9.2.2 Reservoir of Holes - Classical vs. Quantum Mechanical Description

Previously [111], the holes were assumed to be energetically located at the valence band edge of the substrate with the defects being filled corresponding to this energy level. Unfortunately this approximation is questionable for a more complex structure like pMOSFETs with a high-k dielectric layer and a SiGe-layer inside the silicon substrate. This necessitates the incorporation of quantum mechanical (QM) confinement. Instead of assuming all holes to be fixed at , it is now distinguished between the contributing subbands, i.e. their different eigenenergies and hole occupancies are considered. To obtain the wave functions of the subbands in the channel of the MOSFET, the Schr¨odinger and Poisson equation were solved self-consistently using the Vienna Schr¨odinger Poisson solver (VSP2) [171]. The carrier concentration is calculated by treating the quasi-bound states as a two-dimensional electron gas in equilibrium and the continuum states as a 3D electron gas. The three X valley sorts of the conduction band as well as the heavy hole, light hole, and split off band are taken into account.

In Fig. 9.7 the first five subbands are displayed based on their corresponding eigenenergies of which four are localized in the SiGe-layer. In Fig. 9.8 the first two subbands are depicted for two stress and relaxation conditions. When switching from relaxation to stress the maxima of the subband wave functions move towards the interface, which raises the “hole concentration” in the oxide. The penetration of the wave functions is plotted on a log-scale to show the transmission probability. It can be clearly seen that the contribution of the subbands decreases with increasing order. As the bandbending inside the oxide due to the charged defects can be neglected, the WKB approximation is valid here and closely matches the transition probability of the wave function. Due to the lower computational efforts, the WKB approach is therefore used in the following.

In order to calculate the occupancy of the entire defect band during a certain bias condition in time, the effective rates in (9.5) and (9.6) have to be evaluated for each subband for each individual defect. The effective rates of a single defect have then to be summed up over all subbands and determine the single defect occupancy. The occupancy of the entire defect band finally gives the observable degradation.