g., ultrasound) stimulation (Courdurier et al., 29 Jan 2025). Explicitly modeling $C(t)$ in the membrane equations (FitzHugh–Nagumo, Hodgkin–Huxley) reveals that abrupt, sufficiently large changes in the capacitance can induce action potentials even from rest, while slow or frequent changes fail to do so. This establishes theoretical and computational foundations for neuromodulation based on dynamically altering membrane capacitance.

6. Engineering and Design Implications

Dynamic capacitance models have become indispensable in a variety of engineering contexts:

• Circuits: Accurate timing, delay, and signal-integrity analysis in advanced nodes hinges on dynamic, data-driven capacitance modeling (Wu et al., 16 Jan 2024, Dogan et al., 4 Jul 2025, Li et al., 23 Aug 2024).
• Analog/RF devices: Explicit dynamic capacitance modeling is required for accurate prediction of high-frequency figures of merit (e.g., $f_T$, current gain), accommodating nontrivial operating regimes and parasitic effects (Jiménez, 2011, Zebrev et al., 2011).
• Ferroelectric and Spintronic Devices: Exploiting transient negative capacitance and domain-dependent responses enables steep-swing logic and energy-efficient memory, with direct dependencies on material microstructure and device geometry (Chang et al., 2017, Hoffmann et al., 2017, Koduru et al., 20 Jul 2024, Orlova et al., 2022).
• Sensors and Quantum Devices: High dynamic range ASICs for detector readout model the effective input capacitance pulse-by-pulse, enabling high-accuracy measurements across hundreds of pF dynamic range with precise S/H and ToT techniques (Cheng et al., 2019).
• Neuromodulation: Dynamic changes in biological capacitance underlie certain modes of noninvasive stimulation, offering a physical mechanism for nonlinear neural excitation (Courdurier et al., 29 Jan 2025).

7. Open Challenges and Future Directions

• Nonlocal and Quantum Effects: For atomically-thin or strongly coupled systems, the interplay of intra- and interlayer polarizabilities, and the onset of collective phenomena (plasmonics, cross quantum capacitance) demand further theoretical and experimental scrutiny (Berthod et al., 2021).
• Automated, Adaptive Modeling: Incorporating dynamic, ML-informed capacitance models directly into EDA tool flows and physical extraction engines is rapidly advancing, but integrating uncertainty quantification and physical constraints remains a key research direction (Abt et al., 2022, Li et al., 23 Aug 2024, Dogan et al., 4 Jul 2025).
• Multiphysics Coupling: The simultaneous modeling of electromechanical, electrodiffusive, and electrochemical effects (e.g., negative capacitance MEMS, electrolyte gating, membrane phenomena) requires multi-domain dynamic capacitance frameworks that can address strong cross-coupling in realistic geometries (TR et al., 2019, Sandu et al., 2014).

Table: Key Approaches in Dynamic Capacitance Modeling

Approach	Physical System/Domain	Notable Features
Ward–Dutton Partition + Drift–Diffusion (Jiménez, 2011)	Graphene FETs/RF	Unified DC/AC/transient, velocity saturation, closed-form for all regions
Charge–Dipole Approximation (Lu et al., 2013)	Nanogaps/CNT, molecular scale	Accounts for full electrode/device system, electrode length crucial
Integral Equations/Shape Factors (Sandu et al., 2014)	Arbitrary conductors, bio/mem systems	Curvature corrections, analytical for spheroids/rods, unifies static/dynamic
Phase-Field/TDGL (Koduru et al., 20 Jul 2024)	Ferroelectric HZO, memory	Domain bulk and wall resolved; butterfly C–V curves; nonmonotonic scaling
ML Surrogates, GNNs, NAS-CNN (Abt et al., 2022, Dogan et al., 4 Jul 2025, Li et al., 23 Aug 2024)	VLSI, detectors, 3D IC	Surrogate and batched learning, physical constraints, transferability
RC Time-Constant in Branching (Madshaven et al., 2019)	Streamer propagation, liquids	RC propagation ruled by dynamic tip capacitance and time-dependent breakdown

In summary, the modern dynamic capacitance model is a high-dimensional, parameter-rich, rigorously formulated framework adapted to system geometry, quantum and nonlinear phenomena, multi-domain coupling, and data-driven extraction. Its comprehensive development has been catalyzed by explicit physical modeling, advanced numerical methods, and, increasingly, machine learning architectures that encode both physics and empirical data. These advances underlie precise prediction, robust control, and optimization of electronic, quantum, biological, and hybrid systems where capacitance is not simply a static property but a dynamical, state-dependent functional critical to system behavior and design.