Dielectric Discontinuity at Interfaces in the Atomic-Scale Limit: Permittivity of Ultrathin Oxide Films on Silicon

Using a density-functional approach, we study the dielectric permittivity across interfaces at the atomic scale. Focusing on the static and high-frequency permittivities of ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ films on silicon, for oxide thicknesses from $12\AA$ down to the atomic scale, we find a departure from bulk values in accord with experiment. A classical three-layer model accounts for the calculated permittivities and is supported by the microscopic polarization profile across the interface. The local screening varies on length scales corresponding to first-neighbor distances, indicating that the dielectric transition is governed by the chemical grading. Silicon-induced gap states are shown to play a minor role.

Figure 1

Permittivities ${ϵ}_{\infty }$ (circles) and ${ϵ}_0$ (disks) of the oxide overlayer vs its thickness, compared to the results of a classical three-layer model (solid). The classical two-layer model gives a constant permittivity corresponding to that of bulk ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ (dashed). Inset: ionic contribution to ${ϵ}_0$.

Figure 2

High-frequency (upper panel) and static (lower panel) microscopic polarization along $x$ (solid lines) compared to the classical three-layer model (dashed lines). The polarization is normalized by its average value in the cell. For the static polarization we used a Gaussian broadening of $0.9\AA$ to smear out the pointlike ionic contributions. The suboxide region is shaded.

Figure 3

Electronic (upper panel) and ionic (lower panel) polarizability of the Si-centered polarizable units defined in the text vs their position along the $x$ axis. The Si and ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ bulk values are shown (dashed lines). Vertical lines represent planes of Si (solid lines) and O atoms (dashed lines). The suboxide region is shaded.

Figure 4

Microscopic electronic polarization [solid, from Fig. 2 (upper panel)] and the contribution from gap states (dots). The shaded area corresponds to the Si layer. For comparison, we also show a decomposition in silicon and oxide contributions on the basis of localized Wannier functions (dashed): the silicon contribution results from Si-Si bonds, the oxide one from Si-O bonds and O nonbonding orbitals.