Understanding the intrinsic framework of the Hall-Petch relationship of metals from the view of the electronic-structure level

The relationship between grain size and yield strength of metals follows the Hall-Petch relationship {\sigma} = {\sigma}0 + kd^-0.5; however, the specific physical factors that affect the coefficients {\sigma}0 and k of this relationship remain unclear. Here we propose the intrinsic descriptors to determine the Hall-Petch relation across different metals and alloys. Inspired by the tight-binding theory, we find that {\sigma}0 strongly depends on the group and period number, the valence-electron number and electronegativity, while k is determined by the cohesive energy. Our framework establishes a predictive structure-property relationship for the size-dependent yield strength of various metals, and unravels that both the coefficients of the Hall-Petch relationship physically originate from the d-band properties. This novel correlation provides a new perspective for understanding the mechanical strength of metals, which is useful for the design of high-performance materials.