Pump-probe Spectroscopy Study of Ultrafast Temperature Dynamics in Nanoporous Gold

TL;DR

This study investigates how nanoporous gold structures affect ultrafast electron temperature dynamics, revealing slower relaxation and higher peak electron temperatures, with implications for thermal control in nanostructured metals.

Contribution

It demonstrates that nanoporous geometry significantly influences electron-lattice relaxation times and peak electron temperatures, advancing understanding of ultrafast thermal processes in nanostructured metals.

Findings

Nanoporous gold slows electron-lattice relaxation.

Higher peak electron temperatures in nanoporous gold.

Electron-phonon coupling scales with metal filling factor.

Abstract

We explore the influence of the nanoporous structure on the thermal relaxation of electrons and holes excited by ultrashort laser pulses ($\sim 7$ fs) in thin gold films. Plasmon decay into hot electron-hole pairs results in the generation of a Fermi-Dirac distribution thermalized at a temperature $T_{\mathrm{e}}$ higher than the lattice temperature $T_{\mathrm{l}}$. The relaxation times of the energy exchange between electrons and lattice, here measured by pump-probe spectroscopy, is slowed down by the nanoporous structure, resulting in much higher peak $T_{\mathrm{e}}$ than for bulk gold films. The electron-phonon coupling constant and the Debye temperature are found to scale with the metal filling factor $f$ and a two-temperature model reproduces the data. The results open the way for electron temperature control in metals by engineering of the nanoporous geometry.

Tables

Table 1. Table 1: Geometrical scaling of the TT model parameters.

Quantity	$C_e,C_l$	$v_s$	$g$	${\Theta }_D$	${\tau }_{p,\mathrm{relax}}$
scaling ()	$f^1$	$f^{3/2}$	$f^3$	$f^{3/2}$	$f^{-3/2}$

Equations

$$\begin{array}{r}{C}_{\mathrm{e}}\frac{d{T}_{\mathrm{e}}}{dt}=-g(T_{\mathrm{e}}-T_{\mathrm{l}})-\frac{e^{-\left({\tau }_{\mathrm{e},\mathrm{relax}}^{-1}+{\tau }_{\mathrm{p},\mathrm{relax}}^{-1}\right)t}}{t^2}\left[t+{\tau }_{\mathrm{e},\mathrm{relax}}\left(1-e^{t/{\tau }_{\mathrm{e},\mathrm{relax}}}\right)\right]\cdot P_a\\ C_{\mathrm{l}}\frac{T_{\mathrm{l}}}{dt}=g(T_{\mathrm{e}}-T_{\mathrm{l}})-\frac{e^{-\left({\tau }_{\mathrm{e},\mathrm{relax}}^{-1}+{\tau }_{\mathrm{p},\mathrm{relax}}^{-1}\right)t}}{t{\tau }_{\mathrm{p},\mathrm{relax}}}\left[{\tau }_{\mathrm{e},\mathrm{relax}}\left(1-e^{t/{\tau }_{\mathrm{e},\mathrm{relax}}}\right)\right]\cdot P_a\end{array}$$

$$g=\frac{{\pi }^2{m}_e{n}_e{v}_s^2}{6T_{\mathrm{e}}\tau (T_{\mathrm{e}},T_{\mathrm{l}})}$$

$$\frac{1}{\tau (T_{\mathrm{e}},T_{\mathrm{l}})}=\frac{1}{{\tau }_{\mathrm{ee}}}+\frac{1}{{\tau }_{\mathrm{ep}}}+\frac{1}{{\tau }_{\mathrm{B}}}=A{T}_{\mathrm{e}}^2+B{T}_{\mathrm{l}}+\frac{v_{\mathrm{F}}}{d_{\mathrm{wire}}}$$

$${\Theta }_{\mathrm{D}}=\frac{\hslash k_{\mathrm{D}}{v}_s}{k_{\mathrm{B}}}$$

Full text

nanoporous gold, pump&probe, hot electrons, electron-phonon interaction, nano-thermal models.

††preprint: APS/123-QED

The optical excitation of electrons and holes at high energy levels in metal nanostructures has been the subject of considerable attention in the last decade Atwater and Polman (2010); Manjavacas et al. (2014); Brongersma et al. (2015); Wu et al. (2015); Benz et al. (2016); Cortés et al. (2017), with the aim of enabling chemical reactions and charge transfer from the metal to adjacent materials at ambient temperature for energy harvesting and storage Atwater and Polman (2010); Wu et al. (2015), most notably H2 production by water splitting Lee et al. (2012); Mukherjee et al. (2012); Zhang et al. (2017, 2018). In particular, gold nanostructures have been investigated because of the relative ease of obtaining plasmonic field enhancement at their surfaces Naik et al. (2013). The absorption of optical energy by free carriers in a metal implies collective oscillation of electron currents (plasmons) Ruello et al. (2015); Besteiro et al. (2017); Deacon et al. (2017). Such coherent plasmons rapidly decay into non-thermalized electron-hole (e-h) pairs occupying high kinetic energy states. The e-h pairs decay via electron-electron scattering on the femtosecond time scale into hot carriers, which can be represented by a Fermi-Dirac distribution at an electron temperature $T_{\mathrm{e}}$, much higher than the lattice temperature $T_{\mathrm{l}}$. Subsequently, electron-phonon interaction leads to equilibrium defined as $T_{\mathrm{e}}\approx T_{\mathrm{l}}$ on the picosecond timescale Baida et al. (2011); Della Valle et al. (2012).

Very recently, ab-initio calculations of all electron and phonon states of gold have been employed to confirm the above interpretation of ultrafast pump-probe spectroscopy in the case of spherical nanoparticles of $60\text{\,}\mathrm{n}\mathrm{m}$ diameter in aqueous solution Brown et al. (2017). For such a simple geometry, electron and phonon distributions may be taken as constant in space, and the introduction of statistical thermal baths for electrons at $T_{\mathrm{e}}$ and phonons at $T_{\mathrm{l}}$ may not be conceptually necessary any more. The present work, however, explores the opposite limit of an extended nanoscale filament network, also called nanoporous gold (NPG). In NPG, geometrical parameters such as gold filling factor and filament diameter play a key role in determining the electron-phonon thermalization time due to spatially inhomogeneous excitation intensity at the nanoscale, therefore the previous simplified approach of two coupled statistical thermal baths (so called two-temperature (TT) model Anisimov et al. (1974); Carpene (2006); Della Valle et al. (2012)) will be followed in this work so as to effectively include the geometrical parameters of the nanoporous gold structure in the model.

Hot electron plasmonics experiments have been mostly conducted on nanoparticles dispersed in solutions Hodak et al. (1998); Hrelescu et al. (2010); Lee et al. (2012); Mukherjee et al. (2012); Aruda et al. (2013); Manjavacas et al. (2014); Zhang et al. (2017, 2018), and the ultrafast temperature dynamics are poorly understood due to an extremely varied experimental landscape Wu et al. (2015); Keller and Frontiera (2018); Yu et al. (2018).

NPG Biener et al. (2008); Ding and Chen (2009); Lang et al. (2011); Detsi et al. (2014) represents an interesting system for applications, as it allows liquid and gas samples to fill the empty spaces among gold ligaments Lee et al. (2012); Mukherjee et al. (2012); Zhang et al. (2017, 2018) where the radiation field is strongly enhanced by cusp-like geometries of the fractal structure (see Fig. 1 (a)-(b)) Lang et al. (2011); Detsi et al. (2014); Garoli et al. (2018). Nanoporous materials of different kinds (e.g. glass Costescu et al. (2002), silicon Hopkins et al. (2010); Wang et al. (2011) and polymers Costescu et al. (2002)) are also well known for their thermal and acoustic insulation properties. The nanoporous structure should then impact on the ultrafast electron temperature dynamics following the absorption of optical energy by plasmons in NPG. If compared to bulk gold, the decrease in the thermal conductivity at the interior of the effective material constituted by the nanoporous metal should then lead to higher maximum temperatures and slower local energy relaxation, in a way similar to what observed in gold nanoparticles Brown et al. (2017) and clusters Hrelescu et al. (2010). In this work, we present an ultrafast pump-probe spectroscopy study and related thermal modeling of plasmon energy relaxation in NPG. Interestingly, relevant fundamental quantities of the TT model such as the speed of sound, the Debye temperature and the electron-phonon coupling constant are found to follow a simple power scaling law with the metal filling fraction $f$ in NPG, which quantitatively explains both the longer time scales and the higher electron temperatures observed in our experiments.

NPG samples were prepared by chemical dealloying from an Ag0.67Au0.33 thin film following the procedure reported in Ref. Garoli et al. (2018). The two films studied in this work are characterized by different dealloying times (3 hours for NPG3 and 9 hours for NPG9) and have a similar $f$ (mainly related to the composition of the initial alloy). Different dealloying times lead to different average diameter of the gold ligaments $d_{\mathrm{wire}}$ Garoli et al. (2018). In particular, by numerical analysis of the SEM images of Fig. 1(a),(b) Garoli et al. (2018), we found $f=0.39$ and $d_{\mathrm{wire}}\sim$50\text{,}\mathrm{n}\mathrm{m} for NPG3, $f=0.37$ and $d_{\mathrm{wire}}\sim$80\text{\,}\mathrm{n}\mathrm{m} for NPG9. In Fig. 1(c),(d) the optical reflectance $R$ and transmittance $tr$ of the two NPG films in the infrared and visible ranges are reported. A redshift of the plasma edge is observed from $0.5\text{\,}\mathrm{e}\mathrm{V}$ in NPG3 to $0.2\text{\,}\mathrm{e}\mathrm{V}$ in NPG9 Lang et al. (2011); Detsi et al. (2014); Garoli et al. (2018). The dielectric resonance of gold at $2.5\text{\,}\mathrm{e}\mathrm{V}$ is clearly visible in all samples. The broad peak barely seen in the spectra of NPG9 around $1.8\text{\,}\mathrm{e}\mathrm{V}$ is due to an effective medium resonance Garoli et al. (2018).

In a simplified model of optical excitations of gold, the lowest-energy interband transition is the 5$d$-6$sp$ transition at the L-point, which leads to the lowest-energy resonance in the dielectric function of gold. The spectral lineshape of this resonance is a Lorentz function centered at $2.5\text{\,}\mathrm{e}\mathrm{V}$ Della Valle et al. (2012); Brown et al. (2017); Olmon et al. (2012). In this work, in order to focus on the geometrical effect of the nanoporous structure rather than on the details of electromagnetic interactions, we will make use of a corresponding simplified model for ultrafast pump-probe spectroscopy of gold: the infrared pump pulse spectrum, being located at photon energies well below the L-point transitions at $2.5\text{\,}\mathrm{e}\mathrm{V}$ (see Fig. 2(a)), mainly excites the intraband transitions within the 6$sp$ band. As a 6$sp$ intraband transition of gold can be seen as a pure free-electron excitation, it can also be interpreted as a plasmon excitation. The plasmon then decays into a 6$sp$ electron-hole pair that subsequently thermalizes in a hot carrier population in the 6$sp$ band, which we model with a simple Fermi-Dirac distribution thermalized at $T_{\mathrm{e}}$. The white-light probe pulse, instead, encompasses a broader spectral range including the dielectric function resonance at $2.5\text{\,}\mathrm{e}\mathrm{V}$, here used as a qualitative probe of $T_{\mathrm{e}}$ as a function of pump-probe delay. Fig. 2(b) is a sketch that summarizes the simplified model for ultrafast pump-probe spectroscopy of gold. However, it has been recently established, both theoretically Sundararaman et al. (2014); Brown et al. (2015) and experimentally Tagliabue et al. (2018), that 5$d$-6$sp$ interband transitions at the $X$-point can actually be excited by pump photons with energy higher than a threshold approximately set at $1.8\text{\,}\mathrm{e}\mathrm{V}$. The effect of $X$-point transitions is to depress plasmon excitation in the 6$sp$ band taking place at pump photon energies higher than $1.8\text{\,}\mathrm{e}\mathrm{V}$, therefore the simplified picture described above and sketched in FIg. 2(b), which implies pure plasmonic excitation in gold for all pump photon energies below the dielectric function resonance at $2.5\text{\,}\mathrm{e}\mathrm{V}$, has to be rigorously rejected Tagliabue et al. (2018). At odds with the $L$-point transitions, however, the weaker $X$-point transition oscillator does not produce a true resonance in the dielectric function of gold at $1.8\text{\,}\mathrm{e}\mathrm{V}$ Olmon et al. (2012) so our probe pulse will not be sensitive to hot holes in the 5$d$ band at that energy. Also, the pump pulse spectrum in our experiment extends between $1.4\text{\,}\mathrm{e}\mathrm{V}$ to $1.9\text{\,}\mathrm{e}\mathrm{V}$ as shown in Fig. 2(a), so it overlaps only marginally with the $X$-point transitions at $1.8\text{\,}\mathrm{e}\mathrm{V}$. Therefore, the simplified model of Fig. 2(b) can be fairly employed for the scopes of the present work hence allowing us to describe the electron system, after e-h pair thermalization, with the single parameter $T_{\mathrm{e}}$.

Transient absorption experiments were performed with an ultrafast laser system based on a Yb:KGW regenerative amplifier operating at a repetition time of $20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$. A home-built noncollinear optical parametric amplifier delivers excitation pulses with a bandwidth of $0.53\text{\,}\mathrm{e}\mathrm{V}$ at a central energy of $E_{\mathrm{p}}\sim$1.65\text{,}\mathrm{e}\mathrm{V} as reported in Fig. [2](#S0.F2)(a) hence excluding the 5$d$-6$sp$ transition (see Fig. [2](#S0.F2)(b)). Dielectric chirped mirrors compress the pulses to a duration of $7\text{\,}\mathrm{f}\mathrm{s}$. In Fig. [2](#S0.F2)(c) the evolution of the Fermi-Dirac distribution following the excitation of the pump pulse is sketched. At $t=0$ the pulse excites a non-equilibrium distribution whose shape is determined by the pulse energy spectrum in Fig. [2](#S0.F2)(a), which can be roughly approximated by a multiple step function (black dashed curve in Fig. [2](#S0.F2)(c)) Besteiro *et al.* ([2017](#bib.bib13)); Della Valle *et al.* ([2012](#bib.bib16)). The non-equilibrium e-h pair distribution generated by the pump pulse thermalizes to a Fermi-Dirac distribution at $T_{\mathrm{e}}$ on a timescale of the order of hundreds of fs, mainly through electron-electron interactions. At this stage, $T_{\mathrm{e}}$ is still much higher than $T_{\mathrm{l}}$ (red curve in Fig. [2](#S0.F2)(c)). On a longer timescale on the order of ps, the carriers cool down through electron-phonon interactions to a new lattice temperature $T_{\mathrm{l}}=T_{\mathrm{e}}$ (orange curve in Fig. [2](#S0.F2)(c)) higher than the environment temperature $T_{\mathrm{env}}\simeq$300\text{\,}\mathrm{K}.

The pump-induced optical transmission change $tr(t)$ is probed by a synchronous white light pulse obtained from supercontinuum generation in a $2\text{\,}\mathrm{m}\mathrm{m}$ thick sapphire crystal Grupp et al. (2017). Probe pulses cover a spectral range between 1.55 and $2.64\text{\,}\mathrm{e}\mathrm{V}$ including the 5$d$-6$sp$ transition. Spectra of subsequent probe pulses are used to calculate the differential transmission signal $\Delta tr(t)/tr=[tr(t)-tr(t\lesssim 0)]/tr(t\lesssim 0)$ with a modulation of the excitation pulses at half the repetition rate. In Fig. 2(d)-(f), color plots of $\Delta tr(t)/tr$ as a function of pump-probe time delay $t$ and probe wavelength $\lambda$ are shown for a reference bulk gold thin film and for the NPG3 and NPG9 samples. By comparing the three plots of Fig. 2(d)-(f), one immediately sees a strongly increased transmittance around $\lambda=$560\text{,}\mathrm{n}\mathrm{m} in both NPG samples which is almost absent in the bulk gold film Lang *et al.* ([2011](#bib.bib28)); Detsi *et al.* ([2014](#bib.bib29)), accompanied by a decay of $\Delta tr(t)/tr$ slower than that of the gold film at all wavelengths. For probe wavelenghts shorter than $\sim550\text{,}\mathrm{n}\mathrm{m}$the sign of$\Delta tr(t)/trchanges to negative because of pump-induced interband absorption Rotenberg *et al.* ([2009](#bib.bib38), [2010](#bib.bib39)); Baida *et al.* ([2011](#bib.bib15)); Della Valle *et al.* ([2015](#bib.bib40)); Di Mario *et al.* ([2018](#bib.bib41)). High-energy non-thermalized carriers impact on the transmittance of gold films and nanostructures only fort\ll0.5\text{\,}\mathrm{p}\mathrm{s}$$ Della Valle *et al.* ([2012](#bib.bib16), [2015](#bib.bib40)). The transmittance dynamics for probe delays above 0.5\text{,}\mathrm{p}\mathrm{s}$, instead, can be almost entirely attributed to thermalized carriers and to changes in their$T_{\mathrm{e}}, displaying a relaxation time scale independent on the probe wavelength Qiu and Tien ([1992](#bib.bib42)). In this perspective the strongly increased transmittance observed in NPG (positive areas in Fig. [2](#S0.F2)(e),(f)) indicates a much higher value of T_{\mathrm{e}}^{\mathrm{Max}}$ if compared to that reached in bulk gold (Fig. 2(d)). These facts demonstrate that NPG is a very promising candidate for hot-electron plasmonics applications.

Numerical evaluation of $T_{\mathrm{e}}(t)$ and $T_{\mathrm{l}}(t)$ dynamics is performed within the two-temperature model, in which energy relaxation to the lattice from the free carriers, heated by e-h pair thermalization via the fast electron-electron interaction, is mediated by the relatively slow electron-phonon interaction Anisimov et al. (1974). In an improved version of the TT model Carpene (2006), e-h pairs produced by plasmon decay act as the external heat source for both the Fermi-Dirac free carrier distribution and the lattice via electron-electron and electron-phonon scattering processes respectively, resulting in the following coupled equations:

[TABLE]

where $C_{\mathrm{e}}$ and $C_{\mathrm{l}}$ are the electronic and lattice heat capacities per unit volume, $g$ is the electron-phonon coupling constant, $\tau_{\mathrm{e,relax}}$ and $\tau_{\mathrm{p,relax}}$ are characteristic times related to the electron-electron and electron-phonon energy relaxation Carpene (2006). The pump pulse power in the instantaneous pump-pulse approximation is $P_{a}=F_{a}/d$, with $d$ the film thickness, $F_{a}=(1-R-tr)F$ and $F=$180\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{J}\mathrm{/}\mathrm{c}\mathrm{m}^{2} the pump fluence. For bulk gold thin films, the values of the parameters used in the extended TT model are $C_{\mathrm{e}}=\gamma T_{\mathrm{e}}$, $\gamma=$68\text{\,}\mathrm{J}\mathrm{m}^{-3}\mathrm{K}^{-2}, $C_{\mathrm{l}}=2.5\cdot${10}^{6}\text{,}\mathrm{J}\mathrm{m}^{-3}\mathrm{K}^{-1}, $g=2.2\cdot${10}^{16}\text{\,}\mathrm{W}\mathrm{m}^{-3}\mathrm{K}^{-1}, $E_{\mathrm{F}}=$7.3\text{,}\mathrm{e}\mathrm{V}, $\tau_{\mathrm{e,relax}}=$136\text{\,}\mathrm{f}\mathrm{s}, $\tau_{\mathrm{p,relax}}=$1650\text{,}\mathrm{f}\mathrm{s}, $E_{\mathrm{P}}=$1.65\text{\,}\mathrm{e}\mathrm{V} Della Valle et al. (2012). In the inset of Fig. 2(c), the $T_{\mathrm{e}}$ curve obtained from Eq.1 fits to the $\Delta tr(t)/tr(0)$ data for bulk gold, provided that the delay scale is normalized by the relative change factor $\xi=\mathrm{ln}(\Delta tr^{\mathrm{Max}}/tr)/\mathrm{ln}(\Delta T_{\mathrm{e}}^{\mathrm{Max}}/T_{\mathrm{e}}(t\lesssim 0))\simeq 3$.

In order to analyze the ultrafast temperature dynamics of NPG within the extended TT model, we scale all quantities of Eq.1 by $f^{\beta}$, where $\beta$ is the corresponding scaling exponent, as summarized in Table 1. For $C_{\mathrm{e}}$ and $C_{\mathrm{l}}$, the scaling exponent is a trivial $\beta_{C}=1$ as they scale linearly with the mass density. For the thermal conductance, the problem is considerably more complex due to the NPG network connectivity. Previous works have employed the Asymmetric Bruggeman Theory (ABT) McLachlan et al. (1990) to calculate the electron thermal conductivity in NPG Hopkins et al. (2008); Makinson (1938) and the lattice thermal conductivity of nanoporous glass Costescu et al. (2002). In both cases, the results point toward an experimental value of $\beta_{k}=3/2$ for thermal conductivities of nanoporous solids. The lattice thermal conductivity is written as $k_{l}=1/3C_{\mathrm{l}}v_{s}l_{ph}$, where $C_{\mathrm{l}}$ is the lattice specific heat, $v_{s}$ is the speed of sound and $l_{ph}$ is the phonon mean free path. Since $l_{ph}$ and $C_{\mathrm{l}}$ are microscopic quantities that should not depend on the geometry, $v_{s}$ should scale with the exponent $\beta_{v}=\beta_{k}=+3/2$ as well Costescu et al. (2002). There are two quantities in the TT model of Eq.1 that depend on $v_{s}$. The first quantity is $g$ Kaganov et al. (1957):

[TABLE]

where $n_{e}$ is the microscopic electron density, $m_{e}$ is the electron mass, and $\tau(T_{\mathrm{e}},T_{\mathrm{l}})$ is the total electron scattering time including electron-electron $\tau_{\mathrm{ee}}$ and electron phonon $\tau_{\mathrm{ep}}$ scatterings. Following Matthiessen’s rule and assuming momentum-independent scattering, the effect of electron scattering at physical boundaries in NPG ligaments can be included in the model by considering an additional scattering time $\tau_{\mathrm{B}}=v_{\mathrm{F}}/d_{\mathrm{wire}}$, where $v_{\mathrm{F}}=1.40\cdot${10}^{6}\text{,}\mathrm{m}\mathrm{/}\mathrm{s}$$ is the Fermi velocity in gold Hopkins et al. (2008):

[TABLE]

In Eq.3 $A$ and $B$ are temperature-independent coefficients that in gold can be taken equal to $A=1.2\cdot${10}^{7}\text{,}\mathrm{K}^{-2}\mathrm{s}^{-1}, $B=1.23\cdot${10}^{11}\text{\,}\mathrm{K}^{-1}\mathrm{s}^{-1} Wang et al. (1994). In bulk gold $d_{\mathrm{wire}}\to\infty$ and the contribution of $\tau_{\mathrm{B}}$ is negligible. The case of gold nanoparticles can also be obtained by using $f=1$ and $d_{\mathrm{wire}}$ similar to the value of the nanoparticle diameter supplmat. In Eq.2 the only quantity that scales with $f$ is the speed of sound $v_{s}$, therefore for $g$ we obtain a scaling exponent $\beta_{g}=2\beta_{v}=+3$.

The second quantity of the TT model proportional to $v_{s}$ is the Debye temperature $\Theta_{\mathrm{D}}$:

[TABLE]

where $k_{\mathrm{D}}=(6\pi N_{a})^{1/3}$ ($N_{a}$ is the atomic density) and $k_{\mathrm{B}}$ is the Boltzmann constant. $k_{\mathrm{B}}\Theta_{\mathrm{D}}$ represents the average phonon energy and, as such, enters in the definition of the electron-phonon energy relaxation time as $\tau_{\mathrm{p,relax}}=\tau_{\mathrm{ep}}E_{\mathrm{P}}/k_{B}\Theta_{\mathrm{D}}$. Therefore, from $\beta_{\Theta}=\beta_{v}=+3/2$ we obtain $\beta_{\tau}=-\beta_{\Theta}=-3/2$ for $\tau_{\mathrm{p,relax}}$.

Using the scaling exponents of Table 1, we can describe the ultrafast electron dynamics of NPG by solving the extended TT model of Eq.1 as a function of $f$. It is important to notice that the scaled quantities are effective quantities purposely defined for the nanoporous solid, and do not correspond to an actual variation of the microscopic quantities of bulk gold. In Fig. 3(a)-(c) the model results are reported, highlighting the effect of $f$ and $d_{\mathrm{wire}}$ on $T_{\mathrm{e}}$. In the model, the temperature dynamics is clearly slowed down for low $f$ and $T_{\mathrm{e}}^{\mathrm{Max}}$ is considerably increased. Electron scattering at physical boundaries, which is almost absent in bulk gold, becomes relevant only when the electron mean free path in gold $\ell\sim$40\text{,}\mathrm{n}\mathrm{m}$$ Hopkins et al. (2008); Gall (2016) is of the same order of the mean ligament diameter $d_{\mathrm{wire}}$ (as it is in our samples NPG3 and NPG9 with $d_{\mathrm{wire}}$ of $50\text{\,}\mathrm{n}\mathrm{m}$ and $80\text{\,}\mathrm{n}\mathrm{m}$, respectively).

In Fig. 3(d)-(i) we compare cuts of the experimental data of Fig. 2(d)-(f) at fixed $\lambda=$600\text{,}\mathrm{n}\mathrm{m} and $\lambda=$500\text{\,}\mathrm{n}\mathrm{m} with the prediction of the TT model scaled by $f=0.39$ for NPG3 and $f=0.37$ for NPG9. The relaxation dynamics for $t>$0.5\text{,}\mathrm{p}\mathrm{s} is fairly reproduced by the TT model in all plots of Fig. [3](#S0.F3). The much higher $\Delta tr(t)/tr$ for NPG if compared to bulk gold at $\lambda=$600\text{\,}\mathrm{n}\mathrm{m} is indicative of the much higher $T_{\mathrm{e}}^{\mathrm{Max}}$ reached in NPG. The TT model accounts only for the dynamics of thermalized electrons and therefore it cannot reproduce the ultrafast variations of $\Delta tr(t)/tr$ at very short $t\geq 0$. Especially at $\lambda=$600\text{,}\mathrm{n}\mathrm{m}, a strong induced absorption signal can be seen for $t<$100\text{\,}\mathrm{f}\mathrm{s} (see insets of panels (e) and (f)) and it can be attributed to the excitation of non-thermalized high-energy carriers Rotenberg et al. (2009, 2010); Baida et al. (2011); Della Valle et al. (2015); Di Mario et al. (2018). Hot carriers are almost absent in bulk gold for the same excitation conditions as for NPG, as expected due to the high density of field-enhancement hotspots in NPG and to the high surface/volume ratio Besteiro et al. (2017) of the NPG fractal structure Garoli et al. (2018). At $\lambda=$500\text{,}\mathrm{n}\mathrm{m}$$ the contribution of non-thermalized carriers to $\Delta tr(t)/tr$ is much smaller Della Valle et al. (2012); Brown et al. (2017) and it does not impact on the fitting of the model to the data as seen in Fig. 3(h),(i). It has been observed Aruda et al. (2013) that surface functionalization of gold nanostructures leads to similar slowdown of the temperature dynamics. Further studies of functionalized nanoporous gold for future hot electron chemistry applications will be required to understand the combination of the two different slowdown effects.

In conclusion, the predictions of a geometrical scaling theory of nanoporous gold, concerning the reduced thermal capacitance, the weaker thermal link between electrons and phonons and the longer electron-phonon energy relaxation time if compared to bulk gold, could quantitatively account for the ultrafast temperature dynamics experimentally observed by pump-probe spectroscopy. On the basis of these results, higher electron temperatures and longer plasmon decay times can be engineered in gold nanostructures for future applications of hot-electron plasmonics.