Proximate transition temperatures amplify linear magnetoelectric coupling in strain-disordered multiferroic ${\mathrm{BiMnO}}_3$

We report a giant linear magnetoelectric coupling in strained ${\mathrm{BiMnO}}_3$ thin films in which the disorder associated with an islanded morphology gives rise to extrinsic relaxor ferroelectricity that is not present in bulk centrosymmetric ferromagnetic crystalline ${\mathrm{BiMnO}}_3$. Strain associated with the disorder is treated as a local variable, which couples to the two ferroic order parameters, magnetization $\vec{M}$ and polarization $\vec{P}$. A straightforward “gas under a piston” thermodynamic treatment explains the observed correlated temperature dependencies of the product of susceptibilities and the magnetoelectric coefficient together with the enhancement of the coupling by the proximity of the ferroic transition temperatures close to the relaxor freezing temperature. Our interpretation is based on a trilinear coupling term in the free energy of the form $\vec{L}·(\vec{P}\times \vec{M})$, where $\vec{L}$ is a hidden antiferromagnetic order parameter, previously postulated by theory for ${\mathrm{BiMnO}}_3$. This phenomenological invariant not only preserves inversion and time-reversal symmetry of the strain-induced interactions but also explains the pronounced linear magnetoelectric coupling without using the more conventional higher order biquadratic interaction proportional to ${(\vec{P}·\vec{M})}^2$.

Figure 1

Temperature dependence of remanent polarization $P_R$ (solid black) and magnetization $M$ (dashed red) demonstrating the near overlap of the ferroelectric and magnetic transitions. The vertical arrows show the temperatures where the maxima in the slopes occur.

Figure 2

Polarization loops at 90 K acquired on the time scales indicated in the legend show relaxor ferroelectric behavior. On the longer time scales, the remanent polarization has more time to interact with the thermal background and decrease in magnitude.

Figure 3

Temperature and magnetic field dependence of electric polarization loops. (a) Hysteretic polarization loops at the indicated temperatures span a temperature range of $\approx 100$ K with a maximum remanent polarization, $P_R$, of $\approx 23$  $\mu$ C/${\mathrm{cm}}^2$ at 5 K. (b) A magnetic field of 7 T (red) is shown to decrease the FE polarization by $\approx 10$% (red) from its value in zero field (black).

Figure 4

The remanent polarization $P_R$ decreases at all temperatures with a linear dependence on magnetic field. The magnetoelectric coupling constants $\tilde{\alpha }\left(T\right)$ for the three selected temperatures in the legend are extracted from the slopes of each of these curves. The maximum $\tilde{\alpha }\left(T\right)$ with a value of $-0.1\mu \mathrm{C}/\mathrm{cm}{}^2\mathrm{T}$ ($-1.25$ ns/m) occurs near 55 K.

Figure 5

The temperature derivatives of the magnetization (red) and ferroelectric polarization (black) are calculated from Fig. 2 and provide a measure of the magnetic and electric susceptibilities (see text). The ME coupling coefficient, $\tilde{\alpha }\left(T\right)$, is shown in blue, and is independently calculated from the slopes of $P_R$ vs $B$ curves at each temperature (see Fig. 4). The green line representing the normalized product ${\tilde{\chi }}_M\left(T\right){\tilde{\chi }}_P\left(T\right)$ [determined by the independently measured temperature derivatives of Eq. (2)] correlates well with the measured values of $\tilde{\alpha }\left(T\right)$ in accordance with Eq. (17).

Figure 6

Schematic showing evolution of polarization $P$ (blue arrows) and magnetization $M$ (red arrows) as a function of decreasing temperature $T$ (vertical axis). For $T>T_M$, where $T_M=105$ K is the unrenormalized bulk transition temperature, the rapidly fluctuating electric dipoles of the polar microregions of the relaxor ferroelectric dominate. As $T$ decreases, the dipoles increase in size and relaxation times (fluctuation rates) increase (decrease). At $T=T_M$, magnetic moments appear, which because of the trilinear coupling, $\vec{L}·(\vec{P}\times \vec{M})$, fluctuate at rates comparable to the relaxor rates. As $T$ is lowered still further towards the relaxor freezing temperature $T_F\approx 70$ K, the electric and magnetic dipoles simultaneously maintain their coupled orientations on laboratory time scales at comparable transition temperatures, $T_F\approx T_P^{*}\approx T_M^{*}$, where the magnetoelectric coupling coefficient, $\tilde{\alpha }\left(T\right)$, is maximum. As $T$ decreases further, both $P$ and $M$ continue to increase as $\tilde{\alpha }\left(T\right)$ begins to decrease (see Figs. 1 and 5). As discussed in the text the free energy is minimized when the alignment between $\vec{P}$ and $\vec{M}$ is $3\pi /2$; the misalignment shown in the figure schematically illustrates the canting away from $3\pi /2$ introduced by the presence of an antiferromagnetic component, $\vec{L}$.