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 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.

Figure 1

(a) Left panel: Layer dependence of the Hubbard $U$ parameter for fcc Al. The $S$ denotes the surface layer in the slab model. Right panel: Total density of states (DOS) for the (100) and (110) surface of fcc Al. For comparison, the bulk DOS (shaded area) is included; (b) and (c) the same for bcc Cr and rock-salt NaCl.

Figure 2

(a) Partially screened ($U_B$) and fully screened (${\tilde{U}}_B$) Coulomb interaction for the bulk $3d$ TM series in the nonmagnetic state. With open diamonds we show the $U$ for ferromagnetic Fe, Co, and Ni. (b) The difference between the surface and bulk $U$ values. Open (filled) diamonds show $U_S-U_B$ for the (100) [(110)] surface of ferromagnetic Fe, Co, and Ni. (c) Bare Coulomb interaction $V$ for bulk and surfaces. (d) The same as (b) for the fully screened Coulomb interaction $\tilde{U}$.

Figure 3

(a) Frequency dependence of the bulk and surface Hubbard $U$ parameter for bcc Cr. In the inset we expand the low frequency region. (b) The same for fcc Ni.