flux from the interior of the sample. Above Bc it is in the normal state, where the magnetic
flux penetrates the sample as it would normally and the conductivity is finite.
In the case of Type II superconductors, the transition does not occur sharply from
the Meissner state to the normal state but goes through an intermediate phase in which
the applied field is able to pierce through certain local regions of the sample. As the
magnetic field increases, initially the sample behaves as a perfect diamagnet exhibiting
the Meissner effect and rejecting all the magnetic flux. When the applied field increases
beyond a critical field denoted as Bcu the lower critical field, the magnetic
flux lines are no longer totally expelled from the sample. The overall magnetizationM
in the sample opposes the field, but its magnitude does not cancel the field everywhere.
As the field increases, M gets smaller and more flux lines pierce through the sample
until at BC2, the upper critical field, all field lines penetrate the sample and superconductivity
disappears. This behavior is shown in Figure 8.49. Type II superconductors
therefore have two critical fields Bc and Z?C2.
When the applied field is between Bci and BC2, the field lines pierce through the sample
through tubular local regions, as pictured in Figure 8.50. The sample develops local
small cylindrical (filamentary) regions of normal state in a matrix of superconducting
state and the magnetic flux lines go though these filaments of local normal state, as
shown in Figure 8.50. The state between Bc and BC2 is called the mixed state (or vortex
state) because there are two states-normal and superconducting-mixed in the same
sample. The filaments of normal state have finite conductivity and a quantized amount
of lux through them. Each filament is a vortex of flux lines (hence the name vortex
state). It should be apparent that there should be currents circulating around the walls of
vortices. These circulating currents ensure that the magnetic flux through the superconducting
matrix is zero. The sample overall has infinite conductivity due to the superconducting
regions. Figure 8.51 shows the dependence ofBci and BC2 on the temperature and
identifies the regions of Meissner, mixed, and normal states. All engineering applications
of superconductors invariably use Type II materials because BC2 is typically much
greater than Bc found in Type I materials and, furthermore, the critical temperatures of
Type II materials are higher than those of Type I. Many superconductors, including the
recent high-7 superconductors, are of Type II. Table 8.7 summarizes the characteristics
of selected Type I and Type II superconductors.