A Verilog-A-Based Redox-Signal Transduction Model for Co-Simulating Surface-Bound Electrochemical Biosensors and Circuits

Abstract

Surface-bound electrochemical aptamer-based (E-AB) sensors are a promising approach for continuous in-vivo and in-vitro biomolecular monitoring because they offer high selectivity, sensitivity, and real-time detection. However, accurately co-simulating E-AB sensors with readout circuits remains challenging due to the redox reporter’s position-dependent electron-transfer kinetics and the electrical double layer’s (EDL) complex behavior at the electrode-electrolyte interface. Here, we present a compact, SPICE-compatible electrochemical cell model that combines a Verilog-A implementation of the Marcus-Hush-based electron-transfer (ET) kinetics with a fractional-order RC-ladder representation of the EDL’s non-ideal capacitance. The conventional Butler-Volmer model is replaced by Marcus-Hush kinetics, which features bounded and quantum mechanically derived ET rate constants, improving not only the model’s physical interpretability but also numerical stability in circuit simulations. The model was validated with two E-AB sensors using square-wave voltammetry (SWV) across a range of excitation frequencies and target concentrations to confirm that the simulated transient currents accurately capture ET kinetics, thermodynamics, and the Langmuir isotherm’s concentration response. When co-simulated with a transimpedance amplifier constructed with the TI OPA4354, the model produced electronic noise spectra that more closely matched experimental data when compared with spectra simulated using the simplified Randles circuit model. These results demonstrate that the proposed model provides a physically grounded framework for simulating surface-bound redox-based electrochemical biosensors and enables accurate co-simulation with readout circuits.

I. Introduction

Aptamers are single-stranded oligonucleotides that can be evolved to bind with a broad range of targets, from small-molecule drugs and metabolites to proteins and viruses, with high affinity and specificity [1], [2], [3], [4], [5]. These aptamers can be further engineered to undergo a reversible conformational change upon target binding [6], [7], [8], which can be transduced into an electrical signal by attaching a redox reporter, such as methylene blue (MB) or ferrocene (Fig. 1(a)). Target-induced structure switching modulates the average electron transfer (ET) rate of the electrochemical aptamer-based (E-AB) sensor, and, consequently, the measured redox current. Compared to homogeneous assays, where aptamers are freefloating in the biological medium, heterogeneous E-AB sensors feature aptamers immobilized on the electrode surface, making them more suitable for in-vivo and in-vitro deployment. As such, heterogeneous E-AB sensors have been applied to implantable devices [9], [10] and wearable platforms, such as sweat sensors [11] and microneedles [12], [13], as well as diagnostic assays [14] and bioreactor monitors [8].

Fig. 1.

(a) Signal transduction of structure-switching E-AB sensors; (b) Conventional application of simplified Randles circuit model for simulating readout circuits.

Designing readout circuits for E-AB sensors is challenging because the redox current is low, limited by the approximately 1% aptamer packing density on the electrode surface [15], and detection requires sensitivity to subtle variations in the ET rate. Moreover, our prior analysis has shown that in low-drift biosensors, the detection limits are dominated by electronic noise rather than chemical shot noise [16]. It is therefore crucial to design electrochemical readout circuits with low-noise, low-power, and adequate dynamic range. Achieving such a goal requires accurate co-simulation of the electrochemical cell and readout circuit. However, conventional design flows generally employ a simplified Randles equivalent circuit to represent the electrochemical cell between a potentiostat and the currentreadout analog front-end (AFE) (Fig. 1(b)). The simplified Randles circuit model, akin to a Voigt circuit, includes a constant charge-transfer resistance ($R_{ct}$), which represents the linearized redox current, placed in parallel with an ideal capacitor ($C_{dl}$) that models the behavior of the electrical double layer (EDL), which are in series with a resistor ($R_s$) that captures the conductivity of the ionic biofluid. Although the simplified Randles circuit provides a useful first-order approximation, it deviates significantly from the physical behavior of the electrochemical cell due to the following reasons.

First, the redox labels conjugated to the E-AB sensors have distinct oxidation and reduction potentials, resulting in a potential-dependent redox current that cannot be captured by the linear current-voltage relationship produced by $R_{ct}$ (Fig. 2(a)). Since this potential dependence forms the basis of many voltametric techniques, including square-wave voltammetry (SWV), an electrochemical cell model that accurately captures this behavior during simulation is essential for circuit optimization and serves as the main motivation of this work. As a note, SWV is one of the most common methods for interrogating E-ABs [17]. In SWV, a stepwise, rising potential with a superimposed square wave at a user-defined frequency $\left(f_{SWV}\right)$ is applied between the working electrode (WE) and reference electrode (RE) (Fig. 3(a)). The principle is similar to cyclic voltammetry (CV), but rather than linearly sweeping the potential, a series of non-monotonic potential steps are applied. The applied potential causes the redox reporter to undergo an oxidation or reduction reaction, and the resulting redox current, as a function of the applied potential, is shown in Fig. 3(b). The redox current is sampled at the end of each half-cycle after the exponentially decaying capacitive currents have largely decayed, a process akin to chronoamperometry (CA). The sampled points produce the forward and reverse voltammograms, and the net SWV voltammogram is the difference between the two (Fig. 3(b)) [18]. Our aim is to accurately model the potential-dependent ET kinetics observed in SWV and to correlate the model’s critical parameters to the physical behavior of E-AB sensors across different target concentrations. Second, the EDL impedance exhibits non-ideal impedance and phase, producing a flattened Nyquist plot shown in Fig. 2(b), which cannot be represented by a single ideal capacitor $\left(C_{dl}\right)$. Since the charging and discharging dynamics of the EDL directly affect the settling behavior of SWV redox currents, oversimplification causes the sampled currents to deviate from their true values and leads to inaccurate simulations. These shortcomings of the simplified Randles model highlight the need for a circuit-simulator-compatible electrochemical model that captures the E-AB sensor’s large and small signal properties.

Fig. 2.

Limitations of the simplified Randles circuit in modeling (a) redox reaction I-V curves and (b) EDL’s frequency-dependent impedance.

Fig. 3.

(a) SWV potential applied between WE and RE; (b) Resulting redox current and the voltammogram sampling and extraction flow.

It is worth mentioning that multiphysics simulations have been utilized to capture the electrochemical cell’s behavior in detail [19], but these models are incompatible with SPICE-based circuit simulators. Regarding the EDL, a constant-phase element (CPE) has been used to replace $C_{dl}$ to improve frequency-domain accuracy [20], [21], [22], but the model does not include non-linear ET responses. On the other hand, analog circuit realizations of the CPE and ET kinetics have been demonstrated [23], but their device-level characteristics are process-dependent and must be re-tuned when migrating across technology nodes. Recent models address ET kinetics by coupling partial differential equations (PDE) from Fick’s second law of diffusion with the Butler-Volmer (BV) formalism [24]. For example, Jog [25] developed a generalized Verilog-A model that solves the associated PDE using a finite difference method (FDM) with discretized spatial and temporal steps. However, for high-frequency voltammetry over large voltage ranges, very small time-steps are required to maintain model stability and accuracy, which comes at the cost of drastically increased simulation times.

In this work, we present a compact electrochemical cell model for heterogeneous electrochemical reactions that is compatible with both frequency-domain and transient SPICE-class circuit simulators. The models expands upon the simplified Randles circuit by incorporating ET kinetics through a Verilog-A implementation of the Marcus-Hush (MH) formalism, a model better suited for heterogeneous E-AB biosensors, and by representing the non-ideal EDL capacitor with a fractional pole 13th-order RC-ladder (Fig. 4). We validated the model using experimental E-AB measurements across a wide range of frequencies, applied voltages, and target concentrations, and compared noise simulations obtained from the model with measurements from a custom transimpedance amplifier (TIA). To our knowledge, this is the first work to compare circuit noise simulations using a full electrochemical-cell model against those using a simplified Randles representation while incorporating physicsinformed potential-dependent redox reactions. The framework also generalizes to other techniques beyond SWV, including CV and CA.

Fig. 4.

(a) Proposed electrochemical cell model with the (b) Marcus-Hush Verilog-A ET model and (c) 13th-Order RC-ladder CPE model.

The remainder of this paper is organized as follows: Section II details ET kinetics modeling, Section III describes the full electrochemical cell’s model, Section IV compares simulation results with experimental measurements, and Section V provides concluding remarks.

II. Modeling Electron Transfer Kinetics

Electrochemical redox reactions occur when electrons are transferred between a redox-active species and electrode in response to an applied driving potential, akin to a voltage-controlled current-source (VCCS). Near the equilibrium potential, the population of reduced and oxidized species is balanced, enabling maximal ET and producing the current peaks observed in SWV and CV. At more negative potentials, the species becomes predominantly reduced, whereas at more positive potentials, they are predominantly oxidized. In the following sections we first review conventional electrochemical models using Fick’s second law and the BV formulation, and then introduce the MH model, which is more physically relevant to E-AB sensor and more readily implemented in Verilog-A.

A. Advantages of the Marcus-Hush Model

As mentioned, previous works have paired Fick’s second law with the BV formalism to model ET kinetics [24], [25]. Fick’s second law of diffusion is

$$\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x^2},$$ (1)

where C(mol/m³) and D(m²/s) are the concentration and diffusivity of the redox species, respectively. Fick’s law describes the diffusion of redox species for homogenous (solution-phase) and heterogeneous (surface-bound) reactions and leads to a second-order PDE. Solving this PDE implicitly in Verilog-A is challenging, and while explicit integrators can be implemented, they are prone to numerical instability unless the spatial and temporal steps are discretized to be very small, which significantly increases simulation time [25]. The key observation is that for heterogeneous E-AB sensors, the redox species are surface-bound and do not require the mass-transport mechanisms described by Fick’s law, making the PDE unnecessary.

Moving onto the BV model, the BV ET rate expressions for reduction and oxidation are

$$k_{red}^{BV}=k_0\text{exp}\left[\frac{-\alpha nF\eta }{RT}\right]$$ (2)

$$k_{ox}^{BV}=k_0\text{exp}\left[\frac{(1-\alpha )nF\eta }{RT}\right],$$ (3)

where $k_0(\approx 100)$ is the standard ET rate constant $\left(s^{-1}\right),\alpha (\approx 0.5)$ is the ET coefficient, $n$ is the number of electrons transferred per redox molecule, $F$ is the Faraday constant, $\eta$ is the overpotential (V), $R$ is the ideal gas constant, and $T$ is temperature (K) [26]. In the BV model, either Eq. (2) or (3) will grow exponentially without bound with overpotential $\left|\eta \right|=\left|E-E_0\right|$ (V), where $E$ is the applied potential between WE and RE, and $E_0$ is the formal potential of the redox species [27], [28]. Under large overpotentials or high-frequency interrogation, this exponential growth can lead to unphysically large rates and numerical instability in transient circuit simulations, which is the major limitation of the BV formulation.

As an alternative to the BV model, the MH formalism applies Marcus ET theory to predict the probability of ET based on molecular reorganization and Fermi-Dirac statistics [29]. It has been shown to accurately describe heterogeneous redox reactions, including those relevant to E-AB sensors, and provides physical insight because its parameters are rooted in quantum-mechanical descriptions of reorganization energy, driving force, and electronic coupling [30], [31], [32]. The MH ET rates are given by

$$k_{red/ox}^{MH}=k_0\text{exp}\left[-\frac{(\lambda \pm \text{nF}\eta {)}^2}{4\lambda RT}\right],$$ (4)

where $\lambda$ is the reorganization energy required to bring reactants and products into the same nuclear configuration (J/mol) [26], [27]. Unlike the BV expression, Eq. (4) predicts a maximum ET rate at high overpotentials because once the driving force exceeds the reorganization energy, additional bias does not further accelerate the reaction [26]. This physically-rooted saturation behavior avoids unbounded exponential growth and improves numerical stability in circuit simulations. Fig. 5 compares the two models on a Tafel plot, where $I\propto k_{red}-k_{ox}$ is normalized to the exchange current, $I_0$, found at $\eta =0$, illustrating the divergence between the two models at high $\left|\eta \right|$. Specific to the MH model, the standard rate constant, $k_0$, is also a function of the distance between the redox site and electrode [33], [34], making the MH framework especially applicable for representing the structure-switching transduction behavior of heterogeneous E-AB sensor.

Fig. 5.

Tafel plot comparing normalized-log current produced by the Butler-Volmer (blue) and Marcus-Hush (red) ET models.

B. Marcus-Hush Electron Transfer Model

We now derive the MH ET model and explain how it relates to our Verilog-A implementation. In an n-electron heterogeneous redox reaction,

$$O+n{e}^{-}\rightleftharpoons R,$$ (5)

with oxidized species (O) and reduced species (R), the total redox current is given by

$$I=-nFA\frac{d{\Gamma }_{\text{R}}}{dt}$$ (6)

$$\frac{d{\Gamma }_{\text{R}}}{dt}=k_{red}{\Gamma }_o-k_{ox}{\Gamma }_{\text{R}},$$ (7)

where $A$ is the electrode area ($m^2$) and ${\Gamma }_R$ and ${\Gamma }_O$ are the surface-concentrations of reduced and oxidized species, respectively, within the Debye layer near the electrode $\left(mol/m^2\right)$. As expected, Eq. (7) shows that the surface-concentration of reduced species is determined by the difference between the reduction and oxidation rates. For surface-bound E-AB sensors, the concentration of redox species is conserved when converted between O and R, so the total surface concentration must satisfy

$${\Gamma }_{tot}={\Gamma }_O+{\Gamma }_R.$$ (8)

Therefore, Eq. (7) can be expressed as a 1-D ordinary differential equation (ODE) [35]

$$\frac{d{\Gamma }_{\text{R}}}{dt}=k_{red}\left({\Gamma }_{tot}-{\Gamma }_{\text{R}}\right)-k_{ox}{\Gamma }_{\text{R}}.$$ (9)

To further capture the asymmetric redox behavior arising from differences in reduction and oxidation potentials, which is commonly observed in experiments, we separate Eq. (4) for reduction and oxidation and implement a simplified form of the asymmetric MH model [27], [36]. Specifically, the reduction and oxidation rate constants,

$$k_{red}=k_{0,red}\text{exp}\left[-\frac{{\left({\lambda }_{red}+\text{nF}\eta \right)}^2}{4{\lambda }_{red}RT}\right]$$ (10)

$$k_{ox}=k_{0,ox}\text{exp}\left[-\frac{{\left({\lambda }_{ox}-\text{nF}\eta \right)}^2}{4{\lambda }_{ox}RT}\right],$$ (11)

have distinct pre-factors ($k_{0,red},k_{0,ox}$) and reorganization energies (${\lambda }_{red},{\lambda }_{ox}$), which are fitted to electrochemical measurements. Although this technique does not fully represent asymmetric MH-theory, as it breaks the de Doner relation [28],

$$\frac{k_{red}}{k_{ox}}=\text{exp}\left[\frac{-nF\eta }{RT}\right],$$ (12)

it significantly reduces model complexity for Verilog-A implementation while introducing extra degrees of freedom for fitting to experimental data. Note that the reported values of ${\lambda }_{red},{\lambda }_{ox}$ are expressed in eV, corresponding to eV/molecule, and can be converted from J/mol by dividing by Faraday’s constant.

C. Verilog-A Implementation

To enable cell-circuit co-simulation, we implement the Marcus-Hush model in Verilog-A to create a VCCS that accurately reproduces the redox currents of E-ABs during voltammetry. To solve the redox species’ time-dependent dynamics with high temporal resolution, Eq. (9) is expressed as a continuous-time differential equation and integrated using the simulator’s native differential-algebraic equations (DAE) solver [37]. Prior to integration, ${\Gamma }_R$ is initialized to its steady-state solution

$${\Gamma }_R={\Gamma }_{tot}\frac{k_{red}}{k_{red}+k_{ox}},$$ (13)

which is obtained by setting $d{\Gamma }_R/dt=0$. To reliably integrate differential equations in Verilog-A, it is important to understand the capabilities of SPICE simulators. Most SPICE simulators are architected to solve Kirchhoff’s current law (KCL) equations using modified nodal analysis (MNA), which effectively represents the circuit as a system of coupled DAE of the form

$$\mathit{f}[\mathit{x}(t)]+\frac{d}{dt}\mathit{q}[\mathit{x}(t)]=\mathit{u}(t),$$ (14)

where $\mathit{x}\left(t\right)$ is the vector of unknowns, $\mathit{f}$ is the current through memoryless branches of the circuit, $d\mathit{q}/dt$ is the current through dynamic elements (e.g. capacitors and inductors), and $\mathit{u}\left(t\right)$ is the input stimulus [38]. To build a compact electrochemical model suitable for co-simulation with already computationally intensive mixed-signal systems, we strategically minimized the number of additional variables included in $\mathit{x}\left(t\right)$ by only employing time-derivative operators, as opposed to Verilog-A’s built-in integration operator $idt\left(\right)$. We formulate the integral with time-derivative operators by representing the surface concentration $\left({\Gamma }_R\right)$ as a voltage across a 1-farad unit capacitor. The simulator naturally enforces KCL at this node, defines the current as

$$I\equiv C\frac{dV}{dt}=1\frac{d{\Gamma }_{\text{R}}}{dt},$$ (15)

and integrates the DAE to solve for $V\left(t\right)\equiv {\Gamma }_R\left(t\right)$. Since the capacitor natively participates in both the DC and AC operating point calculations, the Verilog-A model is consistent across transient and AC simulations. This capacitor-based integration technique leverages the simulator’s native enforcement of KCL to avoid the need for $idt\left(\right)$ or finite-difference integration schemes and to ensure more stable and efficient simulation results. The code snippet in Fig. 6 shows our Verilog-A implementation.

Fig. 6.

Verilog-A implementation of the Marcus-Hush ET model.

The MH model parameters depend on the properties of the redox reporter and the aptamer sequences. These include the surface density of immobilized aptamers, the ET rates, and the average distance of the redox reporter from the electrode. For surface-bound structure-switching aptamer sensors conjugated to a single MB redox reporter, the surface concentration of redox species can be calculated as follows: at a 1% aptamer surface density, a $250\mu \text{m}$ diameter gold WE with an exposed surface area of $A=0.049{\text{mm}}^2$ will yield $1\text{MB}/100{\text{nm}}^2$, or a total surface concentration of ${\Gamma }_{\mathit{\text{tot}}}=16\times {10}^{-9}\text{mol}/{\text{m}}^2$. MB undergoes a two-electron redox process with a formal potential around $E_0=-0.35V$ vs. Ag/AgCl RE [39]. In addition to these quantities, the MH model requires four key parameters, $k_{0,red},k_{0,ox},{\lambda }_{red},{\lambda }_{ox}$, which are extracted by fitting the MH model to experimental data using nonlinear least-squares regression. To isolate the faradic currents described by the MH model from the non-faradic contributions, we sample the redox current at the end of each half-cycle to construct standard SWV I-V voltammograms, which are then used for model fitting.

To operate the electrochemical cell model in circuit simulators, a voltage source is required to control the potential difference between WE and RE. In most electrochemical sensors, the WE potential $\left(V_{WE}\right)$ is held at a fixed bias by the readout circuit, while the RE potential $\left(V_{RE}\right)$ is swept by the potentiostat. To replicate this behavior, parameterized Verilog-A voltage sources were developed to generate different voltammetry waveforms, including those for SWV, CV, and CA.

D. Model Range and Robustness

Physics-based models can risk being overfit, causing the parameter values to be disconnected from the underlying phenomena that they represent. To verify the model’s stability and clarify the role of each parameter, we independently varied $E_0,k_{0,ox/red}$, and ${\lambda }_{ox/red}$ over wide ranges while holding all other parameters constant. The model was driven with the Verilog-A SWV voltage source, and the resulting current was measured. The current values at the end of each half-period were sampled and processed into the net SWV voltammograms.

Simulation results in Fig. 7(a) show that the SWV peak shifts proportionally with $E_0$ and remains centered near the formal potential where the redox process is most favorable. Fig. 7(b) varies $k_{0,ox/red}$ together. Increasing $k_{0,ox/red}$ raises the SWV peak amplitude without changing its width, which is consistent with MH theory, where higher rate constants increase the probability of ET events and thus produce larger current. Fig. 7(c) varies ${\lambda }_{ox/red}$ together and shows that larger ${\lambda }_{ox/red}$ values reduce the peak amplitude while broadening its width. This behavior is congruent with MH theory, because larger reorganization energies (${\lambda }_{ox/red}$) extend the potential window over which ET occurs, thereby decreasing the maximum ET probability at the $E_0$.

Fig. 7.

SWV voltammograms simulated using the Marcus-Hush Verilog-A model with varying (a) formal potentials $E_0$, (b) rate constants $k_0$, and (c) reorganization energies $\lambda$.

E. Other Voltammetry Techniques

To illustrate the model’s generalizability to other voltammetries, we applied simulated CV and CA potentials and vary $k_{0,ox/red}$ to emulate changes in the ET rate. For E-AB sensors, adjusting $k_{0,ox/red}$ is analogous to modeling concentration-dependent conformational changes. Fig. 8(a) shows simulated CV responses for a potential sweep from −0.6 V to −0.1 V at 5 V/s. The model produces the characteristic forward and reverse CV peaks, with peak separation changing systematically as $k_{0,ox/red}$ increases. This trend mirrors CV measurements reported by Pellitero [40] for E-AB sensors.

Fig. 8.

(a) CV I-V curve and (b) CA transient plot simulated using the Marcus-Hush Verilog-A model with varying rate constants $k_{r/o}$.

Similarly, Fig. 8(b) shows simulated CA responses for potential steps between −0.3 V and −0.5 V. The resulting current exhibits changes in amplitude and RC-decay rate as a function of $k_{0,ox/red}$, consistent with experimental measurements of E-AB sensors interrogated with CA in [16], [41]. These results indicate that the model can reproduce key signal-transduction behavior in other voltametric techniques and could be applied to fit CV and CA measurements for quantitative analysis.

III. Modeling the Electrochemical Cell

Two electrode systems consist of a working electrode (WE) and a combined counter/reference electrode (RE/CE), while three electrode systems decouple the RE and CE. In either configuration, the electrochemical interface at the WE can be modeled with the simplified Randles circuit [26]. For a fully-symmetric electrode system, Lai [42] demonstrated that a single simplified Randles circuit can adequately capture the cell’s frequency response. Building on this foundation, our objective is to integrate the proposed MH-based ET kinetics with a circuit model that accurately represents the EDL impedance, thereby creating a complete electrochemical cell model suitable for co-simulation with electronic circuits.

A. Fundamentals of Electrode-Electrolyte EDL

At the electrode-electrolyte interface, the EDL arises from the separation of charge between the electrode surface and the surrounding ionic solution. In the case of the E-AB sensor, the inner Helmholtz plane is shaped by the self-assembled monolayer, while the outer region is formed by the aptamers, targets, adsorbed counter-ions, and polarized molecules (Fig. 9). When the applied potential changes, ions within a few nanometers of the surface redistribute to re-establish electrochemical equilibrium. This redistribution stores energy electrostatically but, unlike an ideal capacitor, involves a distribution of relaxation times due to variations in ion mobility, binding interactions, and interfacial geometry. The resulting impedance ($Z_{dl}$) exhibits a nearly constant phase angle, as observed in experimental Nyquist plots, and produces a non-faradaic charging current that decays more slowly than the exponential response predicted by an ideal capacitor $\left(C_{dl}\right)$. This behavior is modeled using the constant phase element (CPE), which has an impedance

$$Z_{CPE}=\frac{1}{Q_{dl}{\omega }^n}\text{exp}\left(-j\frac{\pi }{2}n\right),$$ (16)

where $Q_{dl}$ is the pseudo-capacitance $\left(F\cdot s^{n-1}\right),n$ is a dimensionless exponent between 0 and 1, and $\omega$ is angular frequency (rad/s) [43]. We highlight that when $n=1$, the CPE simplifies to an ideal capacitor. Because the electrode-electrolyte’s interfacial impedance often dominates the input load seen by currentsensing front-ends, accurate representation is essential for predicting noise and transient responses. In our full cell model, the electrolyte resistance ($R_s$) is in series with the parallel combination of $Z_{dl}$ and the faradaic current branch ($Z_{ct}$), which is implemented with the aforementioned MH-based VCCS [22], [26]. In the Randles circuit, a Warburg element is occasionally included in series with the ET element to account for the diffusion-limited transport of freely diffusing redox species. However, heterogeneous E-AB sensors are not diffusion-limited so the Warburg element can be omitted.

Fig. 9.

EDL structure at the electrode-electrolyte interface and its equivalent circuit representation.

B. Implementing the Constant Phase Element

To implement the CPE in a circuit simulator, we propose a compact model by using a 13th-order RC-ladder network (Fig. 4(c)) that approximates the fractional-order impedance over the frequency range of interest. As a note, the RC-ladder represents the electrode’s overall EDL impedance, rather than a combination of individual sub-segments for capturing spatial inhomogeneity. Although the CPE can also be implemented directly in Verilog-A, Hou [44] highlights the challenges of implementing the CPE’s irrational fractional transfer function in Verilog-A, leading them to devise an adaptive algorithm that fits the CPE to a rational function approximation. For user simplicity, we opted for the simpler RC-ladder formulation to take advantage of the Fracpole Suite [45], which automatically generates an optimized RC-ladder network using parameters extracted from electrochemical impedance spectroscopy (EIS).

The modeling workflow begins by performing EIS on our aptamer-functionalized and MCH-passivated electrodes to characterize impedance and phase from 1 Hz to 1 MHz, which spans the bandwidth of most electrochemical sensors. The EIS data is then fitted to the equivalent circuit shown in Fig. 10, to extract $Q_{dl},n$, and $R_S$. Based on the extracted CPE values, the Fracpole Suite is then used to synthesize the 13th-order RC-ladder, which was selected to improve accuracy by providing two pole-zero pairs per decade across the six-decade frequency range. We also tested RC-ladders with lower orders, such as those with one and one-and-a-half pole-zero pairs per decade, but they introduced noticeable ripples in frequency-domain impedance simulations. For the $250\mu \text{m}$ diameter gold WEs, the RC-ladder’s component values are listed in Table I, and the extracted values of $Q_{dl},n$, and $R_S$, along with a comparison of the measured and modeled Bode plots, are shown in Fig. 10. The 13th-order RC-ladder-based model closely matches both the electrode’s measured EIS spectrum and a separately fitted simplified Randles circuit employing an ideal CPE, achieving coefficients of variation of 3.08% for impedance and 2.44% for phase.

Fig. 10.

(a) Impedance and (b) phase frequency responses comparing experimental EIS measurement (red) to data modeled with an ideal CPE (green) and the 13th-order RC-ladder-based CPE (blue).

TABLE I.

Capacitor and Resistor Values Used in the 13th-Order CPE Depicted in Fig. 4(c)

Capacitor (nF)		Resistor ($𝛀$)
${\mathbf{C}}_{\mathbf{0}}$	1.653	${\mathbf{R}}_{\mathbf{0}}$	35.127 × 106
${\mathbf{C}}_{\mathbf{1}}$	0.258	${\mathbf{R}}_{\mathbf{1}}$	195.117 × 106
${\mathbf{C}}_{\mathbf{2}}$	0.303	${\mathbf{R}}_{\mathbf{2}}$	49.707 × 106
${\mathbf{C}}_{\mathbf{3}}$	0.296	${\mathbf{R}}_{\mathbf{3}}$	15.299 × 106
${\mathbf{C}}_{\mathbf{4}}$	0.276	${\mathbf{R}}_{\mathbf{4}}$	4.931 × 106
${\mathbf{C}}_{\mathbf{5}}$	0.254	${\mathbf{R}}_{\mathbf{5}}$	1.610 × 106
${\mathbf{C}}_{\mathbf{6}}$	0.233	${\mathbf{R}}_{\mathbf{6}}$	527.903 × 103
${\mathbf{C}}_{\mathbf{7}}$	0.214	${\mathbf{R}}_{\mathbf{7}}$	173.291 × 103
${\mathbf{C}}_{\mathbf{8}}$	0.196	${\mathbf{R}}_{\mathbf{8}}$	56.932 × 103
${\mathbf{C}}_{\mathbf{9}}$	0.179	${\mathbf{R}}_{\mathbf{9}}$	18.737 × 103
${\mathbf{C}}_{\mathbf{10}}$	0.162	${\mathbf{R}}_{\mathbf{10}}$	6.204 × 103
${\mathbf{C}}_{\mathbf{11}}$	0.144	${\mathbf{R}}_{\mathbf{11}}$	2.111 × 103

IV. Experimental Results and Model Verification

The proposed electrochemical cell model was validated using aptamer-based biosensing measurements across a wide range of voltammetry frequencies and target concentrations with a commercial potentiostat. To demonstrate the model’s ability to predict electronic noise spectra, a transimpedance amplifier (TIA) was implemented with TI’s OPA4354, and the co-simulated noise spectra were compared with experimental measurements. All simulations were performed using the Cadence Spectre circuit simulator [37].

A. Aptamer-Based Biosensor Device Preparation

The E-AB biosensors were prepared with gold WEs functionalized with either an 8-base pair anti-tetracycline aptamer (5’-HS-(CH2)6-CGG-TGG-TG-(CH2)6-NH-MB-3’) [46] or a 26-base pair anti-kanamycin aptamer (5’-HS-(CH2)6-GGG-ACT-TGG-TTT-AGG-TAA-TGA-GTC-CC-(CH2)7-NH-MB-3’) [2], [16]. The immobilization protocols are adapted from [16] and proceed as follows: 1) WEs were prepared from $250\mu \text{m}$ diameter gold wire (Thermo Fisher Scientific 000725-G1) insulted with heat-shrink tubing, and the area was defined by exposing the cross-section through razor blade cutting; 2) The electrodes were solvent cleaned via sequential sonication for five minutes each in bleach, acetone, isopropanol, and deionized (DI) water; 3) Following this, the electrodes were electrochemically cleaned by cycling in 200 mM H₂SO₄ with 10 CV scans at 1 V/s from −0.2 V to 1.5 V, followed by 50 mMH₂SO₄ with 3 CV scans at 0.1 V/s from −0.2 V to 1.5 V, and then rinsed with DI water; 4) The aptamers were prepared by mixing and incubating $1\mu \text{L}$ of $100\mu \text{M}$ aptamer solution with $2\mu \text{L}$ of 28.7 mg/mL Tris(2-carboxyethyl) phosphine (TCEP) for 40 minutes, and then diluting the solution to $1\mu \text{M}$ with 1x saline sodium citrate (SSC) buffer; 5) The WEs were incubated in the $1\mu M$ aptamer solution for 1 hour at room temperature in the dark; 6) Finally, we passivate the WEs’ surface by incubating them in 6-mercapto-1-hexanol (MCH) diluted to 6 mM with DI-water for 2 hours at room temperature, and store the WEs in 1x SSC at 4°C for 12 hours to stabilize the self-assembled monolayer [47].

Electrochemical measurements were carried out using a PalmSens4 potentiostat. The built-in SWV measurement mode automatically extracts the current-voltage voltammogram. To directly measure the transient redox currents as a function of time, SWV was implemented using a multi-step amperometry (MSA) protocol ranging from −0.6 V to −0.1 V with a 4 mV step size and a 36 mV pulse amplitude.

B. Varying SWV Frequencies

SWV measurements at $f_{SWV}=10\text{Hz},250\text{Hz}$, and 500 Hz, were performed on WEs functionalized with anti-tetracycline aptamers. Fig. 11 compares the measured and simulated transient redox currents at $f_{SWV}=250\text{Hz}$ and shows the Verilog-A model closely matches the experimental results over the entire 0.5 V scan range. Due to the potentiostat’s finite sampling rate of 1kS/s, the measurements are unable to capture the fast transient spikes at the start of each half-cycle. Nevertheless, the slower RC-settling components are clearly captured and closely matches the model, as highlighted in the zoomed-in plot.

Fig. 11.

Modeled (blue) and measured (red) SWV redox current transients at $f_{swv}=250\text{Hz}$. Zoomed in region details five periods.

For functional verification and interpretability, we sampled the transient redox currents at the end of each half-cycle to generate the SWV voltammogram (I–V). This approach follows the same data-processing workflow as the PalmSens4’s native SWV mode and provides the effective analytical output of SWV for comparing modeled and measured responses across all three frequencies. These results, shown in Fig. 12, demonstrate that the model accurately reproduces the peak amplitudes and widths observed experimentally. The normalized root mean square error (NRMSE) and coefficient of determination $\left({\text{R}}^2\right)$ were 6.06% and 0.73 at 10 Hz, 5.24% and 0.84 at 250 Hz, and 3.67% and 0.90 at 500 Hz, respectively.

Fig. 12.

Modeled (blue) and measured (red) SWV voltammograms at (a) $f_{swv}=10\text{Hz}$, (b) $f_{swv}=250\text{Hz}$, and (c) $f_{swv}=500\text{Hz}$.

For each SWV frequency, the MH parameters $k_{0,ox/red}$ and ${\lambda }_{ox/red}$ are fitted to the measurements, with the values summarized in Fig. 12(a)–(c). The rate constants, $k_{0,ox/red}$, set the speed of ET, thus governing both the transient responses and current amplitudes. The reorganization energies, ${\lambda }_{ox/red}$, control the width and symmetry of the voltametric peaks. Notably, ${\lambda }_{ox/red}$ vary slightly with frequency due to its dependence on the dielectric constants of the solvent and self-assembled monolayer [26], which themselves exhibit frequency-dependent behavior because of ion redistribution [48].

C. Varying Biomolecular-Target Concentrations

SWV measurements at $f_{SWV}=250\text{Hz}$ were performed on electrodes functionalized with anti-kanamycin aptamers, and titration experiments were carried out to increase the kanamycin-target concentration from 0 mM to 8 mM. The transient redox currents were recorded using MSA and processed into SWV voltammograms. As shown in Fig. 13(a), the SWV peak amplitude increases with target concentration, consistent with the signal-on response of anti-kanamycin aptamers, where target binding alters the diffusion distance between redox labels and the WE and thereby modulates the rate constants, $k_{0,ox/red}$, in our MH model. To preserve the model’s physical interpretation, only $k_{0,ox/red}$ were varied across concentrations, while all other parameters are held constant. Fig. 13(a) shows that our model accurately reproduces the observed signal transduction.

Fig. 13.

Modeled (blue) and measured (red) (a) SWV voltammograms $(f_{swv}=250\text{Hz})$ at various kanamycin antibiotic target concentrations, and (b) the corresponding Langmuir Isotherm plot.

Fitting the SWV voltammograms to the Langmuir isotherm equation yields nearly identical dissociation constants for the modeled and measured data: $K_d=1.79\text{mM}$ and 1.78 mM, respectively (Fig. 13(b)). Fig. 14 shows that the average rate constant, $k_{0,avg}=\left(k_{0,ox}+k_{0,red}\right)/2$, at different target concentrations follows a similar Langmuir dependence. The trend is expected since the SWV peak amplitude is directly related to $k_{0,ox/red}$, further supporting the physical interpretation of the model.

Fig. 14.

Average of the ET rate constant $\left(k_{avg}\right)$ used to fit the Marcus-Hush model at various concentrations compared to a Langmuir fit.

D. Modeling Electrical Noise

A single-ended TIA was constructed using a Texas Instruments (TI) OPA 4354 operational amplifier (Fig. 15). The inverting input was connected to a WE functionalized with anti-tetracycline aptamers, and the non-inverting input was grounded, setting $V_{\mathit{\text{Bias}}}=0V$. The RE/CE were also grounded $\left(V_{RE/CE}=0V\right)$, setting an overpotential of $\eta \approx 0.3V$ (vs. Ag/AgCl) and strongly oxidizing MB. Since no AC perturbations were applied and the redox species were depleted under static DC conditions, the large-signal faradic current subsides, allowing for an effective comparison between the small-signal response of the conventional $C_{dl}$ to our 13th-order RC-ladder CPE model. The output noise voltage was measured with a PicoScope 5444D 16-bit oscilloscope, sampling at 8.93 MHz, with a full-scale voltage of $V_{FS}=40$ mV and 1 MHz anti-aliasing digital filter. FFT analysis was performed on $N=\mathrm{89,285,717}$ samples, giving a frequency resolution of $\Delta f\approx 0.1\text{Hz}$. A Hann window was applied, and the spectra from 10 independent recordings were averaged.

Fig. 15.

Schematic and photo of TIA used to model and measure noise spectra.

We co-simulated the OPA 4354 PSPICE macro-model, provided by TI, with the full electrochemical model developed in this work. For comparison, we also simulated a conventional simplified Randles circuit representation of the same WE, with fitted values $R_s=300\Omega ,C_{dl}=10\text{nF},R_{ct}=60\text{M}\Omega$. Fig. 16(a) shows the measured output-noise spectra alongside simulations from both models. Two observations are made. First, the spectra exhibit a pronounced noise peak, which arises from the interaction between the EDL impedance and feedback $R_f$ and $C_f$. Second, the full electrochemical model accurately reproduces the overall spectral shape in the simulation, including the slopes set by the non-linear EDL impedance in the in-band frequencies $(f<10kHz)$. In contrast, the simplified Randles circuit’s linear pole-zero behavior limits its slope to ±20 dB/dec, causing dramatic reductions in accuracy at low frequencies.

Fig. 16.

(a) Output noise spectra and (b) input-referred noise spectra comparing experimental measurements (red) against simulations using the proposed model (blue) and the simplified Randles model (green).

We input-referred the output-noise spectra to current noise using the modeled noise transfer function (NTF):

$$\frac{v_{n,\mathit{\text{out}}}}{i_{n,\mathit{\text{in}}}}=Z_f\frac{RR}{1+RR}+\left(Z_s+Z_f\right)\frac{1}{1+RR}$$ (17)

$$RR=A_0\frac{Z_s}{Z_s+Z_f},$$ (18)

where $Z_s$ is the EDL impedance, $Z_f$ is the feedback impedance from $R_f$ and $C_f$, and $A_0$ is the OPA 4354’s open-loop gain. Fig. 16(b) shows the resulting IRN current spectra. The full electrochemical model again tracks the measured curve more closely, consistent with the NTF’s dependence on input impedance. At frequencies above the noise peak, both models diverge slightly from the measurement, but this region lies beyond the amplifier’s usable bandwidth and is typically suppressed by post-amplification filtering. Table II summarizes measured and simulated IRN integrated over various bandwidths and highlights two observations. First, our full electrochemical model accurately simulates IRN in the 1 Hz to 1 kHz range, whereas the simplified Randles circuit underestimates the measured result by 60.7 to 82.2%. Second, frequency-domain and transient-noise simulations using the full electrochemical model yield integrated noise values that match measurements, confirming the model’s compatibility with both simulation modes.

TABLE II.

Input-Referred RMS Noise Integrated Across Various Bandwidths

	1 Hz – 1 kHz	1 Hz – 10 kHz	1 Hz – 100 kHz
Simulated with Full Electrochemical Model
Frequency-domain Noise $\left({\mathbf{n}\mathbf{A}}_{\mathbf{\text{rms}}}\right)$	0.189	0.878	4.18
Transient Noise $\left({\mathbf{n}\mathbf{A}}_{\mathbf{\text{rms}}}\right)$	0.327	1.49	5.33
Simulated with Simplified Randles Model
Frequency-domain Noise $\left({\mathbf{n}\mathbf{A}}_{\mathbf{\text{rms}}}\right)$	0.086	0.836	4.57
Transient Noise $\left({\mathbf{n}\mathbf{A}}_{\mathbf{\text{rms}}}\right)$	0.039	1.41	3.36
Experimentally Measured with OPA4354
Measured Noise $\left({\mathbf{n}\mathbf{A}}_{\mathbf{\text{rms}}}\right)$	0.219	0.919	2.95

V. Conclusion

This work presents a compact model for heterogeneous electrochemical reactions that is compatible with both frequency and time-domain simulations in SPICE-based circuit simulators. The model was designed to be accurate, stable, and computationally efficient to enable direct co-simulation with readout circuitry. Electron transfer kinetics were implemented in Verilog-A using a simplified asymmetric MH model and integrated in continuous-time via a capacitor to maintain accuracy and stability under high frequency voltammetry scans, like SWV. The non-linear EDL was represented using a fractionalpole RC-ladder.

The model was verified against electrochemical measurements performed with anti-tetracycline and anti-kanamycin aptamers, demonstrating the accurate reproduction of experimental results across a wide range of voltammetry frequencies and target concentrations. To confirm robustness and avoid overfitting, key parameters were individually swept, showing adherence to MH theory across relevant electrochemical operating ranges. In addition, a TIA was constructed, and its electrical noise was measured experimentally. Noise simulations employing the proposed model as the input impedance closely tracked the measured noise PSD and provided more accurate predictions than the commonly used simplified Randles circuit.

These results suggest that our proposed full electrochemical model offers a more physically meaningful and accurate alternative to the simplified Randles circuit for analyzing and quantifying the noise performance of electrochemical sensors. Its physical interpretability, numerical stability, and SPICE compatibility makes this model a practical tool for co-designing heterogeneous electrochemical biomedical sensors and their readout electronics.

Biographies

Wei Foo (Graduate Student Member, IEEE) received the B.S. degree in electrical engineering from California Institute of Technology, Pasadena, in 2023. He is currently working toward the Ph.D. degree in electrical engineering and computer science with the University of California at Berkeley, Berkeley. His research interests include lowpower mixed-signal ICs and aptamer development for biomedical applications. He is the recipient of the 2024 National Science Foundation Graduate Research Fellowship Program Award.

Jun-Chau Chien (Senior Member, IEEE) received the B.S. degree in electrical engineering and the M.S. degree in electronics engineering from the National Taiwan University, Taipei, Taiwan, in 2004 and 2006, respectively, and the Ph.D. degree in electrical engineering from the University of California at Berkeley, Berkeley, in 2015. From 2008 to 2014, he was with Berkeley Wireless Research Center. He received Postdoctoral training with Stanford University, from 2017 to 2018 and served as a Research Engineer, from 2018 to 2020. He was an Assistant Professor with the Department of Electrical Engineering, Graduate Institute of Electronics Engineering, and the Graduate School of Advanced Technology with National Taiwan University, from 2021 to 2022. He is currently an Assistant Professor with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley. He has served on the International Technical Program Committee of the International Solid-State Circuits Conference, from 2021 to 2025. He has held industrial positions with HMicro (now LifeSignals Inc.), Xilinx Inc., InvenSense (now TDK Corporation), and INanoBio Inc. His research interests include integrated microsystems for in vitro and in vivo biosensing, high-speed millimeter-wave circuits and systems, and silicon photonics. He is the recipient of the 2025 NSF CAREER Award, the 2024 NIH Trailblazer Award, the 2024 MediaTek Outstanding Research Award, the 2022 IEEE BioCAS Conference Best Paper Award, the 2019 CICC Outstanding Regular Paper Award, the 2014 Solid-State Circuit Society (SSCS) Predoctoral Award, the 2010 ISSCC Jack Kilby Outstanding Student Paper Award, the 2007 ISSCC Silk Road Award, and the 2006 Best Thesis Award from the Graduate Institute of Electronics Engineering at National Taiwan University. He was a Guest Editor for the IEEE Journal of Solid-State Circuits, in 2024 and is currently an Associate Editor for the IEEE Solid-State Circuits Letters and the IEEE Transactions on Biomedical Circuits and Systems.

Data Availability Statement

Source code for the Verilog-A model is publicly available through the Chien Lab website (https://chienlab-bioic.github.io) at github.com/chienlab-bioic/2025_foo_edlc_model and IEEE Xplore/Code Ocean.