Page 143 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 131
G
Since ∇ ⋅ B ≡ 0 , (167) becomes,

G 4π e 2 n G
∇ 2 B − s B = 0 , (168)
m c 2
e

which may be rewritten as,

G 1 G
∇ 2 B − B = 0, (169)
δ 2 L

with the London penetration depth given by,

m c 2 c
δ = e = , (170)
L 4π 2 ω
e n
s p
where ω is the plasma frequency in the material. Taken along one
p
direction, say z, (170) becomes,
d 2 B 1
x − B = 0, (171)
dz 2 δ 2 z
L
where B () 0 is the magnetic field at the surface of the superconductor.
x
The solution stipulates that the magnetic field decays inside the
superconductor with a characteristic length δ . Assuming a plasma
L
frequency of 10 15 s / , the approximate value of the London penetration
depth is 300Å. This means that at distances greater than ~300Å from the
surface, the magnetic field and, per (165), the current, vanish inside a
superconductor, see Fig. 3-22.

Vacuum x x Superconductor
Vacuum
Superconductor
B B
x x z () 0 ( − − / z δ / z δ
B B () = Bz = B ( )e )e0 L L
x x x x
z z
δ δ L L

Figure 3-22. Decaying magnetic field in superconductor.