Introduction

The magnetoelectric (ME) effect, that is, the induction of either magnetization M by an electric field E (MEE effect) or electric polarization P by a magnetic field H (MEH effect), is a sensitive probe of the magnetic symmetry of magnetic insulators1,2,3. The terms of the free energy F to be considered are F=−MH−PE−αHE–βHHE–γHEE…. Here the first and second terms are the Zeeman and Stark energies, respectively. The third and fourth (fifth) terms give the linear and quadratic ME effects in which P (M), linear or quadratic, in the applied field strength is induced by H (E), respectively. On the basis of symmetry analysis, the linear effect (P=αH and M=αE with non-zero linear ME coefficient α) is allowed only in magnetic insulators in which both space inversion and time reversal symmetries are broken. The toroidal moment t can be generated either by persistent orbital currents or certain spin orderings4,5. In the latter case, t is known as a multi-spin variable breaking both space inversion and time reversal symmetries, and is defined as tΣi ri × Si, where ri denotes the location of spin Si (refs 4, 5). When t is non-zero, an off-diagonal and antisymmetric linear ME effect is allowed, that is, P−t × H and Mt × E, which are obtained from the free energy expansion analysis6,7. Thus, magnetic insulators showing a ferrotoroidic order (for example, Ga2-xFexO3 and LiCoPO4) can exhibit the antisymmetric linear ME effect8,9,10.

Recently, an XY-like spin glass NixMn1-xTiO3 (x≈0.42)11,12 was found to show the off-diagonal and antisymmetric linear MEH effect13, although a linear ME effect is rarely seen in magnetically disordered systems, such as spin glasses. (A higher-order ME effect is seen in perovskite (Sr,Mn)TiO3, so-called ‘multiglass’, in which dipole and spin glass states coexist14.) In NixMn1-xTiO3 with the ilmenite structure (centrosymmetric space group R), a magnetic (Ni,Mn) plane and a non-magnetic Ti plane are alternately stacked along the hexagonal c axis. The Ni2+ and Mn2+ ions are randomly distributed in the (Ni,Mn) plane and form a honeycomb lattice (inset of Fig. 1b). Ni0.4Mn0.6TiO3 (hereafter NMTO) exhibits successive spin glass transitions at TSG1≈9.5 K and TSG2≈6 K11,13, and is classified as an XY-like spin glass with an easy-plane anisotropy15. The recently discovered antisymmetric linear MEH effect in NMTO possesses ME tensor components α12=−α21≠0 below about 9 K13. The MEH effect was reasonably explained by considering that the presence of the cross-product of E and H, (E × H), during the spin-freezing process polarizes net t in the direction perpendicular to the honeycomb lattices. Thus, NMTO is a rare example showing that disordered frozen spins purely induce a linear ME coupling. This study examines cooling history and dynamical features, which are the most intriguing characteristics of the glassy frozen states, in the ME effect of NMTO and provide a new concept, ‘toroidal glass’.

Figure 1: ME-cooling-condition dependence on P induced by ME H effect in NMTO.
figure 1

T dependence of P[1–10] measured at μ0H[110]=9 T and E[1–10]=0 MV m−1 after several ME cooling procedures (a) at fixed E[1–10]=1 MV m−1 and various μ0H[110] from 1 to 9 T (b) at various E[1–10] from 0.2 to 1 MV m−1 and μ0H[110]=9 T. (For example, ‘FC 1 MV m−1’ denotes the data ME-cooled at E[1–10]=1 MV m−1.) Inset of b, crystal structure of NMTO projected along the hexagonal c axis. Randomly distributed Ni2+ and Mn2+ ions form stacked honeycomb lattices.

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Results

Cooling-condition dependence of ME effect

First, we examine the ME-cooling-condition dependence on P induced by the MEH effect in a single crystal of NMTO. Figure 1 shows temperature (T) profiles of P parallel to the [1–10] direction (P[1–10]) at μ0H=9 T along [110] (H[110]). This configuration allows measurement of the ME tensor components α12 (or α21). Before the respective measurements, the specimen was cooled from 20 K (>TSG1) to 2 K in finite |E × H| with selected E along [1–10] (E[1–10]) and H[110] (ME field cooling). Then, P[1–10] was taken at μ0H[110]=9 T in the absence of E[1–10]. All the data shown in Fig. 1a were taken after ME field cooling with E[1–10]=1 MV m−1 and selected H[110]. Below 9 K, magnetoelectrically induced P[1–10] develops in all the data. It is noteworthy that the magnitude of P[1–10] strongly depends on the strength of cooling H[110]. Figure 1a clearly demonstrates that P[1–10] monotonically increases with increasing the cooling H[110]. We also measured the cooling E dependence of P[1–10]. The results are shown in Fig. 1b. Before these measurements, μ0H[110]=9 T and selected E[1–10] were applied during the ME-field-cooling process. The magnitude of P[1–10] monotonously increases as the cooling E[1–10] increases. These results indicate that the magnitude of P is very sensitive to both cooling E and H, that is, ME-field-cooling condition. The observed remarkable cooling H and E effects on P are reminiscent of the cooling H effect on M in spin glasses16 and/or the cooling E effect on P in dipole glasses17.

In ref. 13, it was proposed that the MEH effect, that is, the induction of P[1–10] by applying H[110], observed in NMTO originates from the presence of net toroidal moment t along [001] through the relation P−t × H. If this scenario is correct, we can also expect the MEE effect, the induction of M along [110] (M[110]) induced by applying E[1–10], with the relation Mt × E from phenomenological consideration7. Thus, we examined the MEE effect in NMTO. Figure 2b shows the time dependence of M[110] at μ0H[110]=0 T and T=2 K. Before the measurement, the ME cooling procedure was done with μ0H[110]=7 T and E[1–10]=1 MV m−1. As is common with spin glasses, a remnant magnetization caused by the magnetic-field cooling appears and shows a long-time relaxation16. During the relaxation process, various strengths of E[1–10] were applied, as plotted in Fig. 2a. Comparing Fig. 2a it is evident that the magnitude of M[110] systematically changes, depending on the strength and sign of applied E[1–10]. Such an effect of E[1–10] on M[110] was not observed without the ME-field-cooling process. This result clearly confirms the presence of the MEE effect in NMTO.

Figure 2: Relaxation of M and ME E effect in NMTO.
figure 2

Temporal evolution of (b) isothermal M[110] responding to (a) periodically applied E[1–10] at T=2 K and μ0H[110]=0 T. Before the measurement, ME cooling was done with μ0H[110]=7 T and E[1–10]=1 MV m−1.

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Figure 3a displays E[1–10] profiles of the change in remanent magnetization [ΔM[110]=M[110](E)−M[110](0)] measured at μ0H[110]=0 T and T=2 K. Before each measurement, the ME-field cooling was performed at various E[1–10] and H[110]. As seen in Fig. 3a, ΔM[110] shows the linear E[1–10] dependence, ΔM[110]=αm E[1–10], and the sign of αm is reversed by changing the sign of the ME field (E × H) upon cooling. (Compare solid and broken lines in the same colour.) In addition to the linear E dependence, the magnitude of ΔM[110] (or αm) strongly depends on that of (E × H) upon cooling. As |E × H| increases, αm monotonically increases. To compare the MEE effect with the MEH effect, we also show in Fig. 3b the H[110] dependence of P[1–10] measured at E[1–10]=0 MV m−1 and T=2 K after the selected preparatory ME cooling procedures. As seen in Fig. 3a, the MEH and MEE effects show a similar ME-field-cooling dependence. The ME coefficient αp (=dP/dH) obtained from the P–H curve cooled at μ0H[110]=7 T and E[1–10]=1 MV m−1 is ~0.2 ps m−1 at around 0 T. Meanwhile, αm obtained from the M–E curve for the same ME-field-cooling condition is ~0.2 ps m−1. This good agreement indicates that the origin of the MEH and MEE effects is the same, and supports the presence of net t13.

Figure 3: Comparison of ME E and ME H effect in NMTO.
figure 3

(a) E[1–10] dependence of ΔM[110]=M[110](E)−M[110](0) and (b) H[110] dependence of P[1–10] at T=2 K after ME cooling with selected E[1–10] and H[110]. (For example, ‘FC7T 1 MV m−1, denotes the data ME-cooled at μ0H[110]=7 T and E[1–10]=1 MV m−1.) Solid and broken lines represent the data taken after ME cooling with (+E, +H) and (–E, +H), respectively.

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ME memory effect

We also found that NMTO shows the ME version of ‘memory effect’, which is common in glassy systems18,19,20,21. First, we took P[1–10] by a usual measurement process: P[1–10] was taken upon warming at μ0H[110]=9 T and E[1–10]=0 MV m−1 after ME cooling at μ0H[110]=9 T and E[1–10]=1 MV m−1. The obtained P[1–10] is referred to as the reference curve (Pref) (a black solid curve in Fig. 4a). Then, we again measured P[1–10] by the same procedure, except for including a halt for th=3,600 s at an intermediate temperature Th (=6 K or 5 K) during cooling. During the halt, E[1–10] was turned off. The schematic measurement protocol is illustrated in the inset of Fig. 4b. For all the measurements, the cooling and warming rate is constant at 1 K min−1. As seen in Fig. 4a, P[1–10] after a halt on cooling (Ph) slightly decreases compared with Pref below Th. A more distinct difference is observed in between −dPref/dT and −dPh/dT (see Fig. 4b). The T dependence of −dPh/dT with Th=6 K shows a ‘dip’-like structure around Th=6 K, and −dPh/dT almost overlaps −dPref/dT below 5.5 K and above 7 K. In the data of Th=5 K, −dPh/dT clearly shows a dip around Th=5 K. We also measured th dependence of P[1–10]. As seen in Fig. 4c, P[1–10] measured at 5 K for Th=6 K shows a slow decay over th=1 h. These results clearly indicate that the long-time relaxation of P[1–10] occurs during the halts and NMTO memorizes the halt during cooling. Thus, we successfully observed the slow dynamics and memory effect by use of ME effect for the first time.

Figure 4: ME memory effect in NMTO.
figure 4

Temperature dependence of (a) P[1–10] and (b) −(dP/dT)[1–10] taken upon warming at μ0H[110]=9 T and E[1–10]=0 MV m−1. The solid black line shows a reference curve taken after continuous field cooling at μ0H[110]=9 T and E[1–10]=1 MV m−1. The open blue circles and closed red circles were measured after a halt at Th=5 K and 6 K, respectively, during the same field-cooling procedure. During the halt time, E[1–10] was turned off. (c) The th dependence of P[1–10] measured at 5 K for Th=6 K. Inset of b, schematic representation of field-cooling and warming procedure, including a halt. The upper panel shows the temporal evolution of T, and the lower panel shows those of E and H during a ME cooling and a measurement.

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Discussion

Let us discuss the cooling-(E × H) dependence of the ME effect in NMTO. According to the consideration of the free energy in a magnetic system in which spin ordering breaks the time reversal and space inversion symmetries, the free energy related to the ME effect is expressed as FME=–a (E · H) –t · [E × H] …, where the pseudoscalar a is coupled to the divergence of magnetic field, and is zero in the absence of magnetic monopoles7. This means that the free energy is stabilized by the presence of (E × H) and a toroidal moment t along (E × H). Meanwhile, in spin glass systems with replica symmetry breaking22, multi-valley structures in their free energies are considered to develop and be responsible for their characteristics, such as slow relaxations, dependence on initial conditions and remanent effects. Thus, it is reasonable to consider that one of the multi-valley states having net t is stabilized due to the presence of a certain (E × H) during the spin-freezing process and that the change in the magnitude of (E × H) during the freezing makes the system into a different valley state with a different net t.

Figure 5a–c depicts proposed schematic spin configurations of the spin-frozen states after ME cooling processes at zero, low and high ME fields (E × H), respectively. Free energy related with toroidal moment t can be expressed as Ft=−t · (E × H)7. Thus, when |E × H| is absent upon cooling, there is no energy gain by the presence of t, and then no net t develops, which means spins with an easy-plane anisotropy are randomly oriented within the plane (Fig. 5a). However, by cooling the system in the finite ME fields |E × H|, the induction of net t gives rise to energy gain (Fig. 5b). As |E × H| upon cooling increases, freezing spins are gradually oriented in a vortex-like manner, each local t polarized along the same direction, and then net t increases (Fig. 5c). This scenario well explains the observed ME-field-cooling dependence of MEE and MEH effects (Fig. 3). Figure 5d display contour plots of the magnitude of P[1–10] and ΔM[110] versus the strength of E[1–10] and H[110] upon cooling, respectively. Both ΔM[110] and P[1–10], that is, αm and αp, monotonically increase with increasing |E × H|. These plots further confirm the above scenario and demonstrate ME control of the spin-frozen state (or net t) by changing |E × H|. In addition, the observed ME-field-cooling dependence indicates that there are a number of metastable toroidal frozen states. Such huge degeneracy is discussed as the multi-valley structure in free energy for conventional spin glasses with replica symmetry breaking22 and can be observed as the cooling H dependence of M16,23. In other words, toroidal moments in NMTO appear to behave as glass. Furthermore, the ME slow dynamics, as observed for the memory effect, strongly suggest the validity of the glass nature of toroidal moment.

Figure 5: Toroidal frozen state controlled magnetoelectrically.
figure 5

(a–c) Proposed relationships among the strength of ME cooling field (E × H), the induced spin (gray arrow) configurations and the concomitant toroidal moment t in the spin-frozen state for NMTO with the honeycomb lattice. (d) Contour plot of P[1–10] measured at T=2 K, μ0H[110]=9 T, and E[1–10]=0 MV m−1 versus E[1–10] and H[110] upon the ME-field-cooling process. (e) Counter plot of ΔM[110] at T=2 K, μ0H[110]=0 T, and E[1–10]=1 MV m−1 versus E[1–10] and H[110] upon ME-field-cooling process.

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When looking at the above-mentioned ME-related free energy FME, the development of net the pseudoscalar a, which is related to magnetic monopoles, can also be expected in the presence of (E · H). However, we did not observe any ME signals after the ME-field-cooling only with finite (E · H). Hence, such a spin-frozen state may not exist in the present system, although we do not have a clear reason at this time.

Thus, we conclude that there exists a novel glass state in solids, termed ‘toroidal glass’, which is regarded as the glass of the fourth ordered form, ‘(anti)ferrotoroidic ordered state’6,7. Measurements of ME-cooling-condition dependence of magnetochiral dichroism, non-reciprocal X-ray gyrotropy and magnetic diffuse scattering can be other complementary methods to verify the toroidal glass state7,24. Other XY-like spin glass systems (for example, BaCo6Ti6O19 (ref. 25)) can be promising candidates having the toroidal glass state.

Methods

Crystal growth

A single crystal of NMTO was grown by the floating zone method, as reported previously26. First, a polycrystalline feed rod was prepared by a solid-state reaction. Stoichiometric quantities of NiO, Mn3O4 and TiO2 powders with 99.9% purity were mixed and heated at 1,000 °C for 10 h in air. The resulting polycrystalline sample was ground into powders again, and then pressed into a rod with the dimension of about 6 mm in diameter and 100 mm in length. The rod was sintered again at 1,100 °C for 15 h in air. The crystal growth was carried out on the sintered rod with the use of a halogen-lamp image furnace in flowing air at a rate of 500 ml min−1. The growth rate was 1–3 mm h−1. Thus, a single crystal with clear facets parallel to the c-plane (hexagonal setting) was obtained.

Measurements of ME effects

The obtained crystal was oriented with use of Laue X-ray diffraction patterns and was cut into two rectangular plates with the widest faces parallel to the (1–10) plane. The dimensions of these specimens were 17.5 mm2 × 0.1 mm and 18.5 mm2 × 0.4 mm for measurements of electric polarization (P) and magnetization (M), respectively. Silver electrodes were vacuum-deposited onto the widest faces of these specimens. To obtain P parallel to [1–10] (P[1–10]) as functions of temperature (T) and magnetic field (H) applied along [110] (H[110]), pyroelectric and ME currents were measured during T- and H-sweeping procedures, respectively, in a superconducting magnet (Quantum Design, PPMS) by using an electrometer (Keithley Instruments 6517A). To measure M along [110] (M[110]) as a function of electric field (E) applied along [1–10] (E[1–10]), a home-made insert, which allows the application of E during M measurements was installed in a magnetometer (Quantum Design, MPMS SQUID VSM). On the home-made insert, the silver electrodes vacuum-deposited onto the sample were electrically connected with an external voltage source (Keithley Instruments 6517A) via copper wires. All the ME measurements were done after ME-field-cooling procedures at selected E[1–10] and H[110].

Additional information

How to cite this article: Yamaguchi, Y. & Kimura, T. Magnetoelectric control of frozen state in a toroidal glass. Nat. Commun. 4:2063 doi: 10.1038/ncomms3063 (2013).