Phi Josephson junction

A φ Josephson junction (pronounced "phi Josephson junction") is a particular type of the Josephson junction, which has a non-zero Josephson phase φ across it in the ground state. A π Josephson junction, which has the minimum energy corresponding to the phase of π, is a specific example of it.
Introduction.
The Josephson energy formula_1 depends on the superconducting phase difference (Josephson phase) formula_2 periodically, with the period formula_3. Therefore, let us focus only on one period, e.g. formula_4. In the ordinary Josephson junction the dependence formula_5 has the minimum at formula_6. The function 
where is the critical current of the junction, and formula_8 is the flux quantum, is a good example of conventional formula_5.
Instead, when the Josephson energy formula_5 has a minimum (or more than one minimum per period) at formula_11, these minimum (minima) correspond to the lowest energy states (ground states) of the junction and one speaks about "φ Josephson junction". Consider two examples.
First, consider the junction with the Josephson energy formula_5 having two minima at formula_13 within each period, where formula_14 (such that formula_15) is some number. For example, this is the case for
formula_16,
which corresponds to the current-phase relation
formula_17.
If and , the minima of the Josephson energy occur at formula_13, where formula_19. Note, that the ground state of such a Josephson junction is doubly degenerate because formula_20.
Another example is the junction with the Josephson energy similar to conventional one, but shifted along formula_2-axis, for example formula_22,
and the corresponding current-phase relation
formula_23.
In this case the ground state is formula_24 and it is not degenerate.
The above two examples show that the Josephson energy profile in φ Josephson junction can be rather different, resulting in different physical properties. Often, to distinguish, which particular type of the current-phase relation is meant, the researches are using different names. At the moment there is no well-accepted terminology. However, some researchers use the terminology after A. Buzdin: the Josephson junction with double degenerate ground state formula_13, similar to the first example above, are indeed called φ Josephson junction, while the junction with non-degenerate ground state, similar to the second example above, are called formula_26 Josephson junctions.
Realization of φ junctions.
The first indications of φ junction behavior (degenerate ground states or unconventional temperature dependence of its critical current) were reported in the beginning of the 21st century. These junctions were made of d-wave superconductors.
The first experimental realization of controllable φ junction was reported in September 2012 by the group of Edward Goldobin at University of Tübingen. It is based on a combination of 0 and π segments in one superconducting-insulator-ferromagnetic-superconductor hybrid device and clearly demonstrates two critical currents corresponding to two junction states formula_13. The proposal to construct a φ Josephson junction out of (infinitely) many 0 and π segments has appeared in the works by R. Mints and coauthors, although at that time there was no term φ junction. For the first time the word φ Josephson junction appeared in the work of Buzdin and Koshelev, whose idea was similar. Following this idea, it was further proposed to use a combination of only two 0 and π segments.
In 2016, a formula_26 junction based on the nanowire quantum dot was reported by the group of Leo Kouwenhoven at Delft University of Technology. The InSb nanowire has strong spin-orbit coupling, and magnetic field was applied leading to Zeeman effect. This combination breaks both inversion and time-reversal symmetries creating finite current at zero phase difference.
Other theoretically proposed realization include geometric φ junctions. There is a theoretical prediction that one can construct the so-called geometric φ junction based on nano-structured d-wave superconductor. As of 2013, this was not demonstrated experimentally.