Abrikosov vortex lattice

A periodic array of Abrikosov vortices. In the isotropic local theory near the upper critical magnetic field, the triangular array has a lower Helmholtz free energy than the square array.^[ref]

The useful distinction is between vortex existence, array geometry, and vortex mobility. Quantized winding does not force every array to be triangular or defect-free. A stationary array and vortices moving under an applied electric current also have different transport consequences; flux pinning matters for carrying useful electric currents.

Table of contents

Flux-lattice spacing ( $a = 2 Φ_{0} / (3 \overline{B})$ )

 0  0

Assume a triangular array of straight, singly quantized Abrikosov vortices with nearest-neighbor distance

a

. Here

\overline{B} > 0

is the magnitude of the spatially averaged internal magnetic field, and

Φ_{0} = h / (2 e)

is the superconducting flux quantum;

e > 0

is the elementary charge. Two lattice vectors of length

a

meeting at

6 0^{\circ}

span area

A_{c} = 3 a^{2} /2

. One flux quantum per cell gives

\overline{B} A_{c} = Φ_{0}, n_{v} = \frac{B}{Φ _{0}}, a = \frac{2 Φ _{0}}{3 B} .

(1)

Thus multiplying

\overline{B}

by five decreases

a

5

; it does not multiply the flux of each vortex by five. Using the exact SI Planck constant and elementary charge,

Φ_{0} ≃ 2.06783 \times 1 0^{- 15} Wb

.^[ref]

For chosen internal fields

0.20 T

and

1.0 T

, the spacings are approximately $109 nm$ and $48.9 nm$ . These are geometric model calculations, not measurements from a particular sample. The local magnetic field varies within a cell; replacing

\overline{B}

by the field at a core would not give this counting relation.

Figure 1.
More internal magnetic flux means more vortices per unit area
. Original triangular-array geometry for stipulated $ξ = 10 nm$ and $λ = 100 nm$ . The two core-amplitude maps use a chosen nearest-core hyperbolic-tangent profile, not a solution of the Ginzburg-Landau equations or an experimental image. The spacing curve counts one flux quantum per cell; its endpoint at $B_{c 2}$ does not establish that field by geometry alone. **Details**.

Abrikosov parameter ( $β_{A} = ⟨ ∣ ψ ∣^{4} ⟩ / ⟨ ∣ ψ ∣^{2} ⟩^{2}$ )

 0  0

A measure of spatial nonuniformity in the order parameter, with averages taken over one lattice cell. Set

X = ∣ ψ ∣^{2}

. Then

β_{A} = 1 + \frac{Var ( X )}{⟨ X ⟩ ^{2}} \geq 1.

(1)

Equality requires uniform

∣ ψ ∣^{2}

, so a nonconstant core pattern costs more in the quartic term than a uniform amplitude with the same mean squared amplitude. This identity alone does not compare two different vortex arrays: their admissible profiles must also satisfy the field equations.

Magnetic decoration of superconducting vortices

 0  0

U. Essmann and H. Träuble reported triangular vortex arrays, including defects, using electron microscopy on lead-indium and niobium samples in the remanent state.^[ref] Such a prepared state need not display the equilibrium array at a uniform applied field. Images of regular positions and a proof of unrestricted vortex motion would answer different questions.