The last section has been devoted to the possible physical explanations for the charge trapping process in NBTI. The involved mechanisms, such as quantum mechanical tunneling for instance, are characterized by their stochastic nature. This means that one must deal with probabilities instead of pre-determined transition times for the trapping events. Such problems can be best handled within the framework of homogeneous continuous-time Markov chain theory [131], which rests on the assumption that the transition rates do not depend on the past of the investigated system.
The continuous time Markov chain theory presumes a set of discrete states
:
In the case of charge trapping, a single defect is represented by one vector .
When considering a large ensemble of defects, the stochastic behavior vanishes and
and
can be replaced by their corresponding expectation values
and
, respectively. Then the equations (3.19) and (3.22) read
When applying Markov theory to charge trapping in NBTI, each state must be
assigned to a certain configuration and a certain charge state of a defect. The rates
linking the states
and
must be related to certain defect transitions, such
as charge transfer reactions or thermally activated rearrangements of the defect
structure. The trapping dynamics are then governed by equation (3.23), which must
be solved as a function of time. The details of the applied numerical procedures are
outlined in Fig. 3.2.
Before the real device is subject to stress, the device is assumed to be in equilibrium.
The corresponding band diagram is computed by a P/SP-solver (A) for the
equilibrium conditions and
. Based on this information, the
transition rates
(B) can be evaluated for each defect
. With the
rates
at hand, the equilibrium occupation probabilities
are calculated using the equation (3.24) and must be subsequently stored
(C). When stress sets in, the gate voltage
and the temperature
are
changed to
and
, respectively. Since this alters the band diagram and
in consequence the transition rates
, the S/SP-solver step (D) and
the calculation of the rates (E) must be repeated. Then the implicit Euler
method is employed for the numerical time integration of equation (3.23).
This iteration scheme must be continued until the end of the stress time
has been reached. At each
, the change in the defect occupancies
(G) and the corresponding threshold voltage shift
(H) are
computed. With the beginning of the relaxation phase,
and
are
modified again and the whole iteration loop including the steps D, E, F, G, and
H must be repeated. This can be continued for several stress/relaxation
cycles with different stress/relaxation conditions. The steps E, F, and G
of numerical procedure in Fig. 3.2 must be repeated for each individual
defect, where the calculation of the rates requires most of the computation
time.
For the evaluation of , the charge sheet approximation is employed.
Small-area devices often contain only a handful of defects. Then the steps B, C, E, F, G, and H can be performed at computationally feasible costs. But since the calculation time increases with the number of defects, the simulation of large-area devices can become time-consuming. In order to reduce computation time for these simulations, a certain number of defects with similar properties are grouped together and replaced by one representative trap. This approach implies that the channel area covers several defects with similar properties including the trap level and the spatial location among others1. Furthermore, it is important to note here that the value of trap density is often quite inaccurate and can differ by some orders of magnitude. Therefore, the number of traps must be treated as a variable in the simulations. This means that the calculated degradation curves can be scaled to the experimental data in order to achieve reasonable agreement of the simulations with the measurements.