Bilayer Metal–Organic Framework Altermagnets with Electrically Tunable Spin-Split Valleys

Bilayer altermagnets featuring layer-mediated spin-valley locking hold significant promise in spintronics and valleytronics. In this study, we perform a comprehensive symmetry analysis of bilayer altermagnets and identify seven spin point group candidates with spin-valley-layer coupling, including ²2, ²2²2¹2, ²4̅, ¹4²2²2, ²4̅²2¹<i>m</i>, ¹3²2, and ¹6²2²2. Focusing on the platform of bilayer metal-organic frameworks, we theoretically design materials with <i>S</i>₄ symmetry through chemical modification, achieving spin-splitting in the valence band. Furthermore, the spin valleys from different layers exhibit tunable responses to external static gate electric fields, enabling precise control of spin-splitting. Our findings presented a framework that integrates spin, valley, and layer degrees of freedom in bilayer altermagnets, paving the way for nanoscale spintronics and valleytronics applications.