Hydrodynamic heat transport regime in bismuth: a theoretical viewpoint
Maxime Markov, Jelena Sjakste, Giuliana Barbarino, Giorgia Fugallo, Lorenzo Paulatto, Michele Lazzeri, Francesco Mauri, Nathalie Vast
HHydrodynamic heat transport regime in bismuth : a theoretical viewpoint
Maxime Markov , ∗ Jelena Sjakste , Giuliana Barbarino , Giorgia Fugallo ,Lorenzo Paulatto , Michele Lazzeri , Francesco Mauri , and Nathalie Vast ´Ecole Polytechnique, Laboratoire des Solides Irradi´es, CNRS UMR 7642,CEA-DSM-IRAMIS, Universit´e Paris-Saclay, F91128 Palaiseau c´edex, France, CNRS, LTN UMR 6607, PolytechNantes, Universit´e de Nantes,Rue Christian Pauc, 44306 Nantes c´edex 3, France Sorbonne Universit´es, UPMC Univ. Paris 06,CNRS UMR 7590, MNHN, IRD UMR 206 Institut de Min´eralogie,de Physique des Mat´eriaux et de Cosmochimie, 75005 Paris, France and Dipartimento di Fisica, Universit`a di Roma La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy. (Dated: August 7, 2018)Bismuth is one of the rare materials in which second sound has been experimentally observed.Our exact calculations of thermal transport with the Boltzmann equation predict the occurrence ofthis Poiseuille phonon flow between ≈ ≈ PACS numbers:
Currently a lot of attention is devoted to the studyof phonon-based heat transport regimes in nanostruc-tures [1–4]. Of particular interest is the hydrodynamicregime in which a number of fascinating phenomena suchas Poiseuille’s phonon flow and second sound occur, andwhere temperature fluctuations are predicted to propa-gate as a true temperature wave of the form e i ( k · r − ωt ) [5].The theoretical study of the hydrodynamic regime hasencountered a renewed interest in graphene nanoribbons,where the breakdown of the diffusive Fourier law in favorof the second sound propagation has been predicted [6–9].Bismuth has the particularity to be a semimetal with rel-atively low carrier concentrations so that the dominantmechanism for heat conduction at low temperatures is via phonons [7, 11]. Together with solid helium [12] andNaF [13], it is one of the rare materials that are suffi-ciently isotopically pure so that second sound could beobserved. The degree of physical and chemical perfectionthat has been achieved in Bi crystals is so high that also transitions between the various regimes have been experi-mentally observed with the increase of the (yet cryogenic)temperature : from heat transport via ballistic phonons,to the regime of Poiseuille’s flow with second sound, tothe diffusive (Fourier) propagation [7].Neither the conditions for the occurrence of the hy-drodynamic regime nor the transition temperatures haveever been supported by a theoretical work in one of theabove-cited 3-D materials. So far, phonon hydrodynam-ics has been studied with the lattice Boltzmann formal-ism for a model dielectric material with an ad hoc three-phonon collision term and no resistive processes [14].The transition to the kinetic regime has been modeledin group IV semiconductors through a hydrodynamic-to-kinetic switching factor proportional to the ratio of nor- mal and resistive scattering rates [15–17]. A review onadvances in phonon hydrodynamics points out the lackof a widely applicable hydrodynamic model which wouldconsider all of the normal and resistive processes [5].In this work, a major advance consists in accounting forthe phonon repopulation by the normal processes in theframework of the exact variational solution of the Boltz-mann transport equation (V-BTE) [1, 18], coupled tothe ab initio description of anharmonicity : three-phononcollisions turn out to be particularly strong at low tem-peratures, and lead to the creation of new phonons in thedirection of the heat flow (normal processes) which en-hance the heat transport. This induces time- and length-scales over which heat carriers behave collectively andform a hydrodynamic flow that cannot be described byindependent phonons with their own energy and lifetime.In other words the single mode approximation (SMA),valid for the phonon gas model, breaks down. The re-sistive processes are entirely controlled by few phonon-phonon anharmonic processes which lead to the creationof phonons in the direction opposite to the heat flow(Umklapp processes), and by extrinsic processes comingfrom phonon scattering by the sample boundaries.The characterization of heat transport regimes, and inparticular of the transition between the hydrodynamicand kinetic regimes, is the main focus of present work.We discuss several methods to define the hydrodynamicregime, and provide the link with macroscopic scale quan-tities [5] like Knudsen number and drift velocity. Inparticular, we extract the heat wave propagation length(HWPL) directly from the lattice thermal conductiv-ity (LTC) calculated with V-BTE. We argue that ourmethod to extract the HWPL from the LTC in samplesof different sizes, combined to a measurement of the aver- a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n age phonon mean free path, can be viewed as a Gedankenexperiment which could allow to determine the transitionfrom the hydrodynamic to kinetic regime in any material.Several criteria are used in order to identify the hydro-dynamic to kinetic transition. First, the picture of theheat carried by single (uncorrelated) phonons with finitelifetimes, is valid in the kinetic regime only. Thus, asignificant difference between the LTC obtained by a so-lution of V-BTE and the one obtained in the single modeapproximation (SMA-BTE) is the indication that the hy-drodynamic regime is achieved. Second, we compare thethermodynamic averages of the phonon-scattering ratesfor normal and resistive processes Γ n and Γ U , and thehydrodynamic regime occurs when [20]Γ Uav (cid:28) Γ nav . (1)Then, we address the question of the occurrence ofPoiseuille’s flow inside the hydrodynamic regime. Here aswell, various methods are employed, which now accountfor the additional scattering rate by sample boundariesΓ b . We first use Guyer’s conditions [20],Γ Uav < Γ bav < Γ nav (2)and find the temperature interval in which second-soundis calculated to be observable. In the second method,we extract the heat wave propagation length directlyfrom the LTC calculated with V-BTE and compare it tothe sample size which sets the threshold for the secondsound observability. Above the threshold, the heat-waveis damped before reaching the sample boundary.The thermodynamic averages of phonon scatteringrates for normal, Umklapp and boundary collisional pro-cesses that condition the transport regime readΓ iav = (cid:80) ν C ν Γ iν (cid:80) ν C ν (3)where C ν is the specific heat (see below) of the phononmode ν = { q j } and the index i = n, U, b stands for nor-mal, Umklapp and extrinsic (boundary) scattering re-spectively. Besides the scattering rate (or inverse relax-ation time), the quantities characterizing heat transportare the drift velocity v of the heat carriers defined below,and the phonon propagation length λ = v Γ − av which isthe characteristic distance that heat carrying phononscover before damping. As a source of damping, we con-sider, in infinite samples, either Umklapp processes only λ hydro ( ∞ ) = v/ Γ Uav , (4)or their combination with normal processes throughMatthiessen’s rule λ gas ( ∞ ) = v/ (Γ Uav + Γ nav ) . (5)When scattering by sample boundaries is accounted for,the phonon propagation length reads λ ( L Cas ) instead of
FIG. 1: Temperature dependence of the LTC in the bi-nary direction for a single crystal without (solid lines) or with(dotted lines) millimeter-sized sample boundaries (MSSB).Black curves : exact variational calculation (V-BTE). Redcurves : single mode approximation (SMA-BTE). MSSB mod-eled with the wire geometry and L Cas = 9.72 mm [21]. Greendashed lines : LTC extracted by us from expt. of Ref. 22 for
T >
20 K; for
T <
20 K, LTC from a sample having a rect-angular cross-section 8 . × . (Ref. 11). We used theT-independent bulk value of 6 W(K.m) − [2, 3] of the elec-tronic contribution to extract the LTC from the total thermalconductivity of Bi [11]. The error bar in our calculations re-sults from the variation of the geometrical factor F =2 ± λ ( ∞ ) in eqs. 4 −
5, where Casimir’s length L Cas representsthe smallest dimension of the sample or nanostructure.In bismuth the transport is anisotropic and has compo-nents along the trigonal axis ( (cid:107) ) and perpendicular ( ⊥ )to it, i.e. along the binary and bisectrix directions. Thedrift velocity in these directions reads [9] v j = (cid:80) ν C ν c ν j · c ν j (cid:80) ν C ν , (6)where j stands for (cid:107) or ⊥ direction and c ν is thephonon group velocity. In the thermodynamic aver-ages the specific heat of a phonon mode is calculated as C ν = n ν ( n ν + 1) ( (cid:126) ω ν ) k B T , where n stands for the temper-ature (T) dependent Bose-Einstein phonon occupationnumber and ω ν is the phonon frequency.The LTC, third-order anharmonic constants of the nor-mal and Umklapp phonon interactions, and thermody-namical averages have been calculated on a 28x28x28 q -point grid in the Brillouin zone, but for the drift velocitybelow 2 K, which required a 40 × ×
40 grid. Details ofthe calculation are given in the supplemental material.We have used the wire geometry for boundary scatteringwith Casimir’s model, Γ b = n ν ( n ν +1) | c bν | F L
Cas , where c bν is thegroup-velocity in the direction of the smallest dimension.Specularity [11–13] is neglected and F accounts for thegeometrical ratio of L Cas over the finite (yet large) di-mension along the heat transport direction [1–3, 8, 31].
FIG. 2: Temperature dependence of the thermodynamic av-erage of the anharmonic scattering rates for normal and Umk-lapp processes (resp. black solid and blue dashed line) andof the (boundary) extrinsic scattering rates (ESR) (dashedblack and dot-dashed red and green lines). ESR have beencalculated for a wire geometry using L Cas = 3 .
86 mm [25]and L Cas = 9 .
06 mm [26], and with L Cas = 9 .
72 mm [21]as in Fig 1. ESR for L Cas = 9 .
06 and L Cas = 9 .
72 mm arehardly distinguishable on the scale of the figure. The shadedregion corresponds to the temperature interval in which a sec-ond sound peak has been reported, 1.5 K < T <
Varying F by 2 ± κ ⊥ above T=2 K.Remarkably, our calculated LTC shows the same evo-lution as the experimental one over three orders of magni-tude (Fig. 1, resp. black dotted and green dashed lines),and the various regimes of heat transport are excellentlydescribed from ambient temperature down to 2 K. TheLTC increases as T − with the decrease of temperaturedown to 10 K. Then in the absence of scattering otherthan phonon-phonon interaction, the LTC shows an ex-ponential growth below 10 K (black solid line). This be-havior is directly due to the weakness of resistive (Umk-lapp) processes.The account for boundary scattering makes the LTCvalue remain finite even in the asymptotic limit. More-over, the theoretical curves satisfactorily explain the ex-perimental behavior of the LTC and in particular, theposition of the conductivity maximum, T max , which isfound to be 3.2 K for the 9.72 mm wire, in extremelysatisfactory agreement with the maximum at 3.6 K ob-served in experiment (Fig. 1, resp. black dotted andgreen dashed lines). Further decrease of temperatureleads to a decrease of the LTC with a decay law grad-ually approaching the T behavior expected for a regimein which boundary scattering dominates. Temperature, K -2 -1 L h , µ m Expt. λ hydro = v ⊥ / Γ Uav λ gas = v ⊥ /( Γ Uav + Γ nav )L h - wires - SMAL h - wires - VAR Binary direction
FIG. 3: Heat wave propagation length L h extracted from theLTC calculations in the binary direction as a function of tem-perature. Solid line with black filled disks : L h obtained withV-BTE, accounting for phonon repopulation. Solid line withempty circles : L h obtained with SMA-BTE. Black dashedline : phonon propagation length λ hydro of eq. 4. Black dot-ted line : phonon propagation length λ gas of eq. 5. The redline segment marks the ranges of temperatures, from 3.0 K to3.48 K, and sample dimension, 9.06 mm, in which a secondsound peak has been reported in the binary direction [7]. The first sign of the transition from the kinetic tohydrodynamic regime around 3 K in infinite samples isdemonstrated in Fig. 1 by a large ( > ) difference be-tween our V- and SMA-BTE results for the LTC (resp.black and red solid lines). This result shows that therepopulation of phonon states due to normal processesplays an important role, invalidating the SMA picture inwhich individual phonons have lifetimes and propagationlengths determined by all of the collisional processes (nor-mal and Umklapp, eq. 5). The same conclusion can bedrawn by considering Fig. 2 where, around 3 K, normalprocesses dominate over the resistive ones (Umklapp) bymore than one order of magnitude, so that eq. 1 is ful-filled.The same difference in the LTC between V- and SMA-BTE is found in presence of sample boundaries (Fig. 1,resp. black and red dotted lines) and, remarkably,the average extrinsic scattering rate Γ bav calculated withCasimir’s length L Cas = 3 .
86 mm (Fig. 2, black dottedline) lays in between the average normal Γ nav and Umk-lapp Γ
Uav scattering rates and thus, satisfy the criterionof eq. 2 for the existence of Poiseuille’s flow and secondsound observability [20]. The temperature interval cal-culated with eq. 2 is 1.5 K < T < < T < L Cas = 9 .
06 mm in the binary direction (red dot-dashedline), the calculated interval is 1.3 K < T < < T < L h that we define by the criterion : κ ( T, L
Cas = L h ) = κ ( T, ∞ ) /e, (7)where κ ( T, ∞ ) denotes the LTC obtained for an infinitesample at a given temperature, and κ ( T, L
Cas ) denotesthe LTC obtained for a sample of finite dimension. Theextracted HWPL L h is the cylindrical wire diameter L Cas needed to reduce κ ( T, ∞ ) by e (Fig. 3, filled disks, andsupplemental material for the trigonal direction [34]).Remarkably, at low temperatures, L h is found to beclose to the phonon propagation length computed withUmklapp processes only (eq. 4). These resistive processesdamp the heat wave, thus defining the wave travelingdistance between the instant of heat wave generation tocomplete diffusion. A strong presence of normal pro-cesses, in turn, favors heat conduction and second soundbehavior. With the increase of temperature, L h becomesclose to the phonon propagation length accounting forboth Umklapp and normal processes of eq. 5, i.e. of anuncorrelated phonon gas (empty circles). We see that thebehavior of L h as a function of temperature is the finger-print of the transition from the hydrodynamic to kineticregime. The temperature range and sample dimension inwhich observations of second sound are available in thebinary direction (red line segment) are in extremely sat-isfactory agreement with the calculations, which supportthe occurrence of second sound at 3.0 K for a 9.72 mmwire. Fig. 3 enables us also to predict the occurrence ofsecond sound at other temperatures and sample sizes, forinstance at 4.1 K in a 1 mm size wire.We emphasize that L h is a measurable quantity, pro-vided that LTC can be measured in samples of many dif-ferent sizes, including very large ones. In that sense, theresults presented in Fig. 3 can be viewed as a Gedankenexperiment in which : (i) First, one need to determine theheat wave propagation length from the thermal conduc-tivity measured in samples of different sizes, as describedwith eq. 7. (ii)
Secondly, its combination with a mea-surement of the average phonon mean free path in a bulksample, given by eq. 5, as done, for example, in attenu-ation measurement experiments [35], could, in principle,lead to the identification of the temperature and samplesize ranges in which Poiseuille’s flow occurs.We turn to the characterization of Poiseuille’s flow,defined in the previous paragraph as the range of tem-peratures and propagation lengths where L h and λ hydro are close to each other. For this purpose we use common (a)(b) FIG. 4: Heat flow characteristics in Bi as a function of thetemperature. Panel (a) : drift velocity v in the binary ( ⊥ )and trigonal ( (cid:107) ) directions (resp. black solid and red dashedlines). Symbol : saturated second-sound velocity measuredat 3 K [7]. Panel (b) : Knudsen number L h / L Cas for a wireof Casimir’s length L Cas = 9 .
72 mm (black solid line). Theratio of the phonon propagation length in the hydrodynamic(resp. gas) regime over L Cas is also given (resp. black dashedand dotted lines). The shaded region 3.0 K < T < hydrodynamic quantities : Knudsen number and driftvelocity. The former is defined as the ratio between theHWPL and the characteristic dimension of transport, Kn = L h L Cas . (8)Interestingly, the transition between the hydrodynamicand kinetic regime is found for a calculated Knudsennumber Kn ≈ .
58 at T = 3.5 K in agreement with thecriteria of phonon hydrodynamics 0 . (cid:46) Kn (cid:46)
10 [5](Fig. 4, bottom panel, black solid line). Our drift veloc-ity calculated with eq. 6 in the binary direction shows amaximum of v ⊥ = 770 m/s at 3.0 K whose value matcheswell with the second sound velocity v = 780 m/s mea-sured in Ref. 7. At variance with the experiment [7], wefind however a dependence on the propagation direction(top panel, black solid and red dashed lines).In conclusion, repopulation of phonon states by nor-mal processes turns out to be particularly strong at lowtemperatures and leads to the occurrence of the hydrody-namic regime in bismuth. We have shown that this effectis remarkably well accounted for in the exact (variational)solution of the BTE. This enables us to extract fromthe lattice thermal conductivity a characteristic length,the heat wave propagation length, whose behavior as afunction of temperature, when compared to the phononmean free path, is a fingerprint of the hydrodynamic tokinetic transition regime. We propose our method as aGedanken experiment. It provides an alternative to astandard heat pulse propagation technique used in liter-ature. Our calculated HWPL matches with macroscopicsample dimensions in which second sound was experi-mentally observed [7]. Finally, our calculated HWPL,Knudsen number and drift velocity allow to make thelink with phonon hydrodynamics.We acknowledge discussions with A. Cepellotti andA. McGaughey. Support from the DGA (France), fromthe Chaire ´Energie of the ´Ecole Polytechnique, from theprogram NEEDS-Mat´eriaux (France) and from ANR-10-LABX-0039-PALM (project Femtonic) is gratefully ac-knowledged. Computer time was granted by ´Ecole Poly-technique through the LLR-LSI project and by GENCI(project No. 2210). ∗ Electronic address: [email protected][1] D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. H.Fan, K. E. Goodson, P. Keblinski, W. P. King, G. D.Mahan, A. Majumdar, et al., Applied Physics Reviews , 011305 (2015).[2] S. Volz, J. Ordonez-Miranda, A. Shchepetov, M. Prun-nila, J. Ahopelto, T. Pezeril, G. Vaudel, V. Gusev, P. Ru-ello, E. Weig, et al., Eur. Phys. J. B , 15 (2016).[3] C. W. Chang, D. Okawa, H. Garcia, A. Majumdar, andA. Zettl, Phys. Rev. Lett. , 075903 (2008).[4] N. Yand, G. Zhang, and B. Li, Nano Today , 85 (2010).[5] Y. Guo and M. Wang, Physics Reports , 1 (2015).[6] J. Zhang, X. Huang, Y. Yue, J. Wang, and X. Wang,Phys. Rev. B , 235416 (2011).[7] G. Fugallo, A. Cepellotti, L. Paulatto, M. Lazzeri,N. Marzari, and F. Mauri, Nano Lett. , 6109 (2014).[8] S. Lee, D. Broido, K. Esfarjani, and G. Chen, NatureCommunications , 6290 (2015).[9] A. Cepellotti, G. Fugallo, L. Paulatto, M. Lazzeri,F. Mauri, and N. Marzari, Nature Communications ,6400 (2015).[7] V. Narayanamurti and R. Dynes, Phys. Rev. Lett. ,1461 (1972).[11] J.-P. Issi, Aus. J. Phys. , 585 (1979).[12] C. C. Ackerman, B. Bertman, H. A. Fairbank, and R. A.Guyer, Phys. Rev. Lett. , 789 (1966).[13] H. E. Jackson, C. T. Walker, and T. F. McNelly, Phys.Rev. Lett. , 26 (1970).[14] R. Guyer, Phys. Rev. E , 4596 (1994).[15] C. de Tomas, A. Cantarero, A. F. Lopeandia, and F. X.Alvarez, J. Appl. Phys. , 164314 (2014).[16] C. de Tomas, A. Cantarero, A. F. Lopeandia, and F. X. Alvarez, Proc. R. Soc. A: Math. Phys. Eng. Sci. ,20140371 (2014).[17] C. de Tomas, A. Cantarero, A. F. Lopeandia, and F. X.Alvarez, J. Appl. Phys. , 134305 (2015).[18] M. Omini and A. Sparavigna, Physica B , 101 (1995).[1] G. Fugallo, M. Lazzeri, L. Paulatto, and F. Mauri, Phys.Rev. B , 045430 (2013).[20] R. A. Guyer and J. A. Krumhansl, Phys. Rev. , 778(1966).[21] The longest dimension is along the binary direction. Weused L Cas = 2 (cid:113) l l π for this rectangular wire, whoseexperimental cross-section is defined by the lengths l and l . Choosing L cas = √ l l = 8 . , 765(1974).[2] M. Markov, J. Sjakste, G. Fugallo, L. Paulatto,M. Lazzeri, F. Mauri, and N. Vast, Phys. Rev. B ,064301 (2016).[3] M. Markov, Ph.D. thesis, Universit´e Paris-Saclay, ´EcolePolytechnique, Palaiseau, France (2016), , URL https://pastel.archives-ouvertes.fr/tel-01438827 .[25] The longest dimension is along the trigonal axis. For thecross-section, we have used a circular one oriented in (bi-sectrix, binary) plane with L Cas = d , where d = 3 .
86 mmis the experimental diameter of the cylindrical wire forsample 1. Choosing a spherical grain with L Cas = 3 . L Cas = d , where d = 9 . L Cas = 9 . , 130 (1955).[12] A. Rajabpour, S. V. Allaei, Y. Chalopin, F. Kowsary,and S. Volz, J. Appl. Phys. , 113529 (2011).[13] O. Bourgeois, D. Tainoff, A. Tavakoli, Y. Liu, C. Blanc,M. Boukhari, A. Barski, and E. Hadji, Comptes RendusPhysique , 1154 (2016).[8] A. Sparavigna, Phys. Rev. B , 064305 (2002).[31] See Supplemental Material for detailed discussion, whichincludes Refs. [9, 10].[32] The idea of using LTC obtained with V-BTE to extractthe phonon mean free paths was recently discussed inRef. [33]. The method of Ref. [33] is different from oureq.7.[33] V. Chiloyan, L. P. Zeng, S. Huberman, A. A. Maznev,K. A. Nelson, and G. Chen, Phys. Rev. B , 155201(2016).[34] See Supplemental Material for detailed discussion, whichincludes Refs. [5, 6].[35] R. Legrand, A. Huynh, B. Jusserand, B. Perrin, andA. Lemaitre, Phys. Rev. B , 184304 (2016).[9] M. Park, I.-H. Lee, and Y.-S. Kim, J. Appl. Phys. ,043514 (2014).[10] A. Sparavigna, Phys. Rev. B , 174301 (2002).[5] A. Collaudin, Ph.D. thesis, Universit´e Pierre et MarieCURIE Paris VI (France) (2014).[6] I. Y. Korenblit, M. E. Kuznetsov, V. M. Muzhdaba, and S. S. Shalyt, Sov. Physics JETP , 1009 (1970). Supplemental material
We provide supplemental material to discuss convergence issues, the modeling of phonon-boundary scattering, andshow that phonon repopulation by normal processes in bismuth at low temperatures leads to the occurrence of thehydrodynamic regime also in the trigonal direction.
DETAILS OF THE CALCULATIONS
The lattice thermal conductivity has been computed with the linearized Boltzmann transport equation and thevariational method (VAR-BTE) on a 28x28x28 q -point grid in the BZ with a Gaussian broadening of the detailedbalance condition taken to be σ =1 cm [1]. Details of the calculation have been reported in Refs. 2, 3. Third-orderanharmonic constants of the normal and Umklapp phonon interactions have been computed on a 4 × × q -pointgrid in the BZ. The 4x4x4 grid amounts to 95 irreducible ( q , q , q ) phonon-triplets [4], where q i , i=1,3 are phononwavevectors, and with q = q ± q + G . G is a vector of the reciprocal lattice. The third-order anharmonic constantswere Fourier-interpolated on the 28 × ×
28 denser grid necessary for converged integrations in Γ. The convergenceof Γ U , Γ n , λ gas , λ hydro and of κ computed within VAR-BTE has been checked at T = 2 K on a 34 × ×
34 grid. Thethermodynamic averages were calculated on the grid 28x28x28 and the convergence was checked on the 34x34x34grid. Below 2 K however, the drift velocity of Fig. 4(a) of the main text required a 40 × ×
40 grid.
LATTICE THERMAL CONDUCTIVITY IN THE TRIGONAL DIRECTION
Fig. S 1: Temperature dependence of the lattice thermal conductivity (LTC) in the trigonal direction for a single crystal without(solid lines) or with (dotted lines) millimeter-sized sample boundaries (MSSB). Black curves : exact variational calculation(V-BTE). Red curves : single mode approximation (SMA-BTE). MSSB modeled with the wire geometry and L Cas = 2.56 mm.Green dashed lines : LTC extracted by us from expt. of Ref. 5 for
T >
10 K; for
T <
10 K, LTC from a sample having a circularcross-section 2 .
56 mm (Ref. 6). We used the T-independent bulk value of 3 W(K.m) − [2, 3] of the electronic contributionto extract the LTC from the total thermal conductivity of Bi [5]. Error bars in our calculations represent variation of thegeometrical factor F from 1 to 3, i.e. F =2 ±
1, in Casimir’s model.
Fig. S 1 is equivalent to Fig. 1 of the main text. It shows the lattice thermal conductivity in the trigonal directioncalculated using the exact solution of the BTE (black curves), the single mode approximation (red curves) and exper-imental lattice thermal conductivity obtained from the measured total conductivity from Ref. 5 with the subtractedT-independent bulk value of 3 W(K.m) − [2, 3] of the electronic contribution (green dashed curve). At low temper-atures, T <
10 K, data were extracted from Ref. 6 for the cylindrical wire with diameter d = 2.56 mm. Solid linesrepresent the calculations with phonon-phonon scattering only. Dotted lines represent the calculations accounting forthe phonon scattering by boundaries in addition to the phonon-phonon scattering. We use L Cas = 2 .
56 mm and thegeometrical F = 2. Varying F from 1 to 3, we introduce an error bar in our calculations. Fig. S 2: Temperature dependence of the thermodynamic average of the anharmonic scattering rates for normal and Umklappprocesses (resp. black solid and blue dashed line) and of the (boundary) extrinsic scattering rates (ESR) (green solid anddot-dashed lines). ESR have been calculated for a wire (green dotted-dashed lines) and a spherical grain (green solid lines)geometry with effective sizes L Cas = 3 .
86 mm (dark green) and L Cas = 9 .
72 mm (light green). Wires are oriented in thetrigonal and binary directions respectively. The difference between two geometries is small. The shaded region corresponds tothe temperature interval in which a second sound peak has been reported, 1.5 K < T <
Temperature, K -2 -1 L h , µ m Expt. λ hydro = v || / Γ Uav λ gas = v || /( Γ Uav + Γ nav )L h - wires - SMAL h - wires - VAR Trigonal direction
Fig. S 3: Heat wave propagation length L h extracted from the LTC calculations in the trigonal direction as a function oftemperature. Solid line with black filled disks : L h obtained with V-BTE, accounting for phonon repopulation. Solid line withempty circles : L h obtained with SMA-BTE. Black dashed line : phonon propagation length λ hydro . Black dotted line : phononpropagation length λ gas . The red line segment marks the ranges of temperatures, from 1.5 K to 3.5 K, and sample dimension,3.86 mm, in which a second sound peak has been reported in the trigonal direction [7]. SAMPLE GEOMETRY
In our study, for the sake of unity, we present only the results for the wire geometry in all of the figures. The wiregeometry corresponds to the one in which the thermal conductivity has been measured (Refs. 11 and 22). Phonon-boundary scattering is thus defined by the shortest dimension of the sample L Cas that is assumed to be perpendicularto the transport direction, and the wire length l is assumed to be much larger than L Cas in the calculations. However,the conclusions based on the results presented in Fig. 2, about the occurrence of the hydrodynamic regime, are exactlythe same with the grain geometry, i.e. if the sample length in the heat propagation direction is taken to be l = L Cas .Indeed, in Fig. S 2, the lines for grains and for nanowires (green solid lines and green dotted-dashed lines) are almostindistinguishable.In samples in which second sound has been measured (Ref. 7), the samples were cut in the heat transport direction.The finite length along the transport direction in real samples determined whether the heat pulse can be detectedor not. If the propagation length is smaller than the wire length, the heat pulse reaching the sample edge is alreadydamped and, thus, can not be registered by a receiver. While in the opposite case, the temperature wave is stillobservable and can be detected. In Fig. 3 of the main text, we evaluate the propagation length of the temperaturewave i.e. the distance at which the amplitude of wave is decreased by a factor e . When λ > l , the second sound isnot dumped at length l and,thus, can be registered by the receiver. HYDRODYNAMIC REGIME IN THE TRIGONAL DIRECTION
Fig. S 3 is equivalent to Fig. 3 of the main text, with the heat wave propagation length (HWPL) L h computed fromthe lattice thermal conductivity (LTC) in the trigonal direction with eq. 7 of the main text. At low temperatures theHWPL L h is found to be close to the phonon propagation length, λ hydro , computed with Umklapp processes only,while with the increase of temperature, L h becomes close to the phonon propagation length, λ gas , accounting for bothUmklapp and normal processes, i.e. of an uncorrelated phonon gas (empty circles). Latter quantities λ hydro and λ gas are computed resp. with eqs. 4 and 5 of the main text. SURFACE ROUGHNESS AND SPECULARITY
In Figs. 1 (main text) and S 1, we used Casimir’s model with respectively L Cas = 9 .
72 mm or L Cas = 2 .
56 mmand the geometrical factor F = 2. The Casimir model has been extensively and successfully employed with thesetwo parameters for the description of phonon-boundary scattering in a wide variety of materials (see Refs. 1, 8 fordiamond; Ref. 9 for silicon; Ref. 10 for silicon carbide), including our recent calculation for polycrystalline thin filmsof bismuth [2]. When larger than the unity, F accounts for the geometrical ratio of L Cas over the finite (yet large)dimension along the heat transport direction [1–3, 8] and also, to a smaller extent, to a (small) specularity of anotherwise almost completely diffusive (rough) surface [11–13]. To demonstrate that our prediction does not dependon the value of the geometrical factor we change F from 1 to 3, increasing and decreasing the role of boundaryscattering correspondingly. We set these values as an error bar for our calculations. ∗ Electronic address: [email protected][1] G. Fugallo, M. Lazzeri, L. Paulatto, and F. Mauri, Phys. Rev. B , 045430 (2013).[2] M. Markov, J. Sjakste, G. Fugallo, L. Paulatto, M. Lazzeri, F. Mauri, and N. Vast, Phys. Rev. B , 064301 (2016).[3] M. Markov, Ph.D. thesis, Universit´e Paris-Saclay, ´Ecole Polytechnique, Palaiseau, France (2016), , URL https://pastel.archives-ouvertes.fr/tel-01438827 .[4] L. Paulatto, F. Mauri, and M. Lazzeri, Phys. Rev. B , 214303 (2013).[5] A. Collaudin, Ph.D. thesis, Universit´e Pierre et Marie CURIE Paris VI (France) (2014).[6] I. Y. Korenblit, M. E. Kuznetsov, V. M. Muzhdaba, and S. S. Shalyt, Sov. Physics JETP , 1009 (1970).[7] V. Narayanamurti and R. Dynes, Phys. Rev. Lett. , 1461 (1972).[8] A. Sparavigna, Phys. Rev. B , 064305 (2002).[9] M. Park, I.-H. Lee, and Y.-S. Kim, J. Appl. Phys. , 043514 (2014).[10] A. Sparavigna, Phys. Rev. B , 174301 (2002).[11] R. Berman, E. Foster, and J. Ziman, Proc. R. Soc. Lond. Ser. A , 130 (1955).[12] A. Rajabpour, S. V. Allaei, Y. Chalopin, F. Kowsary, and S. Volz, J. Appl. Phys. , 113529 (2011).[13] O. Bourgeois, D. Tainoff, A. Tavakoli, Y. Liu, C. Blanc, M. Boukhari, A. Barski, and E. Hadji, Comptes Rendus Physique17