Page 155 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electromagnetic System
x =
λx
E = ΣE t = τt (4.22)
Using the Maxwell equations (4.1a)–(4.1d) and the constitutive equations
(4.5)–(4.6), we apply dimensional analysis to derive the scale factors of
all other quantities in terms of the first three

[] [ εE] ⇒ D = ε ΣD (4.23a)
D ≡
o
ε Σ
o
[ ∇• D] ≡ ρ [] ⇒ ρ = --------ρ (4.23b)
λ
∂D ε Σ
o
J [] ≡ ≡ [] J = --------J
H ⇒
t ∂ τ
(4.23c)
ε Σλ
o
⇒ H = ------------H
τ
∂B τε Σ
o
[ ∇× E] ≡ ⇒ B = -----------B (4.23d)
t ∂ λ
For the material properties, ε o , µ o and a typical conductivity σ o are the
scale factors. We now insert the normalizations above into the Maxwell
equations:

µ ε λ 2 ∂ τ 2 ∂ 2 ∂
o o
em
∇ × E = – ----------------- H = – ---------- H = – β H (4.24a)
x 2 t ∂ 2 t ∂ t ∂
τ τ
τσ ∂ τ ∂
o
∇ × H = --------σE + E = ----σE + E (4.24b)
x
ε
o t ∂ τ e t ∂
∇ • ε E = ρ (4.24c)
x r
∇ • µ H = 0 (4.24d)
x
r
In (4.24b) we have defined the parameter
ε
Charge τ = ----- 0 (4.25)
e σ
Relaxation 0
Time

152 Semiconductors for Micro and Nanosystem Technology