Clément
Panais
a,
Noëlle
Lascoux
a,
Sylvie
Marguet
b,
Paolo
Maioli
a,
Francesco
Banfi
a,
Fabrice
Vallée
a,
Natalia
Del Fatti
ac and
Aurélien
Crut
*a
aUniversité de Lyon, CNRS, Université Claude Bernard Lyon 1, Institut Lumière Matière, F-69622 Villeurbanne, France. E-mail: aurelien.crut@univ-lyon1.fr
bUniversité Paris-Saclay, CEA, CNRS, NIMBE, 91191 Gif-sur-Yvette, France
cInstitut Universitaire de France (IUF), France
First published on 28th May 2024
The thermal dynamics and transient optical response of individual gold nanodisks supported on thin silicon nitride membranes were investigated using optical time-resolved pump–probe spectroscopy and finite-element modeling. The effect of reducing the membrane thickness from 50 nm to 15 nm on the nanodisk thermal dynamics was explored. A significant deceleration of the nanodisk cooling kinetics was observed, and linked to a quasi-two-dimensional heat diffusion process within the 15 nm thick membrane, without detectable modification of its thermal conductivity. Systematic measurements involving different optical probe wavelengths additionally revealed the contribution of indirect membrane heating to the measured time-resolved signals, an effect particularly pronounced in the spectral range where direct optical heating of the nanodisk induces minimal ultrafast modifications of its extinction cross-section.
Optical studies conducted on silicon membranes have for instance revealed that the room-temperature thermal conductivity of these 1D-confined systems is thickness-dependent and considerably reduced as compared to bulk values,27,28 consistently with electrical measurements on silicon films and nanowires.29,30 These observations can be attributed to a phonon mean free path diminishment due to boundary scattering at membrane surfaces. Utilizing small nanoparticles as nanoheaters offers the opportunity to collect quantitative insights into both the thermal conductance of their interface with their local environment and the modalities of heat propagation within this environment. Employing nanoparticles with optical responses exhibiting temperature-dependent resonant features has proved advantageous for sensitively probing nanoparticle heating.24,31,32 This advantage occurs in particular for noble metal nanoparticles, whose absorption and scattering spectra are characterized by localized surface plasmon resonances (SPRs), sensitive to the temperature-induced changes in the dielectric properties of both the nanoparticles and their local environment.33 In this context, conducting thermo-optical measurements on single plasmonic nanoparticles offers additional advantages for fundamental investigations of nanoscale heat transfer, because such measurements establish a precise correlation between the composition and morphological features of nanoparticles, their thermal dynamics, and the transient optical response they induce.32,34
While single-approaches have been used for two decades for studying the electronic and vibrational dynamics of metal nanoparticles,35,36 they have only lately been applied to heat transfer investigations. We have recently reported optical investigations on the thermal coupling between a single, 3D-confined nanoheater (a gold nanodisk (ND)) and a bulk or 1D-confined substrate (thick sapphire substrate or suspended silica or silicon nitride membrane with ≈50 nm thickness).32,37 The work presented here focuses on the case of single NDs supported on a notably thinner Si3N4 membrane (15 nm thickness). A key difference with the previously investigated cases is the reduced time (≈0.5 ns) required for heat to traverse the membrane thickness. This shorter time now falls within our experimentally accessible range of pump–probe delays (extending up to about 3 ns), enabling a more direct observation of finite-thickness thermal effects. We also measured the transient optical responses of membrane-supported single NDs with varying probe wavelength, an experimentally accessible parameter allowing to tune and disentangle the influences of ND/environment heating on the measured time-resolved signals.31
The exploration of the ultrafast dynamics of individual supported gold NDs following their sudden excitation was carried out through time-resolved pump–probe optical experiments, using various probe wavelengths λpr, as detailed in the Methods section. To precisely distinguish the influence of λpr from that of the variations of pump beam attributes (wavelength, spot size and fluence) during the course of the experiments, the measured transient extinction changes Δσext were divided by the estimated initial ND temperature rise ΔT0 (of the order of 10 K) following the partial absorption of each pump pulse and ND internal thermalization by electron–lattice energy exchanges.32 Illustrative normalized transient extinction changes measured on a single ND are presented in Fig. 2. The measured Δσext(t) dynamics exhibit the three characteristic features of the transient optical response of metal nanoparticles subsequent to their sudden optical excitation:33,39 (1) an initial peak linked to ND optical excitation and internal thermalization on few picosecond time scales, (2) oscillations associated to ND acoustic vibrations, showing here a damping time of a few hundreds of ps, and (3) a relaxation over a nanosecond time scale, reflecting the ND cooling dynamics, i.e. the dissipation into the ND environment of energy deposited in the investigated nanoparticle through partial absorption of the pump pulses.
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Fig. 2 Probe wavelength effect on the time-resolved signals measured on a gold ND deposited on a 15 nm thick silicon nitride membrane. The figure shows illustrative time-resolved extinction cross-section changes measured with different probe wavelengths λpr on the ND whose extinction spectrum is shown in Fig. 1 (λSPR = 810 nm), divided by ΔT0, the estimated ND temperature rise following its interaction with a pump pulse with λpp = λpr/2 wavelength and ≈0.5 J m−2 fluence. |
Fig. 2 shows that the average value of the measured signals differs from zero at negative time delays. This feature indicates that the Trep = 1/frep = 12.5 ns time separating two successive pump pulses is not sufficient for complete dissipation far away from the ND of the energy injected by each pump pulse. Additionally, Fig. 2 also illustrates the strong and different λpr-dependencies of the amplitudes of the oscillating and thermal components of the signals. Probing at the SPR central position (λpr ≈ λSPR) leads to a large thermal component and to a nearly undistinguishable oscillatory component. Conversely, the signals measured using λpr − λSPR = 20–30 nm present marked oscillations, while remaining nearly constant after damping of the oscillations (indicating a weak sensitivity to ND heating). This work focuses on the thermal components of the measured time-resolved signals, which were deduced from the raw signals by subtraction of their oscillating part, fitted by a sum of 1–3 damped sinusoids (each corresponding to a distinct detected vibrational mode).38,40
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Fig. 3 Thermal dynamics of single Au NDs on thin Si3N4 membranes. (a) Normalized ΔTND(t) temperature rise dynamics measured on two distinct gold NDs supported by a 15 nm thick Si3N4 membrane (black and red traces) and ΔTND(t) dynamics simulated with (solid purple line) and without (dashed line) inclusion of repetition effects in the modeling (see main text for details). (b) Signals deduced from those of panel (a) after subtraction of their average values at negative time delays ΔT′ND(t) = ΔTND(t) − ΔTND(0−) and normalization, presented on a logarithmic scale (same color code). The signals previously measured for two different NDs (blue and green traces) and modeled (solid and dashed orange lines) for a 50 nm thick membrane are also shown for comparison.37 (c) Temperature profiles computed using FEM at time delays of 0, 0.1, 1 and 10 ns after the sudden optical excitation of a ND located on a 15 nm (left) or 50 nm (right) thick Si3N4 membrane. The Q = ΔT/ΔT0 ratios are plotted using a color scale ranging from 0 to Qmax, which corresponds to the Q value inside the ND at each considered time (indicated). |
The ND cooling dynamics shown in Fig. 3a become very similar if temperature rises are defined relatively to the ND temperature before each pump excitation, rather than to that in the absence of any excitation. This is shown in Fig. 3b, where the normalized ΔT′ND(t) = ΔTND(t) − ΔTND(0−) dynamics are plotted alongside previous measurements on NDs supported on a 50 nm Si3N4 membrane,37 highlighting ΔT′ND(t) strong dependence on membrane thickness. The ΔT′ND(t) dynamics measured for NDs deposited on 15 nm thick Si3N4 membranes exhibit a non-exponential behavior, indicating that they are not solely ruled by heat transfer at the ND-membrane interface.21 While they resemble those previously recorded on 50 nm thick membrane at <0.5 ns delays, they become much slower at larger delays (Fig. 3b), presenting for example 3× higher normalized ΔT′ND(t) values after 2.7 ns, the largest pump–probe delays achievable in our experiments.
The observed membrane thickness effect on ΔT′ND(t) dynamics can be qualitatively understood by considering the solutions of the heat equation (with αm the thermal diffusivity of the membrane) for a point source located at (0,0,0) on the surface of a membrane of thickness hm (0 < z < hm) and of infinite lateral extension, initially at a uniform temperature, with insulation boundary conditions on the membrane surfaces
. The time- and position-dependent temperature rise ΔT in the membrane can be deduced from the solution of the heat equation in a 3D semi-infinite medium
by adding the effect of an infinity of virtual sources at positions (0,0,2nhm), with n an integer,41,42 and writes
![]() | (1) |
The temperature rise ΔT is thus obtained by multiplying the solution for a point source at the surface of an infinite 2D medium (Gaussian distribution with a width increasing as t1/2) by the solution for a point source on a 1D finite slab (Jacobi theta function). At short time scales (αmt ≪ hm2), the sum is dominated by its n = 0 term, and ΔT approximately coincides with the solution for a 3D semi-infinite substrate, with a t−3/2 dependence at the source position. Conversely, at long time scales (αmt ≫ hm2, so that the sum can be approximated by an integral), , which corresponds to a uniform temperature along the membrane thickness and a lateral 2D diffusion of heat. The transition between the two regimes occurs for αmt ≈ hm2, i.e., for a Si3N4 membrane (αm = 5 × 10−7 m2 s−1, assuming a Λm = 1 W m−1 K−1 thermal conductivity) at times of the order of 0.5 ns and 5 ns for hm = 15 nm and hm = 50 nm, respectively. Although based on a simplified analysis, the 0.5 ns transition time for a hm = 15 nm membrane corresponds well with that at which experimental signals measured on the two membranes start to significantly deviate (Fig. 3b).
A more quantitative modeling of ND cooling dynamics necessitates to avoid the simplifications made in the previous analysis and to take into account the actual ND geometry, the non-instantaneous release of heat in the supporting membrane and the periodical character of ND optical excitation. To accomplish this, numerical simulations of the cooling dynamics of supported NDs were conducted using a previously described frequency-domain thermal model32,37 (note that an analytical treatment of the problem, such as that reported for stacked layers with infinite lateral extension,19,43 is much more challenging here because of the mixed boundary conditions at the membrane surface). This model is based on the computation, using FEM, of the frequency-dependent thermal response of a supported ND, i.e., the spatial distribution of the amplitude and phase of the temperature oscillations following a sinusoidal excitation, which summarizes the ND thermal response to any time-dependent excitation q(t).19,44 In such calculations, the amplitude of thermal oscillations decays in a frequency-dependent manner around heated regions, over a characteristic length scale given by the heat penetration length (with Lp ≈ 40 nm in silicon nitride when f = frep = 80 MHz). This implies that for ND excitation by a train of Dirac pulses
, which corresponds to a
frequency content, the membrane properties (size, modalities of heat exchanges at the membrane ends and with air, which are poorly known in our experiments) at scales much larger than Lp only affect the average heating (0-frequency component). Uncertainties about the large-scale membrane properties are therefore not limiting for a quantitative analysis of the ΔT′ND(t) dynamics. Here, a Si3N4 membrane with Dm = 2 μm diameter (much smaller than the actual membrane size, thus largely reducing computational times) with fixed temperature boundary conditions (BCs) at its end and insulating BCs on its circular faces was used, as in our previous work.37D = 100 nm and h = 10 nm values were used for ND diameter and thickness. These dimensions roughly correspond to the average ones deduced from electron microscopy observations and SMS characterization on NDs from the colloidal solution used.
The fitting parameters of our model are the thermal conductance G of the Au–Si3N4 interface and the thermal conductivity Λm of Si3N4 (whose previous measurements have led to scattered results17,45–49). The ΔT′ND(t) dynamics of NDs deposited on 15 nm thick membranes can be accurately reproduced using the same values of G and Λm as in the case of 50 nm thick membranes,37 namely G = 150 MW m−2 K−1 and Λm = 1 W m−1 K−1. This is illustrated in Fig. 3b, which shows the ΔT′ND(t) dynamics computed for NDs supported on Si3N4 membranes using these G and Λm values (solid purple and orange lines). The excellent reproduction of experimental signals at <0.5 ns delays (at which ΔT′ND dynamics are mostly controlled by G and Λm37) using G = 150 MW m−2 K−1 and Λm = 1 W m−1 K−1 indicates that these values constitute accurate estimations of G and Λm (with uncertainties of the order of 10%) for both 50 and 15 nm thick membranes. The slight shift between the measured and modeled ΔT′ND(t) dynamics at >1 ns delays could have several causes, including a slight difference between the nominal and actual values of the membrane thickness. Regarding the ΔTND(t) dynamics (Fig. 3a), one can observe that the ΔTND(0−) temperature rises predicted by the model are higher by about 50% than the measured ones, similarly to previous observations for ≈50 nm thick membranes.37 This difference, which would increase with the use in the modeling of larger Dm values, closer to the actual membrane size, suggests that the thermal coupling of the membrane with ambient air taking place at the top and bottom planar surfaces of the membrane (not included in the modeling where dissipation only occurs at the membrane lateral ends) plays a significant role in large-scale membrane cooling.
The deduction of similar Λm values from the analysis of the ΔT′ND(t) dynamics measured on 50 nm and 15 nm thick membranes stands in stark contrast to findings on silicon membranes of similar thickness, whose thermal conductivity was seen to be strongly thickness-dependent and up to ≈5 times smaller than the bulk silicon one.28 This disparity in the thickness-dependence of the thermal properties of Si3N4 and Si membranes can be attributed to the large difference between the heat conductivity and phonon mean free paths of these two materials in the bulk form. Indeed, amorphous media such as silica and silicon nitride have a ≈1 nm room temperature phonon mean free path, while in crystalline silicon a large fraction of phonons have >100× longer mean free paths.15,22,50 Experimental observations are thus consistent with the fact that membrane thickness reduction in the tens of nm range should significantly diminish the mean free path of phonons in Si, but not in Si3N4.
The ΔTND(t) dynamics were also computed in the case of ND excitation by a single pump pulse, which corresponds to discarding accumulation effects (dashed lines in Fig. 3). These complementary calculations allow to appreciate the respectively strong and relatively weak influences of a repeated ND excitation on the ΔTND(t) and ΔT′ND(t) dynamics. They also enable a clear visualization of the transition from a 3D to a 2D heat diffusion in a membrane heated by a source located at its surface, discussed above based on analytical expressions. This transition is illustrated in Fig. 3c, which presents snapshots of the temperature profiles computed for single pulse excitation of a ND positioned on a 15 nm or 50 nm thick membrane. At t = 0.1 ns heat diffusion presents a distinct 3D character, with only regions less than 10 nm away from the ND being significantly heated. Conversely, at t = 1 ns the temperature profile has become independent of depth for a 15 nm thick membrane (but not for a 50 nm thick one). At t = 10 ns (a delay exceeding the experimentally accessible ones) diffusion has evolved into a 2D process in both cases, resulting in depth-independent temperature distributions.
Δσext(λpr) = AND(λpr)ΔTND + Am(λpr)ΔTm | (2) |
![]() | (3) |
As each pump pulse deposits energy in the ND only, the extinction modification immediately following ND excitation and internal thermalization is solely due to an increase of ND temperature. This enables experimental extraction of the wavelength-dependent AND coefficient appearing in eqn (2) and (3), by evaluating the variation of the thermal components of Δσext(t)/ΔT0 (Fig. 2) following ND excitation by the pump pulse. The AND spectra deduced from measurements on two NDs are plotted in Fig. 4a, as a function of λpr − λSPR to facilitate the comparison between the different measurements, as λSPR slightly varies from one ND to another and presents for some NDs a slight polarization dependence, signature of a small ND elongation.38 They are similar for the two investigated NDs and polarizations. ND heating produces a negative σext variation for λpr − λSPR < 20 nm and a positive one for higher λpr values. The maximal |AND| value, obtained for λpr ≈ λSPR, is of the order of 40 nm2 K−1. The AND spectra resemble those previously measured on lithographed NDs,32 with however different maximal |AND| values, which can be ascribed to the use of NDs with different dimensions in the two studies.
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Fig. 4 Sensitivity of ND extinction cross-section to ND heating. (a) Sensitivity deduced from time-resolved experiments (see eqn (2) and (3) for AND definition). The experimental results were obtained on two distinct NDs (black circles and red squares) for two orthogonal polarizations of the probe beam (solid and hollow symbols). (b) Sensitivity computed using FEM optical simulations and temperature dependence of gold dielectric function from ref. 53 (solid purple line). The computed sensitivity to Si3N4 membrane heating is also shown for comparison (dashed line). |
Due to the large uncertainties on the temperature dependence of gold dielectric function (eqn (3)), whose measurement has led to scattered results,51–57 obtaining a precise match between the measured and modeled AND spectra is hardly achievable. Nevertheless, the AND spectrum computed based on eqn (3), shown in Fig. 4b, presents spectral variations and absolute values similar to the measured ones, mostly differing by the spectral position of the AND sign change, which is located further away from λSPR in the simulation case. The general shape of the AND spectrum results from the much larger temperature-dependence of ε2 (which mostly influences SPR width) as compared to ε1 (which mostly impacts SPR position) in eqn (3). ND heating thus mostly induces a broadening of SPR, leading to a decrease of σext close to λSPR and to an increase further away from λSPR. The computed Am spectrum is also shown in Fig. 4b. Its spectral shape largely differs from the AND one, which is due to the fact that membrane heating mostly affects SPR position. |Am| is usually much smaller than |AND| (Fig. 4b). However, this is not the case in the ≈20 nm broad spectral range around the wavelength where AND vanishes, suggesting that measurements with λpr in this spectral range should be very significantly affected by membrane heating.
Fig. 5a shows the Δσext,norm(t) dynamics measured for different λpr values (obtained by dividing Δσext(t) by its initial value) do not overlap. As λpr is increased on the red flank of the resonance, the Δσext,norm(t) dynamics become initially slightly faster (λpr − λSPR = 10 nm) and then much slower (λpr − λSPR = 25–30 nm). A complementary illustration of the λpr-dependence of the Δσext,norm(t) is shown in Fig. 5b, where the Δσext,norm value at the maximal pump–probe delay (2.7 ns) is plotted as a function of λpr. While 0.15 values are observed for most λpr values, significantly larger values, up to 0.45 (corresponding to much slower dynamics) are observed for λpr − λSPR = 20–30 nm. This λpr-dependence of the Δσext,norm(t) dynamics indicates that the transient ND optical response is not simply proportional to ΔTND(t) when λpr ≠ λSPR, especially when the sensitivity to ND heating is small and changes sign (λpr − λSPR = 20–30 nm, Fig. 4a), and strongly suggests that Δσext is significantly affected by environment heating.
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Fig. 5 Influence of Si3N4 membrane heating on the dynamics of transient ND extinction changes. (a) Examples of normalized thermal components of Δσext measured on a single ND using different λpr values. (b) Average value of the normalized extinction change measured at 2.7 ns pump–probe delay (maximal delay in the experiments) for different NDs and polarizations. The same color code as in Fig. 4 was used. (c) Examples of normalized Δσext changes FEM-computed using different λpr values. (d) Normalized extinction change after 3 ns computed using FEM. |
Comparison of the measured signals with the predictions of a complete thermo-optical model taking into account the inhomogeneous and time-dependent temperature rise inside the membrane is useful to fully demonstrate this hypothesis. The Δσext(t) dynamics were thus simulated using the two-step modeling procedure discussed in ref. 31. It basically consists in complementing the purely thermal FEM computations discussed above, which provide the spatially- and time-dependent temperature rise profiles in the ND and the membrane, with optical FEM simulations estimating the temporal evolution of the ND extinction cross-section σext. To do so, σext was recalculated at various delays using dielectric functions modified in a spatially-dependent way, as (considering the example of the membrane one) . The predictions of the model regarding the Δσext,norm(t) dynamics and its value after 3 ns are shown in Fig. 5c and d. The trends observable in these plots are very close to those of the experimental results, shown in Fig. 5a and b. In particular, the modeled results also feature a considerable slow-down of the dynamics (associated with a much larger final Δσext,norm value) for probe wavelengths slightly higher than that at which AND vanishes. Additionally, the model also predicts much faster dynamics at slightly lower wavelength, an effect which could not be observed experimentally (essentially because the raw signals measured in this range are difficult to analyze, as they present strong oscillations and low-amplitude thermal components).
The transient optical response of NDs induced by their thermal dynamics was computed by reproducing the previous calculations at various stages of the thermal dynamics, using non-uniform dielectric functions locally modified as compared to room-temperature Au and Si3N4 values as and
.
was used for Si3N4, while wavelength-dependent tables from ref. 53 were used for
.
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