Magnetoelectric Effects and Charge-Imbalanced Solenoids: Antiferro Quadrupole Orders in a Diamond Structure

We study magnetoelectric (magneto-current) effects in a diamond structure under antiferro quadrupole (AFQ) orders. The AFQ orders break the spacial inversion symmetry and cause the current-induced magnetization. The current-induced magnetization strongly depends on the types of the order parameters and the direction of the current. This gives a way to the experimental identification of AFQ order parameters. We also discuss the current-induced magnetization under the AFQ orders in the diamond structure can be intuitively understood in terms of charge-imbalanced solenoids.