characterization, modeling, and design of esd protection circuits

2.2. Transmission Line Pulsing 31
where P is the input power. By noting that
erf ( c ⁄ 4 Dt)
≈ tc ⁄ t if t≥tc (2.4)
and erf ( c ⁄ 4 Dt)
≈ 1 if t ≤ tc (2.5)
and setting P = Pffor T = Tc, the failure power can be calculated for each of the time
ranges described above:
Pf = ρabcCp ( Tc – T0) ⁄ t for 0 ≤tf≤ tc , (2.6)
P f
P f
ab πκρCp ( Tc – T0) = ---------------------------------------------------- for tc ≤tf≤ tb , (2.7)
tf – tc ⁄ 2
4πκa( Tc– T0) = ------------------------------------------------ for t , (2.8)
ln ( tf ⁄ tb) + 2 – c⁄ b b ≤ tf ≤ta
2πκa( Tc– T0) and Pf = ----------------------------------------------------------------------- for tf ≥ ta . (2.9)
ln ( a ⁄ b)
+ 2– c ⁄ 2b–
ta ⁄ tf The Pf vs. tf curve is shown graphically in Fig. 2.12. For times less than tc , no heat is lost
from the box, and a constant energy ( Pf ⋅ tf ) is needed to destroy the device. In the region
tc ≤ t≤ tb, failure power is proportional to 1 ⁄ t,
then becomes proportional to 1⁄ ln() t
in the region tb ≤ t≤ ta. For times greater than ta , the failure power approaches a constant
value, which means power dissipation is equal to power generation. Using values of
100µm, 1µm, and 0.1µm for a, b, and c, respectively, the values of ta , tb , and tc are approximately
10µs, 1ns, and 10ps, respectively. Thus in the ESD regime we expect to see a
1⁄ ln() t dependence of Pf . As noted in [23], limitations which affect the accuracy of the
model are assumptions that failure follows instantaneously when the temperature reaches
Tc and that there is an infinite heat sink outside the rectangular box. If there is little resistance
between the depletion region and device contacts, such as in silicided processes,
failure should follow quickly after Tc is reached. The main problem with the heat sink
assumption is that the SiO2 layer above the silicon is a thermal insulator and seriously
degrades heat dissipation in the vertical direction. This means that the power needed to
cause failure is actually lower (by less than a factor of two) than that calculated by the
model. Layout parameters which also affect the dissipation of heat are the closeness of the
f