6
1 Introduction
other hand, ions acquire the kinetic energy of electrons, which leads to heat. It is
possible to derive electrical resistance and Ohm’s law by introducing a phenomenological viscous force into the equation of motion, although the viscous force has not
been proved theoretically.
Now we go back the superconductor. Since the electrical resistance is zero, the
electromagnetic phenomena in the superconductor can be derived from first principles, i.e., the minimization of the relevant free energy. First principles cannot be
applied to irreversible phenomena accompanied by energy dissipation. This means
that the superconductor is a pure material in physics.
1.3 Energy Dissipation in Superconductors
In spite of the explanation above, energy dissipation does occur in superconductors
in reality, and electrical resistance appears. Superconductors are used for magnets.
When a superconducting magnet is operated, the superconductor is subject to a
strong magnetic field, and the magnetic flux penetrates in the form of quantized
magnetic flux lines, which are simply called flux lines. When a current is applied to
the superconductor that contains flux lines, the Lorentz force works on the flux lines.
If flux lines are driven by the Lorentz force, an electromotive force appears. Since the
central area of 10–100 nm in diameter in each flux line is in the normal state with a
finite resistivity, normal electrons in this area are driven by the electric field, resulting
in energy dissipation. This state is called the flux flow. Hence, defects such as normal
precipitates or grain boundaries are introduced in practical superconductors used for
various devices so that such defects prevent flux lines from moving, even under the
Lorentz force. This effect is called flux pinning and defects that are effective for
this are called pinning centers. Since the Lorentz force is proportional to the current
density, the critical current density, i.e., the maximum non-dissipative current density,
is proportional to the strength of flux pinning.
Energy dissipation does not occur in the superconductor due to the flux pinning
effect, if the current with a density below the critical current density is applied in a
static condition. The flux motion still occurs, however, resulting in energy dissipation under a condition varying with time such as an alternating current or varying
external magnetic field. This is similar to friction. In reality flux pinning interactions
work against the driving force on flux lines and are irreversible in nature. This seems
to be contradictory to the above statement that electromagnetic phenomena can be
derived from first principles to minimize the relevant energy. It should be noted,
however, that flux pinning originates from the interaction between flux lines and
the pinning potentials of defects and is reversible in nature. Reversible phenomena
are really observed for very small displacements of flux lines. This corresponds to
a play in friction. The irreversibility appears when displacement exceeds a certain
level of reversibility in both cases. Variation from reversibility to irreversibility takes
place continuously. As will be explained later in this book, therefore, flux pinning
1 Introduction
other hand, ions acquire the kinetic energy of electrons, which leads to heat. It is
possible to derive electrical resistance and Ohm’s law by introducing a phenomenological viscous force into the equation of motion, although the viscous force has not
been proved theoretically.
Now we go back the superconductor. Since the electrical resistance is zero, the
electromagnetic phenomena in the superconductor can be derived from first principles, i.e., the minimization of the relevant free energy. First principles cannot be
applied to irreversible phenomena accompanied by energy dissipation. This means
that the superconductor is a pure material in physics.
1.3 Energy Dissipation in Superconductors
In spite of the explanation above, energy dissipation does occur in superconductors
in reality, and electrical resistance appears. Superconductors are used for magnets.
When a superconducting magnet is operated, the superconductor is subject to a
strong magnetic field, and the magnetic flux penetrates in the form of quantized
magnetic flux lines, which are simply called flux lines. When a current is applied to
the superconductor that contains flux lines, the Lorentz force works on the flux lines.
If flux lines are driven by the Lorentz force, an electromotive force appears. Since the
central area of 10–100 nm in diameter in each flux line is in the normal state with a
finite resistivity, normal electrons in this area are driven by the electric field, resulting
in energy dissipation. This state is called the flux flow. Hence, defects such as normal
precipitates or grain boundaries are introduced in practical superconductors used for
various devices so that such defects prevent flux lines from moving, even under the
Lorentz force. This effect is called flux pinning and defects that are effective for
this are called pinning centers. Since the Lorentz force is proportional to the current
density, the critical current density, i.e., the maximum non-dissipative current density,
is proportional to the strength of flux pinning.
Energy dissipation does not occur in the superconductor due to the flux pinning
effect, if the current with a density below the critical current density is applied in a
static condition. The flux motion still occurs, however, resulting in energy dissipation under a condition varying with time such as an alternating current or varying
external magnetic field. This is similar to friction. In reality flux pinning interactions
work against the driving force on flux lines and are irreversible in nature. This seems
to be contradictory to the above statement that electromagnetic phenomena can be
derived from first principles to minimize the relevant energy. It should be noted,
however, that flux pinning originates from the interaction between flux lines and
the pinning potentials of defects and is reversible in nature. Reversible phenomena
are really observed for very small displacements of flux lines. This corresponds to
a play in friction. The irreversibility appears when displacement exceeds a certain
level of reversibility in both cases. Variation from reversibility to irreversibility takes
place continuously. As will be explained later in this book, therefore, flux pinning