The dossier's account of Boron Arsenide (BAs) has one awkward step. Theory first predicted a room-temperature thermal conductivity far above what was measured. The gap was then closed by adding Four-Phonon Scattering, a higher-order process that most calculations had left out. That raises an obvious question. If four-phonon scattering takes so much off BAs, why is BAs still such a good conductor? This preprint, by Navaneetha K. Ravichandran (Indian Institute of Science, Bangalore) and David Broido (Boston College), posted to arXiv on 19 March 2020 (2003.08893v1), answers by widening the view. Instead of looking at BAs alone, it calculates heat transport in seventeen crystals that share its zinc blende structure.
What the paper set out to test
Heat in an insulating crystal is carried by phonons, which are quantised lattice vibrations. Conductivity is limited by how quickly phonons scatter off one another. The standard calculation, which goes back to Peierls, counts only collisions involving three phonons. The authors ask how often that lowest-order theory fails for strongly bonded, weakly anharmonic crystals. Earlier work had found such failures only in a few cases: BAs, boron antimonide (BSb) and aluminium antimonide (AlSb).
The seventeen compounds were GaAs, AlP, AlAs, GaP, GaSb, InAs, InSb, AlSb, InP, BN, BP, BAs, BSb, SiC, GaN, InN and AlN. For each one the paper solves the phonon Boltzmann transport equation in four versions. These are three-phonon scattering alone, three-phonon plus four-phonon, and each of those with or without scattering from the natural mix of isotopes. The harmonic force constants come from density functional perturbation theory (Quantum ESPRESSO, local density approximation). For the third- and fourth-order anharmonic force constants that set the scattering rates, the paper uses a thermal stochastic snapshot technique, and it solves the Boltzmann equation iteratively.
The main findings
Three-phonon theory fails often. The paper reports that adding four-phonon scattering lowers the calculated conductivity of nine of the seventeen compounds by at least 20% at 300 K and at least 40% at 750 K. Among the failures not flagged before, the authors single out indium phosphide (InP) as "a particularly catastrophic failure"; they also report significant failures in GaSb, InAs and InSb. For InP, four-phonon scattering cuts the pure-crystal value by around 45% at 300 K and almost 70% at 750 K. The authors say the corrected curves agree much better with measured data. The comparison covers twelve compounds; five (AlP, GaN, InN, AlN and BSb) had no measurements to compare with.
Selection rules explain where three-phonon scattering goes quiet. Phonons have to conserve energy and crystal momentum when they collide. The shape of a material's vibration spectrum can therefore rule out whole classes of three-phonon collisions. The paper lists six such rules, two of which it describes as new (labelled AAA#2 and AOO#2). One effect is familiar from the 2013 BAs prediction. Arsenic is roughly seven times heavier than boron, which opens a wide frequency gap between acoustic and optic vibrations. Together with bunched acoustic branches, that leaves heat-carrying acoustic phonons with very few ways to scatter. The new AOO#2 rule adds an explanation for why optic phonons in BAs and AlSb would also live unusually long under three-phonon theory, but not in BSb. Four-phonon processes, by contrast, are found to be barely constrained by selection rules. So wherever three-phonon scattering is suppressed, four-phonon scattering takes over as the main limit.
Compounds containing B, C or N are different. The authors present this as their most striking result. The eight compounds containing boron, carbon or nitrogen (BAs, BSb, BP, BN, SiC, GaN, AlN and InN) have four-phonon scattering rates far below those of the nine that do not. For BAs the paper puts the difference at an order of magnitude compared with InP and AlSb. The authors trace it to bonding. BCN compounds have stiffer bonds, but their fourth-order anharmonicity is weak relative to the harmonic part. For BP, BN, SiC, AlN, InN and GaN, the paper reports only small room-temperature reductions of 3% to 15%.
What this means for boron arsenide
For BAs, the paper reports that four-phonon scattering lowers the room-temperature pure-crystal conductivity from 3100 to around 1400 W m⁻¹ K⁻¹. That three-phonon starting value for the pure crystal is higher than the 2200 W/(m·K) figure given on the Four-Phonon Scattering page, and the paper does not reconcile the two. With natural isotopes included, the paper's figure is around 1300 W m⁻¹ K⁻¹. It describes this as confirmed by the three 2018 measurements by Kang et al., Li et al. and Tian et al. The authors call it two to three times higher than SiC, BP, copper or silver.
The key test is a counterfactual. The authors multiplied the four-phonon rates in BAs by ten, to roughly match the non-BCN compounds. The calculated room-temperature conductivity then fell by over 70%, to 330 W m⁻¹ K⁻¹. On this account, BAs owes its record to two separate features. Its vibration spectrum shuts down most three-phonon scattering, and its chemistry keeps four-phonon scattering weak. The paper calls the second "one of the fortuitous benefits that nature gives to the BCN compounds."
Wider claims and their limits
The paper draws two cautionary conclusions. First, it argues that a high-conductivity material made without B, C or N atoms is unlikely. In none of the nine non-BCN compounds did a window of weak three-phonon scattering coincide with weak four-phonon scattering. Second, it argues that any calculation predicting large heat flow through optic phonons is probably missing four-phonon effects. It also suggests that strong four-phonon decay of optic phonons in BAs, AlSb, InP, AlAs and BSb should show up in temperature-dependent Raman linewidths. That is a proposed test, not one the paper carries out.
All of this is calculation, and the "measured" agreement is with other groups' published data. The authors themselves describe how strongly some of the selection rules apply as qualitative, and say quantitative calculations are needed to pin it down. Some results, such as the BP and SiC values with certain processes switched off (over 6000 and 3800 W m⁻¹ K⁻¹ respectively), are artificial and cannot be realised, though the authors suggest hydrostatic pressure might partly tune them. The paper's claim that five-phonon and higher processes should be negligible is an inference from the uniformity of four-phonon rates, not a calculation. The authors themselves say further calculations are needed to fully establish that four-phonon scattering belongs in routine studies of strongly bonded crystals.
Connections
The paper makes a general point that measured conductivity tends to depend more strongly on temperature than many three-phonon-only calculations show; related measurements on BAs are covered in Strong Temperature Dependence of Thermal Conductivity in High-Purity Cubic Boron Arsenide. The paper's note that boron, not arsenic, carries the isotope disorder in BAs bears on Isotope Enrichment and Thermal Conductivity and Testing the Boron Isotope Effect in Boron Arsenide: Boron-10, Boron-11 and Natural-Boron Crystals Compared (Sun et al.). The broader story is in Ultrahigh Thermal Conductivity in Boron Arsenide.
Source notes & attribution
- https://arxiv.org/pdf/2003.08893