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The Physics of Non–Equilibrium Reliability Phenomena

3.3 Impact of an Electric Field

The metal–oxide–semiconductor structure is routinely used in electronic devices and subjected to external electric fields during normal operation. Previous phenomenological studies [226228] led to diverging results and ensuing discussions concerning the Si–H dipole moment and its response upon an applied electric field. A method particularly designed to account for the effect of an electric field within DFT calculations of 3D periodic systems is the modern theory of polarization [229234]. It explicitly includes a self–consistent response of the electron density to electric fields, thereby rendering it an ideal approach to study processes in solid state systems. By minimizing the electric enthalpy functional introduced in [231, 233], the new field–polarized groundstate of the system is determined:

\begin{equation} \mathcal {F}[\rho ;\bm {F}]=E_\mathrm {KS}[\rho ;\bm {F}]-\Omega \bm {P}_\mathrm {mac}[\rho ;\bm {F}]\cdot \bm {F}, \end{equation}

with \(E_\mathrm {KS}\) being the usual Kohn–Sham energy and \(\Omega \) is the unit cell volume. The field coupling term is given by \(-\bm {P}_\mathrm {mac}[\rho ;\bm {F}]\cdot \bm {F}\), where \(\bm {P}_\mathrm {mac}=\bm {P}_\mathrm {ion}+\bm {P}_\mathrm {el}\) is the macroscopic polarization as defined in [231, 233]. Within this approach the polarization is calculated from the Berry phase of the Bloch wavefunction, as described and implemented in [235]. However, an important feature of this formulation is that the polarization is actually a multi–valued quantity, due to its interpretation as a geometric quantum phase. Thus, it is formally defined within one modulo of a quantum of polarization, \(2\pi \bm R\), where \(\bm R\) is the lattice vector, see [232]. To ensure that the results and extracted dipole moments presented in this work can be compared to each other (i.e. they belong to the same polarization branch), they were analyzed and, if necessary, manually corrected by a polarization quantum.

To gain insight into individual processes within the Si/SiO\(_2\) model when applying an electric field, the representation using Wannier centers has been utilized. Unlike Bloch functions, the localized nature of a Wannier center provides an intuitive atomic-like description of a charge density in a solid. This concept allows one to calculate the polarization by summing over the contributions of point charge ions, plus the corresponding electronic charges centered at the Wannier center of each occupied Wannier function [232]

\begin{equation} \label {eq:wannier} \bm {P} = \frac {1}{\Omega }\big (\sum _i(q_i\bm {r}_i)^\mathrm {ions}+\sum _n^{\mathrm {occ}}(q_n\bm {r}_n)^\mathrm {WFs}\big ). \end{equation}

A detailed comparison of the approach using Wannier centers and the Berry phase methods is given in Appendix C.


Figure 3.13: The effect of an electric field up to \(\SI {10}{MV/cm}\) across the interface structure onto the proposed bond breakage trajectory. Left: The extracted change of the components of the dipole moment vector \(\bm {\mu }\) along the given pathway for zero applied field. These quantities can be further used to obtain an estimate of the change by defining an effective dipole moment \(\mu _\mathrm {eff}\). Right: The results of the self consistent calculation with an electric field suggest that the interaction is extremely weak. Applying a field of \(\SI {10}{MV/cm}\) lowers the barrier by only \(\SI {0.023}{eV}\) and leaves the Si–H bond breaking process virtually unaffected.

The defect creation process under an electric field is calculated by the following approach: Using the CI–NEB result at zero field, see Sec. 3.2, static, homogeneous electric fields up to \(\SI {10}{MV\per \cm }\) with a \(\SI {1}{MV\per \cm }\) step were applied across the interface structure in the \(z\) direction. The results of these calculations are summarized in Fig. 3.13. Each data point was obtained employing a single-point calculation, without including the effect of lattice relaxations. However, as can be seen in Fig. 3.13, the interaction of the neutral Si–H bond with an electric field is extremely weak, rendering errors related to this approximation irrelevant. The electric enthalpy along the bond breaking pathway is virtually unaffected, even for fields as high as \(\SI {10}{MV\per \cm }\). Maximum changes of \(\SI {0.023}{eV}\) for the forward barrier, defined as the enthalpy difference between the initial and the transition state, can be extracted, while the enthalpy of the final state changes by \(\sim \SI {0.03}{eV}\).

Additionally, the components of the dipole moment vector and the respective changes have been extracted along the trajectory, see the left panel of Fig. 3.13, for zero applied field. The largest contribution within this approach results from the dissociating Si–H bond and also includes accompanying relaxation effects. Together with Fig. 3.14, which shows selected frames along the pathway including the Wannier centers, this provides some intuitive understanding of this process. Within the first steps, the trajectory is governed by bending the H in the direction of the adjacent Si\(_2\) atom, see 1 and 1\(_\mathrm {a}\). Thereby, only the H atom and the associated Wannier center change their positions without further lattice relaxations, which does not significantly change the dipole moment. Moving the H further away from the initial Si atom (1\(_\mathrm {b}\)) results in a structural and electronic reconfiguration and eventually breaks the Si–H bond, marking the top of the transition barrier, see 4. This is also reflected in the change of dipole moment in Fig. 3.13. Subsequently, the H and the surrounding Si atoms relax to their final bond–center positions, 4\(_\mathrm {a}\) and 5, forming a Si–H–Si complex.


Figure 3.14: Selected simulation snapshots along the dissociation pathway including the atomistic configurations as well as the associated Wannier centers (pink spheres). The representation using Wannier functions allows to obtain an intuitive, albeit qualitative, understanding of the change of the dipole moment. During the first steps (1-1\(_\mathrm {b}\)) the H bends towards the adjacent Si atom without significant lattice relaxations. Only the Wannier center associated with the Si–H bond changes its position, thereby leaving the dipole moment virtually unaffected. However, breaking of the initial Si–H bond (4) is accompanied with structural and electronic reconfigurations which leads to an increase of the dipole moment. Frames 4\(_\mathrm {a}\) and 5 show the subsequent relaxation into the final position, which possesses a higher dipole moment due to the resulting distorted Si–H–Si complex.

Following [160], one can define an effective dipole moment vector \(\bm {\mu }_\mathrm {eff}=\bm {\mu }_2-\bm {\mu }_1\), where \(\bm {\mu }_2\) is the dipole moment at the transition state and \(\bm {\mu }_1\) is associated with the initial configuration. Both quantities can be extracted at zero field, see Fig. 3.13, and used within \(\Delta E_\mathrm {B}=-\bm {\mu }_\mathrm {eff}\cdot \mathbf {F}\) to estimate the change of barrier due to an applied field. Extracting the \(z\) component of \(\bm {\mu _\mathrm {eff}}\) for the given trajectory yields a value of \(\SI {1.15}{D}\) which leads to a reduction of the barrier of \(\SI {0.024}{eV}\). This is in excellent agreement with the self–consistently calculated DFT results using the Berry phase method.