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

Abstract

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 2 2, 2 2 2 2 1 2, 2 4̅, 1 4 2 2 2 2, 2 4̅ 2 2 1 m, 1 3 2 2, and 1 6 2 2 2 2. Focusing on the platform of bilayer metal–organic frameworks, we theoretically design materials with S 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.