We develop a comprehensive theory for magnetoelectricity in magnetically ordered quasi-2D systems whereby in thermal equilibrium an electric field can induce a magnetization $m$ and a magnetic field can induce a polarization. This effect requires that both space-inversion and time-reversal symmetry are broken. Antiferromagnetic (AFM) order plays a central role in this theory. We define a Neel operator $tau$ such that a nonzero expectation value $langle tau rangle$ signals AFM order, in the same way $m$ signals ferromagnetic (FM) order. While $m$ is even under space inversion and odd under time reversal, $tau$ describes a toroidal moment that is odd under both symmetries. Thus $m$ and $langle tau rangle$ quantify complementary aspects of magnetic order in solids. In quasi-2D systems FM order can be attributed to dipolar equilibrium currents that give rise to $m$. In the same way, AFM order arises from quadrupolar currents that generate the moment $langle tau rangle$. The electric-field-induced magnetization can then be attributed to the electric manipulation of the quadrupolar currents. We develop a $k cdot p$ envelope-function theory for AFM diamond structures that allows us to derive explicit expressions for the operator $tau$. Considering FM zincblende and AFM diamond, 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.
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