PISCES-2ET and Its Application Subsystems - Stanford Technology ...

DUET Carrier Transport Model
N C
⎛
2 2πm * n( T L )k B T
------------------------------------ n ⎞ 3 ⁄ 2
= ⎜
⎟
⎝
⎠
h 2
where T L is the lattice temperature and because in our DUET model, T L is not necessarily the same
*
as T n , we have explicitly indicated the lattice temperature dependence of the effective mass, m n .
Expressing E Fn – E C in terms of n in Eq. (2.3), we obtain an expression of f 0 as function of n, T n ,
, and ε as follows:
T L
(2.4)
2n ε
f 0 ( r,
k) = f 0 ( n( r) , T n ( r) , T L ( r)ε
, ) = ---------------------------exp⎛–-----------
⎞
N C ( T n , T L ) ⎝ k B T n
⎠
This form of distribution function constitutes the basis for our model discussion.
(2.5)
2.2 Energy Balance and Thermal Diffusion
Equations
Temperature is a measure of the kinetic energy for random motion and the equation governing the
carrier/lattice temperature can be derived from the energy balance principle. Because of the free
particle nature of carriers, Fick’s second law is applied to describe the carrier kinetic energy balance
(or continuity). For electrons, the continuity principle dictates
∂w
-------- n
= – ∇ ⋅ s (2.6)
∂t
n + j n ⋅ E n – u wn
where w is the kinetic energy density and s is the energy flux, and both can be defined and evaluated
from the carrier distribution function. The last two terms on the right hand side (RHS) of the above
equation represent the energy conversion and net loss rates, respectively. The familiar Joule heat term,
j where is the electric field acting on electrons 1 n ⋅ E n E n , actually represents the rate of conversion
from the electrostatic potential to the kinetic energy, and u w is the rate of net loss (loss minus
1. In heterostructures, the electric field acting on carriers is in general different for electrons and
holes.
PISCES-2ET – 2D Device Simulation for Si and Heterostructures 7