Theory of atomic-scale dielectric permittivity at insulator interfaces

We develop a theory for investigating atomic-scale dielectric permittivity profiles across interfaces between insulators. A local susceptibility $\chi (x;\omega )$ is introduced to describe variations of the dielectric response over length scales of the order of interatomic distances. The nonlocality of the microscopic susceptibility tensor ${\chi }_{ij}(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$ occurs at smaller distances and therefore does not intervene in our formulation. The local permittivity is obtained from the microscopic charge density induced by an applied electric field. We show that the permittivity can conveniently be analyzed in terms of maximally localized Wannier functions. In this way, we can relate variations of the microscopic dielectric response to specific features of the local bonding arrangement. In addition to a continuous description in terms of the local permittivity, we introduce an alternative scheme based on discrete polarizabilities. In the latter case, electronic polarizabilities ${\alpha }_{\mathrm{elec}}^{\left(n\right)}$ are obtained in terms of the displacements of maximally localized Wannier functions, while ionic polarizabilities ${\alpha }_{\text{ion}}^{\left(I\right)}$ are determined from the induced ionic displacements and the corresponding dynamical charges. The potential of our scheme is illustrated through applications to two systems of technological interest. First, we consider the permittivity of Si slabs of finite thickness. Our approach indicates that the local permittivity in the slab interior approaches the corresponding value for bulk Si within a few atomic layers from the surface. Therefore, the decrease of the average slab permittivity with thickness originates from the lower permittivity of the outer planes and the increasing surface-to-volume ratio. Second, we address the dielectric permittivity across the $\mathrm{Si}\left(100\right)\text{−}\mathrm{Si}{\mathrm{O}}_2$ interface. Using two distinct interface models, we are able to show that the dielectric transition from the silicon to the oxide occurs within a width of only a few Å. The polarizability associated to intermediate oxidation states of Si is found to be enhanced with respect to bulk $\mathrm{Si}{\mathrm{O}}_2$, resulting in a larger permittivity of the interfacial suboxide layer with respect to the stoichiometric oxide.

Figure 1

(a) Permittivity of bulk Si as a function of the electric field (disks). The region where the electronic structure could not be converged is shaded. (b) Permittivity of bulk Si as a function of the size of the simulation cell $L_x$ (disks). The solid line corresponds to the $L_x^{-2}$ limiting behavior. The dotted line indicates the value extrapolated for an extended system. Dashed lines in (a) and (b) are guides to the eye.

Figure 2

Local permittivity profile (solid) along the [100] direction in bulk Si, presenting a defective layer with a larger interplanar separation. The horizontal dashed line corresponds to the permittivity of the regular structure. Contributions to the local permittivity resulting from maximally localized Wannier functions of the defective (dotted) and regular (dash-dotted) Si layers are also shown. Vertical lines represent planes of Si atoms.

Figure 3

Permittivity profiles along the [100] direction for Si(100) slabs with H-terminated surfaces. The numbering refers to the slab size in terms of Si planes. Vertical lines represent planes of Si (solid) and H (dotted) atoms for the largest considered slab. All the slabs are aligned through the positions of the H atoms of the surface on the left. The permittivities of the extended bulk (14.3) and of the vacuum are indicated by horizontal lines.

Figure 4

(a) Permittivity of a Si slab as a function of thickness $d$: local permittivity in the central Si plane (circles) compared to the overall slab permittivity (disks). The horizontal line indicates the permittivity of a slab of infinite thickness (14.3), as determined in Sec. 3B. The dashed lines are guides to the eye. (b) Band gap of a Si slab as a function of thickness $d$ (squares). The horizontal line indicates the band gap of a slab of infinite thickness $\left(0.62\mathrm{eV}\right)$. The dashed line is a guide to the eye.

Figure 5

Planar average of the Wannier-function charge densities corresponding to model I of the $\mathrm{Si}-\mathrm{Si}{\mathrm{O}}_2$ interface: $\mathrm{Si}-\mathrm{Si}$ bonds (black solid), $\mathrm{Si}-\mathrm{O}$ bonds (gray solid), and nonbonding O orbitals (black dotted). The horizontal axis is oriented along the [100] direction. Vertical lines represent planes of Si (solid) and O (dotted) atoms.

Figure 6

Permittivity profiles across (a) model I and (b) model II of the $\mathrm{Si}-\mathrm{Si}{\mathrm{O}}_2$ interface: electronic permittivity (solid) and lattice contribution ${\epsilon }_0\left(x\right)-{\epsilon }_{\infty }\left(x\right)$ (dashed). Vertical lines indicate planes of Si (solid) and O (dotted) atoms. The shaded area represents the suboxide transition layer.

Figure 7

Profile of the local electronic susceptibility across model II of the $\mathrm{Si}\left(100\right)\text{−}\mathrm{Si}{\mathrm{O}}_2$ interface (solid line). Contributions to the susceptibility arising from Wannier functions of the Si slab (dotted), the suboxide (dotted), and the oxide (dash-dotted) are shown separately. The shaded area represents the suboxide transition layer. Vertical lines indicate planes of Si (solid) and O (dotted) atoms.

Figure 8

Effective electronic polarizabilities of Si-centered polarizable units, for model I (disks) and model II (circles) of the $\mathrm{Si}\left(100\right)\text{−}\mathrm{Si}{\mathrm{O}}_2$ interface. The $x$ axis is oriented along the [100] direction. Horizontal dashed lines indicate the values calculated for the corresponding bulk materials. The shaded area indicates the suboxide region. Inset: effective electronic polarizability vs the oxidation state of the central Si atom in the polarizable unit, for both models I (disks) and II (circles).

Figure 9

Effective charges ${\zeta }_{I,xx}$ for model I (open symbols) and model II (filled) of the $\mathrm{Si}\left(100\right)\text{−}\mathrm{Si}{\mathrm{O}}_2$ interface. Circles correspond to Si atoms, squares to O atoms. The $x$ axis is oriented along the [100] direction. Horizontal lines indicate the values of the Born dynamical charges calculated for each bulk material separately. Inset: Si effective charges vs Si oxidation state for both model I (disks) and model II (circles).

Figure 10

Effective ionic polarizabilities of the Si-centered polarizable units, for model I (disks) and model II (circles) of the $\mathrm{Si}\left(100\right)\text{−}\mathrm{Si}{\mathrm{O}}_2$ interface. The $x$ axis is oriented along the [100] direction. Horizontal dashed lines indicate the values calculated separately for the corresponding bulk materials. The shaded area indicates the suboxide region. Inset: effective ionic polarizability vs Si oxidation state, for both models I (disks) and II (circles).

Figure 11

(a) Induced charge density corresponding to a ${\mathrm{Si}}^{+1}-{\mathrm{Si}}^{+2}$ bond in the suboxide of model II of the $\mathrm{Si}\left(100\right)\text{−}\mathrm{Si}{\mathrm{O}}_2$ interface. The magnitude of the electric field applied to the simulation cell is ${10}^{-3}\mathrm{a.u.}$ The vertical bold line indicates the center of the Wannier function. (b) Same as (a) but in absolute value and on a logarithmic scale (solid). We also show the corresponding approximation in terms of decaying exponentials (dashed). (c) Contribution to the susceptibility profile arising from the same bond as in (a), (b). The bold dashed line indicates the center of the distribution, clearly shifted towards the Si slab. In all panels, vertical lines represent planes of Si (solid) and O (dotted) atoms.