Josephson Effect for Superconductors Lacking Time-Reversal and Inversion Symmetries

Because of the absence of a center of inversion in some superconducting compounds, a $p$-wave admixture to the dominant $d$-wave (or $s$) order parameter must exist. If time reversal is also violated, an allowed invariant is the product of the $d$ wave (or $s$ wave), $p$ wave, and an appropriately directed current. We show that this leads to a new and remarkable property of the Josephson current for tunneling into a $s$-wave superconductor along the direction parallel to the axis of the $p$-wave component. These ideas are applied to the heavy-fermion compounds which lack center of inversion due to crystalline symmetry, as well as time-reversal symmetry, such as ${\mathrm{CePt}}_3\mathrm{Si}$. They also apply to the superconducting state of the cuprates in the pseudogap region of the phase diagram where in the normal phase some experiments have detected a time-reversal and inversion symmetry broken phase.

Figure 1

Josephson current patterns as a function of magnetic field. Red dotted line: conventional Josephson pattern expected for a superconductor. Blue solid line: Josephson pattern expected for a single domain $d+ip$ superconductor. The parameters used for the calculation in reference to Eq. (17) are $A_1=0.02\mathrm{A}{\mathrm{m}}^{-1}$, $A_2{\delta }_0=0.1\mathrm{A}{\mathrm{m}}^{-1}$, $A_2{\delta }_1=0.05\mathrm{A}{\mathrm{m}}^{-1}{\mathrm{T}}^{-1}$.

Figure 2

Josephson current patterns as a function of magnetic field for a “$d+ip$” superconductor with domains of the $p$-wave component. The parameters are the same as in Fig. 1. Red dotted and blue dashed curves are for two domains of different configuration whose borders are given by ($-L/2$, $-L/3$, $+L/2$) and ($-L/2$, $+L/4$, $+L/2$), respectively. The black solid curve is for five random domains ($-L/2$, $-0.41\mathrm{L}$, 0.09 L, 0.15 L, 0.23 L, $+L/2$). Parameters are $A_1=0.02\mathrm{A}{\mathrm{m}}^{-1}$, $A_2{\delta }_0=0.1\mathrm{A}{\mathrm{m}}^{-1}$, $A_2{\delta }_1=0.05\mathrm{A}{\mathrm{m}}^{-1}{\mathrm{T}}^{-1}$.