- The paper develops a TD-LDA and semi-classical screening framework showing that heating gold electrons to 2000 K makes Re d⊥ about 10% more negative near 1.5 eV, indicating stronger surface spill-in.
- The paper proposes injecting sub-100 fs ballistic hot electrons through an Au/Fe/MgO heterostructure to modulate the mirror’s optical response while protecting the nanoparticle from direct optical damage.
- The paper’s simulations and circuit model show that non-local effects strongly shift 1 nm gap plasmons, including a reported second-order Feibelman correction and an approximately 50% cancellation between bulk-heating and surface-response contributions.
This paper presents a combined theoretical and numerical framework for actively modulating the quantum (non-local) optical response of plasmonic nanocavities on ultrafast timescales, using laser-induced ballistic hot electron injection as the control mechanism (2603.14905). The central object of study is the nanoparticle-on-mirror (NPoM) geometry, whose sub-nanometre-to-nanometre gaps make the optical response acutely sensitive to the Feibelman parameter d⊥, and whose practical use in time-domain experiments has been obstructed by low optical damage thresholds and strong field enhancement in the gap.
Microscopic model of temperature-dependent non-locality
The authors quantify optical non-locality at a gold surface via the Feibelman parameter d⊥, the first moment of the induced charge density, which encodes spill-out/spill-in behaviour and Landau damping. Rather than a full quantum treatment of the excited state—which they acknowledge as a significant challenge—they compute d⊥ within a TD-LDA jellium framework in the non-retarded limit, with three key refinements: (i) a stabilized jellium model, (ii) static, temperature-dependent screening of the ground-state density, and (iii) a semi-classical s–d model in which the d-band screening dielectric function is taken from a temperature-dependent two-band (Rosei-type) description of gold, including fractional-Umklapp electron–electron scattering and electron–phonon damping.
A notable validation of the model is the computed work function, W=5.37 eV, in close agreement with the experimental range of 5.3–5.5 eV, whereas plain LDA underestimates it by more than 1 eV. The calculated room-temperature d⊥ exhibits the negative real part (spill-in) expected for noble metals, with a pronounced dip near the screened surface plasmon at ~2.5 eV, and quantitatively reproduces the semi-classical specular reflection model (SRM) below the interband threshold—diverging above it, since the free-electron model only accounts for d-band screening.
Upon heating to Te=2000 K, the dip in Red⊥ is smeared out and, critically, in the 1–2 eV window relevant to NPoM gap plasmons, Red⊥ becomes more negative: hot electrons exhibit stronger spill-in. Around the typical gap-mode energy of 1.5 eV, variations of roughly 10% are predicted. The decomposition shows that static and dynamic screening changes dominate, while damping contributes negligibly at this scale. The implication is that electronic temperature is a viable control knob for mesoscopic surface response, with a measurable, sign-definite effect at plasmonically relevant photon energies.
Proposed experimental scheme
To circumvent direct optical heating of the fragile nanoparticle, the authors propose a Au/Fe/MgO heterostructure: a 50–100 nm epitaxial gold mirror atop a 5–10 nm iron layer, with a chemically assembled gold nanosphere forming a 1 nm (alkanethiol or cucurbituril spacer) nanocavity. An ultrashort pump pulse absorbed in Fe injects hot electrons through the Fe/Au interface; owing to the energy-dependent electronic transmission and the ~100 nm ballistic mean free path in gold, sub-100 fs bursts of energetic electrons traverse the full gold layer at vF∼1 nm/fs and perturb the mirror's electronic configuration—modifying εAu and the surface spill-in—while the nanoparticle remains cold. A delayed, weak probe monitors transient dark-field scattering. The key practical advantage is that strong perturbation of the electronic density is achieved without irreversible optical damage, since the pump energy is absorbed away from the nanocavity and the probe can be weak. This builds on established ultrafast spin-polarized carrier transport measurements in Au/Fe/MgO and on recent spectroscopic evidence of transient spill-out modulation in plasmonic bilayers.
The authors are candid that hot electron injection may not be the sole modulation mechanism: molecular motion in the gap, chemical transformation, surface charging, and density variations can also contribute. They also note that their thermodynamic treatment assumes a thermalized Fermi–Dirac electronic distribution at d⊥0, reached a few hundred femtoseconds after a ≤50 fs pump; stronger modulation at earlier, non-thermal delays cannot be excluded.
Electromagnetic simulations of non-equilibrium NPoMs
Using a three-dimensional extension of the mesoscopic Feibelman boundary-condition formalism implemented in COMSOL via weak-form integrals, the authors compute dark-field scattering of 50 nm-radius Au NPoMs with gap widths of 1–5 nm. The model is validated in the d⊥1 limit against purely local calculations. Non-local corrections grow as the gap narrows, and the 1 nm gap is selected as the practical optimum, since below ~0.5–1 nm tunnelling dominates and gap uniformity becomes problematic.
With the reference value d⊥2 Å for gold at 790 nm, the simulations yield a resonance sensitivity of d⊥3 nm/Å, essentially independent of temperature. A discernible second-order correction, d⊥4 nmd⊥5, is reported—claimed to be the first identification of a second-order Feibelman-like contribution, underscoring the exceptional susceptibility of gap plasmons. Both heating of the mirror and increased spill-in redshift the gap mode. Notably, the d⊥6 dependence partially cancels the bulk d⊥7 contribution: the total heating-induced resonance shift is reduced by approximately 50% relative to a local-response treatment. This cancellation is directly observable in pump–probe experiments and implies that non-local corrections must be included to correctly interpret transient spectroscopy of hot-electron-heated nanocavities. The simulations assume d⊥8, justified by the microscopic model below the interband onset, so linewidth modulation is not captured.
Analytic non-local circuit model
To complement the numerics, the authors extend the optical circuit model of coupled plasmonic dimers with Feibelman non-locality, introducing a surface impedance element d⊥9 in series with the nanoparticle inductance (an additional capacitance for negative d⊥0), and renormalizing the gap capacitance through an effective gap width d⊥1. The base model with d⊥2 severely underestimates the non-local effect—a deficiency traced to the violation of the dihedral-capacitor approximation by the in-plane gradient in the boundary condition—while d⊥3 yields excellent agreement with the FEM results across both gap widths and mirror temperatures, despite the dimer model not explicitly incorporating the heated-mirror asymmetry. With no free parameters beyond d⊥4, the model reproduces the 805.6 nm resonance (vs. 804.9 nm in FEM) and the thermal shift via d⊥5 at 2000 K.
A cross-check on sensitivity gives d⊥6 near d⊥7 nm, indicating that gap renormalization in the non-local regime is more complicated than a simple d⊥8-mediated displacement of the gap width. The circuit model thus provides an intuitive, single-parameter analytic tool for non-local NPoMs out of thermal equilibrium.
Limitations and open questions
Several assumptions bound the quantitative predictions. The microscopic model is a free-electron jellium treatment with semi-classical d-band screening, valid only below the interband threshold and at a planar interface; extension to metal/dielectric interfaces is asserted to be straightforward but unverified. The full quantum treatment of non-locality in the optically excited state remains an open problem, and d⊥9 is treated as an independent control parameter in the electrodynamics rather than computed self-consistently. The phenomenological coefficient W=5.370 lacks a first-principles derivation. Competing modulation channels (molecular dynamics, charging, chemistry) are acknowledged but not quantified, and the behaviour of non-thermal electronic distributions at sub-100 fs delays is unexplored. Experimentally, the proposal has not yet been demonstrated; whether the predicted ~10% modulation of W=5.371 and the associated resonance shifts survive in chemically assembled cavities with realistic gap disorder is the central open question the paper leaves to future measurement.
Conclusion
This work connects equilibrium quantum plasmonics to non-equilibrium carrier dynamics: it provides a microscopic TD-LDA-based prediction of temperature-dependent Feibelman non-locality in gold, a damage-free experimental route to impose it via ballistic hot electron injection into NPoM mirrors, three-dimensional FEM simulations quantifying the resulting gap-plasmon shifts (including a reported second-order Feibelman correction and a ~50% cancellation between bulk and surface contributions), and an analytic circuit model reproducing the numerics across a broad parameter space. The framework establishes ultrafast non-local plasmonics as an experimentally accessible target and raises specific follow-up questions concerning non-thermal regimes, asymmetric spill-in/spill-out metal pairs, and dynamic control of tunnelling in sub-nanometre gaps.