Altermagnetic type-II Multiferroics with N\'{e}el-order-locked Electric Polarization

Altermagnetism, an emergent magnetic phase featuring compensated collinear magnetic moments and momentum-dependent spin splittings, has recently garnered widespread interest. A critical issue concerns whether the unconventional spin structures can generate spontaneous electric polarization in altermagnets, thereby achieving type-II multiferroicity. Here, with the combination of symmetry analysis and microscopic theory, we explicitly demonstrate the generation of electric polarization by altermagnetic N\'eel order, and establish a microscopic mechanism of N\'eel-order-locked electric polarization in altermagnetic multiferroics.~We further reveal these pronounced magnetoelectric coupling behaviors and classify them into eight distinct categories for two-dimensional altermagnets governed by layer group symmetries. Then we take monolayer MgFe$_2$N$_2$ as a prototypical example of altermagnetic type-II multiferroics by first-principles calculations. We also propose to identify the N\'eel order and accompanying electric polarization in altermagnetic multiferroics by magneto-optical microscopy. Bridging type-II multiferroics and altermagnets, our work could pave the way for altermagnetic multifunctional spintronics.