Pressure-induced polymorphism in ${\mathrm{SrB}}_6$ and deformation mechanisms of covalent networks

We study the high-pressure structures of ${\mathrm{SrB}}_6$ up to 200 GPa using first-principles structure prediction calculations and high-pressure x-ray diffraction experiments. The computations show that the ambient-pressure cubic phase transforms to an orthorhombic structure $\left(Cmmm\right)$ at 48 GPa, and then to a tetragonal structure $\left(I4/mmm\right)$ at 60 GPa. The high-pressure experiments are consistent with the theoretically predicted tetragonal structure, which was quenched successfully to ambient conditions. Pressure induces simple boron octahedra to form complex networks in which the electrons are delocalized, leading to metallic ground states with large density of states at the Fermi level. Calculated stress-strain relations for the $I4/mmm$ structure of ${\mathrm{SrB}}_6$ demonstrate its intrinsic hard nature with an estimated Vickers hardness of 15 GPa, and reveal a novel deformation mechanism with transient multicenter bonding that results in the combination of high strength and high ductility. Our findings offer valuable insights for understanding the rich and complex crystal structures of ${\mathrm{SrB}}_6$, which have broad implications for further explorations of hexaboride materials.

Figure 1

Polyhedral views of (a) $Pm\overline{3}m$ structure, (c) $Cmmm$ structure, and (e) $I4/mmm$ structure of ${\mathrm{SrB}}_6$. The phase transition mechanisms of (b) $Pm\overline{3}m\to Cmmm$ structure and (d) $Cmmm\to I4/mmm$ structure. The black lines denote the unit cells and large and small spheres represent Sr and B atoms, respectively.

Figure 2

(a) Calculated enthalpy curves for ${\mathrm{SrB}}_6$ (relative to the $Pm\overline{3}m$ structure) as a function of pressure. (b) The relative internal energy $\mathrm{\Delta }\mathrm{U}$ and $\mathrm{p}\mathrm{\Delta }\mathrm{V}$ term for the $Cmmm$ and $I4/mmm$ structures with respect to the $Pm\overline{3}m$ structure. (c) Calculated volumes as a function of pressure for different phases of ${\mathrm{SrB}}_6$.

Figure 3

X-ray diffraction and equation of state of the $I4/mmm$ structure of ${\mathrm{SrB}}_6$. (a) Experimental XRD data (black points) collected at 159(9) GPa with Le Bail fitting (blue line). Green and purple ticks (and selected Bragg reflections) indicate contributions from Re and Ar, respectively. The upper panel shows the cake-type image of the two-dimensional XRD pattern with incomplete powder averaging. (b) Experimental $pV$ data for $I4/mmm{\mathrm{SrB}}_6$ are shown as points, and the solid line is the theoretical result (GGA PBE). The estimated error in volume is smaller than the symbol size. The $pV$ data were fitted using the second order Birch-Murnaghan EoS [56], and the estimated experimental bulk modulus $B_0=148\left(6\right)\mathrm{GPa}$ is consistent with the calculated $B_0=152\mathrm{GPa}$ for $I4/mmm\mathrm{Sr}{\mathrm{B}}_6$. The calculated EoS with $B_0=123\mathrm{GPa}$ for cubic ${\mathrm{SrB}}_6$ is shown for comparison.

Figure 4

The projected density of states (DOS) for (a) the $Pm\overline{3}m$ structure at 0 GPa, (b) the $Cmmm$ structure at 50 GPa, and (c) the $I4/mmm$ structure at 200 GPa. The densities of states at the Fermi level increase with pressure, indicating enhanced metallicity.

Figure 5

(a) Calculated tensile stress-strain relations for the $I4/mmm$ structure. (b) Calculated shear stress-strain relations for the $I4/mmm$ structure in the (100) easiest cleavage plane. (c) Two-dimensional electron localization function (ELF) for the key structural snapshots of the $I4/mmm$ structure in the tensile and shear directions. The small and large spheres represent B and Sr atoms, respectively.