This is the accepted version of the article: Ryan A. Duncan, Giuseppe Romano, Marianna Sledzinska, Alexei A. Maznev, Jean-Philippe M. Péraud, Olle Hellman, Clivia M. Sotomayor Torres, and Keith A. Nelson. Thermal transport in nanoporous holey silicon membranes investigated with optically induced transient thermal gratings. Journal of Applied Physics, (2020). 128. 235106: - . 10.1063/1.5141804. Available at: https://dx.doi.org/10.1063/1.5141804 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. Thermal transport in nanoporous holey silicon membranes investigated with optically-induced transient thermal gratings Ryan A. Duncan,1, a) Giuseppe Romano,2 Marianna Sledzinska,3 Alexei A. Maznev,1 Jean-Philippe M. Peraud,4 Olle Hellman,5 Clivia M. Sotomayor Torres,3, 6 and Keith A. Nelson1 1) Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139, United States 2) Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139, United States 3) Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193, Barcelona, PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 Spain 4) Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, 94720, United States 5) Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, CA, 91125, United States 6) ICREA, PG. Lluı́s Companys 23, 08010, Barcelona, Spain (Dated: 26 November 2020) 1 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. In this study, we use transient thermal gratings—a non-contact, laser-based ther- mal metrology technique with intrinsically high accuracy—to investigate room- temperature phonon-mediated thermal transport in two nanoporous holey silicon membranes with limiting dimensions of 120 nm and 250 nm respectively. We com- pare the experimental results with ab initio calculations of phonon-mediated thermal transport according to the phonon Boltzmann transport equation (BTE) using two different computational techniques. We find that the calculations conducted within the Casimir framework, i.e. based on the BTE with the bulk phonon dispersion and diffuse scattering from surfaces, are in quantitative agreement with the experimental data, and thus conclude that this framework is adequate for describing phonon- mediated thermal transport in silicon nanostructures with feature sizes on the order PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 of 100 nm. a) raduncan@mit.edu 2 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. I. INTRODUCTION Nanoscale thermal transport has become a topic of much recent interest due to the novel transport phenomena that emerge at the micro- and nanoscale1,2 and their relevance to tech- nological fields such as microelectronics and thermoelectrics3,4 . In semiconductor systems with feature sizes comparable to the phonon mean free path (MFP), size effects can lead to strong reductions in thermal conductivity—making thermal management in microelectronic devices a significant engineering challenge5 . In the field of thermoelectrics, nanostructur- ing has emerged as a key strategy for enhancing the thermoelectric figure of merit ZT by reducing the thermal conductivity without significantly affecting the electronic properties of the material4,6 . Traditionally overlooked for thermoelectric applications, silicon has gen- PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 erated recent interest as a material for thermoelectric devices due to the strongly reduced thermal conductivity achievable through nanostructuring7 . Experimental results on silicon nanowires have shown thermal conductivity values two orders of magnitude lower than the bulk value and ZT values approaching unity8–10 . Two-dimensional “holey silicon” nanos- tructures—suspended silicon membranes with a periodic array of nanopores—have exhibited thermal conductivity reductions comparable to nanowires11–16 while retaining superior rela- tive mechanical strength. Such nanostructures hold great promise for thermoelectric applica- tions due to the wide variety of well-established and scalable fabrication and manufacturing techniques available for silicon. Thermal transport at the nanoscale differs significantly from macroscopic, diffusive ther- mal transport. In structures with feature sizes comparable to the MFP of heat-carrying phonons, thermal transport no longer obeys the heat diffusion equation1 . One of the earliest attempts to account for non-Fourier phonon-mediated thermal transport in nanostructures was by Casimir17 , whose model featured particle-like phonon transport with diffuse scatter- ing at boundaries. Although Casimir’s original model was concerned with thermal transport in rods, the broader formalism of semiclassical particle-like phonon transport with diffuse boundary scattering is expected to be valid for any nanostructure for which λth ≪ ℓ and λth /2π . R , where R is the surface roughness, λth is the representative wavelength of heat-carrying phonons, and ℓ is the limiting dimension of the nanostructure. Heat-carrying phonons at room temperature in silicon have single-digit nanometer wavelengths18 , which is on the order of lithographically-realistic surface roughnesses19,20 . Thus, silicon nanostruc- 3 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. tures with feature sizes ℓ > 10 nm should be well described by the Casimir formulation of thermal transport—that is, particle-like phonon transport according to the phonon Boltz- mann transport equation (BTE) with diffuse scattering from surfaces. Studies comparing experimental results with ab initio theory based on the BTE have shown that the Casimir formulation is indeed valid for nanoscale silicon membranes21 and silicon nanobeams22 . How- ever, there have been highly conflicting reports regarding the validity of the Casimir for- mulation for thermal transport in nanoporous holey silicon membranes23 . Several studies have reported room-temperature effective thermal conductivities reduced by up to an order of magnitude relative to Casimir formulation predictions for such structures12,13,24 , while others have found good agreement between the Casimir formulation and experiment25–27 . In some cases, measurements showing deviations from the Casimir formulation predictions for PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 holey silicon nanostructures have been invoked as evidence of “coherent” thermal transport effects at room temperature11,13,28 . This notion, however, has been challenged by recent ex- perimental and theoretical works29–31 , in which no effect of nanopore lattice disorder on the room-temperature thermal transport was found. It should be noted that reports of “below Casimir” thermal conductivity rely on the measurements of the absolute values of thermal conductivity, which are challenging even for bulk samples32 . If far-reaching conclusions are to be drawn from the absolute value of thermal conductivity, then a technique with high absolute accuracy is desirable. Transient thermal gratings (TTG) is a non-contact optical technique that measures the time evolution of an impulsively generated sinusoidal temperature profile33,34 . The experi- mental observable is the amplitude of this sinusoidal temperature profile, which decays as heat spreads from the peaks to the nulls of the grating. For a one-dimensional TTG, the amplitude of the thermal profile and therefore the intensity of the heterodyned TTG signal is given by I(t) ∝ e−t/τ (1) where τ ≡ 1/αq 2, α is the thermal diffusivity, q ≡ 2π/L is the transient grating wavevec- tor, and L is the transient grating period. The only parameter other than α that affects the decay rate is L, which can be measured with high accuracy. Thus the thermal diffusivity can be determined to high accuracy from the decay rate of the TTG signal. Furthermore, TTG’s non-contact nature reduces additional sources of error due to the absence of any 4 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. 120 nm 250 nm (a) 400 nm (b) 400 nm FIG. 1. Scanning electron micrographs of the patterned holey silicon regions—(a) region A (400 nm pitch, 280 nm nanopore diameter), and (b) region B (500 nm pitch, 250 nm nanopore diameter). interfaces with metrological structures. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 In this paper, two 250 nm-thick holey silicon membrane nanostructures are investigated with the TTG technique. The experimental results from TTG measurements are compared to the results of two ab initio numerical Boltzmann transport techniques: the OpenBTE computational framework developed by Romano et al.35 and the energy-based deviational Monte Carlo BTE simulation technique developed by Peraud and Hadjiconstantinou36,37 . Quantitative agreement between numerical calculations and experiment is found for both the unpatterned silicon membrane and holey silicon structures, confirming the validity of the Casimir formulation for room temperature heat transport in silicon nanostructures with feature sizes on the order of 100 nm. II. EXPERIMENTAL A. Sample Fabrication The holey silicon structures were fabricated using electron beam lithography (EBL) and reactive ion etching (RIE) of a 250 nm-thick freestanding silicon membrane 3.2 × 3.2 µm window area, obtained from Norcada Inc.)38 . Each of the two structures was a 100 µm- diameter region of the freestanding membrane patterned with a square lattice of nanopores. SEM micrographs of the regions are shown in Fig. 1. “Region A” had a pitch size (nanopore periodicity) of 400 nm and a nanopore diameter of 280 nm, and “region B” had a pitch size of 500 nm and a nanopore diameter of 250 nm. 5 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. B. Transient thermal grating (TTG) measurements As shown in Fig. 2(a), two “pump” laser pulses are crossed at the sample, where optical interference and subsequent absorption lead to the establishment of a transient sinusoidal temperature profile with spatial period L = λ/2 sin (θ/2), where λ is the pump wavelength and θ is the crossing angle for the two pump beams. Through the temperature dependence of the material’s complex index of refraction ñ ≡ n + ik—where n is the real index of refraction and k is the absorption coefficient—this sinusoidal temperature profile is accompanied by a spatially sinusoidal modulation in ñ as well. A quasi-continuous “probe” beam then impinges on the sample, diffracting from this transient optical grating. As the amplitude of the temperature grating diminishes due to heat transport from the peaks to the troughs, the PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 amplitude of the grating in ñ—and thus the amplitude of the diffracted signal—diminishes accordingly. In this way, the time dependence of the diffracted signal can be directly related to the thermal diffusivity according to Eq. 1. TTG measures the thermal transport dynamics over a length scale set by the period of the transient grating, which can be tuned by changing the crossing angle of the pump beams. Further details regarding this technique can be found in Ref.34 . The pump beams were derived from a 515 nm source with a 60 ps pulse duration and 1 kHz repetition rate, and the probe beam was derived from a continuous-wave 532 nm source. A “reference” beam was derived from the same source as the probe beam, and the relative phase between the two was controlled by tilting a highly parallel optical flat through which the probe beam passes to achieve heterodyne detection39 . At the sample the probe beam diffracts from the transient grating and becomes superposed with the transmitted reference beam, and the combined heterodyned signal is collected by a fast photodiode detector and recorded on an oscilloscope. The 1/e2 -intensity radius of the pump and probe beams were 100 µm and 40 µm respectively. While the pump spot size is commensurate with the patterned regions, the probe spot size is much smaller. Thus, although our pump may be exciting a grating pattern that extends somewhat outside of the patterned region, our experiment is only sensitive to the transport dynamics within the region bounded by the much smaller probe spot. The pump pulse energies ranged from 170 - 340 nJ, and the instantaneous power of the probe beam at the sample ranged from 0.8 - 1.6 mW. The probe beam was shuttered by an electro-optic modulator with a duty cycle of 5% to prevent 6 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. steady-state heating of the sample. The thickness of the membrane was smaller than the optical penetration depth of silicon for the wavelengths involved in the measurements, which permitted measurements in the transmission geometry as shown in Fig. 2(a). The raw TTG data obtained from the two holey regions and the unpatterned silicon membrane at a grating period of 4.25 µm are shown in Fig. 2(b). Measurements were performed under medium vacuum at a pressure of 1 mbar. The maximum amplitude of the temperature grating was determined to have an upper bound of 35 K. Upper bounds on the average heating of the sample due to the pump and probe beams were determined to be 20 K each. The TTG signal for a one-dimensional thermal grating exhibiting diffusive thermal trans- PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 port is given by Eq. 1. τ —or equivalently α—is the only free parameter required to model the normalized TTG signal. In addition, the heterodyne detection scheme further yields a gain in signal-to-noise by a factor of the reference field amplitude, which can be increased arbitrarily up to the saturation threshold of the photodetector. The low-dimensionality of the dynamical parameter space and the fact that neither precise knowledge of the magni- tude of the temperature variation nor of the magnitude of the heat flux is required in the analysis of the data allow for the determination of the thermal diffusivity with high ab- solute accuracy. Further discussion regarding the accuracy of transmission-geometry TTG experiments on nanomembranes can be found in Ref.34 . The traces were truncated such that fitting began 5 ns after pump incidence to ensure that the fitted region corresponds only to thermal transport signal without any potential contribution from the fast electronic response shown in the inset of Fig. 2(b). The acquired fits are plotted alongside the raw TTG data in Fig. 2(b). Fig. 2(c) shows the measured thermal diffusivity values obtained according to Eq. 1 as a function of TTG period for each of the three regions measured. Each raw TTG trace consisted of 50,000 individual measurements. The statistical error of the measurement was determined by partitioning the data into subsets of 10,000 measurements, fitting each subset to Eq. 1, and taking the standard error of the mean of the resulting distribution of τ values. In addition to the statistical error of the measurement, the sys- tematic error due to laser heating effects was also considered. The effects of laser heating were determined by performing each measurement three times—once at a baseline set of pump and probe powers, and two additional times at which the pump and probe powers respectively were doubled. Linearly extrapolating the measured values of τ to zero pump 7 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. (a) Pu Si Membrane mp Pro be al ign +S R ef. Patterned region e nce fer Re mp Pu (b) PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 (c) FIG. 2. (a) Schematic of the TTG measurement in the transmission geometry. (b) Time-domain TTG traces at 4.25 µm transient grating period for the two holey silicon regions and an unpat- terned region, as well as exponential fits to the data. Inset: full transient grating response for the unpatterned membrane, including fast early-time electronic signal. (c) Effective thermal diffusivity values obtained from the time-domain data according to Eq. 1. For the patterned regions the error is smaller than the sizes of the symbols. 8 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. and probe laser power allows us to determine the systematic error due to laser heating, which was then added to the appropriate side of the errorbars for each point to account for this systematic heating effect. We note that the upper bounds on laser heating reported above are non-negligible relative to room temperature. However, since the effect of laser heating is experimentally quantified in our error analysis, we can still compare our exper- imental results with calculations that use room-temperature material properties. Despite the somewhat high upper bounds on laser heating, we nevertheless note that the effect of laser heating on the experimentally-determined values of α was generally found to be less than 10%. For grating periods from 4.25-7.5 µm we find that the experimental values of thermal dif- fusivity are independent of L for both the unpatterned membrane and the holey membranes, PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 consistent with preliminary TTG results on holey silicon structures34 . The exponential form of the TTG data and the invariance of thermal diffusivity as a function of grating period indicates that the transport kinetics are “effectively diffusive” over the TTG experimental length scales, albeit with “effective” thermal diffusivity values αef f reduced relative to the bulk because of the non-Fourier size effect due to nanostructuring. It should be noted that occasionally an additional transient with a characteristic timescale much longer than the acquisition timescale (i.e., approximately a constant offset from the pre-pump baseline) was observed in some of the obtained TTG traces. However, we deter- mined that the presence of this contribution to the signal (which is roughly on the timescale that would correspond to thermal diffusion out of the pump spot) was not associated with any change in the αef f value that was calculated from the time constant of the exponentially decaying contribution to the signal observed on the 10s-100s ns timescale (which we took to be the true TTG signal) that remained after subtracting out this approximately constant offset. This issue is more thoroughly addressed in the Supplementary Material. Experimental values of the effective thermal conductivity κef f were calculated from the data in Fig. 2(c) according to κef f = (1 − φ)cSi αef f (2) where φ is the void fraction of the holey silicon membrane and cSi is the bulk volumetric specific heat of silicon. The resulting experimental values of κef f are shown in Fig. 3, where the effective thermal conductivity values are plotted against the neck width ℓn (i.e., the 9 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. B A FIG. 3. Effective thermal conductivity values experimentally measured and numerically computed using the MC-BTE and OpenBTE methods for the holey silicon regions and the unpatterned PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 membrane. Also plotted are the κef f values obtained by using the Fourier law with the bulk silicon thermal conductivity. ℓn is the neck width. The experimental error was determined to be smaller than the size of the symbols. difference between the pitch size and the nanopore diameter). III. COMPARISON TO FIRST-PRINCIPLES NUMERICAL CALCULATIONS Numerical calculations of the thermal transport through the membranes were performed according to the linearized isotropic phonon Boltzmann transport equation (BTE) under the single-mode relaxation time approximation (RTA), which is given by ∂fkp f0 − fkp + vkp · ∇fkp = (3) ∂t τkp where fkp (r, t) is the occupation function for a mode traveling with wavevector k and  polarization p, vkp is the (isotropic) group velocity, f0 ~ω, TL (r, t) is the Bose-Einstein distribution, TL (r, t) is the local temperature field defined such that energy is locally con- served, ~ω is the phonon energy, and τkp is the (isotropic) single-mode relaxation time (where k ≡ |k|). The simulation domain is one pore-centered unit cell of the nanopore lattice with the cylindrical axis of the pore chosen to be oriented along ẑ. Periodic boundary conditions are 10 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. applied along both the x- and y-axes. The phonon group velocities and relaxation times were determined respectively from the harmonic and anharmonic force constants, which were obtained from density functional theory (DFT) calculations using the temperature dependent effective potential (TDEP) method40 . Naturally occurring isotope disorder was taken into account. Details on the DFT calculations can be found in the Supplementary Material. The OpenBTE computational technique of Romano et al.35 and the energy-based devi- ational Monte Carlo BTE (MC-BTE) technique of Peraud and Hadjiconstantinou36,37 were both used for ab initio calculations of κef f for both the holey and unpatterned membranes. For the OpenBTE case, Eq. 3 is transformed into the following form35 : PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 Λŝ(Ω) · ∇T (r, Ω, Λ) + T (r, Ω, Λ) = TL (r), Z ∞ K(Λ′ ) ′ −1 Z ∞ K(Λ′′ ) (4) TL = dΛ hT (r, Ω, Λ′′)idΛ′′ 0 Λ′2 0 Λ ′′2 where ŝ(Ω) is the unit vector for the propagation direction Ω, T (r, Ω, Λ) is the “effective temperature” of phonons with MFP Λ traveling in direction Ω (i.e., the sum of their energy densities divided by cSi ), K(Λ) is the bulk MFP distribution (i.e., the derivative of the ther- R mal conductivity accumulation function with respect to Λ), and hx(Ω)i ≡ (1/4π) 4π xΩdΩ is the angular average over all propagation directions. Eq. 4 is derived by imposing steady- state conditions on Eq. 3, as well as assuming that δT (r) ≡ TL (r) − T0 (where T0 is the reference temperature, which in this study was 300 K) is small such that f0 [~ω, TL (r)] in Eq. 3 can be expanded to first order in δT (r) and any temperature dependence of material properties can be neglected. The advantage of this approach is that the only input required to solve Eq. 4 is the MFP distribution K(Λ). In OpenBTE, a difference of temperature ∆T is applied at the two opposing faces of the unit-cell along the x-axis, and the first guess for TL was given by the standard diffu- sive equation. Diffuse-scattering boundary conditions were imposed on all surfaces of the computational domain. Diffuse scattering at boundaries was modeled in such a way that phonons of a given value of Λ were diffusely emitted (i.e., such that the distribution of out- going phonons next to the surface is isotropic) from the surface with a total energy equal to the total energy of all incident particles with the same value of Λ. To overcome numerical instability due to small-MFP phonons, OpenBTE switches to a modified Fourier’s law to 11 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. compute the diffusive component to heat transport41 for such modes. Eq. 4 was solved by the finite-volume method while a Delaunay mesh was generated for space discretization42,43 . The deviational energy-based MC-BTE technique36,37 was also used to calculate κef f for the nanostructures investigated. This technique achieves low statistical variance compared to other Monte Carlo techniques by only simulating the trajectories of “deviational” parti- cles which describe the excess/deficit thermal energy in a given mode relative to equilibrium, and achieves high computational efficiency by performing the calculation in an energy-based BTE formulation that lends itself naturally to energy conservation. The diffuse boundary scattering condition was modeled in the same fashion as in the OpenBTE method described above—namely, deviational particles with a given MFP were diffusely emitted from the sur- face with a total energy equal to the total energy of all incident particles with the same PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 MFP. Unlike the OpenBTE technique as described in Ref.35 , the MC-BTE solver applies a constant temperature gradient throughout the simulation domain. To assess the impact of this discrepancy in the applied perturbation we compute κef f with OpenBTE using both approaches on test aligned structures, finding negligible differences in effective thermal con- ductivity values. For both computational techniques, the conductance of one unit cell was calculated by dividing the total heat flux through one end of the simulation domain by ∆T . The effective thermal conductivity κef f was then obtained by dividing this conductance value by the rectangular cross-sectional area of the unit cell normal to x̂ and by multiplying by the unit cell length along x̂44 . Our results for both computational techniques are shown in Fig. 3 for comparison to the experimental TTG results for both holey silicon structures and the unpatterned membrane. A discussion of the uncertainties associated with the computational methods is provided in the Supplementary Material. In a previous paper34 , preliminary TTG results investigating thermal transport in a sim- ilar holey silicon membrane were compared to the values of κef f obtained from ab initio MC-BTE simulations. Agreement between experiment and theory was found to within ∼ 20%. However the sample in that study was patterned over the entirety of the suspended membrane, and as such comparison with an unpatterned region to ensure the intrinsic qual- ity of the silicon membrane was not possible. There thus remained an ambiguity in the previous study as to whether the discrepancy between theory and experiment was due to deviations from the Casimir formulation, or due simply to material quality effects. Our 12 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. computational results for κef f are shown alongside our experimental results in Fig. 3, as well as the values of κef f for the structures calculated using the Fourier law with the bulk silicon thermal conductivity value of 143 W/m.K. We see that the size effect associated with the thickness of the unpatterned membrane alone reduces κef f by nearly a factor of two relative to the value obtained from the Fourier law (which is simply the value for bulk silicon in the case of the unpatterned membrane), in good agreement with previous measure- ments on nanoscale silicon membranes21 . A further reduction of κef f is observed due to the nanopore superlattice patterning, resulting in a reduction of κef f by a factor of 3 relative to the Fourier law prediction for region A and a near order of magnitude reduction in κef f for region A relative to bulk silicon. We observe very good agreement between the experimental and calculated results (with the two computational techniques being in excellent agreement PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 with each other), which indicates that the Casimir formulation is valid for nanostructures of this kind. IV. CONCLUSIONS We have used the non-contact optical TTG method to investigate thermal transport in ho- ley regions and an unpatterned region of the same silicon membrane. We observe effectively diffusive transport at grating periods larger than 4 µm and a reduction in effective thermal conductivity by nearly an order of magnitude relative to the bulk value. Two ab initio nu- merical techniques simulating transport according to the semiclassical phonon Boltzmann transport equation yielded excellent agreement with the measurements. Our results indicate that the Casimir framework of semiclassical particle-like phonon-mediated thermal trans- port with diffuse boundary scattering is adequate for describing thermal transport in holey silicon structures with limiting dimensions of ∼100 nm. SUPPLEMENTARY MATERIAL See supplementary material for discussion regarding the long-time contribution to the obtained TTG signals that appears in some of the measurements, details regarding the density functional theory calculations, and discussion of computational uncertainties. 13 This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. ACKNOWLEDGEMENTS We would like to thank Charles Shi and Jonas Rajagopal for assisting with the TTG measurements. RAD and KAN acknowledge support from the NSF EFRI 2-DARE grant EFMA-1542864. AAM and KAN acknowledge support by the Solid State Solar-Thermal Energy Conversion Center (S3TEC), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Science, under award DE- SC0001299. The ICN2 is funded by the CERCA program / Generalitat de Catalunya and supported by the Severo Ochoa Centres of Excellence program, funded by the Spanish Research Agency (AEI, grant No. SEV-2017-0706). MS and CMST acknowledge support from the Spanish National Project SIP (PGC2018-101743-B-100) and from AGAUR Grant PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 No. 2017SGR806. OH gratefully acknowledges financial support from the VINN Excellence Center for Functional Nanoscale Materials (FunMat-2) Grant 2016-05156 and the Knut and Alice Wallenberg Foundation through Wallenberg Scholar Grant No. 2018.0194. This work has been partly supported by La Caixa Foundation MISTI Global Seed Fund program (LCF/PR/MIT18/11830008). 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However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 (a) 120 nm 400 nm (b) 250 nm 400 nm This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 (c) (a) (b) Re Pro fer be Pu e Pu m m p p nce R ef. Si Membrane +S ign al Patterned region This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5141804 A B