1. Consider a superconducting film of thickness d<<l(T). An ac magnetic field is applied parallel to the top surface of the film, and there is no applied field on the bottom surface.
a) Give a brief argument to show that the currents are uniform in this film.
b) Using this fact, show that the surface impedance per square is
Zs = 1/(ds).
[Hint: This quantity is related to the impedance of a sample of length L and width W via
c) Using the low-frequency two-fluid model complex impedance in this equation, find the surface resistance Rs=Re(Zs) in the low-dissipation limit, that is, at temperatures slightly below Tc and low enough frequencies such that s1<<s2. What is the frequency dependence of Rs?
d) Using the same limit as in part c), find the imaginary part of the surface impedance, Xs=Im(Zs), and show that it is proportional to w. Thus, the imaginary part of the impedance (the surface reactance) is inductive. Find an expression for Ls, defined by wLs=Im(Zs). This is sometimes referred to as the "kinetic inductance" because it contains a contribution to the stored energy (U = LI2/2) from the kinetic energy of the superconducting electrons.
2. In a type-II superconductor, B=0 only up to a lower critical field, HC1, while superconductivity persists up to an upper critical field, HC2. A simple approximation to the observed behavior is:
-M = H H < HC1,
-M = Hc1 (Hc2 - H)/(Hc2 - Hc1) Hc1 < H < Hc2
a) Use these approximations, the definition of Hc (fn - fsc = m0 Hc2/2), and the fact that
fn - fsc = -m0 0∫Hmax M dH, to derive a simple approximate relation between HC1, HC2, and Hc.
b) Sketch M(H) and B(H) for Type I and Type II superconductors out to H > Hc or H > Hc2. Make sure your sketch shows the correct relative positions of Hc1, Hc2 and Hc.
3. The Schrödinger equation for the Macroscopic Quantum Wavefunction Y(r,t) for a superconductor.
a) Under the assumption that the number density n*(r,t) is constant in space and time, derive the energy-phase relationship:
-h ¶q/¶t = (1/2n*) L Js2 + q* f
from the real part of the macroscopic quantum Schrödinger equation. Interpret this equation physically.
b) Now assume that n*(r,t) is NOT constant in either space or time. Show that the imaginary part of the macroscopic Schrödinger equation yields:
¶n*/¶t = - Ñ· (n* vs)
Interpret this result physically (it may help to multiply both sides by q*).
4. A ring having 10-mm inside diameter and a width of 1.0 mm is formed of a superconducting material having l = 50 nm. Roughly estimate the current density at the inner surface of the ring when the ring contains one flux quantum.
5. Using the Cooper wavefunction derived in class, show that the expectation value of the Cooper pair radius squared:
<r2> = ∫|y(r1 - r2)|2 (r1 - r2)2 d(r1 - r2) / ∫|y(r1 - r2)|2 d(r1 - r2)
is given by,
<r2> = (4/3) h2 vF2/ W2,
where W =
-2hwc e-2/NV is the binding energy of the Cooper
pair, and vF is the Fermi velocity.
If we say that W ~ kBTc, then estimate the size of
a Cooper pair for Nb.
PROBLEMS NOT USED:
6. Inclusion of Coulomb Repulsion for a Cooper Pair.
Follow through the Cooper pairing argument for a pairing interaction of this form:
Vkk’ = -V for 0 ≤ E < hwc
Vkk’ = +U d(E-hwc) for E = hwc
where the later term represents the effects of electron-electron repulsion. Find an expression for the energy of a Cooper pair. Under what conditions is there a bound state?
This leads to an intractable algebraic equation for E
7. Problem 3.10c of van Duzer and Turner. R/L time, find L, solve for R, solve for r.
8. Given a cylindrical SC with outer radius b and a hole of radius a. Apply H above Tc, cool below Tc, go into the Meissner state. Calculate B and J everywhere in space. Calculate the fluxoid for arbitrary r between a and b. Similar to 3.13 of Orlando and Delin.