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||It has been suggested that this article be merged into Josephson effect. (Discuss) Proposed since January 2013.|
In superconductivity, the Josephson energy is the potential energy accumulated in the Josephson junction when a supercurrent flows through it. One can think of a Josephson junction as a non-linear inductance which accumulates (magnetic field) energy when a current passes through it. In contrast to real inductance, no magnetic field is created by a supercurrent in Josephson junction—the accumulated energy is a Josephson energy.
For the simplest case the current-phase relation (CPR) is given by (aka the first Josephson relation):
Imagine that initially at time the junction was in the ground state and finally at time the junction has the phase . The work done on the junction (so the junction energy is increased by)
Here sets the characteristic scale of the Josephson energy, and sets its dependence on the phase . The energy accumulated inside the junction depends only on the current state of the junction, but not on history or velocities, i.e. it is a potential energy. Note, that has a minimum equal to zero for the ground state , is any integer.
Imagine that the Josephson phase across the junction is and the supercurrent flowing through the junction is
(This is the same equation as above, except now we will look at small variations in and around the values and .)
Imagine that we add little extra current (dc or ac) through JJ, and want to see how the junction reacts. The phase across the junction changes to become . One can write:
Assuming that is small, we make a Taylor expansion in the right hand side to arrive at
The voltage across the junction (we use the 2nd Josephson relation) is
If we compare this expression with the expression for voltage across the conventional inductance
we can define the so-called Josephson inductance
One can see that this inductance is not constant, but depends on the phase across the junction. The typical value is given by and is determined only by the critical current . Note that, according to definition, the Josephson inductance can even become infinite or negative (if ).
One can also calculate the change in Josephson energy
Making Taylor expansion for small , we get
If we now compare this with the expression for increase of the inductance energy , we again get the same expression for .
Note, that although Josephson junction behaves like an inductance, there is no associated magnetic field. The corresponding energy is hidden inside the junction. The Josephson Inductance is also known as a Kinetic Inductance - the behaviour is derived from the kinetic energy of the charge carriers, not energy in a magnetic field.