Calculation of the effective valence for electromigration in niobium

The effective valence $Z^{*}$ of Nb, Ta, Cr, Fe, Co, and Ni in Nb metal have been calculated using a formalism based on the scattering theory and the relaxation-time approximation of the Boltzmann equation. In this formalism the electron-wind force depends on an integral over the Fermi surface involving the electron group velocities and the electronic wave functions of the host metal, and on the phase shifts of the host and the solute metal atoms. The electronic structure and the electronic wave functions of Nb were calculated using the augmented-plane-wave method from the self-consistent potential generated by Moruzzi et al. The calculated values of $Z^{*}$ (not including the contribution from direct force) are found to be very small, largely due to the high phonon resistivity of Nb. These results are compared with the recent electromigration data in Nb.