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.

Plasmonic nanomaterials are intensively investigated for a variety of applications,[1−3] from innovative photodetection[4,5] and photocatalysis[6−11] to photothermal therapies[12,13] and light harvesting.[14−16] Moreover, optical components based on metallic nanostructures have lately emerged as a new promising platform for the manipulation of light[17−21] with particular interest toward nonlinear nanophotonics applications.[22−25]

Underlying the advances envisaged above is the understanding of the nonequilibrium processes regulating the interaction between electromagnetic radiation and nanostructured plasmonic materials.[26−30] In this framework, ultrafast optical spectroscopy has become the most suitable tool to gain insight into such processes (being noninvasive and of general applicability) and has given significant contributions to a vast literature (among the others, see, e.g., refs (31−36)), referred to as ultrafast plasmonics.[37,38]

In all of these studies, a fundamental hypothesis is the strict correspondence between the dynamics of the optical signal retrieved via the interaction with a low-intensity probe pulse interrogating the excited structure and the dynamics of the hot electron distribution (described by an electronic temperature) induced by absorption of the intense pump pulse. In particular, all studies so far assumed that the dynamics of the transient optical signal on the time scale of a few picoseconds after pump excitation correlates directly with the dynamics of the energy equilibration of hot electrons toward the metal lattice. However, to the best of our knowledge this hypothesis has not been systematically tested, and has even shown to be fulfilled in the presence of considerable effects altering the photoexcited carrier dynamics, such as ultrafast charge injection through metal–semiconductor heterojunctions.[39,40] In this work, we aim at bridging this gap, relying on a combination of transient absorption spectroscopy, quasi-static electromagnetic modeling, and semiclassical theory of light–matter interaction to investigate the correspondence between electronic and optical dynamics in plasmonic nanostructures.

The optical properties of nanostructures are dictated by a complex interplay between the permittivity ε of the constituent material and their nanoscale geometry. Upon excitation with ultrashort light pulses, the latter can however be considered unvaried for a wide range of interaction conditions, and the transient optical response can be exclusively ascribed to the dynamical modification of material permittivity, Δε.[41] As such, the nonperturbative regime of photoexcitation arises from two distinct mechanisms: (i) the nonlinear relationship, specific to the nanostructure geometrical configuration, between the nanostructure permittivity variation and the optical observable under consideration; and (ii) the optical nonlinearity detailing at the nanoscale the dependence of Δε on the excitation local intensity, depending on the material photophysical properties.

To illustrate these phenomena, we considered a prototypical plasmonic system represented by a gold nanoellipsoid embedded in a homogeneous medium. A typical transmission spectrum for such gold nanoparticles (NPs) with an aspect ratio of ∼2 dispersed in water computed from quasi-static formulas (Supporting Information, Section 1) is shown in Figure a (yellow curve). The extinction peak at low (high) photon energy can be ascribed to the excitation of the longitudinal (transverse) surface-plasmon resonance, LSPR (TSPR). Upon a change of the NP constituent medium permittivity Δε, the optical transmittance evolves from T(ε) to T(ε + Δε) . When Δε ≪ ε, a perturbative approach can be applied to determine ΔT = T(ε + Δε) – T(ε) = ∂T/∂ε × Δε (see, e.g., refs (42−44)). Yet, generally the evolution from T(ε) to T(ε + Δε) is not perturbative, and ΔT does not scale linearly with Δε, as depicted in Figure a. To quantify this effect, one can analyze spectra of the differential transmittance, ΔT/T, normalized to the corresponding Δε, and the effects induced by imaginary, Δε″, and real part, Δε′, of the permittivity change can be investigated separately. For the structure considered, the obtained two sets of spectra are reported in Figure b,c, for values of Δε (either imaginary or real) increasing from 0.01 to 2 (assumed constant over the entire spectrum for simplicity). The shape of these spectra can be interpreted as due to either the broadening (Figure b) or the red-shift (Figure c) of the LSPR/TSPR resonances, induced by the increase of material losses (via Δε″) and optical density (via Δε′), respectively. Importantly, since spectra are normalized to Δε, the onset of a nonperturbative effect manifests itself as the nonoverlapping of traces corresponding to different values of the Δε. This condition is more evident for the imaginary permittivity modulation (Figure b), compared to the real one (Figure c), and corresponds in any case to a relatively weak saturation effect even for a permittivity increase as high as 2, which is expected to give the strongest contribution to optical modulation around λ1 ∼ 500 nm (blue vertical line in Figure ).

Figure 1

Nonperturbative photoexcitation of Au nanostructures. (a–c) Nonperturbative effect arising from the nonlinear relation between the system transmission T and the metal permittivity ε, for NPs in aqueous environment, as schematized in (a). The two plasmonic resonances (TSPR and LSPR) are marked by arrows. The differential transmittance signal, obtained for a given constant dispersionless Δε ranging between 0.01 and 2, is reported for either a purely imaginary (b) or real (c) variation. (d–f) Nonperturbative effect arising from the universal nonlinear mechanism of Fermi smearing in Au, triggered by the increase of the electronic temperature ΔΘE and resulting in a change of the metal electron Fermi–Dirac distribution f(E), as sketched in (d), where the arrows point from low to high temperatures. The Δε associated with the change in f(E), normalized to ΔΘE, is displayed in both its imaginary (e) and real (f) parts. Vertical lines identify the three wavelengths analyzed in Figures and 3

Figure 2

Breaking the correspondence between electronic and optical dynamics. (a–c) Experimental pump–probe traces recorded at three different probe wavelengths: (a) λ1 = 515 nm, (b) λ2 = 554 nm, and (c) λ3 = 670 nm. The sample is excited by laser pulses at 400 nm wavelength with 100 fs duration, whereas a broadband probe pulse is focused on the sample at a time delay t with respect to the pump. Further details, together with information on the experimental setup, can be found in the Supporting Information, Section 3. The color of the curves refer to different pump fluences with increasing fluence from lighter to darker shades. Pump fluences in the experiment are (in mJ/cm2), F1 = 0.13, F2 = 0.25, F3 = 1.26, and F4 = 3.12. (d–f) Simulated nonthermalized (d) and thermalized (e) electrons, together with lattice (f) dynamics under different excitation regimes. Inset in (d) is a magnification on the subpicosecond time scale. (g–i) Simulated ΔT/T signal evaluated at (g) λ1 = 505 nm, (h) λ2 = 546 nm, and (i) λ3 = 661 nm. Pump fluences in the simulations are (in mJ/cm2), F1 = 0.05, F2 = 0.11, F3 = 0.64, and F4 = 1.50. Insets show a transmission electron microscopy (TEM) image of the sample used in the measurements (top) and a schematic of the transient absorption scheme modeled in the simulations (bottom).

Figure 3

Disentangling contributions from thermalized hot carriers. (a) Simulated temporal dynamics of the pump–probe signal at λ1 = 505 nm due to thermalizd hot carriers only, namely computed as if ΔΘE was the only energetic variable modifying the metal permittivity. (b–e) Imaginary part of the photoinduced permittivity change from thermalized hot carriers (b) and corresponding contribution to the pump–probe signal trace at λ1, obtained as if Δε″ was the only term of permittivity modulation (c). With the same rationale, the real part Δε′ of permittivity modulation from ΘE (d) and the resulting relative change in transmittance (e) are shown. (f–j) Same as (a–e) for quantities evaluated at λ2 = 546 nm. (k-o) Same as (a-e) at λ3=661 nm. Results from the full model, considering a non-perturbative occupation probability of thermal electrons (solid curves), and from its linearised version, considering a linear dependence on ΘE of the thermal electron energy distribution (dashed curves) are compared

Furthermore, to account for the second aforementioned nonperturbative mechanism, we need to consider the dependence of Δε on the photoexcitation fluence. For noble metals, the most prominent effect following photoexcitation is the generation of hot carriers. These carriers, following an ultrafast electron–electron thermalization,[45] are characterized by a Fermi–Dirac distribution in energy, , at temperature ΘE higher than the room temperature Θ0, E being the electron energy and EF the Fermi energy. The temperature increase ΔΘE = ΘE – Θ0 causes a modification of f(E, ΘE) with respect to the equilibrium distribution, an effect referred to as “Fermi smearing”[46] and schematically depicted in Figure d. In essence, the electronic temperature increase results in a reduction (increase) of the occupation probability of the electron states below (above) the Fermi energy EF. This promotes a variation of the imaginary part of the metal permittivity, accounting for the decreased (increased) absorption for 5d-6sp interband transitions (left panel, Figure d) involving final states above (below) EF. This is accompanied by a variation in the real part of the permittivity, due to the Kramers–Kronig relationship (Supporting Information, Section 2 for details). Because of the highly nonlinear dependence of f(E, ΘE) on the hot electron temperature ΘE inherent in the Fermi–Dirac distribution, the Fermi smearing mechanism can significantly contribute to nonperturbative effects of the photoexcitation. A detailed illustration is provided in Figure e,f, showing respectively the spectra of the imaginary and real part of Δε due to thermalizd carriers only, normalized to the corresponding fixed ΔΘE considered in determining the interband transition modulation, here ranging from 20 to 4000 K (typical values induced by laser pulses with mJ/cm2 fluences). The fact that curves corresponding to different values of ΔΘE do not overlap represents a clear-cut indication of the onset of nonpertubative effects. A complex spectral behavior is also displayed in the two cases of real and imaginary permittivity modulations. In particular, a huge enhancement of Δε″ is predicted for λ > 550 nm (Figure e), whereas sign changes in the modulation of the real permittivity can take place over a relatively broad range of wavelengths from ∼520 to ∼620 nm (Figure f), in the red wing of the Au interband transition, starting at around 505 nm. Nonperturbative effects due to Δε′ should therefore govern the optical modulation at wavelengths as λ2 (green vertical line), while Δε″ is expected to dominate close to λ3 (red line). Note that the nonperturbative effects related to the Fermi smearing are intrinsic to any plasmonic system, that is, they do not depend on the specific resonances of the nanostructure, contrary to the nonperturbative mechanism previously introduced (Figure a–c) which is related instead to the connection between Δε and the nanostructure polarizability.

To reveal the occurrence of effects arising from the two phenomena discussed above, we performed nonperturbative transient absorption experiments to retrieve the normalized differential transmission ΔT/T for a sample of colloidal Au nanorods (NRs) dispersed in water (inset of Figure ) with dimensions comparable to the one considered in the model in Figure . The capability of quasi-static formulas for nanoellipsoids to reproduce the static and transient optical response of small Au NRs has been ascertained in a previous work.[45] The results of these measurements for four different fluences of the pump are reported in Figure a–c at three probe wavelengths, λ1 = 515 nm (Figure a), λ2 = 554 nm (Figure b), λ3 = 670 nm (Figure c), selected after having identified (in Figure ) the spectral regions where nonperturbative effects are the strongest and dominated by the Fermi smearing. Full maps of the transient absorption signal are reported in Supporting Information, Section 4 (Figure S1).

At λ1, nonperturbative effects manifest as a mere saturation of the ΔT/T with increasing pump fluence F (Figure a); apart from a slight modification in the signal dynamics, the ΔT/T remains monotonic with F with a peak reached in the first hundreds of femtoseconds and a slow decay on a picoseconds time scale. A strikingly different behavior is instead retrieved when analyzing the temporal evolution of the modulated signal at λ2 (Figure b). While traces corresponding to F1 and F2 follow, apart from the sign (depending on the spectral dispersion of the photoinduced Δε) the same dynamics as the curves at λ1 (cf. Figure a) a substantially different nontrivial evolution of the transient signal over time is observed for higher fluences. The signal starts rising (in absolute value) then decreases again within the first hundreds of femtoseconds. Because of this behavior, in the case of F4 the signal changes sign, reaching a second ultrafast peak after its initial negative-valued one. Following this abrupt sign reversal, the signal starts increasing (in absolute value) again, featuring a peak at ∼5 ps for F3 and at a time delay longer than 15 ps for F4. This is a remarkable delay effect considering that pump pulses have a duration of 100 fs. Also, the optical signal does not scale monotonically with fluence: by fixing a time delay, the highest modulated signal is not the one at the highest fluence, contrary to what happens at λ1. Such a nontrivial dynamics is not peculiar of λ2 only or restricted to a narrow band, since the same trend with fluence is also observed at a distant wavelength λ3 (Figure c). According to the full transient maps of Figure S1, the most pronounced nonperturbative effects are indeed observable in the red wings of the plasmonic resonances, which are more sensitive to photoinduced changes of the isosbestic lines (black contours in Figure S1). Note also that the onset of the nonperturbative regime is experimentally achieved for a pump fluence of F3 = 1.26 mJ/cm2, which is higher than typical low-perturbation fluences[26,32,47] by a factor of ∼10 (at least) but still readily available and previously reported.[33,48]

To relate the observed optical dynamics to the temporal evolution of the hot carriers distribution, we simulated the transient absorption experiments using the so-called three-temperature model (3TM).[30,49−51] It consists of a rate-equation model describing the energy relaxation processes following photoabsorption in terms of three internal energetic degrees of freedom of the nanostructure: N, the density of excess energy stored in a nonthermalized fraction of the out-of-equilibrium electronic population, the aforementioned temperature ΘE, accounting for the excitation level of thermalizd hot carriers, and ΘL, the Au lattice temperature (see Supporting Information, Section 5). For the plasmonic system under investigation, the temporal dynamics of these three internal variables are reported in Figure d–f for increasing values of the pump pulse fluence. As expected, N (Figure d), ΔΘE (Figure e), and ΔΘL (Figure f) monotonically increase with increasing fluence at any time delay t. To be more precise, ΔΘE(t) scales proportionally although sublinearly with the pump fluence, which is a well-known consequence[41,47] of the fact that the 3TM comprises a time-dependent coefficient, that is, the electronic heat capacity CE ∝ ΘE(t).[52] However, this affects the dynamics of ΔΘE only quantitatively, resulting in an increased time constant for the electronic temperature relaxation, which remains monotonic in time but tends to become linear for high fluences because of the increased electron thermal inertia. Note that the approximation of CE as linearly dependent on ΘE is justified by the predicted range of electronic temperatures we span (see Figure e). For higher fluences inducing temperatures exceeding ∼3000 K, more refined models[53] should be employed to accurately assess the excited electron population dynamics. To retrieve then the transmission change of the ensemble of nanostructures via quasi-static formulas (see Supporting Information, Section 1), one should translate the dynamics of N, ΔΘE, and ΔΘL into the corresponding permittivity modulation terms[25,30] to compute spectra of Δε and T(ε + Δε) at each time delay (see Supporting Information, Sections 2 and 6).

The results of our calculations are reported in Figure g–i for three selected wavelengths: λ1 = 505 nm (Figure d), λ2 = 546 nm (Figure e), and λ3 = 661 nm (Figure f). Remarkably, by admitting a slight rigid shift (by less than 10 nm) and a lower value of fluence, ascribed to the increased Drude damping in Au static permittivity and to the linear model of pump absorption, neglecting saturation effects,[48] simulations are in excellent agreement with measured data. As for experiments, the signal traces at λ1 (Figure g) have almost the same dynamics regardless of F apart from the increase in the decay time with the signal tail evolving from exponential to linear at high fluences. Conversely, the differential transmittance at λ2 (Figure h) and λ3 (Figure i) exhibits nontrivial abrupt changes in the first hundreds of femtoseconds, together with delayed peaks reached at several picoseconds, confirming the observed breaking of the electronic-optical dynamics correspondence for nonperturbative excitations: the dynamics of the optical signal (Figure a–c) cannot be directly employed, in general, to infer the dynamics of the hot carrier temperature (Figure e) upon high-fluence photoexcitation, creating an electron distribution at very high temperatures (ΘE > ∼2000 K).

Our modeling approach allows us to gain further insight into the origin of the observed ΔT/T dynamics and in particular on whether the breaking of the electronic-optical correspondence is ascribable to a mere photonic effect (described in Figure a–c), is dominated by the Fermi smearing mechanism (detailed in Figure d,e), or rather both effects contribute.

We thus disentangled the contribution to the ΔT/T (for the fluence F3 = 640 μJ/cm2) arising from thermalizd carriers. This allows us to isolate the contribution of the electronic temperature, and rule out the possibility that the ultrafast dynamics of ΔT/T is due to nonthermal electrons[34] (see Supporting Information, Section 7 for details). Moreover, to ascertain the role of Fermi smearing in the photoinduced modulation, the computations of the permittivity modulation, and of the ensuing differential transmittance, were performed starting from the same dynamics of ΘE (solution of the 3TM) with the full nonperturbative as well as with a linearized model of variation in the hot electron distribution (see Supporting Information, Section 6). In the latter case, the variation of the Fermi–Dirac energy distribution of thermalizd hot carriers, written as ΔfT(E) = f(E,ΘE) – f(E,Θ0) in the full model, is instead expressed as a linear function of ΔΘE, that is, ΔfT(E) = [∂f(E, ΘE)/∂ΘE]ΘΔΘE with Θ0* an effective room temperature fitted to mimic broadening effects in Au interband transitions.[54] With the same rationale as in Figure , the differential transmission is computed as if only the imaginary Δε″ (Figure c,h,m) or the real Δε′ (Figure e,j,o) components were changed by an increase in the electronic temperature ΔΘE(t).

The main results of our analysis at λ1 are presented in Figure a–e for the two modeling approaches (solid lines for the full model, dashed lines for the linearized one), where the ΔT/T due to ΔΘE (Figure a) can be compared with the temporal evolution of the corresponding imaginary (real) part of permittivity variation Δε″ (Δε′), shown in Figure b (Figure d), and the relative transmission modulation, shown in Figure c (Figure e). As suggested by the regular trend of the pump–probe traces reported in Figure a,g, scaling monotonically with increasing fluence, the disentanglement confirms that the correspondence between optical (Figure a–e) and electronic (Figure e) dynamics at λ1 is preserved, the linearized model being adequate to describe the signal dynamics. At λ2 and λ3, first of all, as for the total ΔT/T (Figure b,c for experiments, Figure h,i for simulations), the signals arising from thermalized carriers are also similar (solid curves in Figure f,k) although the error introduced by the linearized Fermi smearing at λ2 is larger (compare dashed curves in Figure f,k). Most importantly, the onset of the delayed signal peak at around 3–4 ps, the fingerprint of the breaking of correspondence between optical and electronic dynamics, is lacking in the linearized model (dashed curves in Figure f,k). This indicates that such breaking is a signature of nonperturbative effects related to the Fermi smearing mechanism illustrated in Figure d. In more detail regarding λ3, Figure l,o show a dramatic discrepancy between the two models when dealing with imaginary permittivity modulations, where the linearized model largely underestimates the ΔT/T (cf. solid and dashed curves in Figure l,m). This correlates well with the analysis of Figure , since at this wavelength (marked by vertical red lines in Figures e,f) nonperturbative effects of the Fermi smearing, weak in terms of Δε′ (the curves in Figure f almost overlap), result in a sharp increase of Δε″ (Figure e). On the contrary, at λ2 (Figures g–j), the models mismatch is illustrated to be due to Δε′ (cf. solid and dashed curves in Figure i,j) and it is not only quantitative but involves a discrepancy in the sign of the contribution at the early stage of the dynamics. Again, this is fully consistent with the analysis reported in Figure , since at λ2 (marked by vertical green lines in Figure e,f) the Fermi smearing does not provide sizable nonperturbative effects due to Δε″ (the curves in Figure e are superimposed), whereas strong nonperturbative effects, including sign reversal, are predicted for Δε′ (Figure f) at high temperatures.

In conclusion, we investigated the ultrafast hot electron dynamics in plasmonic nanostructures excited with ultrashort laser pulses in a highly nonperturbative regime, reaching absorbed energy densities as high as ∼0.3 aJ/nm3 (yet well below the typical estimated[48] damage threshold of a few aJ/nm3, as ascertained by the absence of morphological changes in the sample after pump irradiation). The theoretical predictions turned out to be in excellent agreement with transient absorption spectroscopy experiments performed on colloidal gold nanorods. Our model provides a consistent explanation of the origin of sign changes and unexpected formation of delayed peaks observed in the pump–probe traces despite the monotonic dynamics of hot electron relaxation taking place on the same time scale. These results indicate that the correspondence between electronic and optical dynamics ceases to be valid beyond the perturbative regime. This behavior is intrinsically related to the Fermi smearing mechanism presiding over hot electron relaxation in any metallic structure and is therefore universal in ultrafast plasmonics. In these terms, our study provides a fundamental argument with general validity on how the optical signal retrieved by ultrafast pump–probe spectroscopy relates to the electron temperature in metallic nanomaterials under high-fluence excitation. Moreover, our results will be relevant for understanding the nonequilibrium optical response of plasmon-enhanced nanophotonic devices, from ultrafast photodetectors to all-optical modulators, where the achievement of a nonperturbative regime of the optical excitation is of crucial relevance to address real-world applications.