Morphology-modified contributions of electronic transitions to the optical response of plasmonic nanoporous gold metamaterial

Domain: physics
Published: 2026-01-20
Editorial outcome: minor_revision

Abstract
Nanoporous metals have emerged as promising functional architectures with tunable optical and electronic properties, high surface areas, and applicability in sensing, catalysis, and biomedicine. While their linear optical behavior and morphological properties have been extensively studied, the electronic properties, and in particular how they are affected by morphology, remain not fully understood. Here we combine experimental and theoretical studies of electronic excitation and relaxation in a nanoporous gold metamaterial. Optical pump–probe experiments show slower electron relaxation dynamics compared to the continuous film, consistent with a higher transient electronic temperature and stronger smearing of the Fermi–Dirac distribution, well reproduced by an extended two-temperature model. Furthermore, cathodoluminescence measurements reveal broadband localized plasmon resonances, and atomistic simulations disentangle intra- and interband effects, demonstrating that nanoscale porosity fundamentally reshapes the electronic response. These findings support nanoporosity as a key design parameter for controlling steady-state and ultrafast optical behavior in plasmonic materials.

Introduction
Nanoporous metals are well established materials with capability to enhance light-matter interactions1. They enable applications in many areas, including nanophotonics2,3, biomedicine4–8, spintronics9, and energy conversion10. Notably, they can be considered metamaterials, as their nanostructured surface volume can be designed on purpose to have properties which are not found in ordinary materials11. In fact, it has been shown that nanoporous structures behave as plasmonic metamaterials whose effective plasma frequency can be tuned by controlling the fractal dimensions, i.e., ligament and pore morphology1,12–14. Furthermore, the presence of nanoscale pore-like structures and grain boundaries introduces additional degrees of freedom for tuning plasmon-induced phenomena15,16, making nanoporous metals attractive platforms for developing efficient hot-electron generators17 and optical filters18,19, where controlled optical absorption is crucial. In this context, nanoporous gold (NPG) films have emerged as promising systems due to their tunable optical properties20, high surface area and interconnected nanostructures, making them highly versatile for applications in sensing21–23 and catalysis24,25. Nevertheless, the processes underlying their functionalities are not yet fully understood, and the possibility of capturing the inner mechanisms might allow for engineering their electronic and optical properties with increased precision.

It is well known that gold (Au) exhibits interband transitions from the 5d band to the conduction 6sp band22 in the visible spectral range close to 2.4 eV (~520 nm). In the context of plasmonics, previous research has thus been primarily focused on how interband and intraband transitions influence plasmonic behavior26–28, and only recently, studies have pointed out the role of interband and intraband transitions on plasmonic excitations at ultrafast timescales29–33. Interestingly, while interband transitions are typically regarded as loss and passive channels in plasmonic systems34, their potential active role in ultrafast carrier generation remains an open question35–37. Similarly, while it is known that nanoporous materials effectively act as better catalysts due to the larger surface to volume ratio, a more comprehensive understanding of their electronic properties can support progress in optimization of ultrafast and/or plasmon-supported catalytic processes.

Here, we perform an integrated experimental and theoretical study unveiling the role that morphology plays on intraband and interband transition contributions to the optical response of an NPG metamaterial. First, we investigate the ultrafast relaxation dynamics of thermalized hot carriers in bulk gold (BG) and NPG via nondegenerate pump-probe spectroscopy. We observe that, in the NPG film, the transient transmission (ΔT/T) signal associated with interband transitions from the 5d band to the 6sp conduction band occurs at photon energies below 2.3 eV, significantly lower than the onset energy observed in BG38. By modeling the transient transmission spectroscopy measurements with an analytical approach based on the extended two-temperature model (e2TM), we attribute this effect to an enhanced generation of hot carriers in NPG providing more empty states in the 6sp conduction band, and consequently requiring lower photon energy for interband electron excitation from the 5d band compared to BG. Furthermore, cathodoluminescence (CL) spectroscopy measurements indicate the presence of localized plasmon excitations in NPG from 2.4 eV (L-point of Au band structure) to 1.8 eV (X-point of Au band structure). These experimental results are underpinned by atomistic electrodynamics simulations based on the fully atomistic frequency-dependent fluctuating charges and dipoles (ωFQFμ) model, which allows us to disentangle intraband and interband transitions contribution to the optical response of both BG and NPG cases.

Our results provide physical insights on how morphology alters the electronic dynamics in NPG. Such understanding is important for developing engineered plasmonic platforms based on NPG where control over carrier generation and relaxation is essential, including in hot carrier driven processes, as for instance in light harvesting and catalysis39,40.

Results
NPG films were synthetized on fused silica substrates using a recently developed dry method2,41 (for more details see “Methods”). This approach enables the preparation of NPG thin films completely free of impurities from other metals, a typical limitation observed in the standard preparation techniques based on dealloying1. As benchmark sample, a BG film thermally evaporated on fused silica was also fabricated. X-ray photoelectron spectroscopy (XPS) measurements were performed on all samples, confirming that no oxidation occurred, in particular in NPG (see Supplementary Note 1 and Supplementary Fig. 1). Representative images of the NPG are reported in Supplementary Note 1 and Supplementary Fig. 2.

Transient response and numerical modeling
First, we performed pump-probe measurements (for more details, see “Methods”, Supplementary Note 2 and Supplementary Figs. 3 and 4) to understand the ultrafast carriers’ dynamics upon photoexcitation by a 12-fs infrared pump pulse (central wavelength 850 nm). The dynamics were probed by a broadband pulse of (duration 11 fs), covering a spectral range from 500 to 750 nm. The ultrafast optical responses of both NPG and BG films are presented in Fig. 1a, b, d, e. ΔT/T measurements showing the evolution of the signal over the time delay between pump and probe pulses and across different wavelengths are displayed in Figs. 1a, d for BG and NPG films, respectively.

Upon pump excitation of Au, hot electrons undergo fast thermalization via electron-electron scattering on timescales on the order of 10 fs42. Electron-phonon scattering leads to heat exchange of the electron bath with the lattice, exhibiting typical relaxation times on the ps timescale34,43. Electronic density redistribution around the Fermi level manifests in a change of the dielectric permittivity and thus of the transmission, which is experimentally measured via the broadband probe pulse. The NPG film shows a more intense ΔT/T signal characterized by a broad negative dip spanning from 540 to 670 nm (Fig. 1d). This feature is absent in the BG film at the same fluence of 3 mJ cm-2, where a negative ΔT/T signal is present only at wavelengths shorter than 570 nm, consistent with the presence of interband transitions at the L-point of the Au band structure, confirming previous studies on BG films38,44.

Figure 1b shows ΔT/T traces of BG (purple curve) and NPG (yellow curve) films at 540 nm, (purple and yellow dashed lines in Fig. 1a, d) up to 10 ps (longer time traces up to 20 ps are shown in Supplementary Fig. 5). A larger variation of the ΔT/T signal together with longer relaxation dynamics can be clearly observed in the NPG film case, consistent with reports in previous studies17,45,46. This observation is a result of a combination of factors. Firstly, the morphology of NPG leads to a decrease of the difference of the effective refractive index of the sample and surrounding air and therefore to an increased absorption at the pump wavelength47 (see Supplementary Note 3 and Supplementary Fig. 6). Consequently, the initial electron temperature rises higher in NPG (see inset in Fig. 1b). The maximum variation of the electronic gas temperature in the BG (purple curve) and NPG (yellow curve) was estimated by the e2TM (see “Methods” for more details) to be ~800 K and ~3200 K, respectively. Secondly, the heat capacity of Au increases with electronic temperature48, and higher heat capacity slows down relaxation dynamics. Thus, the elevated initial electron temperature in NPG compared to BG is responsible for the slower dynamics. A single exponential fit on the pump-probe traces in Fig. 1b quantifies this contrast as an increase of relaxation time of 3.1 ps for NPG compared to only 1 ps for BG, which seems reasonable if we consider that the electronic temperature in NPG is roughly 3 times that reached in BG for the same pump fluence.

We now move our focus to the spectral dependence of the ΔT/T signals of BG (green curve) and NPG (pink curve) films (Fig. 1e), which correspond to the green and pink horizontal dashed lines in the maps in Fig. 1a, d, at a time delay of 430 fs. At this time delay the ΔT/T signal is maximum in both NPG and BG cases. The BG film exhibits a negative signal around 520 nm, indicating increased absorption due to interband transitions. This result is consistent with previous reports on ultrafast dynamics of bulk Au38,44. In comparison, the NPG film displays a broader negative signal, suggesting that transient interband transitions can take place in an extended spectral region following pump excitation. This broadening can be understood as a result of the following mechanism: under steady-state conditions at room temperature, the occupied electronic states follow a Fermi–Dirac distribution around the Fermi level EF. Both intraband transitions involving electronic states around EF and interband transitions between the occupied 5d and empty states in the 6sp band can take place. After pump excitation and fast electronic thermalization, a stronger redistribution around the Fermi level frees electronic states below EF, an effect that is considerably more pronounced in NPG than in BG due to higher electronic temperature, as sketched in Fig. 1c, f.

In BG, interband transitions typically require energies exceeding 2.3 eV (green arrow in Fig. 1c). In contrast, in NPG, the significantly elevated electron temperature results in a greater availability of empty states in the 6sp band below EF, thus enabling increased absorption of photons at energies below 2.3 eV (yellow arrow in Fig. 1f). Therefore, the negative transient transmission signal measured in NPG extends over longer photon wavelength (lower energy).

For a more detailed characterization of the mechanism sketched in Fig. 1c, f, we employed the e2TM considering the geometry of the NPG via the Bruggeman effective medium approximation (EMA). Additionally, we modeled the permittivity variation as a function of the electron and lattice temperatures, in combination with the transfer matrix method (TMM). This model allows for the calculation of transient transmission signals as presented in Fig. 2 (for more details see “Methods” and Supplementary Note 4). Our calculations reproduce the experimental observations accurately, corroborating the conclusion that the mechanism outlined above, which is implemented in our model, is responsible for the transient transmission change. Thus, we can attribute the observed transient effects in NPG to the morphology of the nanoporous structure that is quantitatively well represented by the filling factor, i.e., the volumetric solid fraction of Au in the nanoporous film. It is worth mentioning here that, while the EMA description does not account for the presence of localized plasmonic excitations in NPG, these excitations may contribute to the enhanced absorption47 observed experimentally (see Supplementary Note 3 and Supplementary Fig. 6), thus providing the system with more energy upon pump pulse excitation. Nevertheless, such an increased plasmonic absorption does not significantly affect the physics of the mechanism investigated with our model, which is mainly connected to the porosity of the system43,49 and well described by the EMA.

For our simulations of BG and NPG films, we assume that both systems are excited by a pulse centered at 850 nm with a fluence of 3 mJ cm-2, matching the experiments reported in Fig. 1a, d. We consider a film thickness of 30 nm in both cases, and a filling factor of 0.4 was used in the NPG case according to the SEM images (see Supplementary Fig. 2). In Figs. 2a, c, we display the calculated ΔT/T signal as a function of the pump-probe time delay (vertical axis) and the wavelength of the probe pulse (horizontal axis) for the BG and NPG films, along with the transient transmission data at the vertical and horizontal cuts in Fig. 2b, d, analogous to Fig. 1. The transient transmission of the NPG film displays a longer relaxation time compared to the BG film case, confirming that our model captures the underlying physical mechanisms responsible for the spectral features observed in the experiments. Moreover, the calculated negative ΔT/T response in the NPG film is broader, also reproducing the main experimental features.

In further support of the identified mechanism, where the deposited energy primarily determines relaxation dynamics, we performed additional experiments with varying pump fluence. At higher incident pump power, the electronic temperature directly following excitation increases, leading to increased smearing of the Fermi-Dirac distribution, which in turn raises the probability that probe light with lower photon energy will promote interband transitions below 2.3 eV. In NPG, this effect is expected to be stronger with respect to BG, since its smaller geometric heat capacity drives a larger increase in electron temperature for the same energy deposited onto the system. Consequently, we expect the short wavelength negative ΔT/T feature to deepen and broaden with higher fluence, especially in NPG.

Results from the fluence-dependent study on both films are presented in Fig. 3.

In Fig. 3a, b, we show the measured ΔT/T signal of BG and NPG as a function of the probe light wavelength for different pump pulse fluences spanning from 1 to 4 mJ cm−2, with step of 0.5 mJ cm−2, at fixed pump-probe time delay of 430 fs. Indeed, the magnitude of the negative signal increases with fluence and shifts to longer wavelengths, in particular in the case of the NPG film (see black arrow in Fig. 3b). The experimental results match the predictions obtained from our model, shown in Fig. 3c, d. Notably, the model reproduces both the increase in amplitude and the pronounced broadening of the negative ΔT/T signal with increasing fluence, in particular for the NPG film case, thus supporting our claim that the transient optical response of the NPG film arises mainly from thermal effects, which are increased by the morphology of the NPG film.

It is also worth mentioning here that our simulations confirm that the negative ΔT/T signal is indeed due to an increase of the imaginary part of the dielectric permittivity, which is the quantity associated with absorption. This rules out a more trivial origin of the observed effects, such as positive transient reflection larger than ΔT/T, which would not imply that electronic transitions are promoted by the probe light (see Supplementary Note 5 and Supplementary Fig. 7). The transient change of the real and imaginary part of the permittivity upon pump excitation is shown in Supplementary Fig. 7. The imaginary part is positive for both BG and NPG, indicating that the observed effects arise from probe light absorption and not from ground state bleaching (which would give a negative transient absorption signal). Notably, the absorption in NPG is much stronger also below 1.8 eV (X-point in the Au electronic band structure) also for long time delays (>1 ps).

Linear response
Furthermore, to assess whether a change in morphology affects the band structure of the NPG system, we measured XPS valence-band spectra of both BG and NPG (see Supplementary Note 6 and Supplementary Fig. 8). Within the resolution of our instrument (0.1 eV) we do not detect any change in the electronic structure, thus supporting our conclusion that the observed differences between NPG and BG arise from macroscopic effects, i.e., porous morphology, rather than changes in the electronic band structure itself.

Having shown that nanoporosity modifies the ultrafast optical response of the sample, and that this is connected to a modified interband transition response under femtosecond pump excitation, we investigated whether this is a purely transient effect or whether it also appears in steady-state conditions. We performed CL spectroscopy (see details in “Methods” section), and the results are reported in Fig. 4. In the BG case, the CL map (Fig. 4b), which shows the wavelength of maximum CL intensity, is nearly uniform. Notably, the CL spectra (Fig. 4c) corresponding to the colored crosses in Fig. 4a, show a clear peak near 540 nm, the onset of the 5d to 6sp interband transition in bulk Au (close to the L-point in Au band structure). On the other hand, NPG exhibits a markedly different response: the CL map for NPG in Fig. 4e shows a very broad variation in signal caused by the high anisotropy and dispersion in size of the metallic clusters in NPG films. Figure 4f confirms the spatially dependent emission behavior, displaying the CL spectra at the positions marked by the colored crosses in Fig. 4d, revealing a broad set of localized excitations due to the presence of localized plasmon resonances centered around 600 nm, except from the gap regions with emission in the blue (see blue curve in Fig. 4f, corresponding to the CL signal coming from a region where Au is either almost or entirely absent). Additionally, from Fig. 4d, which is a SEM image of the NPG sample, it can be observed that the interconnecting wire size is comparable to the mean free path of electrons in Au, that is around 50 nm50. In the context of ultrafast dynamics, this may lead to increased electron-surface scattering. However, our model suggests that this effect is negligible compared to the impact of the higher electron temperature in NPG following excitation (for more details see “Methods”). It is worth mentioning here that in our linear optical response measurements (see Supplementary Fig. 6), we collect the signal from a much larger area, thus we probe a response averaged over many of these localized modes, resulting in a nearly flat absorption spectrum. Interestingly, in this spectral range, localized plasmon resonances in Au are known to have contributions from both intraband and interband transitions51,52.

Atomistic modeling
To disentangle these two contributions and understand with more detail the effect of morphology on the optical behavior measured in our pump-probe and CL measurements, we used the fully atomistic frequency-dependent fluctuating charges and dipoles (ωFQFμ)53–56 method. This approach has been shown to accurately reproduce the optical response of metal nanostructures from time dependent density functional theory calculations but at much lower computational cost, which is essential when considering large (more than hundreds atoms) systems57. The model is based on the assignment of time-varying charges and dipole moments to each atom, associated with intraband and interband transitions, respectively. In the frequency domain, the method is reformulated in terms of complex charges and dipole moments, oscillating at a single frequency (for additional details see “Methods”). We have summarized the simulation results in Fig. 5.

We have studied two model systems (8×8×2 nm3) to emulate the response of a BG film and an NPG structure (see Fig. 5a). The NPG model structure (3590 atoms) was obtained by random atom removal of the BG model structure (6844 atoms). The wavelength-dependent polarizability αz is derived from the imaginary part of the total dipole moment induced by an incident electric field Ez and shown in Fig. 5b. The simulated optical behavior agrees well with the CL measurements above (see Fig. 4), as compared to the BG (blue) film, showing a peak around 520 nm, the NPG model system (orange) exhibits several peaks between 550 nm and 720 nm.

Additionally, employing this simulation method, we have introduced a metric to quantify the contributions from intraband (fq, purple) and interband (fμ, red) effects to the optical response (see Fig. 5c) based on the calculated charge and dipole densities (ρq, ρμ) (see “Methods” for more details). We can observe that for both BG and NPG, the interband contributions fμ dominate at shorter wavelengths up to 600 nm (fμ ≫ fq). At longer wavelengths, the NPG system retains a mixed character (fμ ≈ 0.54, fq ≈ 0.46 at 700 nm) that clearly differentiates it from the BG case where fq ≫ fμ. This result highlights a stronger interband role in the optical response of NPG. To visually understand this contrast between the BG and NPG film cases, we display the interband and intraband contributions to the optical response at 700 nm in Fig. 5d. For the BG film case (left panel in Fig. 5c), the charge density ρq due to intraband contributions is almost identical to the total induced charge and dipole moment density, while the interband contribution is oscillating out of phase (fμ < 0), as expected in d-metals nanostructured materials58. These results show how morphology, even when varied at the atomistic level, is affecting the electronic behavior of the samples under investigations, thus pointing out the crucial role of nanoporosity in shaping the relative contributions of intraband and interband transitions to both the transient and steady state optical behavior in NPG metamaterials.

Methods
Sample fabrication
NPG thin films were prepared following a dry process method introduced previously in refs. 2,41. In summary, a Poly(methyl methacrylate) (PMMA) thin layer was spin-coated on a fused silica substrate and baked at 180 °C for 3 min. Then, a thin Au layer was deposited by electron-beam evaporation using a tilted configuration (i.e., placing the sample tilted at 80° with respect to the evaporation source). The process was completed with the etching of the PMMA layer by means of O2 plasma (200 W, 10 min). A BG thin film was prepared using standard electron-beam evaporation on a fused silica substrate. Both the NPG and BG films have a nominal thickness of 26 nm. Due to its morphology changed during fabrication41, the NPG film has a varying thickness around this value (±4 nm, that is a maximum of 8 nm from dip to peak).

Pump-probe measurements
The pump-probe employs a Yb:KGW laser (model: PH2-20W-SP, Light Conversion) to generate both pump and probe pulses. The probe pulse was generated through a 6-mm-thick YAG crystal (undoped YAG, orientation [100]), and compressed using custom-made dielectric chirped mirrors (Ultrafast Innovation). These pulses have a spectrum in the range 500–750 nm (2.5–1.3 eV) with 11 fs duration and repetition rate of 50 kHz. The pump was generated using a commercial Visible to Near-Infrared Optical Parametric Amplifier (VIS-NIR OPA, model: ORPHEUS-N-2H, Light Conversion), producing a spectrum in the range 730–885 nm (1.7–1.4 eV) with 12 fs duration. For more details on the characterization of pump and probe pulses see Supplementary Note 2.

An external pulse picker was adopted to halve the repetition rate of the pump beam to 25 kHz for differential transmission (ΔT/T) measurements. A spectrograph (model: HSVIS camera with sensor HA:S14290Q-1024, Stresing) was used to collect the transmitted probe intensity that provides two-dimensional ΔT/T maps as a function of pump-probe delay and probe wavelength. A tunable intensity filter (model: NDC-25C-4-B, Thorlabs) was also added for fluence-dependence measurements.

Cathodoluminescence measurements
NPG films were fabricated on a 100 nm thick Si3N4 membrane to minimize emission from bulky substrates during measurements, which were performed at room temperature by using a Zeiss Merlin scanning electron microscope equipped with a CL imaging system (model: SPARC, Delmic). CL spectra were collected in the range 400–800 nm with an “Andor Kimera 193i” spectrometer with a focal length of 193 mm and a grating of 300 gr mm−1. The photon emission was captured by an “Andor Newton DU920P-BEX2DD” CCD camera with a maximum quantum efficiency of 90%. The electron beam operated at an acceleration voltage of 30 kV and an emission current of 10 nA. The color-coded map consisted of 20×18 pixels (pixel size: 25 nm). The focused electron beam was scanned across the specimen, dwelling for 10 s (integration time) on each pixel to acquire the CL spectra.

Numerical modeling of transient response
The full simulation of a pump-probe experiment was done in three steps60,61: first by (i) calculating the steady-state pump-matter interaction, followed by (ii) calculating the transient change in permittivity induced by the pump, and lastly (iii) the probe interaction with the excited structure. This requires steady-state calculations of reflection, absorption and transmission, and subsequent modeling of the electron and lattice temperature change induced by the pump at different time delays. Then, the transient permittivity change is calculated as a function of the time delay, after which the steady-state calculations are repeated with the altered permittivity at all time delays. Here below we report all these steps in more detail.

(i) Abeles TMM62 was used to calculate the steady state reflection (R), transmission (T) and thus absorption (A = 1 − T − R) of NPG and BG on a glass substrate (see dashed lines in Supplementary Fig. 6). The geometry is a semi-infinite layer of air, followed by either the BG or the NPG film with a finite thickness (in this case the thickness was assumed to be 30 nm in both cases), and finally a semi-infinite layer of glass. We considered the incident light to imping on the structure at normal incidence from the air side. Au permittivity from a Drude-Lorentz model63 was used in the calculations. For NPG, we utilized the Bruggeman EMA to treat the NPG as a film with an effective permittivity49,64–66. This model is valid for samples with relatively high filling factors (>30%) where there is no clear host material, as is the case for NPG. The effective permittivity is determined by solving the equation64

Σn i=1 fi εi − εBr,p / εi + (1 − νi,p) εBr,p = 0 (1)

where n is the number of different materials in the NPG (in this case n = 2, as the system is made of air and gold). The index i = 1 represents Au, while i = 2 represents air. εBr,p is the effective permittivity in directions p, εAu the bulk Au permittivity and νi,p the depolarization coefficient for the i-th inclusion in direction p. The depolarization coefficients depend on the geometry of the inclusions. We assume that the porosity is the same in all direction, thus we use νi,p = 1/3 for both Au and air, and in all directions p. Therefore, the effective permittivity is isotropic. Equation (1) was solved numerically for each wavelength of interest, and the physical solutions with Im εBr > 0 were selected. In Supplementary Fig. 5, the calculated and measured steady state transmission, reflection and absorption of both samples are shown.

(ii) To model the temperature change induced by the pump, we utilized the extended two-temperature model (e2TM), which was first implemented by Carpene67. However, an instantaneous excitation and a homogeneous spatial distribution of the electron and lattice heat temperatures was assumed for simplicity, as we focused on the slower relaxation dynamics17 governed by electron-phonon coupling. This version of the e2TM is represented by the following system of equations

Ce(Te) ∂Te/∂t = −Gep(Te − Tl) + ∂Uee/∂t
Cl ∂Tl/∂t = Gep(Te − Tl) + ∂Uep/∂t (2)

where t is the time delay between pump and probe, Te,l are the electron and lattice temperatures, Ce(Te) the temperature dependent electronic volumetric heat capacity, Cl the lattice volumetric heat capacity and Gep the electron-phonon coupling factor. We refer the reader to the original work by Carpene67 for a detailed derivation of the source terms ∂tUee and ∂tUep. Under the instantaneous excitation assumption, which in this case is reasonable being the dynamics studied longer than 500 fs and the pump and probe pulses are 15 fs long, the source terms are calculated as

∂Uee/∂t = −exp(−t/τee) (1 + τee/τep)−1 (1 − exp(−t/τee)) t−1 2P (3)

∂Uep/∂t = −exp(−t/τee) (1 + τee/τep)−1 (τee/τep) (1 − exp(−t/τee)) t−1 P (4)

where τ represents the electron-electron and electron-phonon scattering rates. Finally, P = AF/d, where A is the absorption of the sample calculated in step (i) at the pump wavelength, F the fluence of the pump, d the thickness of the sample. Here, the absorbed energy is assumed to be distributed equally across the thickness of the sample. The electron-electron scattering rate was determined as67

τee = τ0 (EF / Epu)2 (5)

where τ0 is a constant, Epu the pump photon energy and EF the Fermi energy of Au. Table 1 show the values used for each parameter. Since we treated the NPG as an effective medium that is a mixture of air and gold, where only the gold volume fraction contributes to the heat capacity and electron-phonon coupling, the parameters in Eq. (2) had to be scaled to account for the sample morphology. The heat capacity of air is negligible compared to that of Au and does not meaningfully contribute to the effective heat capacity of NPG. Since the volumetric heat capacities are used in Eq. (2), the effective values for the NPG are the Au values scaled with the filling factor f, such that only the Au part of the NPG contributes to the heat capacities. The same reasoning is applied to the electron-phonon coupling factor. Electrons only interact with phonons inside the Au, therefore the electron-phonon coupling factor, which is a volumetric quantity, also scales linearly with f. Previous studies of NPG films of smaller ligament size have indicated an increase of the electron-phonon coupling factor compared to the BG counterpart68,69. However, since the electron-phonon coupling primarily involves short-wavelength phonons70, we neglected surface-mediated energy transfer from electrons to the lattice, which we assume is a secondary effect in our NPG sample that has an average ligament size of ~50 nm. The parameters in Eq. (2) were scaled as Ce → fCe, Cl → fCl and Gep → fGep. Mathematically, this is equivalent to scaling the source terms with f−1, which means that the energy absorbed from the pump is distributed over less volume of Au in the NPG case compared to the BG film. Using the temperatures from the e2TM, we can estimate the permittivity change Δε(Te, Tl) induced by the pump. The change in the imaginary part of the Au permittivity induced by the pump was determined through semi-classical modeling of optical transitions in Au using hyperbolic approximations of the band structure60,61,71. This change is related to interband transitions and changes due to Fermi smearing of the bands. The corresponding real part was determined using Kramers-Kronig relations. Additionally, the variation of permittivity induced by intraband changes following pump-excitation was also estimated, see Supplementary Note 4 for more details.

(iii) Since the NPG is represented by the Bruggeman EMA, the procedure of simulating the probe interaction was adjusted for NPG compared to BG. For BG, steady-state simulations using the TMM, as described above, were performed for each time delay t with Au permittivity εAu(t) = εAu + Δε(Te(t), Tl(t)). The permittivity variation Δε is complex and includes both the intraband and interband contribution described above. For NPG, we assumed that the permittivity of the air remains unchanged after pump excitation, and thus only the Au permittivity change contributes to a change in the variation of the effective permittivity. εAu(t) was therefore computed in the same way as for BG but with temperatures change considering the scaling factor f, and the effective permittivity εBr(t) was determined using Eq. (1) with the altered gold permittivity.

Atomistic modeling
We employed the atomistic frequency-dependent fluctuating charges and dipoles (ωFQFμ) model57. In this model, both a complex charge qi associated with intraband effects, and a complex dipole moment μi accounting for interband transitions54, are assigned to each metal atom i. The charge exchange between the atoms is governed by a Drude conduction mechanism and modulated by a phenomenological description of quantum tunneling53, which is essential for the description of gaps, junctions, and defects, as in the case of NPG54. To describe d-metals, the complex dipole moment is introduced, this way taking into account the polarizability of the d-shell54,58. The charge and dipole distributions at each atomic site for an external electric field Eext oscillating at frequency ω are obtained by solving the following complex coupled linear equations54

−iωqi(ω) = 1/2 n0 iω τ−1 Σj h(1 − flij) Alij (ϕelj − ϕeli) (6)

μi = αi(ω) (Eext + Eμi + Eqi) (7)

where n0 is the electron density, τ the friction (damping) time, Alij the effective area connecting atoms i and j and lij their distance. ϕel denotes the electrochemical potential acting on each metal atom. It accounts for interatomic interactions (charges and dipole moments) and the link to the external field. Eμi and Eqi are the local fields generated by induced dipoles and by charges, respectively. Finally, α(ω) is the complex, frequency-dependent atomic polarizability used to represent interband contributions, which can be obtained from the experimental permittivity54. Solving these equations allows to describe the optical response of noble-metal nanoparticles with ab initio–level accuracy54, at a much reduced computational cost. The model can also be reformulated in the time domain, enabling the study of the time evolution of the plasmon decay under a time-dependent excitation72. The ωFQFμ scaling efficiency can be exploited to run simulations of systems containing thousands of atoms20. Here, we build a BG structure (8×8×2 nm3; lattice parameter 4.08 Å) comprising of 6844 atoms. The NPG structure is then generated by randomly removing columns of Au atoms from the BG lattice, yielding a structure containing 3590 atoms (see also Fig. 5a).

Owing to its formulation based on well-defined physical quantities, ωFQFμ allows for dissecting the intra- and interband contributions, which are associated with charge and dipole terms, respectively, to the induced total charge and dipole density, as

ρ(r) = ρq(r) + ρμ(r) (8)

where

ρq(r) = Σi qi / π3/2 Rq3 exp(−|r − ri|2 / Rq2) (9)

and

ρμ(r) = Σi μi · ∇r exp(−|r − ri|2 / Rμ2) / π3/2 Rμ3 (10)

Here, ρq, ρμ represent charge and dipole moment density after summation over all atoms i. Rq and Rμ are the Gaussian widths of the charges and dipoles, respectively, mimicking a diffuse quantum-like density73. To quantify intra- and interband contributions, we introduce the following metric, averaging respective contributions from the complete structure:

⟨fX⟩ = ∫Ω fX(r) |ρ(r)| d3r / ∫Ω |ρ(r)| d3r X = q, μ (11)

where fX(r) is defined as

fX(r) = ρX(r) / ρ(r), ρ(r) ≠ 0 X = q, μ (12)

Note that, by definition fq + fμ = 1. All ωFQFμ calculations were performed by using the open-source code plasmonX57, using the Au parameters defined in ref. 54.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Tapani et al. studied the transient response of a nanoporous metal gold (NPG) substrate using pump-probe spectroscopy,\nand contrast it with that of a flat gold layer (BG). Their experimental study is explored using a computational model\ncombining transfer matrix method (TMM) and a two-temperature model (2TM) to account for the absorbance of the system\nand internal electronic dynamics, respectively. The analysis of the results focuses on the possibility of allowing interband\ntransitions in gold after an initial pulse elevates the effective electronic temperature and consequently creates empty states\nin the conduction band at energies lower than the Fermi level.\nThe article explores an interesting phenomenon, but I think that it does not explore it in sufficient depth: some key aspects of\nthe characterization do not seem to match the discussion and explanation of the results, the theoretical modelling is not\nsufficiently well described and fails to explain one of the significant features of the data. Although I would be very interested in\nseeing a significantly extended version of this study to better understand the phenomenon at hand, I do not think that this\nmanuscript is ready for publication.\n1) The difference between BG and NPG on their initial effective electron temperature after the pulse is argued to arise from a\ndifference in their absorption at the pump's wavelength, 850 nm. However, not only this difference is not shown in the main\ntext or SI, but the spectral data shown in Fig. S2, which is presented as justification for the larger absorption of the NPG,\nsuggests instead that their absorption might not be that different after all. Cathodoluminiscence (CL) data, which could\nperhaps be used as an indirect proxy for the absorbance, does not reach these wavelengths either.\n2) On the 2T model, additional information is needed: (i) explicit details about the values of beta taken from Ref. 19 (for\ninstance, the text indicates that beta is a constant, instead of a different constant for each variable, which is the case in the\ncited reference), particularly that of the coupling constant G, and (ii) a discussion of their validity, given that the 2T models\nused are not identicaly (in Ref. 19 they have additional e-e and e-ph scattering terms reducing both temperatures) and thus\nthe same effective parameters might not be adequate.\n3) The 2T model does not reproduce the long-wavelength features in both BG and NPG, but this is not adequately\nacknowledged and discussed.\n4) The TMM+2TM model shows that the lifetime of the electronic excitation is longer in the NPG, but this is information\nalready coded in the model through the scaling of the coupling constant by a power of the filling factor. At the very least, this\nchoice should be clearly justified.\n5) Interpreting the negative values of transmissivity as arising from interband transitions seems reasonable. However, it\nwould be interesting to check if this is the case through other tests. Could this be perhaps tested by measuring\nphotoemission lifetimes?\n6) Regarding the study of the spectral response at different pulse fluence, several aspects of the modelling and the critical\ncomparison with the experiment are unclear: (i) the model predicts qualitatively different features than those found\nexperimentally for the NPG, resembling instead those of the BG (e.g. Fig. 3d), (ii) it predicts a larger redshift than the one\nseen experimentally which, incidentally, could almost be interpreted as step-wise instead, (iii) it is not clear why Fig. 5b\ntracks the minima, while in the model (Fig. 4d) shows that the wavelengths crossing zero values are the ones\nredshifting, while the minima do not, (iv) note S5 does not provide evidence to show that from fluences of 3.5 mJ/cm2 the response is\nnonlinear, as claimed in the text.\n7) The presentation of the CL data in Fig. S3 is unclear. The square in Fig. S3a does not seem to correspond with the\nzoomed data in panel b. Also, the color legend in panel d is unclear, because it only shows labels one color in a map that\nincludes data on a gradient.\n8) I might have missed it, but I think that the thickness of the gold layer not noted.\n9) The model used for the TMM is not described, except for the choice of permittivities.\n10) The time delay scale is not included in Fig. 3ab.\n11) The discussion on the CL data (Fig. 2) should be clarified. (i) As far as I understood, the CL data from “holes and gaps” is\nnot shown in the figure, yet the text compares that against the profiles shown in Fig. 2c. (ii) This section of the\ncharacterization is not clearly connected with the rest of the discussion; what are the key takeaways of these data and how\nthey connect with the rest of the study?", "reviewer_2": "This paper reports on ultrafast transient absorption spectroscopy experiments on a nanoporous gold film (NPG) and\ncompares the results to those obtained with the well studied continuous bulk gold (BG) film. The two films display\nsignificantly different transient properties, with the NPG having a negative DT/T extending to significantly longer\nwavelengths and a much longer electron-phonon thermalization time, extending to several picoseconds.\nThe experimental results are interesting and novel, but I have the impression that the authors are trying to oversell them.\nFirst of all it is not clear to me why the authors call NPG a metamaterial: to my understanding, a metamaterial is a material\nwith nanostructured surface volume designed on purpose to have properties which are not found in ordinary materials (see\ne.g. the seminal paper DOI: 10.1126/science.1210713). Here there is no on purpose design, but just a random nanoporous\nstructure without any control of the properties. The abstract goes even further and defines NPG as a \"temporal\nmetamaterial\". There is currently a great interest in time-varying materials, which are materials whose physical properties,\nsuch as refractive index, change over time, often in a controlled and dynamic way, typically exploiting nonlinear optical\neffects. This temporal modulation introduces new degrees of freedom for manipulating light waves. Again here I do not see\nhow the NPG can be considered as a time-varying material.\nI find the paper to be not very well organized and written, with also several grammar mistakes. For example, figure 1 starts\nwith simulations of electronic temperature using the two-temperature model (TTM) without even showing the experimental\ndata. Also stationary transmission/extinction spectra of the NPG and BG films are not given.\nThe most striking difference between the BG and the NPG in my opinion is the much longer electron cooling time in the latter\nsample. However, I miss a clear physical explanation of why electron-phonon thermalization is much longer in the case.\nAlso, the dynamics in Fig. 3c stop at 8 ps, how does the relaxation in NPG proceed at longer delays? The comparison\nbetween experimental and simulated DT/T spectra in NPG is not very satisfactory: in the experiment, the positive peak is\nhigher than the negative one, while in theory it is almost zero. This discrepancy is not discussed in the manuscript. The\nauthors have a very good time resolution of sub-15 fs, however they do not focus on the very early stages of electron\ndynamics, monitoring the electron-electron scattering process resulting in the build-up of a thermalized electron distribution.\nIt would be very interesting to compare this process in BG and NPG.\nIn summary, the current version of the manuscript does not make a compelling case for publication in Nature\nCommunications. If the authors provide a better organized manuscript with clearer physical insight and toned down claims,\nthen the manuscript could be reconsidered.", "reviewer_3": "This manuscript focuses on the understanding specific excitation pathways for excited carriers in nonporous gold as\ncompared to bulk gold. It uses ultrafastand caathodoluminescence spectroscopy as well as two-state temperature modeling\nto understand the properties of these films, additionally a model describing the relationship between the interbank and\nplasmonic excitations is included. They also tune the porosity to crate a relationship between fill factor and absorption, and\nthus electronic temperature and excitation. This work is important and interesting for exciting hot carriers in plasmonic,\nhowever I have questions and concerns about its breadth and applicability.\n1. It is clear the porosity is important for this property? But why, is this all driven solely by the changes in the mean free path?\n2. how can these changes be modeled from either first principles or the dielectric function, to give broader physical insights?\nCould this be understood by the reductions in the scattering lifetime due to porosity? Presumably that is interrelated with\nchanges in the absorption.\n3. How does porosity change the extinction cross-section? Meaning, not just the MFP but also the intrinsic ability to\nthermalize photons?\n4. One concern I have is how to understand and generalize this inverse porous design within the context of what is well\nknown in colloidal and assembled systems. For example, nonporous gold in some sense is an assembly of small gold\ninclusions or a mesh of gold nanowires, albeit with slightly higher conductivity. This to me is the main gap of the paper as\nwritten. How can this result be connect with the broader knowledge of nanoparticle systems? Could experiments and\nmodels be designed to integrate these two areas of knowledge? To me, this is absolutely critical for this manuscript.\n5. for example, brining the gap to works such as reference 37 and other similar colloidal works focused on excited electronic\nprocesses in plasmonics."}, "author_response": "Dear Reviewers,\nWe thank you for the careful evaluation of our manuscript. Guided by your comments, we\nhave revised it. The major changes are:\n• New samples and measurements. While re-measuring the old samples to\nanswer the Reviewers’ questions, we realized that those samples were degraded.\nThus, new samples were fabricated and measured to assess the reproducibility of\nthe old measurements and to carefully address the Reviewers’ concerns about\ndata interpretation and discussion on the physical origin of the reported results.\n• Revised effective medium theory calculations. An updated effective medium\ntheory has been implemented by correcting the previous model. We also provide\na more detailed description of the relevant assumptions and parameters used in\nthe Methods section. This ensures a thorough explanation of the main mechanism\naffecting the observed changes in the contributions of electronic transitions to the\noptical response of nanoporous gold.\n• Atomistic simulations. We added atomistic simulations to show the respective\ncontributions of interband and intraband transitions to the optical response in\nboth the porous and the bulk samples. These results support the\ncathodoluminescence data, which were not clearly contextualized in the previous\nversion.\n• Figure order updated. Following the Reviewers’ feedback, in particular that of\nReviewer #2, we revised both the figures and the order of discussion points.\nWe hope that this revised version clarifies that modified interband and intraband\ntransitions contributions observed in nanoporous gold (relative to the bulk gold case)\narise from morphology. This conclusion is supported by complementary measurements\n(ultrafast pump-probe, cathodoluminescence, and ultraviolet photoemission\nspectroscopies) and by modelling the system with both an effective medium theory and\natomistic simulations.\nBelow are our point-by-point answers to the Reviewers’ comments.\nREVIEWER COMMENTS\nReviewer #1 (Remarks to the Author):\nTapani et al. studied the transient response of a nanoporous metal gold (NPG) substrate\nusing pump-probe spectroscopy and contrast it with that of a flat gold layer (BG). Their\nexperimental study is explored using a computational model combining transfer matrix\n1\nmethod (TMM) and a two-temperature model (2TM) to account for the absorbance of the\nsystem and internal electronic dynamics, respectively. The analysis of the results focuses\non the possibility of allowing interband transitions in gold after an initial pulse elevates\nthe effective electronic temperature and consequently creates empty states in the\nconduction band at energies lower than the Fermi level. The article explores an\ninteresting phenomenon, but I think that it does not explore it in sufficient depth: some\nkey aspects of the characterization do not seem to match the discussion and explanation\nof the results, the theoretical modelling is not sufficiently well described and fails to\nexplain one of the significant features of the data. Although I would be very interested in\nseeing a significantly extended version of this study to better understand the\nphenomenon at hand, I do not think that this manuscript is ready for publication.\nAuthors’ answer. We thank the Reviewer for his/her appreciation of our work and for\nstating that our article “explores an interesting phenomenon”. We agree with the\nReviewer that our analysis could have been more detailed, and we hope that our revised\nmanuscript, addresses the Reviewer’s concerns. We also would like to thank the\nReviewer for triggering additional investigations, which prove undoubtedly the role of\nmorphology on the observed physical effects.\nReviewer’s comment #1. The difference between BG and NPG on their initial effective\nelectron temperature after the pulse is argued to arise from a difference in their\nabsorption at the pump's wavelength, 850 nm. However, not only this difference is not\nshown in the main text or SI, but the spectral data shown in Fig. S2, which is presented as\njustification for the larger absorption of the NPG, suggests instead that their absorption\nmight not be that different after all. Cathodoluminiscence (CL) data, which could perhaps\nbe used as an indirect proxy for the absorbance, does not reach these wavelengths either.\nAuthors’ answer. We thank the Reviewer for raising this point. As mentioned above, we\nfabricated new samples and repeated the pump-probe experiments to ensure both full\nreproducibility and understanding of the underlying physics. NPG is clearly absorbing\nmore than BG at the pump wavelength (see revised Supplementary Figure S6). As\ndiscussed later when presenting more details about the two-temperature and the\neffective medium theory models, this higher absorption, due to the morphology of the\nsample, is also affecting the initial electronic temperature after pump excitation (see also\nour answer to Reviewer’s comment #2).\nRegarding the CL data, our aim was to show that we have strong localized excitations at\nthe probe wavelengths due to sample morphology. These excitations are mainly due to\nlocalized plasmon resonances, which contain modified contributions from both\ninterband and intraband transitions, as proved by our new atomistic simulations.\n840 nm (maximum spectral resolution of our instrument). We added an additional\nsentence to clarify why we performed CL measurements (see page 7, lines 235-237):\n‘‘Having shown that nanoporosity modifies the ultrafast optical response of the sample,\nand that this is connected to a modified interband transition response under\nfemtosecond pump excitation, we investigated whether this is a purely transient effect or\nwhether it also appears in steady-state conditions.’’.\nReviewer’s comment #2. On the 2T model, additional information is needed: (i) explicit\ndetails about the values of beta taken from Ref. 19 (for instance, the text indicates that\nbeta is a constant, instead of a different constant for each variable, which is the case in\nthe cited reference), particularly that of the coupling constant G, and (ii) a discussion of\ntheir validity, given that the 2T models used are not identically (in Ref. 19 they have\nadditional e-e and e-ph scattering terms reducing both temperatures) and thus the same\neffective parameters might not be adequate.\nAuthors’ answer. We thank the Reviewer for raising this point. Since the last version, we\nupdated the two-temperature model to the extended two-temperature model (e2TM). We\nalso revisited the assumptions for the scaling of the material parameters for NPG and\nchanged to a linear scaling of both the heat capacities and the electron-phonon coupling\nfactor. All details are now in the revised Methods section and in the SI. Nevertheless, we\nwould like to clarify here one important point. Since we treated the NPG as an effective\nmedium, the parameters in equation (2) in the Methods section in the main text, have to\nbe scaled to account for the porosity. The heat capacity of air is negligible compared to\nthat of Au and does not meaningfully contribute to the effective heat capacity of the NPG\nsample. Since the volumetric heat capacities are used in equation (2), the effective\nvalues for the NPG are the Au values scaled with the filling factor 𝑓, such that only the Au\npart of the NPG contributes to the heat capacities. The same reasoning is applied to the\nelectron-phonon coupling factor. Electrons only interact with phonons inside the Au,\ntherefore the electron-phonon coupling factor, which is a volumetric quantity, also scales\nlinearly with 𝑓. Here, we neglected surface-mediated energy transfer from electrons to\nthe lattice, which we assume is a secondary effect in our NPG sample that has an average\nligament size of approximately 50 nm since the electron-phonon coupling primarily\ninvolves short-wavelength phonons (see also new Reference [71]). So, to represent NPG\nin the e2TM, we scale the parameters as 𝐶 → 𝑓𝐶 and 𝐺 → 𝑓𝐺. Mathematically, this is\n𝑒,𝑙 𝑒,𝑙\nequivalent to scaling the source terms by 𝑓−1, which physically means that the energy\nabsorbed from the pump pulse is distributed on less volume of Au, compared to a film.\nUsing the temperatures from the e2TM, we could estimate the permittivity change\nΔ𝜀(𝑇 ,𝑇) induced by the pump.\n𝑒 𝑙\nmodel and changed to a linear scaling of the effective material parameters for NPG. The\nMethod section and the SI have been updated such that all details are presented,\nincluding the discussion on the mathematical and physical reasons why we scale both\nheat capacity and coupling factors linearly with the filling factor (see page 13, lines 447-\n467).\nReviewer’s comment #3. The 2T model does not reproduce the long-wavelength\nfeatures in both BG and NPG, but this is not adequately acknowledged and discussed.\nAuthors’ answer. As mentioned above, we have fabricated new samples and re-\ncharacterized them. We found that the previously observed positive signal originated\nmainly from strong scattering in both BG and NPG. After re-aligning the beams and\nrepeating measurements at multiple locations, we found that the scattering came from\nspecific spots on the samples, most likely due to dust and/or micrometric surface\ndefects, which are known to give a strong Rayleigh scattering contribution. The new\nresults show that we do not have this effect anymore after new and clean samples were\nfabricated, in particular for the BG film, which was already characterized by other groups\n(see for instance Della Valle et al., Phys. Rev. B 86, 155139 (2012)).\nReviewer’s comment #4. The TMM+2TM model shows that the lifetime of the electronic\nexcitation is longer in the NPG, but this is information already coded in the model through\nthe scaling of the coupling constant by a power of the filling factor. At the very least, this\nchoice should be clearly justified.\nAuthors’ answer. We thank the reviewer for raising this important point. We kindly refer\nto our answer to the Reviewer’s comment #2.\nAction taken. We updated the scaling of the parameters in the e2TM for the NPG. We\nupdated the manuscript with more details and justifications for our assumptions (see\npage 3, lines 111-125, and pages 13, lines 447-467).\nReviewer’s comment #5. Interpreting the negative values of transmissivity as arising\nfrom interband transitions seems reasonable. However, it would be interesting to check\nif this is the case through other tests. Could this be perhaps tested by measuring\nphotoemission lifetimes?\nray photoemission spectroscopy (XPS) to probe the valence band of BG and NPG, as\ndiscussed in Supplementary Note 6. We found no difference between the linewidth\nand/or the peak position of the 5d band in the two cases, leading us to the conclusion\nthat the observed effects are not connected to a change in the band structure but to a\nchange in the morphology, which strongly modifies the contributions from intraband and\ninterband transitions to the optical response of the NPG sample. This is also confirmed\nby the fact that to describe the transient response of the NPG sample we can use an\neffective medium theory which capture the main physical features. Furthermore, in the\nrevised manuscript, both CL measurements and atomistic modelling independently\npoint to the fact that we also have a modified contributions from intraband and interband\ntransitions due to localized excitations, which are caused by the morphology.\nFinally, we also tested, using our effective medium theory, if the observed effect has a\nmore trivial origin, such as larger positive transient reflectance, implying that no\nelectronic transitions are promoted by the probe light (see Supplementary Note 5). The\ntransient change of the real and imaginary part of the permittivity upon pump excitation\nis shown in Supplementary Figure S7. The imaginary part is positive for both BG and NPG,\nindicating that the observed effects arise from probe light absorption and not from\nground state bleaching.\nAction taken. We performed new XPS measurements focused more on the valence band\nspectral region and added a new discussion in SI (see Supplementary Note 6). We\nupdated the discussion on CL measurements (see page 7-8, lines 237-256) and updated\nold Figure 2, new Figure 4. We performed frequency-dependent fluctuating charges and\ndipoles (ωFQFμ) atomistic simulations and added a discussion (see page 8-9, lines 268-\n297) and new Figure 5. We also added a discussion in the SI about the possibility that the\nobserved effect has a more trivial origin, such as larger positive transient reflectance,\nimplying that no electronic transitions are promoted by the probe light (see\nSupplementary Note 5).\nReviewer’s comment #6. Regarding the study of the spectral response at different pulse\nfluence, several aspects of the modelling and the critical comparison with the experiment\nare unclear: (i) the model predicts qualitatively different features than those found\nexperimentally for the NPG, resembling instead those of the BG (e.g. Fig. 3d), (ii) it predicts\na larger redshift than the one seen experimentally which, incidentally, could almost be\ninterpreted as step-wise instead, (iii) it is not clear why Fig. 5b tracks the minima, while in\nthe model (Fig. 4d) shows that the wavelengths crossing zero values are the ones\nredshifting, while the minima do not, (iv) note S5 does not provide evidence to show that\nfrom fluences of 3.5 mJ/cm2 the response is nonlinear, as claimed in the text.\nrepeated the measurements on new samples and improved the model. As proved by the\nnew Figure 2 and Figure 3, it is now clear that the model captures the main experimental\nfeatures. In a new discussion, we directly illustrate and compare the broadening of the\nnegative signal region observed experimentally (see new Figure 3). Our effective medium\ntheory reproduces very well this behaviour.\nAction taken. We updated Figure 3 and the discussion (see pages 5-7, lines 160-219). We\nremoved the previous discussion on zero crossing value, local minimum and non-\nlinearity of the signal, as they no longer apply.\nReviewer’s comment #7. The presentation of the CL data in Fig. S3 is unclear. The\nsquare in Fig. S3a does not seem to correspond with the zoomed data in panel b. Also, the color legend in panel d is unclear, because it only shows labels one color in a map that\nincludes data on a gradient.\nAuthors’ answer. We thank the Reviewer for raising this point. We agree that the original\nfigure was not sufficiently clear. We have now moved the CL spectra of the BG to the main\ntext to compare more easily the two cases, that is BG and NPG. New Figure 4a and 4d are\nthe SEM images, while Figures 4b and 4e are the CL intensity maps. The confusion,\ndue to a lack of clarity from our side, raised from the fact that we showed a SEM image and\nshowed the magnified region as a CL intensity map, which was not a magnification of the\nSEM image. We have now clarified this in the revised version. Also, we have added CL\nintensity spectra to panel (c) corresponding to the pixels indicated in panel (a). We have\nalso updated the colour bar in panel (b), extending its range to include a wider spectrum\nof wavelengths. As detailed in the manuscript, the colour corresponds to the wavelength\nat the maximum of the emission spectra, which is strictly confined between 520 and 550\nnm, thus confirming that in BG contributions from interband transitions come mainly\naround 2.3 eV (L-point in gold band structure).\nAction taken. We moved the old Figure 2 and related discussion to a later stage in the\ndiscussion of the results (see new Figure 4), and updated its caption (see page 8, line 259-\n266).\nReviewer’s comment #8. I might have missed it, but I think that the thickness of the gold\nlayer not noted.\nAuthors’ answer. The thickness of the BG film is 26 nm. Due to fabrication processes\n(see Kwon, H.; et al. ACS Appl. Mater. Interfaces 2023, 15, 5620–5627, Ref. 41 in the main\n6\ntext) the NPG film has a varying thickness around this value (+/- 4nm). We have added\nthis information to the Methods section.\nAction taken. We added this information in the Methods section (see page 11, lines 362-\n364): ‘‘Both the NPG and BG films have a nominal thickness of 26 nm. Due to its\nmorphology change during fabrication41, the NPG film has a varying thickness around this\nvalue (± 4 nm, that is a maximum of 8 nm from dip to peak).’’\nReviewer’s comment #9. The model used for the TMM is not described, except for the\nchoice of permittivities.\nAuthors’ answer. We thank the reviewer for noticing this. The modelling section in the\nMethods section has been updated accordingly to specify what TMM model we used,\nthat\nis the Abeles’ matrix method.\nAction taken. We updated the Methods section accordingly and added the references\nwe used to build our own model (see page 12, lines 399-418).\nReviewer’s comment #10. The time delay scale is not included in Fig. 3a,b.\nAuthors’ answer. We thank the reviewer for pointing it out. We’ve corrected it.\nAction taken. We updated the old Figure 3, which is now part of the new Figure 1.\nReviewer’s comment #11. The discussion on the CL data (Fig. 2) should be clarified. (i)\nAs far as I understood, the CL data from “holes and gaps” is not shown in the figure, yet\nthe text compares that against the profiles shown in Fig. 2c. (ii) This section of the\ncharacterization is not clearly connected with the rest of the discussion; what are the key\ntakeaways of these data and how they connect with the rest of the study?\nAuthors’ answer. We thank the Reviewer for this observation. Regarding the first point,\nwe added CL data from a region with a gap (no gold) as shown in new Figures 4d and 4f to\nshow that no emission between 500 and 750 nm is happening from the voids/gaps\nregions. Regarding the second point: CL measurements has been performed to highlight\nthe presence of localized excitations in NPG compared to BG. These excitations are\nmainly due to the presence of localized plasmon polaritons which have contributions\nfrom both interband and intraband transitions. Nevertheless, CL cannot distinguish\n7\nbetween intraband and interband contributions. This is why we performed atomistic\nsimulations to disentangle the two effects.\nAction taken. We updated Figure 2, now new Figure 4. We added a sentence to clarify\nwhy we performed CL measurements (see page 7, lines 235-237): ‘‘Having shown that\nnanoporosity modifies the ultrafast optical response of the sample, and that this is\nconnected to a modified interband transition response under femtosecond pump\nexcitation, we investigated whether this is a purely transient effect or whether it also\nappears in steady-state conditions.’’.\nReviewer #2 (Remarks to the Author):\nThis paper reports on ultrafast transient absorption spectroscopy experiments on a\nnanoporous gold film (NPG) and compares the results to those obtained with the well-\nstudied continuous bulk gold (BG) film. The two films display significantly different\ntransient properties, with the NPG having a negative DT/T extending to significantly longer\nwavelengths and a much longer electron-phonon thermalization time, extending to\nseveral picoseconds. The experimental results are interesting and novel, but I have the\nimpression that the authors are trying to oversell them.\nAuthors’ answer. We thank the Reviewer for his/her positive comments and for stating\nthat the “experimental results are interesting and novel”. We also acknowledge the\nimproper use of some terms and claims, which were removed in the revised version (see\nour answers below).\nReviewer’s comment #1. First of all, it is not clear to me why the authors call NPG a\nmetamaterial: to my understanding, a metamaterial is a material with nanostructured\nsurface volume designed on purpose to have properties which are not found in ordinary\nmaterials (see e.g. the seminal paper DOI: 10.1126/science.1210713). Here there is no\non purpose design, but just a random nanoporous structure without any control of the\nproperties.\nAuthors’ answer. We respectfully provide few references to support our claim about NPG\nas a metamaterial:\n• Jalas, D. et al. Effective medium model for the spectral properties of nanoporous\ngold in the visible. Appl. Phys. Lett. 105, 241906 (2014).\n• Jalas, D., et al. Electrochemical tuning of the optical properties of nanoporous\ngold. Sci Rep 7, 44139 (2017).\n8\n• Garoli, D. et al. Fractal-Like Plasmonic Metamaterial with a Tailorable Plasma\nFrequency in the near-Infrared. ACS Photonics 5, 3408–3414 (2018).\n• Rout, S., Qi, Z., Biener, M.M. et al. Nanoporous gold nanoleaf as tunable\nmetamaterial. Sci Rep 11, 1795 (2021).\n• Ebrahimi, F. et al. Size-Dependent Photoemission Study by Electrochemical\nCoarsening of Nanoporous Gold. J. Phys. Chem. C 128, 12960–12968 (2024).\nThese works by different groups show that NPG structures behave as plasmonic\nmetamaterials whose effective plasma frequency can be tuned by controlling the fractal\ndimension (ligament/pore morphology), which matches well with the claim “a\nmetamaterial is a material with nanostructured surface volume designed on purpose to\nhave properties which are not found in ordinary materials”.\nAction taken. We have added the suggested reference by the Reviewer and some of the\nabove-mentioned references in the main text to clarify why NPG can be considered a\nmetamaterial (see page 2, lines 46-49).\nReviewer’s comment #2. The abstract goes even further and defines NPG as a \"temporal\nmetamaterial\". There is currently a great interest in time-varying materials, which are\nmaterials whose physical properties, such as refractive index, change over time, often in\na controlled and dynamic way, typically exploiting nonlinear optical effects. This temporal\nmodulation introduces new degrees of freedom for manipulating light waves. Again here\nI do not see how the NPG can be considered as a time-varying material.\nAuthors’ answer. We appreciate this comment and find it genuinely helpful to help us to\ncontextualize our work better. We agree that the use of the term “temporal metamaterial”\nin the abstract and the main text was not appropriate in the context of this work.\nAction taken. We removed any reference to “temporal metamaterial” in the manuscript\nand SI.\nReviewer’s comment #3. I find the paper to be not very well organized and written, with\nalso several grammar mistakes. For example, figure 1 starts with simulations of electronic\ntemperature using the two-temperature model (TTM) without even showing the\nexperimental data.\nAuthors’ answer. We thank the Reviewer for the feedback on manuscript logic\norganization and writing quality. We have now added experiments in Figure 1 and updated\n9\nthe other figures and related discussions to make more logical connections between the\nsections. Now the logic is the following:\n(i) We study the transient response of NPG and compare it with the ultrafast dynamics of\na continuous gold film using non-degenerate optical pump-probe spectroscopy. We find\nthat the nanoporous sample displays significantly different electron relaxation dynamics\ndirectly following pump excitation. This is attributed to a much higher electronic\ntemperature in the nanoporous material, causing a stronger smearing of the Fermi-Dirac\ndistribution.\n(ii) The experimental results are well described by an extended two-temperature model,\nwhich highlights the role of morphology in enabling the more efficient generation of hot\ncarriers, and the consequent lower-energy photons absorption inducing a modification\nof the contribution of interband transitions to the optical response of the porous sample.\n(iii) Cathodoluminescence spectroscopy reveals the presence of localized plasmon\nresonances across the visible range which we ascribe to morphology-modified\ncontributions from both intra- and interband transitions.\n(iv) Atomistic simulations disentangle intraband and interband contributions to these\nexcitations, pointing out the role of nanoscale porosity in modifying the electronic\nresponse of the system compared to the bulk benchmark sample.\nAction taken. We updated all the figures and revised the manuscript structure. We also\nperformed a full grammar and wording check on the manuscript.\nReviewer’s comment #4. Also, stationary transmission/extinction spectra of the NPG\nand BG films are not given.\nAuthors’ answer. We thank the Reviewer for raising this point. The spectra from 500 nm\nto 840 nm is now in the SI.\nAction taken. We updated Supplementary Figure S6.\nReviewer’s comment #5. The most striking difference between the BG and the NPG in\nmy opinion is the much longer electron cooling time in the latter sample. However, I miss\na clear physical explanation of why electron-phonon thermalization is much longer in the\ncase.\nAuthors’ answer. We thank the Reviewer for pointing this out. We agree that the\nexplanation needed to be stated more clearly. Fundamentally, this difference stems from\na combination of multiple factors: (i) significantly higher absorbed power per Au volume\ndue to the nanoporous morphology and thus higher electron temperature in NPG\n10\ncompared to BG, and (ii) the heat capacity of Au increases with electronic temperature,\nand higher heat capacity slows down relaxation dynamics. Thus, the elevated initial\nelectron temperature in NPG compared to BG is responsible for the slower dynamics.\nSee also our answers to Reviewer #1 comments #2 and #4.\nAction taken. We updated the manuscript with more details and justifications for our\nassumptions (see page 3, lines 111-125, and page 13, lines 447-467).\nReviewer’s comment #6. Also, the dynamics in Fig. 3c stop at 8 ps, how does the\nrelaxation in NPG proceed at longer delays?\nAuthors’ answer. We agree that this is a valuable information for the readers. We have\nextended the time window to 10 ps in the main text. Additionally, we show extended\ntraces for both NPG and BG up to 20 ps in the SI.\nAction taken. We updated old Figure 2 (now part of new Figure 1), and added\nSupplementary Figure 5.\nReviewer’s comment #7. The comparison between experimental and simulated DT/T\nspectra in NPG is not very satisfactory: in the experiment, the positive peak is higher than\nthe negative one, while in theory it is almost zero. This discrepancy is not discussed in the\nmanuscript.\nAuthors’ answer. We kindly refer to our answer to Reviewer #1 comment #3.\nReviewer’s comment #8. The authors have a very good time resolution of sub-15 fs,\nhowever they do not focus on the very early stages of electron dynamics, monitoring the\nelectron-electron scattering process resulting in the build-up of a thermalized electron\ndistribution. It would be very interesting to compare this process in BG and NPG.\nAuthors’ answer. We thank the Reviewer for pointing this out. We compared the two\ncases, and we do not see a significant difference in the rising phase, as shown in the\nfigure below, where we plot normalized transient transmission ΔT/T of NPG (yellow curve)\nand BG (purple curve) films as a function of the time delay (up to 2 ps) between pump\nand probe pulses at 540 nm. This indicates that the rising phase dynamics are governed\nprimarily by Au’s electronic structure rather than geometry, which dominates the\nthermalization process as captured very well by our e2TM. Additionally, we provide in SI\nthe full spectral and temporal characterizations of the pump and probe pulses.\ncharacterizations of the pump and probe pulses).\nReviewer’s comment #8. In summary, the current version of the manuscript does not\nmake a compelling case for publication in Nature Communications. If the authors provide\na better organized manuscript with clearer physical insight and toned down claims, then\nthe manuscript could be reconsidered.\nAuthors’ answer. We hope the revised version, containing a different structure, revised\nclaims and clearer physical insights, addresses the Reviewer’s concerns.\nReviewer #3 (Remarks to the Author):\nThis manuscript focuses on the understanding specific excitation pathways for excited\ncarriers in nonporous gold as compared to bulk gold. It uses ultrafast and\ncathodoluminescence spectroscopy as well as two-state temperature modelling to\nunderstand the properties of these films, additionally a model describing the relationship\nbetween the interbank and plasmonic excitations is included. They also tune the porosity to\ncreate a relationship between fill factor and absorption, and thus electronic temperature\nand excitation. This work is important and interesting for exciting hot carriers in plasmonics,\nhowever I have questions and concerns about its breadth and applicability.\nwork is important and interesting for exciting hot carriers in plasmonics”. We also\nappreciate the Reviewer’s concern about the breadth and applicability of our results. In\nresponse, we have revised the discussion on these aspects in our manuscript (see also\nour responses to the Reviewer’s concern here below).\nReviewer’s comment #1. It is clear the porosity is important for this property? But why,\nis this all driven solely by the changes in the mean free path?\nAuthors’ answer. In our revised manuscript we prove the key role of morphology/porosity\nin affecting electronic transitions in the NPG system. Nevertheless, we have realized that\nthe explanation for the observed differences needed to be clarified better. We kindly refer\nto our answers to question #5 by Reviewer #2 and question #2 by Reviewer #1 for a more\ndetailed explanation. We just want to stress here that there are 2 main factors involved\nfor the observed effects in NPG compared to the BG case: (i) a significantly higher\nabsorbed power per Au volume due to the nanoporous morphology and thus higher\nelectron temperature in NPG compared to BG, and (ii) the decrease of the electronic heat\ncapacity and electron-phonon coupling due to the nanoporosity.\nReviewer’s comment #2. How can these changes be modelled from either first\nprinciples or the dielectric function, to give broader physical insights? Could this be\nunderstood by the reductions in the scattering lifetime due to porosity? Presumably that\nis interrelated with changes in the absorption.\nAuthors’ answer. We agree that first-principles simulations would give deeper\nmicroscopic insight, but they are not practical at the scale of our NPG networks\n(thousands of atoms). Instead, we model porosity with a Bruggeman effective-medium\ndescription using Rakić’s Au permittivity; the metal filling fraction sets the effective\ndielectric function. Porosity-driven lifetime effects enter automatically through the e2TM\nvia filling-fraction scaling. In parallel, we use compact atomistic ωFQFμ calculations on\nrepresentative ligament–neck motifs to tease apart interband and intraband\ncontributions to the optical response. These atomistic simulations have been shown to\nprovide a very detailed description of plasmonic behavior as detailed as DFT calculations\n(see Giovannini et al., ACS Photonics 9, 3025–3034 (2022); Giovannini et al., Nanoscale\n11, 6004–6015 (2019); Giovannini et al., J. Phys. Chem. Lett. 11, 7595–7602 (2020)).\nReviewer’s comment #3. How does porosity change the extinction cross-section?\nMeaning, not just the MFP but also the intrinsic ability to thermalize photons?\neffectively reducing the difference between the refractive index of air and of the sample,\nwhich leads to a reduced reflectance. Hence, more light interacts with the sample, and a\nbigger fraction of the total intensity is absorbed. This effect is captured by our effective\nmedium theory. Since only the metallic part of the NPG absorbs light, the absorbed\nenergy is distributed on less volume compared to that of BG. This leads to significantly\nhigher electron temperatures (see the inset in new Figure 1b).\nReviewer’s comment #4. One concern I have is how to understand and generalize this\ninverse porous design within the context of what is well known in colloidal and assembled\nsystems. For example, nonporous gold in some sense is an assembly of small gold\ninclusions or a mesh of gold nanowires, albeit with slightly higher conductivity. This to me\nis the main gap of the paper as written. How can this result be connected with the broader\nknowledge of nanoparticle systems? Could experiments and models be designed to\nintegrate these two areas of knowledge? To me, this is absolutely critical for this\nmanuscript.\nAuthors’ answer. We thank the Reviewer for this important point. The model can\ntheoretically be generalized to any system that can be represented by the Bruggeman\neffective medium model, meaning mixtures where the filling factor of each material is of\na similar magnitude. There is no specific limit to the model, but generally it has been used\nsuccessfully for two-mixture samples where the filling factor of one material is at least\n30%. Furthermore, the Bruggeman model can also be adapted to represent samples\nwhere the inclusions are of a specific shape. In the Bruggeman equation (see Methods,\nequation (1), page 12), the depolarization factors can be altered to represent inclusions\nof different shapes. Some examples are inclusions in the shape of vertical rods and disk-\nlike inclusions.\nAction taken. We added Supplementary Note 7 and Supplementary Figure S9, where we\ndiscuss the possibility to represent different systems with Bruggeman model.\nReviewer’s comment #5. For example, bringing the gap to works such as reference 37\nand other similar colloidal works focused on excited electronic processes in plasmonics.\nAuthors’ answer. We thank the Reviewer for this comment. As discussed in the previous\nanswer, the new Supplementary Note 7 has been added to address how the model can\nbe used for different colloidal systems.\n‘‘Interestingly, the applied opto-thermal workflow is controlled by morphology and can\nbe generalized to colloidal or assembled films, suggesting the phenomenon may occur\nacross a broader class of systems (see Supplementary Note 7 and Supplementary Figure\nS9).”\n15"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have responded to all my comments in adequate detail and improved the manuscript significantly. I can\nrecommend it for publication, as is, although I recommend the authors to discuss a couple of points in more detail:\n1) The methodology used to obtain the results shown in the new Fig. 5, especially as to how do you differentiate between\nintraband and interband transitions.\n2) Expand the critical discussion to justify taking the electron-phonon coupling factor as scaling with the filling factor f.\nI personally don't find this intuitive or convincing: the volume occupied by the electrons remains the same as that\noccupied by the phonons. If anything, I would expect the coupling to change with the confinement of the electrons beyond their\nmean free path, or perhaps with changes in the phonon density of states due to confinement. The dependence may end up being\nlineal with the filling factor, but the reasoning behind it may change considerably.", "reviewer_2": "The revised version of the manuscript satisfactorily addresses the comments by myself and by the other reviewers. New\nexperimental and computational data (including atomistic simulations) are added and the explanation of the observed\nphysical phenomena is now much more convincing. Some claims that were not fully justified have now been removed. In the\npresent form, the paper is suitable for publication in Nature Communications.", "reviewer_3": "i now find the manuscript suitable for publication"}, "author_response": "Dear Reviewers,\nWe thank you for reviewing our revised manuscript and are pleased to read that the\nnew version has addressed most of the Reviewers’ comments. Below we provide our\npoint-by-point answers to the remaining remarks.\nWe would also like to mention additional minor corrections: we have edited the\nnumbering of the equations (Eqs. 4–12) and corrected a typing error in Eqs. 3 and 4.\nFurthermore, we added 6 new references that we found during continued discussions\non the topic and we added labels in Figure 5c for clarity.\nREVIEWER COMMENTS\nReviewer #1 (Remarks to the Author):\nThe authors have responded to all my comments in adequate detail and improved the\nmanuscript significantly. I can recommend it for publication, as is, although I\nrecommend the authors to discuss a couple of points in more detail:\nAuthors’ answer. We thank the Reviewer for their assessment of the revised\nmanuscript. We have extended the discussion on the additional aspects as follows.\nReviewer’s comment #1. The methodology used to obtain the results shown in the\nnew Fig. 5, especially as to how do you differentiate between intraband and interband\ntransitions.\nAuthors’ answer. We thank the Reviewer for his/her comment, which allows us to\nbetter clarify this point. As reported in more detail in Refs. (Liebsch, Phys. Rev. B 48,\n11317 and Giovannini et al., ACS Photonics 2022, 9, 9, 3025–3034), interband\ntransitions are associated with the polarizability of the d-shell. In the atomistic model\n𝜔FQ𝜇, such contributions are taken into account by including induced dipole moments\n𝜇, while intraband transitions are described by means of the Drude model of\nconduction and associated to induced charges 𝑞. To quantify the relative weight of\ninter- and intra-band transitions, we have introduced the averaged quantities 〈𝑓 〉, 〈𝑓 〉\n𝑞 𝜇\n(defined in the Methods section, see Eqs. 11–12, page 15, lines 521–528).\nAction taken. We have added the following sentence in the Methods section (see\npage 14, lines 488–489): “To describe d-metals, the complex dipole moment is\nintroduced to take into account the polarizability of the d-shell (Liebsch, Phys. Rev.\nB 48, 11317 and Giovannini et al., ACS Photonics 2022, 9, 9, 3025–3034)”. We also\nclarified the description in the main text (see page 9 lines 285–299).\nReviewer’s comment #2. Expand the critical discussion to justify taking the electron-\nphonon coupling factor as scaling with the filling factor f. I personally don't find this\nintuitive or convincing: the volume occupied by the electrons remains the same as that\noccupied by the phonons. If anything, I would expect the coupling to change with the\nconfinement of the electrons beyond their mean free path, or perhaps with changes in\nthe phonon density of states due to confinement. The dependence may end up being\nlinear with the filling factor, but the reasoning behind it may change considerably.\nAuthors’ answer. We agree with the Reviewer’s opinion that justification of this key\nassumption in our modelling is important. As described in the Methods section (page\n13, lines 450–466), the scaling with the filling factor 𝑓 in the nanoporous gold film is\nbased on the fact that the e2TM considers volumetric quantities for the heat capacity\n(units J/m3/K) and the e-ph coupling (W/m3/K). Thus, when we consider an effective\nmaterial that is a mixture of air and gold, only the gold volume fraction contributes to\nthe heat capacity and e-ph coupling, meaning that the quantities are smaller per unit\nvolume in the effective material than in bulk gold. With this scaling, we assume that\nthe mechanisms from which the electrons and phonons are interacting does not\nchange in the nanoporous sample. This means that we neglect surface-mediated\nenergy transfer from electrons to the lattice, which we assume is a secondary effect in\nour NPG sample that has an average ligament size of approximately 50 nm. This\nassumption is founded on the fact that e-ph coupling is connected to electronic\ntransitions involving mainly short wavelength phonons, see also the derivation of the\nphenomenological e-ph coupling factor (Peierls, R. (1932). Elektronentheorie der\nMetalle. In: Ergebnisse der exakten naturwissenschaften. Ergebnisse der Exakten\nNaturwissenschaften, vol 11. Springer, Berlin, Heidelberg.\nhttps://doi.org/10.1007/BFb0111800). Our results, showing good agreement between\nmodel and experiment, support these assumptions. At even smaller ligament sizes,\nadditional effects may indeed no longer be negligible.\nAction taken. We have rephrased a sentence in the manuscript (see lines 450–452):\n“Since we treated the NPG as an effective medium that is a mixture of air and gold,\nwhere only the gold volume fraction contributes to the heat capacity and electron-\nphonon coupling, the parameters in equation (2) had to be scaled to account for the\nsample morphology.”\nReviewer #2:\nThe revised version of the manuscript satisfactorily addresses the comments by\nmyself and by the other reviewers. New experimental and computational data\n(including atomistic simulations) are added and the explanation of the observed\nphysical phenomena is now much more convincing. Some claims that were not fully\njustified have now been removed. In the present form, the paper is suitable for\npublication in Nature Communications.\nAuthors’ answer. We thank the Reviewer for his/her final and positive assessment of\nour work.\nReviewer #3:\nI now find the manuscript suitable for publication.\nAuthors’ answer. We thank the Reviewer for his/her final and positive assessment of\nour work.\nDear Editor, dear Reviewers,\nWe thank you for reviewing our revised manuscript.\nBelow we summarize the minor changes we have made to the manuscript in order to\nfulfill the formal editorial requirements. The relevant sections are also highlighted in\nyellow in the main manuscript.\n- We have corrected affiliations and addresses.\n- We have shortened the abstract in accordance with the word limit (150 words).\n- We have updated equations, atomic orbital notation, and figures to match the\nstyle requirements. In particular we have corrected the schematics in Figure 1\nso as to not use any potential third-party content.\n- We have corrected formatting of the reference list. We have also updated the\ndata availability statement to include the figshare link and DOI.\n- We have included initials of authors in the funding information.\nWe provide more details regarding all formatting in the attached checklist."}]

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