Magnetism and spin-driven ferroelectricity in the multiferroic material $\alpha -{\mathrm{Cu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$

The magnetic anisotropy, exchange couplings, and spin-driven ferroelectricity in recently discovered multiferroic oxide $\alpha -{\mathrm{Cu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$ have been investigated using first-principles calculation and symmetry analysis. Density functional theory (DFT) calculations predict easy $a$-axis anisotropy, while applying an on-site Hubbard correction (DFT + $U$) converts the easy axis into $b$ direction. These results can explain the discrepancy of easy axis between previous theoretical prediction and experimental results. The third-nearest-neighbor exchange interaction is proved to be dominant for the antiferromagnetic structure. The nearest-neighbor Dzyaloshinskii-Moriya interaction plays an important role in the origins of spin canting, and the canted magnetic moments are oriented predominantly along the $c$ axis. Although the spin-dependent $p-d$ hybridization mechanism induces a sizable electric polarization, the observed large polarization can be mainly attributed to the exchange-striction mechanism including spin-lattice coupling.

Figure 1

Crystal and AFM structure of $\alpha -{\mathrm{Cu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$. The arrows denote the spin orientations of Cu ions. The canting of spins is not shown.

Figure 2

(a) Magnetic anisotropy energy in GGA + $U$ calculation with $U_{\text{eff}}=8$ eV. Each unit cell contains 16 Cu ions. The orientation of spin is denoted by the angle relative to $a$ and $c$ axes within $ab$ ($ac$) and $bc$ planes, respectively. (b) Dependence of $\mathrm{\Delta }E=E_a-E_b$ and magnetic moment on the $U_{\text{eff}}$ value. The energy is calculated with FM order based on relaxed and experimental crystal structures, respectively. The $U_{\text{eff}}=0$ eV means the GGA calculation. (c) Dependence of $\mathrm{\Delta }E$ on the lattice constants obtained by GGA + $U$ ($U_{\text{eff}}=8$ eV) calculation. (d) Dependence of $\mathrm{\Delta }E$ on the $U_{\text{eff}}$ value obtained by PBE + $U$, PBEsol + $U$, and LDA + $U$ calculations. The energy is calculated with experimentally observed collinear AFM order. (e) Magnetic anisotropy energy in GGA calculation with experimentally observed collinear AFM order. (f) Dependence of $\mathrm{\Delta }E$ on the $U_{\text{eff}}$ value with noncollinear canted AFM order obtained by GGA + $U$ calculation.

Figure 3

(a) Schematic illustration of magnetic structure and exchange paths. $J_1$ and $J_5$ denote the intrachain NN and NNN exchange interactions, respectively. $J_2-J_4$ denote the three interchain exchange interactions sorted by the length of bonds. The local geometries of bonds for the three major exchange interactions (b) $J_1$, (c) $J_2$, and (d) $J_3$.

Figure 4

Calculated values of exchange interactions $J_1-J_5$ with different $U_{\text{eff}}$ values.

Figure 5

(a) Exchange paths and corresponding vectors for considered DM interactions, and spin configuration for the calculation of the $D_{1z}$ component. The dashed arrows denote the specified directions of bonds for the DM vectors. (b) The changed spin configuration can be obtained through reversing the bottom spins of zigzag chains in (a) spin configuration.