Multiferroic Magnetic Spirals Induced by Random Magnetic Exchanges

Multiferroism can originate from the breaking of inversion symmetry caused by magnetic-spiral order. The usual mechanism for stabilizing a magnetic spiral is competition between magnetic exchange interactions differing by their range and sign, such as nearest-neighbor and next-nearest-neighbor interactions. In insulating compounds, it is unusual for these interactions to be both comparable in magnitude and of a strength that can induce magnetic ordering at room temperature. Therefore, the onset temperatures for multiferroism through this mechanism are typically low. By considering a realistic model for multiferroic ${\mathrm{YBaCuFeO}}_5$, we propose an alternative mechanism for magnetic-spiral order, and hence for multiferroism, that occurs at much higher temperatures. We show, using Monte Carlo simulations and electronic structure calculations based on density functional theory, that the Heisenberg model on a geometrically nonfrustrated lattice with only nearest-neighbor interactions can have a spiral phase up to high temperature when frustrating bonds are introduced randomly along a single crystallographic direction as caused, e.g., by a particular type of chemical disorder. This long-range correlated pattern of frustration avoids ferroelectrically inactive spin-glass order. Finally, we provide an intuitive explanation for this mechanism and discuss its generalization to other materials.

Popular Summary

Magnetic data storage, in which information is encoded in the magnetic state of a material, underlies much of modern information technology because of its robustness and affordability. However, magnetic data bits in today’s memory devices are written by magnetic fields that are generated by electric currents, which dissipate energy in the form of heat. Multiferroic materials, with their coexisting and cross-coupled magnetism and electric polarization, offer tremendous potential for future energy-saving storage devices since magnetic information can be recorded using low-energy electric fields. The widespread adoption of the cross coupling between electric polarization and magnetism to create electrically writable magnetic bits is currently thwarted by the fact that magnetic spiral states, which often induce electric polarization in multiferroic systems, appear only at low temperatures in the majority of cases. In this work, we present a new mechanism for stabilizing spiral magnetic order at high temperatures.

Using Monte Carlo simulations and electronic structure calculations, we illustrate our stabilization mechanism by studying a recently identified multiferroic magnetic spiral compound, ${\mathrm{YBaCuFeO}}_5$, which displays a magnetic spiral state at temperatures as high as 310 K. The mechanism is based on the diluted presence of magnetically frustrating bonds, which occur because of the disorder between copper and iron ions. This disorder is believed to occur experimentally during the synthesis process and captures the most striking features of the magnetic spiral phase in ${\mathrm{YBaCuFeO}}_5$.

Our results provide a pathway to the design of novel high-temperature multiferroics with magnetism that can be controlled by an electric field.

Figure 1

(a) Two different orderings of ${\mathrm{Cu}}^{2+}$ and ${\mathrm{Fe}}^{3+}$ in the $\sqrt{2}\times \sqrt{2}\times 2$ supercell of ${\mathrm{YBaCuFeO}}_5$. The ${\mathrm{FeO}}_5$ and the ${\mathrm{CuO}}_5$ square pyramids are shown in gold and blue, respectively. The left panel shows a low-energy structure consisting only of ${\mathrm{Fe}}^{3+}\text{−}{\mathrm{Cu}}^{2+}$ bipyramids. The right panel shows a high-energy structure with one ${\mathrm{Fe}}^{3+}\text{−}{\mathrm{Fe}}^{3+}$ and, upon periodic repetition, one ${\mathrm{Cu}}^{2+}\text{−}{\mathrm{Cu}}^{2+}$ bipyramid. (b) Unfrustrated magnetic order in the commensurate phase. (c) Simplified spin model for ${\mathrm{YBaCuFeO}}_5$ with nearest-neighbor exchange interactions. The $a^{\prime }$ and $b^{\prime }$ axes are rotated by 45° with respect to the $a$ and $b$ axes in diagram (a). The magnetic ions corresponding to either ${\mathrm{Cu}}^{2+}$ or ${\mathrm{Fe}}^{3+}$ are depicted as pink spheres. They form bilayers (regions enclosed by the rectangles) obtained by stacking a pair of $x\text{−}y$ square lattices (corresponding to ${\mathrm{YBaCuFeO}}_5$ bilayers) along the $z$ axis. The antiferromagnetic exchange coupling $J_{\parallel }$ within a layer is assumed to be independent of whether it couples ${\mathrm{Fe}}^{3+}$ with ${\mathrm{Fe}}^{3+}$, ${\mathrm{Cu}}^{2+}$ with ${\mathrm{Cu}}^{2+}$, or ${\mathrm{Fe}}^{3+}$ with ${\mathrm{Cu}}^{2+}$. The interlayer coupling $J_{\perp }^{\prime }$ within a bilayer is ferromagnetic for ${\mathrm{Cu}}^{2+}\text{−}{\mathrm{Fe}}^{3+}$ bipyramids (see text). Impurity bonds have a strong antiferromagnetic exchange, $J_{\mathrm{imp}}$, similar in magnitude to the ab initio value for ${\mathrm{Fe}}^{3+}\text{−}{\mathrm{Fe}}^{3+}$ bipyramids. The impurity bonds are randomly distributed, up to the constraint that two impurity bonds in adjacent bilayers cannot be closer than a minimal distance ${\xi }_{\min }$.

Figure 2

Monte Carlo results obtained for systems of size $L=28$. (a) Specific heat $C$, collinear antiferromagnetic order parameter $m_{\mathrm{AF}}$, and spiral order parameter $\psi$, which we estimated by computing $\sqrt{⟨{\psi }^2⟩}$, for ${\xi }_{\min }=4$ and a range of impurity bond concentrations. (b) Square root of the spin-structure factor $S(q_c)$ computed for ${\xi }_{\min }=4$. The data are averaged over the in-plane wave vectors. The crosses represent the wave vector at which $S(q_c)$ takes a maximum value at the given temperature. The data for $n_{\mathrm{imp}}=0.04$ are compatible with a direct transition from the paramagnetic state into an incommensurate magnetic state with $q_{\mathrm{spi}}<1/2$. (c) Phase diagrams obtained for ${\xi }_{\min }=2.5$ and ${\xi }_{\min }=4$.

Figure 3

(a) Sketch of the two ground states of Hamiltonian (1) for a single bilayer, in the presence of a single frustrating impurity bond (red line). For visualization purposes, the spins on every other site in the $ab$ plane have been reversed so that the unfrustrated ground state looks ferromagnetic. The two ground states break the local inversion symmetry and correspond to counterclockwise (left panel) and clockwise (right panel) rotation of the spins as one proceeds along the $c$ axis. They can be labeled by an Ising variable ${\sigma }_{\tilde{\mathit{r}}}$. Spin waves mediate an effective interaction ${\mathrm{\Gamma }}_{\tilde{\mathit{r}},{\tilde{\mathit{r}}}^{\prime }}$ between the local orientations ${\sigma }_{\tilde{\mathit{r}}}$ and ${\sigma }_{{\tilde{\mathit{r}}}^{\prime }}$ of the cantings around two impurity bonds $⟨\tilde{\mathit{r}},\tilde{\mathit{r}}+\mathit{z}⟩$ and $⟨{\tilde{\mathit{r}}}^{\prime },{\tilde{\mathit{r}}}^{\prime }+\mathit{z}⟩$, as given by Eq. (6). (b) The blue cone is the domain of $\mathrm{\Delta }\tilde{\mathit{r}}=\tilde{\mathit{r}}-{\tilde{\mathit{r}}}^{\prime }$ for which the effective interaction ${\mathrm{\Gamma }}_{\tilde{\mathit{r}},{\tilde{\mathit{r}}}^{\prime }}$ is antiferromagnetic, while outside the cone, it is ferromagnetic. The short-range constraint ${\xi }_{\min }$ excludes the domains represented by the red cylinders and thus increases the probability of a ferromagnetic coupling between neighboring Ising variables.

Figure 4

(a) Comparison of a spiral ground state corresponding to an Ising ferromagnetic ground state of the effective Ising Hamiltonian (6) (left) against the spiral ground state of the Heisenberg Hamiltonian (1) (right). The violet arrows represent the average directions of the staggered magnetization for each layer. The red dots lying on the circle represent the direction of each spin in a given plane. The staggered magnetization of layers corresponding to even bilayers have been reversed for visualization purposes. The two central panels depict the ground states in a portion of six layers of the lattice of overall dimensions $14\times 14\times 32$. Here, the color of the arrows indicates the angular difference of each spin from the average staggered magnetization of the layer, and the vertical red lines depict the impurity bonds. For visualization purposes, spins at every other site in the $ab$ plane and at every other bilayer have been reversed. The results were obtained for a superlattice of impurity bonds with lattice constants $\mathit{A}=(4,3,0)$, $\mathit{B}=(0,4,2)$, and $\mathit{C}=(4,0,2)$. The values of the couplings are $J_{\parallel }=28.9\mathrm{meV}$, $J_{\perp }=J_{\perp }^{\prime }=4.1\mathrm{meV}$, and $J_{\mathrm{imp}}=-95.8\mathrm{meV}$. (b) Comparison of the spiral ground state obtained using the effective Ising Hamiltonian (6) (left) against the spiral ground state of the Heisenberg Hamiltonian (1) (right) for the same value of the couplings but with $\mathit{A}=(5,3,2)$, $\mathit{B}=(3,0,4)$, and $\mathit{C}=(2,4,6)$. Panel (c) shows the same comparison for the choice $\mathit{A}=(5,0,0)$, $\mathit{B}=(0,5,0)$, and $\mathit{C}=(0,1,2)$. For such a superlattice of impurity bonds, a fan state (with no net winding along the $c$ axis) is stabilized instead of a spiral. (d) Concentration dependence of the magnitude of $\psi$ defined in Eq. (3) for the ground states obtained from the effective Ising Hamiltonian (6) (red dots) and the Heisenberg Hamiltonian (1) (right) (blue squares) for various superlattices of impurity bonds stabilizing a spiral state.