Using the electrostatics described in the previous section, the defect population can be sampled in the oxide in a subsequent step. Every defect has a certain probability of capturing and emitting a charge, which is determined by the transition rates and in (3.20) from 2-state NMP theory. Using these transition rates it is possible to compute the occupancy of each trap according to the Master equations

where are the probabilities that a defect is in the initial/final state. These equations include the interaction of each defect with all charge reservoirs , which are in the case of a MOSFET the valence and conduction band of the channel and the gate. The solution of the differential equation system is

and can be used to weigh the impact on the potential caused by the defects by

with

being the charge stored at a defect. Subsequently, the surface potential of the device can be computed using (4.2). The shift of the potential compared to the initial
time step then gives the threshold voltage shift. The total threshold voltage shift thus depends on the probability for a charge transition which itself depends on the defect parameters, the applied gate voltage and the temperature
as shown in Fig. 4.1. Note that in (4.8) the
contribution of a defect to the total depends on its depth and is considered according to the charge sheet approximation (CSA). However, recent studies have shown that the CSA considerably underestimates the impact of a single defect on the device
performance, which might give rise to a slight overestimation of the trap densities in the current version of *Comphy* [210, MJJ3, MJC1, MJC2].

Each defect can be described by a set of model parameters: the relaxation energy , the curvature ratio , the trap level , the spatial position inside the oxide , and the configuration coordinate offset . The latter is only relevant when using a quantum mechanical 2-state NMP transition rate. Typically, in large area devices an ensemble of defects is assumed to be distributed across a so called *defect band*. For this,
it has often been assumed that the trap level of defects is normally distributed, that relaxation energies are normally distributed as well and that the spacial position is uniformly distributed [136]. Since the curvature of the
PECs is correlated with the relaxation energy, it is not necessary to treat as a stochastically distributed unit. Finally, a uniform spatial defect concentration across parts of the dielectric is assumed. The parameters can be seen in Fig. 4.2. The sampling of the defects can
either be done by Monte-Carlo sampling or more efficiently, the mean degradation of the defect band can be computed by sampling the parameters on a grid and introducing a weighting scheme.

While the approach of using Gaussian trap bands has proven to be very successful [136], it has the disadvantage that large relaxation energies, compared to values obtained from DFT calculations of suitable oxide defect candidates [185], are required eventually [MJJ2]. This can give rise to unphysically large charge transition times, as exemplary shown for the shallow SiO2 trapband of the 28 nm technology examined in [136] in Fig. 4.3, where the majority of all sampled defects shows transition times above s, which is experimentally inaccessible. The smaller time constants correspond to the tail of the distribution of the relaxation energies. Thus, an alternative approach using an effective single defect decomposition (ESiD) has been developed to avoid artificially high relaxation energies. In this approach no assumptions on the defect parameter distributions are made when the model is calibrated to experimental data [MJJ2].

When using the ESiD approach, the total threshold voltage shift is expressed as superposition of the threshold voltage shifts caused by each defect

where is the weight of each defect. Instead of assuming Gaussian distributions for and , the ESiD algorithm employs a uniform parameter grid sampled for , , and . These parameters span a grid that is considered in the response matrix , where at every point the response is computed. By using the response matrix and defining the observation vector a non-negative linear least square (NNLS) algorithm can be used for computing the weights to

Since this is mathematically an ill-posed problem that can lead to solutions with physically unrealistic high defect densities, a Tikhonov regularization [211] is added to ensure that the least-square solution results in smoother defect densities

where is the regularization parameter. The ESiD algorithm allows an efficient extraction of trap parameters from measurement data which can be compared for their agreement with DFT simulations [MJJ2, 212]