Abstract
Hexagonal boron nitride (hBN) has been predicted to exhibit an inplane thermal conductivity as high as ~ 550 W m^{−1} K^{−1} at room temperature, making it a promising thermal management material. However, current experimental results (220–420 W m^{−1} K^{−1}) have been well below the prediction. Here, we report on the modulation of hBN thermal conductivity by controlling the B isotope concentration. For monoisotopic ^{10}B hBN, an inplane thermal conductivity as high as 585 W m^{−1} K^{−1} is measured at room temperature, ~ 80% higher than that of hBN with a disordered isotope concentration (52%:48% mixture of ^{10}B and ^{11}B). The temperaturedependent thermal conductivities of monoisotopic hBN agree well with first principles calculations including only intrinsic phononphonon scattering. Our results illustrate the potential to achieve high thermal conductivity in hBN and control its thermal conductivity, opening avenues for the wide application of hBN as a nextgeneration thinfilm material for thermal management, metamaterials and metadevices.
Introduction
Hexagonal boron nitride (hBN) is a technologically important layered material used as a dielectric spacer, encapsulant, ultraviolet laser emitter, and hyperbolic material in electronic and photonic applications^{1,2,3}. More recently, hBN has attracted attention for thermal management of electronics as theoretical calculations^{4} predicted an inplane thermal conductivity as high as k_{r}~ 550 W m^{−1} K^{−1} at room temperature, though, highly anisotropic with a two orders of magnitude smaller outofplane thermal conductivity (k_{z}~ 5 W m^{−1} K^{−1}). The high inplane thermal conductivity, as well as atomic flatness, makes hBN an ideal substrate material for nextgeneration thinfilm devices since waste heat can be spread quickly laterally through a large area, avoiding formation of localized hot spots^{5,6}. In addition, hBN could be a good reinforcing filler for thermal interface and encapsulation composite materials due to its high thermal conductivity and electrical resistivity^{7,8}. Despite its predicted favorable thermal properties, experimental results are few and varied. Reported k_{r} values range from 220 to 420 W m^{−1} K^{−1} ^{4,9,10}, well below the predicted maximum value. Developing insight into this discrepancy and driving hBN thermal conductivity to higher values is of great interest both fundamentally and for enabling enhanced thermal engineering.
Quantized lattice vibrations (phonons) in crystals synthesized from elements with natural isotopic concentration scatter due to mass variations of the isotopes in the lattice, thus reducing thermal conductivity^{11}. Enhanced thermal conductivity has been demonstrated in monoisotopic materials (isotopically purified to >99% one isotope), such as in silicon^{12}, germanium^{13}, gallium arsenide^{14}, diamond^{15}, and graphene^{16}. Naturally occurring BN materials are made with two stable B isotopes (19.9% ^{10}B and 80.1% ^{11}B), which present a large mass modulation, and an opportunity to control its thermal conductivity by manipulating the B isotope concentration. Large B isotope effect has been observed in BN nanotubes^{17}, whereas experimental evidence of isotope effects in hBN has not been possible to date because suitable samples have not been available. In terms of theoretical predictions, the conventional Callaway approach^{13,18,19} based on the Boltzmann transport equation (BTE) and formulated within a singlemode relaxation time approximation (RTA) has been widely used to study the isotope effect in numerous material systems, but has challenges in anisotropic layered systems such as hBN. Often, phonon scattering processes in layered systems cannot be treated as independent resistive processes, an assumption of the RTA^{20}. Ab initio approaches based on full solution of the BTE in combination with first principles density functional theory (DFT) have demonstrated accuracy in describing the thermal conductivity of anisotropic layered materials with natural isotopic concentrations^{21,22}, however, experimental data for monoisotopic layered materials are not available for model validations.
Only recently have isotopically engineered hBN crystals become available^{23,24,25}. To date, investigations have focused on fundamental isotope effects related to Raman phonon lifetimes and the electronic bandgap^{23,24}. In this work, we experimentally demonstrate the effect of boron isotope concentration on the thermal conductivity of bulk hBN crystals using a transient thermoreflectance (TTR) technique. The monoisotopic ^{10}B hBN crystals have inplane thermal conductivity as high as 585 W m^{−1} K^{−1} at room temperature, ~ 80% larger than that of hBN with disordered isotope concentrations (52%:48% mixture of ^{10}B and ^{11}B). Our measurements are compared with stateoftheart ab initio thermal conductivity calculations.
Results
hBN crystals and microstructural characterization
hBN crystals were prepared from monoisotopic boron powders (^{10}B and ^{11}B), with the process described in Liu et al.^{25}. This allowed the control of the boron isotope composition from 50%:50% (the most disordered composition) to monoisotopic ^{10}B or ^{11}B. Four hBN crystals were grown, with input isotope compositions of 99% ^{10}B (monoisotopic ^{10}B), 48% ^{11}B (isotopically disordered), 78% ^{11}B (nearnatural), and 99% ^{11}B (monoisotopic ^{11}B), respectively (see Methods). Supplementary Fig. 1 shows an optical micrograph of typical flakelike samples with size around 1 mm. Flake thicknesses were determined by optical microscopy to be 15 ± 2 µm by measuring the height difference between the sample surface and the underlying substrate. Figure 1a shows Raman spectra of the highenergy E_{2g} mode from the different isotopically engineered hBN crystals (see Methods). The energy of this E_{2g} phonon is 1393 cm^{−1}, 1379 cm^{−1}, 1367 cm^{−1}, and 1357 cm^{−1} for monoisotopic ^{10}B, isotopically disordered, nearnatural and monoisotopic ^{11}B hBN, respectively. The Raman shifts of the samples were benchmarked against the established relationship between Raman shifts and the isotope ratios^{24}, verifying that the resulting hBN crystals have the same isotope ratios as the input material (see details in Supplementary Note 1). As expected, the Raman linewidths are much narrower for the monoisotopic ^{10}B hBN (2.9 cm^{−1}) and monoisotopic ^{11}B hBN (3.1 cm^{−1}) than for the isotopically disordered hBN (7.6 cm^{−1}) and nearnatural hBN (7.9 cm^{−1}). The linewidths are in part determined by phonon–isotope interactions in the disordered materials, a feature that was correlated with lowloss phononpolariton modes in monoisotopic hBN previously^{24}.
The crystal microstructure of the hBN samples was characterized with selected area electron diffraction (SAED), transmission electron microscopy (TEM), scanning electron microscopy (SEM), and electron backscattered diffraction (EBSD) (see Methods), with results shown in Fig. 1b–h from a representative hBN specimen (monoisotopic ^{10}B hBN; for results of all samples see Supplementary Fig. 2). The SAED pattern (Fig. 1b) showed a singleoriented hexagonal crystal structure consistent with a [0001] surface. The SEM image (Fig. 1c) and EBSD inverse pole figure (Fig. 1d) confirmed the size of single crystal domains are larger than >150 μm. There were steps between these large domains, which are natural features in hexagonal crystals grown from solutions^{26}. Using TEM, the single crystal domain was found to have areas a few tens of microns across which are free of defects. In some locations, inplane (nearscrew) dislocations with Burgers vectors \({\mathrm{a}}/3\langle 11\bar 20\rangle\) were observed, as seen in the brightfield TEM image in Fig. 1f. Such dislocations describe a rotation about [0001] between successive layers, estimated to be up to ~0.02° in Fig. 1f. In the darkfield TEM in Fig. 1g, taken in weakbeam diffracting conditions, the inplane perfect dislocations are seen to be dissociated into closely spaced partial \({\mathrm{a}}/3\langle 10\bar 10\rangle\) dislocations on a fine scale. In some areas, subgrain boundaries were observed, visible as fringes in the brightfield TEM image in Fig. 1h. The boundary indicates the misorientation (tilt) between the two grains with the misorientation angle estimated to be up to ~ 1°. This conclusion is consistent with that from EBSD (Fig. 1e). The misorientations (tilt) between the grains across the sample were small, about 2° or less. In short, the fabricated hBN samples had highquality large singlecrystalline domains with a very low density of dislocations and tilt.
Anisotropic thermal conductivity characterization
The thermal conductivity of hBN, k_{r}—inplane and k_{z}—outofplane, were measured using a nanosecond laserbased TTR technique^{27,28,29} (see Methods). Figure 2a shows the schematics of the TTR technique. The hBN crystal was coated with a 50 nm Au thin film, which serves as a transducer. A 10 ns, 355 nm pulsed pump laser heats the surface of the Au transducer, creating a temperature response. A continuous 532 nm laser was used to monitor the surface temperature response via the induced change in Au reflectivity. Figure 2b shows an example of the monitored normalized thermoreflectance transient. An analytical photothermal pulsesinduced thermal transport model was built based on the geometric and temporal characteristics of the pump pulse, experimental structure and boundary conditions shown in Fig. 2c to analyze the measured transients. We use \(S_{x_0} = \frac{{\partial \left( {{\mathrm{ln}}T} \right)}}{{\partial \left( {{\mathrm{ln}}x_0} \right)}}\) to quantify the sensitivity of the temperature response (T) to the parameter, x_{0}, which is either of the thermal conductivities of hBN (k_{r}, k_{z}) or the thermal boundary resistance between the Au transducer and hBN (TBR_{eff}) (see Methods). Figure 2d shows the calculated sensitivity results. The sensitivity to k_{z} (S_{z}) and TBR_{eff} (S_{TBR}) increases rapidly from 10 to 100 ns, whereas sensitivity to k_{r} (S_{r}) remains mostly constant. With further increasing time, S_{z} remains relatively constant and S_{TBR} increases slowly, whereas S_{r} gradually increases and exceeds S_{TBR} and S_{z} at ~ 500 ns. Taking advantage of the distinct sensitivity time scales of the thermoreflectance transients, the parameters, k_{z}, k_{r}, and TBR_{eff}, were determined simultaneously via fitting the monitored TTR transients with the analytical thermal transport model (see Methods). The best fit of our model results to an example measured transient for the monoisotopic ^{10}B hBN sample at 300 K is given in Fig. 2b. The ±25% bound curves shown in Fig. 2b illustrate that TTR signals are mainly sensitive to k_{z} at short time scales (10–500 ns) and more sensitive to k_{r} at longer time scales (>500 ns).
Figures 3a, b give the measured values of k_{r} and k_{z} for the four crystals, as a function of temperature. Also shown are the results of (BTE)/(DFT) calculations using threephonon and phonon–isotope scattering from quantum perturbation methods as inputs^{4} (see Methods). The predicted curves for monoisotopic ^{10}B hBN and ^{11}B hBN give the hBN intrinsic thermal conductivities determined solely by threephonon scattering processes. Theory and experiment for k_{r} are in good quantitative agreement for the monoisotopic hBN for temperatures >150 K. Discrepancies for <150 K are likely due to the extrinsic scattering of phonons from crystal imperfections not included in the theoretical calculations. Such extrinsic defects may include subgrain boundaries, dislocations (apparent in TEM micrographs Fig. 1f–h), and point defects such as vacancies and carbon impurities (carbon impurity concentrations of 7.5–27 × 10^{19} cm^{−3} have been measured in hBN crystals grown from identical synthesis methods^{24}). The phonon mean free paths in hBN at 100–150 K range from a few hundred nanometers to 10 μm, and the lower the temperature the longer the phonon mean free paths^{4}. This suggests that subgrain boundaries are a possible contributor to reducing the thermal conductivity of hBN at low temperature considering the long phonon mean free path is comparable to the grain size, as evident in TEM and EBSD micrographs (Figs 1e, h). Point defects, such as carbon impurities, may also scatter phonons as strongly as isotope variations due to the mass and force fluctuations around the defect sites^{30,31}. Such defects, like isotope variations, become more important at lower temperature where the intrinsic phonon–phonon scattering is weak. Figure 3a also shows the theoretical k_{r} of natural and isotopically disordered hBN. The measured results compare well with theoretical calculations >225 K; differences at lower temperatures again may arise from the presence of defects. As shown in Fig. 3b, the outofplane thermal conductivities (k_{z}) for all samples compare favorably with the calculations, over the temperature range measured. Weak van der Waals bonding between hBN planes gives smaller acoustic velocities perpendicular to the planes, whereas strong inplane covalent bonding of the light B and N atoms gives fast phonons along the planes^{4}. Thus, k_{z} is much smaller than k_{r}. There is a relatively small but apparent increase in thermal conductivity for the monoisotopic samples compared with the disordered ones. Extrinsic phonon scattering from grain boundaries and point defects is expected to have a smaller effect on k_{z} due to the much shorter phonon mean free paths (about a few tens of nanometers^{4}) in the outofplane direction, and therefore agreement between simulations and measurements over a broader temperature range is found.
The inplane thermal conductivities k_{r} of 585 ± 80 W m^{−1} K^{−1} and 550 ± 75 W m^{−1} K^{−1} measured at 300 K for the monoisotopic ^{10}B and ^{11}B hBN, respectively, are the highest room temperature values reported to date in the literature for hBN. The measured k_{r} value for nearnatural hBN (78% ^{11}B) is 408 ± 60 W m^{−1} K^{−1} is consistent with previously reported values for natural hBN (80% ^{11}B) by Sichel et al.^{10} and Jiang et al.^{4}, and about twice larger than that measured by Simpson et al.^{9}. Isotopically disordered hBN has the lowest measured k_{r}, 330 ± 42 W m^{−1} K^{−1}. At 300 K, the measured outofplane thermal conductivities, k_{z}, are 3.5 ± 0.8 W m^{−1} K^{−1}, 4.5 ± 1.4 W m^{−1} K^{−1}, 3.3 ± 0.8 W m^{−1} K^{−1} and 2.3 ± 0.5 W m^{−1} K^{−1} for the monoisotopic ^{10}B, monoisotopic ^{11}B, nearnatural and isotopically disordered hBN crystals, respectively. All these values are comparable to those reported for natural hBN in Jiang et al.^{4} and Simpson et al.^{9}. Note that the isotope effect on k_{z} is not clearly distinguishable experimentally due to relatively large error bars of experimental data.
The calculations of k_{r} and k_{z} for monoisotopic ^{10}B hBN predict it to be somewhat larger than those of monoisotopic ^{11}B hBN, despite both systems being free of phonon–isotope scattering. Phonon frequencies roughly scale with mass^{−1/2}. This results in slightly faster acoustic phonons and less scattering from higher frequency optic phonons in monoisotopic ^{10}B hBN compared with monoisotopic ^{11}B hBN. These both lead to larger thermal conductivities (k_{r} and k_{z}) in monoisotopic ^{10}B hBN. This variance is within the error bars of the measured data and therefore not clearly distinguishable experimentally. The enhancement of the thermal conductivity, k_{r}, in monoisotopic ^{10}B hBN and ^{11}B hBN, with respect to the natural BN is 43 and 35%, respectively, at room temperature. This is smaller than monoisotopic enhancements reported in graphene (~ 58%^{16}) and diamond (~ 50%^{15}), although the natural isotopic disorder and resulting mass variance is larger in naturally occurring hBN (19.9% ^{10}B and 80.1% ^{11}B) than in naturally occurring carbon materials (98.9% ^{12}C and 1.1% ^{13}C). One important factor to consider when comparing the carbonbased materials and bulk hBN is the discrepancy between the frequency scales of their phonon dispersions. Covalent CC bonds are stronger than that of BN bonds, which results in “harder” phonons in graphene and diamond. Besides reducing acoustic phonon velocities, the softer phonons in hBN exhibit two important effects, which reduce the k_{r} enhancement: (1) hBN has weaker phonon–isotope scattering as this scales as frequency to the power four^{32}, and (2) hBN has stronger intrinsic phonon–phonon scattering as the phase space for interactions increases as the dispersion frequency scale decreases^{33}. The stronger intrinsic phonon scattering in hBN compared with diamond and graphene is indirectly observed when comparing the isotopically purified thermal conductivities: 585 ± 80 W m^{−1} K^{−1}, ~ 4000 W m^{−1} K^{−1 16}, and ~ 3300 W m^{−1} K^{−1 15} for monoisotopic ^{10}B hBN, graphene and diamond, respectively.
The predicted anisotropic ratio (k_{r}/k_{z}) is as high as 125 at 300 K, displayed in Fig. 3c, in reasonable agreement with measurements. We note that the difference in measured k_{r}/k_{z} between monoisotopic and isotopically disordered samples is large (>40 for T < 200 K) despite the presence of point defects reducing k_{r} at low temperatures. This demonstrates that isotope engineering makes tuning of the thermal conductivity anisotropy possible in hBN over a large range. Recently, tuning thermal anisotropy in materials has been demonstrated to allow precise manipulation of heat flux including for heat shielding, heat concentrators, macroscopic diodes, chip heat management, and energy harvesting^{34,35,36,37,38}. Traditional thermal metamaterials used for these applications are designed from two or more constituent materials with large thermal conductivity contrast, to realize the required anisotropic and inhomogenous conductivity profiles by spatially adjusting their volume filling ratios. Coefficient of thermal expansion contrast can then result in challenges including mechanical instability and complicated fabrication processes^{35}. Clearly, engineering materials via isotope concentration alone is more straightforward, and the resulting product is a single phase material with negligible differences of heat capacity (see Supplementary Fig. 7), density^{23}, and temperaturedependent lattice constant^{23} (leading to similar thermal expansion), providing a promising route to enhanced thermal metamaterials and metadevices.
Discussion
Through isotope engineering, we have experimentally demonstrated the manipulation of the thermal conductivity of hBN. The measured temperaturedependent thermal conductivity of monoisotopic hBN provides a means for the verification of DFT calculations of thermal conductivity as solely impacted by intrinsic phonon–phonon scattering processes. At room temperature, the inplane thermal conductivity (k_{r}) of monoisotopic ^{10}B hBN was measured as high as 585 W m^{−1} K^{−1}, the highest room temperature values for hBN reported in the literature, and in good agreement with theoretical predictions. The enhanced k_{r} in monoisotopic hBN makes it a promising candidate for managing heat in higher power dissipation compact thinfilm devices. A highly tuneable conductivity anisotropy ratio of hBN by isotope engineering, may extend the application of hBN to thermal metamaterial and metadevice applications.
Methods
hBN crystal growth
hBN single crystals were synthesized using the NiCr flux method. Highpurity ^{10}B (99.22 at%) and ^{11}B (99.41 at%) powders were mixed with Ni and Cr powders to give overall concentrations of 4 wt% B, 48 wt% Ni, and 48 wt% Cr. Manipulating the mass ratio of ^{10}B to ^{11}B in the source material renders different isotope compositions of resulting hBN crystals. After loading the crucible, the furnace was evacuated, then filled with N_{2} and forming gas (5% hydrogen in balance argon) to ~850 torr. The N_{2} and forming gases continuously flowed through the system during crystal growth with flow rates of 125 sccm and 25 sccm, respectively. The system was heated to 1550 °C for a dwell time of 24 h. The hBN crystals were formed by cooling at a rate of 0.5 °C h^{−1} to 1525 °C, then quenched to room temperature. Four different mass ratios of ^{10}B to ^{11}B (100%:0, 50%:50%, 20%:80%, and 0:100%) were input as source material, resulting in four different isotope compositions (1, 48, 78, and 99% ^{11}B) in the resulting hBN crystals. Crystals ranged up to a few mm in size.
Raman measurements
Raman measurements were performed using a 532 nm laser line with a Renishaw Raman microscope. In all, <10 mW laser power was directed at the sample through a 50 × 0.75 N.A. objective, with the Raman scattered light collected back through the same objective. The scattered light was dispersed using a 2400 groove mm^{−1} grating onto a silicon chargecoupled device. The spectral positions of the Raman lines were calibrated against a silicon reference sample. For each hBN crystal studied, four measurements were performed, reporting the average spectral position and linewidth.
Microscopic analysis
SAED and TEM planview images were acquired at 200 kV in a Philips EM430 TEM. The darkfield and brightfield TEM was operated in the two beam diffracting conditions with \(g = 11\bar 20\). SEM and EBSD measurements were performed on Zeiss Evo MA10 LaB6 with an instrument probe equipped with EBSD. The EBSD mapping image was constructed by scanning a ~ 600 µm^{2} rectangular area with a 1 µm step size. Lattice constants of hBN were taken from literature reported data^{23} for the EBSD analysis.
For the SAED and TEM analysis, small pieces of hBN were crushed from the asgrown crystals and then adhered onto a holey silicon nitride support membrane for imaging. For the SEM and EBSD analysis, the asgrown crystals were mounted on the carbon tape and the first few layers were exfoliated using Scotch tape to expose the clean surface for imaging.
TTR measurements
The TTR measurement configuration is shown in Fig. 2a. To prepare the samples for the TTR measurements, the hBN flakes were first cleaned with acetone, and then attached to a large glass slide using carbon tape. The first few layers were exfoliated using Scotch tape, to create a clean surface, before depositing a 50 nm (±5%) Au transducer film, with a 10 nm Ti interlayer for good adhesion. The Au film serves as a transducer in the TTR measurements. The schematic of TTR sample is shown in Fig. 2b mounted on a copper disk in the Linkam THMS600 cryostat, which was used to control the sample temperature from 100 K to 300 K during the measurements. The pump beam is a 10 ns, 355 nm frequency tripled Nd:YAG laser with a 30 kHz repetition rate. After passing through a beam expander and dichroic beam splitter, it is directed through a 15 × 0.3 N.A. quartz objective to a defocused spot on the sample with a Gaussian profile (1/e^{2} radius of 41 μm). The pump laser power incident at the sample surface is less than 5 mW (time averaged, peak of 15 W). The transient surface reflectivity change is monitored using a CW 532 nm laser probe beam focused at the sample surface to 2 μm, in the central location of the pump spot. The reflected beam intensity is sampled by a beam splitter and detected by a silicon photodiode transimpedence amplifier (2.3 ns rise time) and a digital oscilloscope (300 MHz bandwidth). To ensure no residual light from the pump beam is detected, a longpass filter is placed before the detector. We note alternative to TTR often time domain thermoreflectance (TDTR)^{39,40} is employed to measure thermal conductivity. However, TTR is more sensitive to anisotropy of thermal conductivities. TDTR, typically uses ultrashort (fs/ps) highfrequency pulse lasers. As the thermal penetration depth of a short laser pulse is much smaller than that of the laser spot size, onedimensional heat transfer is generally assumed making the conventional TDTR insensitive to radial heat conduction. Although, e.g., pump and probe offset TDTR^{41} and variable pump spot size TDTR^{42} has been employed to enable an increased sensitivity to radial heat conduction and hence allow measurement of anisotropic thermal conductivities, the TTR technique is easier to use.
Analytical photothermal pulsesinduced thermal transport model for the analysis of the TTR data
The thermal conductivity of hBN was determined by comparing the measured TTR transients with an analytical photothermal pulsesinduced thermal transport model. The model considers heat conduction in Nlayer films (in this study: N = 3 (Au, Ti and hBN layers)). The ith layer of the film with thickness d_{i}, is taken having an inplane thermal conductivities (k_{ir}), outofplane thermal conductivities (k_{iz}), density (ρ_{i}), and specific heat capacity C_{i}, with i = 1, 2, …N. The material (carbon tape) used for mounting the hBN sample, indexed as N + 1, is considered a thermal insulator due to its low thermal conductivity. At time t = 0 when the system is in thermal equilibrium with ambient temperature T_{0}, an energy pulse is absorbed on the top surface of the film, resulting in heat diffusion in the outofplane (z) direction as well as in the inplane (r) directions. Considering the anisotropic thermal properties, the heat conduction equation for temperature rise ε_{i} = T − T_{0} in layer i is given by
For the heat absorption on the top surface,
where Q(r, t) is the input energy flux, which is spatially and temporally dependent. Here, we adopted the approach described by Hui et al.^{43,44} to solve Eqs. (1) and (2) by using Laplace and Hankel transforms:
The problem defined by (1) and (2) can be recast to obtain the transformed temperature in the spatial and temporal frequency domain (β, s) as
Repeating the solution procedure described in^{43,44} yields identical analytical results for ε_{i}, except \(\gamma _i\) in Eq. (5) is defined by
This analytical model was validated against the solutions obtained by a finite elements method in ANSYS for the same problem, yielding identical results for both cases (see Supplementary Note 2).
Based on the analytical model, the measured transients are a function of the hBN outofplane (k_{z}) and inplane (k_{r}) thermal conductivities, density, specific heat capacity, thickness of each layer/material, and geometrical and temporal characteristics of the pump pulse. Except for the anisotropic thermal conductivity (k_{z} and k_{r}) of hBN, and the thermal boundary resistance between the Au transducer and hBN (TBR_{eff}), all other parameters are input as fixed values (see Supplementary Note 3). Therefore, TBR_{eff}, k_{z}, and k_{r} are treated as free variables, adjusted to fit the analytical model results to the measured traces. A least squares algorithm was built for multiparameter fitting. The uncertainty (error bar) was determined by individually varying each variable about the solution minima and finding the change in the variable that causes a 5% change in the least squares value, i.e., the error bar represents a 95% confidence level. Note that TBR_{eff} is determined by the ratio of Ti thickness to its fitted thermal conductivity. All fitted results of TBR_{eff} are shown in Supplementary Fig. 3. Supplementary Fig. 4 shows the simulated temperature rise (∆T) at the surface of hBN. The maximum ∆T is ~ 45 K. At 100 ns, ∆T drops to 10 K and after 1000 ns, ∆T reduces to ~ 1 K. Thus, the fitted hBN thermal property values approximate the values at ambient temperature. To ensure that the measured thermal conductivity results are reliable and repeatable, at the beginning of each hBN sample measurement, we tested an inhousemade reference sample (Au/Ti coated undoped highpurity single crystal silicon) to make sure the measured silicon thermal conductivity result at room temperature is always equal to the standard value, 150 W/mK^{45,46}.
Sensitivity analysis
The sensitivity of the temperature decay curve (T) to a parameter, x_{0}, which is either thermal conductivity or thermal boundary resistance, is defined as:^{28}
When x_{0} changes by ±10% within the timescale of interest:
Figure 2d shows an example of the sensitivity to k_{r}, k_{z} and TBR_{eff} for a hBN sample (the properties are taken as k_{r} = 400 W m^{−1} K^{−1}, k_{z} = 4 W m^{−1} K^{−1}, C_{p} = 740 J kg^{−1} K^{−1} and TBR_{eff} = 50 m^{2} K GW^{−1}, which are typical values for hBN^{4,42}).
First principles thermal conductivity calculations
Calculations of the thermal conductivity of hBN are derived from Peierls–Boltzmann phonon transport^{32,47,48} with interatomic force constants (harmonic and thirdorder anharmonic) from DFT^{49,50,51} as implemented by the planewave Quantum Espresso package^{51} within the local density approximation using normconserving pseudopotentials. Electronic structure and relaxation (12 × 12 × 8 integration grid and 110 Ryd planewave energy cutoff) gave lattice parameters, a = 2.478 Å and c = 6.425 Å^{4}, somewhat smaller than measurements^{52}. Density functional perturbation theory^{50} (8 × 8 × 6 integration grid) was used to calculate harmonic force constants and long range Coulomb corrections. Γpointonly electronic structure calculations (200 atom supercells, interactions restricted to 2.8 Å within the plane and 4.2 Å for neighboring layers) were used to determine thirdorder anharmonic force constants for constructing threephonon matrix elements^{4}. Thermal resistance from threephonon interactions^{32,47} and phonon–isotope scattering^{11,53,54} is determined from quantum perturbation theory. More details specific to the DFT calculations and phonon properties (e.g., dispersions and scattering rates) are given in^{4}.
Data availability
The data that support the plots and findings of this paper are available at https://doi.org/10.5523/bris.16v9rfpzb3pl221yzel7x5u5ce.
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Acknowledgements
C.Y., J.W.P., and M.K. acknowledge support from the Engineering and Physics Science Research Council Grant EP/P00945X/1, J.L., S.L., and J.H.E. from the Materials Engineering and Processing program of the National Science Foundation, award number CMMI 1538127. First principles calculations were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. We gratefully acknowledge M. Singh for the help of samples preparation for TTR and TEM analysis, C. Jones for support with EBSD, and W. Waller for multiparameter fitting programming and H. Chandrasekar and R. Baranyai for fruitful discussions.
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J.L. and S.L. synthesized the hBN samples under the direction of J.H.E.; C.Y. performed the Raman, SEM, EBSD, and TTR measurements under direction of M.K.; C.Y. and J.W.P. performed the TTR data analysis. D.C. performed the TEM measurements and analysis. L.L. performed the first principles thermal conductivity calculations. All authors discussed results at all stages and participated in the development of the manuscript.
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Yuan, C., Li, J., Lindsay, L. et al. Modulating the thermal conductivity in hexagonal boron nitride via controlled boron isotope concentration. Commun Phys 2, 43 (2019). https://doi.org/10.1038/s4200501901455
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