Ab initio study of the Coulomb interaction in ${\mathrm{Nb}}_x\mathrm{Co}$ clusters: Strong on-site versus weak nonlocal screening

By means of ab initio calculations in conjunction with the random-phase approximation (RPA) within the full-potential linearized augmented plane wave method, we study the screening of the Coulomb interaction in ${\mathrm{Nb}}_x\mathrm{Co}$ ($1\le x\le 9$) clusters. In addition, these results are compared with pure bcc Nb bulk. We find that for all clusters the on-site Coulomb interaction in RPA is strongly screened, whereas the intersite nonlocal Coulomb interaction is weakly screened and for some clusters it is unscreened or even antiscreened. This is in strong contrast with pure Nb bulk, where the intersite Coulomb interaction is almost completely screened. Furthermore, constrained RPA calculations reveal that the contribution of the Co $3d \to 3d$ channel to the total screening of the Co $3d$ electrons is small. Moreover, we find that both the on-site and intersite Coulomb interaction parameters decrease in a reasonable approximation linearly with the cluster size and for clusters having more than 20 Nb atoms a transition from 0D to 3D screening is expected to take place.

Figure 1

The geometry of the NbCo to ${\mathrm{Nb}}_7\mathrm{Co}$ and ${\mathrm{Nb}}_9\mathrm{Co}$ clusters. Between brackets the point symmetry group of the cluster is indicated. The blue spheres correspond to the Nb atoms and the red spheres to the Co atoms.

Figure 2

Calculated spin-resolved total density of states for ${\mathrm{Fe}}_2{\mathrm{O}}_3$ and ${\mathrm{Nb}}_3\mathrm{Co}$ clusters. The Fermi energy (chemical potential) is set to zero.

Figure 3

The averaged partially screened without the Nb $4d \to 4d$ channel and fully screened on-site $U_{\text{avg}}$ (on-site) and nearest-neighbor intersite $V_{\text{avg}}$ (nn) matrix elements between Nb $4d$ electrons as a function of cluster size ($x$) for the ${\mathrm{Nb}}_x\mathrm{Co}$ clusters. Here $x$ corresponds to the number of Nb atoms in the ${\mathrm{Nb}}_x\mathrm{Co}$ cluster. Ignoring the smallest cluster ($x=1$), the blue and red solid and dashed lines represent an extrapolation of the data points. The green solid and dashed lines represent, respectively, the on-site fully screened and partially screened Coulomb interaction for pure Nb bulk.