Direct Magnetization-Polarization Coupling in ${\mathrm{BaCuF}}_4$

Herewith, first-principles calculations based on density functional theory are used to describe the ideal magnetization reversal through polarization switching in ${\mathrm{BaCuF}}_4$ which, according to our results, could be accomplished close to room temperature. We also show that this ideal coupling is driven by a single soft mode that combines both polarization, and octahedral rotation. The later being directly coupled to the weak ferromagnetism of ${\mathrm{BaCuF}}_4$. This, added to its strong Jahn-Teller distortion and its orbital ordering, makes this material a very appealing prototype for crystals in the $AB{X}_4$ family for multifunctional applications. The described mechanism behaves ideally as it couples the ferroelectric and the magnetic properties naturally and it has not been reported previously.

Figure 1

$Cmcm$ and $Cmc{2}_1$ phases of the ${\mathrm{BaCuF}}_4$. The phase transition responsible for the ferroelectric switching involves the octahedral (in blue with Cu at the center) rotations around the $x$ axis as well as the polar Ba sites (in dark gray). The later rotations and displacements, denoted with arrows, belong to the ${\mathrm{\Gamma }}_2^{-}$ mode.

Figure 2

(a) $Cmc{2}_1$ structure where the magnetic exchange constants of the intralayer $J_a$ and $J_c$ and the interlayer $J_b$ constant are depicted. (b) Spin-polarized charge density, where a clear orbital ordering, induced by the strong JT distortion, is observed. Here, the magnetic moment up and down are depicted in red and blue, respectively.

Figure 3

(a) Schematic representation of the four multiferroic states where the ferroelectric polarization and magnetization are shown in ${\mathrm{BaCuF}}_4$. Here, only one of the octahedral layers is shown, the others keep the same octahedral pattern and magnetic ordering. It can be observed that the magnetic moments, depicted as red and blue arrows for up and down orientation, respectively, can be reversed by switching the magnetic ordering and/or the polarization reversal. (b) Double-well energy profile obtained through the ferroelectric switching between the up and down spontaneous polarization orientations. (c) Full polarization reversal going from $-10.9$ to $+10.9\mu \mathrm{C}{\mathrm{cm}}^{-2}$ as well as the octahedral rotation angle $\varphi$, where a full octahedral reversal is observed. (d) Magnetic ($m_z$) and orbital ($l_z$) moments reversal by means of the ferroelectric switching showing the correlation between the switching, rotations, and noncollinear magnetic ordering. Solid and open symbols represent the magnetic moments for $P_s$ up and down orientation, respectively.