characterization, modeling, and design of esd protection circuits

110 Chapter 4. Simulation: Calibration and Results
and does not diminish the value of the simulation because the results of interest all occur
at current levels above 100µA. The thermal boundary conditions consisted of overlapping
the electrical contacts with constant-temperature (297K) thermal contacts with no thermal
resistance. Although the simulations examined here are referred to as calibration
simulations, if the mobility and impact-ionization models have already been fixed by the
MOSFET-characteristic calibration, then comparing the measured and simulated Vt1 , Vsb ,
and Rsb is really a verification procedure rather than a calibration procedure.
An example of the I-V curve of a dc snapback simulation is shown in Fig. 4.42. The
horizontal line in the log curve shows where the solutions began incorporating the
thermal-diffusion equation. Note that although the lattice temperature does not
significantly increase above 300K until after snapback, the breakdown voltage is
substantially lower without including the thermal diffusion equation because the
temperature-dependent impact-ionization model cannot be used. Two things were
immediately noticeable from the initial snapback simulations. First, the snapback
resistance appeared to be a reasonable value (compared to experiment) immediately after
snapback, but the curve quickly rolled over at higher currents, indicating a much higher
resistance than in the experimental structures. Second, even when the snapback voltage
was extrapolated from the initial, steep part of the snapback portion of the curve, i.e.,
using a value of Rsb equal to the measured value, the snapback voltage was about 1.8V too
high. It was apparent from these simulations that calibration of the mobility and impactionization
models using the standard MOSFET curves was inadequate for snapback
simulations and thus that further manipulation of the model coefficients was needed.
Since the problem regarding the high snapback voltage was the simplest to understand, it
was dealt with first. The high Vsb value indicates that the impact-ionization generation rate
is too low for a given electric field in the snapback region of the I-V curve because the
simulated voltage (and electric field) needed to sustain a given current level is too high. As
shown by Eq. (3.27) and Fig. 3.22, the impact-ionization rate for electrons is determined
∞ crit
by two model coefficients, αn and En (or λn , which by Eq. (3.28) is inversely
crit
proportional to En ), assuming βn is constant. In the calibration of the MOSFET
∞
substrate characteristic, αn was held constant and λn was varied until the effective II rate
resulted in the proper amount of substrate current. A good fit of the substrate characteristic
was attained because, as Fig. 3.22 shows, if the spread in peak electric field values
throughout the stress conditions of the substrate-current test is relatively narrow, the