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
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 transition
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 .
Figure 7.7: Left: The calculated hole capture time constants as a function of
the oxide field. The different regimes of (A, B, C, and D) are separated by
the thin vertical lines and labeled by the green circles with the capital letters.
The dotted curves show the capture processes over a metastable state .
The field dependence of within a certain regime is shown by the dashed
curve, which becomes constant if is insensitive to . Right: The same
but for the hole emission time constants with the regimes (E, F, and G).
Figure 7.8: A schematic representation of adiabatic potentials in the regimes
B, C, and D. The arrows show the possible directions of the transitions involved
in the capture process. Their thicknesses indicate the magnitude of their rates,
where the thinner arrows are associated with larger transitions times and thus
governs the oxide field and temperature dependence of the complete capture
process . With higher oxide fields (B D) the blue potential (neutral
defect) is raised relative to the red one (positive defect). This is associated with
an increase of and a decrease of the reverse rate . In contrast to the
charge transfer reactions and , the thermal transition is
not affected by the oxide field.
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
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.
Figure 7.9: The same as in Fig. 7.8 but for the regimes A’ and A” of the oxide
field dependence of .
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).
Figure 7.10: The same as in Fig. 7.8 but for the regimes E, F, and G of the
oxide field dependence of .
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