Plasmonic nanostructures exhibit unique optical properties due to their ability to confine electromagnetic fields at subwavelength scales. One of the most intriguing phenomena in these systems is the generation of hot carriers—high-energy electrons and holes created upon plasmon decay. Theoretical models of hot carrier generation provide insights into the underlying physics, including energy distributions, relaxation mechanisms, and extraction efficiencies. These models rely on frameworks such as Fermi-Dirac statistics, Boltzmann transport equations, and quantum kinetic approaches to describe non-equilibrium carrier dynamics.

The initial step in modeling hot carrier generation involves understanding the plasmon decay pathways. When a plasmon decays, it can generate electron-hole pairs through direct interband or intraband transitions. The energy distribution of these carriers is determined by the plasmon’s spectral characteristics and the electronic structure of the material. Fermi-Dirac statistics play a central role in describing the occupation probabilities of these excited states. At equilibrium, the electron population follows the Fermi-Dirac distribution, but plasmon excitation drives the system out of equilibrium, creating a non-thermal distribution of hot carriers.

The non-thermal nature of hot carriers necessitates the use of kinetic equations to model their evolution. The Boltzmann transport equation is often employed to track the temporal and spatial changes in the carrier distribution function. This equation accounts for carrier generation, scattering, and recombination processes. The collision integral within the Boltzmann equation incorporates electron-electron, electron-phonon, and electron-impurity interactions, which govern the relaxation of hot carriers toward equilibrium. For plasmonic systems, electron-electron scattering dominates at early times, leading to rapid thermalization among the carriers, while electron-phonon scattering becomes significant at longer timescales, dissipating energy into the lattice.

Quantum kinetic models offer a more detailed description by incorporating coherence effects and many-body interactions. These models are particularly relevant for small nanostructures where quantum confinement alters the density of states and scattering rates. The Kadanoff-Baym equations, a quantum generalization of the Boltzmann equation, can be used to describe the non-Markovian dynamics of hot carriers in such systems. These equations account for memory effects and off-shell processes that are neglected in semiclassical approaches. For instance, the impact of plasmon-induced strong light-matter coupling on hot carrier generation can be accurately captured using quantum kinetic theory.

The lifetime of hot carriers is a critical parameter that determines their utility in energy conversion processes. Theoretical studies estimate hot carrier lifetimes in plasmonic nanostructures to range from femtoseconds to picoseconds, depending on the material and excitation conditions. Gold and silver, commonly used in plasmonics, exhibit hot electron lifetimes on the order of 100 fs due to efficient electron-phonon coupling. In contrast, graphene-based plasmonic systems show longer lifetimes, extending into the picosecond regime, owing to weaker electron-phonon interactions and higher carrier mobilities. The lifetime is also influenced by the nanostructure’s geometry, with localized surface plasmons in nanoparticles leading to faster relaxation compared to propagating plasmons in extended films.

Extraction efficiency is another key metric in hot carrier systems, defined as the fraction of generated carriers that can be collected before thermalization. Theoretical models evaluate this efficiency by solving the coupled equations of motion for carriers under external extraction fields. The efficiency depends on the competition between carrier cooling rates and extraction timescales. For instance, in a metal-semiconductor Schottky junction, the extraction efficiency is determined by the hot carrier’s mean free path relative to the barrier width. Materials with longer hot carrier lifetimes and higher mobilities, such as certain transition metal dichalcogenides, are predicted to achieve higher extraction efficiencies.

Spatial nonlocality in plasmonic response further complicates the modeling of hot carrier generation. Nonlocal effects arise when the plasmon’s electromagnetic field varies significantly over length scales comparable to the electron’s mean free path. Hydrodynamic models and density functional theory calculations have been used to incorporate these effects, revealing that nonlocality can alter the hot carrier energy distribution and generation rates. For example, in ultrasmall nanoparticles, nonlocal damping reduces the plasmon’s lifetime and modifies the relative contributions of different decay channels.

The role of interfacial effects is also critical in theoretical descriptions. At metal-dielectric or metal-semiconductor interfaces, the discontinuity in dielectric properties and electronic band structures influences hot carrier generation and transport. Theoretical studies have shown that interface states can act as additional scattering centers, reducing carrier lifetimes, or as selective filters, enhancing extraction efficiencies for specific energy ranges. First-principles calculations combined with nonequilibrium Green’s function methods provide a rigorous framework for modeling these interfacial effects.

Temperature dependence is another aspect explored in theoretical models. Elevated lattice temperatures increase electron-phonon scattering rates, leading to faster hot carrier cooling. However, the interplay between temperature and plasmon resonance broadening can also affect the initial energy distribution of generated carriers. Finite-temperature density functional theory and molecular dynamics simulations have been employed to study these effects, particularly in systems where thermal gradients are present.

Recent advances in computational power have enabled large-scale simulations of hot carrier dynamics in realistic plasmonic nanostructures. These simulations combine electromagnetic finite-difference time-domain methods with electronic transport calculations to provide a comprehensive picture of the entire process, from plasmon excitation to hot carrier extraction. Such approaches are invaluable for designing nanostructures with optimized hot carrier generation and collection properties.

In summary, theoretical models of hot carrier generation in plasmonic nanostructures rely on a combination of statistical mechanics, kinetic theory, and quantum dynamics to describe the complex interplay of light-matter interactions, carrier scattering, and energy dissipation. These models not only deepen the understanding of fundamental processes but also guide the design of plasmonic systems for applications requiring efficient hot carrier utilization. Future directions include the development of multiscale models that seamlessly bridge quantum mechanical descriptions with macroscopic transport phenomena, as well as the integration of machine learning techniques to accelerate the exploration of vast parameter spaces.