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# Pi Josephson junction

A π Josephson Junction is a specific instance of a Josephson Junction where the Josephson phase (φ) is equal to π.

## Background

The supercurrent Is through a conventional Josephson junction (JJ) is given by Is = Icsin(φ), where φ is the phase difference of the superconducting wave functions of the two electrodes, i.e. the Josephson phase. The critical current Ic is the maximum supercurrent that can flow through the JJ. In experiment one usually applies some current through the JJ and the junction reacts by changing the Josephson phase. From the above formula it is clear that the phase φ = arcsin(I / Ic), where I is the applied (super)current.

Since the phase is -periodic, i.e. φ and φ + 2πn are physically equivalent, without loosing generality, we restrict the discussion below to the interval $0\leq\phi<2\pi$.

When no current (I = 0) is passing through the JJ, e.g. when the junction is disconnected, the JJ is in the ground state and the Josephson phase across it is zero (φ = 0). Attentive reader may notice, that according to the above equations the phase can also be φ = π, also resulting in no current through the JJ. It turns out that the state with φ = π is unstable and corresponds to the Josephson energy maximum, while the state φ = 0 corresponds to the Josephson energy minimum and is a ground state.

In certain cases one may obtain a JJ where the critical current is negative (Ic < 0). In this case the first Josephson relation becomes

Is = − | Ic | sin(φ) = | Ic | sin(φ + π)

Obviously, the ground state of such a JJ is φ = π and corresponds to the Josephson energy minimum, while the conventional state φ = 0 is unstable and corresponds to the Josephson energy maximum. Such a JJ with φ = π in the ground state is called π Josephson junction.

π Josephson junctions have quite unusual properties. For example, if one connects (shorts) the superconducting electrodes with the inductance L (e.g. superconducting wire), as shown in Fig.???, one may expect the spontaneous supercurrent circulating in the loop, passing through JJ and through inductance clockwise or counterclockwise. This supercurrent is spontaneous and belongs to the ground state of the system. The direction of its circulation is chosen at random. This supercurrent will of course induce a magnetic field which can be detected experimentally. The magnetic flux passing through the loop will have the value from 0 to a half of magnetic flux quanta, i.e. from 0 to Φ0 / 2, depending on the value of inductance L.

## Technologies and physical principles

• Ferromagnetic Josephson junctions. Consider a Josephson junction with ferromagnetic Josephson barrier, i.e. the multilayers Superconductor-Ferromagnet-Superconductor (SFS) or Superconductor-Insulator-Ferromagnet-Superconductor (SIFS). In such structures the superconducting order parameter inside F-layer oscillates in the direction perpendicular to the JJ plane. As a result, for certain thicknesses of the F-layer and temperatures, the order parameter may become +1 at one at one superconducting electrode and -1 at the other superconducting electrode. In this situation one gets π Josephson junction. Note that inside F-layer the competition of different solutions takes place and the one with the lower energy, wins. Ferromagnetic π junction were fabricated by several groups:
• S-Fi-S JJs 
• Josephson junctions with unconventional order parameter symmetry. Novel superconductors, notably high temperature cuprate superconductors, have anisotropic superconducting order parameter which can change its sign depending on the direction. In particular, so-called d-wave order parameter has a value of +1 if one looks along the crystal axis a and -1 if one looks along the crystal axis b. If one looks along ab direction (45 degree between a and b) the order parameter vanishes. By making Josephson junctions between d-wave superconducting films with different orientation or between d-wave and conventional isotropic s-wave superconductor, one can get a phase shift of π. Nowadays there are several realizations of π JJs of this type:
• tri-crystal grain boundary JJs TsueiKirtley Review
• tetra-crystal grain boundary JJs [Chesca]
• d-wave/s-wave ramp zigzag JJs VanHarlingen Review, SmildeZigZag, HilgenkampSemifluxons, AriandoZigZag
• tilt-twist grain boundary JJs LombardiTiltTwist
• p-wave based JJs [???]
• Superconductor-NormalMetal-Superconductor (SNS) Josephson junctions with nonequlibrium electron distribution in N-layer 
• Superconductor-QuantumDot-Superconductor (S-QuDot-S) JJs [???]

## Historical developments

Theoretically, for the first time the possibility to have π JJs was discussed by Bulaevskii et al. , who considered a JJ with paramagnetic scattering in the barrier. Almost one decade later the possibility of having a π JJs was discussed in the context of heavy fermion p-wave superconductor .

Experimentally, the first π JJ was a corner JJ made of YBCO (d-wave) and Pb (s-wave) superconductors, see VanHarlingen Review. The π JJs based on ferromagnetic barrier were first fabricated and investigated only a decade later.