Article www.acsnano.org A General and Predictive Understanding of Thermal Transport from 1D- and 2D-Confined Nanostructures: Theory and Experiment Albert Beardo,*,⊥ Joshua L. Knobloch,*,⊥ Lluc Sendra, Javier Bafaluy, Travis D. Frazer, Weilun Chao, Jorge N. Hernandez-Charpak, Henry C. Kapteyn, Begoña Abad, Margaret M. Murnane, F. Xavier Alvarez, and Juan Camacho* Cite This: ACS Nano 2021, 15, 13019−13030 Read Online ACCESS Metrics & More Article Recommendations * sı Supporting Information ABSTRACT: Heat management is crucial in the design of nanoscale devices as the operating temperature determines their efficiency and lifetime. Past experimental and theoretical works exploring nanoscale heat transport in semiconductors addressed known deviations from Fourier’s law modeling by including effective parameters, such as a size-dependent thermal conductivity. However, recent experiments have qualitatively shown behavior that cannot be modeled in this way. Here, we combine advanced experiment and theory to show that the cooling of 1D- and 2D-confined nanoscale hot spots on silicon can be described using a general hydrodynamic heat transport model, contrary to previous understanding of heat flow in bulk silicon. We use a comprehensive set of extreme ultraviolet scatterometry measurements of nondiffusive transport from transiently heated nanolines and nanodots to validate and generalize our ab initio model, that does not need any geometry- dependent fitting parameters. This allows us to uncover the existence of two distinct time scales and heat transport mechanisms: an interface resistance regime that dominates on short time scales and a hydrodynamic-like phonon transport regime that dominates on longer time scales. Moreover, our model can predict the full thermomechanical response on nanometer length scales and picosecond time scales for arbitrary geometries, providing an advanced practical tool for thermal management of nanoscale technologies. Furthermore, we derive analytical expressions for the transport time scales, valid for a subset of geometries, supplying a route for optimizing heat dissipation. KEYWORDS: phonon hydrodynamics, non-Fourier heat transport, silicon, high-order harmonic generation, pump−probe spectroscopy A dvances in fabrication have scaled the characteristic design for good thermal management in next-generation dimensions of complex systems to the few nanometer nanodevices. range and even thinner. At these length scales, Theoretical proposals based on truncated Levy flights,13 conventional macroscopic (bulk) models can fail to accurately suppression of phonons,8,14 or relaxons15 have explained describe nanoscale behavior because of the dominance of certain aspects of nondiffusive thermal transport for specific geometries. Phonon hydrodynamics16−24 has been also interfaces and surfaces. Specifically, thermal transport from successfully used to explain thermal transport behavior on nanoscale heat sources on semiconductor substrates strongly 2D materials,25 such as graphene,26 and even in bulk materials deviates from bulk diffusive transport predictions. Experiments at very low temperatures.27 As this behavior is known to occur show that as the heat source size is reduced, the heat transport when “normal” phonon scattering events (i.e., processes that efficiency falls well below what is predicted by bulk diffusion, conserve quasi-momentum) dominate over “resistive” ones, both for structured optical excitation1−4 or for optically excited the existence of hydrodynamic transport in bulk semi- nanostructured transducers.5−12 Moreover, recent experiments have uncovered that both the size and spacing of periodic Received: March 5, 2021 nanoheater arrays strongly influence thermal transport, Accepted: July 23, 2021 resulting in counterintuitive behaviors.8,9 However, there is Published: July 30, 2021 still no consensus on the underlying physicsin large part because there is no comprehensive model to describe these new nanoscale thermal transport regimes. This precludes smart © 2021 American Chemical Society https://doi.org/10.1021/acsnano.1c01946 13019 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article conductors, like silicon, at room temperature has been predicts the heat transport over a wide range of length-scales explicitly discarded.28 and time-scales from 1D- and 2D-confined heat sources on a It is widely accepted that solving the Boltzmann transport silicon substrateincluding the counterintuitive behavior of equation with ab initio calculated parameters is the most the closely spaced geometryand captures the full thermo- precise way to describe the transport of phonons, which are the mechanical response to the system, which is beyond the dominant heat carriers for semiconductor and dielectric capabilities of microscopic models. materials.29−31 However, several difficulties associated with Our mesoscopic hydrodynamic model also provides insight this approach limit its use at a practical level. First, this into the fundamental transport behavior. KCM allows us to equation is challenging to solve in general because of the identify the time scales over which two different transport complexity of phonon collisions. To overcome this challenge, mechanisms are dominant: one characteristic time dominated the relaxation time approximation is often used to simplify the by the thermal boundary resistance and another regime that is collision expression.13,32 However, this approximation does not dominated by hydrodynamic heat transport. The latter guarantee energy conservation, which can lead to invalid mechanism is responsible for the slow thermal decay of small results.13,33 Second, complex geometries are challenging heat sources, and consequently, its reduction is responsible for because one must model how each phonon mode interacts the increased dissipation of close-packed nanoheaters. with every boundary present. Finally, coupling this equation to Furthermore, we develop a two-box model, derived from the other phenomena, such as thermoelectricity or thermo- hydrodynamic equation, which provides a physical interpreta- elasticity, exponentially increases the computational require- tion and specific expressions for the two characteristic ments. These challenges can prevent microscopic models, like dissipation mechanisms. We confirm these findings by the Boltzmann transport equation, from being directly comparing our models to both past 1D-confined and new compared to experimental data. 1D- and 2D-confined experimental data. We conclude that Because microscopic models cannot be applied in many KCMinvolving only a few parametersprovides a predictive complex geometries, experiments often use an intermediate description of the thermal and mechanical response in these layer, or mesoscopic models, to compare results and theory. complex systems with highly nondiffusive behavior and has Most mesoscopic models to date are based on Fourier’s law of specific advantages over the traditional effective Fourier model. heat diffusion with the addition of phenomenological effective This work thus represents a significant advance in both parameters. This approach fits effective parameters to experi- experimental and modeling capabilities opening the door to ments and then formulates theoretical models to connect the improved thermal management in iterative nanoscale device fitted values to ab initio calculations. Recent works have used design, including possible routes to increase clock rates in this effective Fourier model to analyze heat dissipation away nanoelectronics by surpassing what has been called the from metallic nanostructures of varying size and spacing.5−9,11 “thermal wall”.40,41 This can quantify the deviation from the diffusive prediction by We measure the heat dissipation away from 1D-confined fitting either an effective thermal boundary resistance between periodic nanoline heat sources on a silicon substrate using the transducer and substrate7−9 or an effective thermal dynamic EUV scatterometry similar to that of refs 7−9 but conductivity of the substrate.5,6,12,34−36 These techniques with significantly improved signal-to-noise ratioby nearly 2 have significantly advanced our understanding, making it orders of magnitude. These improvements allow us to perform possible to develop new experimental mean free path new measurements on 2D-confined periodic nanodot heat spectroscopy techniques,1 as well as uncovering new transport sources on a silicon substrate, which are more challenging than regimes dominated by the heat source spacing.7,8 However, 1D measurements due the reduced fraction of surface covered using Fourier’s law as a mesoscopic model, even with effective by the heat sources. Both the nanodot and nanoline arrays parameters, can obscure the underlying physics and fails to were fabricated under identical conditions. Our time-resolved predict thermal transport observed for all time and length measurements use an ultrafast infrared pump laser pulse to scales.19,22 Most importantly, this approach is difficult to rapidly excite thermal heating and expansion in the metallic generalize to arbitrary geometries or materials. structures. The resulting thermal and elastic surface In this work, we present a comprehensive set of dynamic deformation is monitored by measuring the change in EUV scatterometry measurements of nondiffusive heat flow diffraction efficiency of an ultrashort EUV probe pulse, as away from 1D- and 2D-confined nanostructures on bulk depicted in Figure 1 (see Methods). Using this technique, we silicon. We use this data to validate and generalize the Kinetic observe the heat dissipation from nanodot arrays in general Collective Model (KCM),37,38 which is a mesoscopic model geometries without complex fabrication and from nanoline which uses a hydrodynamic-like heat transport equation16 with arrays down to 20 nm in size (L) and 80 nm in spacing (P). ab initio parameters. Contrary to conventional understanding, To interpret the experimental data, we implement a we show that heat transport away from nanoscale sources on mesoscopic model using KCM and a thermoelastic set of bulk silicon can be predicted by the hydrodynamic equation. equations (see Methods). For heat transport, we use Fourier’s This generalizes the hydrodynamic framework to situations law for the metal sources (which is dominated by electrons), where phonon momentum is conserved, which applies not and the Guyer and Krumhansl transport equation16 for the only when normal collisions dominate but in regions with size substrate (silicon), which is the material where non-Fourier comparable to the average resistive phonon mean free path behavior is expected: near heat sources and system boundaries.22,38,39 We also dq experimentally observe that closely spaced 2D-confined τ + q = −κ ∇T + S2(∇2 q + α∇(∇·q)) dt (1) (nanodots) on a bulk silicon substrate cool faster than widely spaced ones, and that this effect is larger in 2D-confined than where κ is the bulk thermal conductivity of substrate, τ the in 1D-confined (nanoline) sources observed by previous relaxation time of flux q, and l the nonlocal lengththat can be works.8,9 Moreover, we demonstrate that KCM both fully microscopically interpreted as a weighted average phonon 13020 https://doi.org/10.1021/acsnano.1c01946 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article Figure 1. Schematic of dynamic EUV scatterometry for probing nondiffusive hydrodynamic-like heat flow. An ultrafast laser pulse rapidly heats the nanostructured transducers, which dissipate the thermal energy by transferring heat to the substrate. The heat flows away from the nanoscale heat sources following a nondiffusive hydrodynamic-like behavior, creating a “balloon” shaped temperature profile. The resulting surface deformation of the heated nanostructures and substrate is measured via diffraction of an ultrafast Extreme Ultraviolet (EUV) probe pulse, after a controlled pump−probe time delay. The EUV pulse scatters from the periodic nanostructure arrays into a detector. We reduce the recorded scattering pattern into a single value of diffraction efficiency as a function of time delay between the pump and probe pulses, which precisely tracks the thermal and elastic dynamics in the sample. mean free path. All these parameters are intrinsic properties of 1 is expected to be valid for nanostructure arrays satisfying P − the substrate. Equation (1) resembles the hydrodynamic L > 2l. We term experiments under this condition, where Navier−Stokes equation of fluids; thus, we can build analogies heaters are expected to behave independently, as effectively between heat flow and fluid behavior. Fourier’s law can be isolated heat sources, and those with P − L < 2l as close- easily recovered from eq 1: when the experiment time scales packed heat sources. are much larger than τ, the first term can be neglected, and if spatial scales are much larger than l, the last term can be RESULTS AND DISCUSSION neglected. For sizes comparable to l, however, these viscous terms become important and capture the nondiffusive We first study effectively isolated heat sources for both 1D- transport due to momentum conservation at the scale of the confined (nanolines) and 2D-confined (nanodots) of different phonon mean free paths. In nanoscale regions near the heat sizes and periodicities. Figure 2 compares the experimental sources, the momentum of emitted phonons is conserved due results on nanolines and nanodots with theoretical KCM to the lack of resistive collisions. Hence, hydrodynamic effects solutions obtained using COMSOL. We compare both inertial can locally alter the heat transport even in semiconductors like solutions, which include elastic waves generated by the silicon. In the limit where normal collisions dominate, Guyer impulsive pump laser excitation and quasi-static solutions and Krumhansl16 found α = 2; however, we use α = 1/3 without elastic waves to isolate the effects of the heat flow (see analogous to a fluid with zero volume viscosityin agreement Supplementary Section 1). We use ab initio calculations to with more recent works.22,39,42,43 To solve this equation, compute the intrinsic parameters of bulk silicon at T = 300 K: appropriate boundary conditions are implemented (see κ = 145 W/mK, τ = 50 ps, and l = 176 nm.21,37 For the thermal Methods and Supplementary Section 1). In addition, we boundary resistance, which is an intrinsic property that require the thermal boundary resistance between the metal and depends only on the materials and the fabrication process, the substrate. This boundary resistance is the only parameter in we use R1 = 2.25 nKm2/W for all nanostructure geometries the model that cannot be derived from ab initio calculations (see Methods), which agrees with previous EUV scatterometry since it is highly dependent on the fabrication process rather measurements on these samples9 and is close (∼2×) to the than being an intrinsic material property. However, as all our value obtained from time-domain thermoreflectance.45 We nanograting arrays have been fabricated in identical conditions, extracted this value from the large heater data, where size we use the identical value of the thermal boundary resistance effects are negligible, and it is the only fitted parameter used for for the entire data set. Given the ab initio values for the other this data set. The excellent agreement in Figure 2 between parameters, the model can be solved by using finite elements to experiment and theory demonstrates a significant advance in determine the evolution of the displacement, the temperature, modeling; the nanoline thermal decay has already been shown and the heat flux in the nanostructure and substrate.21 to be highly nondiffusive8,9 and the models employing a The predictability of eq 1 has been recently validated in suppression function are not easily calculable for a nanodot compact and holey films, and thermoreflectance experiments in geometry.5 KCMwhich is based on only a few key silicon, with excellent agreement.21,22,44 As discussed in ref.,21 parametersaccurately predicts the thermal transport and the applicability of the model with ab initio parameters (κ, l, elastic waves in both nanolines and nanodots without any and τ) is restricted to geometries where edge effects produced geometry-dependent fit parameters, which is beyond the by two different boundaries do not overlap, i.e. when current capabilities of microscopic descriptions. boundaries are separated by a distance larger than 2l. Here, In the close-packed situation, (P − L) < 2l, nonlocal effects the distance between heaters is P − L (see Figure 2). Thus, eq are expected to yield interaction between heaters, as phonons 13021 https://doi.org/10.1021/acsnano.1c01946 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article Figure 2. Direct comparison between EUV scatterometry data and KCM modeling in 1 and 2D. Experimental and theoretical normalized change in diffraction efficiency as a function of delay time for different sizes L and periods P for (a) effectively isolated, i.e., where (P − L) > 2l, nanolines (1D) and (b) nanodots (2D). Black lines denote experimental data where the error is represented by the gray bar. Blue lines indicate the inertial KCM predictions, and red lines denote the KCM quasi-static predictions which describe only the thermal transport without the contribution of oscillating elastic waves. Theoretical predictions are computed using the same geometry-independent parameters for all nanostructure sizes and shapes. The theoretical curves are identically normalized in each case so that the initial energy released to the heaters matches experiment (see Methods). Inertial simulations for nanodots are shown just in a short time window because of their high computational cost. We note that the first mechanical oscillation prevents observation of the initial temperature decay described by quasistatic curves (see Supplementary Section 1). Also shown are the experimental and theoretical changes in diffraction efficiency for close-packed, i.e., (P − L) < 2l, (c) nanolines (1D) and (d) nanodots (2D) of different sizes L and periods P. Theoretical results are solutions of eq 1 with leff = (P − L)/2, while the other parameters are the same size-independent values. The only fitting parameter for the entire data set is the intrinsic thermal boundary resistance, which is set to 2.25 nKm2/W for this work. The excellent agreement between KCM and the experimental data for the highly nondiffusive decay for both 1D- and 2D-confined heat source geometries demonstrates the predictive capability of this model. 13022 https://doi.org/10.1021/acsnano.1c01946 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article Figure 3. Hydrodynamic regions in effectively isolated and close-packed situations. Effective thermal conductivity profile on silicon, |q|/|∇T|, predicted by KCM for a nanoheater of width 30 nm at t = 0.5 ns for (left) isolated (P = 600 nm) and (right) close-packed (P = 120 nm) configurations. Similar to fluids, a friction-like reduction of thermal transport appears in the regions of the substrate where heat flux gradients are large. Parameter l defines the characteristic size of the region below heaters where these hydrodynamic effects are important (hydrodynamic region). When sources are separated a distance larger than 2l (effectively isolated lines), one uses the intrinsic value l = 176 nm. When this distance is smaller, i.e., (P − L) < 2l, an effective value leff = (P − L)/2 (100 ps after the pump pulse.5,6 the physical properties of the system. Indeed, if the data from the first hundreds of picoseconds is Equation 2 can be interpreted intuitively as a two-box model excluded from our analysis, the presence of two distinct time as seen in Figure 5. One box represents the heater, while the scales cannot be observed (see Supplementary Section 2). In other box is a region of order L in the substrate below the contrast, our EUV probe is only sensitive to surface heater (referred to as the dam region). The thermal response deformations (see Methods) and thus precisely captures two of the system begins when the heater is filled with thermal distinct time scalesa signature of non-Fourier heat transport. energy from the laser pulse. At short times after the laser pulse, 13024 https://doi.org/10.1021/acsnano.1c01946 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article Figure 5. Two-box model for the thermal decay of heaters for L < l. (a) The energy released by the heater (with heat capacity per unit surface, C1) crosses the interface with the substrate at a rate determined by the thermal boundary resistance, R1. The thermal response of the substrate is determined by a region of size L below the heaterthe dam regionwhich acts both as a heat reservoir of capacity C2, and as a thermal resistance R2 due to viscosity from hydrodynamic effects. An analogy to an equivalent electrical circuit is shown. (b) The temperature as a function of time is shown for the positions indicated in (a) from KCM solutions for L = 30 nm and P = 600 nm. At short times, the dam region retains the energy released by the heater and increases in temperature with a time scale τ1 dominated by the interface resistance. At larger times, a slow joint decay of heater and dam temperatures occurs with a characteristic time τ2 determined by the hydrodynamic resistance R2. (c) Cartoon of the two-box model in analogy with fluids. The two boxes represent the heater and dam with the water level indicating the temperature. For times less than τ1, the excess energy flows out of the heater into the dam through the interface resistance until temperatures equilibrate. For times on scale of τ2, excess energy in heater and dam escapes to the rest of the substrate at a rate ruled by hydrodynamic effects. l 2cs ijj c h yz jj1 + α + B h zzz κ jk cs L z{ the heater releases the energy into the dam region, which retains the energy and rapidly increases in temperature. The τ2 = (4) initial rate of this energy transfer is dominated by the intrinsic thermal boundary resistance between the heater and substrate. Therefore, for small isolated sources, KCM can provide At larger times, when the dam region has equilibrated with the simple analytical expressions for the two different time scales of heater, the dissipation of the thermal energy is dominated by the heat transfer, each one associated with a different resistive the rate of energy transfer out of the dam region into the rest mechanism. This allows accurate experimental validation of the of the substrate. Therefore, the substrate plays two roles in the nonlocal length value for silicon at room temperature (a thermal response of the system: it acts both as an energy sensitivity analysis of various KCM parameters is provided in reservoir with heat capacity C2 and as a thermal resistance R2. Supplementary Section 2). Additionally, the two-box model eq The rate of energy transfer in these later times is controlled by 2 can also be applied to close-packed experiments by the viscous resistance, i.e., hydrodynamic effects. The thermal substituting l by leff; however, the simple expression of eq 4 relaxation of the heaters can be described by an equivalent cannot be used in this case (see Supplementary Section 2). circuit (Figure 5a) and illustrated by a fluid analog (Figure 5c). Although the two-box model has been derived at small sizes, The predicted temperature evolution of the system as a it also characterizes the non-Fourier behavior for all function of time and position are shown in Figure 5b and experimental sizes. To validate the intuition provided by the two-box model, we fit a double-exponential decay (eq 3) to Supplementary Section 2. each of our experimental measurements, as shown in Figure 6a. For small isolated sources, we find simple expressions for the We compare the fits of experiments to fits of numerical KCM characteristic times, namely τ1 = R1 Ceq = R1 C1C2/(C1 + C2), simulations and the analytical two-box model in Figure 6b−d. and τ2 = (C1 + C2)R2. For nanolines of L = 50 nm, these We find that the experimental fit results agree well with both expressions yield τ1 = 50 ps and τ2 = 1050 ps; thus, τ2 is an KCM numerical and analytical calculations. Additionally, we order of magnitude larger than τ1. In this limit, τ1 depends on confirm the existence of a short time scale (τ1 ∼ 100 ps) which the thermal boundary resistance, while the viscous time scale τ2 is dominated by the intrinsic thermal boundary resistance in does not depend on the thermal boundary resistance but Figure 6b and a longer time scale (τ2 ∼ 1 ns) which is mainly on the nonlocal length l and geometry: dominated by the hydrodynamic effects in Figure 6c. Figure 6c 13025 https://doi.org/10.1021/acsnano.1c01946 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article Figure 6. Two characteristic decay times in thermal relaxation of nanoline (1D) experiments. (a) The experimental change in diffraction efficiency, with oscillations removed, for a heater line of L = 250 nm and P = 1000 nm (thin line) can be fitted with a double exponential decay (thick line), from which two characteristic times are extracted: a short time scale (red line region, τ1) and a long time scale (blue line region, τ2). (b,c) Characteristic time τ1 and τ2 versus heater line widths L for effectively isolated (solid triangles) and close-packed (solid circles) experiments. KCM numerical (analytical) results are denoted by open symbols (lines). The color intensity in the symbols indicates the weight of each characteristic time in the overall decay. The short time scale (τ1 ∼ 0.1 ns) is dominated by the interface resistance, while the long one (τ2 ∼ 1 ns) is ruled by the hydrodynamic effects in the substrate. Additionally, the difference between the dissipation of close- packed versus effectively isolated heat sources is demonstrated. (d) The normalized weight of the hydrodynamic characteristic time in the temperature decay, a2 (= 1 − a1), is displayed versus line width for all experiments, showing the transition from interface- to hydrodynamic- dominated decay as source size decreases. These experimental fits include raw data from refs.8,9 and the current study. also displays the splitting of the decay times between effectively uncover two fundamental mechanisms in the thermal isolated and close-packed experiments, i.e., the increase in relaxation of nanoheaters on semiconductor and dielectric dissipation efficiency for close-packed heat sources. In Figure substrates: a first decay ruled by the interface followed by a 6d, we plot the weight of the hydrodynamic dominated decay, second decay controlled by hydrodynamic heat transport in the a2 in eq 3, which shows a transition from a primarily substrate. For large nanoheater sizes, the former mechanism hydrodynamic decay for small heaters, to a decay ruled by the dominates, while for small nanoheater sizes, the hydrodynamic thermal boundary resistance at large sizes. This is expected as effects dominate. We have found compact analytical large sizes should converge to the Fourier prediction, which expressions for the time scales of these mechanisms and for contains a single time scale. Therefore, the size-dependent the general thermal decay of heaters by developing a simple effective boundary resistance extracted by the effective Fourier two-box model. In contrast to the single exponential decay model in refs 8 and 9 can be reinterpreted as capturing the predicted by Fourier for the semi-infinite substrate conditions weighted average of the time-scales (τ1, τ2) generated by a size- of our experiments, the hydrodynamic model yields a two- independent boundary resistance and size-dependent localized exponential decay, which has been confirmed by extensive hydrodynamic effects. experiments in 1D- and 2D-confined source geometries. This two-exponential kinetics thus confirms the non-Fourier CONCLUSIONS behavior of these experiments. In conclusion, we have shown that by adding a hydrodynamic In contrast to previous models, the present hydrodynamic heat transport term, we can explain the thermal transport model contains no geometry-dependent fitting parameters, and behavior of nanoscale metal−semiconductor samples with 1D- it is thus predictive. The excellent agreement between KCM and 2D-confined heat source geometries over a large range of and the experimental data for the highly nondiffusive decay for sizes, allowing a deeper insight of the physical behavior beyond both 1D- and 2D-confined heat source geometries demon- effective Fourier’s law. The hydrodynamic model allows us to strates the versatility and generality of this model to capture 13026 https://doi.org/10.1021/acsnano.1c01946 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article behavior in complex, device-relevant geometries. In addition, the thermal boundary resistance assuming diffusive phonon reflections our formalism enables predictive strategies to reduce the and perfect contact area (see Supplementary Section 1). However, the cooling time of nanoscale heaters. The mesoscopic character of nanogratings fabrication process produces interface defects that the model and the use of intrinsic, geometry-independent increase the actual boundary resistance value. Therefore, a single correcting factor for the boundary parameters is required to predict parameters allows it to be easily extended to the complex the thermal decay of all the gratings (1D and 2D). The obtained architectures required by nanoscale technologies, where the correcting factor is fitted from the thermal decay of the largest lack of control on heat dissipation represents an important experimentally available 1D grating (L = 1 μm) and hence does not limitation for future developments. depend on the model used. For large gratings, the hydrodynamic corrections do not play any role and we obtain the same boundary METHODS resistance correction using KCM or effective Fourier model. EUV Dynamic Scatterometry Measurements. The sample Specifically, we obtained a thermal boundary resistance value 3.1 consists of metallic Ni nanostructure arrays fabricated on the surface times larger than the lower bound. This factor is similar to the one of a silicon substrate using an e-beam lithography technique. The obtained in previous work for a similarly fabricated metal− nanostructure arrays are 150 × 150 μm2 areas consisting of both semiconductor interface.22 periodic nanolines and nanodots with line widths ranging from 1 μm The thermal equations are coupled with the classical elastic down to 20 nm, periods ranging from 4 μm down to 80 nm, and equations to predict the surface deformation of the system in order to average heights of 11.5 nm. The line width and period of the compute the resulting change in diffraction efficiency using numerical nanoline/nanodot arrays is independently controlled in order to Fresnel propagation. Specifically, the stress tensor of the nickel and separate the effects of size and spacing. The dimensions of the various the silicon includes a linear thermal expansion term. Moreover, the arrays are characterized using atomic force microscopy (see thermo-elastic energy exchange term is included in the energy Supplementary Section 4). To launch dynamics in the sample, an conservation equation. For heaters, we use nominal bulk nickel elastic ultrafast infrared (780 nm wavelength, ∼25 fs pulse duration) pump properties. For the substrate, we use an anisotropic stress tensor beam is incident on the sample with ∼ 20 mJ/cm2 fluence and ∼ 275 accounting for the structural defects generated during the fabrication μm spot size. The pump light is preferentially absorbed by the of the nanogratings on the substrate top surface.50 metallic nanostructures which causes rapid heating followed by All the parameter values used and a detailed explanation of the impulsive thermal expansion in the nanostructures. The coherent thermoelastic equations and the boundary conditions can be found in excitation of the periodic arrays launches acoustics waves that Supplementary Section 1. propagate along the surface of the silicon substrate. As the heated Comparison between Experiment and Model Predictions. nanostructures cool down by thermal dissipation into the substrate, Since KCM consists of a linear set of partial differential equations, the they relax back to their original profile. An ultrafast, short wavelength surface deformation and the predicted diffraction efficiency linearly probe beam is generated by focusing an ultrafast infrared pulse into an depends on the amount of energy deposited in the heater by the laser Ar filled glass capillary. A quantum nonlinear process called high pulse. In the simulations, a uniform energy density of 1W/m3 with a harmonic generation converts a portion of the infrared light into a duration of <2.5 ps is introduced in the heater. To compare the model coherent short wavelength (∼30 nm) ultrashort pulse duration (∼10 predictions and the experiments, the diffraction efficiency obtained in fs) extreme ultraviolet (EUV) beam.46 The short wavelength of the KCM inertial simulation is scaled by a factor to match the first probe allows for exquisite picometer sensitivity to the surface experimental peak. This is equivalent to scaling the simulated energy displacement and allows for measurements of 10s nm nanostruc- density by this factor and this same scaling is used to normalize the tures.47 Moreover, these wavelengths interact with core electrons far quasi-static simulations. This procedure is also applied to the effective from the Fermi surface, which are not affected by small temperatures Fourier simulations in order to compare Fourier and experiments in changes as the photon energies are far from resonances in nickel.47−49 Figure 4. Note that a slight correction factor has been added to scaling The probe beam is scattered from the nanostructure arrays at a set of the quasi-static simulation of L = 30 nm with P = 120 nm in Figure time delay, controlled by a mechanical delay stage, relative to the 2d due to a small numerical error in the first few picoseconds of this pump beam and captured on an EUV sensitive CCD camera. Images inertial simulation. of the EUV scattering pattern with and without the pump beam are The quasi-static solutions are obtained by removing inertial elastic subtracted, allowing us to observe the change in the diffraction effects, i.e., dynamic equilibrium is imposed during all the simulation pattern. By subtracting the change in intensity of the reflected EUV (see Supplementary Section 1). These solutions capture the light from the change in intensity of the diffracted EUV light, we can deformations just due to thermal expansion and hence can be used compute the change in diffraction efficiency. This change in to track the temperature evolution of the system (see ref 9). Note that diffraction efficiency is monitored as a function of time delay between the initial peak obtained in the quasi-static simulations is not observed the pump and probe beams and can be directly related to the surface in experiments because the system needs a finite time to expand. deformation of the sample. Double Exponential Fitting to Experimental and Numerical Thermoelastic Modeling. The microscopic expressions required Data. For Figure 6, we performed double-exponential fits to both the for the ab initio calculation of the KCM parameters can be found in experimental data and the numerical simulations. The quasi-static refs 37 and 38. The same parameter values for silicon at room numerical solutions can be easily fit to a double-exponential using temperature have been used to model other experiments.21,22,38,44 nonlinear least-squares; however, due to the noise and inertial elastic The temperature and the heat flux are obtained by solving the effects, a double-exponential function with four free parameters is too energy conservation equation along with the heat transport equation unconstrained to reliably fit to the experimental data. Therefore, we (Fourier’s law for the heaters and eq 1 for the substrate). The second- constrain the number of free parameters in a single fit while still order derivatives in the substrate transport eq (eq 1) require the independently extracting the four parameters of the double- inclusion of extra boundary conditions for the heat flux. A slip exponential. We achieve this by fitting the data in several different boundary condition relating the tangential heat flux in the substrate steps. We determine a cut time, tc, to divide the experimental trace and its derivatives is imposed in the interfaces and in the silicon free into two parts in time to separately fit the two exponentials. We define surfaces (see Supplementary Section 1). In the free surfaces, thermal the cut time as the time when the ratio between the two exponentials insulation is ensured by fixing normal component to zero. In the is 1% and compute tc using the fit values from the numerical solution. interfaces, we impose continuity of the heat flux normal component in We fit the experimental data for times t > tc to extract the longer decay the metal and in the substrate. Finally, we use a generalized boundary time exponential; however, the functional form of the decay for large t condition for the temperature jump in the interface including a is not purely single exponential. Because diffusive transport occurs far Kapitza thermal boundary resistance term along with nonlocal from the heat source at large t, the decay has a power law component terms.22 Using ab initio calculations, we compute a lower bound for superimposed on the exponential. To mitigate the effects of this 13027 https://doi.org/10.1021/acsnano.1c01946 ACS Nano 2021, 15, 13019−13030 ACS Nano www.acsnano.org Article power law on the extraction of the decay constant, we fit an effective Weilun Chao − Center for X-Ray Optics, Lawrence Berkeley Fourier modelwith only two free parameters of effective thermal National Laboratory, Berkeley, California 94720, United boundary resistance (Reff) and the overall normalization (Afit2 )for t States; orcid.org/0000-0002-9752-370X > tc and truncate the fit at roughly 2 times the expected decay Jorge N. Hernandez-Charpak − Department of Physics, JILA, constant. We can then convert Reff to a decay time, τfit2 , since τ2 = fit Reffchh. and STROBE NSF Science & Technology Center, University To extract the other exponential, we correctly set the overall of Colorado and NIST, Boulder, Colorado 80309, United normalization by accounting for the inertial elastic effects in the States experimental data. To do this, we fit the KCM quasi-static simulation Henry C. Kapteyn − Department of Physics, JILA, and to the experimental data with the acoustics waves subtracted for times STROBE NSF Science & Technology Center, University of t < tc, constraining the maxima of the inertial KCM simulation and Colorado and NIST, Boulder, Colorado 80309, United experimental data to be within the experimental noise. The resulting States maximum of the KCM quasi-static simulation, A, allows us to Begoña Abad − Department of Physics, JILA, and STROBE 1 = A − A2 . We can extract the final parameter by fitting a compute Afit fit fit fit NSF Science & Technology Center, University of Colorado −t/τ1 −t/τ2 double-exponential, Afit1e + Afit 2e , for t < tc with only one free and NIST, Boulder, Colorado 80309, United States parameter, τfit 1 . For Figure6b-d, we plot τ1 = τfit 1 , τ2 = τ2 , and a2 = A2 / fit fit Margaret M. Murnane − Department of Physics, JILA, and A, respectively, for both the experimental data and the numerical STROBE NSF Science & Technology Center, University of solutions. The error bars on the experimental data are the standard Colorado and NIST, Boulder, Colorado 80309, United deviation from multiple measurements (if there are no error bars, then only one measurement was included). As the amplitudes of the States exponentials (a1, a2) decrease, the extracted corresponding decay F. Xavier Alvarez − Physics Department, Universitat times (τ1, τ2) becomes more inaccurate. This is partially responsible Autònoma de Barcelona, Bellaterra, Catalonia 08193, for the difference between numerical solutions and experimental data Spain; orcid.org/0000-0001-6746-2144 for τ2 at large L and τ1 at small L. Additionally, inertial effects and Complete contact information is available at: noise affect the extracted values of τ1 as L decreases (see Supplementary Section 5). https://pubs.acs.org/10.1021/acsnano.1c01946 ASSOCIATED CONTENT Author Contributions ⊥ * sı Supporting Information A.B. and J.L.K. contributed equally to this work. M.M.M. and The Supporting Information is available free of charge at H.C.K. conceived the experiment. J.L.K., T.D.F. and B.A. https://pubs.acs.org/doi/10.1021/acsnano.1c01946. performed the measurements and analyzed the experiment data. W.C. fabricated the samples. A.B., L.S., J.B., F.X.A., and Full details of the thermoelastic model (system of partial J.C. performed the theoretical modeling and analysis. A.B. differential equations, boundary conditions, and param- carried out the Finite Element modeling and L.S. the ab initio eter values), analysis of the heater thermal decay and calculations. A.B., J.L.K., M.M.M., F.X.A., and J.C. wrote the derivation of the two-box model, details about Matrix article. All authors contributed to the study, discussed the Pencil Method and Atomic Force Microscopy, and results, and commented on the manuscript. M.M.M., H.K., discussion on the experimental extraction of the heater B.A., J.N.H.C., F.X.A., and J.C. supervised the work. thermal decay times (PDF) A preprint version of the present manuscript is Notes available.51 The authors declare no competing financial interest. AUTHOR INFORMATION ACKNOWLEDGMENTS Corresponding Authors A.B., L.S., J.B., F.X.A., and J.C. acknowledge financial support Albert Beardo − Physics Department, Universitat Autònoma by the Spanish Ministerio de Ciencia, Innovación y de Barcelona, Bellaterra, Catalonia 08193, Spain; Universidades under Grant No. RTI2018-097876−B-C22 orcid.org/0000-0003-1889-1588; Email: albert.beardo@ (MCIU/AEI/FEDER, UE). J.L.K., T.D.F., J.N.H.C., H.C.K., uab.cat B.A., and M.M.M. gratefully acknowledge support from the Joshua L. Knobloch − Department of Physics, JILA, and STROBE National Science Foundation Science & Technology STROBE NSF Science & Technology Center, University of Center, Grant No. DMR-1548924, and a Gordon and Betty Colorado and NIST, Boulder, Colorado 80309, United Moore Foundation EPiQS Award GBMF4538. W.C. gratefully States; orcid.org/0000-0002-4086-3746; acknowledges support by the University of Colorado, Boulder Email: joshua.knobloch@colorado.edu through the U.S. Department of Energy under Contract No. Juan Camacho − Physics Department, Universitat Autònoma DE-AC02-05CH11231. J.L.K. acknowledges support from an de Barcelona, Bellaterra, Catalonia 08193, Spain; SRC Fellowship. Email: juan.camacho@uab.cat REFERENCES Authors Lluc Sendra − Physics Department, Universitat Autònoma de (1) Minnich, A. J.; Johnson, J. A.; Schmidt, A. 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