Advances in modeling plasmonic systems

The field of plasmonics has been in the spotlight in physics,¹ materials science,² chemistry,³ and biology⁴ for more than a decade, with relevant applications in photovoltaics,⁵ biosensing,⁶ and nanoimaging.⁷ Thanks to fascinating progress in nanofabrication, we now have tools to create artificial metal/dielectric interfaces with 1 nm precision,^8,9 while ultrafast optical detection techniques are now offering riveting time resolution as short as 2 fs.¹⁰ Combined together, these tools are used to further advance our understanding of light–matter interaction at the nanoscale when the classical view collides and merges with quantum properties of matter and light.¹¹ The ability of plasmonic materials to sustain surface plasmon-polariton (SPP)¹² resonances results in a very small SPP mode volume. Even though the characteristic quality factors of such modes are relatively low, large local field enhancements make such systems very attractive for numerous applications in chemistry and biology.^6,13 Consequently, such an impressive spatiotemporal resolution, on the one hand, and high local electromagnetic (EM) field enhancement at plasmonic interfaces, on the other, led to the idea of adding physical moieties into plasmonic systems.¹⁴ These include molecular aggregates, transition metal dichalcogenide monolayers, graphene, and various types of semiconductor systems (quantum dots, nanowires, etc.)—all exhibiting optical resonances in the mid-/near infrared (IR) and visible region. Unlike conventional applications in plasmonics for chemical detection, where chemical/biological samples are passive, the research of exciton–plasmon systems is far-reaching.¹⁵ 

Concurrent with experimental nanophotonics developments in fabrication and characterization, theory and computational methods have been developed and applied for interpreting experimental results, gaining insight into the underlying photonic mechanisms and enabling design for specific functional responses. Modeling of plasmonic materials and light–matter interaction phenomena at the nanoscale employ different theoretical methods ranging from classical electromagnetic theory to first-principles electronic structure in a multidisciplinary approach.

The field of computational/theoretical plasmonics has significantly advanced in recent years, in particular, in moving from classical electrodynamics to methods that include atomistic effects^16–21 or quantum effects,^22–29 and in developing methods to treat strong coupling and^30–35 cavity phenomena,^36–43 as well as improvements in treatments for large-scale ab initio computations.^44–47 This Special Issue highlights recent theoretical advances in different plasmonics research areas, from ab initio plasmonics for modeling excitation energy to strong-coupling phenomena in exciton–plasmon systems and to nonlinear effects in phonon-polariton systems.

The name ab initio plasmonics^48–52 has been used in recent years to indicate the theoretical description of plasmonic systems using ab initio quantum-chemistry methods such as the time-dependent density-functional theory (TD-DFT).⁵³ Compared to classical electrodynamics, TD-DFT includes all atomistic and quantum effects but neglects retardation effects, which indeed will not play a significant role, considering the small system size that can be (currently) treated by TD-DFT, i.e., about one thousand atoms.⁵⁴ Despite limitations in the choice of the exchange–correlation (XC) functional and the high computational cost, TD-DFT is often considered as a reference method for the description of quantum effects in plasmonics.⁵⁵ 

In the context of ab initio plasmonics, a relevant recent issue is the characterization of the degree of plasmonicity of each absorption peak. In fact, in the absorption spectrum computed using TD-DFT or similar ab initio methods, all excited states are present, including single-particle transitions; thus, it is important to distinguish the true plasmon peaks from states with other characteristics.

Among the different plasmonicity indices present in the literature,^56–60 two are discussed in this Special Issue: the energy-based plasmonicity (EPI) index developed by Muller and co-workers⁶¹ and the one by Gieseking and co-workers,⁶² which is based on the combination of three indicators: collectivity, dipole-additivity, and super-atomic character. In this Special Issue, the EPI approach is validated in hollow spherical gold nanoparticles,⁶³ while the three-indicator approach is further validated in silver nanowires and nanorods; see Ref. 64.

Another relevant topic for ab initio plasmonics is the development of efficient and accurate TD-DFT approximations, which can be applied for the description of large plasmonics systems. The research group of Visscher has developed a parameter-free tight-binding approximation to linear-response TD-DFT, named TD-DFT+TB,^47,65 which extends the original time-dependent density functional tight-binding (TDDFTB),⁶⁶ and it is similar to other approaches.^45,67,68 The TD-DFT+TB is based on the “exact” ground-state Kohn–Sham orbitals and eigenvalues and on a strongly simplified TD-DFT kernel matrix, reducing the computational cost by about two orders of magnitude while keeping the accuracy very high (deviation of about 0.1–0.2 eV). In this Special Issue, the TD-DFT+TB has been applied to silver and gold dimers,⁶⁹ which represent a very relevant and widely investigated plasmonics configuration to create a very large field enhancement in between the two monomers. The authors show that the TD-DFT+TB can correctly describe both the bonding plasmon and the charge-transfer plasmon for different inter-monomer distances.

At the same time, developments of real-time TDDFT (RT-TDDFT)^70,71 can provide insight into dynamics underlying mechanisms.⁷² When combined with the efficient DFTB scheme,^73,74 the RT-TDDFTB has been used to describe plasmonics systems.⁷⁵ Although RT-TDDFTB can be employed to probe dynamics, inaccuracies can arise due to the approximate DFT method and the adiabatic approximation. Aikens and co-workers⁷⁶ examined in this Special Issue a dimer of Ag₁₄ clusters as a model system for studying excitation energy transfer in a plasmonic dimer, following earlier work,⁷⁵ which enabled new insight and elucidation of general trends, noting that relatively good agreement between TD-DFT and TDDFTB for silver nanorod model systems was previously achieved.⁷⁷ Analyses of simulations up to 1.2 ps as a function of distance between the clusters, assuming the dipole approximation, demonstrated two regimes, where up to 0.7 ps, the dipole moment of the dimer derives mostly from the first nanoparticle, while this trend reverses at longer time dynamics. More efficient excitation energy transfer is simulated for distances smaller than 5 nm, while at distances less than 1.5 nm, the opposite occurs, indicating an interplay between the excitation of the first nanoparticle and the back transfer from the second nanoparticle.

Ab initio TD-DFT methods, as well as related tight-binding approximations, describe all types of excitations, including single-particle ones, as these approaches are based on Kohn–Sham orbitals. For plasmonics applications, it is often enough to compute only the collective excitations (i.e., the surface plasmon resonances). To this end, the quantum hydrodynamic theory (QHT)^23,25 and the time-dependent orbital-free DFT^78,79 are attracting strong interest for modeling plasmonics systems, as these methods are orbital-free, i.e., they require only the ground-state density, and the final accuracy depends only on the approximations made for the (universal) kinetic energy functional.

In the review in Ref. 80, Della Sala has described in detail the various orbital-free methods for plasmonics, ranging from the semiclassical approximation to QHT, covering the past literature as well as the most recent developments.^29,81 The various orbital-free approaches have been derived directly from the linear-response TD-DFT, thus clarifying the origin of the different approximations. The different methods have then been tested and compared for jellium nanospheres, which represent the most widely used theoretical model for metallic nanoparticles.

When the coupling strength of a local EM field and an exciton surpasses all the damping rates in a system, the system enters the strong coupling regime.¹¹ In such a regime, new hybrid states called upper and lower polaritons are formed.¹⁵ Under strong coupling conditions, these states have a mixed exciton–plasmon character, which has been observed experimentally in different systems^56,82,83 and modeled at different levels of theory.^{11,30–35,84,85} Strongly coupled materials are a great testing ground for conventional semi-classical physical models. Additionally, it has been recently argued that such systems can be used to control chemical reactions.^86,87 Conventionally, the strong coupling dynamics is treated using a semiclassical approach that relies on the quantum nature of emitters while approximating EM dynamics by classical Maxwell’s equations. At the heart of this model is the mean-field approximation that neglects all emitter–emitter quantum correlations. Despite these approximations, it has been shown that such a model predicts many experimental observations in both linear⁸⁸ and nonlinear⁸⁹ optical regimes, properly capturing polaritonic state dynamics on a qualitative level. However, it clearly lacks many fascinating features of a fully quantum treatment, including quantization of EM radiation, photon statistics, and entanglement, along with the quantum dynamics of the emitters, which can be described within a quantum electrodynamics (QED) framework. This presents theoretical challenges as it usually leads to a set of coupled differential equations that must be solved numerically. Recent QED developments have emerged, based on DFT,^36,37 coupled-cluster theory,^40,42 or other methods.⁴³ 

In this Special Issue, Maitra and co-workers⁹⁰ investigated the photon dynamics for strong light–matter coupling using classical trajectories,^91,92 which can be used to describe the coupling to many photon modes.⁹³ Different quasi-classical methods have been compared to the exact factorization approach.^94,95 The authors found an underestimation of the photon number and intensities, which is related to the approximate description of the matter-photon coupling in the treatment of the independent-trajectory Ehrenfest coefficients.

In Ref. 96, Wang and co-workers combined a previously developed theory,^97,98 which describes a molecular emitter strongly coupled to SPP in the framework of macroscopic quantum electrodynamics (MQED),^41,97–99 with a pseudo-mode approach. The resulting method allows us to estimate the light–matter coupling strength and optical loss without free parameters. In addition, the authors demonstrated, both analytically and numerically, its equivalence with the cavity QED-based Lindblad master equation.

Due to the general complexity of systems that merge several spatial and time scales, analytical work is rare and sparse. The research group of Shahbazyan has been developing semi-analytical tools for quantum plasmonics for many years. They were the first group to consider and describe in detail the effects of super-radiance for quantum emitters driven by the near-field of a plasmonic nanoparticle.¹⁰⁰ In this Special Issue, Shahbazyan and co-workers¹⁰¹ consider coupling between the bright (collective) mode of quantum emitters with an SPP mode. They investigate how the optics of such a system varies when the light–matter coupling goes from weak to strong, giving rise to polaritonic states. The major highlight of this work is the prediction of a Fano interference¹⁰² between SPP and SPP-induced bright states. This has been achieved solely by the authors’ analytical approach, helping to dissect complex contributions from various interaction channels. The authors analytically derived the optically induced dipole moment of a plasmon–exciton system in a linear limit. Several distinct contributions were identified and noted, ranging from dressed plasmons (each quantum emitter is driven by a near-field of SPP), through a bright state coupled to SPP (a collective state of quantum emitters coupled to SPP), to the dark state (response of molecular frequency), and interference term. The latter is responsible for the Fano interference mentioned above.

In Ref. 103, Herrera and co-workers present a semi-empirical phenomenological method to model a coupled molecule–resonator system by the Lindblad form of the master equation that describes driving and dissipation in the evolution of the reduced density matrix. In investigating the Rabi splitting due to strong coupling, the authors show that when probing by a broadband tip when bringing the tip closer to the molecular layer for the particular system modeled, the Rabi splitting would disappear because of an increase in the tip-vibration coupling strength beyond the antenna and vibrational linewidths. Interestingly, the line shape of the Rabi split side bands is modified due to destructive and constructive Fano interference between overlapping response functions. A local anharmonic vibrational Hamiltonian was developed within the framework, showing that a vibrational blockade effect occurs in the weak coupling regime for strong pulses that excite the second vibrational level, which could be considered for implementation of optical phase gates in the mid-IR.

In Ref. 104, Mondal and co-workers consider systems comprising single molecules and gold and silver nanoparticles of several geometries using the Nitzan–Gersten theory.¹⁰⁵ This work studies in detail the molecule–SPP coupling and corresponding relaxation rates. It is shown that if sources of broadening due to radiative and non-radiative relaxation for molecular vibrational transitions including metal-induced effects and relaxation are eliminated, a necessary but not sufficient criterion to observe molecule–SPP strong coupling is often satisfied by many configurations for a wide range of metal-to-molecule distances, implying the possibility to observe Rabi splitting even in a single molecule case.¹⁰⁶ 

Continuing in line with analytical works, the manuscript by Cortese and De Liberato¹⁰⁷ takes a giant step forward in quantum cavity electrodynamics. The authors describe fully analytically a general polaritonic problem involving cavities (with and without losses) and matter described as a set of oscillators in the strong coupling regime. The full diagonalization is performed for arbitrary losses in the Power–Zienau–Woolley representation. Taking the seminal work of Fano,¹⁰⁸ they expand his approach by deriving exact solutions for polaritonic systems. The theory is applied to two special cases: both cavity and matter resonances are described by Lorentzians, and the system has a resolved fixed absorption band. The authors also provide a Matlab code, in which all the results are combined.

Another significant contribution from the De Liberato group¹⁰⁹ is the derivation of a full quantum theory for longitudinal-transverse polaritons (LTP), i.e., with transverse photonic fields and a longitudinal phonon field. These hybrid states are present in polar materials with a non-local (i.e., spatially dispersive) dielectric constant.^110,111 The equations of motion for LTPs are shown to be equivalent to the nonlocal Maxwell equations, allowing us to use conventional numerical tools, and the derivation of the non-local Purcell enhancement factor highlights the relevance of LTPs for mid-infrared optoelectronics devices.¹⁰⁹ 

Notwithstanding tremendous progress in plasmonics and strongly coupled nano-systems, scientific discussions are primarily focused on linear optical phenomena, with much less investigation into nonlinear optics, e.g., second harmonic generation.^89,112,113 Although nonlinear plasmonics has been a subject of intense research for some time,^114,115 nonlinear optics of polariton systems has just recently become a hot field. Combining the nonlinear response from plasmonic materials with the semi-classical description of molecular excitons, for instance,¹¹⁶ it was demonstrated that the second harmonic signal produced by periodic arrays of nanoholes coupled to molecules exhibits pronounced Rabi splitting—the very signature of the strong coupling imprinted in the nonlinear signal.¹¹⁷ Later, this prediction has been experimentally confirmed.⁸⁹ 

In their manuscript, Sugiura and co-workers¹¹⁸ consider phonon-polariton materials and examine nonlinear optics of such systems under pump–probe conditions, when the pump excites the upper polaritonic branch, which in turn changes the overall optical response when probed by a low-intensity pulse. The authors build a self-consistent nonlinear model for phonons using the Floquet formalism and include the second order nonlinearity coupling in the resulting equations with Maxwell’s equations. They examine conditions under which the material generates a nonlinear response analogous to four-wave mixing. It is shown that such a system can also support lasing modes in a deep THz regime. Simulations are performed for experimentally realizable parameters describing the one-dimensional interface of a SiC insulator.

In another article in this Special Issue, Herrera and Litinskaya¹¹⁹ develop a macroscopic model of exciton–plasmon metamaterials composed of nanoparticle dimers with organic molecules (chromophores are considered in the paper) situated inside the gap. The strong coupling is achieved between molecular vibrations and the gap plasmon modes¹²⁰ associated with a dimer. The authors expand on their previous work,¹²¹ in which the theory of the gap plasmon–molecule coupling was established, and consider a macroscopic slab comprising many dimers. Using semiclassical theory, the effective susceptibility of such a system is calculated. The evaluated response depends sensitively not only on the dimer’s geometry but also on the signal field, which in turn can be used as a tuning knob to achieve desired phase modulation. It is demonstrated that under strong coupling conditions, the incident radiation acquires a well-defined phase. The properties of the induced phase are thoroughly analyzed. The optical phase switching mechanism is discussed, at which the probing field going through an ensemble of nanodimers remains coherent and thus can predictably interfere with the reference field.

Computational work in plasmonics is primarily centered on the numerical integration of Maxwell’s equations for a given experimentally realizable geometry, which has been proven many times to be quite reliable.

In many experiments, periodic arrays of nanoparticles (or nanoholes milled in a thin otherwise continuous metal film)¹²² are used to investigate the near-field dynamics of quantum materials driven by SPP.^83,123 The biggest challenge in such simulations is usually associated with Wood’s anomaly (also referred to as Rayleigh’s anomaly)—an optical mode induced at a wavelength that coincides with an array’s period. Such a mode gives rise to plasmonic surface lattice resonances—extremely narrow peaks in transmission/reflection spectra, which find a wide range of applications, including SPP induced lasing.¹²⁴ Numerically, such modes are notoriously difficult to simulate since the electromagnetic waves associated with them span large spatial distances, which in turn require large simulation domains, making simulations very memory-consuming.

In this Special Issue, Han et al.¹²⁵ develop a semi-analytical approach to describe the optical response of two-dimensional arrays of nanoparticles. Their approach is free from numerical problems. It agrees perfectly with several other methods but requires little numerical effort. Moreover, this work paves the way for properly describing the lineshapes of lattice resonances—a notoriously hard task in every numerical simulation.

Modeling the optical response of metallic nano-spheroids, which would not require computationally intensive calculations yet provide accuracy beyond the Rayleigh–Gans theory for spheroids and an efficient method to compare to experiment, particularly for medium-size and larger spheroids, is detailed in this Special Issue by Le Ru and co-workers.¹²⁶ Based on the authors’ recent derivation of a Taylor expansion of the full T-matrix formalism,¹²⁷ a benchmarking study of the extinction spectra of oblate and prolate Ag spheroids in water vs exact T-matrix calculations was performed. The coefficients of the Taylor expansion are expressed in terms of the relative refractive index, including the homogeneous and non-absorbing medium of the spheroid and its aspect ratio. The dipolar approximation was considered, as well as quadrupole and higher order resonances, with a focus on orientation-averaged cross sections, where both the longitudinal and transverse plasmon resonances are included. Calculations with the modified long-wavelength approximation (MLWA) and extended MLWA, as well as other related approximations, demonstrate improved agreement with the exact values, specifically for both oblate and prolate nanoparticles, the transverse resonance, and for small aspect ratios, namely, spherical-like spheroids.¹²⁶ Comparative analyses for gold spheroids demonstrated similar results, and when considering larger spheroids, where higher order resonances become important, resulted in improved agreement with the exact results. Although there are still some discrepancies, the proposed method is suitable for practical applications, for example, refractive index sensing and surface-enhanced Raman scattering.

The contributions presented in this Special Issue cover several aspects of computational plasmonics and demonstrate the vivacity of the field. Example investigations range from first-principles calculations to analyze excitation energy transfer in plasmonic nanoparticle dimers, methods to model strongly coupled molecular-plasmon resonances, including nonlinear effects, e.g., for phonon-polariton systems, to computationally efficient semi-analytical methods to explain phenomena in large plasmonic systems. The development of theoretical methods with a higher accuracy/computational cost ratio, the coupling of different methods for a multiscale description of plasmonics systems, and the derivation of simple models to explain experimental results are very active fields, and many other relevant advances can be expected in the near future. Combining the advances in theoretical and computational plasmonics in this Special Issue is important in providing the basis for motivating exploration of emerging plasmonics concepts, further theoretical advances, and experimental fabrication and characterization.

We acknowledge the authors who contributed original works to this Special Issue, the JCP editor, Professor Lasse Jensen (Pennsylvania State University), and the editorial staff. We also thank Professor Prineha Narang (UCLA) for her contribution as a guest editor of this special issue. M.S. is grateful for the financial support from the Air Force Office of Scientific Research under Grant No. FA9550-22-1-0175.

The authors have no conflicts to disclose.