Ultrafast Plasmonics Beyond the Perturbative Regime: Breaking the Electronic-Optical Dynamics Correspondence

The transient optical response of plasmonic nanostructures has recently been the focus of extensive research. Accurate prediction of the ultrafast dynamics following excitation of hot electrons by ultrashort laser pulses is of major relevance in a variety of contexts from the study of light harvesting and photocatalytic processes to nonlinear nanophotonics and the all-optical modulation of light. So far, all studies have assumed the correspondence between the temporal evolution of the dynamic optical signal, retrieved by transient absorption spectroscopy, and that of the photoexcited hot electrons, described in terms of their temperature. Here, we show both theoretically and experimentally that this correspondence does not hold under a nonperturbative excitation regime. Our results indicate that the main mechanism responsible for the breaking of the correspondence between electronic and optical dynamics is universal in plasmonics, being dominated by the nonlinear smearing of the Fermi–Dirac occupation probability at high hot-electron temperatures.


Static Optical Response
To determine the optical response of an individual nanostructure upon plane wave illumination, its absorption, scattering and extinction cross-sections, σ A , σ S and σ E respectively, should be defined. By considering a monochromatic linearly polarised plane wave in the quasi-static approximation (QSA), these quantities are readily expressed in terms of the particle polarizability, α, most generally a spectrally dispersed and complex-valued tensor. 1 For the specific case of nanoellipsoids, with wavelength dependent permittivity ε(λ), embedded in a homogeneous medium with wavelength independent dielectric constant ε m , the polarizability tensor α = diag(α x , α y , α z ) has diagonal components given by where i = x, y, z, V is the nanoparticle volume, ε 0 the vacuum permittivity. The coefficients L i are geometrical factors accounting for depolarization effects, reading: 1 f i (q) = 1 (a 2 i + q) (a 2 x + q)(a 2 y + q)(a 2 z + q) , for nanoellipsoids of semi-axes a x , a y , a z . The absorption and scattering cross-sections are then computed as σ A = P A /I 0 and σ S = P S /I 0 , with P A and P S the total power absorbed and scattered by the nano-object, respectively, and I 0 = 1 2 √ ε m cε 0 |E 0 | 2 the incident intensity, given the incident (uniform) electric field E 0 . The total extinction cross-section is then simply given by the sum σ E = σ A + σ S . In particular, regarding the absorbed power, P A is defined as a volume integral of the ohmic losses power density over the nanoparticle. Under QSA, assuming that the incident electric field E 0 induces a point-like electric dipole p = ε m α · E 0 , we have P A = πc λ Im p T · E 0 . On the other hand, the power scattered by the nanostructure can be expressed by introducing in the QSA leading order the effects of radiation reaction onto the induced electric currents, which leads to P S = 4 √ εmπ 3 c 3ε 0 λ 4 |p| 2 . Note that, having an anisotropic polarizability (i.e. α y = α z and α x = α y,z ), spheroids with different orientation of the major axis with respect to the polarization of light exhibit different amplitudes of the two plasmonic oscillations. To mimic the random orientation of the nanorods (NRs) in the sample, the incident electric field in the simulations was written as E 0 = [cos(θ x ), cos(θ y ), cos(θ z )]E 0 , with an average polarization angle θ x = 75 • to best fit the experimental absorbance (θ y and θ z being arbitrarily chosen, in view of the degeneracy α y = α z , under the constraint cos 2 (θ x ) + cos 2 (θ y ) + cos 2 (θ z ) = 1).
Finally, when a realistic macroscopic sample is considered, e.g. an ensemble of single-size nanoellipsoids dispersed in aqueous solution with concentration N p , the optical quantity of interest is the transmission of light, T , by the cuvette sample. Being L the cuvette thickness, Lambert-Beer's law gives For gold permittivity, a Drude-Lorentz model 2 fitted on experimental data 3 was adopted.
Finally, in order to mimic the broadening of the plasmonic resonances caused by size inhomogeneities and scattering defects in the NR ensemble, an increased Drude damping Γ was introduced by following the same approach as the one reported in Ref. 4. We found a good agreement with the linear extinction measurements for Γ/Γ 0 2.5, with Γ 0 = 72 meV the Drude damping in bulk gold. 2

Permittivity Modulation
The temporal evolution of the electronic temperature Θ E determines a variation of the occupation probability of thermalized electrons: 9,10 where f (E, Θ E ) is the Fermi-Dirac distribution and Θ 0 is the equilibrium temperature in static conditions. This modulation entails a modification ∆J T of the Joint Density of States (JDOS) for interband optical transitions near the L-point in the irreducible Brillouin zone, which, in turns, results in the variation of the absorption probability. 5,6 Precisely, where D(E, λ) is the energy distribution of the JDOS of the considered transition, with E the energy of the final state and λ the probe wavelength. Following the well-established 9 approach proposed by Rosei and co-workers, 5,6 we computed D(E, λ) under parabolic band approximation and chose E and E , the effective masses, energy gaps and dipole matrix element as in the previously cited work. 6 The following step consisted in deriving the variation of the imaginary part of the interband dielectric function at the probe wavelength λ: 7 with m the electron mass an P L the electric-dipole matrix element. Finally, Kramers-Kronig relations allow to retrieve the corresponding variation of the real part of the permittivity, Regarding the contribution to permittivity modulation arising from N (t), the calculation of ∆ε N (λ, t) proceeds similarly to ∆ε Θ E (λ, t), the only difference being in the modulation of the electron occupation probability induced by N , which is not a Fermi-Dirac function, but rather a double step-like distribution. 8 ∆ε Θ L (λ, t) is instead retrieved from the modification of the Drude permittivity, since a higher Au lattice temperature results in an increased Drude damping factor 12 and a reduced plasma frequency as a consequence of volume expansion. 9 The total modulation of gold permittivity is finally computed as the sum of the three contributions detailed above, i.e. ∆ε = ∆ε N + ∆ε Θ E + ∆ε Θ L .

Experimental Measurements
The NR samples were synthesized by seedless growth methods 11

Broadband Transient Absorption Spectroscopy
Transient absorption spectroscopy allows for determining the ultrafast dynamics of the photoexcited system over a broad spectral region. Specifically, Figure S1 displays the pump-

The Three-Temperature Model (3TM)
The three-temperature model (3TM) 8 reads as follow: Here, the coefficients a and b are coupling constants detailing the relaxation of non-thermalized electrons via electron-electron and electron-phonon scattering, respectively. C E = γ E Θ E and C L are the electron and lattice heat capacities, with γ E the electron heat capacity constant, whereas G is the electron-phonon coupling coefficient. A detailed discussion on the estimation of all the parameters of the 3TM is reported in Ref. 9. Note that the linear dependence of C E on Θ E , as outlined in the main text, affects the electronic temperature dynamics in the nonperturbative regime. Indeed, such dependence results in an increased time constant for the hot electron temperature relaxation, which tends to become linear in time for very high fluences. The higher is the temperature, the higher is the heat capacity, i.e. the longer is the time required to exchange energy, because of the increased thermal inertia of the electronic population. Furthermore, P abs (t) is the pump pulse absorbed power in unit of volume, representing the driving term modelling the ultrashort pump pulse photoexcitation and expressed as a function of the pump fluence F by In this formula, σ A eff is the effective absorption cross-section at the pump wavelength (λ p = 400 nm) and τ p = 200 fs the half width of the pump signal at 1/e 2 power.

Modelling Au optical nonlinearities
In order to reproduce numerically the transient absorption spectroscopy measurements, we implemented a multi-step model for the photoexcitation and the subsequent energy relaxation processes taking place in plasmonic nanostructures upon illumination with ultrashort laser pulses. Figure Figure S2: Modelling Au optical nonlinearities upon ultrashort laser pulse illumination. Schematic of the algorithm for the implementation of the numerical model employed to predict the dynamical evolution of the differential transmittance signal ∆T /T retrieved by transient absorption spectroscopy experiments.
The photoexcitation level of the structure is set by the fluence of the pump, its value being used to determine P abs (t), the power density (Eq. S11) absorbed by the nano-objects and following the time evolution of the pump pulse. Such quantity, as outlined in Supporting section 5, acts as the drive term of the 3TM, which in turn, when integrated, provides the dynamics of the energetic degrees of freedom of the plasmonic nanostructure, i.e. N (t), Θ E (t) and Θ L (t). With those three time-dependent quantities at hand, the permittivity modulation of Au driven by each of them is then computed, following the procedure mentioned in the Supporting section 2. Specifically, (from top to bottom of the central panel in Fig. S2) the energy density stored in the nonthermal fraction of electrons N (t) results in a modification of the electronic energy distribution ∆f N , the excited electron temperature Θ E determines a change of the hot electron distribution ∆f T , and finally, as a consequence of an increased lattice temperature Θ L , the Drude damping Γ and the plasma frequency ω p are modified. Importantly, based on the same temporal evolution of the electronic temperature, two approaches can be pursued to determine the variation in the thermalised hot electron distribution ∆f T . According to the model we refer to as full in the main text, i.e. fully nonperturbative, ∆f T is at each time instant is given by the rigorous difference between the Fermi-Dirac distribution corresponding to an excited Θ E and the static f (E, Θ 0 ), as written in Eq. S5. Conversely, when the linearised model is considered, such ∆f T is, indeed, linearised with respect to the electronic temperature variation. Its expression, written in the central box as well as in the main text, is given by the product of the temperature increase, ∆Θ E , by an energy-dependent constant factor, computed as the partial derivative of f (E, Θ E ) with respect to Θ E , evaluated at a given temperature Θ corresponding to the modified time-dependent optical properties of the structure, an excited polarizability (Eq. S1) is then computed and employed to determine (by iteratively applying Eq. S4) the dynamical evolution of the system transmission and the differential signal ∆T /T .

Nonthermal electron contribution
To ascertain the role of thermalised and nonthermal electrons as the origin of the non-trivial dynamics of the pump-probe signal experimentally observed and theoretically predicted, we employed our model to disentangle the contributions to the optical modulation arising from the three energetic degrees of freedom of the plasmonic nanostructure, that is, N , Θ E and Θ L . Figure S3 reports such disentanglement of the differential transmittance ∆T /T computed for the wavelength λ 2 = 546 nm examined in the main text, for the simulated fluences F 1 = 0.05 mJ/cm 2 (Fig. S3a-S3d) and F 3 = 0.64 mJ/cm 2 (Fig. S3e-S3h). Consistently with the discussion presented in the main text, the total signal (Fig. S3a,  S3e) is shown to be dominated for both fluences by the effect arising from thermalised electrons (Fig. S3c, S3g), which contribute to the optical modulation to a much larger extent than nonthermal carriers, mostly acting on an ultrafast (sub-ps) timescale (Fig. S3b, S3f) and the lattice, much slower (Fig. S3d, S3h). Moreover, also in the nonperturbative regime, thermalised hot carriers are demonstrated to be at the origin of the peculiar dynamics of the ∆T /T . Indeed, by inspecting the dynamics of the disentangled signals at high fluence (right panels), the contribution arising from N (Fig. S3f) cannot explain the abrupt changes in the total differential transmittance (Fig. S3e), both for relative magnitude and sign (N gives a positive transmission modulation). On the contrary, the signal due to Θ E (Fig. S3g, as well as Fig. 3f in the main text) precisely follows the non-trivial temporal trend of the total ∆T /T , which can thus be understood by considering the non-perturbative effects discussed in the main text. Moreover, note that at low fluence, the ∆T /T due to thermalised electrons  Figure S4: Ultrafast dynamics of the differential transmittance signal. a, Experimental pump-probe trace recorded at λ 4 = 650 nm for a pump fluence of F 1 = 0.13 mJ/cm 2 . b-e, The total simulated ∆T /T signal (b) is disentangled in terms of contributions arising from nonthermal electrons (c), thermalised hot carriers (d) and the metal lattice (d). In the simulation, λ 4 = 635 nm and F 1 = 0.13 mJ/cm 2 .
By disentangling the contributions to the ∆T /T arising from nonthermal electrons, thermalised carriers and the lattice, each of the main features of the dynamics can in fact be ascribed to one specific of the three terms. The first ultrafast peak could be interpreted as the fingerprint of nonthermal electrons, giving a positive contribution to modulation at 635 nm during the early times following photoexcitation (Fig. S4c). After hundreds of fs, the thermal electron contribution, negative at this wavelength, starts being the dominant one ( Fig. S4d), explaining the change in sign at 300 fs. Then, at longer times, the contribution due to the lattice heating arises, bringing the signal from negative to positive values (Fig. S4e).