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PHYS I CS

1Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551,Japan. 2Centre de Physique Théorique, École Polytechnique, CNRS, Université Paris-Saclay, F-91128 Palaiseau, France. 3Collège de France, 11 place Marcelin Berthelot,75005 Paris, France. 4The Institute for Solid State Physics, University of Tokyo, Kashiwa,Chiba 277-8581, Japan. 5Graduate School of Material Science, University of Hyogo,Kamigori, Hyogo 678-1297, Japan. 6Laboratoire Physique et Etude de Matériaux(CNRS-UPMC), ESPCI Paris, PSL ResearchUniversity, 75005 Paris, France. 7II. PhysikalischesInstitut, Universität zu Köln, 50937 Köln, Germany.*Corresponding author. Email: machida@ap.titech.ac.jp (Y.M.); kamran.behnia@espci.fr (K.B.)†Present address: Department of Physics, Gakushuin University, Toshima, Tokyo171-8588, Japan.‡Present address: Division of Materials Physics, Osaka University, Toyonaka, Osaka560-8533, Japan.

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Observation of Poiseuille flow of phononsin black phosphorusYo Machida1*†, Alaska Subedi2,3, Kazuto Akiba4, Atsushi Miyake4, Masashi Tokunaga4,Yuichi Akahama5, Koichi Izawa1‡, Kamran Behnia6,7*

The travel of heat in insulators is commonly pictured as a flow of phonons scattered along their individual trajec-tory. In rare circumstances, momentum-conserving collision events dominate, and thermal transport becomeshydrodynamic. One of these cases, dubbed the Poiseuille flow of phonons, can occur in a temperature windowjust below the peak temperature of thermal conductivity. We report on a study of heat flow in bulk black phos-phorus between 0.1 and 80 K. We find a thermal conductivity showing a faster than cubic temperature depen-dence between 5 and 12 K. Consequently, the effective phonon mean free path shows a nonmonotonic temperaturedependence at the onset of the ballistic regime, with a size-dependent Knudsen minimum. These are hallmarks ofPoiseuille flow previously observed in a handful of solids. Comparing the phonon dispersion in black phosphorusand silicon, we show that the phase space for normal scattering events in black phosphorus ismuch larger. Our resultsimply that themost important requirement for the emergence of Poiseuille flow is the facility ofmomentumexchangebetween acoustic phonon branches. Proximity to a structural transition can be beneficial for the emergence of thisbehavior in clean systems, even when they do not exceed silicon in purity.

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INTRODUCTIONThe finite thermal resistivity of an insulating solid is a manifestationof the anharmonicity of the underlying lattice. The most commonapproach to calculate the thermal conductivity of a given solid isto assume the harmonic approximation and introduce a finite life-time for phonons that captures the effects beyond the harmonic approx-imation. In the Boltzmann-Peierls picture, heat-carrying phonons arescattered by other phonons or by crystal imperfections and boundaries.The rate of collisions relaxing the momentum of the traveling phononset the magnitude of thermal resistivity. Only a subset of phonon-phonon scattering events, called a Umklapp (or U) process, degradesthe heat current. The initial and final momenta in a U process differby a multiple of a reciprocal lattice vector. On the other hand, a normal(or N) phonon-phonon collision cannot lead to thermal resistance inthe absence of Umklapp scattering. In an infinite sample, if all collisionswere normal, then one expects the flow of phonons to be undamped inthe absence of any external field applied to sustain it (1).

Black phosphorus (BP) has attracted much recent attention as acleavable solid with a promising exfoliating potential toward two-dimensional phosphorene (2, 3). It has an orthorhombic crystal structurewith puckered honeycomb layers in the ac planes and van der Waalscoupling between the layers (Fig. 1, A and B). Unlike graphene, it has atunable direct bandgap ranging from0.3 eV in bulk to 2 eV in amono-layer (4). These features make BP a promising material for applica-tions. In addition, the presence of a significant in-plane anisotropy may

induce spatially anisotropic transport (5) absent in other graphene-likematerials. Although electronic conduction in BP has been extensively in-vestigated (6–9), few studies have been devoted to its thermal transport(10–15). They are restricted to relatively high temperatures anddidnotexplore the low-temperature range, the focus of the present study.

Here, we report on a study of thermal conductivity along the a and caxes of BP single crystals and establish the magnitude and temperaturedependence of the thermal conductivity down to 0.1 K.We find a mod-erately anisotropic thermal conductivitymainly reflecting the anisotropyof the sound velocity. Unexpectedly, we find that below its peak tem-perature, the thermal conductivity evolves faster than the ballistic T 3.This feature, combined with size dependence of the Knudsen minimumin the effective phononmean free path, provides compelling evidence forPoiseuille phonon flow, a feature previously restricted to a handful ofsolids (16). We argue that this arises because of the peculiar phonondispersions enhancing the phase space for normal scattering events inBP. This observation has important consequences for the ongoing re-search in finding the signatures of hydrodynamicphonon flow in crystals.

RESULTSOur samples are insulators, as seen in Fig. 1C, which shows the tem-perature dependence of the electrical resistivity r along the a andc axes. For both directions, r shows a peak around 250 K, reflecting thechange in the number of thermally excited carriers across more thanone gap. The magnitude of the resistivity, its temperature dependence,and its anisotropy are in agreement with an early study (6). Below 50 K,r shows an activated behavior, as seen in the upper inset of Fig. 1C.The distance in the energy between the top of the valence band andthe acceptor levels set the activation energy of hole-like carriers (6).Using the Arrhenius equation r = exp(Eg/2kBT) between 40 and 20 K, onefinds an activation energy of Eg ~ 15 meV, in agreement with the pre-vious result (6). Below 20 K, the temperature dependence of r becomesappreciably weaker, indicating that the system enters the variable rangehopping (VRH) regime where the electrical transport is governed bythe carriers trapped in local defects hopping from one trap to another.One can describe the resistivity by the expression rº exp[(T/T0)

−1/(d+1)]

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Fig. 1. Crystal structure and resistivity. (A) Side view of the crystal structure of BP. (B) Top view of single-layer BP. (C) Temperature dependence of the electricalresistivity r of BP along the a and c axes in a logarithmic scale. The upper inset shows an activated behavior of r for T > 20 K, where solid lines represent an Arrheniusbehavior. The lower inset shows the ratio ra/rc, which quantifies charge transport anisotropy. At low temperatures, VRH governs, as shown in the bottom panels wherer is plotted against T −1/2 (D), T −1/3 (E), and T−1/4(F).

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in a system with dimension d. As seen in Fig. 1 (D to F), it is hard todistinguish the hopping space dimensionality from our results. Theresistivity anisotropy ratio, ra/rc (lower inset of Fig. 1C), shows lit-tle temperature dependence in the activated regime. The ratio, al-most constant (~3.5) in the activated regime and reflecting theintrinsic anisotropy in the carrier mobility (6), sharply increases inthe VRH regime and attains ~10.

Because of the extremely low electrical conductivity, phonons al-most entirely transport heat, andwe are going to focus on them. Figure 2Ashows the temperature dependence of the thermal conductivity,k, alongthe a and c axes in a BP single crystal. Four different groups recentlymeasured the thermal conductivity of bulk BP (12–15). Sun et al. (13)found a large and anisotropic thermal conductivity, much larger thanwhat was found in an early polycrystal study by Slack (10) and asingle-crystal study by other groups (12, 15). As one can see in the figure,our data for the samples of ~100-mm thickness (solid circles) is in goodagreement with the data reported by Sun et al. (13), including theanisotropy essentially due to the sound velocity (see the SupplementaryMaterials for details). Both sets of data point to an intrinsic bulk con-ductivity much larger than what was found in a polycrystal (10). Thelarge thickness dependence of the thermal conductivity in BP foundby our study (see below), in agreement with what was recently reported

Machida et al., Sci. Adv. 2018;4 : eaat3374 22 June 2018

by Smith et al. (14), provides a clue to the striking discrepancy betweenthe single-crystal and polycrystal data. If, even at temperaturesexceeding 100K, the phonon thermal conductivity is affected by sam-ple dimensions, then the attenuation of thermal conductivity in thepresence of grain boundaries is not surprising.

In the absence of anisotropy, the lattice thermal conductivity canbe represented by the following simple expression

k ¼ 13C⟨v⟩lph ð1Þ

where C is the phonon specific heat, ⟨v⟩ is the mean phonon velocity,and lph is the phonon mean free path. Note that the 1/3 prefactor, aconsequence of averaging over the whole solid angle in isotropic me-dium, can be different in the presence of anisotropy. At temperaturesexceeding the Debye temperature, where the Dulong-Petit law applies,the specific heat saturates to a constant value. In this regime, the tem-perature dependence of the phonon mean free path, which becomesshorter with increasing temperature, governs the temperature depen-dence of the thermal conductivity. At the other extreme, that is, at lowtemperature, the mean free path saturates to its maximum magni-tude, of the order of the sample dimensions, and the T 3 behavior of

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Fig. 2. Thermal conductivity. (A) Temperature dependence of the thermal conductivity along the two high-symmetry axes for the samples with a comparable thickness(solid circles). The a-axis thermal conductivity measurement is extended down to 0.1 K using the thicker sample (open circles). We compared our data to three sets ofprevious reports (10, 12, 13). We also show the thermal conductivity of P-doped Si obtained by using the same experimental setup. Top inset illustrates a schematic of themeasurement setup for the thermal conductivity. Bottom inset shows k divided by T 3 as a function of temperature. By plotting the thermal conductivity as a function of T 3,one can see how the ballistic regime, where k º T 3, evolves to the Poiseuille regime where k changes faster than T 3. Such a behavior can be seen in BP (B) and in theliterature data of Bi (20) (C), but it is absent in P-doped Si (D). (E) Schematic representation of Poiseuille flow: the temperature dependence of mean free path of normalscattering lN (dashed-dotted line) and resistive scattering l R (solid line) and the sample dimension d in the three regimes of thermal transport. Poiseuille flow of phononstakes place in the limited temperature range between ballistic and diffusive transport when the inequality of lN ≪ d ≪ lR is fulfilled. In the center of samples, the phononmean free path (dashed line) becomes lphº d2/lN in these conditions. The three small panels represent three distinct scattering mechanisms. Normal scattering betweentwo phonons does not lead to any loss of momentum and does not degrade the heat current. In contrast, Umklapp scattering and collision with impurities lead to a loss inheat current and amplify thermal resistance. In any finite-size sample, phonons are also scattered by the boundaries.

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the phonon specific heat sets the temperature dependence of ther-mal conductivity. Between these two extreme regimes, dubbed thekinetic (at high temperature) and the ballistic (at low temperature), thethermal conductivity passes through a peak. As demonstrated by ex-tensive studies on LiF (17), introducing isotopic impurities dampskmax, the peak value of the thermal conductivity. On the other hand,decreasing the size of the sample shifts the position of kmax with littleeffect on the magnitude of thermal conductivity in temperatures ex-ceeding the peak.

Our key finding is the observation of a k evolving faster than T 3

just below the peak temperature in BP. This can be seen in Fig. 2B. Asa consequence, k divided by T 3 shows a maximum or a shoulderaround 10 K (see the inset of Fig. 2A). Upon further cooling, the heattransport eventually becomes ballistic as evidenced by the T3 behav-ior of k (saturation of k/T 3) observed in the low-temperature data ofthe thickest sample (open circles in Fig. 2A). As a consequence, theeffective phononmean free path, lph, extracted from Eq. 1, presents alocal maximum (around 10 K) and a local minimum (around 4 K;solid circles in Fig. 3, C and D). Here, we calculated lph using ⟨va⟩ =0.536 × 104 m/s, ⟨vc⟩ = 0.354 × 104 m/s [these are calculated velocitiesin agreement with previous calculations (18) and neutron scatteringmeasurements (19)], and the measured specific heat of the crystal(see the Supplementary Materials for details). In milliKelvin temper-atures, lph becomes comparable to the sample thickness (see the insetof Fig. 3C).Decades ago, the phononPoiseuille flow (16)was detected in

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Bi (20) by observing these two features, namely, a faster than T3 evolu-tion of k (Fig. 2C) and a peak in the temperature dependence of lph justbelow the maximum thermal conductivity (Fig. 3E). Similar featureshave also been found in crystalline 4He (21), 3He (22), and H (23).

To exclude any experimental artifact, wemeasured the thermal con-ductivity of P-doped Si using the same experimental setup.As seen fromFig. 2A, the temperature dependence of k for P-doped Si conforms towhat was found previously (24). k shows a cubic temperature depen-dence between 2 and 5 K. (Fig. 2D). To show this point more explicitly,we plotted lph that was calculated from Eq. 1 as a function of tem-perature in the inset of Fig. 3D. The physical parameters used in thecalculation are qD = 674 K and ⟨v⟩ = 6700 m/s (25). As expected, lphdoes not show a local maximum, and at low temperature, its magni-tude is in reasonable agreement with the sample dimensions (0.3 mm×1.4 mm × 4.0 mm).

Only Umklapp and impurity scattering events degrade the momen-tumof the traveling phonons. Normal scattering events conserve crystalmomentum and do not contribute to the thermal resistance. Whennormal scattering is sufficiently strong and the momentum-degradingscattering events are significantly rare, normal scattering can even en-hance the heat flow (1, 16). The essential condition for this is given bythe inequality

lN≪ d≪ lR ð2Þ

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Fig. 3. Thickness dependence of thermal conductivity. Temperaturedependence of the thermal conductivity in sampleswithdifferent thicknesses along the a axis (A) andalong the c axis (B). We also show the data from the thin BP flakes (14). Thermal conductivity shows a thickness dependence in the whole temperature range. The extractedphonon mean free path lph in samples with different thicknesses along the a axis (C) and along the c axis (D). The local maximum andminimum are present in all samples.The insets of the (C) and (D) show plots of lph versus T in a logarithmic scale for BP including the thicker sample and for P-doped Si, respectively. (E) The effective phononmean free path of Bi for various average diameters from (20).

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Here, d is a characteristic sample dimension, and lN and lR arenormal and resistive mean free paths, respectively. In such a situation,the phonons lose their momentum only by diffuse boundary scattering,and they flow freely while continuously exchanging momentum in thecenter of substances, analogous to the Poiseuille flow in fluids (Fig. 2E).As a result,moving like a Brownian particle, a phonon traverses the pathlength of the order of lph ~ d2/lN between successive collisions with theboundaries, and in the ideal case, lph can reach a value larger than thecharacteristic sample dimension d. If lN increases with decreasing tem-perature as T−5, then since k ~ C⟨v⟩d 2/lN, one can expect thermal con-ductivity to follow as T 8 (1). Experimentally, in all systems in whichPoiseuille flow has been reported (namely, Bi, solid He, and H), whathas been observed is a faster than T3 thermal evolution of k and a non-monotonic lph(T ) (20–23), not a superlinear size dependence. This isalso the case in the present study on BP.

In the narrow temperature window of the Poiseuille flow, the domi-nance of normal scattering creates an unusual situation where warmingthe system enhances the mean free path because it enforces momen-tumexchange among phonons. In otherwords, the temperature depen-

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dence of thermal conductivity is set by the temperature dependenceof the phonon viscosity and not exclusively by the change in the pop-ulation of thermally excited phonons or the frequency of momentum-relaxing events, as it is the case in the ballistic and diffusive regimes. Atsufficiently low temperatures, the phonons start to behave just like aKnudsen flow of gas (26). This transition from the Poiseuille flow tothe Knudsen flow, lph is accompanied by a mean-free-path minimum,dubbed Knudsenminimum, which occurs when d/lN ~ 1. This is wherethe diffuse boundary scattering rate is effectively increased because ofnormal scattering (Fig. 4A).

By changing the thickness of the samples along the b axis, we alsoexamined a size dependence of the thermal conductivity and con-firmed that the effect is more prominent in larger samples (Fig. 3, Aand B). As expected, in thicker BP samples, the maximum lph is longer.Moreover, as seen in the case of Bi (Fig. 3E) (20), both the local maxi-mum and the local minimum in lph(T) persist (Fig. 3, C and D). Inaddition, note that despite the anisotropic thermal conductivity, lph at-tains values comparable to the sample dimensions at low temperatures,irrespective of the orientation of the heat current, meaning that lph is

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quasi-isotropic along the three principle axes and is set by the aver-age sample size. We derived this from the low anisotropy in the soundvelocity between the ac-plane and out-of-plane orientations despite thelayered structure of BP (see fig. S4 and table S2). The peak thermal con-ductivity in larger BP crystals attains amagnitude as large as 800W/Km,much higher than the values found in thinner samples (Fig. 3, A and B)(14) but still smaller than the maximum thermal conductivity of large(~cm) crystals of Si (3000 W/Km) (24) or Bi (4000 W/Km) (20).

In the original Gurzhi (1) picture, the quadratic dependence of lphon the sample dimension, d, would lead to a superlinear size depen-dence of the thermal conductivity in the Poiseuille regime. As one cansee in Fig. 5B, this is absent in BP, as it was in the case of Bi (20). Now,thePoiseuille flowof phonons is expected to emerge concomitantlywiththe second sound, a wavelike propagation of entropy (16, 27). Both re-quire a peculiar hierarchy of scattering events (16, 28), where the normalscattering rate ismuch larger than the boundary scattering and the lattermuch larger than resistive (Umklapp and impurity) scattering. Its ex-perimental observation in Bi in the same temperature range (29)confirms the interpretation of static thermal conductivity data.

The absence of superlinear size dependence can be traced back tothe expression for phonon kinematic viscosity, proposed by Gurzhi(1), n = vTl

N, which implies that the phonon fluid is non-Newtonian.Here, vT is the phonon thermal velocity whose magnitude is compa-rable to the velocity of the second sound, so that vT ~ ⟨v⟩/

ffiffiffi

3p

= 2000 to3000m/s and lN is estimated to be of the order of 10 mmat the Knudsenminimum. This yields n ~ 0.02 to 0.03 m2/s, several orders of magni-tude larger than the kinematic viscosity of water. This comparison isto be contrasted with the case of electrons in PdCoO2 (30) whose dy-namic viscosity was found to be comparable to the dynamic viscosityof water. Since the phonon viscosity depends on the normal scatter-ing rate and local velocity, it is not a constant number at a given tem-perature. The velocity profile of such a non-Newtonian fluid is muchflatter than parabolic (31). Thismakes the size dependencemuch lesstrivial than in the parabolic andNewtonian case. A recent theoreticalstudy (32) has shown that the thickness dependence can be either

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sublinear or superlinear according to the relative weight of boundaryand normal scattering rates.

There is, however, another signature of the Poiseuille flow in thethickness dependence, which our experiment detected. In the Poiseuilleregime, boundary scattering is rare. Therefore, with increasing thick-ness, phonons can travel a longer distance between successive collisionswith the boundaries, giving rise to an enhancement of the thermal con-ductivity and the maximum in lph with the thickness. On the otherhand, at the onset of the ballistic regime, the fraction of phonons thatcontribute to the diffuse boundary scattering due to numerous normalscattering increases with the thickness. As a consequence, the Knudsenminimum shifts to lower temperatures in the thicker samples (Fig. 4A).One can see both these features in a plot of lph normalized by itsminimum value lmin

ph , clearly along the c axis (Fig. 4C) but not clearlyalong the a axis (Fig. 4B), as it was in the case of Bi (Fig. 4D) (20).We conclude that the Poiseuille flow ismost prominent along the c axis.This suggests that the relative weights of normal and resistive scatteringrates depend on orientation and one needs to go beyond a Boltzmannpicture with a scalar scattering time (33).

The Ziman regime is another hydrodynamic regime (16 ). Here,normal scattering still dominates, but resistive scattering becomes largerthan boundary scattering. In this regime, expected to occur just abovethe peak temperature, one expects k to decrease exponentially with in-creasing temperature before showing a T−1 temperature dependence athigher temperatures. Our measurements do not detect a regime wherethe thermal conductivity follows an exponential behavior in BP. In-stead, all samples showed a robust T −1 behavior in a wide tempera-ture window extending down to 40 K, an order of magnitude lowerthan theDebye temperature (Fig. 5A).Moreover, the slope of theT −1

temperature dependence was larger in the thicker samples (see theinset of Fig. 5A).

Changing the thickness from6(15) to 20(30) mmof the a(c)-axis sam-ple is to multiply the number of parallel heat-conducting phosphorenelayers by a factor of 3(2). Surprisingly, while no change in phonon dis-persion is expected to occur in such a context, the enhancement in ther-mal conductivity persists up to 80 K, the highest temperature of ourrange of study (Fig. 5B). Such a large thickness dependence [first re-ported in submicronic samples by Smith et al. (14)] is unusual, as shownby a comparisonwith themuchmoremodest effect observed inGe (34).In Bi, a comparable enhancement is seen in its Poiseuille regime (Fig.5B), but it rapidly dies awaywith heating. A plausible explanation is thatin BP, some heat carriers remain ballistic at high temperature. Previoustheoretical treatments of the thermal conductivity in graphene andgraphite have argued for the presence of normal processes up to roomtemperature leading to collective phononic modes with a mean freepath on the order of hundreds of micrometers (28, 35, 36). Our obser-vation may also indicate the presence of substantial normal processesabove the peak temperature of thermal conductivity in BP.

DISCUSSIONOur theoretical calculations confirm that phonon-phonon scattering inBP is large at low energies. Figure 6 shows the phonon dispersions ofBP and Si, which agree well with previous calculations (37, 38). Thephonon dispersions of Si show highly dispersive acoustic branches thatlead to high thermal conductivity. The acoustic branches of BP are lessdispersive because of a relatively weaker bonding than the one in Si,which has strong sp3 bonds. This leads to a higher phonon density ofstates (PDOS) in the low-frequency region, which allows for a larger

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phase space for the momentum-conserving normal three-phononscattering events in BP.

Figure 6 (G and H) shows the calculated PDOS of BP and Si. ThePDOS of BP is significantly larger than that of Si in the low-frequencyregion below 100 cm−1 as a consequence of the smaller dispersion ofthe acoustic branches. The phonon scattering processes involvingthe linearly dispersive acoustic modes conserve momentum and arenot resistive to heat flow. The large low-frequency PDOS in BP providesmuchmore phase space for these nondissipative scattering events. Thiscan favor the emergence of Poiseuille flow by establishing the requiredhierarchy of time scales as discussed above.

In summary, by detecting that phonons present a faster than T 3

thermal conductivity in a finite temperature window, we have identifieda hallmark of Poiseuille flow of phonons in BP, which was previouslyseen only in four other solids (39). Theoretical calculations indicate thatthe peculiarities of phonon dispersion lead to a significant enhancementof the phonon-phonon normal scattering in BP compared to an arche-typal insulator such as Si. We note that BP is close to a structural in-stability. Like Bi and other column V elements, its crystal structureresults from a slight distortion of the cubic structure (40, 41). Our resultsindicate that hydrodynamic phonon flow can be observed in a systemclose to a structural instability (39).

MATERIALS AND METHODSSingle crystals of BP were synthesized under high pressure (42). Mag-netophonon resonance, which requires reasonable purity, was observedin a sample from the batch used in the present experiments (7). A singlecrystal of P-doped Si was provided by the Institute of Electronic Ma-terials Technology (Warsaw). The electrical resistivity and the thermalconductivity measurements were performed along the a and c axes ofBP. Each sample has a rectangular shape with edges parallel to the three

Machida et al., Sci. Adv. 2018;4 : eaat3374 22 June 2018

high-symmetry axes: the a, b, and c axes. Lengths of the samples aresummarized in table S1. The a(c)-axis sample has the largest lengthalong the a(c) axis, respectively. The thickness dependence of thethermal conductivity was investigated by using the same sample.The thickness (number of layers) along the b axis was decreased dur-ing the course of the investigations by cleaving.

The thermal conductivity was measured by using a home built sys-tem. We used a standard one-heater-two-thermometers steady-statemethod. A very similar setup was used to measure the thermal conduc-tivity of cuprate (43) and heavy-fermion samples of a comparable size(44, 45). The measurements were performed in the temperature rangebetween 0.1 and 80K. The thermometers, the heater, and the cold fingerwere connected to the sample by gold wires of 25-mm diameter. Thegold wires were attached on the BP sample by Dupont 4922N silverpaste and were soldered by indium on the P-doped Si sample. Thecontact resistances were less than 10 ohm at room temperature. Thetemperature difference generated in the sample by the heater was de-termined by measuring the local temperature with two thermometers(Cernox resistors in the 4He cryostat and RuO2 resistors in the dilutionrefrigerator). The heat loss along manganin wires connected to the twothermometers and the heater is many orders of magnitude lower thanthe thermal current along the sample including the thinner ones (fig.S1). The heat loss by radiation is negligible in the temperature rangeof our study (T < 80 K), since it follows a T4 behavior. The heat lossby residual gas is also negligible because themeasurements were carriedout in an evacuated chamber with a vacuum level better than 10−4 Pa.Moreover, the external surface of the chamber is directly in the he-lium bath at 4.2 K for the measurement above 2 K, so that residualgas is condensed on the cold wall. The experimental uncertainty inthe thermal conductivity arising from the measurements of the ther-mometer resistances is less than 1% in the whole temperature range.The main source of uncertainty results from uncertainty in themeasured

A D

EB

C F

H

G

Fig. 6. Phonon dispersions. Calculated phonon dispersions of BP (A) and Si (D). (B) and (E) show the blow-up in the region below 100 cm−1. (C) and (F) show the Brillouinzones along with the high-symmetry points. The out-of-plane direction in BP corresponds to the path along G-Y. (G) Calculated PDOS of BP and Si. (H) PDOS below 100 cm−1.The acoustic modes of BP are less dispersive than that of Si, leading to higher PDOS in BP in the low-frequency region. At low temperatures, BP should have much more phasespace for the momentum-conserving phonon scattering processes among the linearly dispersive acoustic modes that are relevant for Poiseuille heat flow.

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thickness of the samples, which is about 5% in the thinnest sample. Theincrease of thermal conductivity with the sample thickness, however,dominates over the experimental uncertainty.

The same contacts and wires were used for the electrical resistivitymeasurements by a four-point technique. ADC electric current was ap-plied along the sample using the manganin wire attached to the heater.AKeithley 2182Ananovoltmeter was used for themeasurement of elec-trical voltage. The input impedance of the nanovoltmeter is larger than10 gigohms, which is well above the resistance of the sample even at thelowest temperatures (R ~ 5 megohms).

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SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaat3374/DC1Supplementary Texttable S1. Size of samples.table S2. Phonon velocity along the high-symmetry axes.fig. S1. Thermal resistance of BP samples and manganin.fig. S2. Reproducibility of thermal conductivity data.fig. S3. Specific heat.fig. S4. Phonon dispersions at low energies.References (46–55)

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Acknowledgments: We acknowledge R. Nomura for fruitful discussions. Funding: This workwas supported by the Japan Society for the Promotion of Science Grant-in-Aids KAKENHI

Machida et al., Sci. Adv. 2018;4 : eaat3374 22 June 2018

16K05435, 15H05884, and 17H02920; Fonds ESPCI; and the European Research Councilgrant ERC-319286 QMAC. The calculations were performed at the Swiss National SupercomputingCenter, project s575. Author contributions: Y.M. and K.B. conceived the project and plannedthe research. Y.M. performed transport measurements with the help of K.I. and analyzed data.K.A., A.M., and M.T. carried out the specific heat measurement. Y.A. synthesized samples.A.S. performed theoretical calculations. Y.M., A.S., and K.B. wrote the manuscript. All authorsdiscussed the results and reviewed the manuscript. Competing interests: The authors declarethat they have no competing interests.Data andmaterials availability: All data needed to evaluatethe conclusions in the paper are present in the paper and/or the Supplementary Materials.Additional data related to this paper may be requested from the authors.

Submitted 15 February 2018Accepted 8 May 2018Published 22 June 201810.1126/sciadv.aat3374

Citation: Y. Machida, A. Subedi, K. Akiba, A. Miyake, M. Tokunaga, Y. Akahama, K. Izawa, K. Behnia,Observation of Poiseuille flow of phonons in black phosphorus. Sci. Adv. 4, eaat3374 (2018).

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Observation of Poiseuille flow of phonons in black phosphorus

BehniaYo Machida, Alaska Subedi, Kazuto Akiba, Atsushi Miyake, Masashi Tokunaga, Yuichi Akahama, Koichi Izawa and Kamran

DOI: 10.1126/sciadv.aat3374 (6), eaat3374.4Sci Adv 

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