Based on the existence of oxide defects and the band-to-trap transition
possibilities, depicted in Fig. 8.1, already a single defect system has to consider
all transitions originating from various band states. This means that the whole
conduction or valence band has to be considered, instead of only or
. On the basis of the statistical description of the recombination of
electrons and holes under the release of energy in terms of lattice vibrations
(Shockley-Read-Hall theory [122]), the determination of effective rates in and out
of a specific defect system is possible. The corresponding rate equations are
Assuming detailed balance [122], which means that each process is balanced by its reverse process, both rates have to equal within (8.1) and (8.2). This yields
![]() | (8.3) |
Combining (8.3) with (8.2)1
and evaluating the integral finally gives the capture time constant of the
holes
![]() | (8.5) |
with the rate to fill the defect at
and
for the reverse rate.
Furthermore,
gives the probability that the defect is actually filled.
Consequently, the capture and emission rates can be written as
![]() |
In addition to a tunneling coefficient of to account for the oxide
trap depth after [121], the cross section is considered to be thermally activated
with a bias independent barrier
[124]. Putting these assumptions together
yields
![]() | (8.8) |
with a constant prefactor [131, 124]. With the knowledge that whether the
defect level lies below or above
, different barriers are obtained after
Fig. 8.1, equations (8.6) to (8.8) are now used to calculate the capture rates
![]() | (8.11) |
with as effective valence band weight,
![]() |
The trapping barrier can further be written as a superposition of the
energy distance during flatband
and the applied field
which changes the relative barrier between semiconductor and oxide, cf. Fig. 8.1
(right) and Fig. 8.2 (right)
![]() | (8.12) |
With the help of (8.11) and (8.12) the time constants in (8.9) and (8.10) finally read as
At first only the part of (8.13) and (8.14), which depends on the relative position of In a typical BTI stress/relaxation sequence all defects are in thermal
equilibrium prior to stress. Due to stress the Fermi level is shifted
below
. For defects with
the resulting barrier
can
only be balanced by the
term in (8.13). After (8.12) this means
that energetically deeper defects also need to be located deeper in the
oxide in order to become charged during stress, i.e. only defects with
, where
denotes the potential at the interface,
are accessible during stress [130]. When the stress is completely removed,
is shifted back above
and the previously charged defects will by
moved back below
. According to (8.14) they can be emptied over a
small barrier if there is any. Thereby accessible oxide defects now feature
during relaxation [130]. Thus, the exact defect level
is not of particular interest for the capture and emission process.
only has to lie inside the accessible energy region, i.e. above
for
stress and below
for relaxation. This means that the conditional
part of (8.13) and (8.14) only exhibits a small temperature and field
dependence.
It is important to realize that it is the thermal barrier in (8.13) and
(8.14) introduced by Kirton and Uren, which gives the required temperature
dependence, though this dependence is not fully correct, as will be shown
later. To first order, the capture
and emission times
of the
defects are determined by
and
, making another fact visible:
and
are correlated, while measurement results determining
these times during BTI revealed uncorrelated behavior [111, 116]. This
rules out the possibility of describing oxide defects by an extended SRH
theory.