### 7.4 Analytics Derivation of the Capture and Emission Time Constants

In order to promote the understanding of the eNMP model, and will be derived analytically in the following. The time constants observed in TDDS can be calculated on the basis of first passage times of a two-step process (see Fig. 7.5).

This quantity corresponds to the mean time it takes the considered system to arrive at the state , provided that it was in the state but not in state B at the beginning. In the eNMP model, one is only interested in the transition times between the stable states and , in which the defect dwells most of the time. Since the metastable states and are energetically higher than their corresponding stable counterparts and , the defect only remains temporarily in these metastable states. This is in agreement with the condition for the first passage time that the system must not be in state B at the beginning. As a result, the transition rates between the states and can be reasonably described as the inverse of first passage times.

The various transition pathways allowed in the eNMP model are summarized in the state diagrams of Fig. 7.6. The corresponding first passage times for the hole capture or emission read

For studying the field dependence of these capture and emission times, the definition () is used for and in the expression for the NMP barriers (7.17):

Using the definitions
the mean time constants (7.54)-(7.57) can be expressed as
with

Recall that the hole capture process can proceed from state over one of the metastable states or to the final state according to the state diagram of Fig. 7.3. The corresponding capture time constants are denoted as and , respectively, and will be discussed in the following. When the transition pathway is preferred, the capture time constant in the form of (7.52) is given by

Each of the summands in the nominator can be dominant so that is characterized by three distinct regimes, namely B, C, and D in Fig. 7.7.
• At extremely high negative oxide fields (regime D), is the dominant rate meaning that the transition2 proceeds much faster than (cf. Fig. 7.8). Thus the pace of the complete capture process () is determined by the second transition , which is much slower and has a time constant of . Since this second step is only thermally-activated, does not depend on the oxide field. This is consistent with equation (7.65) at extremely high negative oxide fields, at which both exponential terms become negligible compared to .
• At high negative oxide fields (regime C), the rate approaches the order of magnitude of and even falls below . Then the transition over the thermal barrier is undergone immediately after the defect has changed from the state to . Thus the kinetics of the hole capture process are governed by the forward rate of the NMP process . As a result, shows an exponential oxide field dependence, which is reflected in the first term of equation (7.65). Note that the second term is negligible due to its steeper exponential slope within this regime.
• At low negative oxide fields (regime B), is already outbalanced by its reverse rate (see Fig. 7.8) and the ratio of both rates determines the oxide field dependence. This gives an increased exponential slope originating from the second term of equation (7.65).

The transitions between these three regimes are smooth so that a curvature appears in the time constant plots of .

However, when the transition over the metastable state is favored (regime A), the capture time constant can be again formulated as a first passage time:

Since the metastable state is situated above the state by definition, holds. Therefore, the expression (7.75) can be approximated by
which is characterized by only two regimes (A’ and A”) now.
• At negative oxide fields (regime A’), the state is located relatively high (see Fig. 7.9) and the transition rate exceeds . Therefore, the first term of expression (7.76) vanishes and the field-insensitive transition with a time constant of dominates .
• When reducing the oxide field, the state is shifted downwards in the configuration coordinate diagram, thereby decreasing the transition rate . At a certain oxide field, falls below and the first term of the expression (7.76) becomes dominant (regime A”). As a consequence, governed by the field-dependent transition , which causes the exponential term of the expression (7.66). Depending on the value of , there exists a crossing point between the curves and , marking the transition between the regime A and B. It is noted that the NMP transitions in the regimes A, B, and C also involve a nearly negligible field dependence, which has already been present in the TSM for instance.

The transition between A’ and A” yields a kink, which is visible in of Fig. 7.7 (dotted line) but not in the overall hole capture time given by

As a result, this transition has not been observed in TDDS experiments, the regimes A’ and A” are not differentiated in Fig. 7.8.

Also the hole emission process has the possibility to proceed over either the state or , with and being the corresponding emission time constants (see Fig. 7.10). For the transition pathway over the emission time constant can be expressed as:

Since applies, has only two regimes, labeled with the capital letters F and G in Fig. 7.7.
• At extremely high negative oxide fields (regime G), the state is shifted upwards so that dominates and the field-dependent NMP transition determines the pace of . The sensitivity of to the oxide field is reflected in the exponential term of equation (7.67).
• At high negative oxide fields (regime F), the transition proceeds much faster than over the purely thermal barrier . Thus, is determined by the field-insensitive transition with a time constant of . It is important to note that the field independence of this regime is experimentally observed in the time constant plots of ‘normal’ defects (cf. Fig. 7.4 left).

At a low oxide field (regime E), the state is further shifted down, which speeds up the NMP transition and allows the pathway over the metastable state . The corresponding emission time constant is given by

For a sufficiently large barrier the rate is negligible compared to and and the above equation simplifies to
In this case, the state diagram reduces to a subsystem which includes the states and and is marginally disturbed by the rate . Then the states and can be assumed to be in quasi-equilibrium.
In this subsystem the condition holds so that the trap occupancy is given by
In the above equation, it becomes obvious that the condition is equivalent to . Furthermore, this equation can be used to simplify the equation (7.68) as follows:
If falls below at a certain relaxation voltage, the state becomes occupied and the emission time is determined by the field-independent transition with the time constant . By contrast, if is raised above , the state is underpopulated thereby slowing down the hole emission process. This occupancy effect is reflected in the second term, which reacts sensitive to changes in .

The overall hole emission time follows from

and is depicted in Fig. 7.7. At a certain oxide field, when the state is shifted below state , reaches its minimum value and falls below . The resulting drop in is observed as the field dependence characterizing ‘anomalous’ defects at weak oxide fields in TDDS experiments. As pointed out in Fig. 7.4, the drop of occurs when the minimum of the state passes that of state , and is thus related to the exact shape of the configuration coordinate diagram.