Type-II Superconductors

Derivation: Vortex Line Energy

We derive the energy per unit length of an isolated Abrikosov vortex in the London limit ($\kappa \gg 1$), where $|\psi|$ is suppressed only in a small core of radius $\sim\xi$.

Step 1: Outside the core ($r > \xi$), the London equation for a vortex line along $\hat{z}$ with a source term is:

Step 2: The solution in cylindrical coordinates is:

where $K_0$ is the modified Bessel function of the second kind.

Step 3: The energy per unit length has contributions from the magnetic field energy and the kinetic energy of supercurrents:

Step 4: Using the London equation to simplify the integrand: $h^2 + \lambda^2|\nabla h|^2 = h \cdot (h - \lambda^2\nabla^2 h)$. Integrating by parts and using $K_0(\xi/\lambda) \approx \ln(\lambda/\xi)$for $\xi \ll \lambda$:

where $c \approx 0.12$ is a numerical constant from the core contribution. For$\kappa \gg 1$, the logarithmic term dominates, giving the standard result$\varepsilon_1 \approx (\Phi_0^2/4\pi\mu_0\lambda^2)\ln\kappa$.

Derivation: Lower Critical Field $H_{c1}$

Step 1: The Gibbs free energy change upon introducing a single vortex into a superconductor in an applied field $H$ is:

The first term is the vortex self-energy (positive cost). The second term is the energy gained by admitting one flux quantum into the sample, reducing the magnetization energy.

Step 2: The first vortex enters when $\Delta G = 0$:

Step 3: Substituting $\varepsilon_1$:

Note that $H_{c1} \propto 1/\lambda^2$ and is small for materials with large penetration depth. The ratio $H_{c1}/H_{c2} = \ln\kappa/(2\kappa^2) \ll 1$ for $\kappa \gg 1$, showing that the mixed state extends over a wide field range.

Derivation: Abrikosov Lattice Spacing

Step 1: Each vortex carries exactly one flux quantum $\Phi_0$. If the average magnetic induction is $B$ and the vortex density is $n_v$, then:

Step 2: For a triangular (hexagonal) lattice with nearest-neighbor spacing $a_\triangle$, the unit cell area is:

Step 3: One vortex per unit cell gives$n_v = 1/A_{\text{cell}}$, so:

At $B = 1$ T, this gives $a_\triangle \approx 49$ nm. At $B = H_{c2}$, $a_\triangle \approx 2.7\xi$, meaning the vortex cores begin to overlap, consistent with the destruction of superconductivity.

Derivation 2: Abrikosov Vortex Solution from GL Theory

We derive the single-vortex solution of the Ginzburg–Landau equation in cylindrical coordinates $(r, \theta)$. The key physical insight is that a vortex carries a quantized phase winding of $2\pi n$ around its core, where $n$ is the winding number (topological charge).

Step 1: Ansatz. We write the order parameter as:

where $f(r) \geq 0$ is a real radial amplitude and $n \in \mathbb{Z}$ is the winding number. Single-valuedness of $\psi$ requires $n$ to be an integer, and flux quantization gives $\Phi = n\Phi_0$. Energetically, $|n| = 1$vortices are stable; higher-$|n|$ vortices are unstable to splitting.

Step 2: GL equation in cylindrical coordinates. The time-independent GL equation (neglecting the vector potential for clarity near the core) is:

Here we have normalized $f$ so that the bulk equilibrium value is $f = 1$ (i.e., $f = |\psi|/|\psi_\infty|$).

Step 3: Near-core behavior ($r \to 0$). Near the vortex center, we seek a power-law solution $f(r) \sim A r^p$. Substituting into the GL equation, the dominant terms at small $r$ are:

This gives $p^2 = n^2$, so $p = |n|$ (taking the regular solution). Thus:

For $n = 1$, the order parameter vanishes linearly at the core:$|\psi| \propto r$. This linear suppression defines the vortex core size $\sim\xi$ — the scale over which $f(r)$ recovers from zero to its bulk value.

Step 4: Far-field behavior ($r \to \infty$). Far from the core, $f \to 1$ and the centrifugal term $n^2/r^2$ becomes negligible. Writing $f = 1 - g(r)$ with $g \ll 1$, linearizing gives:

The solution is $g(r) \propto K_0(r\sqrt{2}/\xi)$, so the order parameter approaches unity exponentially: $f \to 1 - \text{const}\cdot K_0(r\sqrt{2}/\xi)$.

Step 5: Supercurrent profile. The circulating supercurrent around a vortex with $n = 1$ is:

In the London limit ($\xi \ll r \ll \lambda$), where $f \approx 1$, this reduces to $J_s \propto 1/r$. At $r \lesssim \xi$, the current is suppressed because $f^2 \propto r^2$, and at $r \gg \lambda$, screening causes exponential decay.

Step 6: Vortex energy per unit length. The total energy per unit length receives contributions from the kinetic energy of supercurrents and the magnetic field energy. The $1/r$ current profile for$\xi < r < \lambda$ gives a logarithmically divergent integral:

Evaluating the integral:

The logarithmic dependence on $\kappa = \lambda/\xi$ reflects the$1/r$ supercurrent profile between the two characteristic length scales. The energy is dominated by the kinetic energy of circulating supercurrents, not the magnetic field energy or the condensation energy lost in the core.

Derivation 3: Lower Critical Field $H_{c1}$

We derive $H_{c1}$ from the thermodynamic condition for the first vortex entry and show that $H_{c1} < H_c$ when $\kappa > 1/\sqrt{2}$.

Step 1: Gibbs free energy of a single vortex. Consider a superconductor in an applied field $H$. Introducing one vortex line (along$\hat{z}$, length $L$) changes the Gibbs free energy by:

The first term $\varepsilon_1$ is the vortex self-energy per unit length (positive — the cost of creating the vortex core and supercurrents). The second term $-\mu_0 H\Phi_0$ is the energy gained by admitting one flux quantum $\Phi_0$ into the sample, reducing the diamagnetic energy.

Step 2: Critical condition. The first vortex enters when$\Delta G = 0$, i.e., when the energy cost equals the energy gain:

Step 3: Substitution of vortex energy. Using the result from Derivation 2, $\varepsilon_1 = (\Phi_0^2/4\pi\mu_0\lambda^2)\ln\kappa$:

Step 4: Comparison with $H_c$. The thermodynamic critical field is $H_c = \Phi_0/(2\sqrt{2}\pi\mu_0\lambda\xi)$. Taking the ratio:

For $H_{c1} < H_c$, we need $\ln\kappa < \sqrt{2}\,\kappa$. This is satisfied for all $\kappa > 0$, but the physically meaningful condition is that the mixed state is thermodynamically stable, which requires $\kappa > 1/\sqrt{2}$ — exactly the type-II criterion from the surface energy analysis.

Physical interpretation: For type-II superconductors ($\kappa > 1/\sqrt{2}$), there is a field range $H_{c1} < H < H_{c2}$ where partial flux penetration via vortices is energetically preferred over both complete Meissner screening and the normal state. The ratio $H_{c1}/H_{c2} = \ln\kappa/(2\kappa^2)$ becomes very small for large $\kappa$, meaning the mixed state extends over a vast field range in high-$\kappa$ materials like the cuprates.

Derivation 4: Upper Critical Field $H_{c2}$ and Landau Level Analogy

We derive $H_{c2} = \Phi_0/(2\pi\mu_0\xi^2)$ by showing that the linearized GL equation maps exactly onto the Schrödinger equation for a charged particle in a uniform magnetic field, and $H_{c2}$ corresponds to the lowest Landau level.

Step 1: Linearized GL equation. Near the normal-to-superconducting transition, the order parameter $\psi$ is small, and the GL equation can be linearized by dropping the cubic term $|\psi|^2\psi$:

where $\alpha < 0$ below $T_c$ (so $-\alpha > 0$),$e^* = 2e$ is the Cooper pair charge, and $m^* = 2m$. Here$\mathbf{A}$ is the vector potential of the applied field.

Step 2: Mapping to the Landau level problem. This equation has exactly the form of the Schrödinger equation for a particle of charge $e^* = 2e$ and mass $m^*$ in a uniform magnetic field $\mathbf{B} = B\hat{z}$:

with eigenvalue $E = -\alpha$. The eigenvalues of this problem are the Landau levels:

Step 3: Critical field condition. A superconducting solution exists when at least one Landau level satisfies $E_n \leq -\alpha$. The highest field that allows superconductivity corresponds to the lowest Landau level ($n = 0$):

Step 4: Express in terms of $\xi$. The GL coherence length is defined by $\xi^2 = \hbar^2/(2m^*|\alpha|)$, so $|\alpha| = \hbar^2/(2m^*\xi^2)$. Substituting:

Solving for $B_{c2}$ with $e^* = 2e$:

where we used $\Phi_0 = h/(2e) = 2\pi\hbar/(2e)$.

Physical interpretation: The upper critical field is reached when the magnetic length $\ell_B = \sqrt{\Phi_0/(2\pi B)}$ equals the coherence length $\xi$. At this point, the Landau level wavefunction has a spatial extent comparable to $\xi$, meaning that the “orbits” of the Cooper pair center-of-mass motion fit within a single coherence length — superconducting correlations can no longer be sustained. Since $H_{c2} = \sqrt{2}\kappa H_c$, type-II superconductors with large $\kappa$ can sustain superconductivity to very high fields.

Derivation 5: Abrikosov Vortex Lattice and Magnetization near $H_{c2}$

We derive the triangular vortex lattice structure near $H_{c2}$ and show that Abrikosov's $\beta_A$ parameter selects the triangular lattice as the ground state.

Step 1: Lowest Landau level expansion. Just below $H_{c2}$, the order parameter is small and can be expanded in the degenerate lowest Landau level (LLL) eigenstates. In the Landau gauge$\mathbf{A} = (0, Bx, 0)$, the LLL wavefunctions are:

The general LLL solution is a superposition: $\psi = \sum_n C_n \phi_{k_n}$ with$k_n = 2\pi n / a_y$ for a lattice periodic in $y$ with period $a_y$.

Step 2: GL free energy in the LLL. The GL free energy density near $H_{c2}$ reduces to:

where $\alpha_{\text{eff}} \propto (B - B_{c2})$ changes sign at $H_{c2}$. Minimization gives:

The Abrikosov parameter $\beta_A \geq 1$ characterizes the spatial uniformity of $|\psi|^2$. A smaller $\beta_A$ corresponds to a more uniform order parameter and lower free energy.

Step 3: Lattice structure selection. The free energy is minimized when $\beta_A$ is minimized. Abrikosov originally considered a square lattice, but the correct minimum is the triangular (hexagonal) lattice:

For the triangular lattice, the coefficients satisfy $C_{n+2} = C_n$with $C_1/C_0 = ie^{i\pi/4}$, producing a lattice with basis vectors $\mathbf{a}_1 = a_\triangle\hat{x}$and $\mathbf{a}_2 = (a_\triangle/2)\hat{x} + (\sqrt{3}a_\triangle/2)\hat{y}$.

Step 4: Magnetization near $H_{c2}$. The equilibrium magnetization is obtained by minimizing the Gibbs free energy$G = F - \mathbf{B}\cdot\mathbf{H}/\mu_0$. Near $H_{c2}$:

Key features of this result:

• • The magnetization vanishes linearly as $H \to H_{c2}$, in contrast to the discontinuous jump at $H_c$ in type-I superconductors.
• • The slope $dM/dH$ at $H_{c2}$ is determined by $\kappa$ and $\beta_A$.
• • For large $\kappa$, $(2\kappa^2 - 1)\beta_A \approx 2\kappa^2\beta_A$, and the magnetization is small — the transition is nearly second-order.
• • The $\beta_A$ factor confirms that the triangular lattice (smaller $\beta_A$) yields a larger equilibrium magnetization and lower free energy.