The Flux Creep
Field Note — Thermal Activation in a Pinned State
Nothing is ever truly still.
The vortex glass is pinned. The vortices sit on their defects. The critical current flows without resistance. This is the ideal picture, the zero-temperature picture, the picture that appears in the scaling theory and the T=0 phase diagram.
But T is not zero. And at finite temperature, the vortices remember that they are quantum objects in a thermal bath, and the thermal bath has energy, and that energy is enough, given enough time, to move things.
The Hopping Mechanism
A vortex sits in a pinning potential. This potential has a bottom — the defect — and a barrier — the energy required to leave the defect and move through the crystal. The barrier height is U, the pinning potential depth. At zero temperature, the vortex cannot cross this barrier. It is stuck.
At temperature T, the vortex is coupled to phonons, to quasiparticles, to the thermal environment. And the thermal environment fluctuates. Sometimes, by random chance, the fluctuations add up. The vortex gains energy U. It crosses the barrier. It hops to a nearby location — another defect, or just a new position in the landscape.
This is flux creep. It is slow. It is probabilistic. It is the difference between a truly pinned state and a state that is pinned but slowly leaking.
The Arrhenius Law
The hopping rate follows an Arrhenius law:
Γ ∝ exp(−Ueff/kBT)
where Ueff is the effective barrier height. This barrier is not constant — it depends on the Lorentz force, which depends on the current density. As the current increases, the Lorentz force tilts the pinning potential, lowering the barrier. The barrier decreases, the hopping rate increases exponentially, and the resistance grows.
This is why the resistance is so sharply dependent on current. A small increase in J tilts the potential just enough that the exponential factor changes by orders of magnitude. The resistance turns on like a switch.
Collective Flux Creep
Individual vortex hopping is the simplest picture. The real system is more complex. Vortices interact. They form lines. They form bundles. And the barriers they must cross are collective barriers — you don't move one vortex, you move a chunk of the vortex lattice.
The collective pinning theory gives Ueff ∝ J₀/J^μ where μ depends on the collective regime:
- μ = 1/2 for small bundle creep
- μ = 1 for large bundle creep
- μ = 3/7 for creep of a single vortex line
Each regime corresponds to a different length scale at which the vortex line is pinned coherently. The small bundle is a handful of vortices pinned together. The large bundle spans the entire sample. A single line is a vortex line pinned by individual defects along its length.
The result is always the same: Ueff decreases as J increases, the hopping rate increases, and the electric field E ∝ exp(−Ueff/kBT) turns on exponentially.
The Logarithmic Decay
Here is the thing that experimentalists notice and theorists try to explain: the magnetic relaxation. If you trap a magnetic flux profile in a superconductor and let it sit, the profile decays. Logarithmically.
M(t) ∝ −ln(1 + t/τ₀)
where τ₀ is a microscopic time scale, typically 10⁻¹⁰ to 10⁻⁸ seconds. Over experimental time scales — seconds, minutes, hours — the decay is approximately logarithmic. It is extremely slow. The magnetic profile barely moves.
This logarithmic decay is the fingerprint of collective flux creep. It tells you that the system is exploring its energy landscape through thermally activated hops, each one small, each one nearly reversible, each one contributing to an irreversible rearrangement that you can only see over long times.
The Implications
Flux creep matters because it limits the performance of superconducting devices. A magnet that slowly loses field. A cable that slowly develops resistance. A quantum computer whose flux qubits slowly decohere because trapped vortices slowly move.
But it also matters because it tells you about the landscape. The logarithmic decay rate gives you Ueff. Ueff gives you the pinning potential. The pinning potential gives you the defect structure. And the defect structure tells you whether your material is good or bad.
Flux creep is the slowest dissolution of a glass. The glass is still there — the vortices are still pinned — but it is leaking, drop by drop, barrier by barrier, hop by hop. And if you wait long enough, the glass flows.
Not all at once. Not catastrophically. But inevitably. Because temperature is patient, and entropy always collects its debt, and the pinned state is not the true ground state — it is a metastable state, and metastability has an expiration date.
The vortex does not want to move. The vortex is held by the defect. The defect is held by the crystal. The crystal is held by nothing. And at finite temperature, nothing is enough.