Theory of orbital magnetic quadrupole moment and magnetoelectric susceptibility

We derive a quantum-mechanical formula of the orbital magnetic quadrupole moment (MQM) in periodic systems by using the gauge-covariant gradient expansion. This formula is valid for insulators and metals at zero and nonzero temperature. We also prove a direct relation between the MQM and magnetoelectric (ME) susceptibility for insulators at zero temperature. It indicates that the MQM is a microscopic origin of the ME effect. Using the formula, we quantitatively estimate these quantities for room-temperature antiferromagnetic semiconductors ${\mathrm{BaMn}}_2{\mathrm{As}}_2$ and ${\mathrm{CeMn}}_2{\mathrm{Ge}}_{2-x}{\mathrm{Si}}_x$. We find that the orbital contribution to the ME susceptibility is comparable with or even dominant over the spin contribution.

Figure 1

Chemical potential dependence of the nonzero orbital MQM (a) for $\vec{h}=h\vec{z}$ and (b), (c) for $\vec{h}=h\vec{x}$ in the unit of $qh/\hslash$. The black star shows the total orbital MQM, the red square shows the itinerant contribution given by the first term in Eq. (11), and the blue circle and magenta triangle show the local contributions given by the second and third terms. The gray region shows that the system is an insulator.