Homework assignment 5: Superconductivity (due Oct 15)

PHYS-E0551
Homework assignment 5: Superconductivity (due Oct 15)
Homework assignment 5: Superconductivity (due Oct 15)
1. At T = 0, find the minimum voltage V = µL − µR above which quasiparticle can flow for a) NSN, b)
SNS, and c) SSS superconducting tunnel structures. Consider gate charge ng = 0 so that the island
charge states involved are n = 0 and n = 1.
Current can flow if both transition rates ΓL→I (n) and ΓI→R (n + 1) are finite. Remember the charging
energy of the center island I.
2. Consider a superconductor carrying supercurrent. Take ∆(r) = |∆|eiq·r with constant |∆|. q is the
momentum of the superflow.
(a) Find the quasiparticle energy spectrum k from the Bogoliubov de Gennes equation:
!
~2
∇2 − µ
∆(r)
− 2m
u(r)
u(r)
=
~2
v(r)
v(r)
∆(r)∗
−[− 2m
∇2 − µ]
(1)
Try Ansatz uk (r) = Uk ei(k+q/2)·r , vk (r) = Vk ei(k−q/2)·r . You can simplify things by assuming
|q| |k| so that |k + q|2 ≈ |k|2 + 2k · q. (For each k there are two eigenenergies k,+ and k,− ,
since it’s a 2 × 2 matrix. For q = 0, k,+ > 0 and k,− < 0. k,+ are the excitation energies.)
(b) Determine the critical superflow velocity vs,c = ~|q|
2m above which for some k, k,+ < 0.
q
Since |q| |k|, it’s enough to check the condition only at wavevectors k = ±kF |q|
, where kF =
p
2mµ/~2 is the Fermi wavevector.
(c) If some excitation energies are negative, k,+ < 0, is the BCS vacuum ground state still the state
with the lowest energy? What will happen experimentally when the superflow is too large?
(d) The charge current density is jc = ens vs , where ns is superconducting electron density.
Estimate the order of magnitude of maximum supercurrent density (= critical current density) in
aluminum. At T = 0 for aluminum, ns ≈ n = free electron density ∼ 1029 m−3 , and |∆| ∼ 200 µeV.
For 3D metals, kF = (3π 2 n)1/3 . What is the critical current of an aluminum microstrip with
cross-section 1 µm × 100 nm?
~
3. Estimate the average voltage V¯ = 2e
∂t ϕ across a Josephson junction (overdamped) at |I| Ic caused
by thermal fluctuations. Assume the dynamics of the junction in this regime is described by motion of
~
ϕ in the “washboard” potential U (ϕ) = −EJ cos(ϕ) − 2e
Iϕ, with EJ = ~Ic /(2e).
(a) Thermal activation follows Arrhenius law: rate Γ± = ω0 e−U± /(kB T ) . Determine the barriers U± for
transitions between neighboring washboard potential minima (ϕ 7→ ϕ ± 2π). Simplify by assuming
I Ic so the minima are at ≈ 0, ±2π, ±4π, . . ..
(b) Evaluate the average voltage based on the estimated transition rates.
(c) Estimate ω0 by requiring that at kB T ≈ EJ ,
dV
dI
|I=0 is equal to the normal-state resistance RN .
(d) Sketch the resulting current-voltage relation.
(updated 2014-10-03 17:31:25)

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