Linear and bilinear magnetoelectric effects in magnetically biased magnetite

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Abstract
In a synthetic magnetite (Fe3O4) crystal at 4.2°K the macroscopic anisotropy coefficients depend1 on an external electric field E↘. We now show that this dependence gives rise to linear as well as to two kinds of bilinear magnetoelectric (ME) effects provided the crystal is biased by means of an external static magnetic field H↘0. Upon expanding the previously proposed1 free energy density to F̃ to third order in the perturbing fields E↘ and h↘=H↘−H↘0, we obtain terms (among others) of the form −PoiEi, −αijEihj, −1/2βijkEihjhk and −1/2γijkEiEjhk, where, i,j,k denote a,b or c. By requiring the magnetization M↘ to be in equilibrium in each order of the perturbation, we calculate, without adjustable parameters, the dependence on H↘0 of P0i (obtained previously1) and of the ME susceptibilities αij, βijk and γijk. We measure the dependence on H↘0 of these three susceptibilities by using 1 kHz fields for E↘ and h↘ and observing the 2 kHz as well as the 1 kHz parts of both P↘ (=−∂F̃/∂E↘) and M↘ (=−∂F̃/∂H↘). For h↘∥c↘ (= easy axis) and H↘0 oriented arbitrarily in the bc plane, the measured H↘0‐dependence of αac agrees very well with the single‐domain‐rotation version of the theory whenever H0 is sufficiently large and H↘0 is not parallel to b↘. For h↘∥H↘0∥b↘ (=intermediate axis), the measured H0b‐dependences of αab and βabb all agree reasonably well with the two‐domain‐rotation version of the theory at all values of H0b. Our measured ‖βabb‖ at H0=0 is also calculable and is the largest ‖βijk‖ ever observed in any material.

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