Abstract
We investigate the supercurrent in a silicene-based Josephson junction under external-field modulations spatially. Employing the qualitative analysis and solving the Dirac--Bogoliubov--de Gennes equation, it is found that, for the bulk states, a $\ensuremath{\pi}$ junction is generated from the valley polarization by combining an antiferromagnetic exchange magnetization and spin-orbit coupling. In contrast, for the topologically protected edge states, a $\ensuremath{\pi}$ as well as a ${\ensuremath{\varphi}}_{0}$ junction can be obtained by adjusting ferromagnetic exchange field or antiferromagnetic exchange magnetization to shift the edge states in wave vector space; or alternatively by modulating electric and light fields to modify the Fermi velocity of the edge states. It is proposed that a direct current superconducting quantum interference devices can be used to observe these $\ensuremath{\pi}$ and ${\ensuremath{\varphi}}_{0}$ junctions in experiment.
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