Superconductivity Simulator: Phase Transitions at Cryogenic Temperatures

simulator intermediate ~10 min
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R = 0 Ω — superconducting state

At 9.2 K with no applied field, niobium is below its critical temperature — resistance drops to exactly zero and magnetic flux is expelled via the Meissner effect.

Formula

R/Rn = 0 for T < Tc(B), 1 for T ≥ Tc(B)
Tc(B) = Tc₀ × √(1 − (B/Bc₀)²)
λ(T) = λ₀ / √(1 − (T/Tc)⁴)

Zero Resistance Below Tc

In 1911 Heike Kamerlingh Onnes cooled mercury to 4.2 K and watched its electrical resistance vanish entirely — not just decrease, but drop to unmeasurably zero. This phase transition is one of the most dramatic in all of physics: below the critical temperature, electrons form Cooper pairs that glide through the lattice without scattering. A current set flowing in a superconducting ring persists for years without decay.

The BCS Theory

Bardeen, Cooper, and Schrieffer explained superconductivity in 1957 through a quantum many-body theory. An electron distorts the positive ion lattice, creating a region of slightly higher positive charge density that attracts a second electron. This phonon-mediated attraction binds electrons into Cooper pairs with opposite momenta and spins. The pairs condense into a collective ground state separated from excited states by an energy gap, making scattering impossible at low temperatures.

Magnetic Field Effects

Applied magnetic fields compete with the superconducting state. In type-I superconductors, the Meissner effect completely expels flux until the critical field Bc is reached, causing an abrupt transition to normal state. Type-II superconductors allow partial flux penetration through quantized vortices in a mixed state between lower critical field Bc1 and upper critical field Bc2. This simulation shows how the effective Tc decreases with increasing field and visualizes Cooper pair density.

From MRI to Quantum Computing

Superconducting magnets generate the powerful, stable fields needed for MRI scanners and particle accelerators like the Large Hadron Collider, which uses 1,232 NbTi dipole magnets cooled to 1.9 K. SQUID sensors detect magnetic fields a billion times weaker than Earth's, enabling magnetoencephalography. Today, superconducting transmon qubits operating at 15 millikelvin form the basis of leading quantum computers from IBM and Google.

FAQ

What is superconductivity?

Superconductivity is a quantum mechanical phenomenon where certain materials exhibit exactly zero electrical resistance and expel magnetic fields when cooled below a critical temperature Tc. Discovered by Heike Kamerlingh Onnes in 1911 in mercury at 4.2 K, it arises from Cooper pairing of electrons mediated by lattice vibrations.

What determines the critical temperature?

The critical temperature depends on the material's electron-phonon coupling strength, density of states at the Fermi level, and crystal structure. BCS theory predicts Tc from these microscopic parameters. High-temperature superconductors (cuprates) can reach Tc above 130 K through mechanisms still debated.

How does a magnetic field destroy superconductivity?

Applied magnetic fields supply energy that breaks Cooper pairs. The effective critical temperature decreases as Tc(B) = Tc₀√(1 − B/Bc). Type-II superconductors allow partial flux penetration via quantized vortices in the mixed state between Bc1 and Bc2.

What are practical applications of superconductors?

MRI magnets, particle accelerator dipoles (LHC uses NbTi at 1.9 K), SQUID magnetometers for brain imaging, maglev trains, and lossless power transmission cables. Quantum computers use superconducting qubits operating at ~15 mK.

Sources

Embed

<iframe src="https://homo-deus.com/lab/cryogenics/superconductivity/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub