Strength of the Effective Coulomb Interaction at Metal and Insulator Surfaces

The effective on-site Coulomb interaction (Hubbard $U$) between localized electrons at crystal surfaces is expected to be enhanced due to the reduced coordination number and reduced subsequent screening. By means of first principles calculations employing the constrained random-phase approximation (cRPA) we show that this is indeed the case for simple metals and insulators but not necessarily for transition metals and insulators that exhibit pronounced surface states. In the latter case, the screening contribution from surface states as well as the influence of the band narrowing increases the electron polarization to such an extent as to overcompensate the decrease resulting from the reduced effective screening volume. The Hubbard $U$ parameter is thus substantially reduced in some cases, e.g., by around 30% for the (100) surface of bcc Cr.