We develop a comprehensive quantitative theory for magnetoelectricity in magnetically ordered quasi-two-dimensional (quasi-2D) systems whereby in thermal equilibrium an electric field can induce a magnetization and a magnetic field can induce an electric polarization. This effect requires that both space-inversion and time-reversal symmetry are broken. Antiferromagnetic order plays a central role in this theory. We define a Néel operator $\mathit{\tau }$ such that a nonzero expectation value $\langle \mathit{\tau }\rangle$ signals collinear antiferromagnetic order in the same way a magnetization signals ferromagnetic order. While a magnetization is even under space inversion and odd under time reversal, the operator $\mathit{\tau }$ describes a toroidal moment that is odd both under space inversion and under time reversal. Thus the magnetization and the toroidal moment $\langle \mathit{\tau }\rangle$ quantify complementary aspects of collinear magnetic order in solids. Focusing on quasi-2D systems, itinerant-electron ferromagnetic order can be attributed to dipolar equilibrium currents that give rise to a magnetization. In the same way, antiferromagnetic order arises from quadrupolar equilibrium currents that generate the toroidal moment $\langle \mathit{\tau }\rangle$. In the magnetoelectric effect, the electric-field-induced magnetization can then be attributed to the electric manipulation of the quadrupolar equilibrium currents. We develop a $\mathrm{k}·\mathrm{p}$ envelope-function theory for the antiferromagnetic diamond structure that allows us to derive explicit expressions for the Néel operator $\mathit{\tau }$. Considering ferromagnetic zincblende structures and antiferromagnetic diamond structures, we derive quantitative expressions for the magnetoelectric responses due to electric and magnetic fields that reveal explicitly the inherent duality of these responses required by thermodynamics. Magnetoelectricity is found to be small in realistic calculations for quasi-2D electron systems. The magnetoelectric response of quasi-2D hole systems turns out to be sizable, however, with moderate electric fields being able to induce a magnetic moment of one Bohr magneton per charge carrier. Our theory provides a broad framework for the manipulation of magnetic order by means of external fields.

• Received 7 January 2020
• Accepted 26 August 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043060

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Published by the American Physical Society