Magnetoelectric Behavior from $S=1/2$ Asymmetric Square Cupolas

Magnetoelectric properties are studied by a combined experimental and theoretical study of a quasi-two-dimensional material composed of square cupolas, ${\mathrm{Ba}(\mathrm{TiO}){\mathrm{Cu}}_4({\mathrm{PO}}_4)}_4$. The magnetization is measured up to the field above the saturation, and several anomalies are observed depending on the field directions. We propose a $S=1/2$ spin model with Dzyaloshinskii-Moriya interactions, which reproduces the full magnetization curves well. Elaborating the phase diagram of the model, we show that the anomalies are explained by magnetoelectric phase transitions. Our theory also accounts for the scaling of the dielectric anomaly observed in the experiments. The results elucidate the crucial role of the in-plane component of Dzyaloshinskii-Moriya interactions, which is induced by the noncoplanar buckling of a square cupola. We also predict a “hidden” phase and another magnetoelectric response, both of which appear in a nonzero magnetic field.

Figure 1

(a) Schematic of the lattice structure, which includes a pair of upward and downward square cupolas. The spheres and black dots represent Cu cations and O ions, respectively. The yellow arrows on the blue bonds represent the DM vectors ${\mathbf{D}}_{ij}$ in the model in Eq. (1). The intercupola couplings $J^{\prime }$ and $J^{\prime \prime }$ are also shown. (b) Top view. The blue, green, and red dashed lines represent $J_1$, $J_2$, and $J^{\prime }$ bonds in Eq. (1), respectively.

Figure 2

Magnetic field dependence of magnetization and its field derivative in the (a),(b) experiment for ${\mathrm{Ba}(\mathrm{TiO}){\mathrm{Cu}}_4({\mathrm{PO}}_4)}_4$ at $T=1.4\mathrm{K}$ ($dM/dB$ for $\mathbf{B}\parallel [001]$ is shifted by 0.02 for visibility) and the (c),(d) theory for the model (1) in the ground state with $J_1=1$, $J_2=1/6$, $J^{\prime }=1/2$, $J^{\prime \prime }=1/100$, $\theta =80°$, and $D=0.7$.

Figure 3

Ground-state and finite-$T$ phase diagrams with the magnetic fields (a),(b) $\mathbf{B}\parallel [001]$ and (c),(d) $\mathbf{B}\parallel [100]$. In the ground-state phase diagrams (a),(c), the angle $\theta$ between ${\mathbf{D}}_{ij}$ and $c$ axis is varied, and the other parameters are set to be the same as Figs. 2 and 2. In the finite-$T$ phase diagrams (b),(d), we set $\theta =80°$ as in Figs. 2 and 2. The labels I, II, III, Z, and Y identify different antiferromagnetic ordered phases where ${\mathbf{m}}_{\mathrm{AF}}\ne 0$. FF and FF’ are for the forced ferromagnetic phases above the saturation fields. The dashed line in (a) represents the peak position of the hump in $dm/dB$. Representative spin configurations in each phase are also shown [8].

Figure 4

$B$ and $T$ dependence of the antiferromagnetic order parameter ${\mathbf{m}}_{\mathrm{AF}}$ and the interlayer-antiferroic electric polarization ${\mathbf{P}}_{\mathrm{AF}}$ with the parameter set used in Figs. 2 and 2. The $B$ dependence are computed at $T=0$ for (a) $\mathbf{B}\parallel [001]$ and (b) $\mathbf{B}\parallel [100]$, while the $T$ dependence of (c),(d) are computed at $B=1.4$ ($\mathbf{B}\parallel [001]$). In addition to ${\mathbf{m}}_{\mathrm{AF}}$, the magnetization $m$ is also plotted in (c); $m_{\mathrm{AF}1}^{xy}=\max [m_{\mathrm{AF}}^x,m_{\mathrm{AF}}^y]$ and $m_{\mathrm{AF}2}^{xy}=\min [m_{\mathrm{AF}}^x,m_{\mathrm{AF}}^y]$. The inner and outer products of ${\mathbf{m}}_{\mathrm{AF}}$ and ${\mathbf{P}}_{\mathrm{AF}}$ are plotted in (d). The insets of (b) show ${\mathbf{m}}_{\mathrm{AF}}$ and ${\mathbf{P}}_{\mathrm{AF}}$ for $\mathbf{B}=(2,0,\delta )$ with varying $\delta$.

Figure 5

The scaling plot for the field-induced component of the dielectric constants: (a) $\mathrm{\Delta }{ϵ}_{[100]}$ and (b) $\mathrm{\Delta }{ϵ}_{[110]}$ with $\mathbf{B}\parallel [100]$ and $\mathbf{B}\parallel [1\overline{1}0]$, respectively. The insets show (a) ${ϵ}_{[100]}(B)$ and (b) ${ϵ}_{[110]}(B)$.