McWhorter Model

6.1 McWhorter Model

In the middle of the last century, McWhorter [182] dealt with the spectrum of $1∕f$ -noise observed at germanium-oxide interfaces. This kind of noise is attributed to fluctuations in the trap occupancy $ft$ due to charge carriers tunneling forth and back between the bulk and the defects. McWhorter described these fluctuations using a simple SRH-based model, which can be considered as a prototype for other charge trapping models. His model extends the conventional SRH theory by the effect of charge carrier tunneling, which is accounted for by the factor $exp(xt∕xp,0)$ . Thus, the simplified time constants read as

( ) τ = τSRH exp xt-- Nv-, (6.1) cap,h p,0 xp,0 p SRH ( xt ) τem,h = τp,0 exp xp,0- exp(βΔEt ) exp(- βq0Foxxt) . (6.2)

The model presumes that all traps are energetically located within the substrate bandgap so that none of them will be found below $E v$ . Furthermore, they are assumed to be spatially distributed over the entire dielectric.

In the following, the McWhorter model will be evaluated against the findings of the TDDS experiments (see Section 1.3.4).

After equation (6.1) $τcap,h$ shows a weak field dependence following $1∕p$ . The drop in the hole concentration $p$ at small fields leads to a sharp peak, which is inconsistent with the field acceleration observed in TDDS experiments.
Since the hole concentration weakly varies with changes in the temperature, the hole capture process is not thermally activated.
The field dependence of $τ em,h$ is governed by the exponential term $exp(- βq F x ) 0 ox t$ and thus is inconsistent with the behavior of ‘normal’ defects. But one should keep in mind that some RTN investigations [125, 55] have revealed that there exist defects whose emission times increase exponentially with the gate bias.
Consequently, the McWhorther model does also not agree with the linear dependence on $p$ seen for $τem,h$ in the case of the ‘anomalous’ defects. Beyond that, it does not give an explanation for the two distinct kinds of defects in general.
The McWhorther model predicts $τem,h$ to be temperature-activated in agreement with the experimental findings.

The term $exp(xt∕xp,0)$ in $τcap,h$ and $τem,h$ accounts for the trap depth dependence of tunneling and leads to an upwards shift of the entire $τcap,h$ and $τem,h$ curves with an increasing trap depth $xt$ . Due to the wide distribution of $xt$ , the McWhorter model allows a wide range of capture and emission times in thick oxides. In modern device technologies, however, the time constant of the devices with an oxide thickness of $2nm$ would be limited to $1ms$ after the model. As such, this model cannot explain time constants larger than $1ms$ for devices with an oxide thickness of $2nm$ . This is in contrast to the experimental results (cf. Fig. 1.2), in which $τem,h$ extends well into the kilosecond regime. In conclusion, this model cannot be reconciled with the findings of the TDDS and is thus inadequate to describe the traps involved in NBTI.

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