Altermagnetic Even-Odd Effects in CsV$_2$Te$_2$O Josephson Junctions

Abstract: The interplay between conventional superconductivity and unconventional magnetism offers an exciting platform for realizing exotic superconducting phenomena. Here, we investigate Josephson effects in planar and vertical junctions based on CsV$_2$Te$_2$O-family materials, which host hidden $d$-wave altermagnetism with G-type antiferromagnetic order. In monolayer-based planar junctions, the quasi-1D, nearly flat, spin-polarized bands of the altermagnet, when coupled to $s$-wave superconductors, produce a \textit{fully} spin-polarized supercurrent with strong directional anisotropy -- a spin-selective Josephson effect. In multilayers, we uncover an \textit{altermagnetic even-odd effect}: spin-polarized supercurrents persist only in odd-layer planar junctions but cancel exactly in even layers. Thus, layer parity acts as a switch for spin-polarized supercurrent. In vertical junctions, odd-layer barriers enhance equal-spin triplet transport while even layers favor opposite-spin transport, yielding a robust period-two oscillation in the total supercurrent with layer number. These layer-parity-dependent responses represent a general even-odd effect in hidden altermagnets, applicable to diverse magnetic and transport phenomena.