Thermodynamics of Superconductivity: Phase Transition, Entropy & Free Energy

Discover the thermodynamic framework behind superconductivity, including phase transitions, entropy changes, and the role of free energy.

Written by: Ajay Kumar

Posted: 6/15/2025

Superconductivity
Superconductivity phase transition diagram

🔁 Previously in This Series (Part 4)

In Part 4, we compared Type I and Type II superconductors. While Type I completely expel magnetic fields below a critical field, Type II allow magnetic flux to penetrate via quantized vortices. We explored their classification through the Ginzburg–Landau parameter κ\kappa and discussed why Type II superconductors dominate modern tech applications.

🌡 What Does Thermodynamics Tell Us About Superconductivity?

Superconductivity isn’t just about zero resistance or expelling magnetic fields—it also involves a dramatic shift in thermodynamic state. This transformation is governed by:

  • Free energy
  • Entropy
  • Specific heat
  • Phase transitions

Let’s break these concepts down as they apply to superconductors.

🌀 The Superconducting Phase Transition

Superconductivity arises below a material’s critical temperature TcT_c. This is not a chemical reaction or structural shift—it’s a second-order phase transition.

🧾 What is a Second-Order Phase Transition?

In thermodynamics, a second-order (continuous) transition is marked by:

  • No latent heat (no sudden energy jump)
  • Continuous first derivatives (entropy, volume)
  • Discontinuous second derivatives (specific heat, susceptibility)

Superconductors undergo this kind of transition at TcT_c, where:

  • Entropy changes gradually
  • Specific heat shows a finite jump

This is different from first-order transitions (like melting), which involve abrupt state changes and latent heat.

🧮 Free Energy: The Thermodynamic Foundation

In equilibrium thermodynamics, the Gibbs free energy GG tells us which phase is stable.

Below TcT_c, the Gibbs free energy of the superconducting state (GsG_s) is lower than that of the normal state (GnG_n):

ΔG=Gs−Gn<0\Delta G = G_s - G_n < 0

This negative difference explains why the system prefers to enter the superconducting state—it’s thermodynamically favorable.

When an external magnetic field is applied, this energy balance shifts. The system transitions back to the normal state when it becomes more energetically favorable.

🔥 Specific Heat: A Signature Jump at TcT_c

The specific heat (CC) measures how much energy is required to raise the temperature of a system.

For superconductors:

  • Below TcT_c: Cs<CnC_s < C_n (for some range)
  • At TcT_c: Sharp discontinuity

In BCS theory, this jump can be expressed as:

Cs−CnCn≈1.43\frac{C_s - C_n}{C_n} \approx 1.43

This specific heat anomaly is a classic fingerprint of superconductivity and is often measured to confirm the transition.

🧊 Entropy: Ordered State Below TcT_c

Entropy (SS) quantifies disorder. Since superconductors exhibit a more ordered quantum state (with coherent Cooper pairs), their entropy is lower than in the normal metallic state:

Ss<SnS_s < S_n

This behavior aligns with the second law of thermodynamics—the system evolves toward a lower-energy, lower-entropy state.

Importantly, the entropy difference vanishes at TcT_c, ensuring continuity across the transition:

ΔS=Ss−Sn→0asT→Tc\Delta S = S_s - S_n \to 0 \quad \text{as} \quad T \to T_c

🧲 Magnetic Field and Free Energy

In the presence of a magnetic field HH, the difference in Gibbs free energy becomes field-dependent:

Gs(H)=Gs(0)+H28πG_s(H) = G_s(0) + \frac{H^2}{8\pi}

At the critical field HcH_c, the energies of the superconducting and normal states become equal:

Gs(Hc)=GnG_s(H_c) = G_n

This defines the point of magnetic breakdown—a field strong enough to suppress superconductivity thermodynamically.

📊 Ginzburg–Landau Free Energy Functional

In Ginzburg–Landau (GL) theory, the superconducting state is described by a complex order parameter ψ(r)\psi(\mathbf{r}), and the free energy density takes the form:

F=Fn+α∣ψ∣2+β2∣ψ∣4+12m∗∣(−iℏ∇−2eA)ψ∣2+∣B∣28πF = F_n + \alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*}|(-i\hbar\nabla - 2e\mathbf{A})\psi|^2 + \frac{|\mathbf{B}|^2}{8\pi}
  • α\alpha changes sign at TcT_c
  • β>0\beta > 0 ensures stability
  • The last two terms account for kinetic energy of supercurrents and magnetic field energy

Minimizing this functional leads to the Ginzburg–Landau equations, which predict many thermodynamic properties and vortex behavior.

We’ll explore this further in the next post.

🧠 Key Takeaways

  • Superconductivity involves a second-order phase transition
  • The superconducting state has lower entropy and free energy
  • A specific heat jump is a critical signature at TcT_c
  • External magnetic fields modify the free energy landscape, leading to breakdown at HcH_c

Thermodynamics provides a macroscopic lens through which we understand why superconductors emerge and how they behave under stress.

🔮 Coming Up Next (Part 6)

In Part 6, we will explore the London equations, which describe how magnetic fields decay inside superconductors and give rise to the Meissner effect. We’ll derive these equations and explain what they mean for real-world materials.

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