This is the submitted version of the journal article: Mukherjee, Samik; Wajs, Marcin; la Mata, Maria de; [et al.]. «Disentangling phonon channels in nanoscale heat transport». Physical review B, Vol. 104, issue 7 (August 2021), art. 75429. DOI 10.1103/PhysRevB.104.075429 This version is available at https://ddd.uab.cat/record/271935 under the terms of the license Disentangling Phonon Channels in Nanoscale Thermal Transport Samik Mukherjee1,*, Marcin Wajs1, Maria de la Mata2,¥, Uri Givan4, Stephan Senz4, Jordi Arbiol2,3, Sebastien Francoeur1, and Oussama Moutanabbir1,* 1 Department of Engineering Physics, École Polytechnique de Montréal, C. P. 6079, Succ. Centre- Ville, Montréal, Québec H3C 3A7, Canada 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Catalonia, Spain 3 ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Catalonia, Spain 4 Max Planck Institute of Microstructure Physics, Weinberg 2, Halle (Saale) 06120, Germany ¥ Present Address: Departamento de Ciencia de los Materiales, Ing. Met. y Qca. Inorg., IMEYMAT, Universidad de Cádiz, 11510 Puerto Real, Spain * Email: samik.mukherjee@polymtl.ca; oussama.moutanabbir@polymtl.ca Abstract: Phonon surface scattering has been at the core of heat transport engineering in nanoscale structures and devices. Herein, we demonstrate that this phonon pathway can be the sole mechanism only below a characteristic, size-dependent temperature. Above this temperature, the lattice phonon scattering co-exist along with surface effects. By artificially controlling mass disorder and lattice dynamics at the atomic-level in nanowires without affecting morphology, crystallinity, chemical composition, and electronic properties, the temperature-thermal conductivity-diameter triple parameter space is mapped, and the main phonon scattering mechanisms are disentangled. This led to the identification of the critical temperature at which the effect of lattice mass-disorder on thermal conductivity is suppressed to an extent that phonon transport becomes governed entirely by the surface. This behavior is discussed based on Landauer-Dutta-Lundstrom near-equilibrium transport model. The established framework provides the necessary input to further advance the design and modelling of phonon and heat transport in semiconductor nanoscale systems. 1 Phonon transport in a crystal is controlled by surface as well as lattice effects1,2. The latter dominate in bulk materials and include isotopic-disorder, charge carrier, defect, and anharmonic scattering channels. Surface scattering is, however, prevalent in nanoscale materials and has been at the center of the recently expended efforts to engineer thermal transport at the nanoscale3–9. In this regard, semiconductor nanowires (NWs) have been extensively studied as a unique playground for phonon engineering that combines shape anisotropy and large surface-to-volume ratio10–16. In such quasi-one-dimensional nanostructures, the phonon-surface interaction renormalizes the phonon lifetime and/or the non-equilibrium phonon population distribution3,9,17. The interaction with the rough and amorphous surface showed evidence of Anderson localization of phonons in NWs4,5. It was also demonstrated that surface interaction shortens the dominant phonon wavelength and mean-free path in the NW heat spectra relative to bulk values6. Additionally, the surface roughness and its spatial correlation length were also found to tune the NW thermal conductivity below the Casimir limit7–9. Despite the expected overwhelming dominance of surface scattering, lattice effects must also persist because of the NW finite diameter. For example, calculations showed that a small degree of lattice disorder in alloyed NWs can act like a low-pass filter, suppressing the high- frequency phonon modes while leaving the low-frequency modes unaffected18. A separate calculation revealed a complementary effect, wherein a short-range spatial correlation in the mass- disorder of one-dimensional structures acts like a high-pass filter, showing a dramatic increase in the transmittance of high-frequency phonon modes and suppression of the low-frequency modes19. Thermal transport in NWs was also reported to be anisotropic, depending on the crystal orientation20. For NWs of same diameter, such behavior could only be attributed to the anisotropy 2 of the acoustic phonon group velocity along different crystal orientations. The aforementioned studies indicate that along with surface scattering, thermal transport in such quasi-one-dimensional structures underlies a subtle but important contribution from the crystalline lattice as well. However, the individual contributions of these mechanisms remain hitherto undiscovered and experimentally inaccessible. This work addresses this fundamental question by establishing an experimental framework to track and distinguish the inherently intertwined lattice and surface scatterings. By tailoring the mass-disorder in silicon (Si) NWs, while keeping the chemical composition, crystallinity, electronic properties, and morphology intact, the work demonstrates that the temperature-dependent effects and the size-dependent effects on phonon transport can indeed be disentangled. Nanoscale engineering of lattice disorder Mass-disorder engineering is achieved during the NW growth in ultrahigh vacuum chemical- vapor-deposition (UHV-CVD) chamber using isotopically enriched silane precursors 28SiH4 and 30 SiH4 with an isotopic purity higher than 99.9%21 (see Methods for details on NW growth). The 30 NWs discussed in this work are either isotopically pure Si NWs or isotopically mixed 28 Six 30Si1−x (x~0.4) NWs. The two sets of NWs are therefore chemically, electrically, and morphologically identical. The NWs are ~ 10 μm-long with an average diameter of ~55 nm. Fig. 1(a) shows a representative high-angle annular dark field (HAADF) image of an isotopically mixed 28 Six 30Si1−x NW analyzed in an aberration-corrected scanning transmission electron microscope (STEM) (see Methods). The panels 2 and 3 in the figure show the HAADF/STEM images at progressively increasing magnification. The fast Fourier transform (FFT) indicates the NW growth 3 orientation. The STEM analysis of isotopically lattice-ordered 30Si NWs (not shown here) exhibits similar microstructure. Fig. 1(b) displays the Si-Si longitudinal optical (LO) phonon mode of single 30Si (left) and 28 Six 30Si1−x (right) NWs, recorded at three different base temperatures. The data indicates a consistent redshift of the peak position in both sets of NWs, as the base temperature increases. Fig. 1(c) summarizes this evolution over a broader temperature range of 4-300 K. The reasons for the observed redshift lie in the anharmonic effects (phonon-phonon coupling) and the quasi-harmonic effects (changes in crystal volume), both being temperature-dependent. The frequency of a mode at a temperature T is given by ω(T) = ω0 + [∆ω(T)vol ] + [∆ω(T)anh ], where ‘ω0 ’ is the frequency as T approaches 0 K, ∆ω(T)vol is the shift in frequency caused by changes in crystal volume, and ∆ω(T)anh is the frequency shift caused by the anharmonic phonon-phonon coupling. ω(T) is expressed as22,23 2 3 3 ω(T) = ω0 + [ω0 {e−3AT − 1}] + [B {1 + } + C {1 + z + 𝑧 }], (1) e𝑦 −1 e − 1 (e − 1)2 where y = ℏω0 ⁄2k B T and z = ℏω0 ⁄3k B T. B and C are cubic and quartic anharmonic constants denoting the strength of the three- and four-phonon anharmonic coupling processes, respectively. Details related to the parameter ‘A’ can be found in Section 2 of the Supplementary Information. The data in Fig. 1(c) was fitted using equation (1) (solid lines) yielding ω0 of (512.3 ± 0.3) cm-1 and (521.7 ± 0.3) cm-1 for 30Si and 28Six 30Si1−x NWs, respectively. ω0 is a key parameter for the subsequent analysis. The data in Fig. 1(c) also indicate that quartic anharmonicity can be excluded from the observed temperature-induced redshift of the LO modes, since the coupling constant C 4 was found to be negligible for both sets of NWs. The total redshift in ω from 4 to 300 K is insensitive to the isotopic content. This is expected, since neither the anharmonic effect nor the quasi-harmonic effect depend on the lattice-disorder24. Local heat in mass disorder-engineered NWs Next, the evolution of LO mode peak positions (ωPD ) were measured as a function of the incident laser power density, at fixed base temperatures. Examples showing the evolution of ωPD as a 30 function of incident power density at 140 K are displayed in Figs. 2(a) and (b) for Si NWs and 28 Six 30Si1−x NWs, respectively. ωPD typically undergoes a redshift with increasing power density, indicative of laser-induced local heating of the NWs25,26. At a first glance, it is clear that for the same change in the laser power-density, the total redshift in ωPD of 28Six 30Si1−x NWs (Fig. 2(b)) is larger than that of 30Si NWs (Fig. 2(a)). This means that 28Six 30Si1−x NWs are heating up much 30 faster than Si NWs. The effect can be best summed up by extracting the NW local temperature rise rise (TNW ) as a function of the laser-power density. TNW is estimated from equation (1), after expanding the exponential terms in a Maclaurin series and terminating at the linear term. Note that rise rise ω(T) in equation (1) is replaced with ωPD and the base temperature (T) by TNW . TNW is the sum of the base temperature (T) and the increase in the NW local temperature due to laser-induced rise heating. Combining all these factors, the expression of TNW is obtained: rise (ω0 + B) − ωPD TNW = (2) 4k B [3Aω0 − B ] ℏω0 5 Note that ω0 as well as the parameters A and B in the right-hand-side of equation (2) are rise known for both NW types. The evolution of TNW as a function of the laser power density for 30Si NWs and 28Six 30Si1−x NWs is also displayed in Figs. 2(a) and (b), respectively. The rate of change rise in TNW with incident power density is inversely related to κT (thermal conductivity)21,27. Higher rise the κT , easier it is for the laser-induced heat to flow away and slower is the rate of increase in TNW rise with incident power density. Note that only the linear regime of TNW vs. power density graph has rise been used in the subsequent analysis. Fig. 2(c) compares this linear regime of the TNW vs. power 30 28 density graph for Si NWs and Six 30Si1−x NWs at 140 K. The ratio of the respective slopes rise 28 shows that TNW of Six 30Si1−x NWs increases ~1.30× faster with power density than that of 30 28 Si NWs. This indicates that at 140 K, κT of Six 30Si1−x NWs is ~30 % lower than that of 30 Si NWs. This reduction in κT results from the isotope scattering in lattice-disordered 28 rise Six 30Si1−x NWs. Fig. 2(d) shows the evolution of TNW at 60 K. Note that the ratio of the slopes reaches unity which indicates similar κT for both isotopically mixed and isotopically pure NWs. This means that the effect of isotope-induced mass-disorder has been suppressed at 60 K, leaving surface scattering as the dominant mechanism determining the overall phonon lifetime. Also, note rise 30 the difference in the slope of TNW at 140 K and that at 60 K. For example, for Si NWs at 60 K the slope is ~1.45× larger than that at 140 K, indicating that κT at 60 K is reduced by ~45 % w.r.t. that at 140 K. The temperature evolution of κT of both the NW sets is discussed next. Fig. 3(a) exhibits the evolution of κT with temperature of both NW sets. In this work, κT is 30 always given relative to that of Si NWs at 300 K. The temperature evolution of κT is quite different from bulk materials28–30. At low temperatures, when the phonon mean-free path is long and comparable to the sample dimensions, surface scattering (independent of temperature) is 6 dominant and κT of a bulk semiconductor essentially follows the ~T 3 dependence of the phononic specific heat31. At higher temperatures, anharmonic phonon-phonon scattering starts to dominate the phonon lifetime and κT scales as ~T −1. A bulk sample reaches the maximum value of κT (κmax T ) between these two temperature regimes. Tmax , the temperature at which a bulk sample reaches κmax T is material specific, varying from ~16.5 K for Ge29, to ~26 K for Si28, to ~75 K for diamond32. It is around Tmax that the isotope effect on κT of bulk semiconductors is the most prominent. This difference in the evolution of κT with temperature in NWs is related to phonon surface scattering that competes with all other scattering mechanisms, even at elevated temperatures33–35. In the following, this behavior is further discussed based on the Landauer-Dutta- Lundstrom (LDL) transport model36,37. According to this model, the lattice thermal conductance (K T ) under near-equilibrium conditions is given by37,38 k 2B T ph ph KT = ∑ ∫ Mj (ω)W ph (ω, T)𝒯j (ω, T)dω , (3) h j=LA,TA j where the summation is over the longitudinal acoustic (LA) and the doubly degenerate transverse ph acoustic (TA) phonon modes. Mj (ω) is the total number of channels available to the mode ‘j’ for heat conduction at an energy of ℏω and is given by the cross-section (S) of the conductor times ph the phononic density of states, Dj (ω). W ph (ω, T) is the energy- and temperature-dependent window function that selects only those channels, out of all the available channels, that can take part in thermal conduction and is given by W ph (ω, T) = (ℏω⁄k B T)2 (− ∂𝓃Bose ⁄∂ω), where 𝓃Bose −1 ph is the equilibrium Bose-Einstein factor given by 𝓃Bose = [eℏω⁄kB T − 1] . 𝒯j (ω, T) is the 7 transmittance function which defines the probability that the jth mode is transmitted along the ph length (L) of the heat conductor, without backscattering. 𝒯j (ω, T) can be approximated as λj (ω, T)⁄L, where λj (ω, T) is the backscattering mean-free path. λj (ω, T) can in turn be expressed as 𝒻𝓋j τj (ω, T), where 𝓋j is velocity of the mode ‘j’ along the direction of transmission, τj (ω, T) is the mode lifetime at an energy ℏω and temperature T, and 𝒻 is a pre-factor which accounts for the backscattering mean-free path. Mapping the temperature-thermal conductivity-diameter space Various scattering mechanisms were considered in order to calculate the total phonon lifetime (see details in Section 3 of the Supplementary Information). This includes the normal anharmonic scattering of phonons39, the Umklapp anharmonic scattering34, scattering of phonons from isotopic mass-disorder40, and the surface phonon scattering29,41. Additionally, the decay of high-energy optical phonons into acoustic phonons was also considered. The scattering of optical phonons is usually neglected due to the flatness of their dispersion curves in Si, making them rather ineffective heat carriers. However, earlier works on theoretical formalism of thermal transport in Si NWs argued that the decay of optical phonons into acoustic phonons at an energy ℏω can be considered as a generation rate of acoustic phonons, which partially counteracts their scattering rate at the same energy42,43. Finally, the total acoustic phonon scattering rate was calculated according to the Matthiessen rule. Phonon confinement is neglected due to the relatively large diameter of NWs. ph To this end, a Debye-like phonon spectrum was considered, wherein Dj (ω) = ω2 ⁄2π2 𝓋j3 shows a parabolic dependence on ω44. The experimental data in Fig. 3(a) was fitted using equation (3), M(P) with the specularity factor (P) and the constants 𝒞LA(TA) (see Section 3 of the Supplementary Information for details) as free parameters. The upper limit of the integral in equation (3) was 8 initially set at Debye frequencies [ωD,LA(TA) = k B θD,LA(TA) ⁄ℏ] in Si: 76.7 THz for the LA mode and 31.4 THz for the TA mode. The validity of this approach was also confirmed by comparing Nat 30 simulated κT vs. T of bulk Si and isotopically pure bulk Si to experimental data28,45. The results are shown and discussed in Section 4 of the Supplementary Information. The parameters extracted from the simulated data of the bulk materials match the experimental values28,45 with 5% uncertainty, justifying the approximations and with it the accuracy of the LDL model in predicting temperature evolution of κT . Next, the analysis was carried out on the data for 30Si NWs in Fig. 3(a). Clearly, the model (solid black line) disagrees with the experimental data, more so above 50 K. The reasons for this discrepancy lie in the bulk-like Debye phonon spectrum, which might overestimate the ph contribution of phonon Umklapp scattering rate41 and that of Mj (ω)46 in equation (3). The optimization was done by choosing a lower cut-off frequency on a trial-and-error basis41. The solid green line in Fig. 3(a) was obtained after the cut-off optimization, which yielded a cut-off frequency (ωC,LA ) of 51.7 THz for the LA mode. The optimization also found that lowering the cut-off frequency of the TA mode impacts the quality of the fit only nominally (see Section 5 of the Supplementary Information). The upper limit of the integral for the TA mode in equation (3) was therefore maintained at its Debye frequency of 31.4 THz all throughout, to avoid an additional P undetermined parameter in the model. The values of P and 𝒞LA(TA) for 30Si NWs were found to be 28 0.26 and 14.86 × 104 s2 (22.01 × 106 s4), respectively. For Six 30Si1−x NWs, the value of P was M fixed to the same value as that of 30Si NWs, while only 𝒞LA(TA) were kept as free parameters. The two sets of NWs were grown under identical conditions and mass-disorder is the only difference between them. Consequently, there is no reason that the specularity factor of surface phonon 9 scattering will be any different between them. The fit to the experimental data of 28Six 30Si1−x NWs M is shown by the solid red line in Fig. 3(a), which gives 𝒞LA(TA) = 17.34 × 104 s2 (20.21 × 106 s4). With all parameters optimized, equation (3) was then used to evaluate κT by varying the diameter (dNW ) from 25 nm up to 250 nm. The results are shown in Fig. 3(b) at the two extreme ends of the range (dNW of 25 nm and 250 nm). See examples of the simulated κT vs. T data for NWs at a few intermediate values of dNW in Section 6 of the Supplementary Information. The temperature behavior of κT in Fig. 3(b) is remarkably sensitive to dNW , which can be attributed to the diameter dependence of the contribution of surface scattering to the phonon lifetime in the NWs. From the simulated data, four parameters related to κT of the NWs are evaluated next, all of which were found to depend strongly on dNW . Of these four, one key parameters that is crucial d here is the temperature limit of mass disorder-induced phonon scattering (TL NW ), or the temperature at which any effect of the lattice on κT vanishes due to the overwhelming influence d of phonon surface scattering. Fig. 4(a) shows that the evolution of TL NW as a function of dNW . The dependence on dNW is very close to being hyperbolic, as extracted from the power-law fit. The hyperbolic dependence is also confirmed by the log-log plot displayed in the inset of Fig. 4(a), d which shows a slope of −1.1. Given that TL NW is the temperature below which lattice mass- disorder effect on κT is totally suppressed, it is also interesting to investigate how it varies as a 28 function of the mass-variance. To this end, κT vs. T data of Six 30Si1−x NWs was also simulated by varying the fractional composition of 28Si (see an example in Section 6 of the Supplementary d 28 Information) and the dependence of TL NW on the Si concentration is shown in Fig. 4(b), d indicating a linear behavior wherein TL NW decreases by ~ 0.85 K for every 10.0 at.% increment 10 28 d in the Si concentration47. Interestingly, the influence of the mass-variance on TL NW is fairly independent of dNW , as reflected by the same slope of the NWs at the diameter extremities of 25 nm and 250 nm. Shown in Section 7 of the Supplementary Information are the other three 28 parameters which include: the reduction in room temperature κT of Six 30Si1−x NWs relative to 30 NW d NW d Si NWs (R RT ); the temperature at which κT reach the maximum value (Tmax ); and the 28 d d d reduction in κmax of Six 30Si1−x NWs relative to 30 NW Si NWs at Tmax (R TNW max NW ). Tmax was found d d to have a ~(dNW )−1.4 dependence on dNW , while R RT NW and R TNW max were found to depend on dNW as ~exp(−dNW )0.65 and ~exp(−dNW )0.72, respectively. Conclusion The analyses above outline a framework to evaluate the temperature-𝜅𝑇 -diameter triple-parameter space by artificially controlling mass-disorder and lattice dynamics in NWs. The work also demonstrates that phonon transport in NWs is indeed not all about the surface and there exists a temperature regime where the lattice starts contributing to the heat transport in a nanoscale structure. The framework identifies the boundary between this regime and the purely surface- dominated one and elucidates how this boundary evolves as a function of the NW diameter and the extent of mass-disorder in the crystalline lattice. By unravelling these heretofore unexplored fundamental aspects, this work disentangles the primary phonon scattering channels that strongly influences the thermal behavior at the nanoscale. This knowledge is crucial to guide the design and modelling of future nanoscale phononic and thermoelectric devices and heat management platforms. 11 ACKNOWLEDGEMENTS O.M. acknowledges support from NSERC Canada (Discovery, SPG, and CRD Grants), Canada Research Chairs, Canada Foundation for Innovation, Mitacs, PRIMA Québec, and Defence Canada (Innovation for Defence Excellence and Security, IDEaS). ICN2 acknowledges funding from Generalitat de Catalunya 2017 SGR 327. ICN2 is supported by the Severo Ochoa program from Spanish MINECO (Grant No. SEV-2017-0706) and is funded by the CERCA Programme / Generalitat de Catalunya. The HAADF-STEM microscopy was conducted in the Laboratorio de Microscopias Avanzadas at Instituto de Nanociencia de Aragon-Universidad de Zaragoza. Authors acknowledge the LMA-INA for offering access to their instruments and expertise. AUTHORS CONTRIBUTION STATEMENT S.M. analyzed the data and carried out all systematic experimental and theoretical studies. M.W., supervised by S.F., helped in the low-temperature Raman studies. M.d.l.M. performed TEM analysis under the supervision of J.A.. U.G. and S.S. were involved in the initial nanowire growth work. O.M. conceived the project. S.M. and O. M. wrote the manuscript and all authors comment it. COMPETING INTERESTS The authors declare no competing financial interests. 12 References 1. Cahill, D. G. et al. Nanoscale thermal transport. J. Appl. Phys. 93, 793–818 (2003). 2. Cahill, D. 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Jeong, C., Datta, S. & Lundstrom, M. Full dispersion versus Debye model evaluation of lattice thermal conductivity with a Landauer approach. in Journal of Applied Physics 109, 073718 (American Institute of PhysicsAIP, 2011). d 47. Note, this rate of decrease in TL NW is only true up to 28Si fractional composition of 50.0 at.%. At this concentration the isotopically mixed NWs attain their maximum mass- varience. The lattice would start to become more ordered with further increase in 28Si concentration 15 Methods i) Growth of isotopically engineered NWs: NatSiF4 gas was first enriched into isotopically enriched jSiF4 (j= 28, 29, and 30) inside a Zippe-centrifuge. This was followed by reduction of the isotopically enriched jSiF4 gases into the jSiH4 precursors, following the chemical reaction: jSiF4 + 2CaH2 → jSiH4 + CaF2. The vapor-liquid-solid (VLS) method was implemented to grow the NWs, with gold (Au) acting as the catalyst. For this, Si substrates underwent degreasing in acetone, followed by a dip in 2% hydrofluoric (HF) acid for 2 mins to remove the native oxide, followed by rinsing in de-ionized water. Next, 2 nm thick Au films were then deposited on the substrates before loading them into the UHV-CVD chamber. Au-Si eutectic droplets were formed by annealing the substrate at 450℃ for 20 min. The NWs started to grow with the injection of the enriched jSiH4 precursors, at a flow rate of 2 sccm. The growth temperature and the chamber pressure were maintained 480 ℃ and 1.5 mbar, respectively. ii) Scanning transmission electron microscopy: The HAADF-STEM images of the NWs were recorded in a probe-corrected FEI Titan 60-300, equipped with a high brightness field emission gun and a CETCOR corrector from CEOS. iii) Raman Spectroscopy: The Raman measurements were performed in a custom micro- Raman setup at λex = 532 nm laser excitations. The laser was focused on the sample with a 100X (NA 0.85) objective with a spatial resolution approaching the diffraction limit. The laser power density incident on the sample was measured using a hand-held power meter. The spectra were acquired using a liquid-nitrogen cooled charged-coupled device camera (JY Symphony) mounted on a Jobin-Yvon Triax iHR550 spectrometer (grating 1,800 g·mm–1) with a precision of 0.2 cm–1. During the laser exposure, the sample was mounted inside a liquid-He cryostat that enabled adjusting the base temperature from 300 K down 16 to 4 K, with a temperature stability of 80 mK. The samples were held under high vacuum of below 10-6 Torr. The spectrometer calibration was done using the Raman peak position of bulk silicon at 520.8 cm–1. Prior to the Raman measurements, a suspension was first prepared by putting a small piece of as-grown sample in acetone under ultrasonic vibration for 10 min. The suspension was then dispersed on two different type of substrates: first, an Au-coated Si substrate and second, a commercial Au-grid with 4 µm sized holes. The first substrate was used for investigating such intrinsic properties of the NWs (the ω vs. T analysis in Fig. 1(c)), wherein any laser-induced heating effect is undesirable. The Au- coated Si substrate underneath the NWs provide a good thermal contact and act as a heat sink that can quickly dissipate any laser-induced heat. The second substrate was used for the transport measurements within the framework of the power dependent study (the ωPD vs. P.D. analysis in Fig. 2). This pronounced the laser-induced heating effect of the NWs and suppressed any underlying effect of the substrate on the transport measurements. 17 Fig. 1│HAADF image and Raman spectra of the isotopically programmed NWs. (a) HAADF image from HR-STEM analysis of a 28Six 30Si1−x NW at different magnifications. (b) Back- scattering micro-Raman spectra, collected at different ambient temperatures, showing the LO phonon mode of a 30Si NWs (left) and a 28Six 30Si1−x NWs (right). The Voigt fits are shown as solid lines. The vertical red dotted line marks the peak positions of the LO modes at 4 K. See section 1 of the supplementary information for details on laser power optimization. (c) The evolution of Si-Si peak position (ω) as a function of ambient temperature. The fit to the ω vs. T data is done using equation (1) (solid lines). From the fits, the parameters A in equation (1) was found to be 2.63 ×10-3 K-1 and 2.70×10-3 K-1, while the parameter B was found to be −3.96 cm-1 and −3.91 cm-1, for the 30Si NWs and the 28Six 30Si1−x NWs, respectively. 18 Fig. 2│ Evolution of Raman peak position and the local temperature of the NWs as a function rise of laser power density (a) Evolution of ωPD and TNW of 30Si NWs as a function of laser power rise density at 140 K; (b) Evolution of ωPD and TNW of the 28Six 30Si1−x NWs, as a function of laser power density, at an ambient temperature of 140 K. The data for ωPD displayed in (a) & (b) were averaged over 10-15 individual NWs. The corresponding error bars are the standard deviation of rise the data. The error bars associated with the TNW comes from two sources: the standard deviation (error bars) associated with the mean value of ωPD and the uncertainty associated with the value rise of ω0 estimated from the fit; (c) Evolution in the linear regime of TNW as a function of laser power 30 28 30 density of Si NWs and the Six Si1−x NWs at 140 K. The lines indicate the linear fit to the corresponding data. The data for the 30Si NW has also been used to demonstrate how the rise uncertainty associated with the slope of linear fit to the TNW vs. power density data was calculated. The errors bars were used to estimate the maximum possible slope (smax ) and the minimum rise possible slope (smin ), associated with the TNW vs. power density data. The uncertainty associated 19 rise with the slope was calculated as ∆s = (smax − smin )⁄2; (d) Evolution of TNW as a function of laser power density of the 30Si NW and the 28Six 30Si1−x NWs at 60 K. 20 Fig. 3│Experimenal and simulated 𝛋𝐓 of the NWs (a) Evolution of κT of 30Si NWs and 28 Six 30Si1−x NWs as a function of the ambient temperature. The data are normalized to κT of 30 Si NWs at 300 K. The solid black line is the fit to the data of the 30Si NWs using equation (3) with [0, ωD,LA ] as the limits of the integration. The solid blue line is a fit to the data of the 30Si NWs using equation (3) with [0, ωC,LA ] as the integration boundaries. The solid red line is a fit to the data of the 28Six 30Si1−x NWs using equation (3) with [0, ωC,LA ] as the integration boundaries; (b) Simulated κT of 30Si NWs and 28Six 30Si1−x NWs as a function of ambient temperature. The figure displays two sets of simulations corresponding to two different diameters of 250 nm and 25 nm. In order to be consistent with the experimental data in (a), the simulated data in (b) was also normalized to κT of 30Si NWs at 300 K. 21 𝐝 Fig. 4│ Evolution of 𝐓𝐋 𝐍𝐖 as a function of NW diameter and mass-variance (a) Evolution of d TL NW as a function of the NW diameter (dNW ). The solid red line is a power-law fit to the simulated d data. Inset: Log-log plot of the TL NW vs. dNW data. The data has been fitted with a straight line which shows a slope of −1.1. The simulated data point corresponding to dNW = 55 nm (the average diameter of the NWs investigated experimentally in this work) is highlighted using the d colored circle; (b) Evolution of TL NW as a function of the fractional composition of 28Si isotope within a 28Six 30Si1−x NW. The fractional composition is indicative of the mass-variance, the maximum mass variance being at 50.0 at.% composition of the two isotopes. The simulated data is shown only for the thinnest and the thickest NWs with diameters of 25 nm and 250 nm, respectively. The solid lines are linear fit to the respective data. 22 SUPPLEMENTARY INFORMATION Disentangling Phonon Channels in Nanoscale Thermal Transport Samik Mukherjee1,*, Marcin Wajs1, Maria de la Mata2,¥, Uri Givan4, Stephan Senz4, Jordi Arbiol2,3, Sebastien Francoeur1, and Oussama Moutanabbir1,* 1 Department of Engineering Physics, École Polytechnique de Montréal, C. P. 6079, Succ. Centre- Ville, Montréal, Québec H3C 3A7, Canada 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Catalonia, Spain 3 ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Catalonia, Spain 4 Max Planck Institute of Microstructure Physics, Weinberg 2, Halle (Saale) 06120, Germany ¥ Present Address: Departamento de Ciencia de los Materiales, Ing. Met. y Qca. Inorg., IMEYMAT, Universidad de Cádiz, 11510 Puerto Real, Spain * Email: samik.mukherjee@polymtl.ca; oussama.moutanabbir@polymtl.ca 1) Optimization of laser power density: Before Raman studies, a suspension was first prepared by putting a small piece of as-grown sample in acetone under ultrasonic vibration for 10 min. The suspension was then dispersed on two different type of substrates, as shown in Fig. S1(a):left. During the (ω, Γ) vs. T measurements of the 30Si NWs and the 28Six 30Si1−x NWs (Fig. 1(c) of the main manuscript), the incident laser power density was optimized in the following manner to ensure the minimum laser induced heating of the NWs. The optimization was performed at 4 K. The reason being that the κT of NWs is expected to be the minimum1, and therefore the NWs have a much higher sensitivity to laser-induced heating effect at 4 K. So, if the laser power density is optimized at 4 K, it can be assumed to stay optimized for T > 4 K, all the way up to 300 K. Fig. S1(b) shows the Raman peak position as a function of power density for 3 different 28 Six 30Si1−x NWs, at an ambient temperature of 4 K. As can be seen in Fig. S1(b), the measured peak position of the NWs shifts at power densities above 0.32 mW/μm2. This blueshift is due to 23 the laser-induced heating. The peak position stays essentially the same for power densities below 0.32 mW/μm2, meaning the laser power density is low enough to induce any additional heating. The optimal power density for 30Si NWs was identified using the same procedure and found to be the same (0.32 mW/μm2). The (ω, Γ) vs. T measurements shown in Fig. 1(c) of the main manuscript were carried out using this optimal power density of 0.32 mW/μm2. The power optimization for the peak position vs. power density plots (Fig. 2 of the main manuscript) was performed in a similar manner, with NWs suspended atop holes on Au-grid (commercially purchased) as shown in Fig. S1(a): right. Fig. S1│Sample preparation for Raman measurements and laser power optimization. (a) Left: Scanning electron microscope (SEM) image of a single 28Six 30Si1−x NW, dispersed on a Au- 24 coated Si substrate. While investigating such intrinsic properties of the NWs (like the ω vs. T analysis in Fig. 1(c) of the main manuscript), any laser-induced heating effect is undesirable. The Au-coated Si substrate underneath the NWs provide a good thermal contact and act as a heat sink that can quickly dissipate any laser-induced heat. Right: SEM image of a single 28Six 30Si1−x NW, dispersed atop a hole on an Au-grid. This pronounced the laser-induced heating effect and suppressed any underlying effect of the substrate. (b) Measured Raman peak position as a function of laser power density of 3 individual 28Six 30Si1−x NWs, at an ambient temperature of 4 K. 2) Details related to the expression of ∆𝛚(𝐓)𝐯𝐨𝐥 : ∆ω(T)vol is the shift in frequency caused by changes in crystal volume given by2,3 T ∆ω(T)vol = ω0 [exp {∫0 −𝓃γj (T′)α(T′) dT′} − 1], (E1), where γj is the mode-specific Grüneisen parameter, the degeneracy factor (𝓃) is 3 for the LO mode in Si, and α is the linear thermal expansion coefficient. Both γj and α evolve non-linearly with temperature4,5, making the product of γj and α rather complicated to describe using any phenomenological analytical expression. Hence, for practical purposes, mean isothermal values of 〈γ〉T and 〈α〉T are often considered in literature for fitting the temperature evolution of phonon energies6,7. Here, the product of 〈γ〉T and 〈α〉T has been considered as a single fitting parameter, A, which can be taken outside the integral in equation (E1). Equation (E1) was therefore simplified to ∆ω(T)vol = ω0 [e−3AT − 1] (E2) 25 3) Description of the phonon scattering mechanisms: Table 1: Summary of the scattering processes considered for calculating τj (ω, T) Scattering Phonon Scattering rates (s-1) Description of parameters Reference mechanism mode LA τ−1 2 3 N,LA (ω, T) = BN,LA ω T BN,LA = k 3B γ2LA V⁄m 5 ̅ ℏ2 𝓋LA Normal Ref 8,9 anharmonic TA τ−1 N,TA (ω, T) = BN,TA ωT 4 BN,LA = k 4B γ2TA V⁄m 2 ̅ ℏ3 𝓋TA Umklapp τ−1 U,LA(TA) (ω, T) BU,LA(TA) Ref 9,10 anharmonic LA(TA) = BU,LA(TA) ω2 Te−θD,LA(TA)⁄3T = ℏγ2LA(TA) ⁄m 2 ̅ 𝓋LA(TA) θD,LA(TA) Isotopic mass- LA(TA) τ−1 I,LA(TA) (ω) = A LA(TA) ω 4 3 ALA(TA) = Vg 2 ⁄4π𝓋LA(TA) Ref 11 disorder Surface LA(TA) τ−1 S,LA(TA) = 𝓋LA(TA) ⁄ℒ eff ℒeff = Ref 12–14 [(1 − P)⁄(1 + P)Ws + (1⁄L)]−1 τ−1 GM (ω, T) M 𝒞LA(TA) is the proportionality LA(TA) Acoustic phonon M = 𝒞LA(TA) ω3 g 2 [coth(ℏω⁄2k B T)]1⁄2 constant for the isotopically mixed NWs generation rate due to τ−1 GP (ω, T) P 𝒞LA(TA) is the proportionality anharmonic LA(TA) LA(TA) scattering of P = 𝒞LA(TA) ω5 [coth(ℏω⁄2k B T)] constant for the isotopically pure Ref9 optical NWs phonons γLA(TA) is the Grüneisen parameter for the LA(TA) mode which were taken to be 1.04 (0.56)8; 𝓋LA(TA) is the velocity of the LA(TA) mode which were taken to be 8430 m⁄s (5840 m⁄s)8; m ̅ is the average isotopic mass which is 29.973 amu. for the 30Si NWs and 29.015 amu. for the 28Six 30Si1−x NWs; V is the volume per atom which was taken to be 2.0 × 10-23 cm3 for Si15; θD,LA(TA) is the Debye temperature of the LA(TA) mode which were taken to be 586 K (240 K)8; g 2 = {∑i(ci mi )2 − (∑i ci mi )2 }⁄(∑i ci mi )2 is the second-order moment of mass-fluctuation. ci and mi are the fractional composition and the atomic mass of the ith isotope; P is the specularity factor for phonon surface scattering; Ws and L are the physical width and length of the conductor, respectively. Ws in the expression of τ−1 S,LA(TA) was set equal to the average NW diameter of 55 nm and the contribution of L was neglected due to high NW aspect-ratio. ℏ and k B have their usual meaning. Scattering mechanisms: The first scattering mechanism highlighted in Table 1 is the anharmonic phonon-phonon scattering mechanism, which shows a frequency and temperature dependence of the form τ−1 2 3 −1 4 N,LA (ω, T) = BN,LA ω T for the LA mode and τN,TA (ω, T) = BN,TA ωT for the TA mode8,9. The constants BN,LA and BN,TA can be expressed as k 3B γ2LA V⁄m 5 ̅ ℏ2 𝓋LA and 2 k 4B γ2T V⁄m ̅ ℏ3 𝓋TA , respectively9. Note, that the normal anharmonic scattering of the acoustic phonons is not a resistive process. Hence, combining this effect with the other scattering 26 mechanisms while fitting the data for isotopically enriched and disordered bulk Ge13, called for an additional corrective term in the expression for κT . However, the contribution of this corrective term to the total κT was found to be below 1% at all temperatures. Even in the Callaway’s original calculations, the corrective term was initially included after having taken the normal anharmonic scattering into account, only to be subsequently neglected once its contribution was found to be insignificant16,17. Consequently, the corrective term has been neglected here. Second, the Umklapp scattering of phonons whose scattering rate can be shown to be9,10 τ−1 U,LA(TA) (ω, T) = BU,LA(TA) ω2 Texp(− θD,LA(TA)⁄3T), where BU,LA(TA) = ℏγ2LA(TA) ⁄m 2 ̅ 𝓋LA(TA) θD,LA(TA). Third, the scattering rate of phonons from isotopic mass-disorder which is given by11 τ−1 I,LA(TA) (ω) = 3 Vg 2 ω4 ⁄4π𝓋LA(TA) . Fourth, is the phonon-surface scattering which was first investigated by Casimir18 and later extended by Berman et. al.19 to include the effect of non-zero specularity factor (P). The phonon surface scattering rate can be expressed as12–14 τ−1 S,LA(TA) = 𝓋LA(TA) ⁄ℒeff . ℒeff is the effective width of the heat conductor. Generation mechanisms: As mentioned in the main text, the earlier works on theoretical formalism of thermal transport in Si NWs argued that the decay of the optical phonons into acoustic phonons at an energy ℏω can be considered as a generation rate of acoustic phonons, which partially counteracts their scattering rate at the same energy9. Now, in majority of materials including Si, the anharmonic scattering and the isotope scattering of the optical phonons are linked to one another. It was argued that mass-disorder can open up additional scattering channels for the optical phonons, leading to an enhancement in their anharmonic scattering rate20–22. Therefore, 28 while treating this mechanism in the isotopically mixed Six 30Si1−x NWs, a convolution of the two mechanisms (three-phonon anharmonic scattering and scattering from mass-disorder) needs 27 to be considered. The analytical expression for the generation rate of acoustic phonons (negative scattering rate of optical phonons) in presence of the convolution is given by9 τ−1 GM (ω, T) = LA(TA) M 𝒞LA(TA) ω3 g 2 [coth(ℏω⁄2k B T)]1⁄2 (M is the superscript denoting isotopically mixed NWs). In the 30 isotopically pure Si NWs, devoid of any isotope scattering (so no convolution with isotope scattering), the generation rate of the LA(TA) phonons can be expressed as9 τ−1 GP (ω, T) = LA(TA) P 𝒞LA(TA) ω5 [coth(ℏω⁄2k B T)]. Finally, the total acoustic phonon scattering rate was calculated according to the Matthiessen rule as 1 1 1 1 1 1 =[ + + + ]− (E3) τj=LA(TA) (ω, T) τN,j (ω, T) τU,j (ω, T) τI,j (ω) τs,j τGM(P) (ω, T) j 28 4) Simulated 𝛋𝐓 vs. T data for bulk 𝐍𝐚𝐭𝐒𝐢, bulk 𝟑𝟎𝐒𝐢, and bulk 𝟐𝟖𝐒𝐢𝟎.𝟒 𝟑𝟎𝐒𝐢𝟎.𝟔 To simulate the κT vs. T data for the bulk semiconductors, the upper limit of the integral in equation (3) of the main manuscript was set to ωD,LA for the LA phonons. The result of the simulations is 30 bulk shown in Fig. S2. It shows that the bulk Si sample reach κmax at a temperature Tmax of 26.5 K. Nat 30 Nat κmax of Si is reduced by a factor of ~7.9 relative to Si at 26.5 K, and κT of Si is reduced by ~9.5 % relative to 30Si at 300 K. The isotope effect on κT of NatSi vanishes at ~5.5 K (see pink dotted line). These values lie within ±5% of the experimentally measured values in literature15,23. Note that no theoretical or experimental investigation regarding the evolution of κT of isotopically mixed bulk Si exists in the literature. The model has therefore been extended to display the temperature evolution of κT of bulk 28Si0.4 30Si0.6 (same isotopic composition as the NWs used in this work), as shown using the solid red curve in Fig. S2. κmax of bulk 28Si0.4 30Si0.6 is reduced by 30 Nat bulk a factor of ~45.8 relative to Si and by a factor of ~5.8 relative to Si, at Tmax of 26.5 K. At 28 300 K, κT of bulk Si0.4 30Si0.6 is reduced by ~32 % relative to 30 Si. The isotope effect on κT of bulk 28Si0.4 30Si0.6 vanishes at ~2.6 K (see orange dotted line). 29 Fig. S2│Simulated 𝛋𝐓 of bulk isotopically engineered Si. Simulated κT of bulk 30Si, and bulk Nat Si, and bulk 28Si0.4 30Si0.6 as a function of ambient temperature, using equation (3) of the main manuscript. The temperature limit to the isotope induced phonon scattering is the temperature at which κT of bulk NatSi, and bulk 28Si0.4 30Si0.6 merges with the κT of bulk 30Si, and are shown using the pink and orange dotted lines, respectively. The relative changes in κT between the bulk 30 Si, bulk NatSi, and bulk 28Si0.4 30Si0.6 at 26.5 K and at 300 K are highlighted in the Figure. 30 5) Optimization of the cut-off frequency As shown in Fig. S3 and discussed in the main manuscript, the quality of the fit to the experimentally obtained κT vs. T data is rather poor when equation (3) of the main manuscript is integrated all the way up to the Debye frequencies of the modes. Note, during the fits, the M(P) parameters P and 𝒞LA(TA) were used as fitting parameters and a bound of [0,1] was imposed on P. It is worth mentioning here that no amount of parameter adjustments led to any significant improvements in the quality of the fit when integrated up to ωD,LA(TA) . The reasons for are two ph fold: first, Mj (ω) in equation (3) calculated using the Debye approximation can be shown to agree excellently with that calculated from the full phonon band-structure in most semiconductors, but only up to a certain energy, beyond which the approximation starts to divert24. Second, contribution that the Umklapp scattering makes to the total phonon lifetime in NWs. In bulk samples, the Umklapp scattering makes a dominant contribution to the total lifetime. The presence of phonon surface scattering in NWs at all temperatures means that the contribution of Umklapp scattering is relatively suppressed than in bulk samples. Consequently, considering a bulk-like phonon spectrum and integrating all the way up to the Debye frequency might overestimate the contribution of Umklapp scattering to the total phonon life-time in NWs12. The solution is therefore to chose a lower cut-off frequency of the modes which would get rid of the excess contribution of ph both Mj (ω) as well as that of the Umklapp scattering to the total phonon life-time. The optimization was done on a trial and error basis by lowering the cut-off frequency of the modes by 1.0 THz. Fig. S3 shows a selected few examples, demonstrating the gradual improvement to quality of the fit as a result of the optimization. Also note that, that lowering the cut-off frequency of the TA mode leads to only a nominal change in the quality of the fit (see the pink and light green dot-dash-dot lines corresponding to ωD,LA = 51.7 THz, ωD,TA =26.4 THz and ωD,LA = 51.7 THz, 31 ωD,TA =21.4 THz, respectively) that too in the high-temperature range (> 180 K). For this reason, during the cut-off optimization, the maximum frequency of the TA mode was maintained at ωD,TA =31.4 THz, while only ωD,LA was altered. This prevented having an additional unknown parameter (that is, ωC,TA ) into the theoretical model. Note, every time the cut-of frequency was altered, the cut-off temperature, in the expression of τ−1 U,LA(TA) (ω, T) was also changed accordingly, from θD,LA(TA) to θC,LA(TA) = ℏωC,LA(TA) ⁄k B . Fig. S3│Cut-off optimization for the LDL model. The cut-off optimization process with the experimentally obtained κT vs. T data of the 30Si NWs. Selected few demonstrations are displayed using the colored lines (solid and dot-dashed), which are the best possible fits to the experimental data, obtained by choosing various values of ωD,LA and ωD,TA (as mentioned in the Fig. legend). The solid green line (ωD,LA = 51.7 THz, ωD,TA =31.4 THz) is the one depicted in Fig. 3(a) of the main manuscript. 32 6) Simulated 𝛋𝐓 vs. T data for NWs at intermediate values of 𝐝𝐍𝐖 Fig. S4│𝛋𝐓 vs. T data for NWs at intermediate values of 𝐝𝐍𝐖 . The simulated κT vs. T data for isotopically pure and isotopically mixed NWs with diameters (a) 30Si NWs and 28Si0.4 30Si0.6 NWs 30 28 with dNW = 205 nm. (b) Si NWs and Si0.1 30Si0.9 NWs, 28 Si0.2 30Si0.8 NWs, 28 Si0.3 30Si0.7 NWs, 28Si0.4 30Si0.6 NWs, 28Si0.5 30Si0.5 NWs, with dNW = 160 nm. (c) 30Si NWs and 28Si0.4 30Si0.6 NWs with dNW = 100 nm. (d) 30Si NWs and 28Si0.4 30Si0.6 NWs with dNW = 40 nm. 33 𝐝 𝐍𝐖 𝐝 𝐍𝐖 𝐝 7) Variation of 𝐑 𝐑𝐓 , 𝐓𝐦𝐚𝐱 , and 𝐑 𝐓𝐍𝐖 𝐦𝐚𝐱 as a function of 𝐝𝐍𝐖 34 𝐝 𝐍𝐖 𝐝 𝐝 Fig. S5│Evoltuion of 𝐑 𝐑𝐓 , 𝐓𝐋 𝐍𝐖 , and 𝐓𝐦𝐚𝐱𝐍𝐖 as a function of the NW diameter. (a) The dNW evolution R RT as a function of dNW . The solid green line is an exponential fit to the simulated dNW data, showing a ~exp(−dNW )0.65 dependence on dNW . (b) The evolution of Tmax as a function of dNW . The solid blue line is the fit to the simulated data, showing a ~(dNW )−1.4 dependence. (c) d The evolution of R TNW max as a function of dNW . The solid brown line is the fit to the simulated data, showing a ~exp(−dNW )0.72 dependence. Note, that the NWs with dNW of 25 nm never reaches Tmax within the temperature range of 300 K considered in this work (see Fig. 3(b) of the main manuscript). Consequently, the data in (b) and (c) starts at dNW = 40 nm and does not show the data point corresponding to dNW = 25 nm. In (a)-(c), the simulated data point corresponding to dNW = 55 nm (the average diameter of the NWs investigated experimentally in this work) is dNW highlighted using the colored circles. Note that Tmax of the 28Six 30Si1−x NWs and the 30Si NWs dNW are not exactly at the same point. Tmax of the 28Six 30Si1−x NWs was always found to be at a dNW d slightly lower temperature (4−6 K) than that of the 30Si NWs. Tmax in (b) and R TNW max in (c) are 30 relative to that of the isotopically pure Si NWs. References 1. Li, D. et al. Thermal conductivity of individual silicon nanowires. Appl. Phys. Lett. 83, 2934 (2003). 2. 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