Strength of the Effective Coulomb Interaction at Metal and Insulator Surfaces

Abstract

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. The effective on-site Coulomb interaction (Hubbard U ) between localized electrons at surfaces of solids is expected to be enhanced since the effective screening volume of the surface is reduced with respect to the bulk. As a consequence, the electron polarization decreases at the surface, which reduces the effect of screening and gives rise to a larger U value. These arguments are underscored by interpolating between the Hubbard U values of an isolated atom and an atom in a bulk solid, the former being 3 to 5 times larger than the latter [1]. Neither experimental nor theoretical works have been reported so far that would address the strength of the surface U parameter explicitly. However, a large number of phenomena observed in solids indicates an enhancement of the U at surfaces. For instance, the metal-insulator transition at the surface of correlated materials [2], the appearance of magnetism at the surface of paramagnetic transition metals [3], and the enhanced exchange splitting at the surface of 3d ferromagnets [4] have been attributed to an increase of the correlation strength, which is defined by the ratio U/W , where W is the bandwidth. In the theoretical description of surfaces, the U is usually assumed to be unchanged [5][6][7] so that the enhancement of correlation at the surface (S) with respect to the bulk (B) is provided by the effective band narrowing, i.e., W S < W B . In principle, depending on the relative values of the surface U and the bandwidth W , the correlation strength can further increase or decrease even below the bulk value. However, the latter case is considered to be unlikely because it is believed that U always increases at surfaces. In this Letter, we show by means of first-principles calculations that contrary to this conventional wisdom, this is not always the case. It decreases at many transition-metal (TM) surfaces and insulator surfaces with pronounced surface states, as a result of additional screening channels that open up due to surface-related changes in the electronic structure.Recently, the calculation of the Hubbard U parameter in solids from first principles has been addressed by several authors [8][9][10][11][12][13][14][15]. A nu...

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