Ferroelectric Dynamic‐Field‐Driven Nucleation and Growth Model for Predictive Materials‐To‐Circuit Co‐Design

ABSTRACT

Real ferroelectric devices operate under mixed and distorted time‐varying voltages, yet the standard nucleation‐growth frameworks used to interpret ferroelectric switching, most notably the Kolmogorov‐Avrami‐Ishibashi (KAI) and nucleation‐limited switching models (NLS), are derived under the critically limiting assumption of a constant electric field. Thus, the prevailing interpretation of ferroelectric switching dynamics fails under real operating conditions. Here we introduce a compact dynamic‐field‐driven nucleation and growth (DFNG) model that enables quantitative fits to switching transients across multiple ferroelectric materials to extract time‐varying domain wall velocity and growth dimensionality, even under arbitrary voltage waveforms. This capability then motivates its use in device modeling under complex signals spanning disparate time and frequency scales. Coupling the compact model to application‐related waveforms and a circuit‐level simulation platform facilitates a predictive materials‐circuit co‐design framework by linking nucleation and growth parameters to memory window, disturb error, speed, and energy dissipation for next‐generation ferroelectric technologies.

This study presents a compact dynamic‐field‐driven nucleation and growth (DFNG) model that captures ferroelectric switching behavior under arbitrary voltage waveforms. It enables extraction of time‐dependent domain wall velocity and growth dimensionality, which can then be extended to device‐level modeling. By coupling the model to application‐related waveforms, it establishes a predictive materials–circuit co‐design framework linking nucleation dynamics to technology metrics.

1. Introduction

The low‐power, high‐speed, and non‐volatile characteristics of ferroelectric materials have enabled device concepts [1, 2, 3] that can surpass conventional CMOS (complementary metal‐oxide semiconductor) technologies [4, 5, 6]. Key to the ferroelectric computing technologies is the manipulation of polarization under ultrafast mixed‐frequency and time‐dependent voltage waveforms (Figure 1a). The time evolution of polarization transient composes the intricate metrics of ferroelectric materials and devices, such as latency, energy, disturb errors, and readout signals. To assist high‐throughput material design and to optimize device operation protocols, a compact model is needed to predict the physical response of ferroelectrics under the dynamic electric fields.

FIGURE 1.

Ferroelectric switching transients under dynamic electric fields. (a) A schematic of a ferroelectric capacitor driven by various time‐dependent voltage waveforms (V_in ). Assuming the ferroelectric material is preset to an up‐polarized state (blue) before t = 0, the waveform induces partial switching via the nucleation and growth of down‐polarized domains (purple). The voltage across the ferroelectric capacitor (V_FE ) and the current transient, including ferroelectric switching current (I_FE ) and linear dielectric current (I_DE ), are measured. By integrating I_FE and normalizing it to spontaneous polarization (P_S ), the instantaneous switched (transformed) fraction (f) can be determined. (b) Illustration of the error from a naive voltage‐dependent KAI model contrasted with the DFNG model presented here. Given a constant applied voltage (solid lines) and a voltage ramp to the same set point (dashed lines), the KAI model produces a polarization transient insensitive to the ramp history, in contrast to the DFNG model.

Existing models, however, are unable to describe the nucleation and growth processes which drive ferroelectric switching under a time‐varying voltage, largely because ferroelectric polarization evolution is hysteretic and depends on the full history of applied electrical stimuli. For example, the seminal Kolmogorov‐Avrami‐Ishibashi (KAI) model [7, 8, 9] provides an explicit description of the polarization transient under a constant applied voltage. Even when a voltage dependence is inappropriately forced, the polarization is only dependent on the instantaneous voltage value and ignores the prior voltage path (Figure 1b; Note S1). This deficiency is also found in the nucleation‐limited switching (NLS) model [10], which simulates a switching event from a distribution of local nucleation times under a constant applied field and neglects domain growth. In addition, the distribution function in the NLS model is ad hoc and thus usually leads to overfitting of experimental transients, thereby overlooking physical rationalization. On the other hand, time‐dependent Ginzburg‐Landau‐Devonshire [11, 12, 13], Preisach, or other equivalent‐circuit models, can be numerically driven by any waveform, but they neglect the necessary nucleation and growth physics or are generally too complex to serve as compact, closed‐form fitting tools for experimental transients.

In this work, we present a dynamic‐field‐driven nucleation and growth (DFNG) model that is universally applicable to distinct materials and arbitrary waveforms (Figure 1b). We first demonstrate the capability by self‐consistently fitting experimental polarization switching transients for polycrystalline Hf_0.5Zr_0.5O₂ (HZO), single‐crystalline BaTiO₃ (BTO), and Al_0.92B_0.08N (AlBN) thin films with as few as 3 parameters. The model robustly extracts intrinsic material parameters (growth dimension, nucleation density, and Merz activation field), even when there are mixed time‐scale changes in the voltage profile due to the rise time of electronics (10–100 ps) and slower circuit parasitic effects (1–10 ns), making materials‐level inference possible even under practical measurements with non‐idealities. We then derive domain wall velocity and domain size of HZO, which motivates a speculation of a domain‐growth‐limited switching mechanism with instantaneous nucleation corroborated by experimental findings, contrary to the proliferated belief of nucleation‐limited switching. Furthermore, the model generates realistic material response to time‐varying mixed‐frequency waveforms, which can readily incorporate with circuit‐level modeling to optimize circuit design and operation protocols. Finally, the model is used to map material parameters to hysteresis loops and technology metrics, providing insight into design principles based on application requirements. We further demonstrate the capability of large‐scale circuit simulation by implementing the DFNG model and including other circuit elements with SPICE (Simulation Program with Integrated Circuit Emphasis). Together, our study for the first time establishes a universal, physics‐grounded framework that links nucleation and growth dynamics to device‐level performance metrics, offering a foundation for elucidating material fundamentals and a rational co‐design platform for next‐generation ferroelectric technologies.

2. Results and Discussion

2.1. A Compact yet Generalized Nucleation and Growth Model

Our DFNG model extends the traditional Avrami theory of nucleation and growth to capture the ultrafast voltage profiles driving a ferroelectric phase transformation by including two important physical considerations: (1) Nucleation is either instant or adopts a changing steady‐state nucleation rate governed by a thermodynamic barrier that is dependent on the time‐varying voltage profile; (2) a non‐linear voltage‐dependent growth rate based on Merz law [14].

We build our derivation from Cahn's time‐cone model [15], which relates the transformed fraction, f(t), of a system undergoing a phase transition to the number of nuclei, 〈N(t)〉, in a cone of space‐time that precedes the time and point of space where nucleation may occur (reverse time cone) by:

$$f\left(t\right)=1-\exp \left[-〈N\left(t\right)〉\right]$$ (1)

Where,

$$〈N\left(t\right)〉=\gamma \underset{0}{\overset{t}{\int }}J\left(\tau \right){\left[\underset{\tau }{\overset{t}{\int }}v\left(t^{\prime }\right)d{t}^{\prime }\right]}^{d}d\tau$$ (2)

Here d is the domain growth dimension, and γ is the geometric factor of domain growth, e.g., γ  =  1 for 1D, π for 2D, and $\frac{4}{3}\pi$ for 3D. γ and d can be applied to mixed dimensions. v(t′) is the domain wall velocity at time t′. We introduce Merz law [14] to describe the velocity, namely $v(t^{\prime })=v_0\exp \left[\frac{-V_a}{V(t^{\prime })}\right]$, where v ₀ is the domain wall velocity at infinite voltage, approximated as the speed of sound, V(t′) is the instantaneous voltage and V_a is the activation voltage, corresponding to an average potential that the domain wall depins from a defect site. J(τ) is the nucleation rate at time τ, which is then discussed in two cases: instantaneous nucleation and homogeneous nucleation.

For instantaneous nucleation, J_Inst  (τ) = ρ₁ δ(0). Substituting into Equation (2), 〈N(t)〉 is acquired.

$$〈N\left(t\right)〉=A^d{\left[\underset{0}{\overset{t}{\int }}\exp \left[-\frac{V_a}{V\left(t^{\prime }\right)}\right]d{t}^{\prime }\right]}^d,$$ (3)

Here, $A^d=\gamma {\rho }_1{v}_0^d$, and ρ₁ is the density of pre‐existing nuclei. Thus, parameter A is an indicator of instantaneous nucleation density.

For homogeneous nucleation, we adopt an expression from the Janovec‐Kay‐Dunn (JKD) model [16, 17] to calculate the nucleation rate. In the JKD model, the probability of nucleation per site is given by $\exp \left(-\frac{\mathrm{\Delta }{G}^{\ast }}{k_{B}T}\right)$, where ΔG* is the critical Gibbs free energy to create a nucleus. Assuming a cylindrical nucleus that extends across the thickness and ignoring the depolarization field, ΔG* is given by $\frac{\pi t_{FE}^2{\sigma }^2}{2P_{S}V(\tau )}$. t_FE is the thickness of the ferroelectric layer, σ is the domain wall energy, P_S  is the spontaneous polarization, and V(τ) is the instantaneous voltage when nucleation happens at time τ. Another physical process involved in homogeneous nucleation is the attachment of dipoles, the rate of which is calculated by the product of dipole flipping frequency ω₀ and the probability of a successful flip given by $\exp \left(-\frac{V_a}{V(\tau )}\right)$. Then the homogeneous nucleation rate at time τ is the product of site density, nucleation probability, and dipole attachment rate [18],

$$J_{Homo}\left(\tau \right)={\rho }_2{\omega }_0\exp \left[-\left(\frac{\pi t_{FE}^2{\sigma }^2}{2P_{S}V\left(\tau \right)k_{B}T}+\frac{V_a}{V\left(\tau \right)}\right)\right],$$ (4)

where ρ₂ is the homogeneous nucleation density. At constant temperature, the expression of 〈N(t)〉 becomes:

$$〈N\left(t\right)〉=B^d\underset{0}{\overset{t}{\int }}\exp \left[-\frac{V_a+{{\sigma }^{\prime }}^2}{V\left(\tau \right)}\right]{\left[\underset{\tau }{\overset{t}{\int }}\exp \left[-\frac{V_a}{V\left(t^{\prime }\right)}\right]d{t}^{\prime }\right]}^{d}d\tau ,$$ (5)

where$B^d=\gamma {\rho }_2{\omega }_0{v}_0^d$, ${{\sigma }^{\prime }}^2=\frac{\pi t_{FE}^2{\sigma }^2}{2P_S{k}_{B}T}.$ We note that the integral forms in Equations (3) and (5) preserve voltage path‐dependent switching transients.

2.2. Material Parameters From Measurements in Dynamic Field Across Materials

We begin by applying the model to the HZO system. We experimentally measure ferroelectric switching transients in 3‐µm diameter capacitors by simultaneously acquiring the voltage across the ferroelectric capacitor (V_FE ), the ferroelectric switching current df/dt (normalized by 2P_S ), and the polarization (transformed fraction f when normalized by 2P_S ) as a function of time (Figures 1a and 2a). Details of the transient measurements are given in the Methods section and elsewhere [19]. We find that the model best fits the experimental data when an instantaneous nucleation rate is used (Figure S1), indicating the presence of preexisting nuclei. These nuclei are possibly permanent dipoles formed by bound charges at the electrode‐insulator interface, grain boundaries, and phase boundaries (hence heterogeneous). The fit yields d  =  0.699  ±  0.003, V_a =  12.75 ±  0.09 V, A  =  3236  ± 191 ns⁻¹. These parameters are extracted via a global fit across four different supply voltage profiles while feeding the measured real‐time voltage as an input. This ensures the extracted parameters are intrinsic to the material, independent of the variance in applied voltage. The values of these parameters reflect their physical origins and will be discussed in a few paragraphs.

FIGURE 2.

Self‐consistent fitting of the DFNG model across distinct material systems and various dynamic waveforms. Experimental V_FE , f, normalized I_FE (df/dt), and Avrami exponent over time of a polycrystalline HZO (a), single‐crystalline BTO (b), and wurtzite AlBN (c) capacitors along with the DFNG model fit. The green dashed vertical line marks the end of the fast voltage ramp of the supply (200 ps for HZO and BTO, 100 ns for AlBN). The region between the green and grey dashed lines indicates waveform distortion due to a circuit effect. The grey circles mark the corresponding transformed fraction when the voltage saturates to the setpoint. Notably, most switching occurs under a time‐varying voltage beyond the validity of traditional nucleation and growth models. The DFNG model accurately reproduces these transients and captures the evolving Avrami exponent, enabling physical interpretation beyond fixed growth dimensionality.

The expression of Equation (3) for HZO is compact, as it only involves three parameters A, V_a , and d, yet able to capture the full experimental switching transient, particularly through the rapid voltage variations that occur at sub‐ns and ns scales (Figure 2a). In real circuits, the rise of the voltage pulse is skewed by the finite rise time of the pulse generator and the dielectric current, which here is ∼ 200 ps. Interestingly, ferroelectric switching occurs during this rapid voltage rise and produces a dynamic Avrami exponent (exceeding 4) at the beginning. The DFNG model captures this behavior and outputs a large initial Avrami exponent because the domain wall velocity accelerates non‐linearly upon the rapid rise of the voltage. Afterward, the Avrami exponent relaxes back to an interpretable growth dimension value as the voltage settles. This result indicates that apparent super‐physical Avrami exponents frequently reported in the literature naturally arise from dynamic‐field effects. After the initial voltage rise, significant ferroelectric switching current is generated and flows through the series resistor in the circuit under larger voltage amplitudes, leading to a sluggish rise to the set voltage (between the green and grey dashed lines in Figure 2a). This distortion of the voltage waveform spans almost the entire switching process, which highlights the necessity of incorporating a time‐dependent voltage for modeling ferroelectric switching transients. The deviation from the ideal square pulse also implies the common misuse of the conventional KAI and NLS model where constant voltage is assumed. Contrarily, our model allows for robust physical parameter extraction even when there are ultrafast changes in the voltage profile due to the rise time of the electronics (10–100 ps scale) and slower circuit parasitic effects (1–10 ns scale), making possible materials‐level inference under unideal measurement conditions.

To demonstrate universality, we apply the model to single‐crystalline BTO with both heterogeneous and homogeneous nucleation included, by summing Equations (3) and (5), and involving 5 fit parameters. The model fits the whole transient of a 3‐µm BTO capacitor despite rapid voltage variations and for all voltage profiles (Figure 2b). A low V_a of 0.536 V, sparse heterogeneous nucleation density of ∼ 10⁻⁴ nm^−d, and domain wall energy of 4.4 mJ m⁻² are determined and consistent with the single‐crystalline nature of BTO (Note S2). Additionally, the model is applied to an ultrathin AlBN capacitor (10 nm thick, 100 µm in diameter) as a representative wurtzite ferroelectric system (Figure 2c; Note S3 and Figures S2 and S3). The fitting results indicate that in this thickness regime, AlBN switches predominantly via a heterogeneous nucleation mechanism with a high activation voltage (66.0 V). Although heterogeneous nucleation typically implies defect‐assisted switching, the defects that can initiate nucleation are rare (∼10⁻⁵ nm^−d). The combination of low nucleation density and high activation voltage accounts for the large switching field (7–9 MV/cm) and long switching times (10s µs), in good agreement with reported literature values [20, 21, 22, 23].

2.3. Uncovering Microscopic Switching Mechanisms From Model Analysis

Nucleation sites, domain wall velocity, and domain sizes that can be visualized or inferred with modern techniques such as piezoresponse force microscopy (PFM) and advanced transmission electron microscopy [24, 25, 26, 27], serve as observable quantities to verify physics postulated in theoretical models. With the DFNG model, we derive physical values for these quantities and further posit a growth‐limited switching mechanism, which can be corroborated by domain imaging techniques and simulations.

By assuming a v ₀ value of 6000 m/s [28], domain wall velocity v, nucleation rate J, and domain radius r can be directly output from the model (Figure 3). The domain wall velocity of HZO (Figure 3b) during switching increases from 0 to 1.2 m/s (low voltage, light purple) and 14.4 m/s (high voltage, dark purple), resulting from the high activation field for depinning in a defective medium. Despite the slow growth rate, HZO exhibits a comparable switching time to BTO because of a much higher nucleation density (Figure 3c). The heterogeneous nucleation density of HZO is estimated to be ∼1 nm^−d. Note that the heterogeneous nucleation density is the volumetric number of nucleation sites averaged over the device area. Therefore, multiple nucleation defects can exist in one grain as they reside at the grain boundaries, intragranular phase boundaries, and the ferroelectric‐electrode interface. The domain wall velocity also leads to distinct profiles in time cones and domain sizes (Figure 3d). In contrast to the straight time cone profile in the conventional KAI model, real materials possess a curved lateral surface due to a time‐varying domain wall velocity. The domain radius r of HZO is estimated to be several nanometers. The small domain size of HZO provides an effective alternative to the NLS model's claim that domain wall motion is confined within a grain. Our model posits that nucleation is instantaneous and domain walls are mobile across grain boundaries, albeit encountering a high Merz barrier. Therefore, a switching paradigm that is limited by domain growth (Figure 3d, top right) is proposed in contrast to the NLS picture (Figure 3d, bottom right), where domain wall mobility is bounded by polycrystallinity and granularity. Our picture is commensurate with a recent electron holography experiment [25], which shows a domain size of a few nanometers and the lateral growth of domains to the neighboring grain, validating the physical foundation of our model. Additionally, multiple PFM studies also emphasized the non‐negligible role of lateral domain growth in the switching in HfO₂‐based ferroelectrics [26, 29, 30]. On the other hand, complementary experimental and computational works report a transition between KAI‐type and NLS‐type switching kinetics via thickness scaling, crystallinity tuning, and temperature control [22, 29, 31, 32] within one material system, indicating a competition in the dominance of nucleation and growth, as well as a potential competition between different nucleation mechanisms. These findings motivate a physics‐based analytical model to cover a continuum of switching kinetics and extract the key parameters governing the interplay between nucleation and growth, rather than relying solely on phenomenological descriptions tied to a single limiting regime.

FIGURE 3.

Time‐dependent voltage‐driven domain nucleation and growth. (a) Representative experimental time‐dependent voltage waveforms across the HZO ferroelectric capacitor. The darker curves deviate more from a square pulse than the lighter curves due to the more pronounced circuit effect. (b) Extracted domain wall velocity (normalized). The domain wall velocity of HZO roughly increases linearly in time. (c) Extracted nucleation rate (normalized) of HZO in comparison with a qualitative nucleation profile derived from the NLS (nucleation‐limited switching) model. (d) Extracted reverse time cones and transformed domain radii. In real materials, the time‐varying domain wall velocity leads to a curved lateral surface of the reverse time cone. In the idealized KAI model (green), the cone has a straight profile due to its limiting assumptions. Our model reveals that the transformed domain can grow to a radius of a few nanometers, which is comparable to the grain size, but is not restricted inside a grain (top right schematic). In contrast, the NLS model assumes that the domain wall is stagnated by the grain boundary (bottom right schematic). The colored areas are the transformed regions.

2.4. Polarization Behavior Under Realistic Complex Circuit Voltages

Nowadays, ferroelectric materials have attracted significant interest for neuromorphic computing as a compute‐in‐memory (CiM) technique to overcome the von Neumann bottleneck, as the stored polarization charge naturally encodes synaptic weights. FeFET (ferroelectric field effect transistor), FeCAP (ferroelectric capacitor), and FTJ (ferroelectric tunnel junction) crossbar arrays are developed for non‐destructive readout of the states [33, 34, 35]. The operation of these devices requires deliberately designed mixed‐signal waveforms to achieve correct and efficient functionality [36]. The previous void of a compact and physical model of polarization switching under dynamic electric field hinders the incorporation of ferroelectric components into circuit‐level modeling and thus optimization. Here we show that the DFNG model can produce realistic material responses to a potentially complex dynamic waveform.

We use the mixed‐signal nature of small‐signal capacitance‐voltage C(V) hysteresis loop measurements as an example (Figure 4), where a low‐amplitude AC signal superimposed on a DC bias is used. This measurement well represents the complexity of the waveform used in real‐world circuits, and the small‐signal capacitance is recently proposed as an efficient readout mechanism for charge‐based neuromorphic computing [37, 38]. In Figure 4a, the model produces the ideal hysteresis, where leakage and internal fields are ignored. It replicates the empirical trend that increased AC frequency leads to a higher coercive field and lower peak capacitance defined by the C(V) loops. Moreover, the polarization (f) and current (df/dt) traces in Figure 4b essentially indicate the accumulated charges and current flow, which are potential readout signals in a circuit with a ferroelectric element. We note that the polarization and current drift during the AC oscillation causes a time‐dependent variation in the readout capacitance values and may compromise the read margin if multi‐level operation is performed on the FeCAP. The variance of capacitance is most significant around the DC voltage where polarization switching occurs most rapidly.

FIGURE 4.

Mixed‐signal simulation: small‐signal capacitance. (a) Simulated frequency‐dependent capacitance‐voltage C(V) hysteresis. Red dots with error bars are output from the model. Grey curves are a guide to the eye. (b) Assumed waveform (V_FE ) used to acquire peak capacitance, transformed fraction (f), and normalized current (df/dt), corresponding to C(V) data points at 0.5 GHz in (a). The first three oscillations in the voltage and current are used to generate mean capacitance values and standard deviations. The model captures C(V) hysteresis as well as empirically known behavior such as the decrease in the DC voltage at peak capacitance with decreasing frequency and decreasing peak capacitance with increasing AC frequency. The model also illustrates that polarization drift may lead to significant variance in C(V) data. The model can handle arbitrary mixed voltages spanning orders of magnitude in time and frequency scales.

2.5. Materials‐To‐Circuit Co‐Design Platform Enabled by the Model

We next present a potential routine for evaluating material eligibility for technologies and accelerating synthesis optimization. From a ferroelectric material synthesis prospective, polarization‐voltage P(V) hysteresis measurement is the most typical diagnosis. We therefore vary the material parameters and map the corresponding P(V) loops in Figure 5a (Note S4 and Figures S4 and S5). It qualitatively shows that decreasing heterogeneous nucleation sites or increasing the depinning barrier leads to wider loops, and that decreasing growth dimension leads to skewed loops. Meanwhile, pulse operation (Figure S6) and high‐frequency ramps, as frequently used waveforms in applications, are performed to extract metrics of interest. We then grade the material parameters following the procedure below and Note S5:

• (1) Determinism under CMOS conditions: 99% switching fraction should be completed within 10 ns (delay) with a pulse amplitude (V_min ) ≤ 1.5 V [4].

• (2) Memory stability against disturb: characterized by the polarization loss fraction Δf under a write disturb pulse in the V_DD /2 scheme [39]. Nominally, Δf ≤ 5% is acceptable for memory devices such as FeRAM (ferroelectric random‐access memory), and Δf ≤ 1% for CiM devices as they encounter more frequent write operations.

• (3) Energy and delay for CiM under write operation.

• (4) Memory window for FeCAPs [40]: defined by normalized peak differential capacitance, . ε _FE is the dielectric constant of the ferroelectric (subtracting the linear part).

• (5) Read margin/Memory window for FeFETs [41]: defined by the coercive voltage, V_C = V_{f = 0.5.}

FIGURE 5.

Mapping material design space to application metrics. (a) Hysteresis loops generated in a segment of model parameter space using a 1 MHz triangular waveform. The same material parameter sets are used to extract application metrics. We grade each loop according to nominal CiM and embedded memory application standards, illustrated in the legend. The values for energy and capacitance are normalized by polarization and capacitor area. (b) The correlation between application metrics for CiM devices. The orange box selects parameter sets that render a relatively large V_C and hence sufficient read margin for FeFETs. The green box selects parameter sets that allow a relatively large C_norm , which is the memory window in FeCAP devices. Orange stars in (b) label a representative parameter set suitable for FeFET and correspondingly in (a) indicate a possible modification direction of the current HZO sample (purple star) for such application. However, modifying it for FeCAPs requires demanding conditions and hence may not be ideal. Notations used: Δf: normalized polarization loss due to V_DD /2 write disturb; C_norm : normalized peak differential capacitance, calculated by ${(\frac{df}{dV})}_{f=0.5}$; V_C : coercive voltage; V_min : minimum pulse amplitude to reach 99% switched fraction within 10 ns.

We locate the loop of the current HZO sample according to its fit parameters (Figure 5a, purple star). This immediately suggests that a modest increase in nucleation density and growth dimension would adapt the material for a CiM application (orange star). Physically, such tuning could be achieved by introducing small dipole clusters and removing some grain boundaries, as charged defects act as heterogeneous nucleation sites while mesoscopic grain boundaries truncate the time cone and reduce growth dimension. The corresponding ranges of energy and delay are indicated by the red dots beside the loop. The orange star matches the loop in Figure 5a to a FeFET application with corresponding metrics in Figure 5b. Furthermore, Figure 5b provides the tunable range and pairwise correlation of the metrics, which allow an inverse design strategy and fast estimation of performance. For example, FeFETs favor materials with large V_C , outlined by the orange box. These selected sets are bound to good control of memory stability and tight distribution of energy dissipation, leaving delay the only variable of concern. Similarly, FeCAP devices favor high capacitances, outlined by the green box. However, achieving these capacitance values requires high growth dimension (d > 1.6) and low Merz barrier (V_a < 4 V), which can be extremely demanding in polycrystalline materials. More discussion is attached in Note S5, Tables S1 and S2. Other general trends can also be concluded. The correlation between V_min , delay, and energy indicates that material parameters enabling lower operating voltages tend to reduce the switching energy, as the switching time decreases exponentially. The relation between Δf and other metrics is less defined as it is subject to a different pulse amplitude, suggesting that disturb immunity cannot be inferred from other measurements and requires a dedicated examination.

To facilitate a materials‐circuit co‐design framework and enable large‐scale circuit‐level modeling, we further establish a SPICE model that incorporates the dynamic ferroelectric elements described by the DFNG model (Figure 6a). On a single device level, the equivalent FeCAP sub‐circuit consists of an intrinsic linear capacitor in parallel with a voltage‐controlled current source representing the ferroelectric elements. Details of the SPICE FeCAP cell model can be found in Note S6 and Figure S7.

FIGURE 6.

Scalable circuit modeling platform with physical material parameters. (a) A schematic of the equivalent circuit of a single ferroelectric device implemented in sub‐circuits in SPICE. The FeCAP SPICE model accurately implements the DFNG model to simulate the dynamics of ferroelectric polarization, as discussed in detail in Note S6. (b) Polarization and total current transients measured experimentally (purple) and simulated by the FeCAP SPICE model (red). (c) Schematic of a 32 × 32 FeCAP crossbar array under the V_DD /2 write scheme (left) and the polarization transients of the selected, half‐selected, and unselected cells in this array (right) produced by SPICE simulations, taking account of realistic parasitic effects in the crossbar. The SPICE simulation verified that the selected cell can be successfully programmed while the half‐selected cell and unselected cells maintain negligible polarization (< 1.4%).

As shown in Figure 6b, the dynamic switching current as well as the polarization state produced by the SPICE FeCAP model accurately match the experimentally measured results under similar waveforms, demonstrating the capability of transitioning the DFNG model to SPICE device models for practical circuit simulations. Building upon this single‐device foundation, the SPICE simulation framework is expanded to evaluate the FeCAP crossbar array architecture that takes into account the dynamic ferroelectric switching behaviors and parasitic effects in the crossbars (Figure 6c, left). The simulation produces a switching delay of 15.6 ns on a 3 × 3 array and 15.7 ns on an expanded 32 × 32 array, due to RC loading effects from the parasitic interconnects. To verify the robustness of programming against write disturb, we show the SPICE simulation results of the polarization states of a selected, half‐selected, and unselected cell in the 32 × 32 array under the standard V_DD /2 write scheme (Figure 6c, right). The polarization switching transients confirm that the target cell can be successfully programmed while the half‐selected cells maintain a negligible polarization of 1.4%. This circuit modeling platform effectively evaluates the performance of both the material and the circuit architecture.

3. Conclusion

While the thriving of complex atomic‐scale computational methods has greatly enriched fundamental understanding of ferroelectric switching mechanisms [31, 42, 43, 44], the DFNG model presented in this work is compact yet able to serve as a complementary framework that links physical parameters related to microscale structure to experimental observables such as realistic switching transients, domain wall motion, and domain size. It enables extraction of material parameters under arbitrary time‐dependent waveforms and fast evaluation of technology metrics. We envision the model to be a powerful tool for performance‐guided materials design and circuit‐level optimization, thereby accelerating the development of next‐generation ferroelectric computing technologies.

4. Methods

4.1. Thin Film Growth

The Hf_0.5Zr_0.5O₂ (W 20 nm/HZO 10 nm/W 50 nm) film was grown using plasma‐enhanced atomic layer deposition on an undoped (001)‐oriented silicon substrate, with the bottom and top tungsten layers deposited by DC magnetron sputtering [45]. The film was crystallized at 600°C for 30 s via rapid thermal annealing in a dynamic N₂ atmosphere at atmospheric pressure. The epitaxial BaTiO₃ (SrRuO₃ 20 nm/BTO 20 nm/SrRuO₃ 50 nm) was grown by pulsed laser deposition on a (110)‐oriented DyScO₃ single crystal substrate. The Al_0.92B_0.08N (TiN 50 nm/AlBN 10 nm/Ti 100 nm) film was grown on low‐doped (100) Si via RF reactive magnetron co‐sputtering [46].

4.2. Ferroelectric Capacitors Fabrication

The HZO and BTO ferroelectric capacitors were fabricated using a combination of E‐beam lithography and optical photolithography. A negative ma‐N 2405 E‐beam resist was spun on the trilayer film stack and exposed using a JEOL JBX‐6300FS electron beam lithography system to serve as an etch mask to define the capacitor size. Then the sample was etched down to the bottom electrode with an ion mill (Nanoquest II) to create self‐aligned top electrode/ferroelectric islands. The sample was backfilled with 140 nm SiO₂ using magnetron sputtering (PVD 75 Proline, Lesker) for low κ isolation between the top and bottom electrodes. Next, a series of photolithography steps were performed to pattern bottom electrodes and lift‐off ground planes. Finally, gold (300 nm)/titanium (10 nm) top contact pads were deposited with an E‐beam evaporator. The AlBN capacitors were fabricated by first depositing Pt top contacts with a shadow mask and then etching the TiN layer with 30% H₂O₂.

4.3. Ultra‐Fast Ferroelectric Switching Measurements

Ultra‐fast measurements were performed in an RF probe station. Ground‐Signal‐Ground probes (Model 40A, GGB) and low‐loss RF cables (SF526S/11PC35, HUBER+SUHNER) were used to transmit waveforms to and receive current response from the device. The voltage pulse train was generated by a pulse generator with a rise time down to 100 ps, connected to the top electrode, while the signal was recorded with a 13 GHz oscilloscope with a 50 Ω input, connected to the bottom electrode. With this setup, the current in the series RC circuit can be determined using the oscilloscope's 50 Ω internal loading. For real‐time voltage monitoring, the high‐impedance probes were landed at the top and bottom electrodes to capture the differential voltage across the capacitor. All measurements were conducted in real‐time with averaging over three single‐shot measurements. To probe the ferroelectric switching transients, a reset pulse with negative polarity was first applied, followed by a positive‐up (PU) pulse train. Each pulse width was 1 µs. The ferroelectric current was then extracted by subtracting the current during the U pulse (dielectric current and resistive current) from the current during the P pulse (ferroelectric current, dielectric current, and resistive current).

4.4. Numerical Model and Fitting

Equations (1)–(3) and (5) and Equation S3 were used to generate f and df/dt as a function of time t and ln (− ln (1 − f)) as a function of ln t. Time and real‐time voltage involved in the equations were experimental values. The slope of the dataset (ln (− ln (1 − f)), ln t) is the classical Avrami exponent. The sum of squared errors between model‐generated outputs and experimental measurements under four applied voltage waveforms was minimized using a differential evolution algorithm to obtain the global optimum parameter set. After convergence, the Jacobian matrix of residuals with respect to the fitted parameters was evaluated, and the associated covariance matrix was obtained via nonlinear least‐squares refinement. The fit error was determined as the square root of the diagonal elements of this covariance matrix. We added a weight of 2 to the data points during 0–500 ps for HZO and BTO, as the experimental data points are taken every 7 ps and the experimental voltage rise time is ∼200 ps, to reduce the intrinsic bias of the least‐squares fit at low time and f values. This weight was chosen to not overbias this region as well. For AlBN, a weight of 10 000 is added to the df/dt data, as the original values are 10 000 times smaller than the other datasets. This weight is to ensure all datasets contribute comparably to the overall fit. An analysis of the weighing method is provided in Note S7 and Figure S8.

Funding

This work was funded by the Intel Corporation FEINMAN 2.0 program and the ONR grant N000142612047.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File: adma73722‐sup‐0001‐SuppMat.docx.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request. A preprint version of this work (https://arxiv.org/abs/2602.02957) has previously been made publicly available.