As previously mentioned, a superconductive material in a weak applied magnetic field cooled below its transition temperature will enter a diamagnetic state called Meissner state. At low enough temperatures the Meissner state is characterized by the total expulsion of magnetic fields from the bulk of the superconductive domain.
Let’s consider the simple situation illustrated Fig. 2.1 where a magnetic field of intensity is applied parallel to the surface of a superconductive infinite domain extended in the half space defined by .
Figure 2.1 Magnetic field distribution in a superconductive semi-infinite domain under uniform applied field B0
In this case, London equation 1.10 can be written as:
Given the symmetry of the problem we can impose the boundary conditions ( ) to arrive at the solution for the magnetic field inside the superconductive domain:
( ) 0 1 ( 2.2 )
This simple case demonstrates how that the magnetic flux density is exponentially attenuated within a distance comparable to , as it enters the superconducting domain, vanishing deep inside the bulk of the superconductor. The length is characteristic to each material and is a fundamental parameter in superconductors known as the London penetration depth.
The same attenuation is experienced by electric fields inside a superconductor. Making use of the Maxwell relation and the continuity equation for currents expressed by:
( 2.3 )
Considering the conservation of electric charges, it is easy to show that, inside a superconductor, screening currents perpendicular to the field are formed at the surface in order to shield the external applied magnetic field. For the case considered above, the supercurrent density has the following spatial dependence:
( ) 0 1 ( 2.4 )
The screening currents will generate a total magnetic moment, similar to the one created by the bound currents in magnetic materials. The induced magnetic moment has enough magnitude so that it produces a response magnetic field to cancel the applied field inside. The magnetic moment will also alter the field distribution outside the superconductive material. This induced magnetic moment can be associated with a magnetization (magnetic moment per unit volume) expressed by the relation:
( 2.5 )
where, as in the case of magnetic media, the total magnetic moment can be calculated from:
∫ ( 2.6 )
In magnetic materials, considering a relation of the form:
( 2.7 )
will result in total magnetic moment created by the induced currents that can be expressed as:
∫ ( 2.8 )
Although the relation described by Eq. 2.8 does not locally apply to supercurrents, considering that the only physically meaningful quantity is the total magnetic moment (as we will show in Chapter III), both definitions will lead to the same result for in superconductors. Introducing Eq. 2.5 in
London’s second equation (Eq. 1.9) one arrives at the constitutive equation for the magnetization in a superconductor:
( 2.9 )
If we consider the case of the semi-infinite superconductor described above, using the expression in Eq. 2.9 with the calculated field dependence in Eq. 2.2, the spatial variation of the magnetization can be found to be:
( ) . 0 1/ ( 2.10 )
In the bulk of the superconductor ( ), the magnetization reaches its maximum value:
( 2.11 )
Similar to the magnetic intensity introduced in the case of magnetic media to account for the contribution of free current sources, an auxiliary magnetic field intensity can be introduced for superconductors defined in the same manner:
( 2.12 )
which satisfies the following relation, where is the free external current density,
( 2.13 )
For the case of the semi-infinite superconductive domain, looking at Eq. 2.2 and Eq. 2.10, it is straightforward that
( )
( ) 0 1 . 0 1/ ( 2.14 ) We can thus see that the magnetic intensity is uniform and has the same value, outside and inside the superconductor, equal to the applied field created by external currents. This is however not the case for finite size superconductive domains where demagnetizing effects will alter both the magnetic induction and intensity inside and outside the superconductor, as we will see later.
The magnetic susceptibility describes how a material behaves in an applied magnetic field and in its simplest form can be locally defined as
( 2.15 )
Considering the situation described above, one can easily integrate to find the total susceptibility of a superconductive domain. Since the domain is extended to infinity, the total magnetization will converge to its value in the bulk thus, considering , the value of susceptibility will tend to the perfect diamagnetic value of . The same value is obtained if we consider that the penetration
depth is much smaller than the geometric dimensions of the domain. Consequently, perfect diamagnetism is a unique property of superconductors as evidentiated by the Meissner effect.
Fig. 2.2 illustrates the Meissner effect in a superconductive sphere of radius obtained by solving the London equation with appropriate boundary conditions (uniform applied field). Although analytical solutions exist for a sphere, we show the results obtained through numerical simulations. The details of the simulations will be presented in the next section where the simple case of a sphere constitutes a straightforward way to corroborate the numerical results of our simulation method.
Figure 2.2 Meissner effect in a superconductive sphere. Calculated magnetic flux density and field lines (top) and supercurrent density (bottom) for (left), ⁄ (middle) and (right).
Above the critical temperature the sphere is in a normal state so the magnetic flux will penetrate its volume unhindered. As temperature is decreased, the sphere enters a superconductive state
characterized by a gradual expulsion of magnetic flux lines. At the lowest temperature, the London penetration values become very small (typically in the micrometer to nanometer range) so the magnetic
field in completely expelled from the bulk of the superconductor. The supercurrent density is proportional to the vector potential according to London equation (Eq. 1.15). As we can see from the illustrated results, as the penetration depth decreases the current density becomes more concentered near the surface of the superconductor. At low penetration depth values, supercurrents exist only in a thin layer near the surface.