Page 150 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Basic Equations of Electrodynamics
In vacuum, and for the semiconductor silicon, the permeability µ =
–
2
–
2
2
⁄
= 1 c ε . Note that, by convention, µ = 4π × 10 – 7 Js C m µ 1 0 ,
0 0
which immediately fixes the value for ε . In vacuum the permittivity
0
1
–
ε = ε = 8.854 × 10 – 12 AsV m – 1 . For magnetically active materials
0
the dependence of the relative permeability on the magnetic field is in
fact highly nonlinear and frequency dependent. The electric field also
E
drives the electric displacement D , and is hampered thereby by the
ε
dielectric permittivity of the material
D = εE = ε ε E (4.7)
0 r
For solids, the relative permittivity ε is a function of the spatial distribu-
r
tion of atomic charge, as well as the charge’s mobility. In fact, its value is
strongly frequency dependent. A more detailed discussion of the cause of
permittivity can be found in Section 7.2.2. (Solving the Maxwell equa-
tions is outlined in Box 4.1).
Taking the curl (∇× ) of the Faraday law (4.1a) and the time derivative
(∂∂t⁄ ) of the Ampére law (4.1b), we obtain
∂B ∂
∇× ∇× E = – ∇× = – ∇× µH (4.8a)
t ∂ t ∂
2 2
∂H ∂J ∂ D ∂ ∂
∇× = + = σE + εE (4.8b)
t ∂ t ∂ t ∂ 2 t ∂ t ∂ 2
Assuming vacuum conditions, so that σ = 0 , µ = µ , and ε = ε , and
0 0
requiring ρ = 0 , we obtain

2
∂H ∂ E
∇× ( ∇× E) = – µ ∇× = – µ ε (4.9)
0 t ∂ 0 0 t ∂ 2

Using the identity ∇× ( ∇× E) = ∇ ( ∇• E) ∇ 2 E , and the Gauss law
–
(4.1c), we obtain the second order partial differential Helmholtz equation

˙˙
∇ 2 E µ ε E = 0 (4.10)
–
0 0
Semiconductors for Micro and Nanosystem Technology 147