characterization, modeling, and design of esd protection circuits

3.1. Lattice Temperature and Temperature-Dependent Models 59
effects to a reasonable degree. For example, some of the low-field/transverse-field models
which do include temperature dependence use only a simple scaling factor to model surface
mobility.
In the high-field mobility region, the empirical Caughey-Thomas expression [48] is used
to account for velocity saturation. For electrons, the high-field mobility is
µ n
=
µ S, n
⎛µ Sn , EII⎞ 1 ⎜---------------- sat ⎟
⎝ ⎠
β ⎛ n⎞
⎜ +
⎟
⎝ ⎠
1
v n
– β ⁄ n
, (3.24)
where µ S,n is the low-field mobility, E || is the electric field in the direction of current flow,
sat
vn is the saturation velocity, and βn is a fitting parameter. An analogous equation is used
for hole mobility. Degradation of mobility at high electric fields is due to high-energy
carriers interacting with optical phonons rather than acoustic phonons. Inherent in this
situation is that the carriers are no longer in thermal equilibrium with the lattice, i.e.,
electrons and holes have their own characteristic temperatures. However, since the carrier
temperature is related to the local electric field [42], an expression such as Eq. (3.24)
allows us to calculate mobility degradation without solving for carrier temperature (such
modeling still neglects the non-local effects of extremely high fields on carrier transport).
This mobility model is implicitly dependent on the lattice temperature through the
temperature-dependent saturation-velocity [29],
sat
vn 7
= 2.4×10
⁄ ( 1 + 0.8exp ( T ⁄ 600)
) . (3.25)
Modeling of impact-ionization (II) generation of carriers is essential for the simulation of
breakdown and snapback phenomena in ESD protection MOSFETs. The II generation rate
can be expressed as
G II
=
Jn Jp αn ⋅ ------- + α
q p ⋅ ------q
, (3.26)
in which αn and αp are the electron and hole ionization coefficients, respectively, with
units of cm -1 . An expression for these coefficients commonly used in numerical
simulation is [46]
αnp ,
=
∞
αnp ,
crit
β np
⎛ ⎛E⎞ , ⎞
np ,
⋅ exp⎜–
⎜--------- ⎟ ⎟ , (3.27)
⎝ ⎝ EII ⎠ ⎠