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Unusual high thermal conductivity in boron arsenide bulk crystals

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Science  10 Aug 2018:
Vol. 361, Issue 6402, pp. 582-585
DOI: 10.1126/science.aat7932

Moving the heat aside with BAs

Thermal management becomes increasingly important as we decrease device size and increase computing power. Engineering materials with high thermal conductivity, such as boron arsenide (BAs), is hard because it is essential to avoid defects and impurities during synthesis, which would stop heat flow. Three different research groups have synthesized BAs with a thermal conductivity around 1000 watts per meter-kelvin: Kang et al., Li et al., and Tian et al. succeeded in synthesizing high-purity BAs with conductivities half that of diamond but more than double that of conventional metals (see the Perspective by Dames). The advance validates the search for high-thermal-conductivity materials and provides a new material that may be more easily integrated into semiconducting devices.

Science, this issue p. 575, p. 579, p. 582; see also p. 549

Abstract

Conventional theory predicts that ultrahigh lattice thermal conductivity can only occur in crystals composed of strongly bonded light elements, and that it is limited by anharmonic three-phonon processes. We report experimental evidence that departs from these long-held criteria. We measured a local room-temperature thermal conductivity exceeding 1000 watts per meter-kelvin and an average bulk value reaching 900 watts per meter-kelvin in bulk boron arsenide (BAs) crystals, where boron and arsenic are light and heavy elements, respectively. The high values are consistent with a proposal for phonon-band engineering and can only be explained by higher-order phonon processes. These findings yield insight into the physics of heat conduction in solids and show BAs to be the only known semiconductor with ultrahigh thermal conductivity.

Materials with high thermal conductivity (κ) can help to address a range of grand technological challenges, such as keeping nanoelectronic devices cool by removing the high-density heat generated within them. At room temperature (RT), diamond and graphite, the two carbon allotrope bulk crystals, have a record high κ of about 2000 W/m·K (14). However, high-quality natural diamond is scarce and expensive. Although future technological advances may help to alleviate the cost of high-quality synthetic diamond, the large mismatch in the coefficient of thermal expansion between diamond and common semiconductors can introduce large thermal stresses. The κ of graphite, moreover, is highly anisotropic, with the cross-plane value being two orders of magnitude smaller than the corresponding in-plane value (1). The thermal anisotropy and the weak interlayer bonding have limited the use of graphite for thermal management. In addition, the large bandgap of diamond and semimetallic behavior of graphite prevent their use as active electronic materials. Common electronic materials such as copper and silicon have a RT κ of about 400 and 150 W/m·K, respectively (1), which are well below the diamond value. The highest measured RT κ values for semiconductors are about 490 W/m·K in silicon carbide (5) and 460 W/m·K in boron phosphide (6). Although these values are comparable to the highest electronic contribution to κ in metals, it is desirable to discover semiconductors with values of κ comparable to the ultrahigh value for diamond.

In semiconductors and nonmagnetic insulators, the thermal conductivity is dominated by the phonon contribution. Thermal conductivity is typically limited by the lowest-order process arising from the anharmonicity of the interatomic potential, three-phonon scattering, at and above RT (7). According to the criteria established by Slack about half a century ago (2), only crystals composed of strongly bonded light elements should exhibit ultrahigh κ. However, Lindsay, Broido, and Reinecke recently proposed that ultrahigh κ could be achieved in compounds that combine a light and a heavy atom if (i) the frequency gap between heat-carrying acoustic phonons and optic phonons is sufficiently large and (ii) some of the acoustic phonons with different polarizations have regions of similar frequencies away from the Brillouin zone center. First-principles calculations supported this phonon-band engineering concept in predicting that cubic boron arsenide (BAs) should have a RT κ of around 2000 W/m·K when only three-phonon interaction is considered (8, 9). Subsequent theoretical calculations found that four-phonon scattering would lower the calculated RT κ in BAs to about 1400 W/m·K (10), which is still exceptionally high, but surprising, because three-phonon scattering accurately describes the measured κ data for many semiconductors and insulators, and higher-order processes are expected to be weak at RT.

Synthesis of high-quality BAs bulk crystals has proved challenging, which has prevented experimental verification of the unusual predicted transport properties. Several measurements have captured RT κ values of only 200 to 350 W/m·K in small BAs particles (1113). The inability to measure an ultrahigh κ for BAs has limited adoption of the phonon-band engineering strategy as a viable route for achieving ultrahigh κ, and the possibility of higher-order phonon processes suppressing κ has remained.

We report experimental evidence that validates the phonon-band engineering route. We grew BAs bulk crystals from seed microparticles in a chemical vapor transport (CVT) process. Local measurements of low-defect regions obtained a RT κ exceeding 1000 W/m·K, whereas multiple local and bulk transport measurement methods yielded average RT κ values of about 800 and 900 W/m·K for two bulk crystal samples. The bulk crystal has a high κ despite twin boundaries and other defects known to decrease κ. Both the peak and average κ values show a rapid decrease with increasing temperature, which is a signature of lattice anharmonicity. This behavior agrees with our detailed first-principles theoretical model that included both three- and four-phonon interactions.

Previously reported efforts to synthesize BAs yielded only particles with maximum dimensions less than about 500 μm (1113). Because bulk-size crystals are required for device applications, we investigated a seeded CVT growth mechanism for the synthesis of bulk BAs crystals. In this approach, we used small single BAs crystals with a lateral dimension of a few micrometers as seeds to ensure that the nucleation centers were sparse and under control during the growth process (13, 14). We optimized seed-crystal quality and distribution to obtain BAs crystals as large as about 4 mm by 2 mm by 1 mm within a 14-day period of seed growth followed by another 14-day period of crystal growth from the chosen seed crystals (15). This increased crystal size allowed us to use transport measurement techniques established for bulk samples. Increasing the growth time would increase the crystal size. To probe the crystal structure of the BAs, we obtained an aberration-corrected, annular dark-field scanning transmission electron microscopy (STEM) image (Fig. 1A) and a low-magnification bright-field TEM image (Fig. 1B) of representative crystals. We found planar defects (Fig. 1B) that were mirror twin boundaries (Fig. 1, C to E).

Fig. 1 STEM characterizations of BAs.

(A) Annular dark-field STEM image within one grain of BAs, looking down the [110] zone axis. (B) Low-magnification bright-field TEM image near the surface of the BAs crystal. Horizontal lines indicate the locations of mirror twin boundaries. (C) Annular dark-field STEM image showing the atomic structure of the mirror twin boundary from the region highlighted by the red box in (B). (D) Electron diffraction pattern of BAs within a single grain. (E) Electron diffraction pattern of BAs across the grain boundary, showing the presence of the mirror twin.

We discovered unusually high but nonuniformly distributed κ in these BAs crystals by using time- and frequency-domain thermoreflectance (TDTR and FDTR) techniques with micrometer resolution (1618). We used a large 58-μm-diameter pump laser spot and a small 9-μm-diameter probe laser spot in conjunction with a relatively low modulation frequency of 3 MHz to improve the TDTR measurement accuracy (15). For both the pump beam and probe beam, the diameter quoted here is the 1/e2 diameter of the Gaussian beam. We used the same TDTR platform and parameters to measure the κ of a synthetic diamond crystal (Figs. 2A and 3). The values that we measured for diamond are in good agreement with theoretical calculations and literature values (14). Among the single-spot measurements at five locations on BAs sample 1, the highest and lowest RT κ values were 1160 ± 130 and 640 ± 70 W/m·K, respectively. Among the 10 single-spot TDTR measurements on sample 2, RT κ values ranged from 790 ± 100 to 450 ± 60 W/m·K. We found a sharp decrease in κ as temperature increased to 500 K at the location on sample 1 where we found our maximum κ (Fig. 3). We found the same behavior at a location on sample 2 with a RT κ of 740 ± 110 W/m·K (Fig. 3). This temperature behavior is consistent with dominant anharmonic phonon-phonon scattering. At the same sample 1 spot where our TDTR measurements yielded the highest κ, our single-spot FDTR measurements obtained a κ of 1310 ± 740 W/m·K, where the large uncertainty is due to the use of a small 3.36-μm-diameter pump beam and 2.60-μm-diameter probe spot to measure the high-κ region (19). We observed no spot-size dependence of κ when we increased the pump and probe laser spot sizes to 5.60 and 4.80 μm, respectively.

Fig. 2 TDTR and FDTR measurements.

(A) Representative TDTR phase signals and best-fitted curves for a diamond crystal acquired from Element Six and a BAs crystal at different temperatures. The diamond sample has the natural carbon abundance (1.1% 13C) and a low level of boron [<0.05 parts per million (ppm)] and nitrogen (<1 ppm) impurities. The 300 K data are averaged over 200 and 140 runs at the same location for diamond and BAs, respectively. The data for BAs at higher temperatures are averages of about 10 runs and show slightly increased noise. (B) Representative FDTR signal phase as a function of the pump modulation frequency measured on a BAs crystal, diamond, sapphire, and fused silica. The phase lag between the probe and the pump increases with decreasing sample κ.

Fig. 3 Measured thermal conductivity of BAs in comparison with values from theoretical calculations and other crystals.

Calculated κ versus temperature for BAs (black) and diamond (green) including only three-phonon scattering (dashed lines) and both three- and four-phonon scattering (solid lines); measured κ for diamond by TDTR (green diamonds); measured κ for BAs samples 1 (solid red symbols) and 2 (open red symbols) by TDTR; measured κ for sample 3 by FDTR (solid orange star, mean value), steady-state (SS; open blue squares), and lock-in Raman (open brown square) methods; and measured κ for sample 5 by the steady-state method (solid blue squares). Also shown are the fits to measured steady-state and TDTR κ for BAs (blue and red solid lines, respectively) and reported measured κ for GaN (21) and GaAs (22) (magenta and purple triangles, respectively). The error bars for the TDTR and FDTR data represent one standard deviation and were obtained via Monte Carlo simulations and derivative matrix-based analysis of uncertainty propagation, respectively (15). The error bars for the steady-state and lock-in Raman measurements were calculated by propagating random errors at 95% confidence and combining them with systematic errors (15).

The increased size of the bulk BAs crystals allowed us to make steady-state comparative measurements of the bulk κ (Fig. 4A) (15). Without accounting for the contact thermal resistance errors between the thermocouples and the sample, we obtained a κ of 770 ± 100 W/m·K at 305 K on a 0.1 mm by 0.2 mm by 2 mm bar cut from sample 3. The thermal conductivity increased with deceasing temperature. In comparison, we obtained an average value of 820 ± 140 W/m·K with FDTR at 14 locations on another piece cut from sample 3 (Fig. 3). We addressed the uncertainty due to contact resistance by using a lock-in Raman thermometry approach with a sinusoidally modulated heating current at a low modulation frequency (υ) of about 1 mHz (15). A fast Fourier transform of the measured Raman peak shift shows clear modulation at the second harmonic frequency corresponding to the Joule heating frequency (Fig. 4B), which we used to measure the temperature drops along the Si and BAs bars (Fig. 4A). The Raman measurements obtained similar temperature gradients as the thermocouple measurements in both silicon and BAs (Fig. 4C) and a κ of 690 ± 120 W/m·K at 338 K (Fig. 3). On samples 4 (fig. S19B) (15) and 5 (Fig. 3), we used the steady-state method to measure a bulk κ at 300 K of 570 ± 70 and 920 ± 120 W/m·K, respectively, and found a similar temperature dependence as for samples 1 to 3.

Fig. 4 Steady-state comparative and lock-in Raman thermometry measurements.

(A) Temperature modulation amplitudes (ΔT) measured by Raman thermometry at two locations on the Si bar and two locations on the BAs bar. The lines are linear fittings to the measurement data. The inset is a schematic diagram of the experimental setup for thermocouple (TC) and Raman measurements. (B) Amplitude spectrum of the measured Raman peak modulation for BAs at locations x = 3.38 mm (location 1) and 4.39 mm (location 2). The curve for x = 4.39 mm is shifted manually by +0.2 along the x axis so that it can be distinguished from the other curve. The inset shows the modulation of the Raman peak frequency of BAs at location x = 3.38 mm as a function of the cycle number during the first six cycles. (C) Temperature gradients (∇T) on the Si and BAs bars obtained from TC and Raman measurements. The ambient temperature was 308.9 K, and the heater power amplitude was 0.081 W. The TC measurement error bars include random uncertainties with 95% confidence and systematic uncertainty, and the Raman measurement error bars consist of random uncertainties from the signal-to-noise ratio and systematic error (15).

The measurement results agree with first-principles calculations of the κ of BAs, including both three- and four-phonon scattering, scattering of phonons by the natural boron isotope mix, and phonon scattering by point defects and grain boundaries. Although the κ of most high-quality insulating crystals is well described by lowest-order three-phonon scattering, in BAs, the phase space for three-phonon scattering is unusually small (8). We show that four-phonon scattering is necessary to accurately capture the intrinsic κ of BAs (10). We implemented several changes in our calculation from that of Feng et al. (10) to improve the accuracy (15). In addition, we found hole scattering of phonons to be negligible at the hole concentration of 7.6 × 1018 cm−3 that we measured in the p-type BAs semiconductor sample (15).

Our calculated BAs κ at 300 K including three-phonon, four-phonon, and phonon-isotope scattering is 1260 W/m·K, about half that obtained without four-phonon scattering (2330 W/m·K) and about 10% smaller than that obtained by Feng et al. (10). The BAs κ that we calculated including only three-phonon and phonon-isotope scattering, κ3 (dashed black curve, Fig. 3), lies well above all measured data. The calculation including four-phonon scattering as well, κ3+4 (solid black curve), suppresses κ and brings the calculated values close to the measured local high TDTR values (solid red circles). It also provides a strong temperature dependence, as previously found (10). Importantly, the TDTR temperature behavior follows the temperature dependence of κ3+4, which is stronger than that of κ3. We fit the steady-state and TDTR data by including additional scattering from assumed point defects and grain boundaries (Fig. 3) (15). Defect scattering mechanisms are typically much less sensitive to temperature change than phonon-phonon scattering, so increasing defect scattering to match the measured RT value weakens the temperature dependence of κ. The large defect concentrations needed to match the magnitudes of the measured data when including only three-phonon scattering cannot produce the steep observed temperature dependence (figs. S30 and S31) (15). In contrast, the best fit of κ3+4 is excellent for the steady-state data (fig. S19B) (15) and reasonably good for the TDTR data.

The comparison between the measurements and theoretical calculations provides strong evidence that BAs is distinct from other known high-κ materials in achieving κ through the phonon-band engineering concept and in having higher-order phonon-phonon interactions play such a large role. By breaking the conventional theoretical criteria, these findings have firmly established a different route to ultrahigh κ and highlighted the rich physics of phonons. Our strategy for the growth of bulk BAs crystal is an important step toward implementation in future applications of BAs, which is now the only known semiconductor with a bandgap (20) comparable to silicon and an ultrahigh room-temperature thermal conductivity.

Supplementary Materials

www.sciencemag.org/content/361/6402/582/suppl/DC1

Materials and Methods

Figs. S1 to S33

References (2330)

References and Notes

  1. Supplementary materials.
Acknowledgments: We thank S. Huberman, J. Zhou, and Y. Xu for assistance with the experiments. Funding: The work was supported by the Office of Naval Research under MURI grant N00014-16-1-2436. Author contributions: F.T. and Z.R. designed the crystal growth process. F.T. grew the crystal samples. B.S., K.C., M.G., T.-H.L., Z.D., A.J.S., and G.C. performed the thermal conductivity measurements and analyses by microprobe. X.C., S.S., J.K., Y.Z., J.Z., and L.S. performed the thermal conductivity measurements by steady-state and Raman methods and the electrical transport property measurements. N.K.R. and D.B. performed the thermal conductivity calculations. Y.L. and P.Y.H. performed electron microscopy characterization in the Fredrick-Seitz Material Research Laboratory Central Facilities. J.S., G.A.G.U.G., H.S., H.Z., L.D., S.H., S.C., and C.-W.C. performed structural characterizations. F.T., B.S., X.C., D.B., L.S., G.C., and Z.R. wrote the paper. All authors contributed to the discussion of the results and writing of the manuscript. The project was directed and supervised by Z.R., G.C., L.S., and D.B. Competing interests: A provisional patent application has been filed based on this work. D.B. is an inventor on U.S. Patent 9,986,663. Data and materials availability: All data are available in the manuscript and supplementary materials.
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