Persistent Current in a Mesoscopic Ring

Threaded Flux

Flux Φ / Φ₀ 0.00

Ring Population

Electrons N 12

Ring radius R (nm) 400

Dephasing (T / T_φ) 0.00

Live Readouts

Persistent current I

0.0 nA

Ground-state energy E

0.0 μeV

Direction

— none —

Parity

N even

Period

Φ₀

How it works

Thread a magnetic flux Φ through a metal or semiconductor ring whose circumference is shorter than the electron phase-coherence length ℓ_φ. The vector potential shifts the ring's periodic boundary condition, so the single-particle energy levels become

E_n(Φ) = (ħ²/2mR²) · (n − Φ/Φ₀)²,   Φ₀ = h/e,   n = 0, ±1, ±2, …

Filling the N lowest levels and summing gives a total ground-state energy E(Φ) that is periodic in Φ with period Φ₀ — not the superconducting h/2e, but the normal-metal h/e, since single electrons (not Cooper pairs) carry the phase. Because the state is a true equilibrium ground state, it carries a current that never decays despite scattering:

I(Φ) = − dE(Φ)/dΦ   (sawtooth, period Φ₀)

Φ/Φ₀ slider — thread more flux through the loop; watch the sawtooth trace fill in on the strip chart and the current arrows on the ring reverse each time Φ crosses a half-integer multiple of Φ₀.
Electrons N — sets how many single-particle levels are occupied. Flipping N by one electron (odd ↔ even) shifts the sawtooth by half a period — the famous parity effect: an even-N ring is diamagnetic near Φ=0, an odd-N ring is paramagnetic.
Ring radius R — sets the level spacing ħ²/2mR² and hence the current amplitude; smaller rings give a larger, more sharply-sawtoothed current, matching why the effect needs genuinely mesoscopic (sub-micron) samples.
Dephasing — models finite temperature / residual inelastic scattering (ℓ_φ shrinking toward the ring size) by exponentially damping the current amplitude, reproducing why the effect vanishes above ~1 K in real devices.

Real-world relevance: exactly this h/e-periodic circulating current was measured in isolated gold and GaAs rings (Lévy et al. 1990; Bleszynski-Jayich et al. 2009, using a nano-SQUID cantilever) and is a direct, textbook demonstration that quantum phase coherence — not just interference visibility as in the Aharonov–Bohm effect — can drive a real, measurable, dissipationless equilibrium current at the nanoscale.

Conduction ring Flux tube (Φ) Circulating current

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