The amount and the distributions of the carriers generated by tunneling are
investigated at different gate, drain and bulk voltages, assuming graded (doped
by P) and abrupt (doped by As) drain junctions. The junctions are single
implanted. The resulting distributions of the generated electrons and holes
for a graded drain junction are presented in Figures 4.4 and
for an abrupt junction in Figures 4.5. In all four bias
conditions the gate-drain voltage is hold constant at 
, while the
drain-bulk bias is varied. The complete characteristics, i.e. the leakage
current versus the drain-bulk bias with a constant gate-drain bias as a
parameter, for these junctions are shown in Figures 4.6
and 4.7. These characteristics nicely reflect the
two-dimensional nature of the potential distribution in the critical area.
Although the quantitative conclusions are different for every specific
technology, some general observations can be made:
and 
, is smaller in the
1D than in the 2D model for the following two reasons:
, the current is smaller
in the 1D than in the 2D approach.
is much layer in the 1D model. If we assume a
model or a variable field 
model, the
becomes much smaller in the 1D approach in comparison with the
2D method. Consequently, the 1D approach to find the tunneling path
cannot be applied due to a strongly two-dimensional distribution of
the potential and the field in the gate/drain overlap
region [347][326][260][189][170][73][4].
, the field
distribution in the critical area close to the interface
changes weakly with 
. The magnitude of the
surface field is only determined by , because the lateral
component of the field is small compared to the component perpendicular
to the interface (graded junction). Deeper in the bulk (about 
),
where the transverse field component is significantly reduced, the field
strength changes as a function of as well, due to the lateral
field which is no more negligible in comparison with the transverse field
component. Therefore, the field 
at the starting points which are
located close to the interface, depends strongly on , but weakly
on , while the endfield 
increases with increasing .
These findings enable us to explain the roll-off of the tunneling
current with decreasing at a constant , obtained
experimentally in [347] and found in our calculations as well,
Figure 4.6. At 
, equilibrium holds in
the bulk and tunneling vanishes. When 
increases, the
band-bending which is necessary for tunneling occurs already at a low
drain-bulk bias, Figure 4.4(d). However, if we assume
for the generation model, the total tunneling current
is still small, although is very large due to a large 
. By
further increasing the lateral field component increases, an
average across the active area increases, the average is
reduced and 
increases. The tunneling paths are skew with
respect to the interface, which is also obtained in [4]. These
effects are clearly shown in Figures 4.4 (b and c). At
the same time the remains approximately constant while changing
. Therefore, if the model of the form is assumed (so
that the tunneling rate depends solely on the local field), the roll-off
of the curves is much less pronounced, as can be seen in
Figure 4.6.
, the total tunneling
current also increases with because of a small broadening of
the active area.
and 
,
respectively and do not change with .
are about 
and 
in Figure 4.4(b) and
Figure 4.4(c), respectively. In general, we found that
the electric field varies typically from 
to 
times in common
cases. The ratio can also be larger than 
, as is the case
for tunneling towards the interface in the BBISHE injection in
-channel devices (Section 4.1) and for tunneling
in the gate/drain overlap region in 
-channel devices. The impact of
the field variations on the generation rate is studied by different
direct tunneling models in Appendix H. In these calculations
a simple one-dimensional analytical model is assumed for the
totally-depleted uniformly doped bulk.
also yields an increasing current with increasing
and constant , as is obtained in
Figure 4.7.
and clearly show that the local
field-dependent approaches cannot be applied to calculate the
tunneling rates, because the electric field strength varies too much
along the tunneling paths.
-gate/-channel devices is
found to be smaller that in -gate/-channel devices. The
differences between -channel and -channel devices have been
ascribed to three effects [58]:
-channel devices is more graded than P or As
profiles in -channel devices (compare the characteristics for
P and As profiles in Figures 4.6
and 4.7). As a consequence, the leakage is
smaller in -channel than in -channel MOSFETs, regarding
the influence of the doping profile.
-channel and -channel devices is larger for the dual-gate
CMOS technology than for the single -gate technology. The
difference of about 
times can be estimated for a
dual-gate CMOS technology presented in [5].
is smaller in
-channel devices than in -channel devices, causing
different tunneling rates [58]. This explanation cannot
be accepted because the tunneling path and the average field
along the tunneling path are the same in both type of devices
( and are only exchanged). The penetration
probability for an incident electron is the same, but the number
of incident electrons per unit time differs at the starting
point from that at the endpoint. Consequently, the tunneling
rate in -channel devices differ from those in -channel
devices because of different values of the pre-exponential
factor in the expression for the tunneling rate. The value of
the exponential factor is the same in both cases
(cf. in Appendix H).
Several conclusions follow:
