Abstract
The densification of integrated circuits requires thermal management strategies and high thermal conductivity materials^{1,2,3}. Recent innovations include the development of materials with thermal conduction anisotropy, which can remove hotspots along the fastaxis direction and provide thermal insulation along the slow axis^{4,5}. However, most artificially engineered thermal conductors have anisotropy ratios much smaller than those seen in naturally anisotropic materials. Here we report extremely anisotropic thermal conductors based on largearea van der Waals thin films with random interlayer rotations, which produce a roomtemperature thermal anisotropy ratio close to 900 in MoS_{2}, one of the highest ever reported. This is enabled by the interlayer rotations that impede the throughplane thermal transport, while the longrange intralayer crystallinity maintains high inplane thermal conductivity. We measure ultralow thermal conductivities in the throughplane direction for MoS_{2} (57 ± 3 mW m^{−1} K^{−1}) and WS_{2} (41 ± 3 mW m^{−1} K^{−1}) films, and we quantitatively explain these values using molecular dynamics simulations that reveal onedimensional glasslike thermal transport. Conversely, the inplane thermal conductivity in these MoS_{2} films is close to the singlecrystal value. Covering nanofabricated gold electrodes with our anisotropic films prevents overheating of the electrodes and blocks heat from reaching the device surface. Our work establishes interlayer rotation in crystalline layered materials as a new degree of freedom for engineeringdirected heat transport in solidstate systems.
Main
Anisotropic thermal conductors, in which heat flows faster in one direction than in another, can be characterized by the thermal conductivity anisotropy ratio ρ (= κ_{f}/κ_{s}) between the thermal conductivities along the fast axis (κ_{f}) and the slow axis (κ_{s}). One common way to engineer ρ in fully dense solids is via nanostructuring^{6}, such as fabricating inorganic superlattices^{7,8,9,10,11} or designing symmetrybreaking crystal architectures in a single material^{12}. However, such engineered materials have relatively small ρ values of less than 20 at room temperature. Conversely, some natural crystalline materials have an intrinsically large ρ (for example, graphite^{1} and hexagonal boron nitride (hBN)^{13}, with ρ ≈ 340 and 90 respectively), but they are often difficult to process scalably for thin film integration. Some of these films may also lack the electrical or optical properties necessary for functional device applications.
To design materials with higher ρ that are also suitable for realworld applications, an approach needs to be developed to include three key features: (1) a candidate material with intrinsically high κ_{f}, usually one with efficient phononmediated thermal transport; (2) a method to substantially reduce κ_{s} without affecting κ_{f}; and (3) facile, scalable production and integration of such a material with precise control of the material dimensions (for example, film thickness). Layered van der Waals (vdW) materials such as graphite and transition metal dichalcogenides (TMDs) provide an ideal material platform for designing such highρ materials. They generally have excellent intrinsic inplane thermal conductivities (κ_{}) in singlecrystalline form. Previous studies have also measured recordlow thermal conductivities in nanocrystalline vdW films (for example, WSe_{2})^{14,15,16,17} and heterostructures^{18}. One currently missing capability, however, is an approach for significantly decreasing the outofplane thermal conductivity (κ_{⊥}) while maintaining high κ_{}.
TMD films with interlayer rotations
Here we show that such capability is provided by interlayer rotations, as illustrated in Fig. 1a. Interlayer rotation breaks the throughplane translational symmetry at the atomic scale while retaining inplane longrange crystallinity in each monolayer, thereby providing an effective means for suppressing only κ_{⊥}. For this, we produce largearea TMD films without interlayer registry (referred to here as rTMD), which possess longrange crystallinity inplane and relative lattice rotations at every interlayer interface (Fig. 1b). The films are produced in largescale using two steps: waferscale growth of continuous TMD monolayers (polycrystalline; domain size D) and layerbylayer stacking in vacuum using previously reported methods^{19,20} (see Methods).
The transmission electron microscopy (TEM) diffraction (Fig. 1c, left) and darkfield (inset) images from a representative MoS_{2} monolayer show that it comprises large (D ≈ 1 μm), randomly oriented crystalline domains, which connect laterally to form a continuous polycrystalline film. The vacuum stacking generates rTMD films with a precise layer number (N) and highquality interfaces^{20} with interlayer rotation at every stacked interface. The TEM diffraction pattern of N = 10 rMoS_{2} (Fig. 1c, right) shows a ringlike pattern due to the significant increase in the number of diffraction spots, emphasizing the random crystalline orientation in the throughplane direction. Clean and welldefined interfaces can be seen from the crosssectional highangle annular darkfield scanning TEM (HAADFSTEM) images of rMoS_{2} (Fig. 1d and Extended Data Fig. 1; see Methods). The monolayers have a uniform interlayer spacing d ≈ 6.4 Å (see Methods and Extended Data Fig. 2), which is close to the expected value (6.5 Å) for twisted MoS_{2} multilayers^{21}. Both the growth and stacking steps are scalable, as shown by the optical images of N = 1 and N = 10 rMoS_{2} films (~1 cm^{2}) in Fig. 1e and as demonstrated later in Fig. 4. The largescale uniformity of these films also enables precise and reproducible measurements with minimal spatial variation (Extended Data Fig. 3a–c). In our experiments, rMoS_{2} or rWS_{2} films with different N (up to 22) are transferred onto a sapphire wafer for the measurements of κ_{⊥} or suspended over a holey TEM grid (Fig. 3a) for the measurements of κ_{}.
Ultralow outofplane conductivity
In Fig. 2, we illustrate κ_{⊥} of rTMD films, which is measured using time domain thermoreflectance (TDTR; Fig. 2a, inset; see Methods). A stream of laser pulses (pump) heats up the surface of an Al pad deposited on an rTMD film on sapphire and produces a temperaturesensitive thermoreflectance signal (−V_{in}/V_{out} in Fig. 2a), which is measured with a probe pulse after a varying time delay (for cooling). Figure 2a shows three representative curves measured from rMoS_{2} with N = 1, 2 and 10. The curves flatten with increasing N, suggesting that heat dissipation slows down significantly. Fitting these curves using a heat diffusion model (solid lines, Fig. 2a) enables us to obtain R_{TDTR}, the total thermal resistance between the Al transducer layer and sapphire across the rTMD film for different N.
Figure 2b shows R_{TDTR} versus N for rMoS_{2} (N ≤ 22; solid circles) and rWS_{2} (N ≤ 10; open circles) measured under ambient conditions. We make two observations. First, R_{TDTR} monotonically increases with N. Second, R_{TDTR} varies linearly with N for N ≥ 2. These observations confirm that the throughplane thermal transport in rTMD films is diffusive in nature, in contrast to the ballistic transport reported in fewlayer singlecrystalline MoS_{2} (as thick as 240 nm)^{22,23}. A single parameter κ_{⊥} characterizes the thermal resistance across rMoS_{2} (or rWS_{2}) using the equation R_{TDTR} = R_{0} + Nd/κ_{⊥}\(,\) where Nd is the total film thickness, and R_{0} is a constant corresponding to the total interface resistance (rTMD/Al and rTMD/sapphire; see Extended Data Table 1). Therefore, we apply linear fitting to the data (N ≥ 2) in Fig. 2b (solid lines) to determine κ_{⊥} of rMoS_{2} alone, regardless of the quality and chemical nature of the top and bottom interfaces (see Methods and Extended Data Fig. 3d), which can potentially be altered by metal deposition^{24,25}. We measure κ_{⊥} = 57 ± 3 mW m^{−1} K^{−1} for rMoS_{2} and κ_{⊥} = 41 ± 3 mW m^{−1} K^{−1} for rWS_{2}, which are similar to the lowest value ever observed in a fully dense solid^{15} and comparable to the thermal conductivity of ambient air (~26 mW m^{−1} K^{−1}). These values are approximately two orders of magnitude smaller than those of singlecrystalline MoS_{2} (2–5 W m^{−1} K^{−1})^{26,27} or WS_{2} (~3 W m^{−1} K^{−1})^{27}, despite the rTMD films having the same chemical composition as their bulk counterparts as well as clean interfaces (Fig. 1d). This strongly suggests that the main difference, the interlayer rotation, is the principal cause for the ultralow κ_{⊥} in these rTMD films. Furthermore, repeating similar TDTR experiments on rMoS_{2} at different temperatures (T) produces a relatively flat κ_{⊥}(T) curve (green stars, Fig. 2c), a behaviour different from the decreasing κ_{⊥} with T seen in bulk MoS_{2} (blue squares, lower Fig. 2c).
To understand the microscopic mechanisms that give rise to the dramatic reduction in κ_{⊥}, we carry out homogeneous nonequilibrium molecular dynamics (HNEMD) simulations for the model structures of rMoS_{2} and bulk MoS_{2} (see Methods and Extended Data Table 2)^{28,29,30}. Figure 2c shows κ_{} and κ_{⊥} of rMoS_{2} (solid circles) and bulk MoS_{2} (empty circles) calculated from our molecular dynamics (MD) simulations at different temperatures. The calculated κ_{⊥} drops by a factor of more than 20, from 3.7 ± 0.5 W m^{−1} K^{−1} in bulk MoS_{2} to 0.16 ± 0.04 W m^{−1} K^{−1} in rMoS_{2} at 300 K, and also does not decrease with T, suggesting a transition away from the phononlimited thermal transport mechanism observed in bulk MoS_{2}.
Further analysis of the vibrational spectrum of rMoS_{2} enables us to break down the reduction in κ_{⊥} in terms of the changes in the group velocities (v_{g}) and lifetimes (τ), which are the two factors that determine the thermal conductivity according to Boltzmann transport theory. Figure 2d shows that the v_{g} of the throughplane longitudinal acoustic (LA) mode in rMoS_{2} remains similar to that of bulk MoS_{2} (dashed lines), but the transverse acoustic (TA) modes in rMoS_{2} undergo extreme softening with their v_{g}s practically vanishing^{31}. This implies a loss of resistance with respect to lateral shear, consistent with the lowfrequency Raman spectra of rMoS_{2} films (see Methods and Extended Data Fig. 4) and previous calculations^{32,33}. In addition, the τ of both the LA and the TA modes (Fig. 2e) in rMoS_{2} are more than one order of magnitude smaller than in bulk MoS_{2}, with the LA lifetimes being close to the period of the LA mode vibration (dashed line). From these results, the median mean free path \(\widetilde{l}\) = v_{g}τ for the LA modes is estimated to be 2 nm, suggesting that the heatcarrying LA modes are strongly scattered and that a larger D is unlikely to significantly affect κ_{⊥} since D >> \(\widetilde{l}\). Overall, the strongly suppressed TA modes, indicating a loss of resistance to lateral shear, and the overdamping of the LA modes as the main heat carriers, lead to extremely inefficient thermal transport along the throughplane direction in rMoS_{2}. Along with the nearly temperatureindependent κ_{⊥}, this result suggests a glasslike conduction mechanism.
Inplane conductivity and anisotropy
In contrast to κ_{⊥}, κ_{} remains high in our simulations with only a modest reduction compared to the ideal bulk crystal (less than a factor of two at 300 K; Fig. 2c). This is indeed what we observe in our Raman thermometry experiments as discussed in Fig. 3 (see Methods). We direct a focused laser spot (λ = 532 nm) at the centre of a suspended rMoS_{2} film (Fig. 3a; hole diameter of 5 μm, at 15 torr), which increases its temperature (ΔT) locally upon absorbing laser power P_{abs}. ΔT is then measured using the temperaturesensitive Raman peak shift (Δω) using a sensitivity factor (dω/dT) independently measured for each sample. Examples of Raman spectra measured for N = 2 are shown in Fig. 3b.
Figure 3c plots Δω versus P_{abs} for rMoS_{2} with different N (2 to 5). The slope of the linear fit (d(Δω)/dP_{abs}), which is inversely proportional to the inplane thermal conductance of the film, is plotted in the inset (solid dots; D ≈ 1 μm). We again observe a linear relation, which indicates that κ_{} is well defined for rMoS_{2} independent of N, similar to the case of κ_{⊥}. Using a simple diffusion model with radial symmetry (see Methods and Extended Data Fig. 5c for calculation details and other input measurements), we calculate a high κ_{} value of 50 ± 6 W m^{−1} K^{−1}. This value is similar to the predictions of our MD simulations (Fig. 2c) and consistent with previous reports of Raman thermometry on singlecrystalline monolayer MoS_{2} (35–84 W m^{−1} K^{−1}) at room temperature^{34,35,36,37}. The κ_{} of these rMoS_{2} films is close to the intrinsic phononlimited value despite the films being made of polycrystalline monolayers. This result is further supported by our additional measurements on continuous rMoS_{2} films with a smaller D ≈ 400 nm (open dots, dashed lines, Fig. 3c inset; Extended Data Fig. 5e). The measured value of κ_{} ≈ 44 ± 6 W m^{−1} K^{−1} is within the margin of error of that of the D ≈ 1 μm films. This suggests that the phonon mean free path is smaller than 400 nm, which is consistent with previous reports^{23,38,39,40,41,42}. Furthermore, the measured inplane conductance decreases with T (Extended Data Fig. 6a). This further confirms the phononmediated thermal transport mechanism inplane, in contrast to the glasslike thermal conduction along the throughplane direction.
Our experiments and calculations confirm that interlayer rotation in rTMD films results in highly directional thermal conductivity and a directiondependent thermal conduction mechanism. The rotation significantly reduces κ_{⊥} while maintaining high κ_{}, leading to an ultrahigh value of ρ. We estimate ρ ≈ 880 ± 110 at room temperature for the rMoS_{2} films, higher than that of pyrolytic graphite (PG), which is considered to be one of the most anisotropic thermal conductors (ρ ≈ 340)^{43}. In Fig. 3d, we compare our result with other previously reported values of ρ in phononbased solids^{15,27,43,44} (for a full comparison, see Extended Data Fig. 6b). Compared to a bulk MoS_{2} crystal (ρ ≈ 20)^{27} or disordered layered WSe_{2} (ρ ≈ 30)^{15,44}, rMoS_{2} has a significantly larger ρ because interlayer rotation reduces only κ_{⊥}, as denoted by the grey arrow parallel to the equiκ_{f} lines. This also suggests that ρ can be made even larger by starting with the monolayers of a layered vdW material with a higher κ_{} value such as graphene.
Anisotropic vdW heat diffuser
In Fig. 4, we show that the extreme anisotropy of our rMoS_{2} films can lead to excellent heat dissipation inplane from a heat source and drastic thermal insulation in the throughplane direction. Using the COMSOL software, we perform thermal finiteelement simulations of a 10nmthick rMoS_{2} film draped over a nanoscale Au electrode (15 nm tall, 100 nm wide) on a 50 nm SiO_{2}/Si substrate (Fig. 4a). Our simulation results show that for a fixed power of 8 mW supplied to the Au electrode (near thermal breakdown), the temperature rise ∆T of the Au electrode covered by rMoS_{2} is 50 K lower than that of the bare electrode, thereby demonstrating our film’s effectiveness at spreading heat due to its excellent κ_{} (Fig. 4b, c). Interestingly, the extreme thermal anisotropy of our rMoS_{2} films provides thermal insulation in the throughplane direction, with much lower MoS_{2} surface ∆T values that are only onethird of the value of the bare Au electrode. While singlecrystal MoS_{2} displays similar properties, the insulation effect is stronger in rMoS_{2} (Extended Data Fig. 7a). This implies that heat is efficiently directed away from the hot Au electrode laterally through rMoS_{2} but not to the surface of rMoS_{2}, making the surface of the entire device significantly cooler.
Our experiments corroborate these simulation results. For this, we fabricate nanoscale Au electrodes with the same geometry and substrate as in our simulation (image shown in Fig. 4d, inset) and transfer N = 16 rMoS_{2} (~10 nm thick) using the vacuum stacking process. Both bare and coated Au electrodes show similar resistance at low currents. At higher currents, currentinduced Joule heating leads to the thermally activated electromigration process, which causes the electrodes to fail^{45}. Figure 4d compares representative current–voltage (I–V) curves measured from a bare and coated Au electrode, which shows that the Au electrode with rMoS_{2} can carry a larger current without breaking. The histogram of critical current I_{c} (maximum current a Au electrode sustains for at least 20 s) measured from 20 electrodes (10 bare and 10 with rMoS_{2}) reveals a ~50% increase in the median I_{c} values (Fig. 4e). These results demonstrate our rMoS_{2} film’s ability to efficiently dissipate Joule heat and keep the electrodes cool, as our simulation predicts. As the electromigration process is dominated by the temperature, the observed increase of I_{c} and maximum power before breaking is in good agreement with our simulation in Fig. 4c. Furthermore, we note that the rMoS_{2} film can be integrated with the Au electrodes using mild conditions that do not affect their electrical properties (Extended Data Fig. 7b).
Outlook
We expect interlayer rotation to be an effective and generalizable way to reduce κ_{⊥} and potentially engineer anisotropic thermal properties in a variety of layered materials. Our results call for a systematic study of the exact relation between κ_{⊥} and rotation angle, which could reveal unexpected relationships analogous to the studies of electrical transport in twisted bilayer graphene^{46}. Interlayer rotations can be combined with other parameters (such as pressure or interlayer spacing^{47,48}) and advanced structures (superlattices and heterostructures^{18}) to realize highly tunable ρ, allowing for the customization of thermal transport properties with an unprecedented level of directional and spatial control.
Methods
Sample preparation
Largearea, polycrystalline transition metal dichalcogenide (TMD) (MoS_{2} and WS_{2}) monolayers were grown on SiO_{2}/Si substrates in a hotwalled tube furnace via metalorganic chemical vapour deposition adapted from a previously reported protocol^{19}. The growth conditions were optimized to produce highquality monolayer materials with structural characteristics necessary for thermal measurements. These characteristics include large domain size (D ≈ 1 μm and 0.4 μm), full monolayer coverage, and laterally stitched grain boundaries.
Briefly, Mo(CO)_{6} and W(CO)_{6} (diluted in N_{2} to 15 torr) were used as the metal precursors for the MoS_{2} and WS_{2} growths, respectively. (C_{2}H_{5})_{2}S was used as the chalcogen source. All precursors were kept at room temperature. N_{2} and H_{2} were used as carrier gases. Typical growth times were 15–20 h for MoS_{2} at a growth temperature of 525 °C. Typical growth times for WS_{2} were 2 h at a temperature of 650 °C.
To make the rTMD films, a TMD monolayer was spincoated with PMMA A8 (polymethyl methacrylate, 495 K, 4% diluted in anisole) at 2,800 rpm for 60 s, then baked at 180 °C for 3 min. The PMMAcoated monolayer was stacked onto TMD monolayers layer by layer to a target layer number (N) and transferred to the desired substrates using a previously reported programmed vacuum stacking method^{20}.
TDTR samples
The stacked rTMD films were transferred onto sapphire substrates (Valley Design, Cplane), which were cleaned with Nanostrip solution for 20 min at 60 °C and then rinsed with deionized water. The PMMA layer on the film was removed by immersing the entire substrate in acetone at 60 °C for 1 h. The film was annealed under a 400/100 SCCM Ar/H_{2} environment at 350 °C for 4 h. After cleaning, ~90nmthick, 90 μm × 90 μm Al pads were deposited onto the TMD films through a holey TEM grid shadow mask using electronbeam evaporation.
Raman thermometry samples
Raman experiments were performed on a different set of films from the TDTRmeasured films. First, holey SiN_{x} transmission electron microscopy (TEM) grids were cleaned in a N_{2}/H_{2} plasma at 100 °C and 180 mtorr for 3 min, followed by the transfer of stacked MoS_{2} films onto the TEM grids. During the transfer process, the PMMAcoated rMoS_{2} was suspended on holey thermal release tape before contacting the TEM grid. The extra PMMAMoS_{2} not on the TEM grid was cut away at 180 °C so the PMMA layer was softened. PMMA was removed from rMoS_{2} on the TEM grid via annealing the film in 400/100 SCCM Ar/H_{2} at 350 °C for 4 h.
Crosssectional STEM
The N = 10 films were coated with Al that was electron beam evaporated onto the surface, whereas the top surface of N = 20 films was bare. The rMoS_{2} crosssection was prepared using a Thermo Scientific Strata 400 focused ion beam. Protective layers of carbon (~200 nm) and platinum (~1 µm) were deposited on the sample. A crosssection was milled at a 90° angle from the sample using a Ga ion beam at 30 kV. The crosssection was then polished to ~150 nm thickness with the ion beam at 5 kV.
The crosssection was imaged in a Thermo Scientific Titan Themis scanning transmission electron microscope at 120 kV with a probe convergence angle of 21.4 mrad. The N = 10 film was imaged at a beam voltage of 120 kV, whereas the N = 20 film was imaged at 300 kV. All images were analysed using the opensource software Cornell Spectrum Imager^{49}. The highangle annular dark field (HAADF) image of the sample (see 'TMD films with interlayer rotation' in the main text; Fig. 1d) shows, from top to bottom, the Al crystal lattice along the [110] zone axis, ten layers of MoS_{2} (bright bands), followed by an AlO_{x} layer.
TDTR
We used TDTR to measure the thermal conductivity of our rTMD films. We used a modelocked Ti:sapphire laser, which produced a train of pulses at a repetition rate of 74.86 MHz, with wavelength centred at 785 nm and a total power of 18 mW. The steadystate temperature rise at the surface of the samples was <4 K for all temperatures. For the low temperature TDTR measurements, an INSTEC stage was used with liquid nitrogen cooling; the other beam conditions were the same. The laser beam was split into pump and probe beams. A mechanical delay stage was used to delay the arrival of the probe with respect to the pump on the sample surface by changing their optical path difference, before they were focused onto the sample surface through an objective lens. The 1/e^{2} radius of the focused laser beams was 10.7 μm. For our measurements, we modulated the pump beam at a frequency of 9.3 MHz so that the thermoreflectance change at the sample surface could be detected by the probe beam through lockin detection. The ratio of the inphase and outofphase signals from the lockin was fitted to a thermal diffusion model. The full details of the TDTR measurement can be found elsewhere^{50,51}.
Calculation of κ _{⊥}
The modelling required material parameters such as heat capacity (C), thickness (h), interface conductance (G) and thermal conductivity (κ) for each layer. Our TDTR samples have three chemically distinct layers with the following structure (from the top): Al/rTMD/sapphire. In our fitting process, the heat capacities of all materials were adopted from literature^{52}. The thickness of Al layer was obtained from picosecond acoustics using a longitudinal speed of sound of 6.42 nm ps^{−1} (Extended Data Fig. 3e). The thickness of the rTMD film was calculated from the product of N and the interlayer spacing (d). The latter was measured by performing grazingincidence wideangle Xray spectroscopy (GIWAXS; see GIWAXS section below in the Methods and Extended Data Fig. 2) on the rTMD films, which gave d ≈ 0.64 nm. The total thicknesses of the rMoS_{2} films were <15 nm; thus, this layer was treated as part of the Al–sapphire interface as a single thermal layer characterized by a single thermal conductance value G_{1}. We used the bulk value of the volumetric heat capacity of 1.89 J K^{−1} cm^{−3} for the rMoS_{2} layer. The thermal conductivity of the Al layer was calculated from the Wiedemann–Franz law using the electrical resistance of a transducer layer deposited on a bare sapphire substrate as a reference sample. The thermal conductivity of the sapphire substrate, 35 W m^{−1} K^{−1}, was measured using the same reference sample. Thus, the only remaining free parameter to fit for was G_{1}. To obtain κ_{TMD} from G_{1}, we perform TDTR on various Nlayer TMD films, then perform a linear fit on the effective thermal resistance (R_{TDTR}, equal to 1/G_{1}) versus N data points; the slope of the linear fit is inversely proportional to the thermal conductivity, whereas the y intercept yields the total interfacial thermal resistances (R_{0}) of the top and bottom interfaces. In Extended Data Table 1, our R_{0} values match the values reported in literature^{22,27,53}. We note that, although R_{0} changes depending on the chemical nature of the metal–TMD interface, the slope of the R_{TDTR}–N plot (which is used to extract κ_{⊥}) remains constant, despite the use of different transducer metals, as illustrated in Extended Data Fig. 3d.
For highly anisotropic materials, the anisotropy ratio of an inplane thermal conductivity to a throughplane conductivity should be included in the thermal model. Despite the ultrahigh thermal anisotropy expected of our rTMD films, our throughplane thermal conductivity measurements were probably not sensitive to the thermal conductivity anisotropy given the thinness of our rTMD films. Hence, we assumed a onedimensional thermal transport model and neglected the inplane thermal transport in our calculations. We found that the effect of the anisotropy was significant only at a smaller modulation frequency (f = 1.12 MHz) and 1/e^{2} beam radius of ~3.2 μm, and so we deliberately chose a larger f and a 1/e^{2} beam radius to reduce the sensitivity of our TDTR signal to the inplane thermal transport.
Raman thermometry
We followed a similar procedure from previous reports^{34,54,55} with the modification of lower pressures during measurement. All the Raman measurements were performed using a Horiba Raman spectrometer with a laser excitation wavelength of 532 nm and a longworking distance, 50× objective lens (numerical aperture (NA) = 0.5). The rMoS_{2} A_{1g} peak shift (ω) versus temperature (T) relation was calibrated using a temperaturecontrolled, lowvacuumcompatible Linkam stage. For all our Raman measurements, we used the A_{1g} peak since this outofplane vibrational mode is less sensitive to inplane strain^{56}. The ω–T calibration measurements were performed at atmospheric pressure and with low laser powers. The stage was purged with dry N_{2} gas throughout the calibration step to prevent oxidative damage to the film at high temperatures. Extended Data Fig. 5d shows representative ω–T calibration curves for N = 2 and N = 4 rMoS_{2} films, where a linear fit was performed to obtain the temperaturedependent Raman coefficients. This process was repeated for rMoS_{2} films with different domain sizes D (400 nm and 1 μm) for N = 1–3 (Extended Data Fig. 5e).
To measure the inplane thermal conductivity (κ_{}) of our films, the laser power (P) was varied and the corresponding Δω values were recorded. The inplane thermal conductance was obtained from the reciprocal of the slope of the Δω–P linear fit, which is illustrated for rMoS_{2} films with N = 2–5 and D = 1 μm in Fig. 3c and for rMoS_{2} films with N = 1–3 and D = 400 nm in Extended Data Fig. 5b. As thermal conductivity changes with temperature, laser powers were kept below 250 μW to induce a relatively small ΔT in the film and ensure that the value of κ_{} remained relatively constant. This was verified from the observation of a linear Δω–P regime for P < 250 μW. Any higher laser powers caused the Δω–P curve to deviate from the linear regime with \(\frac{{{\rm{d}}}^{2}\omega }{{\rm{d}}{P}^{2}} < 0\). This indicates that the local film temperature increased faster at higher P > 250 μW, which signified that the thermal conductivity could no longer be assumed to be constant. Instead, the thermal conductivity decreased with increasing temperature, consistent with the Tdependent Raman measurements.
The Δω–P measurements were conducted at a pressure of 15 torr to eliminate any heat loss to air. We verified that a lower pressure down to 4 mtorr gave rise to similar Δω values as the measurements at 15 torr (Extended Data Fig. 5a), weighted by the beam spot size.
The other relevant input quantities for our thermal calculations were obtained as follows: the beam spot radius (r_{0}) was estimated using the knifeedge method, whereby a onedimensional Raman map was taken across a gold step edge on an Aupatterned silicon chip, and the spatial distribution of the integrated peak intensities was fitted to an error function. We measured r_{0} = 0.71 ± 0.09 μm. The laser powers were measured using a Thorlabs standard silicon photodiode power sensor. The rMoS_{2} absorbance \(A=\frac{{\rm{Absorbed\; light\; intensity}}}{{\rm{Incident\; light\; intensity}}}\) was measured at room temperature on a whitelight microscope with a 532 nm bandpass filter and a lowNA condenser aperture. We measured the light intensity transmitted through and reflected from a rMoS_{2} film suspended on a TEM grid, then compared it against a blank TEM grid. The data were collected using a 12bit SensiCam QE CCD camera. The pixel intensities were analysed using ImageJ. The values for A were calculated using the formula A = 1 − T − R. We measured A(N) for N = 1–5, then fitted A to a power law. A(N) was found to follow the relation \(A=1{0.92}^{N}\) (Extended Data Fig. 5c), which matched previous reports^{20}. We use the value A measured at room temperature for our Raman analysis.
Calculation of κ _{}
To obtain the value of κ_{}, we used the twodimensional thermal diffusion equation with a radial symmetry, following previous reports of Raman thermometry of twodimensional films^{54,55}. We assumed a Gaussian laser profile \(q(r)=\frac{{PA}}{({\rm{\pi }}{{r}_{0}}^{2})t}\,{\rm{\exp }}\left(\frac{{r}^{2}}{{{r}_{0}}^{2}}\right)\). We solved for κ_{} numerically using the following equations:
applying the boundary conditions
where \(r\) is the distance from TEM hole centre, \(P\) is the laser power, \(t\) is the film thickness, \(R\) is the TEM hole radius, \(T\) is the film temperature where \({T}_{{\rm{susp}}},{r}\le R\) and \({T}_{{\rm{supp}}},{r}\ge R\), \({T}_{{\rm{a}}}\) is the ambient temperature, \(A\) is the fraction of laser power absorbed, and G = 10 MW m^{−2} K^{−1} is the interfacial thermal conductance between rMoS_{2} and SiN_{x}.
We solved for the expression of T(r) and obtained an expression for the average temperature measured by the Raman shift
κ_{} was obtained by substituting the experimentally measured value for \({T}_{{\rm{m}}}\) and solving the above equation numerically for κ_{}. We calculated κ_{} for each N, and we reported the average value in the main text.
The total measurement uncertainty reported in the main text was calculated based on the error assessment for individual parameters. We used an approximate analytical solution
where \(\omega \) is the Raman frequency of A_{1g} peak, \({A}_{0}\) is the absorption of the monolayer, and \(d\) is the thickness of a monolayer. The difference between the full numerical solution and this analytical form is below 3%. We identified the following independent quantities that carry uncertainty for consideration in our overall uncertainty estimation of κ_{}.

1.
\({\frac{\triangle P}{\triangle T}=(\frac{\triangle \omega }{\triangle P})}^{1}\left(\frac{\triangle \omega }{\triangle T}\right)\): the associated uncertainty was derived from the error in the linear fit of \((\frac{\triangle \omega }{\triangle P})\) and \(\left(\frac{\triangle \omega }{\triangle T}\right)\) for every sample measured. The total uncertainty in the average \(\frac{\triangle P}{\triangle T}\) value was 9% for D = 1 μm and 8% for D = 400 nm.

2.
\(\left(\frac{{A}_{0}}{2{\rm{\pi }}{d}_{0}}\right)\): the uncertainty in A_{0} from the A(N) fit was 4%.

3.
\({\rm{ln}}\left(\frac{R}{{r}_{0}}\right)\): the uncertainty originated from the uncertainty in r_{0}. From 14 repeated measurements of the beam spot size using the knifeedge method, we calculated the standard deviation of r_{0} to be 12%, which translated to an uncertainty in the expression \({\rm{ln}}\left(\frac{R}{{r}_{0}}\right)\) to be 9%.
Total uncertainty in κ_{}: 13% (for both D = 1 μm and D = 400 nm).
Variable pressure Raman thermometry measurements
Previous Raman thermometry measurements on graphene films^{57} and carbon nanotubes^{58} had shown an appreciable difference between measurements performed in air and at lower pressures, as well as in different gaseous environments. We extended the same precaution and repeated our Raman thermometry measurements at low pressures to reduce heat dissipation to air, an extra heat loss channel that would lead to an overestimation of the thermal conductivity of the rMoS_{2} films.
Our Δω–P measurements in Fig. 3 were conducted at a pressure of 15 torr. In Extended Data Fig. 5a, we compared the Δω values of N = 2 rMoS_{2} at three different P (1 atm, 15 torr and 4 mtorr), after correcting for the different laser spot sizes.
Temperaturedependent Raman thermometry for κ _{}
We performed Raman thermometry while varying the ambient temperature T_{a} using a Linkam stage. No oxidation or sample damage was detected for any of the temperatures used. We performed the same Δω–P measurements and calculated the κ_{} value of for each T_{a}. We plot a κ_{} versus T curve, where the x axis is T = T_{a} (Extended Data Fig. 6a).
We note that the measured values of κ_{} here were lower than the room temperature values reported in the main text. We ascribe this to the suboptimal growth conditions for the constituent monolayers used for this sample.
rMoS_{2} heat spreader experiments (electromigration of Au nanoelectrodes)
All Au electrodes were fabricated on Si substrates with 50 nm dry SiO_{2} in three nanopatterning and deposition layers: (A) the nanoelectrodes (10 μm long, 100 nm wide, 15 nm thick); (B) the contact pads that would interface with the external electronics (200 μm long, 300 μm wide, 100 nm thick); and (C) the leads connecting the nanoelectrodes and the contact pads (~1,000 μm long, 50 μm wide, 15 nm thick).
We first defined the leads (B) and then the contact pads (C) using standard photolithography, and electronbeam evaporation of Ti (1 nm)/Au and liftoff. The final step was defining the nanoelectrodes (A) using electronbeam lithography, deposition of 15 nm Au, and liftoff.
Electronbeam lithography
We used a bilayer of resists: copolymer P(MMAMAA 11%) in ethyl lactate and 950 K PMMA A4. The writing was executed with a Raith EBPG 5000 Plus Ebeam writer with the beam conditions of 25 nA current, dose of 1,200 μC cm^{−2}, 300 μm aperture size, 100 kV accelerating voltage.
Film transfer
After the nanoelectrodes, leads and pads were fabricated, the device was cleaned with an O_{2} plasma for 30 s to remove any resist residue and to promote adhesion of the rMoS_{2} film to the Au electrodes and the SiO_{2} surface. A PMMA coated N = 16 rMoS_{2} film was transferred onto the electrodes using the same process as the stacking method as outlined above. The PMMA on the rMoS_{2} film was removed by immersing the entire chip in toluene at 60 °C for 1 h.
Electrical measurements
All measurements were performed in ambient conditions with a homebuilt probe station in a twoprobe geometry. To measure I_{c} in Fig. 4d, we swept the voltage bias in only one direction at a rate such that the rate in current increase is 0.05 mA per 20 s.
For comparison, we deposited SiN_{x} onto Au electrodes (10 μm long, 10 nm thick, 100 nm wide) using plasmaenhanced chemical vapour deposition with the following conditions: 10 s deposition at 90 °C and 10 torr and 1,000 W plasma power, with 25 SCCM/35 SCCM SiH_{4} and N_{2} as the precursors. The film thickness was measured via ellipsometry to be 16 nm.
Computational methodology
Structural models
Structural models were created according to an algorithm previously described in literature^{59}, which was implemented in Python using the atomic simulation environment package^{60}. The structure models were subsequently relaxed using an analytic bondorder potential^{61} and implemented in the LAMMPS package^{62}.
The main rMoS_{2} model used in the simulations described here comprised 10 randomly stacked layers with a total of 10,152 atoms. The 10 layers came in pairs; each pair was related by a 60° rotation. The four primitive angles present in the stack are 16.1, 25.28, 34.72 and 43.9°. Due to strain, each layer contained a different number of atoms in accordance with strains of around 10%.
The bulk structure used in the MD simulations comprised 40 layers (20 conventional unitcells) with a total of 26,880 atoms and cell vectors of 44.44, 43.98 and 243.57 Å.
HNEMD simulations
The interatomic potential in our simulations produces the expected slight increase in the interlayer spacing in rMoS_{2} and yields thermal conductivities and phonon dispersions of bulk MoS_{2} that agree with previous experimental observations and Boltzmann transport calculations based on density functional theory^{38}, confirming our MD model’s suitability for this study. The structures described above were driven by an optimized driving force (Extended Data Fig. 8a) and subsequently relaxed, after which the thermal conductivity was computed using HNEMD simulations^{63} and implemented in the graphics processing units molecular dynamics (GPUMD) package^{29}. We also included the effects of thermal expansion in the simulations (Extended Data Fig. 8b). The calculated κ_{⟂} values of rMoS_{2} are higher than the experimental values. We attribute the discrepancy to our neglecting any quantum effects and all boundary scattering in our simulations. Including such effects could further improve results from simulations. Statistics and averages were gathered from ten independent simulation runs for each system and temperature. The other parameters used in these simulations are compiled in Extended Data Table 2.
Phonon dispersion and lifetimes
We first generated the bulk MoS_{2} phonon dispersion in the harmonic limit as a reference to phonon dispersion calculations using MD simulations. We computed the harmonic (0 K) phonon dispersion using the PHONOPY package^{64}. Forces were computed for 6 × 6 × 2 supercells using the LAMMPS code. Lifetimes were calculated using the lowest applicable order of perturbation theory using the PHONO3PY package^{65}, which also provided us with the thermal conductivity as obtained from a direct solution of the Boltzmann transport equation^{66}. In these calculations, the Brillouin zone was sampled using a 10 × 10 × 10 Γcentred qpoint mesh, which was chosen for consistency with the supercell used in the HNEMD simulations.
Next, we compared the dispersion of bulk MoS_{2} to that calculated using MD simulations to verify the accuracy of our MD simulations for calculating the phonon dispersion of rMoS_{2}. For both bulk and rMoS_{2}, we extracted the phonon dispersions and lifetimes at 300 K by analysing the longitudinal and transverse current correlation functions generated by MD simulations in the microcanonical (NVE) ensemble using dynasor^{67}. The MD simulations details were otherwise identical to the HNEMD simulations. The obtained correlation functions were Fourier transformed and fitted to peak shape functions corresponding to (over)damped harmonic oscillators using the full expressions given in the dynasor paper^{67} to obtain phonon frequencies and lifetimes.
Finiteelement analysis
We used the COMSOL software to simulate the steadystate temperature distribution in a Au electrode on a SiO_{2}/Si substrate. Our geometry contains an Au electrode that is 100 nm wide, 15 nm thick, and 10 μm long on a 50nmthick SiO_{2} layer on a Si substrate. We layer a 10nmthick MoS_{2} film onto the Au electrode and the SiO_{2} layer. For the thermal anisotropy consideration, we define the thermal conductivity slow axis direction to always be perpendicular to the film’s bottom surface in contact with the substrate or the Au electrode, including the Au electrode side walls.
We supply the Au electrode with 8 mW uniformly over the entire volume as the heat source, matching the power conditions at which the Au electrode fails in our experiments. As the boundary condition, we set the bottom surface on the Si substrate to be at 293.15 K. We also account for all the interfacial thermal resistances between heterogeneous surfaces in our calculations, which include rMoS_{2}/Au, rMoS_{2}/SiO_{2}, Au/SiO_{2}, SiO_{2}/Si (refs. ^{68,69,70,71}). All effects of radiation are neglected as they do not affect the temperature values in our simulations.
Lowfrequency Raman measurements
The lowfrequency Raman spectra of N = 2, 3 and 4 rMoS_{2} films, along with the spectrum for MoS_{2}, are shown in Extended Data Fig. 4a. From the layerdependence of the peak positions, we assigned these to be the breathing modes of MoS_{2} (ref. ^{72}). We did not observe any peaks corresponding to shear modes in rMoS_{2}. Our findings agree with theoretical studies of lowfrequency Raman modes of twisted MoS_{2} bilayers, which showed that the shear mode peaks redshift to below the detection capabilities (2 cm^{−1})^{73}. The positions of the breathing mode peaks of the rMoS_{2} films were close to those in exfoliated few layer MoS_{2} (Extended Data Fig. 4b) as reported in literature^{72,74}. This observation agreed with our MD simulations that suggested that the transverse vibrational mode was suppressed by interlayer rotation, while the longitudinal vibrational mode was retained.
GIWAXS
The GIWAXS measurement was performed using SAXSLAB (XENOCS)’s GANESHA (labsource Cu Kα, photon flux ~108 photons s^{−1}) to characterize the interlayer spacing of rMoS_{2} films. An N = 10 rMoS_{2} film was prepared on a SiO_{2}/Si substrate. The incidence angle of the Xray beam was 0.2° and the integration time was ~60 s. Radially integrating the twodimensional diffraction images along the outofplane direction produced the diffraction spectrum along the c axis shown in Extended Data Fig. 2.
The peak position of 14° corresponded to an interlayer spacing of 6.4 Å in the [0 0 1] direction, which matched previous reports of rTMD films^{20}.
Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request. Source data are provided with this paper.
Code availability
All code used in this work is available from the corresponding authors on reasonable request.
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Acknowledgements
We thank D. F. Ogletree, J. Jureller, A. J. Mannix, J.U. Lee, K.H. Lee and M. Lee for their helpful discussions. We also acknowledge J.H. Kang, A. Ye and C. Liang for their help with materials preparation. Primary funding for this work comes from Air Force Office of Scientific Research MURI projects (FA9550–18–10480 and FA9550–1610031). Material growths done by F.M. and C.P. are partially supported by the University of Chicago MRSEC (NSF DMR2011854) and Samsung Advanced Institute of Technology. This work makes use of the characterization facilities of the University of Chicago MRSEC (NSF DMR2011854) and the Pritzker Nanofabrication Facility at the University of Chicago, which receives support from SHyNE Resource (NSF ECCS1542205), a node of NSF’s NNCI network. TDTR measurements are supported by Office of Naval Research MURI grant N000141612436 and are carried out in Frederick Seitz Materials Research Laboratory at the University of Illinois at UrbanaChampaign. F.E., E.F. and P.E. are funded by the Knut and Alice Wallenberg Foundation (2014.0226), the Swedish Research Council (201504153 and 201806482), and the FLAGERA JTC2017 project MECHANIC funded by the Swedish Research Council (VR 201706819); they acknowledge the computer time allocations by the Swedish National Infrastructure for Computing at NSC (Linkӧping) and C3SE (Gothenburg). A.R. and electron microscopy at the Cornell Center for Materials Research are supported by the NSF MRSEC grant DMR1719875. The Titan microscope was acquired with the NSF MRI grant DMR1429155. F.M. acknowledges support by the NSF Graduate Research Fellowship Program under grant no. DGE1746045. Y.Z. acknowledges support by the Camille and Henry Dreyfus Foundation, Inc., under the Dreyfus Environmental Postdoc award EP16094.
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S.E.K., J.S., and J.P. conceived the experiments. S.E.K. produced and performed the structural characterizations of the rTMD films, and ran the finiteelement simulations. F.M. and C.P. grew the TMD monolayers. A. Rai and D.G.C. performed the TDTR measurements. F.E., E.F. and P.E. conducted the atomistic simulations. P.P. fabricated the Au electrodes for the electromigration experiments. A. Ray and D.A.M. performed the STEM imaging. S.E.K. performed the Raman measurements and analysed the data with the help of D.G.C. and Y.Z. S.E.K., F.M. and J.P. wrote the manuscript. J.P., D.G.C. and P.E. oversaw the project. All authors discussed the manuscript and provided feedback.
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Extended data figures and tables
Extended Data Fig. 1 Crosssectional TEM images of N = 20 and 10 rMoS_{2} on AlO_{x}.
Each set of N images are taken from the same sample at different locations. The N = 10 films are coated with Al that was electron beam evaporated onto the surface.
Extended Data Fig. 2 GIWAXS data of N = 10 rMoS_{2}.
The peak position corresponds to a 2θ value of 14˚, which translates to an interlayer spacing of 6.4 Å (scattering direction).
Extended Data Fig. 3 Additional TDTR measurements and details.
a, Microscope image of an N = 10 rMoS_{2} film coated with a square grid of Al pads. b, 4 × 4 TDTR map of R_{TDTR} of an N = 10 rMoS_{2} film. c, Histogram of R_{TDTR} array measurements. d, TDTR measurements of N ≤ 10 rTMD films coated with Au or Al. The error bars denote s.d.; number of TDTR measurements per film sample, n = 3 for Au samples; n = 3–5 for the Al samples. e, Picosecond acoustics of a MoS_{2} monolayer on thick sapphire substrate, coated with an Al transducer layer. The y axis V_{in} is the inphase signal of the lockin amplifier. The red arrows indicate the acoustic waves reflected at the Al/MoS_{2} interface.
Extended Data Fig. 4 Lowfrequency Raman modes of rMoS_{2}.
a, Raman spectra reflecting the breathing modes (BM) of rMoS_{2} (blue) and the shear mode (SM) for MoS_{2}. b, The lowfrequency Raman peak positions of rMoS_{2} and exfoliated MoS_{2}. The filled squares indicate the BM peak positions of rMoS_{2}. The open squares indicate the BM peak positions of exfoliated MoS_{2}, and the open circles indicate the SM peak positions of exfoliated MoS_{2}^{72}.
Extended Data Fig. 5 Raman thermometry on rMoS_{2} films.
a, Δω–P_{abs} curves of representative N = 2 rMoS_{2} films at different pressures. The P_{abs} values along the x axis are normalized to account for the slight differences in beam spot sizes (Δr = 20%). The results at 15 torr and 4 mtorr signify no effect of reducing the pressure to below 15 torr. Δω was approximately fivefold smaller at atmospheric pressure due to the extra heat loss channel by air. b, Δω–P_{abs} curves of rMoS_{2} films made up of D = 400 nm (grain size) monolayers. c, Optical absorption of suspended rMoS_{2} films, which follows the trend \(A=1{(1{A}_{0})}^{N}\), where A_{0} is the monolayer absorptance. From the fit, A_{0} = 0.08 ± 0.003. d, ω–T calibration measurements of suspended rMoS_{2} films (D = 1 μm), with the N = 2 and N = 4 data as the representative curves. e, ω–T slopes versus layer number for all films, with D = 400 nm or 1 μm.
Extended Data Fig. 6 κ(T) and ρ of rMoS_{2}.
a, κ(T) of rMoS_{2}, with κ_{} measured using Raman thermometry of N = 4 rMoS_{2}, and κ_{⊥} measured via TDTR as reported in Fig. 2c. The error bars are the propagated uncertainties from the calculation of the conductance value for each N. The error bars denote the propagated uncertainty of the calculations from the input parameters. We observed that κ_{} decreased with T, alluding to phononmediated heat transport and attesting to the longrange crystallinity of the rMoS_{2} films inplane. This was in contrast with the κ_{⊥}(T) behaviour (Fig. 2c), which showed a slightly increasing trend. b, Catalogue of experimentally measured anisotropy ratios at room temperature versus slowaxis thermal conductivity (κ_{s}) of thermally anisotropic materials from literature, by category.
Extended Data Fig. 7 rMoS_{2} efficacy as a heat spreader.
a, Finite element simulations of the linear temperature profiles of Au electrodes covered with MoS_{2} and rMoS_{2}. b, SiN_{x} as heat spreaders for Au electrodes. Electrical properties of 10nmthick, 100nmwide and 10μmlong Au electrodes before and after 16 nm SiN_{x} film deposition onto the electrodes using plasmaenhanced chemical vapour deposition. In contrast to rMoS_{2}, the direct deposition of an ultrathin inorganic film such as SiN_{x} with a comparable κ to κ_{} of rMoS_{2} negatively affects the performance of the Au electrodes.
Extended Data Fig. 8 Optimization of the MD simulations for κ calculations.
a, Optimization of the driving force of the system, where the grey zone denotes the error. b, Effect of thermal expansion on κ.
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Kim, S.E., Mujid, F., Rai, A. et al. Extremely anisotropic van der Waals thermal conductors. Nature 597, 660–665 (2021). https://doi.org/10.1038/s41586021038678
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DOI: https://doi.org/10.1038/s41586021038678
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