Page 170 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Summary for Chapter 4
1 ∂Θ
r ∂R
r ∂ R
2
2
2
---- 2 2 + --- – r k –[ 2 k ε r()] = p = – ---- 2 (4.64)
0
∂
R r ∂ 2 R r Θ ∂ θ 2
Since radial and angular components are separated, we can consider the
radial terms alone
2 2
∂ R 1 ∂ 2 2 p
+ --- R () – Rk – k ε r() – ----- = 0 (4.65)
0
∂
r ∂ 2 r r r 2
2
where p is the constant linking the radial and angular equations. We
2 2
make two further substitutions, setting U = k – k A , α = k b , to
0 0
obtain
2 2
∂ R 1 ∂ 2 p
+ --- R () – RU – ( αr) – ----- = 0 (4.66)
∂
r ∂ 2 r r r 2
This is exactly the form of the 2D harmonic oscillator (in cylindrical
coordinates) with total energy U and potential V = ( αr) 2 , the well-
known solutions of which are of the form
m
2
Rr() = exp – ( αr ⁄ 2) ∑ a r j (4.67)
j
j = m 0
and where the summation limits come from the boundary conditions.

4.4 Summary for Chapter 4

We encounter electromagnetic radiation as light from the sun and other
sources, as radio and television signals, mobile telephone communica-
tions, fibre-optic laser light rays, and of course in the form of background
radiation from deep space. The very successful model for the propagation
of radiation is the set of wave equations of Maxwell. Solid state interac-
tions expose the particle nature of light, where a quantum-mechanical

Semiconductors for Micro and Nanosystem Technology 167