_{1}

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A detailed analysis of the magnetic response of field-cooled type-I superconducting hollow cylinders shows that the so-called “paramagnetic Meissner effect” can take place in opportunely devised multiply connected superconductors. Adopting simple circuital analogs of the latter superconducting systems, the magnetic susceptibility of micro-cylinders with one or two holes is studied by means of energy considerations.

Expulsion of magnetic field from the inner region of simply connected type-I superconductors was first observed by Walther Meissner and Robert Ochsenfeld in 1933, 22 years after the discovery of superconductivity [_{c} in the presence of low magnetic fields, behaved like perfect diamagnetic materials. With the discovery of high-T_{c} superconductivity in layered perovskites by Johannes Bednorz and Karl Müller [_{c} granular superconductors was observed to be positive for low measuring fields. It thus became evident that the polycrystalline structure in sintered high-T_{c} materials could play a role in explaining this experimental outcome [

In the present work, after having briefly recalled the results obtained for the hollow cylinder, we study the field-cooled magnetic behavior of multiply connected superconductors consisting of type-I superconducting cylinders in which two holes are present. By adopting a simple circuital model and by taking into account the mutual coefficient between the circuits representing the current loops in the system, we derive a method to calculate the field-cooled magnetic susceptibility of multiply connected type-I superconducting cylinders.

Let us first consider the hollow superconducting cylinder shown in _{S} flows in the outer surface of the cylinder; a second current I_{1} shields the inner part of the superconductor from the field h inside the hole.

By applying Ampere’s law, following path C_{S}, and by noticing that the magnetic induction in the superconducting region is zero, we can write

where d is the height of the cylinder and C_{S} goes through the superconductor in a region sufficiently far from the outer surface. The latter hypothesis is necessary in order to avoid considering the decaying magnetic field inside the superconductor, due to the existence of finite penetration lengths in these materials [_{1}, we can write

By now applying (1), we see that Equation (2) reduces to the following:

Let us now write down the magnetic energy E_{M} due to the circulating currents as follows:

where, denoting the permeability of vacuum as,

and are the inductance coefficients pertaining to the two virtual loops followed by and, respectively, and M is the mutual inductance coefficient between these same loops. By taking and by substituting Equations (1)-(3) into Equation (4), we have:

where is a constant. By now introducing the flux numbers and, being the elementary flux quantum, we can rewrite Equation (5) in the following final form:

where and. The possible quantized values of the trapped field in a field cooling experiment has been given by Goodman and Deaver in 1970 [

where the function Ω, when applied to a real number x, gives the closest integer to x. This function can be easily interpreted by considering the minima of the energy. In fact, by fixing the value of the applied flux number, the system arranges itself in the quantized flux state with n trapped fluxons inside the hole of area S_{1} in such a way to minimize the energy. In this way, only the lower parts of all parabolas in (6) are chosen as possible magnetic state in the system. The result of this procedure, by which one chooses the possible magnetic energy states as varies, is shown in

Having specified the value of the flux number n in (7), the field distribution inside the cylinder can be summarized as follows:

In order to determine the field-cooled susceptibility

, we need to find the average value of the magnetic induction inside the hollow cylinder. By applying (8), we find, where is the normal fraction of the sample. Therefore, by setting the field-cooled magnetic susceptibility equal to, we find:

In

intervals. In fact, by considering, for which, we have if. On the other hand, for, for which

, we cannot have. Therefore, we argue that, for α > 1/2, the only field interval for which is given by the following simple relation:

In order to detect the range in which this effect can be measured, we may notice that in a micro-cylinder (with a hole of inner radius of about 20 μm) the ratio is of 1.64 μT.

Let us now consider the multiply connected superconductor shown in

We may now write down the magnetic energy as follows:

where the inductance coefficients are,

, and, and where the mutual inductance coefficients between the different current loops are denoted as, , and. By proceeding as in the previous section, we define the following flux numbers, , , and, with. By now taking and, we may write down the energy, with, as follows:

where and. In Equation (13) we notice that, depending on the choice of and, we obtain different parabolic dependence of as a function of.

As before, the magnetic state for a given value of the forcing term is the one which minimizes the energy. Therefore, by collecting the different parabolas, we shall choose only the low-lying states at a fixed value of. The representation of these states is given in

the red curve pertaining to the low-energy states. By a similar algorithm, we can choose to register, for a fixed value of, the couple giving the parabola on which the minimum of the energy lies. In this way, by again calculating the average value of the magnetic induction over the whole sample, we define the fieldcooled magnetic susceptibility as follows:

where now. Therefore, by knowing the quantities and, for a given value of, we can plot the vs. curves.

By implementing the algorithm for finding the couple, for a given value of, giving the minimum energy value, we find the vs. curves in Figures 6(a)-(c).

In the curves shown in Figures 6(a)-(c) one notices that the inequality on the minimum value of α giving positive field-cooled susceptibility found in the case of a hollow cylinder with a single hole (namely, α > 1/2) does not hold anymore. In fact, we find intervals of for which even for α = 1/2, as shown in all three curves in Figures 6(a)-(c). From the same curves it can be argued that, by choosing a more negative value of the mutual inductance coefficient in

The field-cooled magnetic susceptibility of type-I superconducting hollow cylinders is studied by means of energetic considerations. Starting from the case of a hollow superconducting cylinder with a single hole, we interpret the classical Goodman and Deaver experiment by means of simple concepts on energy minimization. In fact, we see that the magnetic flux trapped inside a hollow superconducting type-I superconductor cooled in the presence of an axial external field of magnitude H can be derived by considering the minima in the magnetic energy states. This energy is written, under elementary assumptions, by considering the magnetic energy generated by the currents flowing in a classical equivalent circuit. In this picture, the superconductor is seen as a perfectly diamagnetic entity. The field cooled magnetic states of the system are described in terms of the applied flux number, taken to be proportional to the externally applied field magnitude H.

By extending this concept to the case of a multiply connected superconductor containing two holes, we are able to derive the vs. curves, detecting finite

intervals of the field magnitude H in which the susceptibility is positive. We therefore argue that in these systems the so called “paramagnetic Meissner effect” is linked to topological as well as to electromagnetic effects.