Abstract

CHAPTER 1. OVERVIEW 22
get
K(f)
(xm,kj)
⎧
⎪⎨ −
≈
⎪⎩
hkj
2m∗ (−3 fmj +4 fm−1,j − fm−2,j
), if kj > 0
2∆x
− hkj
2m∗ (3 fmj − 4 fm+1,j + fm+2,j
), if kj < 0
2∆x
(1.18)
We use a differencing scheme that depends on the sign of kj to respect the physics of
the problem and thereby keep this approximation from being unstable. If kj > 0, this
corresponds to electrons with positive momentum, moving from the left side of the
RTD to the right side of the RTD. This means information is propogating from the
left when kj > 0, and therefore to compute an approximation to the spatial derivative,
we want to use functions values that are to the left of the current point. Similarly,
when kj < 0, information is propogating from the right side of the RTD to the left
side of the RTD, and we use function values that are to the right of the current point
where we are computing an approximation to the spatial derivative.
The P (f) term is discretized with the composite trapezoidal rule, which is second-
order accurate. So we have the approximation:
P (f)
(xm,kj)
≈− 4
Nk
fmj
h
′T (xm,kj − kj ′)wj ′ (1.19)
where wj ′ are the weights of the composite trapezoidal rule, given by:
wj ′ =
⎧
j ′ =1
⎪⎨ ∆k, for j ′ =2, 3,...,Nk − 1
⎪⎩
∆k
2 , for j′ =1,Nk