Tuning order-by-disorder multiferroicity in CuO by doping

The high Curie temperature multiferroic compound CuO has a quasidegenerate magnetic ground state that makes it prone to manipulation by the so-called “order-by-disorder” mechanism. First principle computations supplemented with Monte Carlo simulations and experiments show that isovalent doping allows us to stabilize the multiferroic phase in nonferroelectric regions of the pristine material phase diagram with experiments reaching a 250% widening of the ferroelectric temperature window with 5% of Zn doping. Our results allow us to validate the importance of a quasidegenerate ground state on promoting multiferroicity on CuO at high temperatures and open a path to the material engineering of multiferroic materials. In addition we present a complete explanation of the CuO phase diagram and a computation on the incommensurability in excellent agreement with experiment without free parameters.

Figure 1

Spheres represent Cu atoms in even planes (orange), odd planes (red), or impurities (black). Brown arrows represent spins. (a) Low-temperature AF1 state. We also show some of the exchange interactions considered. $J_a,J_b,J_c,J_d$ connect sites defined as “first neighbors.” Bonds of the same color are equivalent and have the same exchange. Because the Cu ion is at an inversion center in the undistorted structure, the Weiss field due to the orange Cu's on the red Cu cancels (and vice versa). More interactions are shown in Fig. 2. (b) First-rank Henley effect. In the presence of an impurity (nonmagnetic in the example) inversion symmetry is locally broken (dashed vs full orange bond) and the cancellation of the Weiss field at atom 1 is no longer valid. (c) Second-rank Henley effect. If the wave function at atom 1 is perturbed due to the impurity, all the intersublattice interactions indicated by thick bonds get affected.

Figure 2

The exchange couplings used in the calculations. Bonds of the same color are equivalent. Notice that the orientation of the bonds bridged by $J_a$ (violet) and $J_d$ (gray) alternate on the [101] direction. The same alternation occurs in the $\left[10\overline{1}\right]$ and [010] directions. $J_b$ and $J_c$ instead are equivalent by symmetry. The lattice defined by the Cu atoms resembles a body centered cubic structure with atoms on even planes (orange) at the vertices of the cuboids and atoms in odd planes at the center (red). Not shown in the picture are the second neighbor couplings $J_{a2},J_{b2},J_{c2}$, and $J_{d2}$ which act, respectively, on the same direction of $J_a,J_b,J_c$, and $J_d$, i.e., $J_{a2}$ bridges the two even atoms at opposite vertices of the cuboid that are connected by $J_a$ with the odd atom (red) at the center. $J_{a2}$ and $J_{d2}$ follow the same alternation as $J_a$ and $J_d$.

Figure 3

Order parameters for undoped CuO from classical Monte Carlo simulations. The static structure factor $S\left(\mathbf{q}\right)$ for ${\mathbf{q}}_{\text{AF}1}$ and ${\mathbf{q}}_{\text{AF}2}$, and the out of plane polarization $P_y$, for a $36\times 4\times 36$ cell. Simulation cell sizes are stated in terms of repetitions $L_a\times L_b\times L_c$ along the lattice vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ of the conventional unit cell with four Cu atoms.

Figure 4

DFT computation of the stabilization of the ferroelectric phase with different impurities. We show the energy difference per Cu atom between the AF1 and the AF2 configurations as a function of doping for different dopants. The arrows show the magnitude of Henley relaxation energy at 6.250% doping for (from left to right) Mg, Zn, and Cd. The legend reports the magnetic moment we find at the impurity site in the DFT computations.

Figure 5

$b$-axis electric polarization as a function of temperature. (a) Monte Carlo computations for undoped CuO (blue) and for 3.125% nonmagnetic doping without enhancement of interactions (dashed, red line) and with an enhancement factor $\eta =3$ (full red line). (b) Experiment using the same protocol as in Ref. [2].

Figure 6

Experimental and theoretical phase diagram as a function of doping. We plot the Néel temperatures as a function of doping for the experiment with Zn (dots) and Co (diamonds) together with the theoretical results for $\eta =3$ (triangles).

Figure 7

The incommensurate AF2 configuration. Cu ions in an even (odd) plane are represented by orange (red) spheres. Brown arrows represent spins. The pitch of the spiral has been increased by a factor of 40 to allow the visualization of the spin rotations. $\mathbf{Q}$ is the ordering wave vector of the spiral distortion.