en>VossBC: /* Informal fallacies */ Changed double quotes to italics.

2014-01-26T17:14:02Z

Informal fallacies: Changed double quotes to italics.

New page

{{Unreferenced|date=November 2006}}
{{mergeto|Josephson effect|date=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.

== Derivation ==

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

<math>I_s = I_c \sin(\phi)\,</math>

where <math>I_s\,</math> is the supercurrent flowing through the junction, <math>I_c\,</math> is the critical current, and <math>\phi\,</math> is the [[Josephson phase]], see [[Josephson junction]] for details.

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

<math>
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).
</math>

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

== Josephson inductance ==

Imagine that the Josephson phase across the junction is <math>\phi_0\,</math> and the supercurrent flowing through the junction is

<math>I_0 = I_c \sin\phi_0\,</math>

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

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

<math>I_0+\delta I = I_c \sin(\phi_0+\delta\phi)\,</math>

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

<math>\delta I = I_c \cos(\phi_0) \delta\phi\,</math>

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

<math>
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)}.
</math>

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

<math>
V = L \frac{\partial I}{\partial t}
</math>,

we can define the so-called Josephson inductance

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

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

One can also calculate the change in Josephson energy

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

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

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.

[[Category:Superconductivity]]
[[Category:Energy (physics)]]
[[Category:Josephson effect]]

List of fallacies - Revision history

en>VossBC: /* Informal fallacies */ Changed double quotes to italics.