Geometrical frustration in an elemental solid: An Ising model to explain the defect structure of $\beta$-rhombohedral boron

Recently, it was reported that $\beta$-rhombohedral boron has a negative defect formation energy, which explains the presence of a macroscopic amount $\left(4\text{at}\text{.}\mathrm{\%}\right)$ of intrinsic defects. In this work, it is shown that the defects in boron have geometrical frustration described by an antiferromagnetic Ising model on an expanded kagome lattice, which is responsible for the reported macroscopic residual entropy. We suggest that the reported anomalies in the transport properties of $\beta$-boron are due to the hopping of boron atoms between nearly degenerate configurations.

Figure 1

(Color) The lattice structure of POSs in $\beta$-rhombohedral boron. (a) The POS forms a layered lattice structure at $z=0$ (not drawn) and $z=c/2$ (drawn), where $c$ is the lattice parameter of a rhombohedral crystal. Two clusters made of 28 boron atoms (${\text{B}}_{28}$, gold bonds) are connected by an interstitial atom and B13 vacancies (red spheres), and they are located in the middle of the rhombohedral cell. (b) and (c) depict the connectivity between B13 and B16–B20. B13 (red), B16 (blue), B17 (green), and B18 (yellow) are displayed in (b), and the rest, B19 (light green) and B20 (gray), are displayed in (c) separately for the sake of clarity. The viewpoint is along the $z$ axis, and the size of spheres and bonds correspond to the offset in the $z$ axis. (d) A single layer of this lattice, an expanded kagome lattice, is depicted, where the corners of the hexagonal lattice are decorated by triangles. Each POS lattice consists of two layers of expanded kagome lattices connected to each other through the B13 site. For a more detailed explanation, see Refs. 5, 19.

Figure 2

(Color) In (a), one of the ground-state spin configurations of the AF Ising model on a kagome lattice is realized, which is described as “two-up, one-down” or “two-down, one-up” spin configurations that maximize the number of opposite spins on a triangle. The expanded kagome lattice is made by decorating the corners of the hexagonal rings with triangles (Ref. 13). If one duplicates the spins on the kagome lattice (a) onto the spins on an expanded kagome lattice, (b) is obtained. Choose half of the triangles in an alternating manner and flip the spins on the shaded triangles to get (c). The resulting spin configuration gives the ground state of the AF Ising model on an expanded kagome lattice, where, in addition to the AF intertriangle bonds, the “two-up, one-down” ground-state rule for the intratriangle bonds is preserved.

Figure 3

(Color) (a) Specific heat of the Ising model fit to ab initio data, including error bars, as a function of temperature. (b) Entropy as a function of temperature calculated by thermodynamic integration from the high-temperature limit. The data were obtained for a $4\times 4\times 6$ supercell (4032 sites) derived from the rhombohedral cell. The solid lines in (a) and (b) are guides for the eyes. The horizontal dotted line in (b) is the high-temperature limit of the entropy $S\left(T\to \infty \right)=-R\left(p_{occ}\text{log}{p}_{occ}+p_{unoc}\text{log}{p}_{unocc}\right)$, where, $p_{occ}$ is the occupation rate (23/126), $p_{unocc}=1\text{−}{p}_{occ}$, and $R$ is the gas constant. We obtained 1280 data points, and only 10% (with error bars) are plotted for clarity. (c) Measured specific heat (crosses) (Ref. 24), together with calculated phonon specific heat (blue line) and the sum of phonon and configurational specific heats (red line). The temperature range below 300 K is magnified in the inset. See the supporting material of Ref. 5 for the specific details of the phonon calculations.

Figure 4

(Color) The total energy (eV) per rhombohedral cell along the reaction coordinates of the two lowest-energy barrier paths. $\text{B}16\to \text{B}16$ diffusion (blue arrow) corresponds to the intratriangle diffusion along a symmetric direction with respect to the B19 atom. The $\text{B}19\to \text{B}20$ diffusion (the purple arrow) takes place with the far-side B16 site occupied (as drawn).