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Thermal Conductivity of BAs under Pressure

Boron arsenide (BAs) is an ultrahigh thermal conductivity material with special phonon-phonon scattering properties. At ambient pressure, the bunching of acoustic phonon branches in BAs leads to a small phase-space for three-phonon scattering. Density functional theory predicts that this acoustic phonon bunching effect is sensitive to pressure. To explore this physics, we measure the thermal conductivity of BAs from 0 to 25 GPa using time-domain thermoreflectance in a diamond anvil cell. We characterized two BAs samples with ambient thermal conductivities of 320 and 480 W m-1 K-1. Our experiments show that the thermal conductivity of both samples depends weakly on pressure from 0 to 25 GPa. We attribute the weak dependence of the thermal conductivity of BAs on pressure to the weak pressure dependence of phonon-phonon scattering rates. Our experimental results are consistent with DFT predictions that three-phonon scattering rates increase from 0 to 25 GPa, while four-phonon scattering rates decrease.


Introduction
High-thermal-conductivity materials are desirable for thermal management applications. Power electronic devices operate at power densities higher than 100 W/cm 2 , roughly three orders of magnitude larger than the irradiance of the Sun 1,2 . Discovery and integration of high thermal conductivity materials into electronics offer a route for increasing performance. However, discovery of such materials requires a detailed understanding of what properties lead to high thermal conductivity.
In the 1970s, Slack came up with four rules for finding non-metallic crystals with high thermal conductivity.
These rules are: 1) low average atomic mass, 2) strong interatomic bonding, 3) simple crystal structure, and 4) low anharmonicity 3 . In 2013, Lindsay et al. used density functional theory (DFT) predictions to study the effects of atypical phonon dispersion and expanded Slack's rules 4,5 . Phonon dispersion can have significant effects on three-phonon scattering rates by selection rules 6 . A large frequency gap between acoustic and optic phonon branches (a-o gap) reduces the phase space for three-phonon scattering processes involving two acoustic phonons and one optic phonon (aao processes). A large a-o gap exists in materials made up of heavy and light atoms, e.g., boron and arsenic. Additionally, bunching of acoustic phonon branches also increases the thermal conductivity of a material. Acoustic phonon bunching happens when longitudinal and transverse phonons have similar frequencies across a significant volume of the Brillouin zone. The similar frequencies cause weak three-phonon scattering between longitudinal and transverse acoustic modes (aaa processes). A small phonon scattering phase space leads to a high thermal conductivity. The aim of our experimental study is to use pressure to investigate phonon scattering mechanisms in BAs.
We perform time-domain thermoreflectance (TDTR) measurements of BAs in a diamond anvil cell (DAC).
TDTR is a well-established tool for measuring thermal conductivity 17 . Diamond anvil cells can generate pressures on the scale of GPa. We report the thermal conductivity as a function of pressure for two BAs samples with ambient thermal conductivities of 350 ± 40 and 480 ± 40 W m -1 K -1 . We also measure the thermal conductivity of an MgO single crystal as a control experiment. The thermal conductivity of both BAs samples depends weakly on pressure between 0 and 25 GPa. Alternatively, for MgO, we observe a monotonically increasing thermal conductivity with increasing pressure.
A pressure independent thermal conductivity between 0 and 25 GPa is atypical behavior for nonmetallic materials. Simple models for phonon thermal conductivity predict a monotonic increase in thermal conductivity with increasing pressure 3 . At high pressures, usually, atomic bonds stiffen and phonon frequencies increase. Both effects favor a higher thermal conductivity. Furthermore, three-phonon scattering rates are governed by phonon anharmonicity. Anharmonicity typically decreases with increasing pressure. This is evidenced by the fact that most material's Grüneisen parameter decrease upon compression 18 . Therefore, in most nonmetallic materials, three-phonon scattering rates are expected to decrease with increasing pressure, which favors a larger thermal conductivity at a higher pressure.
To better understand the atypical pressure dependence of BAs's Λ, we use a relaxation time approximation (RTA) model to analyze our experimental results. The RTA model examines how pressure induced changes in phonon group velocities, phonon-phonon scattering, and defect scattering affect Λ vs. P. Our RTA analysis suggests the weak pressure dependence of Λ on P is because total phonon-phonon scattering rates are pressure independent. Therefore, we conclude that our experiments are consistent with DFT predictions for how acoustic bunching, three-phonon scattering, and four-phonon scattering govern Λ vs. P in BAs 14 .

Results
In Fig. 3, we show the thermal conductivity vs. pressure of the two BAs samples. As described in Methods, we performed multiple TDTR measurements at various locations on the BAs samples at each pressure. The thermal conductivity reported in Fig. 3 is the average value from all measurements for a given sample and pressure. The error bars in Fig. 3 denote the standard deviation in thermal conductivity observed between measurements. We show the thermal conductivity vs. pressure for each measured spot in Supplementary   Fig. S6 and S7.
For both samples, we observe that the thermal conductivity has a weak pressure dependence. At nearly all pressures, we observe a thermal conductivity variation within 10% of the value we observe at 0 GPa. We show raw data and thermal model fits at a variety of pressures in Supplementary Fig. S9.
In both samples, we observed Λ values that deviate from the overall trend between 4 and 6 GPa. We are not certain what the origin of this off-trend behavior is. However, the off-trend behavior may be related to the significant changes in Al's optical properties at 1.58 eV across this pressure range. The optical properties of Al near 1.5 eV are strongly affected by absorption due to interband transitions 19 . Upon pressurization, the interband transition energy threshold increases, which causes the thermoreflectance at 1.58 eV crossing through zero and changing sign 20 . We observe significant changes in the amplitude of thermoreflectance and picosecond acoustic signals in our data in this pressure range. We also observe the Al thermoreflectance flips its sign at ~6 GPa.
The standard deviation in Λ we observe across multiple locations is ~10-15% at most pressures. At some pressures, standard deviation is larger, e.g., >20%. We are not certain about why variance increased in these measurements, but believe it is related to the quality of the sample surface. Variance with laser position on the sample is smaller in Sample B than Sample A [see Supplementary Fig. S6 and S7]. For Sample B, we coated the sample with Al after polishing to reduce thickness. This modification in sample preparation procedure improved surface quality. Another possibility is all measurements on Sample B are localized to a smaller region than on Sample A (see Supplementary Fig. S1). We note that, regardless of whether the pressures with large error bars are included or excluded from the data set for Λ vs. P, overall trends are unaffected.
The weak pressure dependence we observe in BAs is in stark contrast with our observations for MgO, see Our results on MgO agree with prior reports 21 .

Discussion and Analysis
In the absence of contextualizing information, the dramatic difference in Λ(P) for BAs vs. MgO (Fig. 3 vs.
Here B is a constant, ̅ is the average mass of an atom in the crystal, 3 is the average volume occupied by one atom in the crystal, θ is the Debye temperature, T is temperature, and γ is the Grüneisen parameter.
We take the pressure dependence of these quantities for BAs from Refs. 25,33. Not surprisingly given BAs' special phonon properties, the LS equation drastically overestimates the pressure dependence of Λ in BAs, see Fig. 3 at P < 20 GPa, as shown in Fig. 3(b). DFT predicts that increases in phonon group velocity and phonon number density are offset by decreases in the phonon-phonon scattering rates 5,14,15 . We believe this is the most likely explanation for our data. However, there is an important difference between our BAs samples and the ideal material evaluated in Ref. 14. Our samples have defects 35,36 . The thermal conductivity of Sample A and B are 50~70% lower than the ~1000 W m -1 K -1 values previously reported for high purity single crystals [7][8][9] . The most likely source of the lower thermal conductivity is phonon-defect scattering.
Previous studies suggest there could be many kinds of defects in BAs 35,37,38 . A boron or arsenic vacancy concentration of ~1.5×10 19 cm -3 would be sufficient to explain the reduced Λ of our samples 37 . AsB-BAs antisite pairs of ~1.5×10 19 cm -3 concentration would explain Λ ~500 W m -1 K -1 of BAs 38 . A recent research suggests ~10 20 cm -3 carbon impurity concentration also could result in the Λ reduction of our samples 35 .
Another contributor to the lower thermal conductivity is the laser spot sizes in our TDTR measurements are comparable to phonon mean-free-paths 39,40 . We report the spot size dependent thermal conductivity of Sample B in the Supplementary Fig. S5, along with a comparison with previously reported data 7 . The ambient thermal conductivity of Sample B is ~620 W m -1 K -1 when using a ~13-μm-radius spot size in the TDTR measurement.
We believe that phonon-defect scattering does not explain the unusual Λ vs. P trend we observe in our BAs samples. Phonon-defect scattering rates may reduce the pressure-dependence of phonon relaxation times.
However, a survey of the literature reveals Λ⁄ remains large and positive, even in materials where phonon-defect scattering is the primarily limiter of phonon mean-free-paths 26 . For example, alloying MgO with 8% Fe causes the ambient thermal conductivity to reduce from 53 W m -1 K -1 to 5 W m -1 K -1 , but the Λ⁄ change from ~4 % per GPa to 8 % per GPa 26 .
For Λ be pressure independent, phonon relaxation times need to decrease with increasing pressure. To show this, we constructed a simple relaxation time approximation model to quantitatively evaluate how various vibrational properties govern Λ vs. P dependence. In our model, we consider the pressure dependence of (1) phonon group velocity, (2) phonon number density, (3) total rates of phonon-phonon scattering, and (4) phonon-defect scattering rates. We set the pressure dependence of phonon group velocities and density of states to mimic DFT predictions 14,25,41 . We assume the total phonon-phonon scattering rate is −1 = ( ).
Here, is a pressure dependent scattering amplitude, and ( ) is a function chosen to mimic the frequency dependence of −1 predicted by DFT calculations accounting for both 3-and 4-phonon processes 14 . We set the magnitude of at 0 GPa so that Λ ≈ 1300 W m -1 K -1 . Finally, to model phonon-defect scattering rates, we assume a frequency dependent point-defect scattering rate like the one described in Ref. 42. Further details of our RTA model are provided in Supplemental Materials.
In Fig. 3(b), we show the prediction of our RTA model with two different assumptions on how depends on pressure. The purple curve assumes ∝ 2 ( ). Alternatively, for the yellow curve, we assume A is independent of pressure. The pressure dependent γ is from DFT calculations 25 . We find that the pressure independent A does a reasonable job reproducing our experimental observation of Λ⁄ ≈ 0 for P < 25 GPa. In the DFT prediction, the relaxation times due to all phonon-phonon scattering processes change very little at pressures below 30 GPa 14 .
In conclusion, we measured the pressure dependent thermal conductivity of two BAs samples between 0 and 25 GPa. In contrast to the typical behavior for nonmetallic materials, we observe weak pressure dependence. We attribute the weak pressure dependence of Λ to pressure independent phonon-phonon scattering rates at P < 25 GPa. Our results are consistent with DFT models for how the pressure dependence of acoustic phonon bunching affects three-and four-phonon scattering rates. Our results improve fundamental understanding of the complex interplay between phonon dispersion, phonon scattering, and thermal transport in high thermal conductivity materials.
Details about the synthesis can be found in Ref. 43. We include X-ray diffraction and Raman scattering data on BAs in Supplementary Fig. S2.

Sample preparation
We prepared two pieces of BAs for DAC experiments. The first sample, which we label as Sample A, was first coated with a ~90-nm-thick Al film by electron beam evaporation. After coating, we polished the BAs from the uncoated side down to ~7 ± μm. The final thickness was measured with an optical microscope.
We loaded one 50 × 80 μm sample into a DAC with a culet size of 00 μm. For the second sample, labelled Sample B, we changed our procedure to improve the surface quality. We first polished a piece of BAs crystal down to ~ 7 ± μm. Then, we deposited a ~ 90-nm-thick Al film on the sample. We loaded ruby spheres alongside the samples as pressure indicators. We used silicone oil (Polydimethylsiloxane, CAS No. 63148-62-9 from ACROS ORGANICS) as the pressure medium for both measurements.
We used 50 μm thick stainless-steel gaskets and pre-indented them in our DAC to a thickness between 30 to 60 μm. Then we drilled holes with a diameter of 170 μm at the center of the indentations by a laser drill system. The holes serve as containers for the samples, ruby spheres, and pressure medium.

Time-domain thermoreflectance (TDTR) in diamond anvil cells
We measured the thermal conductivity of BAs at ambient and high pressures by TDTR. TDTR is a well- loaded inside a DAC. The pressure of the system is calibrated using the pressure dependent shift of the R1 line in the ruby fluorescence spectrum 44 . We also use the Brillouin frequency of silicone oil as a second measure of pressure 45 . Fig. 2(c) shows a Brillouin oscillation that we observe in our experimental TDTR signals. When the pump beam heats the Al surface, it launches a strain wave into the silicone oil medium.
The strain wave front moves at the speed of sound of silicone oil. Both the strain wave and Al can reflect the subsequent probe beam. These two reflected probe beams interfere with each other and cause Brillouin oscillations in the Vin signal 46 .
We used the beam-offset method to measure the laser spot size 47 . The 1/e 2 radius was 5.1 μm and 4.5 μm for the measurements on Sample A and B, respectively.
We observed ~10-15% variation in the TDTR derived thermal conductivity depending on the measured locations on the samples. We attribute these variations to two factors. First, sample defects or inhomogeneities can affect TDT signals. We tried to avoid visible defects on the sample's surface, but processing of the sample for DAC measurements led to less than perfectly smooth and clean surfaces. A second reason for variation is likely intrinsic to the crystal. Prior studies of BAs crystals report a Λ variation of ~10-15% across the crystal surface 7 . To minimize the effect of this location sensitivity in our Λ( ) measurements, we initially tried to restrict our experiments to a single location at all pressures.
However, this proved challenging for several reasons. Changing the DAC pressure requires removing the DAC from the TDTR setup, and then resetting up our experiment. Furthermore, the sample undergoes a small amount of deformation upon pressurization. To cope with these challenges, we changed our approach.
Instead of trying to restrict our experiment to a single location, we performed TDTR scans at multiple locations on the samples at each pressure. We measured four spots which were 8 μm away from each other on Sample A. We measured five spots which were μm away from each other on Sample B (see Supplementary Fig. S1). The thermal conductivity values we report for Sample A and B are the average from all four or five spots, respectively.
As a control experiment, we measured the pressure dependent thermal conductivity of a MgO sample. The pressure dependence of MgO's thermal conductivity is well studied experimentally 21 and theoretically 14,48 .
We prepared the MgO sample following similar procedures as Sample A (coat with Al first, then polish to reduce thickness). Then we performed TDTR measurements at pressures between 0 and 19 GPa. The 1/e 2 beam radius for measurements of MgO was μm.

Data analysis of TDTR under pressure
We use a bidirectional heat diffusion model to analyze the collected TDTR data 17 . The bidirectional model accounts for heat flow from the Al transducer into both the BAs and silicone oil. The thermal conductivity, heat capacity and thickness of each layer are the input parameters in the heat diffusion model. Therefore, we must estimate how these parameters evolve with pressure to interpret our TDTR data. Below, we describe how we account for the pressure dependence of all parameters.
Prior to loading the sample into the DAC, we measure the Al film thickness by picosecond acoustics 46 . At high pressures, we assume BAs shrinks equally in every direction since BAs is a cubic crystal 33 . If the volume of BAs at pressure P is VP, and the in-plane area is SP, then = 0 ⋅ ( ∕ 0 ) 2 3 . Here, V0 and S0 are volume and area of BAs at 0 GPa. We assume the in-plane area of Al is equal to Sp. Then the thickness of Al at pressure P will be ℎ ≈ / . Here, is the Al volume at pressure P based on Al's equation of state 49 .
To estimate the pressure dependence of Al's heat capacity, we follow Ref. 45, and use a Debye model. For silicone oil, we use previously reported pressure dependent heat capacities and thermal conductivities 50 .
To model the pressure-dependence of BAs's heat capacity, we use a simple isotropic model for the phonon dispersion. We assume = − 2 .
Here ω is the phonon frequency, is the longitudinal or transverse speed, k is the wavevector magnitude, and A is a constant. The value of A is determined by the phonon frequency at the Brillouin zone boundary. We set the values of and A to mimic DFT predictions for phonon dispersion relations vs. pressure 14,51 . Fig. 1