Though extensively studied, hardness, defined as the resistance of a material to deformation, still remains a challenging issue for a formal theoretical description due to its inherent mechanical complexity. The widely applied Teter's empirical correlation between hardness and shear modulus has been considered to be not always valid for a large variety of materials. Here, inspired by the classical work on Pugh's modulus ratio, we develop a theoretical model which establishes a robust correlation between hardness and elasticity for a wide class of materials, including bulk metallic glasses, with results in very good agreement with experiment. The simplified form of our model also provides an unambiguous theoretical evidence for Teter's empirical correlation.

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Conventional MAX phases (M is an early transition metal, A represents a p-block element or Cd, and X is carbon or nitrogen) have so far been limited to carbides and/or nitrides. In the present work, a series of stable layered ternary borides were predicted by combining variable-composition evolutionary structure search and first-principles calculations. The predicted Hf2InB2, Hf2SnB2, Zr2TlB2, Zr2PbB2, and Zr2InB2 show a Ti2InB2 type of structure (space group P6̅m2, No. 187, Nat. Commun. 2019, 10, 2284), and the structures of Hf3PB4 and Zr3CdB4 share the same space group with Ti2InB2 but belong to a new structure type. These two structural prototypes, M2AB2 and M3AB4 (M is Zr or Hf), have the composition and local structures of MAB phases, but inherit a hexagonal symmetry of MAX phases. Moreover, Hf2BiB and Hf2PbB exhibit a typical structure of conventional MAX phases (M n+1AX n , space group P 63/mmc, No. 194). These findings suggest that boron-based ternary compounds may be a new platform of MAX phases. The functionalized two-dimensional (2D) borides derived from the predicted ternary phases are calculated to be with improved mechanical flexibility and adjustable electronic properties relative to the parent ones. In particular, the 2D Hf2B2T2 and Zr2B2T2 (T = F, Cl) can transform from metal to semiconductor or semimetal under appropriate compressive biaxial strains. Moreover, the 2D Zr2B2 exhibits a high theoretical lithium-ion (Li+) storage capacity and low Li+ migration energy barriers. These novel properties render 2D boron-based materials promising candidates for applications in flexible electronic devices and Li+ battery anode materials.

Elemental gallium possesses several intriguing properties, such as a low melting point, a density anomaly and an electronic structure in which covalent and metallic features coexist. In order to simulate this complex system, we construct an ab initio quality interaction potential by training a neural network on a set of density functional theory calculations performed on configurations generated in multithermal–multibaric simulations. Here we show that the relative equilibrium between liquid gallium, α-Ga, β-Ga, and Ga-II is well described. The resulting phase diagram is in agreement with the experimental findings. The local structure of liquid gallium and its nucleation into α-Ga and β-Ga are studied. We find that the formation of metastable β-Ga is kinetically favored over the thermodinamically stable α-Ga. Finally, we provide insight into the experimental observations of extreme undercooling of liquid Ga.