8 Latest Modeling Attempts

Chapter 8
Latest Modeling Attempts - Hole Trapping

In Chapter 3 it was tried to explain BTI by using either the diffusion of hydrogen or a dispersive bond breaking mechanism. In both cases interface states are involved. Unfortunately, the theoretical and experimental analysis of the on-the-fly interface traps (OFIT) technique presented in the last chapter revealed that the aberrations leading to the assumption of fast interface state stress and recovery are due to an artifact of the measurement routine. Since the recovery of BTI, especially its short-term behavior, is not explicable with interface states only, hole trapping models have been added [69, 40, 119, 120]. Today the BTI community does still not agree on how holes contribute in detail. The earliest hole modeling attempts date back to the 1950s, where McWhorter used hole trapping to describe $1∕f$ -noise at germanium surfaces [121]. More precisely, $1∕f$ -noise was considered as oscillations of the trap occupancy of individual defects caused by capture and emission of carriers. McWhorter’s attempt is based on the Shockley-Read-Hall (SRH) theory which was originally developed to model the recombination of bulk defects with an energy $ET$ inside the bandgap [122]. He extended this theory to also model oxide defects, which feature a trap level within the semiconductor bandgap. The local depth of the oxide defect $xT$ , measured from the interface, enters the model as a tunneling WKB factor $exp(− xT∕x0 )$ , where $x0$ acts as scaling factor.

When assuming a defect at $ET$ capturing a hole from the reservoir in the substrate, e.g. from $Ev$ , the hole does not have to surmount a barrier because of $ET > Ev$ . For the opposite process, namely the hole emission from the defect, the transition probability is reduced by the Boltzmann factor $exp(− β (E − E )) T v$ . However, the application of this approach to a defect level $ET < Ev$ , which can be assumed for oxide defects, makes the above Boltzmann factor larger than unity in the simplest picture. The hole emission barrier rather vanishes in the case of $ET < Ev$ . In turn the corresponding capture process is now affected by an additional Boltzmann factor $exp(− β (Ev − ET ))$ [123]. The hole capture $cp$ and emission $ep$ barriers for both kinds of defect, leveling above $Ev$ for the simple SRH and below $E v$ for the extended SRH, are all depicted in Fig. 8.1 (left).

Figure 8.1: Band diagram including a single defect in the oxide. Left: Depending on whether the defect level $ET$ lies above or below the valence band edge $Ev$ , different barriers are obtained. For the characterization of $Ec > ET > Ev$ the original SRH picture is used, while for defects with $ET < Ev$ an extension is necessary. Right: Applying an additional oxide electric field $F ox$ respective to a reference of $F ox,ref$ shifts the defect level because $ET$ has to be kept constant within the bandgap. Moreover, the further away from the interface, the more $ET$ is affected, cf. the oxide bandedges changing from dashed to solid.

When an additional oxide electric field $Fox$ is present, the defect level is shifted with respect to $Ev$ . Since the barrier $EvT = Ev − ET − q0xTFox$ is linearly dependent on $Fox$ , the defect may now effectively lie below or above $Ev$ , cf. Fig. 8.1 (right). Unfortunately, the McWhorter model was originally developed for $1∕f$ -noise in thick oxides and not designed to explain the strong temperature and bias dependence observed during BTI stress in modern devices with oxide thicknesses of only a few nanometers, e.g. $2 − 3nm$ . In such devices the McWhorter model only gives time constants smaller than a millisecond, which contradicts the measurement results [55].

About thirty years later Kirton and Uren used a modified McWhorter model to explain their random telegraph noise or signal (RTN/RTS) measurements, which characterize the change in the drain current of small-area MOSFETs as a function of time. The times where the signal randomly jumps into the high- and low-current were identified to be Poisson distributed around the expectation value of the capture $τc$ and emission $τe$ time constants of individual defects respectively. To link this capture and emission kinetics to the observed $1∕f$ -spectra, Kirton and Uren proposed the existance of many defects with uniformly distributed time constants on a log scale ranging from milliseconds to days [124]. Since they expected a multi-phonon emission (MPE) process to be responsible for their experimental findings, they added a thermal barrier $ΔEB$ to the existing SRH model [125, 126, 127, 128]. This approach will be continued in the next chapter, where a mathematical description is presented.

Chapter 8Latest Modeling Attempts - Hole Trapping

Chapter 8
Latest Modeling Attempts - Hole Trapping