Page 159 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electromagnetic System
Equation (4.35) is amenable to numerical solution by so-called fast-Pois-
son solvers, computer programs that exploit the following fundamental
property of the Poisson equation. If we consider a region of homoge-
ε
neous , then inside this region equation (4.35) becomes a Poisson equa-
tion ∇ 2 ψ = ρε . We consider the situation where ρε⁄ represents a
⁄
point charge at position which we represent by a Dirac delta function:
r
ρ
(
--- = δ r′ – r) (4.36)
ε
Green For this case, we obtain the so-called Green function for the Poisson
Function equation:

1 1
ψ = ----------------------- (4.37)
(
4π r′ – r)
that satisfies equation (4.35) exactly. Since we are, in principle, able to
represent any spatial charge distribution as a linear superposition of indi-
vidual point charges, we can obtain the associated potential by a linear
superposition of the appropriate fundamental solutions via equation
(4.37). In reality this is not very practical, as we would require large sums
n
of terms at each point of interest in space: for charges and m evalua-
tion points we would require of the order of n × m evaluations.

Multipole An important simplification technique is the so-called multipole expan-
Expansion sion. The idea is remarkable. Consider a group of charges in space within
an enclosing sphere of radius r g . Clearly, if we are far enough from the
group of charges, i.e., r>>r g , the values of the Green functions of the
individual charges will not indicate strongly the spatial separation of the
charges, merely their number and charge polarity. Algebraically, we can
represent the combined Green function of a group of charges using a
multipole expansion. The power of the method is that a group of groups
of charges can again be represented in this manner. What this ultimately
means is that a single sum over the Green function of many charges can
be made much more efficient. We first spatially partition the charges into
a hierarchy of clusters, then, starting inside the first groups in the hierar-

156 Semiconductors for Micro and Nanosystem Technology