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.

Figures (5)

• Figure 1: Crystal and electronic structure of the $d$-wave altermagnetic CsV$_2$Te$_2$O. (a) Magnetic lattice structure showing C-type or G-type antiferromagnetic orders. (b) Nonmagnetic band structure of the monolayer; states near the Fermi level arise from V $3d$ orbitals (light teal region). (c) Four-band projection near the Fermi level highlighting $d_{xz}$ and $d_{yz}$ orbitals. (d) The effective V$_2$O Lieb lattice model with orbital-sublattice locking. The hopping parameters are labeled with black arrows. (e) $d$-wave altermagnetic spin splitting bands from Eq. \ref{['eq:H_AM']}, with spins denoted by $\uparrow$ and $\downarrow$. (f) Spin-resolved Fermi surface at the Fermi level [dashed line in (e)] featuring quasi-1D flat segments near $k_x, k_y \approx \pm \pi/2$.
• Figure 2: Planar junctions in monolayer CsV$_2$Te$_2$O. (a) and (b) illustrate the spin-selective Josephson effect in $s$-wave superconductor/altermagnet/$s$-wave superconductor Josephson junctions. It is along the (a) $x$ and (b) $y$ directions. The proximity-induced pairing correlations in the altermagnet are anisotropic: (a) purely spin-up triplet; and (b) purely spin-down triplet. (c) For a short $x$-oriented junction ($L_\text{AM}=25$), the spin-singlet supercurrent $I^x_s$ is significantly smaller than the spin-triplet supercurrent $I^x_{\uparrow\uparrow}$, while $I^x_{\downarrow\downarrow}$ is vanishingly small. (d) Dependence of $I^x_{s}$, $I^x_{\uparrow\uparrow}$, and $I^x_{\downarrow\downarrow}$ on the junction length $L_\text{AM}$. (e) Comparison of the spin supercurrent for four different magnetic configurations: ferromagnetic order with $\pm M_\textbf{FM}\hat{\sigma}_0\hat{s}_z$ and altermagnetic order with $\pm M_\textbf{AM}\hat{\sigma}_z\hat{s}_z$.
• Figure 3: Planar junctions in bilayer CsV$_2$Te$_2$O. (a) Schematic of layer-resolved spin-split bands in the top and bottom layers. Proximity-induced superconducting pairings include intra-layer equal-spin triplet and inter-layer opposite-spin triplet pairing, the latter mediated by the inter-layer hopping $t_z$. (b)-(c) Supercurrent components in a long junction ($L_\text{AM}=50$) for two cases: (b) For $t_z=0.01$ eV, the spin-singlet supercurrent $I_s^x$ is negligible. (c) For $t_z=0.1$ eV, $I_s^x$ becomes comparable to the equal-spin triplet components $I_{\uparrow\uparrow}^x=I_{\downarrow\downarrow}^x$. (d) Dependence of all supercurrent components on $t_z$ at $\alpha=0.4$ and $\phi_J = \pi/2$. (e) Normalized supercurrent difference $\Delta I^x_\text{JJ}/I^x_\text{tot}$ in the $t_z$-$\alpha$ plane, where $\Delta I^x_\text{JJ}$ is defined in Eq. \ref{['eq-Ix-JJ-diff']}.
• Figure 4: Altermagnetic even-odd effect in CsV$_2$Te$_2$O with small inter-layer hopping $t_z=0.01$ eV. (a) Schematic illustration of the planar junction for the multilayer structure. A gate voltage is applied perpendicular to the altermagnetic planes. (b) Layer-parity dependence of $I^x_{\uparrow\uparrow}$ and $I^x_{\downarrow\downarrow}$ at $\phi_J=\pi/2$. (c) The corresponding spin current $I^x_\text{spin} = I^x_{\uparrow\uparrow}-I^x_{\downarrow\downarrow}$, showing on-off switching by adding layers ($V_G=0$, black curve). A finite gate voltage ($V_G = 0.1$ eV) induces a spin supercurrent in even-$N_z$ systems (red curve). (d) Nonmonotonic dependence of the spin current $I^x_\text{spin}$ on gate voltage $V_G$ for a four-layer junction ($N_z = 4$).
• Figure 5: Altermagnetic even-odd effect in vertical Josephson junctions. (a) Layer dependence of the equal-spin triplet supercurrent $I^z_{\text{eq}}$ and opposite-spin supercurrent $I^z_{\text{op}}$ at $\phi_J=\pi/2$ and $\alpha=0.1$. Gray-shaded rectangles represent the total supercurrent. Inset: schematic of the vertical junction. (b) Same as (a) but for $\alpha=0.4$. Inset: current-phase relation for $N_z = 4$ (solid curve) and $N_z = 5$ (dotted curve). Here, $t_z = 0.2$ is used to emphasize the even-odd effect.