This is an open access article published under a Creative Commons Non-Commercial No Derivative Works (CC-BY-NC-ND) Attribution License, which permits copying and redistribution of the article, and creation of adaptations, all for non-commercial purposes. pubs.acs.org/NanoLett Letter Ballistic Phonons in Ultrathin Nanowires Daniel Vakulov,⊥ Subash Gireesan,⊥ Milo Y. Swinkels,⊥ Ruben Chavez, Tom Vogelaar, Pol Torres, Alessio Campo, Marta De Luca, Marcel A. Verheijen, Sebastian Koelling, Luca Gagliano, Jos E. M. Haverkort, F. Xavier Alvarez, Peter A. Bobbert, Ilaria Zardo, and Erik P. A. M. Bakkers* Cite This: Nano Lett. 2020, 20, 2703−2709 Read Online ACCESS Metrics & More Article Recommendations * sı Supporting Information ABSTRACT: According to Fourier’s law, a temperature difference across a material results in a linear temperature profile and a thermal conductance that decreases inversely proportional to the system length. These are the hallmarks of diffusive heat flow. Here, we report heat flow in ultrathin (25 nm) GaP nanowires in the absence of a temperature gradient within the wire and find that the heat conductance is independent of wire length. These observations deviate from Fourier’s law and are direct proof of ballistic heat flow, persisting for wire lengths up to at least 15 μm at room temperature. When doubling the wire diameter, a remarkably sudden transition to diffusive heat flow is observed. The ballistic heat flow in the ultrathin wires can be modeled within Landauer’s formalism by ballistic phonons with an extraordinarily long mean free path. KEYWORDS: Nanowires, GaP, heat transport, ballistic phonons, Raman spectroscopy eat flow in macroscopic systems is traditionally described H by Fourier’s law for diffusive heat flow.1 The advance- ment of micro and nanotechnology has led in the past few In this letter, we demonstrate room-temperature axial ballistic heat flow in ultrathin, 25 nm diameter, GaP NWs up to wire lengths of at least 15 μm, and a sharp transition to years to observations of nondiffusive heat flow. Deviations diffusive heat flow when the diameter is doubled to 50 nm. from diffusive heat flow can occur due to the wave nature of Room-temperature ballistic transport over such large distances phonons,2 hydrodynamic effects,3−10 and ballistic behav- and a diffusive-ballistic transition with decreasing lateral ior.11−18 In nanowires (NWs), scattering of phonons at the dimension have not been reported thus far. The temperature NW surface reduces the thermal conductivity as compared to profile along the NWs is obtained by Raman thermometry, the bulk.19 However, when the surface is atomically flat, confirming the presence of a temperature gradient in the phonons may reflect specularly from the surface20,21 and diffusive flow regime and its absence in the ballistic regime. ballistic heat flow in the axial direction may occur. We grow GaP NWs with 25−140 nm diameters by metal− Realizing ballistic flow over large distances at room organic vapor-phase epitaxy using the vapor−liquid−solid temperature requires a long phonon mean free path (MFP), growth technique (see Supplementary Section SI1). The NWs which is challenging due to different scattering mechanisms. are grown in arrays by substrate/catalyst patterning. Within an Only a few studies have reported the observation of ballistic array, the NWs have a uniform diameter along their length transport at room temperature. Ballistic transport at room with a <5 nm diameter spread. We analyze the structural temperature up to a length of 260 nm was reported in short properties of the NWs by transmission electron microscopy and wide graphene ribbons.12 The heat transport is diffusive in (TEM). TEM images show stacking-fault-free NWs, grown short and narrow ribbons due to the presence of edge disorder. along the ⟨0001⟩ direction (see Figure 1a). The NWs have Room-temperature ballistic heat flow up to a length of 8.3 μm atomically flat {110̅0} side facets covered by an approximately was reported for SiGe NWs with diameters in the range 50− 2 nm-thick amorphous oxide layer (see Figure 1b). Electron 200 nm.13 Its occurrence was attributed to the filtering out of diffraction patterns taken in the TEM further show that NWs high-frequency short-MFP phonons by alloy scattering. Beyond 8.3 μm, the heat flow in these NWs was found to be Received: January 23, 2020 diffusive. Ballistic phonon transport at room temperature was Published: February 24, 2020 also reported in holey silicon up to a length of 200 nm.15 The explanation in this case was that surface disorder filters out high-frequency phonons, leaving the ballistic transport of low- frequency phonons. © 2020 American Chemical Society https://dx.doi.org/10.1021/acs.nanolett.0c00320 2703 Nano Lett. 2020, 20, 2703−2709 Nano Letters pubs.acs.org/NanoLett Letter Figure 1. GaP NW properties and device for thermal conductance measurements. (a) TEM image of a 25 nm diameter GaP NW, grown along the ⟨0001⟩ direction. (b) High-resolution TEM image taken along the ⟨112̅0⟩ direction of a NW, showing a defect-free WZ crystal structure and a layer of oxide (orange arrows). (c) APT analysis of a 25 nm diameter NW showing its high chemical purity. (d) 45°-tilted SEM image of the thermal conductance measuring device, with suspended heating (red) and sensing (blue) membranes. (e) Zoom-in of the GaP NW bridging the suspended membranes. Electron beam-induced deposited Pt/C improves the thermal contact. with different diameters used in our study all have a wurtzite a current to one of the coils and measuring the temperature on (WZ) crystal structure (see Figure S1). Atom-probe both sides of the NW. From these measurements, the NW tomography (APT) demonstrates the chemical purity of the thermal conductance is extracted through a heat balance NWs and the absence of heavy element impurities (see Figures calculation for the sensing membrane, where the heat 1c and S2). The NWs are therefore atomically clean and transported through the wire is put equal to the heat that defect-free. Our GaP NWs are thus ideal systems to look for leaves the membrane through the SiNx beams, as is commonly ballistic heat transport. Limitations of the APT technique done when employing this kind of devices (see Section SI3 and prevent detection of the oxide layer.22 We fabricate micro- ref 23.). devices23,24 to study the thermal conductivity of individual The extracted κ at room temperature T = 300 K is plotted in NWs (see Section SI3). The devices consist of two suspended Figure 2a as a function of diameter d for NWs with membranes with Pt meander structures on top, which can be approximately the same length L = 6.7 ± 0.1 μm, with d used as heater and thermometer; see the scanning electron ranging from 25 to 140 nm. These results were fitted to expose microscopy (SEM) picture in Figure 1d. Different spacings the influence of the thermal contact resistance, Rc.24 In a NW between the membranes allow measuring NWs of different with heat flow dominated by diffuse scattering at the NW lengths. Using a micromanipulator under an optical micro- surface, κ increases with d according to Mathiessen’s rule κ ∝ scope, we place as-grown single NWs with selected diameter (1 + lbulk/d)−1, where lbulk is the bulk MFP, as indicated by the between the membranes. Pt/C contacts are deposited on top dashed line. A reduced κ with respect to this line, which is of the NWs to improve the thermal contact (see Figure 1e). observed for d ≥ 75 nm, indicates that Rc is no longer We determine the thermal conductivity κ from the measured negligible compared to the NW resistance.24 For thin wires (d thermal conductance G and the NW dimensions. The thermal < 50 nm), on the contrary, an enhanced κ is observed, conductance of the NW, which is defined as the amount of indicating that a nondiffusive transport mechanism, not heat flux going through a material when one Kelvin dominated by a contact resistance, becomes progressively temperature difference is applied, is determined by applying important with decreasing diameter. To investigate the 2704 https://dx.doi.org/10.1021/acs.nanolett.0c00320 Nano Lett. 2020, 20, 2703−2709 Nano Letters pubs.acs.org/NanoLett Letter Figure 2. Transition from diffusive to ballistic heat flow. (a) Room-temperature (300 K) thermal conductivity κ of GaP NWs of length L = 6.7 ± 0.1 μm as a function of diameter d. Full curve: Fit of results for d ≥ 50 nm to κ = (1/c1d + c2πd/4L)−1 (diffusive heat flow with MFP l = d ≪lbulk and contact resistance Rc = c2/d, with c1 = 1.7 × 108 W/m2·K and c2 = 4.6 m·K/W). Dashed line: κ = c1d (diffusive heat flow without contact resistance and d ≪ lbulk). (b) Symbols: Measured length dependence of resistance R of NWs with d = 25 and 50 nm. Lines: Modeling results based on Landauer’s formalism, using the numbers of specular phonon modes Ns (d = 25 nm, solid line: MFP l = ∞, dashed line: l = 100 μm) and diffusive phonon modes Nd (d = 50 nm, MFP l = d) from Figure S12. For d = 25 nm, the curve for l = 100 μm is obtained by dividing the result for l = ∞ by the transmission coefficient l/(l + L).30 Inset: Conductance G of a device without a NW (see Section SI4) and with 1, 2, and 3 NWs with d = 25 nm and L = 6.7 ± 0.1 μm. Dashed line: Linear fit. (c) Same results as in main panel of (b), but for the conductivity κ. Inset: Results for d = 40 nm NWs measured at 300 and 50 K (measurements on different devices at the different temperatures). Dashed line: Guide to the eye. (d) Measured (symbols) and modeled (curves) T-dependent thermal conductance G of d = 25 and 50 nm NWs with length L = 12.7 ± 0.2 μm. transport mechanism in the thin wires, the thermal resistance R transition from diffusive to length-independent ballistic heat = 1/G has been plotted as a function of NW length L for flow in 25 nm diameter NWs over unprecedented distances. diameters d = 25 and 50 nm in Figure 2b (main panel). For the To further investigate the features of the diffusive-to-ballistic 50 nm NWs, R is linearly dependent on L, in accordance with transition, we plot in Figure 2c (main panel) κ vs L for d = 25 diffusive heat flow. For the 25 nm NWs, however, R is and 50 nm NWs at 300 K, clearly showing the remarkable fact independent of L for all measured NWs up to L = 15 μm, that for long wires, the thermal conductivity of the ultrathin 25 which is proof of ballistic transport. We attribute this resistance nm NWs exceeds that of the thicker 50 nm NWs. The inset to a small number of ballistic modes contributing to transport displays results for 40 nm NWs at 300 and 50 K, showing that (see below). We note that the extrapolation of the 50 nm data at both room and low temperature, this is a transition case: goes through the origin, corroborating a negligible Rc, in The heat flow is ballistic up to L ≈ 11 μm and then becomes agreement with a model where Rc is taken inversely diffusive. Figure 2d shows a very weak T dependence of the proportional to the contact area between contact and NW thermal conductance G for both 25 and 50 nm NWs of 12.7 ± (solid curve in Figure 2a). Although we cannot estimate Rc for 0.2 μm length down to about 100 K (below this temperature, the 25 nm NWs directly from the measurements, a negligible G steeply decreases due to phonon freeze-out). Results for 75 contact resistance follows from this model. The measured nm NWs are comparable to those for 50 nm NWs, showing linear dependence of the device conduction with the number also a very weakly T-dependent diffusive heat flow (see Figure of NWs for d = 25 nm and L = 6.7 ± 0.1 μm (inset in Figure S6). The T insensitivity rules out explanations for the diffusive- 2b) excludes the influence of a possible background to-ballistic transition based on phonon−phonon interactions, conduction. We note that these data have been collected which are strongly T dependent. In particular, Umklapp from a device with a gap between the SiNx membranes of 4 scattering, which leads to a strong decrease of κ in bulk GaP μm, where we put down the NWs under an angle to measure above 50 K,25 appears to play no significant role. Also, the NWs with L = 6.7 ± 0.1 μm. We have also characterized involvement of phonon-polaritons as heat conducting species devices with a smaller gap size of 2 μm. Also for these devices, is unlikely, because they would yield a contribution to G that is the influence of a possible background conduction can be linear in T.26 The thermal conductivities shown in Figure 2 are excluded. The data therefore unequivocally point at a significantly smaller than the room-temperature thermal 2705 https://dx.doi.org/10.1021/acs.nanolett.0c00320 Nano Lett. 2020, 20, 2703−2709 Nano Letters pubs.acs.org/NanoLett Letter conductivity of bulk GaP. For bulk GaP, experimental thermal modes, since the frequency shift of the Raman peak is the conductivities are only available for the equilibrium zinc blende result of higher-order phonon effects and thus involves all structure, with a room-temperature value of 75.2 W/m·K.27 phonon modes. In other words, our Raman experiments The calculated bulk thermal conductivity of WZ GaP is 92 ± 5 measure an average lattice temperature. The temperature W/m·K.28 profile along the 50 nm NW (Figure 3c) is linear, in agreement To substantiate the ballistic transport in the thin NWs, we with diffusive transport. Importantly, the Raman thermometry probe the thermal profile of 25 and 50 nm NWs with Raman measurements on the 50 nm NW further confirm the absence thermometry, which is a contactless method, in the same of a significant contact resistance, since the measured devices as used for the thermal conductivity measurements temperature gradient agrees with the applied temperature (see Figure 3a). Here, the local average lattice temperature is difference, which was calibrated in an independent manner. In the 25 nm NW (Figure 3e), however, we observe symmetric temperature jumps at the contacts and a constant temperature along the whole wire with δT close to 15 K, exactly as predicted by ballistic heat transport theory.30 We note that the TO mode itself contributes negligibly to the heat flow because of its very short MFP, but probes the average temperature of the heat-carrying phonons. In ballistic heat flow, heat-carrying phonons traveling in either direction through the NW have the temperature of the contact from which they were injected. The lattice thermalizes equally with the “hot” and “cold” phonons, and therefore, the TO mode probes the average of the contact temperatures, that is, δT = 15 K. These results demonstrate the analogy with electrical transport through a ballistic quantum wire, where the electrical potential along the wire is constant and equal to the average of the potentials applied at the contacts.31 To the best of our knowledge, this is the first measurement of the theoretically predicted temperature profile for ballistic heat transport. Together with the heat transport measurements of Figure 2, this provides strong experimental proof for ballistic heat flow. Interestingly, these measurements hint at the existence of a residual scattering between heat- carrying and non heat-carrying phonons. We are not aware of an existing model in the literature that Figure 3. Temperature profiles in the diffusive and ballistic regimes. can capture a diffusive-to-ballistic transition for decreasing (a) Illustration of the microdevice with NW used for the dimensions with the length scales involved in this work. To measurements of the thermal profile. A laser spot is moved along give a tentative explanation, we apply a model based on the wire, and Raman spectra are acquired for T1 = T2 = 300 K and T1 Landauer’s formalism for phonon transport. It neglects = 300 K, T2 = 300 K + ΔT. (b) Spectra measured at the position anharmonic effects that lead to phonon−phonon scattering, indicated by the arrow in (c) for a NW with d = 50 nm and L = 14.7 in accordance with the weak T dependence of our ± 0.3 μm. Solid lines: Lorentzian fits to the spectra. (c) Resulting experimental data (see insets of Figure 2c,d). The model we temperature profile along the NW while applying ΔT = 30 K using apply has been developed by Murphy and Moore20 and Chen the Pt meander heaters. Solid line: Modeled profile for purely diffusive heat flow. (d) and (e) The same as (b) and (c) but for a NW et al.32 (see Section SI7) and was successfully used to describe with d = 25 nm and L = 14.0 ± 0.3 μm. Solid line in (e): Modeled heat flow in silicon NWs32 and holey silicon.15 It distinguishes profile for purely ballistic transport with MFP l = ∞. Dashed line: between two different types of phonon scattering at the NW Profile for l = 100 μm with a transmission coefficient l/(l + L), having surface: The surface appears sharp (diffuse) to phonons with a slope [1 − l/(l + L)]ΔT/L.30 The bars at the left and right y-axis in an inverse perpendicular wave vector component longer (c) and (e) show the temperature of the contacts. (shorter) than the surface oxide layer thickness, leading to mostly specular (diffuse) scattering with long (short) measured using the Raman shift of the transverse optical (TO) MFP.20,32 This description is also consistent with Lévy walk mode29 (see Section SI6 for experimental details). We keep the dynamics.18 We obtain the numbers of modes Ns and Nd of the system temperature at T = 300 K and heat the hot contact by “specular” and “diffusive” phonons at each frequency from ab ΔT = 30 K. Swapping the contacts yields the same results (see initio calculations of the phonon band structure of WZ GaP Figure S7). The frequency shift, δω, of the TO mode (at about and phonon subband formation in the NW (see Section SI8). 363 cm−1 in GaP) with respect to the case without applying a In Figure S12, these numbers are plotted for d = 25 and 50 nm temperature difference provides an estimate of the local lattice and an oxide layer thickness h = 2.45 nm (fitted to obtain temperature increase, δT, since δω ∝ δT (see Figure 3b,d). agreement with the d = 25 nm thermal transport data, see This differential measurement cancels out changes in the TO below). This value of h is close to the measured thickness (see mode frequency along the wire caused by other effects than the Figure 1b). We note that the number of specular phonon temperature difference, such as small laser heating effects and modes Ns is only a small fraction (∼ 1%) of the number of local strains. It is worth highlighting here that in the case of diffusive modes Nd. We only consider phonons with a Stokes scattering (as we are employing), the measured frequency below the Debye frequency. Higher-frequency spectrum reflects the average temperature of all phonon phonons are hardly excited at the considered temperatures 2706 https://dx.doi.org/10.1021/acs.nanolett.0c00320 Nano Lett. 2020, 20, 2703−2709 Nano Letters pubs.acs.org/NanoLett Letter and have too short MFPs to significantly contribute to the heat weak T dependence of the transition, is explained in Section flow. SI10. A model where both contributions of specular phonons with These qualitative explanations of the reduction of the a strongly increased MFP and of diffusive phonons with an contribution of diffusive phonons to the heat flow and the MFP l = d are considered equally does not agree with our data enhancement of the contribution of specular phonons should (see Figure S13). The contribution of diffusive phonons in the be quantified, such that they can be experimentally verified. 25 nm NWs would lead to an offset in the thermal conductivity Ballistic phonon transport may find application in phonon κ, and the contribution of specular phonons in the 50 nm NWs transistors,38,39 phonon waveguides,40 and in novel cooling would lead to a linear increase in κ, neither of which is solutions for computer chips, which require rapid removal of observed. A suppression of the contribution to the heat flow of heat from ever decreasing volumes. Ballistic heat flow could diffusive phonons in the 25 nm NWs accompanied by an possibly be realized more easily in other compound semi- enhancement of the contribution of specular phonons is in our conductors composed of lighter elements than Ga and P, such as Al and N, because of even longer phonon MFPs. ■ model a necessary ingredient to explain the experimental data. As shown by the solid lines in Figure 2b−d, we obtain agreement with experiment both for 25 and 50 nm NWs if we: ASSOCIATED CONTENT (1) neglect for the 50 nm NWs the heat flow carried by * Supporting Information sı specular phonons in Landauer’s expression for G (eq S4 in The Supporting Information is available free of charge at Section SI7) and (2) disregard for the 25 nm NWs the https://pubs.acs.org/doi/10.1021/acs.nanolett.0c00320. contribution of diffusive phonons, taking only into account the Nanowire growth, elemental analysis, thermal conduc- heat flow carried by specular phonons with strongly increased tance measurements, background conductance, diffusive MFP. A finite but strongly increased MFP l up to 100 μm heat flow in 75 nm diameter nanowires, Raman would still agree with the heat transport data (dashed lines in thermometry, application of Landauer’s formalism for Figure 2b,c) and would result in a small temperature gradient heat flow, calculation of the numbers of specular and in the 25 nm NW that is within the experimental uncertainty of diffusive phonon modes, contributions of specular and the Raman measurements (dashed line in Figure 3e). Phonons diffusive phonons to thermal conductivity, decoupling of in the 25 nm NWs therefore travel practically unimpeded and specular and localized diffusive phonons (PDF) ■ can truly be called ballistic. The weak T dependence of the heat transport shown in Figure 2d is explained by the fact that the main features in both Ns and Nd appear at phonon energies AUTHOR INFORMATION significantly below the thermal energy (see Figure S12). Corresponding Author Although the original model of Murphy and Moore20 and Erik P. A. M. Bakkers − Department of Applied Physics, Chen et al.32 makes a distinction between specular and Eindhoven University of Technology, 5600 MB Eindhoven, The diffusive phonons, it contains no mechanism for the Netherlands; orcid.org/0000-0002-8264-6862; suppression of the heat flow carried by diffusive phonons Email: e.p.a.m.bakkers@tue.nl and the emergence of ballistic character of the specular Authors phonons. In their model, specular phonons still undergo a Daniel Vakulov − Department of Applied Physics, Eindhoven weak scattering into diffusive phonon modes caused by the University of Technology, 5600 MB Eindhoven, The disordered NW surface, preventing ballisticity. Netherlands A possible cause for the suppression of the heat flow carried Subash Gireesan − Department of Applied Physics, Eindhoven by diffusive phonons is Anderson localization33 of these University of Technology, 5600 MB Eindhoven, The phonons in the 25 nm NWs. The phonon mean free paths of Netherlands; Center for Computational Energy Research, 5600 the diffusive phonons in the bulk WZ GaP crystal are in the HH Eindhoven, The Netherlands range of hundreds of nanometers to several micrometers.28 Milo Y. Swinkels − Department of Physics, University of Basel, The dominant scattering of these phonons is therefore by the 4056 Basel, Switzerland amorphous surface of the NW. Recent works show that in Ruben Chavez − Department of Applied Physics, Eindhoven NWs with rough or amorphous surfaces, Anderson localization University of Technology, 5600 MB Eindhoven, The could take place.34,35 The effect was shown to be increasingly Netherlands important for thinner wires.20 Another cause could be the Tom Vogelaar − Department of Applied Physics, Eindhoven coupling of the localized modes in the amorphous oxide layer University of Technology, 5600 MB Eindhoven, The to propagating modes in the crystalline core.36,37 Such Netherlands coupling could result in an effective layer near the boundary ́ Pol Torres − Departament de Fisica, Universitat Autònoma de of the wire where the thermal conductivity is greatly reduced, Barcelona, Bellaterra 08193, Spain an effect that can be much stronger for thinner wires. Finally, Alessio Campo − Department of Physics, University of Basel, within the hydrodynamic framework,3−5,10 the so-called no-slip 4056 Basel, Switzerland condition could also lead to a strong reduction of the thermal Marta De Luca − Department of Physics, University of Basel, conductivity near the boundary of the NW, quenching the 4056 Basel, Switzerland; orcid.org/0000-0001-6061-3261 contribution to the heat flow of diffusive phonons. A possible Marcel A. Verheijen − Department of Applied Physics, origin for the enhancement of the contribution of specular Eindhoven University of Technology, 5600 MB Eindhoven, The phonons and the emergence of ballisticity is a strongly reduced Netherlands scattering of these phonons into diffusive modes caused by Sebastian Koelling − Department of Applied Physics, Eindhoven localization-induced decoupling of specular and diffusive University of Technology, 5600 MB Eindhoven, The phonons. This scenario, which would agree with the observed Netherlands 2707 https://dx.doi.org/10.1021/acs.nanolett.0c00320 Nano Lett. 2020, 20, 2703−2709 Nano Letters pubs.acs.org/NanoLett Letter Luca Gagliano − Department of Applied Physics, Eindhoven ductivity of Semiconductors. Phys. Rev. B: Condens. Matter Mater. University of Technology, 5600 MB Eindhoven, The Phys. 2017, 95 (16), 165407. Netherlands; orcid.org/0000-0003-4214-8639 (8) Ziabari, A.; Torres, P.; Vermeersch, B.; Xuan, Y.; Cartoixà, X.; Jos E. M. Haverkort − Department of Applied Physics, Torelló, A.; Bahk, J.-H.; Koh, Y. R.; Parsa, M.; Peide, D. Y.; et al. Full- Field Thermal Imaging of Quasiballistic Crosstalk Reduction in Eindhoven University of Technology, 5600 MB Eindhoven, The Nanoscale Devices. Nat. Commun. 2018, 9 (1), 255. Netherlands; orcid.org/0000-0003-3051-673X (9) Martelli, V.; Jiménez, J. L.; Continentino, M.; Baggio-Saitovitch, ́ F. Xavier Alvarez − Departament de Fisica, Universitat E.; Behnia, K. Thermal Transport and Phonon Hydrodynamics in Autònoma de Barcelona, Bellaterra 08193, Spain; Strontium Titanate. Phys. Rev. Lett. 2018, 120 (12), 125901. orcid.org/0000-0001-6746-2144 (10) Beardo, A.; Calvo-Schwarzwälder, M.; Camacho, J.; Myers, T. Peter A. Bobbert − Department of Applied Physics, Eindhoven G.; Torres, P.; Sendra, L.; Alvarez, F. X.; Bafaluy, J. Hydrodynamic University of Technology, 5600 MB Eindhoven, The Heat Transport in Compact and Holey Silicon Thin Films. Phys. Rev. Netherlands; Center for Computational Energy Research, 5600 Appl. 2019, 11 (3), 34003. HH Eindhoven, The Netherlands (11) Siemens, M. E.; Li, Q.; Yang, R.; Nelson, K. A.; Anderson, E. Ilaria Zardo − Department of Physics, University of Basel, 4056 H.; Murnane, M. M.; Kapteyn, H. C. Quasi-Ballistic Thermal Basel, Switzerland; orcid.org/0000-0002-8685-2305 Transport from Nanoscale Interfaces Observed Using Ultrafast Coherent Soft X-Ray Beams. Nat. Mater. 2010, 9 (1), 26−30. Complete contact information is available at: (12) Bae, M.-H.; Li, Z.; Aksamija, Z.; Martin, P. N.; Xiong, F.; Ong, https://pubs.acs.org/10.1021/acs.nanolett.0c00320 Z.-Y.; Knezevic, I.; Pop, E. Ballistic to Diffusive Crossover of Heat Flow in Graphene Ribbons. Nat. Commun. 2013, 4, 1734. Author Contributions (13) Hsiao, T.-K.; Chang, H.-K.; Liou, S.-C.; Chu, M.-W.; Lee, S.-C.; ⊥ Chang, C.-W. Observation of Room-Temperature Ballistic Thermal These authors contributed equally to this work. Conduction Persisting over 8.3 $μ$m in SiGe Nanowires. Nat. Notes Nanotechnol. 2013, 8 (7), 534. The authors declare no competing financial interest. ■ (14) Wilson, R. B.; Cahill, D. G. Anisotropic Failure of Fourier Theory in Time-Domain Thermoreflectance Experiments. Nat. ACKNOWLEDGMENTS Commun. 2014, 5, 5075. We would like to acknowledge Rob van der Heijden, Riccardo (15) Lee, J.; Lim, J.; Yang, P. Ballistic Phonon Transport in Holey Rurali, Michel de Jong, Bernd Gotsmann, Ali Shakouri, and Silicon. Nano Lett. 2015, 15 (5), 3273−3279. Chris Palmstrøm for valuable feedback on the manuscript. This (16) Hu, Y.; Zeng, L.; Minnich, A. J.; Dresselhaus, M. S.; Chen, G. work has been supported by the European Research Council Spectral Mapping of Thermal Conductivity through Nanoscale Ballistic Transport. Nat. Nanotechnol. 2015, 10 (8), 701. (ERC 617256 and 756365), the Dutch Organization for (17) Anufriev, R.; Ramiere, A.; Maire, J.; Nomura, M. Heat Guiding Scientific Research (NWO), the Foundation for Fundamental and Focusing Using Ballistic Phonon Transport in Phononic Research on Matter (FOM), and the Swiss National Science Nanostructures. Nat. Commun. 2017, 8, 15505. Foundation research grant (project grant no. 165784). We (18) Anufriev, R.; Gluchko, S.; Volz, S.; Nomura, M. Quasi-Ballistic acknowledge Solliance, a solar energy R&D initiative of ECN, Heat Conduction Due to Lévy Phonon Flights in Silicon Nanowires. TNO, Holst, TU/e, imec and Forschungszentrum Jülich, and ACS Nano 2018, 12, 11928. the Dutch province of Noord-Brabant for funding the TEM (19) Hicks, L. D.; Dresselhaus, M. S. Thermoelectric Figure of Merit facility. This work is part of the Industrial Partnership of a One-Dimensional Conductor. Phys. Rev. B: Condens. Matter Programme (IPP) “Computational sciences for energy Mater. Phys. 1993, 47 (24), 16631. research” of The Netherlands Organization for Scientific (20) Murphy, P. G.; Moore, J. E. Coherent Phonon Scattering Research Institutes (NWO-I). This research program is co- Effects on Thermal Transport in Thin Semiconductor Nanowires. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76 (15), 155313. financed by Shell Global Solutions International B.V. 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Nanotechnol. 2014, 9 (7), 520. ■ NOTE ADDED AFTER ASAP PUBLICATION Due to a production error, this paper was published ASAP on March, 3, 2020, with an incorrect version of Figure 2. The corrected version was reposted on March 5, 2020. 2709 https://dx.doi.org/10.1021/acs.nanolett.0c00320 Nano Lett. 2020, 20, 2703−2709