Primary ferrotoroidicity in antiferromagnets

P. Tolédano, M. Ackermann, L. Bohatý, P. Becker, T. Lorenz, N. Leo, and M. Fiebig

Phys. Rev. B 92, 094431 – Published 18 September 2015

Abstract

Antiferromagnetic ordering does not give rise to a measurable macroscopic symmetry-breaking order parameter such as the magnetization in a ferromagnet. An exception is the case of antiferromagnets with a vortexlike toroidal alignment of the magnetic moments in the unit cell because this gives rise to the formation of a spontaneous macroscopic toroidal moment which can be measured as a magnetoelectric effect. It is a long-standing question whether the toroidal moment is merely a side effect of the antiferromagnetic state or whether it can become the primary order parameter in a ferroic phase transition. Here we report the magnetoelectric properties of ${LiFeSi}_{2} O_{6}$ and show that they point to the role of the toroidal vector moment as the primary order parameter. Based on Landau theory we distinguish it from primary antiferromagnetic order. We thus justify the proposal that along with ferromagnets, ferroelectrics, and ferroelastics the ferrotoroidics constitute a new class of primary ferroics.

Received 25 June 2015

DOI:https://doi.org/10.1103/PhysRevB.92.094431

Authors & Affiliations

P. Tolédano¹, M. Ackermann², L. Bohatý², P. Becker², T. Lorenz³, N. Leo⁴, and M. Fiebig⁴

¹Laboratoire de Systèmes Complexes, Université de Picardie, 80000 Amiens, France
²Institut für Kristallographie, Universität zu Köln, Greinstraße 6, 50939 Köln, Germany
³II. Physikalisches Institut, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany
⁴Department of Materials, ETH Zürich, Vladimir-Prelog-Weg 4, 8093 Zurich, Switzerland

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Vol. 92, Iss. 9 — 1 September 2015

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Images

Figure 1
Primary ferrotoroidic order in ${LiFeSi}_{2} O_{6}$ . (a) View of the magnetic unit cell with edge-sharing chains of ${Fe}^{3 +} O_{6}$ octahedra (brown), $Si O_{4}$ tetrahedra (blue), and Li ions (green). Gray arrows depict the ${Fe}^{3 +}$ magnetic moments [11, 12]. (b) Magnetic order within the $a c$ , respectively $x z$ , plane with spins $s_{i}$ ( $i = 1 ... 4$ ). Spins on sites 1 and 3, respectively 2 and 4, are antiparallel, while the in-chain pairs (1-4, 2-3) are mainly coupled ferromagnetically. Due to the vortexlike arrangement a toroidal moment $T$ emerges along the $b ∥ e_{y}$ direction (blue circle). (c) Depiction of the spin sums contributing to the antiferromagnetic exchange interaction. We have $L_{1} = s_{1} - s_{2} - s_{3} + s_{4}$ and $L_{3} = s_{1} + s_{2} - s_{3} - s_{4}$ . The leading exchange contribution is given by a term $\propto {(L_{1} \cdot L_{3})}^{2}$ which is suppressed because of the approximate relation $L_{1} ⊥ L_{3}$ .
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Figure 2
Linear magnetoelectric effect in ${LiFeSi}_{2} O_{6}$ . (a)–(i) Temperature dependence of the electric polarization $P^{x}, P^{y}, P^{z}$ (top to bottom) for magnetic fields $μ_{0} H$ applied parallel to the $e_{x}, e_{y}$ , or $e_{z}$ axis (left to right). (j) Relative magnitude (proportional to circle area) of the magnetoelectric polarization at 14 K and 3 T. $Γ_{1}^{-}$ - and $Γ_{2}^{-}$ -related contributions are bright and dark gray, respectively.
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Figure 3
Measured inclination angle of the extinction angle $θ$ relative to the $e_{y}$ axis of a ${LiFeSi}_{2} O_{6}$ (100) plate, using an incident wave with wave normal along $e_{x}$ . A polarizing microscope with crossed polarizers was used for the observation. Above $T_{N} = 18 K$ , we get $θ = 0^{\circ}$ in accordance with monoclinic symmetry. Below $T_{N}$ the rotational degree of freedom expressed by $θ \neq 0^{\circ}$ indicates the loss of the monoclinic symmetry elements $2 ∥ e_{y}, m ⊥ e_{y}$ and, hence, a triclinic symmetry of the crystal. The line is a guide to the eye.
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Figure 4
Possible magnetic space groups obtained from the $P 2_{1} / c 1^{'}$ parent group by activation of irreducible representations $Γ_{i}$ defined in Table 1. A single active representation can induce the phases highlighted by blue circles. The magnetic space groups in rectangles are acquired if two representations are nonzero. The only phase in agreement with the observed magnetoelectric effect in ${LiFeSi}_{2} O_{6}$ is the space group $P {\bar{1}}^{'}$ (orange square). It requires both $Γ_{1}^{-}$ and $Γ_{2}^{-}$ to be active.
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Figure 5
Schematic phase diagrams associated with the free-energy $F = a_{1} {(T^{y})}^{2} + a_{2} {(T^{y})}^{4} + b_{1} {(T^{x, z})}^{2} + b_{2} {(T^{x, z})}^{4} + b_{3} {(T^{x, z})}^{6} + δ {(T^{y})}^{2} {(T^{x, z})}^{2} - H^{2} (δ_{1} {(T^{x, z})}^{2} + δ_{2} {(T^{y})}^{2} + δ_{3} T^{x} T^{z})$ . Hatched and solid curves represent second- and first-order transitions curves. $T_{1}$ to $T_{4}$ are triple points. $N_{1}$ and $N_{2}$ are four-phase points. Arrows show the thermodynamic paths assumed in our description. (a),(b) Phase diagrams without applied magnetic fields ( $H = 0$ ). (a) Phase diagram for a strong negative $δ$ coupling promoting the triggering mechanism, so that the $P {\bar{1}}^{'}$ phase is reached in a single step [ $δ < - 2 (| a_{2} {| b_{2})}^{0.5}, a_{2} < 0$ ]. (b) Phase diagram for weak coupling of the antiferromagnetic order parameters [ $δ > 2 (| a_{2} {| b_{2})}^{0.5}, a_{2} < 0$ ]. Now the $P {\bar{1}}^{'}$ phase is reached by two successive phase transitions across the region of stability of the $P 2_{1} / c^{'}$ phase. (c),(d) Projections of the field-induced phase diagrams on the $(b_{1}, H^{x, z})$ and $(b_{1}, H^{y})$ planes for strong coupling of the order parameters. In (c) the $P 1$ phase is reached continuously above threshold fields $H_{th}^{x}$ and $H_{th}^{z}$ . In (d) the threshold field $H_{th}^{y}$ decouples the $T^{y}$ order parameter and the $P {\bar{1}}^{'}$ phase transforms discontinuously into $P c$ , which is a polar subgroup of $P 2_{1}^{'} / c$ .
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