characterization, modeling, and design of esd protection circuits

58 Chapter 3. Simulation: Methods and Applications
To create thermal boundary conditions, thermal electrodes are placed anywhere along the
edges of a device in the same manner as electrical contacts and act as infinite heat sinks by
enforcing a constant temperature at the contact (Dirichlet boundary conditions). Noncontacted
edges obey homogeneous Neumann boundary conditions, i.e., there is no heat
flow across non-contacted edges. Lumped linear thermal resistance, in K/W, and
capacitance, in J/K, may be placed on a thermal contact to simulate the conduction of heat
away from the part of the device defined by the simulation. For example, a lumped
resistance may be placed on a thermal contact along the bottom of a structure to simulate
the dissipation of heat into the substrate.
3.1.1 Mobility and Impact Ionization Models
Since the lattice temperature is no longer constant throughout a simulated device, the
mobility and impact-ionization models must be dependent upon the local temperature. The
Lombardi surface mobility model [47] is chosen for low-field and transverse field mobility
modeling because it accounts for parallel and perpendicular fields needed to simulate
MOSFETs and because it includes lattice-temperature dependence. It is a semi-empirical
model with separate terms which account for surface-roughness scattering,
surface acoustical-phonon scattering,
and bulk mobility,
µ ac
=
µ sr
DN
= -------- , (3.21)
2
E⊥ BN CN N
------- , (3.22)
E⊥ ⋅ total
+ --------------------------
T3E⊥ µ b ≠
function ( T E ) , (3.23)
where Ntotal is the local total doping concentration, T is the local temperature, E⊥ is the
local electric field perpendicular to carrier flow, and BN, CN, DN, and EN are coefficients
with different values for electrons and holes. These mobility terms are added in parallel to
calculate the overall mobility (Mathiessens’s rule) at each point in the simulation space.
Other mobility models are available which account for transverse-field and/or temperature
effects, but the Lombardi formulation was judged to be the only model which treats both
EN
, ⊥