In order to validate the approach used within Sec. 3.3 to include the effect of an electric field, the Berry–phase method as implemented in
Cp2k will be thoroughly tested in the following. Fundamental material properties, such as the polarization and the permittivity for bulk amorphous silicon dioxide (\(a\)–SiO\(_2\)), bulk \(c\)–Si
as well as an Si/\(a\)–SiO\(_2\)/Si interface structure will be calculated. For all calculations 3D periodic boundary conditions are applied. Two types of calculations were run with field strengths of up to \(\SI {10}{MV\per \cm
}\): one with fixed atomic positions, which gives the high–frequency permittivity \(\epsilon _\infty \), and another where the effects of lattice relaxations are included, yielding the static permittivity \(\epsilon \).
The Figs. C.1 and C.2 summarize the results for bulk \(a\)–SiO\(_2\) as well as for crystalline silicon. To ensure that the created
\(a\)–SiO\(_2\) model possesses an isotropic polarization, the field has been applied along the \(x\), \(y\) and \(z\)–axis, whereas for the Si model, the field was only considered along the \(z\) direction within the calculations.
Quite reassuringly, the calculated values for the static and high–frequency permittivities for both materials are in excellent agreement with the experimental values. While for silicon the permittivity is virtually independent of the
frequency [327], the dielectric constant for SiO\(_2\) decreases from \(\SI {3.81}{}\) at static electric fields to \(\SI {1.96}{}\) for \(\epsilon
_\infty \) [328, 329]. Furthermore, as is shown in the insets, the approach
using Wannier functions and (3.2) gives the same results as the Berry phase method.
Figure C.1: The extracted polarization using the Berry phase method of \(a\)–SiO\(_2\) as a function of increasing field strengths. Additionally, the inset shows the calculated polarization by means of the Wannier center
approach. Both methods agree extremely well with each other.
Figure C.2: The extracted polarization using the Berry phase method of \(c\)–Si as a function of increasing field strengths. Additionally the inset shows the calculated polarization by means of the Wannier center approach.
Both methods agree extremely well with each other.
Unfortunately, for a heterostructure, i.e. a Si/\(a\)–SiO\(_2\)/Si interface, the above mentioned approaches are not applicable. However, one possibility to evaluate the spatially dependent dielectric constants is given by the
induced charge density (ICD) method [330–333]. Within this approach
the local microscopic polarization \(p(\bm {r})\) can be evaluated as
with \(\overline {p}(z)\) and \(\overline {\rho }(z)\) being the planar averaged polarization and the induced charge density, respectively and \(\overline {p}_{-\infty }\) is a boundary condition constant, as will be
discussed below. Thereby, \(\overline {\rho }(z)\) is defined as the difference in charge density upon the application of different electric field strength (\(\Delta E=\SI {1}{MV\per \cm }\) is used here). The resulting
polarization profile can be further used to determine the spatially dependent relative permittivity of linear dielectrics by using the following relation
where \(\varepsilon _0\) is the vacuum permittivity and \(E_\mathrm {ext}\) is the externally applied field.
Two different realizations of the atomistic interface structure were utilized: the Si/\(a\)–SiO\(_2\)/Si unit cell, as well as a slab model, where a \(\SI {5}{\angstrom }\) vacuum gap was introduced on either side. The Si atoms
facing the vacuum were passivated by H, which creates two surfaces. Thus, in the latter case \(\overline {p}_{-\infty }\) can be set to zero due to the vanishing induced charge, while for the interface cell the constant is set to
the polarization of Si, assuming bulk Si at \(-\infty \). Again, single point calculations as well as geometry relaxations have been conducted for both models.
The resulting polarization together with the permittivity profiles across the structures is shown in Fig. C.3.
Overall, a very good agreement with experimental values as well as with the results presented above is achieved. However, by taking a closer look, some particular features and differences between the two atomistic models can be
highlighted.
Figure C.3: Optical and static dielectric constant profiles across the interface slab using the ICD method with a H passivated Si/a–SiO\(_2\)/Si model.
First, lattice relaxations have a strong impact on the results of the slab model. Whereas the unrelaxed structure shows an increasing permittivity trend towards the Si/\(a\)–SiO\(_2\) interface regions, the optimized variant
actually shows the opposite trend. This result can be explained by an expansion of \(\SI {0.1}{\angstrom }\) for the whole structure during the relaxation procedure. Thus, particularly in the direct interfacial regions, atoms
relax into new equilibrium positions, thereby balancing the shift of the electronic density. In contrast, the unit cell model shows a consistently enhanced permittivity directly at the interface, albeit for the relaxed structure this effect
is reduced. Since the atomistic structure within the cell is not able to expand in the \(z\) direction, structural reconfigurations affect the position and the width of the interfacial region, as can be seen in Fig. C.3.
Second, both models clearly show an interfacial transition region with a width of \(\SI {4}{}-\SI {6}{\angstrom }\) to resemble the bulk properties of the respective material. Such a region, with altered electronic
properties, was confirmed by experiments [192, 193], constraining the minimum
thickness for a SiO\(_2\) gate dielectric to a theoretical value of \(\SI {7}{\angstrom }\), which is in good agreement with the results presented here.
