Pump-probe Spectroscopy Study of Ultrafast Temperature Dynamics in Nanoporous Gold
Michele Ortolani, Andrea Mancini, Arne Budweg, Denis Garoli, Daniele Brida, Francesco de Angelis
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Pump-probe Spectroscopy Study of Ultrafast Temperature Dynamics in NanoporousGold
Michele Ortolani , Andrea Mancini , Arne Budweg , Denis Garoli , Daniele Brida , , and Francesco de Angelis Department of Physics Sapienza University of Rome-00185 Rome-Italy. Department of Physics and Center for Applied Photonics University of Konstanz-78457 Konstanz-Germany. Plasmon Nanotechnology Department Istituto Italiano di Tecnologia (IIT)-16163 Genoa-Italy. Physics and Materials Science Research Unit University of Luxembourg-L-1511 Luxembourg-Luxembourg. (Dated: January 7, 2019)We explore the influence of the nanoporous structure on the thermal relaxation of electrons andholes excited by ultrashort laser pulses ( ∼ T e higher than the lattice temperature T l . The relaxation times of the energy exchange betweenelectrons and lattice, here measured by pump-probe spectroscopy, is slowed down by the nanoporousstructure, resulting in much higher peak T e than for bulk gold films. The electron-phonon couplingconstant 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 controlin metals by engineering of the nanoporous geometry. Keywords: nanoporous gold, pump&probe, hot electrons, electron-phonon interaction, nano-thermal models.
The optical excitation of electrons and holes at highenergy levels in metal nanostructures has been the sub-ject of considerable attention in the last decade [1–6],with the aim of enabling chemical reactions and chargetransfer from the metal to adjacent materials at ambi-ent temperature for energy harvesting and storage [1, 4],most notably H production by water splitting [7–10].In particular, gold nanostructures have been investigatedbecause of the relative ease of obtaining plasmonic fieldenhancement at their surfaces [11]. The absorption of op-tical energy by free carriers in a metal implies collectiveoscillation of electron currents (plasmons) [12–14]. Suchcoherent plasmons rapidly decay into non-thermalizedelectron-hole (e-h) pairs occupying high kinetic energystates. The e-h pairs decay via electron-electron scat-tering on the femtosecond time scale into hot carriers,which can be represented by a Fermi-Dirac distributionat an electron temperature T e , much higher than the lat-tice temperature T l . Subsequently, electron-phonon in-teraction leads to equilibrium defined as T e ≈ T l on thepicosecond timescale [15, 16].Very recently, ab-initio calculations of all electron andphonon states of gold have been employed to confirmthe above interpretation of ultrafast pump-probe spec-troscopy in the case of spherical nanoparticles of 60 nmdiameter in aqueous solution [17]. For such a simple ge-ometry, electron and phonon distributions may be takenas constant in space, and the introduction of statisticalthermal baths for electrons at T e and phonons at T l maynot be conceptually necessary any more. The presentwork, however, explores the opposite limit of an extendednanoscale filament network, also called nanoporous gold(NPG). In NPG, geometrical parameters such as goldfilling factor and filament diameter play a key role in de-termining the electron-phonon thermalization time due to spatially inhomogeneous excitation intensity at thenanoscale, therefore the previous simplified approach oftwo coupled statistical thermal baths (so called two-temperature (TT) model [16, 18, 19]) will be followedin this work so as to effectively include the geometri-cal parameters of the nanoporous gold structure in themodel.Hot electron plasmonics experiments have been mostlyconducted on nanoparticles dispersed in solutions [2, 7–10, 20–22], and the ultrafast temperature dynamics arepoorly understood due to an extremely varied experimen-tal landscape [4, 23, 24].NPG [26–29] represents an interesting system for ap-plications, as it allows liquid and gas samples to fill theempty spaces among gold ligaments [7–10] where the ra-diation field is strongly enhanced by cusp-like geome-tries of the fractal structure (see Fig. 1 (a)-(b)) [28–30].Nanoporous materials of different kinds (e.g. glass [31],silicon [32, 33] and polymers [31]) are also well knownfor their thermal and acoustic insulation properties. Thenanoporous structure should then impact on the ultrafastelectron temperature dynamics following the absorptionof optical energy by plasmons in NPG. If compared tobulk gold, the decrease in the thermal conductivity atthe interior of the effective material constituted by thenanoporous metal should then lead to higher maximumtemperatures and slower local energy relaxation, in a waysimilar to what observed in gold nanoparticles [17] andclusters [21]. In this work, we present an ultrafast pump-probe spectroscopy study and related thermal modelingof plasmon energy relaxation in NPG. Interestingly, rele-vant fundamental quantities of the TT model such as thespeed of sound, the Debye temperature and the electron-phonon coupling constant are found to follow a simplepower scaling law with the metal filling fraction f in Figure 1. (a), (b) Scanning electron micrographs (SEM) ofthe two NPG samples characterized by different f and d wire .(c) Scheme of the solid thin-film samples with optical beams.(d),(e) Reflectance and transmittance spectra of the NPGfilms at equilibrium. The transmission dip around 0 . . d -6 sp interband transition atthe L -point of the first Brillouin zone. NPG, which quantitatively explains both the longer timescales and the higher electron temperatures observed inour experiments.NPG samples were prepared by chemical dealloyingfrom an Ag Au thin film following the procedurereported in Ref. [30]. The two films studied in this workare characterized by different dealloying times (3 hoursfor 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 diam-eter of the gold ligaments d wire [30]. In particular, by nu-merical analysis of the SEM images of Fig. 1(a),(b) [30],we found f = 0 .
39 and d wire ∼
50 nm for NPG3, f = 0 . d wire ∼
80 nm for NPG9. In Fig. 1(c),(d) the opticalreflectance R and transmittance tr of the two NPG filmsin the infrared and visible ranges are reported. A redshiftof the plasma edge is observed from 0 . . . . d -6 sp transi-tion at the L-point, which leads to the lowest-energy res-onance in the dielectric function of gold. The spectrallineshape of this resonance is a Lorentz function centeredat 2 . . sp band. As a 6 sp intraband transition of gold can beseen as a pure free-electron excitation, it can also be in-terpreted as a plasmon excitation. The plasmon thendecays into a 6 sp electron-hole pair that subsequentlythermalizes in a hot carrier population in the 6 sp band,which we model with a simple Fermi-Dirac distributionthermalized at T e . The white-light probe pulse, instead,encompasses a broader spectral range including the di-electric function resonance at 2 . T e as a function of pump-probe delay. Fig.2(b) is a sketch that summarizes the simplified model forultrafast pump-probe spectroscopy of gold. However, ithas been recently established, both theoretically [34, 35]and experimentally [36], that 5 d -6 sp interband transi-tions at the X -point can actually be excited by pumpphotons with energy higher than a threshold approxi-mately set at 1 . X -point transitionsis to depress plasmon excitation in the 6 sp band takingplace at pump photon energies higher than 1 . . L -point transitions, however, theweaker X -point transition oscillator does not produce atrue resonance in the dielectric function of gold at 1 . d band at that energy. Also, the pump pulse spec-trum in our experiment extends between 1 . . X -point transitions at 1 . T e .Transient absorption experiments were performed withan ultrafast laser system based on a Yb:KGW regenera-tive amplifier operating at a repetition time of 20 µ s. Ahome-built noncollinear optical parametric amplifier de-livers excitation pulses with a bandwidth of 0 .
53 eV at acentral energy of E p ∼ .
65 eV as reported in Fig. 2(a)hence excluding the 5 d -6 sp transition (see Fig. 2(b)).Dielectric chirped mirrors compress the pulses to a dura-tion of 7 fs. In Fig. 2(c) the evolution of the Fermi-Diracdistribution following the excitation of the pump pulse issketched. At t = 0 the pulse excites a non-equilibriumdistribution whose shape is determined by the pulse en-ergy spectrum in Fig. 2(a), which can be roughly approx-imated by a multiple step function (black dashed curvein Fig. 2(c)) [13, 16]. The non-equilibrium e-h pair dis-tribution generated by the pump pulse thermalizes to aFermi-Dirac distribution at T e on a timescale of the or- DOS at L -point5d6sp E = E F
1 t<0; T e =T l =T env
2 t=0 Nonequilibrium distr.3 t ∼ tens of fs Te>>Tl4 t ∼ e =T l >T env P e l e c t r on ( E ) (c)(d) E = E F -E p E = E F +E p E ne r g y E E F (b)(a) ∼ energy (eV)1.90 1.65 1.46 1.30 NPG3 NPG9 wavelenght (nm)
Time delay (ps) (cid:1) T r a s m / T r a s m x -
010 1 2 3
Bulk Gold
Au T e T e ( K ) w a v e l engh t ( n m ) (cid:1) t r / t r x - -2-1012 w a v e l engh t ( n m ) (cid:1) t r / t r x - (cid:1) t r / t r x - w a v e l engh t ( n m ) (cid:2)(cid:3) = 600nm Pump spectrum (e) (f)
Figure 2. (a) Spectrum of the pump pulse used in the experiments (duration is 7 fs). (b) Simplified sketch of the density ofstates (DOS) of gold at the L -point employed in this work for interpretation of the pump-probe data. (c) Simplified sketchof the evolution of the Fermi-Dirac distribution following the pump pulse excitation. The shift of the chemical potential withtemperature is neglected for clarity. (d-f) ∆ tr ( t ) /tr maps for a reference bulk gold thin film (thickness 30 nm) (d) and for thetwo NPG samples (e),(f). Inset of panel (d), green curve: cut of the map in (d) at λ = 600 nm; red curve: the T e ( t ) obtainedfrom the extended TT model. der of hundreds of fs, mainly through electron-electroninteractions. At this stage, T e is still much higher than T l (red curve in Fig. 2(c)). On a longer timescale onthe order of ps, the carriers cool down through electron-phonon interactions to a new lattice temperature T l = T e (orange curve in Fig. 2(c)) higher than the environmenttemperature T env ≃
300 K.The pump-induced optical transmission change tr ( t )is probed by a synchronous white light pulse obtainedfrom supercontinuum generation in a 2 mm thick sap-phire crystal [37]. Probe pulses cover a spectral rangebetween 1.55 and 2 .
64 eV including the 5 d -6 sp transi-tion. Spectra of subsequent probe pulses are used tocalculate the differential transmission signal ∆ tr ( t ) /tr =[ tr ( t ) − tr ( t . /tr ( t .
0) with a modulation of theexcitation pulses at half the repetition rate. In Fig. 2(d)-(f), color plots of ∆ tr ( t ) /tr as a function of pump-probetime delay t and probe wavelength λ are shown for a ref-erence bulk gold thin film and for the NPG3 and NPG9samples. By comparing the three plots of Fig. 2(d)-(f), one immediately sees a strongly increased transmit-tance around λ = 560 nm in both NPG samples whichis almost absent in the bulk gold film [28, 29], accompa-nied by a decay of ∆ tr ( t ) /tr slower than that of the goldfilm at all wavelengths. For probe wavelenghts shorter than ∼
550 nm the sign of ∆ tr ( t ) /tr changes to negativebecause of pump-induced interband absorption [15, 38–41]. High-energy non-thermalized carriers impact on thetransmittance of gold films and nanostructures only for t ≪ . . T e , displaying a relaxation time scale independent on theprobe wavelength [42]. In this perspective the stronglyincreased transmittance observed in NPG (positive areasin Fig. 2(e),(f)) indicates a much higher value of T Maxe if compared to that reached in bulk gold (Fig. 2(d)).These facts demonstrate that NPG is a very promisingcandidate for hot-electron plasmonics applications.Numerical evaluation of T e ( t ) and T l ( t ) dynamics isperformed within the two-temperature model, in whichenergy 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 slowelectron-phonon interaction [18]. In an improved versionof the TT model [19], e-h pairs produced by plasmondecay 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 respec-tively, resulting in the following coupled equations: C e dT e dt = − g ( T e − T l ) − e − ( τ − , relax + τ − , relax ) t t h t + τ e , relax (cid:16) − e t/τ e , relax (cid:17)i · P a C l T l dt = g ( T e − T l ) − e − ( τ − , relax + τ − , relax ) t tτ p , relax h τ e , relax (cid:16) − e t/τ e , relax (cid:17)i · P a (1) T e ( K ) (cid:1) t r / t r ( x - ) (cid:2) = 600 nm (cid:2) = 500 nm x-0.125x-1x-1 Bulk Gold f = 1.00NPG3 f = 0.39 d wire = 50 nmNPG9 f = 0.37 d wire = 80 nm (a)(e)(f) (g)(h)(i)Time delay (ps) T e ( K ) T e ( K ) (b) (c)(d) f f d wire = (cid:1) f =0.4 Figure 3. (a) Effect of varying f on the T e ( t ) dynamics ( f = 1 corresponds to bulk gold). (b) Same curves as (a) normalized at T Max e to highlight the different temperature dynamics. (c) Effect of d wire on the T e ( t ) decay for f = 0 .
4. (d-i) Comparison ofthe spectra obtained from the ∆ tr ( t ) /tr color plots of Fig. 2 at λ = 600 nm (d)-(f) and at λ = 500 nm (g)-(i) with the electrontemperatures obtained from the extended TT model (dark blue curves). (d),(g): bulk gold; (e),(h): NPG3; (f),(i): NPG9.Inset of panels (b),(c) show the full undershoot at short delay due to the generation of non-thermalized carriers in NPG. Notethat ∆ tr ( t ) /tr in panels (g)-(i) is reported with negative multiplication factors. where C e and C l are the electronic and lattice heat ca-pacities per unit volume, g is the electron-phonon cou-pling constant, τ e , relax and τ p , relax are characteristic timesrelated to the electron-electron and electron-phonon en-ergy relaxation [19]. The pump pulse power in the in-stantaneous pump-pulse approximation is P a = F a /d ,with d the film thickness, F a = (1 − R − tr ) F and F = 180 µ J / cm the pump fluence. For bulk gold thinfilms, the values of the parameters used in the extendedTT model are C e = γT e , γ = 68 Jm − K − , C l =2 . · Jm − K − , g = 2 . · Wm − K − , E F = 7 . τ e , relax = 136 fs, τ p , relax = 1650 fs, E P = 1 .
65 eV [16]. Inthe inset of Fig. 2(c), the T e curve obtained from Eq.1fits to the ∆ tr ( t ) /tr (0) data for bulk gold, provided thatthe delay scale is normalized by the relative change factor ξ = ln(∆ tr Max /tr ) / ln(∆ T Maxe /T e ( t . ≃ f β , where β is the corresponding scalingexponent, as summarized in Table 1. For C e and C l , thescaling exponent is a trivial β C = 1 as they scale linearlywith the mass density. For the thermal conductance, theproblem is considerably more complex due to the NPGnetwork connectivity. Previous works have employed theAsymmetric Bruggeman Theory (ABT) [43] to calcu-late the electron thermal conductivity in NPG [44, 45] and the lattice thermal conductivity of nanoporous glass[31]. In both cases, the results point toward an exper-imental value of β k = 3 / k l = 1 / C l v s l ph , where C l is the lattice spe-cific heat, v s is the speed of sound and l ph is the phononmean free path. Since l ph and C l are microscopic quan-tities that should not depend on the geometry, v s shouldscale with the exponent β v = β k = +3 / v s . The first quantity is g [46]: g = π m e n e v s T e τ ( T e , T l ) (2)where n e is the microscopic electron density, m e is theelectron mass, and τ ( T e , T l ) is the total electron scat-tering time including electron-electron τ ee and electronphonon τ ep scatterings. Following Matthiessen’s rule andassuming momentum-independent scattering, the effectof electron scattering at physical boundaries in NPGligaments can be included in the model by consider-ing an additional scattering time τ B = v F /d wire , where v F = 1 . · m / s is the Fermi velocity in gold [44]:1 τ ( T e , T l ) = 1 τ ee + 1 τ ep + 1 τ B = AT + BT l + v F d wire (3)In Eq.3 A and B are temperature-independent coeffi-cients that in gold can be taken equal to A = 1 . · K − s − , B = 1 . · K − s − [47]. In bulk gold d wire → ∞ and the contribution of τ B is negligible. Thecase of gold nanoparticles can also be obtained by using f = 1 and d wire similar to the value of the nanoparticlediameter [ ? ]. In Eq.2 the only quantity that scales with f is the speed of sound v s , therefore for g we obtain ascaling exponent β g = 2 β v = +3.The second quantity of the TT model proportional to v s is the Debye temperature Θ D :Θ D = ~ k D v s k B (4)where k D = (6 πN a ) / ( N a is the atomic density) and k B is the Boltzmann constant. k B Θ D represents the averagephonon energy and, as such, enters in the definition ofthe electron-phonon energy relaxation time as τ p , relax = τ ep E P /k B Θ D . Therefore, from β Θ = β v = +3 / β τ = − β Θ = − / τ p , relax . Table I. Geometrical scaling of the TT model parameters.Quantity C e , C l v s g Θ D τ p , relax scaling ( f < f f / f f / f − / Using the scaling exponents of Table I, we can de-scribe the ultrafast electron dynamics of NPG by solvingthe extended TT model of Eq.1 as a function of f . Itis important to notice that the scaled quantities are ef-fective quantities purposely defined for the nanoporoussolid, and do not correspond to an actual variation ofthe microscopic quantities of bulk gold. In Fig. 3(a)-(c) the model results are reported, highlighting the effectof f and d wire on T e . In the model, the temperaturedynamics is clearly slowed down for low f and T Maxe isconsiderably increased. Electron scattering at physicalboundaries, which is almost absent in bulk gold, becomesrelevant only when the electron mean free path in gold ℓ ∼
40 nm [44, 48] is of the same order of the mean lig-ament diameter d wire (as it is in our samples NPG3 andNPG9 with d wire of 50 nm and 80 nm, respectively).In Fig. 3(d)-(i) we compare cuts of the experimentaldata of Fig. 2(d)-(f) at fixed λ = 600 nm and λ = 500 nmwith the prediction of the TT model scaled by f = 0 . f = 0 .
37 for NPG9. The relaxation dy-namics for t > . tr ( t ) /tr for NPG if compared to bulk gold at λ = 600 nm is in-dicative of the much higher T Maxe reached in NPG. TheTT model accounts only for the dynamics of thermalizedelectrons and therefore it cannot reproduce the ultrafastvariations of ∆ tr ( t ) /tr at very short t ≥
0. Especially at λ = 600 nm, a strong induced absorption signal can beseen for t <
100 fs (see insets of panels (e) and (f)) and it can be attributed to the excitation of non-thermalizedhigh-energy carriers [15, 38–41]. Hot carriers are al-most absent in bulk gold for the same excitation con-ditions as for NPG, as expected due to the high densityof field-enhancement hotspots in NPG and to the highsurface/volume ratio [13] of the NPG fractal structure[30]. At λ = 500 nm the contribution of non-thermalizedcarriers to ∆ tr ( t ) /tr is much smaller [16, 17] and it doesnot impact on the fitting of the model to the data as seenin Fig. 3(h),(i). It has been observed [22] that surfacefunctionalization of gold nanostructures leads to similarslowdown of the temperature dynamics. 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