Theory

The macroscopic wavefunction of a superconductor has the form

, (1)

where n is the number density of Cooper pairs and φ is the phase of the wavefunction. The behavior of superconductors 1 and 2 (with wavefunctions

and ) separated by a weak link, is determined by solving the system of coupled Schrödinger’s equations, which gives the two relationships

, (2)

and

, (3)

where e is the electron charge,

is Planck’s constant, h, divided by 2π, V is the potential difference between the two superconductors, and T, having dimensions of a rate or frequency, is a measure of the “leakage” of into the region of and vice versa [5, 7]. Since only a comparatively small number of Cooper pairs are exchanged between the superconductors, the reasonable assumption can be made that

. Therefore, the current density of Cooper pairs crossing the weak link is [8]. Further assuming J to be uniform across the junction, which is also reasonable since the barrier is very thin, the total current across the junction can be found by integrating J over the area of the junction, yielding [8]

, (4)

where

, and is the maximum current supported by the junction, also called the critical current. From Eq. 3 we see that when V=0, the phase difference between superconductors is constant and so a steady current can develop, and this is just the DC Josephson effect. The I-V characteristic of a theoretical Josephson junction is shown in Fig. 1.

Figure 1: Simplified theoretical I-V characteristic of a Josephson junction. Current will be between and for |V| ≤ V_c, depending on the phase difference of the superconductors. Above V_c, the junction behaves ohmically. The dashed line represents a normal metal. Figure taken from citation [5].

In the presence of a magnetic field, the current in Eq. 4 must be gauge-invariant, which means that it should not change with the addition of an arbitrary constant to the vector potential of the magnetic field. Since the phase difference (

) is not uniquely determined in this instance, it must be replaced by the gauge-invariant phase difference [7]

, (5)

where A is the magnetic vector potential, s is the path along the length of the weak link connecting the superconductors, and is the flux quantum given by [6]

, (6)

for h, Planck’s constant. The path integral of A around a closed loop encompassing the junction gives the magnetic flux, , through the junction. Replacement of the original phase difference in Eq. 4 with δ yields (under the same assumptions) the total current [9] through the junction

. (7)

The implication of Eq. 7 is that for every value of

, there is a maximum current which the junction can support, namely, when , which is also true in the absence of a magnetic field. In this case, plotting the maximum junction current vs. flux through the junction will yield a relationship which looks very similar to the single-slit (Fraunhofer) diffraction pattern for light intensity (Fig. 2), the only difference being that the single-slit intensity is proportional to the square of a sinc function. The maximum current becomes zero when the flux takes integer values of the flux quantum, and so it is by determining these minima that will allow us to measure the flux quantum.

Figure 2: Graph of the maximum current through a Josephson junction when in a magnetic field vs. the magnetic flux through the junction (blue line). The Fraunhofer diffraction pattern is shown as the orange line. The maximum current equals the critical current for zero flux and is zero when the flux is an integer multiple of the flux quantum. Figure adapted from citation [6].