Page 144 - Principles and Applications of NanoMEMS Physics

132 Chapter 3

The vanishing of the magnetic field inside a superconductor is called the
Meissner effect, and has certain practical consequences. For instance, if a
superconducting wire is turned into a ring, then the fact that its bulk
magnetic field and current are zero implies that,

G
G
'
'
= ∇ χ = 2e A Ÿ ³ ∇ χ ⋅ dl = χ 2 − χ 1 = 2e ³ A ⋅ dl . (172)
c = c
C C
Therefore, the Cooper pair wave function may be written as,
e 2 A ⋅dl i e 2 φ
i
Ψ = Ψ e χ i = Ψ e c = ³ = Ψ e c = , (173)
A 0 0 0
where φ is the magnetic flux inside the hollow part of the ring. Since the
phase must equal an integer multiple of π2 , however, we have,

e 2
φ = 2 n π
c = , (174)
or,

φ = π= n = 1 § ¨ 2π hn · = 1 φ n . (175)
¸
ec 2 © e2π c ¹ 2 0

Thus, the magnetic flux confined by the superconducting ring is quantized in
units of flux φ = h 2e , called a fluxoid.
0
The phase of the Cooper pair wave function and the fluxoid are at the
heart of two effects of fundamental import for applications, namely, the
Josephson effect and the nonlinear Josephson inductance.
The Josephson effect refers to the fact that, whenever two
superconductors at the same temperature are brought in proximity to one
another, separated by a thin insulating layer (so thin that tunneling of Cooper
pairs may occur), Fig. 3-23, a supercurrent I flows, which depends on the
J
V V
I I
J J
ψ 1 ψ 1 ,χ ,χ 1 1 ψ 2 ,χ ,χ 2 2
ψ 2
S S I I S S

Figure 3-23. Schematic of Josephson junction.