Spin-triplet superconductor–quantum anomalous Hall insulator–spin-triplet superconductor Josephson junctions: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mtext>−</mml:mtext><mml:mi>π</mml:mi></mml:math> transition, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ϕ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> phase, and switching effects
Qiang Cheng, Qing Yan, Qing‐Feng Sun
Abstract
We study the Josephson effect in spin-triplet superconductor\ensuremath{-}quantum anomalous Hall insulator\ensuremath{-}spin-triplet superconductor junctions using the nonequilibrium Green's function method. The current-phase difference relations show strong dependence on the orientations of the $\mathbf{d}$ vectors in superconductors. We focus on two $\mathbf{d}$-vector configurations, a parallel one with the left and right $\mathbf{d}$ vectors being in the same direction and a nonparallel one with the left $\mathbf{d}$ vector fixed at the $z$ axis. For the parallel configuration, the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition can be realized when one rotates the $\mathbf{d}$ vectors from the parallel to the junction plane to the perpendicular direction. The ${\ensuremath{\phi}}_{0}$ phase with nonzero Josephson current at zero phase difference can be obtained as long as ${d}_{x}{d}_{z}\ensuremath{\ne}0$. For the nonparallel configuration, the $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition and the ${\ensuremath{\phi}}_{0}$ phase still exist. The condition for the formation of the ${\ensuremath{\phi}}_{0}$ phase becomes ${d}_{Rx}\ensuremath{\ne}0$. The switch effects of the Josephson current are found in both configurations when the $\mathbf{d}$ vectors are rotated in the $xy$ plane. Furthermore, the symmetries satisfied by the current-phase difference relations are analyzed in detail by the operations of the time-reversal, mirror-reflections, the spin-rotation, and the gauge transformation, which can well explain the above selection rules for the ${\ensuremath{\phi}}_{0}$ phase. Our results reveal the peculiar Josephson effect between spin-triplet superconductors and the quantum anomalous Hall insulator, which provide helpful phases and effects for device designs. The distinct current-phase difference relations for different orientations may be used to determine the direction of the $\mathbf{d}$ vector in the spin-triplet superconductor.