Fig. 4

d-band bimodality and solute–defect interaction energies in bcc Ta. a Projected LDOSs of d orbitals of a Ta atom on the interface of the Σ3\(\left( {11\bar 2} \right)\) TB (dashed line) and in bulk lattice (solid line), respectively. b, c DFT-calculated \(E_{{\mathrm{int}}}^{{\mathrm{fix}}}\) in comparison with the predictions from the linear regression model in the cases of the Ta–Hf and Ta–Os systems, respectively. The DFT-calculated \(E_{{\mathrm{int}}}^{{\mathrm{fix}}}\) refers to the solute–defect interaction energies calculated based on fixed atomistic structures that are already fully relaxed in pure Ta. In the legend, Dumb refers to the abbreviation of the dumbbell defects, S-Dis refers to the abbreviation of the \(\frac{1}{2}\left\langle {111} \right\rangle\) screw dislocation, TB refers to twin boundaries, and GB refers to grain boundaries. The values of Δdip and x_sp of each defect site used for the linear regression are listed in Supplementary Table 3. The regression parameters for each matrix–solute element pair are summarized in Table 1