Page 145 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 133

phase difference δ = χ − χ of the respective wave functions in the
1 2
superconductors. Since the velocity of a Cooper pair is proportional to the
phase gradient of its wave function, i.e., v ~ ∇ χ , and since the phase has a
period of π2 , it is not difficult to accept that the supercurrent be periodic.
Indeed, it can be shown [28] that the Josephson junction current is given by,

I = I sin δ , (176)
J 0
where,

φ dδ
V = 0 , (177)
2π dt

is the voltage across the junction.
The Josephson inductance, in turn, derives from substituting (176) and
(177) in the definition of inductance voltage, namely,

dI
V = L J J . (178)
dt
Thus,

dI dδ 2π
J = I cos ⋅δ = I cos δ V , (179)
dt 0 dt 0 φ
0
and, from (178) we obtain,
φ
L = 0 . (179)
J δ
2 πI cos
0
Clearly, the denominator, cos δ makes the inductance nonlinear, becoming
large as δ → π 2 , and negative in the range π 2 < δ < 3π 2. The

nonlinearity of the Josephson inductance gives rise to the formation of the
Josephson qubit, which is a nonlinear LC resonator consisting of the
Josephson junction’s inductance, L , and capacitance.
J
To conclude our exposition on superconductivity, we point out that there
are two types of superconductors according to how the Meissner effect
manifests in them [28]. In particular, type I superconductors are
characterized by a magnetization versus applied magnetic field curve that
increases up to a critical field, H , where it drops to zero and, concurrently,
c