Several recent experiments revealed a change of the sign of the first harmonic in the current-phase relation of Josephson junctions (JJs) based on, e.g., $d$-wave superconductors or JJs with ferromagnetic barrier. In this situation, the role of the second harmonic can become dominant; in this case, it determines the scenario of a $0\text{−}\pi$ transition. We discuss different mechanisms of the second harmonic generation and its sign. If the second harmonic is positive and the first harmonic changes sign as a function of any control parameter, e.g., temperature, the ground state phase changes abruptly between 0 and $\pi$. If the second harmonic is negative, the ground state phase changes continuously passing all values between 0 and $\pi$ and the realization of a so-called $\phi$-junction is possible in the vicinity of the point where the first harmonic vanishes. We study the unusual properties of this kind of Josephson junction such as critical currents, magnetic field penetration, plasma gap, microwave response, and zero field steps. We also analyze the possible experimental techniques for the observation of predicted effects.

We fabricated high quality $\mathrm{Nb}/{\mathrm{Al}}_2{\mathrm{O}}_3/{\mathrm{Ni}}_{0.6}{\mathrm{Cu}}_{0.4}/\mathrm{Nb}$ superconductor-insulator-ferromagnet-superconductor Josephson tunnel junctions. Using a ferromagnetic layer with a steplike thickness, we obtain a $0\mathrm{\text{−}}\pi$ junction, with equal lengths and critical currents of 0 and $\pi$ parts. The ground state of our $330\mu \mathrm{m}$ ($1.3{\lambda }_J$) long junction corresponds to a spontaneous vortex of supercurrent pinned at the $0\mathrm{\text{−}}\pi$ step and carrying $\sim 6.7\%$ of the magnetic flux quantum ${\Phi }_0$. The dependence of the critical current on the applied magnetic field shows a clear minimum in the vicinity of zero field.

We investigate experimentally a Josephson vortex ratchet—a fluxon in an asymmetric periodic potential driven by a deterministic force with zero time average. The highly asymmetric periodic potential is created in an underdamped annular long Josephson junction by means of a current injector providing an efficiency of the device up to 91%. We measured the ratchet effect for driving forces with different spectral content. For monochromatic high-frequency drive the rectified voltage becomes quantized. At high driving frequencies we also observe chaos, subharmonic dynamics, and voltage reversal due to the inertial mass of a fluxon.

We propose, implement, and test experimentally long Josephson $0\mathrm{\text{−}}\pi$ junctions fabricated using conventional $\mathrm{N}\mathrm{b}\mathrm{\text{−}}{\mathrm{A}\mathrm{l}\mathrm{O}}_x\mathrm{\text{−}}\mathrm{N}\mathrm{b}$ technology. We show that by using a pair of current injectors one can create an arbitrary discontinuity of the Josephson phase and, in particular, a $\pi$ discontinuity, just as in $d$-wave/$s$-wave or in $d$-wave/$d$-wave junctions, and study fractional Josephson vortices which spontaneously appear. Moreover, using such junctions, we can investigate the dynamics of the fractional vortices—a domain which is not yet available for natural $0\mathrm{\text{−}}\pi$ junctions due to their inherently high damping. We observe half-integer zero-field steps which appear on the current-voltage characteristics due to the hopping of semifluxons.

We investigate analytically long Josephson junctions with phase $\pi$-discontinuity points. Such junctions are usually fabricated as a ramp between a superconductor such as ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ with d-wave symmetry of the order parameter and an s-wave superconductor such as Nb. From the top, they look like zigzags with $\pi$ jumps of the Josephson phase at the corners. These $\pi$ jumps, at certain conditions, lead to the formation of half integer flux quanta, which we call semifluxons, pinned at the corners. We derive a version of sine-Gordon equation which describes the dynamics of the Josephson phase in such structures, and obtain an explicit formula that describes the shape of a semifluxon. Some properties of semifluxons are discussed. We propose a way to construct artificial 0-π-junctions using only s-wave superconductors.

We investigate numerically a long Josephson junction with several phase π-discontinuity points. Such junctions were recently fabricated as a ramp between an anisotropic cuprate superconductor such as ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ and an isotropic metal superconductor such as Nb. From the top, they look like zigzags with $\pi$ jumps of the Josephson phase at the corners. These $\pi$ jumps, at certain conditions, lead to the formation of half-integer flux quanta, which we call semifluxons (SFs), pinned at the corners. We show that (i) the spontaneous formation of SFs depends on the junction length, (ii) the ground state without SFs can be converted to a state with SFs by applying a bias current, (iii) the SF configuration can be rearranged by the bias current. All these effects can be observed using a superconducting quantum interference device microscope.