Exact Theory of Fermi-Energy Response at Metallic Interfaces

TL;DR

The paper addresses the challenge of determining how the Fermi level $E_F$ of a metal shifts under external perturbations at metallic interfaces, which is governed by interfacial electrostatic screening. It introduces an exact finite-field linear-response formulation based on conceptual DFT to compute the Fukui function $f(\mathbf{r})$ and relates it to a local surface-charge derivative, $f(\mathbf{r}) = \dfrac{1}{2}\dfrac{\partial \rho(\mathbf{r})}{\partial Q}$, enabling electroneutral calculations that avoid Hartree background artifacts. The method yields strictly quadratic error scaling in the Fermi-level shifts with respect to the applied field, achieving sub-meV accuracy up to $\varepsilon_z = 0.1$ V/Å and is validated on Pt(111) surfaces, on Pt surfaces with islands and CO adsorbates, and on Ag(111)/NaF(aq) interfaces to reproduce work-function changes and C–V curves consistent with experiment. The results establish a rigorous, local theory linking electrostatic screening and Fermi-energy variations at metallic interfaces and offer a practical route to predict electrode potentials and surface dipole effects from first principles.

Abstract

The response of the Fermi energy to external perturbations governs key physical observables at metallic interfaces. Although this response admits a local formulation in terms of the Fukui function, its evaluation has traditionally been limited by inherent approximations, fundamentally rooted in the difficulty of adding a finite charge in a periodic system. We present an exact resolution to this problem that leverages the screening properties of electronic conductors to compute Fukui functions via a finite electric field. The resulting linear-response theory yields strictly quadratic error scaling of Fermi-level shifts across representative platinum surfaces, achieving sub-meV accuracy up to fields of 0.1 V/Å. The approach is further validated by reproducing work-function changes under molecular perturbations, and by providing mean-field estimates of electrode potentials that yield capacitance--voltage curves consistent with experiment. Our findings establish a rigorous foundation for a local theory relating electrostatic screening and Fermi-energy variations at metallic interfaces.

Figures (4)

• Figure 1: Comparison between finite-charge ($\pm \Delta N$) and finite-field ($\pm \varepsilon_z$) methods for computing Fukui functions at the interface between an electronic conductor (gray area) and an electronic insulator (white area). Highlighted regions of equivalent color and number label indicate physically analogous interfaces. No spatial symmetry of the electronic charge distributions is assumed between left and right surfaces.
• Figure 2: Computed Fukui functions for three relaxed Pt(111) surfaces, reported as 3D isovalues: a) bare metal slab, b) metal slab with a 3 Pt-atoms island, c) metal slab with an adsorbed CO molecule on a top Pt-site. d) Signed error of Fermi energy variations $\Delta E_\text{F}$ computed through our linear-response theory, for systems subject to linear potentials $v(z) = -\varepsilon_z z$ of increasing electric field $\varepsilon_z$. Full lines are obtained from a monomial quadratic fit of the data. Inset: magnified view for $\varepsilon_z\le0.1$ V/Å, showing an overall sub-meV accuracy.
• Figure 3: a) Representation of ten lateral displacements $R^\text{O}_{xy}$ over the $xy$-plane of a water molecule at 3 Å distance from a defective Pt(111) surface. b) Dipolar potential drop $\Delta V_0$ associated with the planar $xy$-average of the molecular external potential, $\Delta \bar{v}_\text{ext}(z)$, for an example upright orientation. c) Empty circles: $\Delta W$ response induced by the field of three molecular orientations. Black dots and lines: DFT reference. Black dashed lines: external potential drops $\Delta V_0$.
• Figure 4: Computed capacitance ($C_\text{diff}$) versus potential ($U$) curves of the Ag(111)/NaF(aq) interface at increasing ionic concentrations. Predicted values of electrode potentials are shifted by the PZC value of $-0.695$ V (dashed line), used to compare directly with the experimental saturated calomel electrode (SCE) potential scale reported in Ref. Valette1989. Inset: planar integral of the thermally averaged Fukui function of the Ag(111)/water interface used as reference PZC state.