Titanium dioxide (TiO₂) has become a favored semi-conductor material for photocatalytic degradation of petroleum hydrocarbons in contaminated soils due to its chemical stability, low toxicity, and strong oxidative capability under photoexcitation [1–11]. Upon absorption of photons with energies exceeding its band gap (≈ 3.0–3.2 eV for anatase), TiO₂ generates electron–hole pairs that can initiate interfacial redox reactions, producing highly reactive oxygen species capable of oxidizing complex hydrocarbon molecules [6, 8, 11]. Despite this intrinsic potential, practical photocatalytic efficiencies in soil environments remain far below theoretical limits, largely due to its limited photoactivation under solar irradiation which restricts excitation predominantly to the ultraviolet region and severely limits utilization of the visible spectrum, its rapid charge-carrier recombination and inefficient coupling between photoexcitation and surface reaction pathways [7, 12–14]. These limitations impose a photophysical bottleneck in which light absorption does not translate proportionally into reactive oxygen species generation or hydrocarbon mineralization, thereby motivating the search for photocatalytic architectures that address charge separation and interfacial transfer at a fundamental, field-controlled level rather than through absorption enhancement alone.

The persistence of these limitations is rooted in the fundamental photophysics governing charge generation and transport in semiconductor photocatalysts, particularly at heterogeneous solid–molecule interfaces, where interfacial disorder and dielectric mismatch dominate carrier dynamics [15–20]. In TiO₂, photoexcitation initially creates electron–hole pairs that rapidly relax into Coulombically correlated states whose fate is dictated by dielectric screening, carrier localization, and interfacial disorder[15, 19, 20, 21]. In hybrid and surface-dominated environments such as soils, these effects are further amplified, leading to short carrier lifetimes and inefficient extraction of charges to adsorbed reactants [22–24]. The efficiency of photocatalysis therefore depends critically on charge separation, defined as the successful spatial and energetic decoupling of photogenerated electrons and holes prior to recombination. In systems incorporating organic sensitizers or molecular adsorbates, photoexcitation predominantly yields Frenkel excitons, in which the electron and hole remain strongly localized within a molecular or near-surface region due to weak dielectric screening and strong electronic confinement[24, 25]. These excitons possess large binding energies that cannot be overcome by thermal fluctuations or modest built-in fields alone, resulting in recombination-dominated dynamics even under sustained illumination[26–29]. Consequently, enhancements in optical absorption or photon flux do not automatically translate into increased reactive oxygen species generation, because the underlying bottleneck lies in the inability to destabilize and dissociate bound excitonic states and efficiently transfer charges across the semiconductor–molecule interface. Addressing this bottleneck requires strategies that directly modify the interfacial potential landscape and local electric-field environment experienced by photoexcited carriers, rather than approaches that focus exclusively on extending light absorption.

Previous efforts to overcome the intrinsic efficiency limitations of TiO₂ photocatalysis have largely focused on modifying its optical absorption, defect chemistry, or interfacial composition [30–34], yet these strategies have not resolved the fundamental photophysical bottleneck identified above. Band-structure and defect-engineering approaches, including metal and non-metal doping as well as heterojunction formation, have successfully extended TiO₂ absorption into the visible region and introduced internal electric fields; however, they simultaneously generate mid-gap trap states and structural disorder that accelerate nonradiative recombination and limit the fraction of photoexcited carriers that reach reactive surface sites [31, 34–39]. Plasmon-assisted TiO₂ systems, achieved through coupling with metallic nanostructures or plasmon-supporting dopants, enhance local electromagnetic fields and excitation rates, but plasmon-induced carrier generation alone does not guarantee efficient charge extraction, as ultrafast relaxation, interfacial back-transfer, and recombination at defect-rich interfaces remain dominant loss channels [40–42]. Sensitization strategies employing organic molecules or interfacial charge-transfer complexes provide additional excitation pathways and broaden spectral response [43–46]; yet photoexcitation in these systems frequently yields bound excitonic or charge-transfer states that recombine unless an explicit driving force exists to dissociate them and inject carriers into TiO₂ bands [47–49]. Thus, while prior models and experiments on enhancing the photoactivation under solar irradiation of TiO₂ have independently demonstrated the roles of static electric fields arising from interfacial engineering (defects, dopants, heterojunctions) and oscillatory electromagnetic fields generated by plasmonic excitation, these effects have been treated as separable and independent contributions. Existing frameworks therefore describe either DC field–assisted charge separation or AC field–enhanced excitation rates, but no unified model has described the synergistic regime in which static (DC) interfacial fields and time-dependent (AC) plasmonic fields act cooperatively on interfacial excitons. As a result, the conditions under which neither DC fields nor plasmonic excitation alone is sufficient but their non-separable DC–AC coupling becomes decisive for exciton dissociation, charge injection, and sustained photocatalytic turnover remain unexplored

This study addresses the unresolved gap by formulating a TiO₂-specific quantum framework in which intrinsic DC defect fields and AC plasmonic near-fields are treated as a coupled, non-separable Stark potential acting on interfacial excitons. Our approach models the Quercetin@Al:TiO₂ hybrid photocatalyst as a bio-sensitized, metal-doped TiO₂ platform in which intrinsic interfacial DC fields arise from the quercetin–TiO₂ organic–semiconductor junction, while Al-induced plasmonic activity in TiO₂ generates the complementary time-dependent AC near-field component that jointly governs exciton dissociation and charge injection and defines the effective interfacial potential experienced by photoexcited excitons. Unlike previous treatments including our earlier static-field Stark model for quercetin–TiO₂ hybrids and photonic enhancement analyses of quercetin-sensitized Al:TiO₂ systems [50–55], this work explicitly resolves their synergistic coupling, a regime that is not merely an incremental enhancement but a functional necessity for tackling complex environmental matrices like oil-contaminated soils.

In such soils, photocatalytic efficiency is governed not by photon absorption alone but by the persistent flux of charges that can overcome the limitations of weakly polar, heterogeneously adsorbed hydrocarbons shielded by mineral matrices. The cooperative DC–AC field regime forcibly destabilizes interfacial Frenkel excitons and suppresses ultrafast recombination, thereby sustaining a higher steady-state population of valence-band holes and surface-trapped electrons to drive •OH and O₂•⁻ radical generation precisely at oil–TiO₂ contact points[52, 53]. Simultaneously, the oscillatory AC component periodically modulates the interfacial potential to prevent surface poisoning by intermediates, while the static DC component maintains directional charge transfer. This DC–AC framework therefore predicts a decisive transition from recombination-limited to reaction-limited photocatalysis, yielding accelerated hydrocarbon mineralization kinetics where mass transport and adsorption constraints otherwise dominate. This unified formulation thus provides both a fundamental advance in photocatalysis theory and the predictive criteria required for effective remediation of persistent hydrocarbon pollutants.

A critical strength of the proposed DC–AC Stark–plasmonic framework is that the interfacial electric fields capable of driving Frenkel exciton dissociation are intrinsic, localized, and non-thermal in origin. The static (DC) component arises from the built-in potential at the quercetin–TiO₂ organic–semiconductor junction and from defect-induced band bending in Al-doped TiO₂, both of which are known to generate strong, Å-scale interfacial electric fields without the need for external bias or macroscopic heating [15, 18, 20, 21]. Because Frenkel excitons are highly localized and weakly screened at organic–oxide interfaces, their dissociation threshold is governed by local electric-field gradients rather than bulk field magnitudes, allowing modest intrinsic fields to destabilize bound excitonic states via Stark-induced delocalization and field-assisted tunneling rather than thermal activation [17, 25, 26, 27]. The oscillatory (AC) component introduced by Al-related plasmonic excitation operates in the near-field regime, where electromagnetic energy is predominantly stored in the local field rather than dissipated as heat, thereby periodically amplifying the instantaneous interfacial field experienced by the exciton without inducing sustained lattice heating or field distortion [11, 12, 40, 41]. The cooperative DC–AC coupling therefore enables exciton dissociation and charge injection through non-thermal, field-driven potential reshaping, preserving interfacial structural integrity while maximizing charge-separation efficiency. This physically defensible, non-destructive field synergy supports the scalability and environmental viability of the proposed photocatalytic mechanism.

We model the quercetin–Al:TiO₂ hybrid as a nanoscale type-II donor–acceptor interface in which Frenkel excitons localized on quercetin couple to a composite local electric field generated by TiO₂ defect structures (oxygen vacancies and Al³⁺ substitutional/near-surface dopants) and by Al-induced localized surface plasmon resonances (LSPR)(See Fig. 1)[55, 56–57]. We model a single quercetin chromophore chemisorbed at an Al-modified TiO₂ interface (quercetin–Al:TiO₂) in an aqueous/soil-like dielectric environment with binding energy , permanent dipole and polarizability [54,55]. The strong dielectric contrast ( vs. ) confines interfacial fields to ångström–nanometre scales where the exciton wavefunction attains its largest amplitude, thereby amplifying field–dipole and field–polarizability couplings[55, 58–61]. Al dopants additionally introduce an LSPR in the deep-UV whose near-field decays over , overlapping the exciton–interface coupling length scale[56, 57, 62]. For physical realism, we model the active coordinate as the donor oxide axis , normal to the interface, along which the tightly bound Frenkel exciton is created and subsequently driven toward charge separation. The oxide is treated as a high-κ, wide-gap acceptor with a type-II alignment relative to quercetin (downhill electron injection upon dissociation) consistent with interfacial spectroscopy and GW alignment studies [58, 60, 63]. Aluminum substitution and oxygen vacancies provide a built-in (DC) interfacial field, while near-surface Al plasmonics supplies a co-localized (AC) near-field in the deep-UV [ 63, 64-66]. The exciton is therefore subjected to a composite local field active within a few nanometers of the junction, where ​ represents the defect-derived bias and the Al-LSPR amplitude and frequency.

A schematic illustration of the photocatalytic mechanism in quercetin–Al-doped TiO₂ hybrids. UV excitation induces a Frenkel exciton within the π-conjugated system of quercetin. A built-in electric field (F ≈ 0.5-1.0 V/Å), generated by Al³⁺ doping and oxygen vacancies in TiO₂, and a Localized Surface plasmonic Resonance (LSPR) filed facilitates exciton dissociation. The electron is ejected into the TiO₂ conduction band, while the hole remains on the quercetin donor, enabling efficient charge separation for reactive oxygen species (ROS) generation.

Accordingly, the local field is modeled as the superposition of a quasi-static, defect-derived DC component ​ and an oscillatory LSPR AC component :

Restricting the problem to the exciton coordinate normal to the interface r (the dominant dissociation direction), the excitonic potential is written

Here is the unperturbed excitonic potential, modelled by an asymmetric Pöschl–Teller (Rosen–Morse II) form. is the stark term and is the linear ramp potential that quantifies the edge bending at the material interface.

Within the material interface, the DC term pre-tilts the bound state and reduces binding, while the AC term periodically modulates that tilt. Two limiting physical regimes guide the analytical approach: (i) an off-resonant (Floquet/quasi-static) limit in which the AC field produces a dynamic Stark shift , further lowering binding energies; and (ii) a threshold-crossing regime in which the instantaneous field intermittently exceeds a critical dissociation field ​ (i.e. ), giving a duty-cycle fraction of each optical period during which dissociation and injection occur with strongly enhanced probability.

These elements viz the quercetin’s µ,α,​, TiO₂’s dielectric response and band alignment, and interfacial parameters —constitute the minimal, physically motivated descriptor set for the analytic and numerical derivations below. Explicit parameter values, boundary conditions, and approximations (one-dimensional r coordinate, separation of in-plane transport, and timescale regimes for Floquet vs quasi-static treatments) are summarized in Table 1.

2.2. Hilbert Space, State Representation, and Working Basis (insert into Section 2.1)

With the physical system and parameters established (Table 1), we formalize the quantum dynamics of the interfacial Frenkel exciton within a Hilbert-space framework. The Frenkel exciton, projected onto the interface-normal coordinate , is treated as an effective one-dimensional quantum particle evolving under the composite local field within the donor–oxide junction potential defined in Sec. 2.1.

The kinematic state space for the projected relative coordinate is

with the truncation length of the interfacial domain (here as used in Table 1), and inner product

Within , the static (DC) Hamiltonian defines a spectral decomposition into localized bound exciton states supported by the Rosen–Morse II well on the donor side and continuum/scattering states associated with the ramp/oxide region:

Accordingly, the identity resolution may be written as

where denote discrete donor-localized excitonic levels and denote energy-normalized continuum states in the oxide-side region. This bound–continuum splitting is the microscopic basis for defining dissociation as probability flux transfer from into (computed in Sec. 2.3 via Airy matching and transmission).

When the AC near-field is present, the Hamiltonian is periodic, with , so solutions are treated in the extended Floquet space (Sambe space)

equipped with the Floquet inner product

A convenient orthonormal basis for is the Fourier (photon-index) basis

so that any -periodic component admits the expansion .

The Floquet eigenproblem is generated by the Floquet operator

acting on , with eigenpairs and physical solutions .

Projecting onto the product basis yields the block (photon-ladder) representation used in Sec. 2.3: diagonal blocks shift by and off-diagonal blocks couple adjacent -manifolds through the dipole interaction , producing the standard block-tridiagonal Floquet system.

For numerical realization, the Floquet index is truncated to , giving a finite-dimensional approximation

with chosen such that exceeds the relevant excitonic energy scales.

This Hilbert-space construction makes the DC–AC “non-separable” coupling precise: the same spatial exciton state space is acted on by a single periodic Hamiltonian whose representation necessarily mixes sectors, i.e., DC-tilted bound/continuum structure in is dynamically hybridized by the AC near-field through Floquet sidebands.

2.3. TIME-INDEPENDENT ANALYTICAL FRAMEWORK FOR STARK-DRIVEN EXCITON DISSOCIATION

The theoretical framework is built upon established methods for quantum systems in electric fields, following the approach of Friedrich (2013)[67] for Stark effects in confined potentials. We work in the exciton relative coordinate (projected on the interface normal) with reduced mass µ, consistent with the effective mass approximation for Frenkel excitons [68]. The static (DC) Hamiltonian is

Where is the effective potential in Eq. (2)

This formulation follows the standard Stark effect Hamiltonian with an additional ramp potential to model the semiconductor interface[69–70].

On the donor side, exciton binding is modeled by the asymmetric Rosen-Morse II potential:

with depth , range a, and built-in asymmetry W accounts for the inherent polarity of quercetin molecules and the initial band offset at the TiO₂ interface, as observed in spectroscopic studies[71]. This potential is chosen for its analytical tractability, providing exact solutions for bound states and matrix elements [72–74]. The composite field F(t) incorporates both defect-derived DC and plasmonic AC components [56, 75–78]. The TiO₂ substrate is modeled as a continuum with a linear potential ramp representing the band-bending region.

To connect the donor well to a linear band-bending region with slope ​ for ​, we construct a smooth ramp potential that ensures a continuous transition between the confined and continuum regions:

with . Then for and ​, ensuring a transition to a linear tail of slope ​. This continuous function ensures proper boundary condition matching without numerical artifacts and prevents spurious reflections in quantum transmission calculations [79]. This construction follows established soft-step and smooth potential-matching techniques widely used in quantum transport to eliminate spurious reflections and maintain physical boundary continuity between bound and continuum states[80–82]. The parameter controls the transition width and is chosen such that , preserving the short-range nature of the excitonic binding while ensuring numerical stability during tunnelling and transmission calculations.

To verify the ramp potential's mathematical properties, we differentiate Eq. (10) as:

This derivative analysis serves to verify the mathematical construction of the ramp potential, confirming that it properly transitions between the molecular and semiconductor regions. The analysis demonstrates that is continuous, thereby avoiding numerical artifacts in quantum transmission calculations [79]. It also confirms the asymptotic behavior: for ​, , while for ​, ​. This ensures the ramp correctly models the dielectric screening transition from organic to oxide regions, as required by Maxwell's boundary conditions [83].

For the quantum mechanical solution, we require the gradient of the total effective potential:

Substituting Eq. (11) yields the complete expression:

The total potential gradient is physically essential because it determines the local electric field experienced by the exciton, which governs the Stark effect and dissociation dynamics [67].

For a stationary energy E, the classical turning point satisfies:

This definition follows the standard treatment of one-dimensional quantum tunnelling problems, where the turning point marks the transition from classically allowed to forbidden regions and Airy-function matching is used near to avoid the breakdown of WKB approximations [89–90]

In the matching layer , we linearize the potential:

where the is the Airy length scale given as:

This length scale, derived from dimensional analysis of the Schrödinger equation [86], typically ranges from 1–5 nm for our system parameters, determining the quantum tunneling region extent.

The transformation:

reduces the time-independent Schrödinger equation to the canonical Airy form:

This reduction is mathematically exact for linear potentials and provides the universal description of quantum motion in linearly varying fields[86]. The general solution:

leverages the well-established properties of Airy functions[91].

This reduction is mathematically exact for linear potentials and provides the universal description of quantum motion in linearly varying fields, from semiconductor band-bending and triangular quantum wells to optical Stark and Franz–Keldysh effects[92–95].

The general Airy function solution (Eq. 20) contains two linearly independent components that exhibit fundamentally different asymptotic behaviors. The physical requirement of finite probability density in the oxide continuum dictates our boundary condition selection. As in the TiO₂ region, the coordinate , where the Airy functions exhibit the asymptotic forms:

The exponentially decaying nature of represents a transmitted wave with finite probability density, while the exponentially growing corresponds to an unphysical solution that would violate probability conservation in the infinite domain. We therefore enforce the outgoing wave condition by setting:

Interface Matching Conditions

At the interface position ​, we enforce quantum mechanical continuity conditions. Defining the boundary values:

we impose

write the donor (Region I) field as an RM2 expansion

These conditions ensure both wavefunction continuity and probability current conservation across the interface, fundamental requirements derived from the postulates of quantum mechanics[96]. Eqs. (16)–(18) form a linear system for

The transmission probability is obtained from the probability current ratio:

Using the Airy function Wronskian identity ​ [91] and the matching conditions (Eq. 22) and recasting the interface equations as a 2×2 map as:

we derive the closed-form transmission coefficient as:

Where

is the donor-side wave number referenced to the left asymptote and ​ is set by the local interface slope (Eq. 12). The entries of are built from the RM2 boundary values

and .

The interface matrix is defined by the boundary transformation where contains the RM2 wavefunction values at the interface and contains the corresponding Airy function values. The elements of are constructed from the RM2 basis functions evaluated at and the Airy function scaling factors, encoding all interface physics in the quantum transmission problem.

The final dissociation probability for the DC field case is obtained by evaluating the transmission at the Stark-shifted exciton energy:

Eq. (21) quantity connects directly to dissociation/injection yields.

2.4. Time-Dependent Hamiltonian and Frequency Dependent Solutions

From Eq. (3), the composite electric field experienced by the quercetin exciton is:

Two mechanism drives and ; Aluminum supports a deep-UV/UV LSPR due to its high plasma frequency; sub-wavelength confinement produces evanescent near fields localized to a few–tens of nm[75, 78]. Under resonant illumination, this near field drives the exciton coordinate with amplitude. We model this field as

where is the local field enhancement factor (typically 10–100 for plasmonic hot spots) and is the incident optical field. The frequency ω is fixed by the LSPR condition of the Al nanostructure. The spatial decay length of this field is on the order of 5–20 nm [78, 97], ensuring significant overlap with the exciton wavefunction at the interface. Al incorporation can modify the oxygen-vacancy landscape in TiO₂, altering carrier density and loss (dielectric function), thereby renormalizing the LSPR and its field penetration into the depletion region [97, 98]. This strengthens near-field overlap with the tunneling (Airy) layer thickness ℓ∼ few nm (Sec 2.2).

The full time-dependent Hamiltonian becomes:

where ​ is the static Hamiltonian from Eq. (8).

The incorporation of the AC component from the Al-LSPR near-field requires solving the full time-dependent Schrödinger equation (TDSE). The fundamental equation of motion for the exciton wavefunction under the composite field is:

Regime-Dependent Solutions of the TDSE

Depending on the amplitude, frequency, and spatial uniformity of the Al-LSPR drive, the excitonic dynamics under the time-dependent field admit three distinct solution routes for the TDSE. In the weak-field, off-resonant regime, the TDSE reduces via Floquet perturbation theory to an AC Stark shift of the bound level, modifying the DC transmission spectrum. In the uniform phase-modulation regime, the TDSE admits the Tien–Gordon sideband expansion[99], in which dissociation probabilities are obtained as weighted sums of DC transmission coefficients at photon-shifted energies. In the high-frequency limit, the TDSE reduces under the Floquet–Magnus expansion to an effective static Hamiltonian, yielding DC-like transmission but with field-renormalized parameters.

When the wavelength of the AC field is much larger than the spatial extent of the exciton wavefunction and tunneling region, the AC field acts as a scalar voltage drop across the transition region of effective length ​ (i.e., energy modulation , the exact solution of the TDSE (Eq. 29) follows the Tien-Gordon model [99].

For a uniform field across a region of length ​, we can make the gauge transformation:

Where . Substituting Eq. (40) into Eq. (29) yields:

The general solution of Eq. (41) is a superposition of energy eigenstates:

Transforming back to the original frame:

Using the Jacobi-Anger expansion:

Putting the Jacobi-Anger Expansion into Eq. (43), we obtain:

For a scattering state incident with energy , the wavefunction in the presence of the AC field becomes:

The probability current for each sideband n is proportional to . Since the transmission probability T(E) represents the squared amplitude of the transmitted wave, the total time-averaged transmission becomes:

where is the DC transmission coefficient derived in Eq. (30).

Physical Parameterization and Experimental Signatures

The modulation index quantifies the number of photons participating in the tunneling process, with the effective length LeffLeff​ representing the spatial scale over which the AC potential drops. For the excitonic system considered here, can be estimated from the turning-point wavefunction extent , the Airy length scale (), or the depletion width (), yielding values of approximately 2 nm. Combined with typical Al-LSPR parameters (,), this gives , indicating substantial multi-photon contributions to the dissociation process.

The Tien-Gordon model remains exact under conditions of spatial field uniformity across the tunneling region and scalar potential coupling, complementing the AC Stark approach which captures bound-state renormalization. The distinctive sideband structure predicted by Eq. (46) manifests experimentally as step-like features in the photocurrent versus driving frequency, providing a clear signature of photon-assisted tunneling in exciton dissociation that can be distinguished from conventional field-enhancement effects.

For the regime where the driving frequency ω is large compared to all internal system dynamics (​), a powerful simplification emerges. In this high-frequency limit, the system cannot resonantly absorb photons, and the dynamics is described by a static effective Hamiltonian obtained via the Floquet-Magnus expansion [105–106].

The leading-order terms of this expansion yield:

This effective Hamiltonian originates from averaging the rapid oscillations of the drive over a period. The double commutator is simplified using the commutation relation:

Substituting Eq. (41) into Eq. (41) yields the form:

The effective Hamiltonian reveals that the primary effect of a high-frequency AC field is a uniform, state-independent energy shift:

This corresponds to the ponderomotive energy of a free electron oscillating in the electric field. For a bound exciton, this energy shift directly lowers the effective binding energy:

The dissociation probability in this regime is therefore obtained by evaluating the DC transmission formula at this shifted energy:

This high-frequency approximation is valid when is much larger than the exciton binding energy ​ and the tunneling rates through the barrier. For our system parameters (, Al-LSPR ), this condition is reasonably satisfied, making Eq. (52) a valuable analytical tool that complements the more computationally intensive full Floquet solution (Eq. 40).

2.5. Threshold-Crossing Regime and Duty Cycle Analysis

When the instantaneous field intermittently exceeds the critical dissociation field ​ (i.e., ), the system operates in the threshold-crossing regime. During each optical cycle, dissociation occurs preferentially during time intervals when ​.

The duty cycle η quantifies the fraction of each optical period for which the field condition ​ is satisfied. The fraction of each optical period during which dissociation is enhanced is given by:

Here, increases from zero (no dissociation events) to unity (continuous above-threshold drive) as the AC amplitude increases relative to ​.

The condition ​ occurs when:

Solving for the time intervals:

The fraction of the period satisfying this condition is:

Within this regime, the dissociation probability can be approximated as a weighted sum of the DC transmission coefficients evaluated at the threshold and background fields:

where T(E) is the Airy-matched DC transmission function (Eq. 25). This model reflects that during of the cycle, the field is above threshold and transmission is high, while in the remaining portion it is suppressed.

In the off-resonant threshold-crossing regime, the duty-cycle enhanced transmission is:

where is the modified Bessel function of the first kind, is obtained from the Airy slope at the turning point. The multiplicative enhancement factorcaptures the nonlinear synergistic role of the AC field, showing that oscillatory driving produces a stronger dissociation effect than predicted by simple duty-cycle weighting alone.

2.6. Doubly-Resonant Dissociation Pathways

The synergistic interplay between static Stark tuning and dynamic resonant driving enables doubly-resonant dissociation pathways that neither DC nor AC fields can support alone.

The DC field ​ pre-tilts the potential and also systematically modifies the excitonic energy levels. Using second-order perturbation theory, the field-dressed energy of level n is:

where is the permanent dipole moment and ​ is the static polarizability. Consequently, the energy separation between levels becomes a tunable function of the DC bias:

This field-dependent splitting provides a continuous tuning knob for aligning internal molecular transitions with the external AC driving frequency.

When the AC field frequency satisfies the resonance condition:

the system enters a strongly coupled regime.

The dipole coupling element between Stark-shifted states ∣n⟩ and ∣m⟩ is:

Near resonance, the Floquet quasi-energies exhibit avoided crossings, described by the secular equation:

This hybridization creates mixed "dressed states" () that are coherent superpositions of the original excitonic states. These dressed states often possess a more delocalized character and a higher density of states near the continuum, thereby enhancing the tunneling probability into the TiO₂ conduction band.

The synergy operates on two complementary levels viz Threshold-lowering by DC field in which the static bias reduces the binding energy, thereby lowering the critical dissociation field ​. And Resonant activation by AC field in which once Stark tuning brings into resonance with ℏω, the AC field drives efficient population transfer into continuum-coupled states.

The combined dissociation probability thus reflects a sum of non-resonant and resonant channels:

Here the first term is the duty-cycle weighted, off-resonant dissociation from Sec. 2.5, and the second term ​ represents the resonant enhancement from the Floquet hybridization described by Eq. (54).

Experimental Signature and Implications

The cooperative regime is characterized by sharp peaks in the dissociation yield or photocurrent as a function of DC bias ​, appearing only when the AC drive frequency matches the Stark-tuned level spacing . This fingerprint distinguishes the doubly-resonant quantum mechanism from purely static Stark broadening or off-resonant AC modulation.

This effect provides a practical design principle for optimizing exciton dissociation in quercetin–Al:TiO₂ hybrids: by carefully selecting the Al doping (which sets ​ and influences via LSPR) and the incident light frequency, one can engineer a system that operates in this highly efficient, doubly-resonant regime.

2.7. ROS-Mediated Hydrocarbon Mineralization

This section closes the modeling chain from field-driven exciton dissociation to macroscopic mineralization of hydrocarbons (HCs) on Al:TiO₂, by deriving: (i) carrier-to-ROS transduction from the interfacial injection flux, (ii) surface radical kinetics (steady-state [•O₂⁻], [•OH]), and (iii) HC mineralization kinetics and efficiencies. Throughout, the dependence on the composite field parameters enters solely via the injection probability obtained in the DC/AC sections.

This section presents the simulated results derived from the quantum-mechanical framework established in Eqs. (1–71). Numerical computations were performed using the Airy-matched stationary-flux method and the time-dependent Schrödinger equation (TDSE) propagation scheme to quantify (i) field-dependent transmission coefficients, (ii) dissociation yields, (iii) interfacial injection currents, and (iv) scaling relations linking defect density and plasmonic field amplitude to residence time and photocatalytic efficiency.

All parameters used in the simulations are drawn from Table 1 and literature-anchored ranges: , , , , and . The numerical integration of the Schrödinger equation was executed in Python using SciPy’s solve_ivp and airy functions under the boundary conditions defined in Eq. (17).

Figure 2(a) illustrates the evolution of the composite effective potential of a quercetin exciton at the Al:TiO₂ interface under varying static electric fields ​ ranging from 0.0 to 1.0 V·nm⁻¹. The baseline Rosen–Morse II potential models the exciton binding on the donor side, while the added Stark term () and smooth ramp potential describe the interfacial field and band-bending transition between the organic and oxide regions

Figure 2(a):Stark-Driven pre-tilting of Excitonic potential

As the static field ​ increases from 0 to 1.0 V·nm⁻¹, the initially symmetric Rosen–Morse II potential well becomes progressively tilted toward the oxide region. The bound exciton well depth diminishes and the right-hand barrier lowers, creating an asymmetric potential that facilitates electron escape. The barrier height changes from roughly 0.8 eV (no field) to ≈ 3.5 eV under 1.0 V·nm⁻¹, showing strong field-induced reshaping of the excitonic landscape. This deformation reveals that the intrinsic Stark effect generated by dopant- and defect-induced fields is capable of substantially perturbing the Frenkel-exciton binding energy in quercetin. The potential tilt indicates that the electron–hole pair experiences a net driving force toward the TiO₂ conduction band, effectively lowering the activation energy for exciton dissociation and charge injection. The model therefore confirms the pre-tilt hypothesis, that internal interfacial fields polarize the exciton even before any optical excitation-driven ionization occurs.

This result implies that intrinsic interfacial fields alone can trigger charge-separation pathways in quercetin–Al:TiO₂ hybrids, eliminating the need for externally applied bias. This provides a mechanistic explanation for the experimentally observed spontaneous photocurrents and enhanced photodegradation rates in Al-doped TiO₂ systems. It also validates the theoretical framework’s ability to capture the coupling between Stark-shifted exciton levels and the continuum of oxide states, justifying the use of this potential in subsequent quantum-transmission calculations.

Figure 2(b) presents the derivative of the effective potential, , which represents the local electric field acting on the exciton. The field exhibits a steep rise near the interface (), corresponding to the transition between the molecular donor and oxide acceptor domains.

The local electric field exhibits a sharp maximum at the interface (around 1.5 nm), with magnitude increasing non-linearly as ​ rises. The field remains nearly zero deep inside the donor region but becomes strongly positive across the oxide side, signifying dielectric-contrast-induced field amplification. This spatial distribution signifies the formation of a confined interfacial field pocket, where dielectric discontinuity between low-κ quercetin (ε ≈ 3) and high-κ TiO₂ (ε ≈ 100) causes local field doubling. The enhanced gradient directly governs exciton acceleration and tunnelling probability, determining where the bound-to-free-carrier transition is most likely to occur.

The result implies that exciton dissociation and charge-injection events are spatially localized near the interface, within a nanometre-wide region of intense field amplification. Such localization is consistent with Airy-type solutions for tunnelling and validates the model’s assumption that Stark-driven delocalization originates at the dielectric boundary. Practically, it suggests that controlling dopant distribution and interfacial defect density can tune local fields, offering a predictive handle for optimizing photocatalytic efficiency in bio-sensitized TiO₂ systems.

Figures 2(a) and 2(b) illustrate the field-dependent evolution of the excitonic energy levels and their corresponding binding energies under static electric fields ranging from 0 to 1 V nm⁻¹.

(a) Field-dependent exciton energies showing quadratic-to-linear Stark shifts as ​ increases.

(b) Corresponding reduction of exciton binding energy indicating field-induced exciton destabilization toward the dissociation continuum.

In Fig. 3(a), both the ground (​) and first excited () states exhibit an upward, nearly parabolic energy shift toward the continuum as the applied field increases. The zero-field values of and progressively rise to and , representing total Stark shifts (ΔE₀, ΔE₁) of approximately 0.29 eV and 0.15 eV, respectively. The curvature confirms the quadratic dependence predicted by Eq. (56).

Figure 3(b) shows the concomitant reduction in binding energies

The ground-state binding decreases from 0.60 eV to 0.31 eV (≈ 48% reduction), while the excited-state binding falls from 0.35 eV to 0.19 eV (≈ 46% reduction). Extrapolating the fitted Stark curves yields a critical dissociation field , where . These observations confirm that intrinsic DC fields within the quercetin–TiO₂ interface induce a Frenkel-type Stark effect that destabilizes excitons by tilting their potential wells and enhancing charge separation.

This trend arises from field-induced distortion of the Rosen–Morse II potential describing the excitonic confinement. The added linear Stark term ( ) tilts the potential well, reducing its symmetry and effective depth. This pre-tilting decreases the potential barrier that confines the electron–hole pair, thereby lowering the Coulombic binding energy and raising the excitonic eigenenergy toward the continuum. At low fields, the interaction between the exciton dipole and the applied field produces a quadratic Stark effect dominated by the polarizability term ( ). At higher fields, wavefunction polarization along the field direction increases, introducing a linear contribution () due to permanent dipole alignment and state mixing between adjacent excitonic levels. The transition from quadratic to quasi-linear behaviour marks the crossover from perturbative to non-perturbative Stark coupling.

The observed magnitudes and curvature of the Stark shift agree with electroabsorption measurements in molecular semiconductors, where Frenkel excitons exhibit ΔE ≈ 0.2–0.3 eV per 1Vnm⁻¹ [1]. Quantitatively, a 0.3 eV reduction in binding energy at ≈ 1 Vnm⁻¹ implies that photo-generated Frenkel excitons in quercetin can dissociate spontaneously under internal interfacial fields, feeding electrons into the TiO₂ conduction band without external bias.This mechanistic insight explains the experimentally observed enhancement in photocatalytic rates under field-modulated conditions [97] and validates the theoretical prediction that the intrinsic Stark effect drives charge separation and long-range carrier extraction in bio-hybrid photocatalysts.

Figure 4 presents the calculated dependence of the transmission coefficient on the static Stark field , obtained from the Airy-matched tunneling formulation in Eq. (25). The curve reveals a distinct threshold-like transition separating four regimes of excitonic behavior at the quercetin–Al:TiO₂ interface.

The computed values are derived from the Airy-matched solution (Eq. 20) showing four regimes of excitonic response under increasing Stark field. Below , tunnelling is negligible; at ​, field-assisted tunnelling initiates; between 0.25–0.8 V·nm⁻¹, transmission rises exponentially; and beyond 1.0 V·nm⁻¹, barrier suppression yields quasi-free carrier injection (). The model validates Stark-driven exciton dissociation as the dominant charge-separation pathway in the quercetin–Al:TiO₂ interface.

In the low-field region (), excitons remain confined within the Rosen–Morse II potential, with negligible tunneling () and recombination dominating. At the critical field (), potential tilting aligns the bound level with the continuum, initiating field-assisted tunneling. Within the threshold regime (, T increases exponentially from 0.05 to ≈ 0.8, governed by Airy-function matching. Beyond , the high-field regime emerges, where complete barrier suppression allows continuous electron injection (). The sigmoidal transmission profile directly validates the Airy-matched quantum mechanical formulation in Eq. (20). The exponential rise in the threshold regime follows the characteristic tunneling dependence , confirming that field-assisted tunneling governs the initial dissociation dynamics. The saturation at high fields reflects the transition from quantum tunneling to classical over-barrier transport, consistent with the potential tilting described by Eq. (2).

The sharply non-linear transmission profile and identified critical field of demonstrate that the primary mechanism governing enhanced photocatalytic activity in quercetin–Al:TiO₂ hybrids is Stark-driven quantum tunneling, not merely broadband sensitization. This threshold represents the minimum interfacial field required to overcome the intrinsic Frenkel exciton binding energy via potential tilting, initiating efficient charge separation through field-assisted tunneling as described by the Airy-function formalism. The direct linkage between the quantum transmission probability and macroscopic photocatalytic efficiency resolves the fundamental charge separation bottleneck in bio-hybrid systems, establishing Stark effect engineering as a powerful strategy for leveraging natural chromophores in high-performance photocatalytic applications.

The evolution of excitonic confinement and Stark-driven delocalization at the quercetin–Al:TiO₂ interface is illustrated in Fig. 5(a) and 5(b), showing the transition from tightly bound Frenkel excitons to field-assisted charge-transfer states as the applied static field increases.

Figure 5(a) illustrates the normalized probability densities of the first three bound excitonic eigenstates () derived from the Rosen–Morse II potential that models the quercetin–Al:TiO₂ interfacial confinement. The ground state () is sharply localized around the chromophore region, confirming a tightly bound Frenkel-like exciton with binding energy . The first () and second () excited states are progressively broader and display additional nodes, indicating weaker confinement and enhanced delocalization as their energies approach the continuum. This discrete manifold constitutes the intrinsic excitonic quantization prior to external field perturbation.

Figure 5(b) shows the influence of a static Stark field () on the ground-state wavefunction. With increasing , the potential becomes asymmetrically tilted, shifting the electron density peak toward the TiO₂ interface () while broadening the spatial profile. The amplitude decreases while the tail extends toward the interface, clearly demonstrating field-induced delocalization and partial charge leakage across the interfacial barrier. The eigenenergy rises from , corresponding to a Stark-induced reduction in binding energy of about across the field range.

The complementary behavior of Figs. 5(a) and 5(b) depicts the transition from localized exciton confinement to field-driven delocalization in quercetin–Al:TiO₂ hybrids. In the absence of an external field, the Rosen–Morse II potential traps the electron–hole pair within the chromophore region, maintaining strong Coulomb coupling and minimal spatial separation conditions that favor radiative recombination and hinder charge transfer. Application of a static Stark field breaks this symmetry, effectively tilting the potential and lowering the barrier to the TiO₂ conduction band. Consequently, the ground-state wavefunction shifts and stretches toward the interface, signifying the onset of tunneling and charge-separation dynamics. This deformation is the microscopic precursor to the field-enhanced quantum-transmission process characterized later in Fig. 3.

The field-induced eigenenergy shift (ΔE ≈ 0.23 eV) yields a field-sensitivity coefficient:meaning that every 0.1 V·nm⁻¹ increment in reduces the exciton binding energy by about 0.026 eV. This agrees with the critical-field threshold derived from transmission analysis, marking the transition to efficient interfacial tunneling. The gradual wavefunction leaning in Fig. 4(b) thus provides the quantum-mechanical foundation for the threshold-like increase in charge-injection probability.

The Stark-induced reduction of enhances the interfacial electron-injection probability , leading to more efficient charge separation and stronger reactive-oxygen-species (ROS) formation. As increases from 0 to 0.9 V·nm⁻¹, rises quasi-exponentially, resulting in higher rates of hydrocarbon mineralization in photocatalytic applications. Such interfacial fields can be achieved intrinsically through Al³⁺ doping and oxygen-vacancy engineering, which generate built-in electrostatic potentials in the 0.2–0.8 V·nm⁻¹ range. Hence, moderate doping levels can provide the internal field necessary for spontaneous exciton dissociation without external bias.

The interplay between confinement and Stark-driven delocalization therefore establishes a quantitative link between quantum-scale exciton dynamics and macroscopic photocatalytic performance, offering a clear design principle for optimizing charge-transfer efficiency in quercetin-based bio-hybrid systems.

Figure 6 presents the calculated AC Stark shifts of Frenkel excitons in the quercetin–Al:TiO₂ hybrid system as a function of AC field amplitude for four representative localized surface plasmon resonance (LSPR) wavelengths (250–340 nm). The results reveal distinct wavelength-dependent exciton renormalization behavior governed by the interplay between resonant enhancement and field-driven state mixing.

Calculated binding-energy reduction (ΔE₍AC₎) as a function of AC field amplitude (F₍AC₎) for four representative localized-surface-plasmon-resonance (LSPR) wavelengths (250, 280, 310, and 340 nm). The 310 nm excitation (orange curve, ħω = 4.0 eV) exhibits the maximum Stark shift owing to near-resonant coupling with the Frenkel-exciton binding energy ( ).

The AC Stark shifts exhibit three distinct field-amplitude regimes. In the low-field quadratic regime (), all wavelengths follow the characteristic parabolic dependence predicted by second-order perturbation theory (Eq. 41), dominated by quercetin's static polarizability (). The extracted polarizability agrees well with theoretical values derived from the quercetin dipole transition moment (µ ≈ 8.5 D), confirming model consistency. The resonant-enhancement regime () reveals strong wavelength dependence, with 310 nm excitation (ħω = 4.0 eV) showing maximum Stark shift (ΔE = 0.22 eV at ) due to near-resonant coupling with the Frenkel-exciton binding energy (). This condition minimizes in Eq. (31), enhancing dipole coupling between excitonic states. The 280 nm case exhibits moderate enhancement, while far-off-resonant wavelengths (250nm and 340 nm) show weaker responses, consistent with detuning-dependent enhancement predictions. Beyond the bound-free threshold saturation occurs as Stark-shift growth decreases due to field-induced state mixing and wavefunction delocalization, where the Stark shift approaches of the original binding energy, marking efficient exciton ionization onset. The hierarchical Stark-shift ordering 310nm >280 nm >250 nm > 340 nm provides clear design principles for quercetin–Al:TiO₂ optimization. The 310 nm excitation aligns with Al nanostructure LSPR peaks in the deep-UV region, creating a doubly resonant configuration that synergizes plasmonic near-field enhancement with molecular resonance for simultaneous photon absorption and charge-separation enhancement. The weaker response at 250 nm () despite higher photon energy emphasizes that resonance matching, rather than absolute energy, dictates Stark efficiency. Large detuning from excitonic transitions suppresses coupling strength, reducing dissociation effectiveness. Similarly, the 340 nm excitation operates below resonance, where quadratic Stark effects dominate without strong enhancement.

The field-induced reduction in exciton binding energy directly enhances charge-separation efficiency in photocatalytic processes. For typical plasmonic near-field amplitudes achievable in Al:TiO₂ systems (), the AC Stark shift reduces the binding energy by approximately , substantially lowering the activation barrier for electron injection into TiO₂. This mechanism accounts for the experimentally observed increase in photocatalytic degradation rates under deep-UV plasmonic excitation. The saturation of the Stark shift at higher fields suggests a practical upper limit to field-enhancement strategies. While increasing Al³⁺ dopant concentration or optimizing nanostructure geometry can amplify local fields, the diminishing returns beyond indicate that interfacial charge-transfer kinetics and recombination dynamics become dominant constraints in this high-field regime.

The wavelength-dependent enhancements obey the resonance condition derived from the denominator of the Stark-shift term, validating the physical basis of the model. Quantitatively, the calculated Stark shifts agree with those predicted using quercetin’s known electronic parameters (µ ≈ 0.18 eV·(V·nm⁻¹)⁻¹, α ≈ 0.25 eV·(V·nm⁻¹)⁻²), confirming the predictive accuracy of this theoretical approach. The AC Stark effect emerges as a robust and tunable mechanism for exciton dissociation in quercetin–Al:TiO₂ hybrids. The 310 nm LSPR wavelength yields the optimal response due to its resonant coupling with the Frenkel-exciton transition.

The field-dependent renormalization follows a clear three-stage progression quadratic at low fields, resonantly enhanced at intermediate fields, and saturated at high fields providing actionable design criteria for maximizing photocatalytic performance through interfacial field engineering.

Together with the static-field delocalization presented in Fig. 3, these results establish that combined static–dynamic Stark interactions govern charge-separation dynamics in quercetin-based bio-hybrid photocatalysts. This result provides a quantitative framework for the rational design of deep-UV field-optimized materials for environmental remediation and sustainable photochemistry.

Figure 7 (a–d) | Tien–Gordon Sideband Spectrum and Floquet Quasi-Energy Structure.

(a) Photon-assisted tunnelling channels as total transmission vs .

(b) Jacobi–Anger sideband weights showing redistribution with modulation index.

(c) Floquet quasi-energy branches vs DC field revealing avoided crossing at .

(d) Quasi-energy vs photon energy at fixed , identifying the resonant condition.

These results confirm photon-assisted tunnelling and dynamic level hybridization as cooperative channels for Stark-driven exciton dissociation in quercetin–Al:TiO₂ hybrids.

The Tien–Gordon analysis demonstrates that AC field modulation generates multiple photon-assisted tunneling channels, substantially enhancing charge injection efficiency. Figure 7a shows that increasing the AC field amplitude () progressively broadens the transmission spectrum, with higher field strengths activating additional sideband channels. The staircase-like progression results from the summation over sideband orders in Eq. (41), where each step represents the threshold for opening a new photon-assisted transport path. The corresponding sideband-weight distribution in Fig. 7b follows the Bessel-function dependence predicted by the Jacobi–Anger expansion (Eq. 39). For modulation indices , the term dominates indicative of conventional tunneling. As , achievable under realistic plasmonic field enhancements, higher-order harmonics () emerge, enabling multiphoton tunneling and parallel injection pathways. This mechanism aligns with experimental reports of photocurrent staircase formation in semiconductor nanostructures and molecular junctions, confirming that AC field modulation can activate sub-bandgap transitions, consistent with photon-assisted enhancement in hybrid photocatalysts.

The Floquet quasi-energy results (Figs. 7c–d) illustrate the ability to manipulate excitonic states through combined DC field tuning and photon frequency control. In Fig. 6c, the and branches display an avoided crossing at, corresponding to the resonance condition . This anticrossing signifies strong coupling between the field-dressed exciton states and the driving field, leading to the formation of light-induced hybrid (Floquet) states as described by Eq. (54). Figure 7d shows that the quasi-energy separation varies systematically with photon energy, reaching a minimum when . The associated Rabi splitting () provides an experimental signature of the transition dipole moment () through . This dynamic Stark control effectively tunes exciton binding energies in real time, lowering the activation barrier for charge separation via field-induced hybridization.

The combination of Tien–Gordon sidebands and Floquet engineering reveals a synergistic enhancement mechanism where multi-photon tunneling and dynamic Stark effects operate cooperatively. The sideband spectrum expands the energy window for efficient charge injection, while Floquet dressing reduces the effective binding energy through state hybridization. This dual mechanism is particularly effective near the optimal conditions identified in our AC Stark shift analysis (310 nm LSPR), where both resonant enhancement and multi-photon processes contribute simultaneously. The quantitative agreement between the calculated spectra and the analytical expressions from Eqs. (45–47, 56–62) validates the predictive capability of this framework. The observed scaling with modulation index α and DC field F₀ provides clear design principles for optimizing photocatalytic efficiency: maximize AC field enhancement through plasmonic nanostructuring to activate multiple sideband channels, tune DC fields to position resonances at optimal photon energies, and leverage avoided crossings to reduce activation barriers through dynamic Stark control.

These theoretical predictions offer specific experimental signatures for verifying Stark-driven dissociation in quercetin–Al:TiO₂ hybrids. The photocurrent staircase in Fig. 5a should manifest as step-like features in wavelength-dependent quantum efficiency measurements, while the avoided crossings in Figs. 6c–d should appear as anticrossings in electroabsorption spectra under AC modulation. The Rabi splitting magnitude provides a direct measure of the effective transition dipole moment, offering a spectroscopic route to quantifying field-matter coupling strengths.

This comprehensive quantum mechanical framework establishes that AC field engineering through Al-LSPR not only enhances local field amplitudes but also creates fundamentally new dissociation pathways through photon-assisted tunneling and Floquet dressing. The demonstrated effects provide a solid theoretical foundation for the superior photocatalytic performance observed in Al-doped TiO₂ systems and offer clear strategies for optimizing next-generation bio-hybrid photocatalysts through synergistic field and frequency control.

The multi-photon tunneling mechanism explains the enhanced photocatalytic activity observed under resonant illumination, where the AC field enables charge injection even when photon energies are insufficient for direct band-to-band transitions. This represents a paradigm shift from conventional sensitization approaches, as it leverages the temporal structure of the optical field rather than merely its spectral content. The Floquet engineering approach further demonstrates that dynamic Stark effects can be harnessed to actively control exciton dynamics in real-time, opening possibilities for optically gated photocatalytic systems where reaction rates can be modulated on ultrafast timescales.

Collectively, these results confirm that Al-induced plasmonic fields in quercetin–TiO₂ hybrids not only amplify local electric fields but also introduce new quantum transport channels through photon-assisted tunneling and Floquet dressing. This unified framework explains the superior photocatalytic efficiency of Al-doped TiO₂ systems and establishes guiding principles for field-optimized bio-hybrid photocatalyst design.

Figure 8 presents the quantitative evolution of the duty-cycle ratio (η) and the corresponding effective transmission (Tₑff) as functions of the normalized AC field amplitude (). These results directly implement the analytical framework of Eqs. (43–45), describing how dynamic field modulation drives the transition from intermittent exciton ionization to continuous dissociation within the quercetin–Al:TiO₂ hybrid system.

(a) Duty cycle η from Eq. (43) showing the transition from sub-threshold confinement to continuous dissociation. (b) Effective transmission from Eqs. (44)–(45) highlighting nonlinear enhancement via . Together the plots quantify how oscillatory fields convert intermittent exciton ionization into continuous Stark-driven dissociation in quercetin–Al:TiO₂ hybrids.

The left panel presents the evolution of the duty cycle derived from Eq. (43). At low modulation amplitudes where , the field never exceeds the dissociation threshold and η remains near zero, signifying complete exciton confinement. As approaches 0.5, η increases sharply, indicating that portions of each optical cycle now exceed the critical field and initiate transient dissociation. When surpasses approximately 2.0, η asymptotically approaches unity, marking the onset of continuous field-driven ionization where the instantaneous field remains above for nearly the entire oscillation period. This behaviour captures the smooth transition from sub-threshold confinement to sustained dissociation as the oscillatory amplitude grows. The right panel reveals the corresponding effective transmission calculated from Eqs. (44)–(45). The dashed curve represents the simple threshold-weighted model where enhancement arises only from temporal duty-cycle expansion, while the solid curves for β = 1.0–2.0 include the modified Bessel term that accounts for nonlinear field–exciton coupling. At β = 2.0, the transmission reaches at , corresponding to a roughly 37% increase over the linear model. The enhancement begins precisely at the threshold condition , confirming the analytical prediction of Eq. (49) and validating the physical interpretation of the threshold-crossing mechanism.

The results delineate three dynamical regimes of field-driven dissociation. In the sub-threshold regime (), excitons remain localized and no effective ionization occurs. In the threshold-crossing regime (), partial dissociation takes place during the fraction of each cycle where the field exceeds the critical value, and η quantifies this active window. In the continuous regime (), dissociation becomes sustained as the system remains predominantly above threshold, although practical saturation effects eventually limit further enhancement. The nonlinear term encapsulates the non-perturbative Stark coupling between the field and excitonic wavefunctions, leading to exponential amplification of the dissociation probability that cannot be captured by temporal weighting alone.

These results also clarify the synergy between duty-cycle modulation and plasmonic near-field enhancement. Aluminum LSPR fields typically reach amplitudes of 0.8–1.5 V·nm⁻¹, corresponding to ratios of 1–2, which coincide with the optimal enhancement regime predicted by the model. The rapid rise of η near unity explains the nonlinear dependence of photocatalytic activity on illumination intensity often observed in experiments on plasmon-enhanced TiO₂ systems. Larger β values, associated with stronger transition dipole moments and tightly bound Frenkel excitons characteristic of quercetin, yield steeper transmission growth, demonstrating that strong binding can in fact facilitate efficient field-driven dissociation when modulated by AC fields.

From a design perspective, these findings establish quantitative principles for optimizing Stark-driven charge separation in bio-hybrid photocatalysts. Plasmonic nanostructures should be engineered to sustain ratios around 1–2 where both duty-cycle and nonlinear effects act cooperatively. Molecular sensitizers should possess large transition dipole moments, corresponding to higher β values, to maximize nonlinear enhancement. Controlled doping or built-in potentials can be used to tune the DC bias so that closely matches , ensuring efficient threshold activation. The analytical framework directly relates measurable parameters such as , , and β to the dissociation efficiency, providing an experimentally testable route to optimization. Enhancement dominated by duty-cycle expansion exhibits linear scaling with η, whereas nonlinear enhancement displays the exponential dependence described by , offering clear spectral and kinetic signatures for experimental verification.

Beyond this system, the duty-cycle enhancement mechanism constitutes a general paradigm for overcoming activation barriers in field-modulated charge separation. Any molecular assembly with field-tunable energy landscapes can exploit this threshold-crossing strategy, whether driven by plasmonic, photonic, or externally applied AC fields. The formalism developed here thus provides a broadly applicable theoretical tool for designing efficient field-modulated systems across organic photovoltaics, molecular electronics, and artificial photosynthesis, where static fields alone cannot achieve continuous charge liberation. The demonstrated interplay between temporal modulation and nonlinear Stark coupling opens pathways to controlling chemical and electronic reactivity through engineered electromagnetic environments, enabling photocatalytic transformations that are otherwise kinetically or thermodynamically inaccessible.

Figure 9(a)-(d) summarize the multi-scale relationship between quantum-level charge injection, reactive oxygen species (ROS) generation, and macroscopic hydrocarbon mineralization in the quercetin–Al:TiO₂ hybrid photocatalyst under deep-UV field modulation.

Panel 9a shows that both superoxide and hydroxyl radical concentrations increase steeply with interfacial field until saturation sets in near . The rise follows Eqs. (61a–b), where the electron injection flux drives radical generation but is counterbalanced by bimolecular recombination. The hydroxyl pathway dominates due to its higher reaction constant and shorter diffusion length, producing roughly twice the steady-state concentration of superoxide under equivalent fields. The dashed line at marks the Stark threshold beyond which exciton dissociation becomes continuous, while the dotted line identifies the recombination-limited regime where additional field strength yields diminishing returns. The correlation between radical concentration and quantum injection yield becomes clearer in Fig. 9b, which plots and against the electron-injection flux Jₑ(F₀). Both radicals exhibit near-exponential growth at low Jₑ, followed by gradual saturation at higher fluxes. This behavior reflects the balance between injection-driven formation and field-independent recombination, consistent with Langmuir–Hinshelwood-type surface kinetics. The curvature of the plots indicates that even modest increases in Jₑ near threshold conditions can produce disproportionate gains in ROS concentration, providing a mechanistic basis for the observed intensity-dependent nonlinearity in plasmon-enhanced photocatalysis. Panel 8(c) presents pseudo-first-order mineralization behavior ( vs ) derived from Eq. (64). Straight-line fits validate the assumption of first-order dependence with respect to hydrocarbon concentration, while the slope increases systematically with . The apparent rate constants rise from 0.40 h⁻¹ at 0.5 V·nm⁻¹ to 0.54 h⁻¹ at 2.0 V·nm⁻¹, demonstrating that interfacial field strength directly modulates oxidation kinetics through enhanced ROS availability. Panel 8(d) quantifies the dependence of on according to Eq. (63). The curve exhibits rapid exponential growth followed by saturation beyond . The threshold at coincides with the onset of the continuous Stark regime in Fig. 8a, confirming that radical production and macroscopic degradation share a common physical origin, the field-driven transition from bound excitons to free carriers.

The observed behaviors collectively establish a coherent mechanism linking quantum-scale field modulation to macroscopic oxidation kinetics. The progression from exciton dissociation to ROS generation and finally to hydrocarbon mineralization is captured quantitatively by the sequence:

confirming that the Stark field acts as the governing parameter for charge injection and chemical reactivity. The predominance of hydroxyl radicals as the reactive species is consistent with prior literature on TiO₂ photocatalysis (Li et al., Appl. Catal. B 2022; Ahmed et al., J. Phys. Chem. C 2023), where •OH radicals are responsible for oxidative cleavage of C–H and C = C bonds in hydrocarbon chains.

The nonlinear field dependence of both and further aligns with the predicted exponential-to-saturated scaling described by Eqs. (61)–(63). Importantly, the saturation regime identified here corresponds to practical limits in plasmon-enhanced photocatalysis, where further increases in field amplitude yield minimal kinetic gain due to recombination constraints. These results confirm that the optimal operation window for quercetin–Al:TiO₂ hybrids lies between 1.0 and 2.0 V·nm⁻¹, where field-assisted injection, ROS yield, and catalytic efficiency are simultaneously maximized.

The comprehensive correlation between interfacial field strength, radical generation, and degradation kinetics provides compelling evidence that the quercetin–Al:TiO₂ hybrid system operates through a Stark-enhanced radical oxidation mechanism. The electric field effectively bridges quantum excitation and chemical transformation, enabling controlled tuning of photocatalytic efficiency. Theoretical predictions from Eqs. (61–64) are quantitatively verified by the observed trends, establishing a predictive framework for designing deep-UV-driven bio-hybrid photocatalysts with optimized field–reactivity coupling.

This study provides a rigorous quantum-mechanical framework that fundamentally advances our understanding of photocatalytic charge separation in bio-hybrid systems, specifically addressing the critical challenge of Frenkel-exciton dissociation in quercetin–Al:TiO₂ composites. By implementing an asymmetric Pöschl–Teller potential to model the excitonic system and incorporating the synergistic effects of intrinsic Stark and plasmonic fields, we have demonstrated a novel pathway for enhancing charge separation efficiency without external bias. The key findings are identification of a critical Stark threshold (~ 0.25 V nm⁻¹), characterization of nonlinear AC field modulation through Bessel function formalism, and quantification of duty-cycle, dependent activation, collectively validate the proposed DC–AC field synergy and provide a comprehensive mechanistic picture of the dissociation process.

The universal scaling law derived from these elements represents a significant advancement beyond conventional doping strategies, offering a predictive tool for designing self-tuning photocatalytic systems. While the model's one-dimensional formulation and exclusion of dynamic environmental effects present limitations, they also delineate clear directions for future research. Extension to multidimensional configurations, experimental validation through electroabsorption spectroscopy, and systematic investigation of defect–performance relationships emerge as critical next steps. This work not only provides a theoretical foundation for understanding field-enhanced exciton dynamics but also establishes a transformative paradigm for developing highly efficient, autonomously driven photocatalytic materials, bridging quantum mechanical principles with practical environmental applications and opening new avenues for sustainable solar-based remediation technologies.