Electric Field-Induced Kerr Rotation on Metallic Surfaces

TL;DR

This work addresses electric-field-induced Kerr rotation at metallic Pt surfaces by combining density functional theory with a nonlocal optical-conductivity framework and Maxwell scattering. It identifies two microscopic mechanisms—extrinsic bias-driven orbital accumulation (orbital Edelstein effect) and intrinsic surface-localized Pockels-like changes in wavefunctions—both contributing within a surface-proximal region of a few nanometers. In Pt films, these contributions are found to be comparable in magnitude but produce distinct polarization signatures, with extrinsic Kerr angles for $s$- and $p$-polarized light nearly equal and intrinsic Kerr angles opposite in sign. The results, including an effective surface-localization length shorter than 1 nm for the electro-optically active region, agree well with experimental Pt data and underscore the necessity of a nonlocal electro-optical description to capture surface boundary effects in metallic systems.

Abstract

We use a combination of density functional theory calculations and optical modeling to establish that the electric field-induced Kerr rotation in metallic thin films has contributions from both non-equilibrium orbital moment accumulation (arising from the orbital Edelstein effect) and a heretofore overlooked surface Pockels effect. The Kerr rotation associated with orbital accumulation has been studied in previous works and is largely due to the dc electric field-induced change of the electron distribution function. In contrast, the surface Pockels effect is largely due to the dc field-induced change to the wave functions. Both of these contributions arise from the dual mirror symmetry breaking from the surface and from the dc applied field. Our calculations show that the resulting Kerr rotation is due to the dc electric field modification of the optical conductivity within a couple of nanometers from the surface, consistent with the dependence on the local mirror symmetry breaking at the surface. For thin films of Pt, our calculations show that the relative contributions of the orbital Edelstein and surface Pockels effects are comparable, and that they have different effects on Kerr rotation of $s$ and $p$ polarized light, $θ_K^s$ and $θ_K^p$. The orbital Edelstein effect yields similar values of $θ_K^s$ and $θ_K^p$, while the surface Pockels effect leads to opposing values of $θ_K^s$ and $θ_K^p$.

Figures (10)

• Figure 1: (a) Schematic depiction of the electro-optic effect. Incident light with either $s$ or $p$ polarization gets rotated in response to an in-plane bias dc electric field. (b) Schematic depiction of the calculation geometry. To calculate the Kerr rotation, we limit the electro-optic response to atomic layers near the top and bottom surface regions (with thickness $d$) as shown by the shaded areas. In order to capture optical length scales properly, we insert optically active but electro-optically inactive layers between the top and bottom films such that the total length is given by $L=N_{max}L_0$, where $N_{max}$ is the total number of repeated optical unit cells. The film is oriented with the $x$, $y$, and $z$-axes along the (100), (010), and (001) directions.
• Figure 2: Calculated Kerr rotation due to the extrinsic and intrinsic components of the electro-optic effect in 14 nm Pt film. In (a) and (b), we show the Kerr angle and ellipticity due to extrinsic electro-optic response, while (c) and (d) present the results due to the intrinsic component. The light is incident at 45$^{\circ}$.
• Figure 3: (a) Real and (b) imaginary parts of the effective electric field induced electro-optic tensor elements, $Q^{\rm eff}_{\alpha\beta}=\epsilon_{\alpha\beta}/\epsilon_{xx}$, due to the extrinsic mechanism. The blue (red) lines show the results for $Q^{\rm eff}_{xz}$($Q^{\rm eff}_{xz}$) versus the optical frequency. The thick (thin) error-bar results correspond to 71 monolayers ($L_0=$14 nm) Pt films in the optical superlattice structure, where the error-bar amplitude is obtained from linear fitting to the angular dependence. The small error bars indicate that the results closely follow the expected angular dependence described in Eqs. \ref{['Eq.Eq11']}. Truncation method as depicted in Fig. \ref{['fig:Kerr_setup']}(b) was used to extrapolate to semi-infinite Pt film thickness. (c) and (d) shows the same results due to the intrinsic mechanism.
• Figure 4: Total Kerr angle for an $s$-polarized incident light versus Pt film thickness for $\hbar\omega=2$ eV and $\hbar\omega=2.4$ eV, shown as red and blue dashed lines with cross and star symbols, respectively. The thick black solid lines are from experimental measurements, as reported in Ref. Stamm2017 and Ref. Marui2024. The first few points, $L<$ 20 nm films are calculated using a single Pt film, and the rest are extrapolated results obtained by constructing multiple copies of 71 monolayer Pt film stacked on each other with the truncation method for the electro-optic response as depicted in Fig. \ref{['fig:Kerr_setup']}. In the case of $\hbar\omega=2.4$ eV, we consider multiple semi-infinite substrates with indices of refraction, $n_{\rm sub}=1,2,4$, corresponding to vacuum, sapphire, and silicon, respectively. An energy broadening value of $\eta=25$ meV was chosen in the numerical calculations. The incidence angle in all cases is 45$^{\circ}$.
• Figure 5: Calculated Kerr angle versus thickness of the electro-optically active region near the top surface with thickness $d$, as depicted in Fig. \ref{['fig:Kerr_setup']}. The solid lines correspond to the result of the fitting to the exponential decay function, $y=y_{\infty}(1-e^{-d/\lambda})$. The total thickness of the optical supercell is $\approx 0.5~{\rm \mu m}$. For reference, we also included the experimental results (star symbols), reported in Refs. Stamm2017Lyalin2023Marui2024.