characterization, modeling, and design of esd protection circuits

78 Chapter 3. Simulation: Methods and Applications
ionization, for ESD simulation its most important application is the modeling of gross
device heating which leads to thermal runaway. In transient simulations, if the conduction
of heat away from a device is accurately modeled by the thermal boundary conditions and
if the defined device geometry and doping profiles produce the proper current densities
and electric fields, electrothermal simulation should be able to predict at what time
thermal failure will occur for a given input pulse and thus to generate a failure power vs.
time to failure curve. Thermal runaway is inherently a three-dimensional phenomenon
because the hot spot always forms at a point in a device, and after formation current rushes
into the spot from all directions. Heat conduction theory predicts that if current is flowing
uniformly across the width of a device and the device is surrounded by a spatially
invariant heat sink, the hot spot will form in the center of the width dimension because this
is the point of peak temperature. (Experimentally, it has been found that thermal runaway
may originate at a “weak spot” where the electric field is slightly higher due to the erose
drain edge of the gate oxide [52].) In contrast, 2D simulation can only model current
rushing in from two dimensions after a hot spot forms. Although it cannot properly model
the runaway itself, if current flows relatively uniformly in a device before second
breakdown and the simulation cross-section is representative of the real cross-section
containing the “weak spot,” 2D simulation should be able to predict the onset of second
breakdown, i.e., the time at which the device voltage drops due to a reduction in overall
device resistance. The simulated voltage does fall off with time after the onset of
breakdown due to the negative differential resistance, but not as sharply as seen
experimentally (e.g., Fig. 2.10) because current cannot rush in from the third dimension.
It is illuminating to apply an analysis like that of the 3D thermal box model in
Section 2.2.2 to 2D device simulation. If the assumptions are analogous, i.e., if all power
generation occurs uniformly within a rectangle in the drain depletion region and second
breakdown follows instantaneously when the peak temperature reaches a critical value,
then it appears that the governing equation for peak temperature is just like that of the 3D
case (Eq. (2.3)) except there are only two dimensions:
t
P′
T() t T0 ---------------------- erf⎛
b
-------------- ⎞ c
=
+
erf⎛--------------
⎞ . (3.33)
ρCp ( bc)
∫
dτ
⎝4 Dτ⎠
⎝4 Dτ⎠
0
Note that the width dimension, a, is omitted and the power, P, has been replaced by P´, the
power per width in W/cm, which is the product of the voltage and the current per width in