Question

Consider an inifinitely long superconducting circular cylinder of radius a in a transverse magnetic field. At large distances from the cylinder the field is unifrom and of magnitude B₀. Compute the fields inside and outside cylinder using the London equations and the phenomenological penetration depth lamda. Assume that the superconducting properties are represented by perfect diamagnetism and perfect conductivity.

Answer 1

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These studies comprise both static and time-dependent analyzes, several different geometries, and even the case of non-homogenous superconductors. However, while the spherical case is studied in and the cylinder case in a parallel magnetic field is presented i, to the best of our knowledge the detailed solution of a superconducting cylinder in a transversal magnetic field cannot be easily found in the academic literature. we derive the solution of the London equations in this paper, for the magnetic field expulsion from an infinite homogeneous superconducting cylinder in a constant external magnetic field in the transverse plane. The dependence of the strength and configuration of the expelled magnetic field with the London penetration depth is examined in detail.

The London equations for the magnetic field flux expulsion out of a superconductor can be expressed, in terms of the magnetic vector potential A, in a single equation:

(1)j=-1λ2A,

where j represents the electric current and λ the London penetration depth. After applying the Maxwell’s equation for the static case, and by taking the Coulomb gauge (∇·A=0), one obtains:

(2)∇2A-1λ2A=0,

or, in terms of the magnetic field,

(3)∇2B-1λ2B=0.

This equation is perhaps the most common form of the London equation for the magnetic field expulsion inside the superconducting region, and, in the cartesian coordinate system, corresponds to the well-known Helmholtz’s equation for a complex wavenumber, for each of the vector components. The magnetic field is, therefore, suppressed in the interior region through negative exponential dependencies, or, in the cylindrical and spherical symmetries, through modified Bessel functions of the first kind [20].

2. Cylindrical symmetry

Consider a cylindrical superconductor with radius R in an external constant perpendicular magnetic field, in such a way the external magnetic field points in the y-direction and the cylinder axis coincides with the z-axis. In order to obtain the final static vector potential configuration resulting from the magnetic flux expulsion, and assuming the Coulomb gauge, one must solve the Maxwell equation for the outer region:

(4)∇×B=∇×(∇×A)=∇2A=0,

and the London Eq. (2) for the inner region. Due to the symmetry of the system, the magnetic field has no component in the z-axis direction,

(5)B=1ρ∂Az∂ϕρˆ-∂Az∂ρϕˆ,

and, therefore, the axial component of the vector potential Az is the only relevant component needed to compute the magnetic field. For the outer region, this component can be determined from Eq. (4):

(6)1ρ2∂2Az∂ϕ2+∂2Az∂ρ2+1ρ∂Az∂ρzˆ=0,

which is simply the Laplace equation for Az. The most general solution in the outer region is, therefore,

(7)Az=∑n=0+∞[Ancos(nϕ)+Bnsin(nϕ)]×Cnρn+Dnρ-n.

Assuming the magnetic field takes the form of the external field in the limit ρ→+∞,

(8)B0=B0yˆ=B0ϕˆcosϕ+ρˆsinϕ,

and in order to respect this boundary condition, the vector potential becomes,

(9)Az=const.+A1(C1ρ+D1ρ-1)cosϕ,

where A1C1=-B0. The remaining parameters must be determined by the boundary conditions at the surface of the superconductor.

The general solution of the London equation in the interior region, for the axial component of the vector potential, is somewhat more complicated:

(10)Az=∑n=0+∞Encosnϕ+Fnsinnϕ×GnIn(ρ/λ)+HnKn(ρ/λ),

where In(ρ/λ) and Kn(ρ/λ) are the cylindrical modified Bessel functions of the first and second kind, respectively. As the modified Bessel functions of the second kind diverge at ρ=0, we can exclude them a priori.

To ensure continuity at the superconducting surface, the Az function must take the following form on the inner region,

(11)Az=E0G0I0(ρ/λ)+E1G1I1(ρ/λ)cosϕ,

where

(12)E1G1I1(R/λ)=-B0R+A1D1/R.

Since the electric current is distributed in volume, j=-1λ2A, there are no pure surface currents at the cylindrical surface, allowing some penetration of the magnetic field. Therefore, the following relation can be used as a boundary condition:

(13)k=nˆ×B+-B-=0,

where k is the surface current, nˆ is the unitary vector normal to the surface and, B+and B- are the magnetic field at the cylinder surface in the outer and inner limits, respectively. As a result, one can establish the following condition:

(14)∂Az∂ρ+=∂Az∂ρ-,

which, with the use of the relation,

(15)∂Iν(x)∂x=Iν-1(x)-νxIν(x)

in the differentiation process, leads to,

Bρ>R=B01-R2ρ2+2λRρ2I1(R/λ)I0(R/λ)sinϕρˆ+B01+R2ρ2-2λRρ2I1(R/λ)I0(R/λ)cosϕϕˆ,

Bρ<R=2B0λρI1(ρ/λ)I0(R/λ)sinϕρˆ+2B0I0(ρ/λ)I0(R/λ)-λρI1(ρ/λ)I0(R/λ)cosϕϕˆ.

In summary, the magnetic field vector was easily determined by imposing that the axial component of the vector potential must a differentiable function in any point of space, in particular, at the superconductor surface. A detailed analysis of these solutions is presented in the next section.

3. Discussion of results

Furthermore, one can see that no dramatic changes are perceptible until the penetration depth becomes one order of magnitude lower than the radius of the cylinder, when the ellipsoid starts to form. It is also worth noting that when the London depth equals the radius of the cylinder the magnetic field expulsion is almost imperceptible, or in other words, the superconductor becomes transparent to the magnetic field. This is also visible in Fig., where the dependence of the magnetic energy density along the x-axis is shown at y = 0 for different penetration depths. In particular, a single jump on the order of magnitude, from λ=0.1R to λ=R, is enough to have a significant phenomenological impact, i.e. the difference between an almost fully expelled field and a considerable field penetration.

Finally, it is also worth discussing the range of applicability of these solutions in both type-I and type-II superconductors. In type-I superconductors, the London penetration length is of the same order of magnitude or smaller than the coherence length. Therefore, the most accurate description is not given by the London equations alone, but by the Pippard’s model instead. However, in type-II superconductors, the London condition indeed provides a very good description of the electromagnetic field inside the superconductor in the Meissner state, i.e. below the lower critical field. Above the lower critical field, the superconductor enters a mixed state, where the magnetic field is allowed to penetrate the superconducting region through quantized vortices, and the London solutions are no longer an accurate description.

4. Conclusions

We presented a detailed solution of the London equation for the magnetic field expulsion from an infinite cylindrical superconductor, which can be easily obtained by using physical and differentiability arguments at the boundary conditions. The field expulsion is analyzed with the use of contour plots of the magnetic energy density to show how it evolves with the value of the penetration length. In this case, the magnetic field penetrates through the formation of an elliptic region which becomes more pronounced as the penetration depth increases.