Orbital Antiferromagnetic Order and Magnetoelectricity in Quasi-2D Paramagnets, Ferromagnets and Antiferromagnets*

In magnetoelectrics, an electric field induces a magnetization and a magnetic field induces a polarization, while the system remains in thermal equilibrium. We present a comprehensive theory [1] for magnetoelectricity in magnetically ordered quasi-2D systems. Considering ferromagnetic (FM) zincblende and antiferromagnetic (AFM) diamond structures, we obtain quantitative expressions for the magnetoelectric responses due to electric and magnetic fields that reveal explicitly the inherent duality of these responses required by thermodynamics. For this, AFM order plays a central role. We define a Néel operator t that describes AFM order, in the same way a magnetization m reflects FM order. While m is even under space inversion and odd under time reversal, t describes a toroidal moment that is odd under both symmetries. Thus m and t quantify complementary aspects of magnetic order. In quasi-2D systems, FM order can be attributed to dipolar equilibrium currents that give rise to a magnetization. In the same way, AFM order arises from quadrupolar currents that generate the toroidal moment.
[1] Phys. Rev. Research 2, 043060 (2020)