Josephson energy

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Derivation[edit]

For the simplest case the current-phase relation (CPR) is given by (aka the first Josephson relation):

I_s = I_c \sin(\phi),

where I_s\, is the supercurrent flowing through the junction, I_c\, is the critical current, and \phi\, is the Josephson phase, see Josephson junction for details.

Imagine that initially at time t=0 the junction was in the ground state \phi=0 and finally at time t the junction has the phase \phi. The work done on the junction (so the junction energy is increased by)


U = \int_0^t I_s V\,dt
= \frac{\Phi_0}{2\pi} \int_0^t I_s \frac{d\phi}{dt}\,dt
= \frac{\Phi_0}{2\pi} \int_0^\phi I_c\sin(\phi) \,d\phi
= \frac{\Phi_0 I_c}{2\pi} (1-\cos\phi).

Here E_J = {\Phi_0 I_c}/{2\pi} sets the characteristic scale of the Josephson energy, and (1-\cos\phi) sets its dependence on the phase \phi. The energy U(\phi) 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 U(\phi) has a minimum equal to zero for the ground state \phi=2\pi n, n is any integer.

Josephson inductance[edit]

Imagine that the Josephson phase across the junction is \phi_0\, and the supercurrent flowing through the junction is

I_0 = I_c \sin\phi_0\,

(This is the same equation as above, except now we will look at small variations in I_s\, and \phi\, around the values I_0\, and \phi_0\,.)

Imagine that we add little extra current (dc or ac) \delta I(t)\ll I_c through JJ, and want to see how the junction reacts. The phase across the junction changes to become \phi=\phi_0+\delta\phi\,. One can write:

I_0+\delta I = I_c \sin(\phi_0+\delta\phi)\,

Assuming that \delta\phi\, is small, we make a Taylor expansion in the right hand side to arrive at

\delta I = I_c \cos(\phi_0) \delta\phi\,

The voltage across the junction (we use the 2nd Josephson relation) is


V = \frac{\Phi_0}{2\pi}\dot{\phi} 
= \frac{\Phi_0}{2\pi}(\underbrace{\dot{\phi_0}}_{=0} + \dot{\delta\phi})
= \frac{\Phi_0}{2\pi} \frac{\dot{\delta I}}{I_c \cos(\phi_0)}.

If we compare this expression with the expression for voltage across the conventional inductance


 V = L \frac{\partial I}{\partial t}
,

we can define the so-called Josephson inductance


  L_J(\phi_0) = \frac{\Phi_0}{2\pi I_c \cos(\phi_0)}
  = \frac{L_J(0)}{\cos(\phi_0)}.

One can see that this inductance is not constant, but depends on the phase (\phi_0)\, across the junction. The typical value is given by L_J(0)\, and is determined only by the critical current I_c\,. Note that, according to definition, the Josephson inductance can even become infinite or negative (if \cos(\phi_0)<=0\,).

One can also calculate the change in Josephson energy


  \delta U(\phi_0) = U(\phi)-U(\phi_0) 
  = E_J (\cos(\phi_0)-\cos(\phi_0+\delta\phi)\,
Making Taylor expansion for small \delta\phi\,, we get 
  \approx E_J \sin(\phi_0)\delta\phi
  = \frac{E_J \sin(\phi_0)}{I_c \cos\phi_0}\delta I

If we now compare this with the expression for increase of the inductance energy dE_L = L I \delta I\,, we again get the same expression for L\,.

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.