PRL 113, 147202 (2014)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
Spin-Chirality-Driven Ferroelectricity on a Perfect Triangular Lattice Antiferromagnet H. Mitamura,1,* R. Watanuki,2 K. Kaneko,3 N. Onozaki,2 Y. Amou,2 S. Kittaka,1 R. Kobayashi,1,4 Y. Shimura,1 I. Yamamoto,2 K. Suzuki,2 S. Chi,4 and T. Sakakibara1 1
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan 2 Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan 3 Quantum Beam Science Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan 4 Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Received 24 February 2014; revised manuscript received 20 June 2014; published 1 October 2014) Magnetic field (B) variation of the electrical polarization Pc (∥c) of the perfect triangular lattice antiferromagnet RbFeðMoO4 Þ2 is examined up to the saturation point of the magnetization for B⊥c. Pc is observed only in phases for which chirality is predicted in the in-plane magnetic structures. No strong anomaly is observed in Pc at the field at which the spin modulation along the c axis, and hence the spin helicity, exhibits a discontinuity to the commensurate state. These results indicate that the ferroelectricity in this compound originates predominantly from the spin chirality, the explanation of which would require a new mechanism for magnetoferroelectricity. The obtained field-temperature phase diagram of ferroelectricity agree well with those theoretically predicted for the spin chirality of a Heisenberg spin triangular lattice antiferromagnet. DOI: 10.1103/PhysRevLett.113.147202
PACS numbers: 75.85.+t, 75.50.Ee, 77.80.-e
The ground state of a triangular lattice antiferromagnet (TLAFM) is known to be a “120° structure”. This spin configuration can be viewed as a staggered order of triangular-spin chirality (hereafter “chirality”) σ t ¼ n123 · ðS1 × S2 þ S2 × S3 þ S3 × S1 Þ=S2 , where the unit vector n123 normal to the triangle plane is defined according to the right-hand screw rule. In particular, the in-plane 120° structure of a planar TLAFM is known to be doubly degenerate with respect to the chirality [1]. Whereas the chirality is microscopically distinguishable by a polarized neutron scattering experiment, whether or not the chirality controls the macroscopic physical properties of TLAFMs remains unanswered. In this Letter, we provide clear evidence that the chirality can indeed induce ferroelectricity. Recently, magnetoferroelectricity (MFE), which deals with the phenomenon of the simultaneous appearance of electrical polarization and magnetic ordering [2], has been observed in some TLAFMs with an electrical polarization P perpendicular to the spin rotational plane and discussed in the context of inversion symmetry breaking by chirality [3–9]. In CuCrO2 [5], P is induced parallel to the triangular plane in the out-of-plane 120° ordering state below T N ¼ 23.6 K [6]. However, in this compound, magnetoelastic coupling induces a lattice distortion [10], and the threefold lattice symmetry is broken in the ordered state. Thus, the spin structure becomes incommensurate [11,12], and MFE can probably be explained in the same framework as proper helical multiferroics [12,13]. On the other hand, RbFeðMoO4 Þ2 (RFMO) is extremely interesting because MFE appears in an in-plane 120° spin structure [3], for which none of the existing microscopic models [13–17] predict the appearance of electrical polarization. 0031-9007=14=113(14)=147202(5)
In RFMO, Fe3þ ions (spin 5=2 with zero orbital angular moment) form equilateral triangular lattice planes, each separated by a MoO4 -Rb-MoO4 layer [Fig. 1(a)] [18]. The crystal structure at room temperature belongs to space ¯ group P3m1 [Fig. 1(b)] [19]. Below 190 K (T s ), the symmetry is lowered to P3¯ by a lattice distortion (rotation of the MoO4 pyramids) that preserves the threefold axis [Fig. 1(c)] [19]. This compound orders antiferromagnetically below T N ¼ 3.9 K, becoming ferroelectric at the same temperature [3]. According to neutron diffraction measurements under zero magnetic field, the ordered moments are confined to the c plane and have the wave vector q ¼ ð1=3; 1=3; 0.458Þ [3]; the magnetic structure is an in-plane 120° structure with helicity along the c axis. Since no evidence of threefold symmetry breaking has been observed [19], the electrical polarization (Pc ) must be parallel to the c axis. In previous pyroelectric measurements [3], it has been reported that Pc ðTÞ is proportional to (a)
(b)
(c)
Fe Rb O4 MoO
FIG. 1 (color online). (a) Crystal structure of RbFeðMoO4 Þ2 (RFMO). Fe sites form simple-stacking perfect triangular lattice layers. Each Fe-Fe nearest-neighbor bond is equivalent. (b) Top ¯ view above T s (space group P3m1). (c) Top view below T s (space ¯ where threefold symmetry remains. Arrows indicate group P3), the 120° spin structure below T N with positive (þ) and negative (−) chirality.
147202-1
© 2014 American Physical Society
PRL 113, 147202 (2014)
PHYSICAL REVIEW LETTERS
the zero-field temperature variation of the magnetic Bragg intensity at (1=3, 1=3, 0.458). When an external magnetic field B is applied along the a axis at 1.5 K, five phases P1–P5 successively appear with increasing B [Fig. 3(a)], before reaching a paramagnetic saturation near 20 T [20–22]. All these phases are characterized by the wave vector (1=3, 1=3, qz ), where qz ≈ 0.458 for P1, 1=3 for P3-P4, and qz ≈ 0.44 for P5 [3,23]. Phase P3 was found to be paraelectric; however, P2, occurring in a narrow field region below P3 [20–22], was not resolved in previous neutron and pyroelectric measurements [3]. On the other hand, a NMR experiment [24] has reported P2 to be a commensurate phase. Accordingly, it has remained controversial whether the switch of qz from 0.458 to 1=3 occurs at the P1-P2 transition or the incommensurate state continues until the P3 phase appears. Owing to weak interlayer magnetic interactions 100 times smaller than the in-plane nearest-neighbor interaction [22], RFMO is considered to be a rather good example of a two-dimensional TLAFM. Hence, the magnetic phases of RFMO can be intimately related to those of the 2D classical Heisenberg spin TLAFM model [25]. P1 and P2 correspond to the low-field noncollinear state (phase I). P3 can be assigned to a collinear phase II with a 1=3-magnetization plateau, while P4 and P5 correspond to a high-field noncollinear state (phase III). According to theoretical predictions, only P1 and P2 have finite chirality. In previous work, MFE was discussed on the basis of a phenomenological model according to which the chirality in RFMO itself breaks the inversion symmetry [3,4]. Later on, however, it was pointed out that the spin helicity along the c axis may also contribute to MFE [8]. The magnetoelectric free energy takes the form Pz ðc1 vσ t þ c2 Aσ h Þ [8], where σ h ¼ rab · ðSa × Sb Þ denotes the helicity (rab is a unit vector connecting spins Sa and Sb along the c axis), and v and A are the z components of the polar and axial vectors v ¼ ð0; 0; vÞ and A ¼ ð0; 0; AÞ, respectively, arising from the crystal structure. Note that σ t in Ref. [8] is equal to vσ t in the present definition. Indeed, σ t and σ h are coupled together in this lattice structure, and it was demonstrated that both switch simultaneously when Pc is reversed by an electric field [8]. As pointed out [8], whether σ t or σ h contributes to MFE more remains undetermined. In order to find out, we examine the field variation of qz and Pc of RFMO in more detail. In particular, we focus on the field-induced P1-P2 transition. If these two phases differ from each other only in their z component of the wave vector qz as suggested by the NMR experiment [24], then only σ h exhibits a large discontinuity across the transition while σ t remains unchanged. In that case, knowing whether Pc ðBÞ exhibits a strong anomaly across the P1-P2 transition is crucial in determining whether σ t or σ h is more important for MFE in this system. Single crystals of RFMO were grown in a platinum crucible. The as-grown crystals had a thin hexagonal shape
week ending 3 OCTOBER 2014
with flat surfaces in the c plane. The typical dimensions were ∼7.5 × 6 × 0.3 mm3 for neutron diffraction and ∼3 × 3 × 0.1 mm3 for other measurements. Elastic neutron scattering measurements were performed in vertical fields ¯ up to 5 T using the wide-angle neutron along ½110 diffractometer (WAND) installed at the high-flux isotope reactor at Oak Ridge National Laboratory (ORNL). Pc and the magnetization M were measured by a nondestructive pulsed magnet system in the Faculty of Engineering, Yokohama National University (YNU). The pulse width was approximately 15 ms. We employed an inductive and a pyroelectric method to measure MðBÞ and Pc ðBÞ [26], respectively. The electrodes for measuring the pyroelectric currents I pyro were made from silver paste on either side of the largest flat surfaces of the sample. Since RFMO has relatively high electrical conductivity, it is very difficult to measure I pyro under a quasistatic magnetic field owing to a leakage current I leak across the sample. Nevertheless, the short measurement time of the pulsed magnetic field makes it possible to measure I pyro . In the pyroelectric method, the total current measured (I total ) is the sum of I pyro and I leak . Under a time-varying magnetic field, I pyro can be expressed as I pyro ¼ dPc =dB⋅dB=dt, which can be much larger than I leak because of a large dB=dt, 106 times as large as the typical value for an ordinary superconducting magnet. Temperature variation of the dielectric constant ϵ was measured using a digital capacitance bridge (AndeenHaagerling 2500 A) in a steady magnetic field up to 14 T produced by a superconducting magnet. dc magnetization was measured by a force magnetometer at temperatures down to 0.25 K [27] and a SQUID magnetometer (S700X-R, Cryogenic Ltd., installed in YNU). Specific heat was measured by a semiadiabatic method in magnetic fields of up to 8.5 T. Neutron scattering intensity maps in the (h, h, l) scattering plane, measured at T ¼ 2 K, are shown in Fig. 2 for [2(a)] zero magnetic field and [2(b)] B ¼ ¯ direction. For B ¼ 0, strong 3.8 T applied along the ½110 Bragg peaks were observed at (1=3, 1=3, l), with l ¼ 1.45 and 1.55, in agreement with the reported ordering vector of q ¼ ð1=3; 1=3; qz Þ with qz ¼ 0.458 [3] or 0.448 [19]. At B ¼ 3.8 T, the peak position moves along l to be l ≈ 1.33 and 1.66, indicating that the ordering vector of the P2 phase is (1=3, 1=3, 1=3). Figure 2(c) shows the field variations of ¯ obtained at T ¼ 2 K. qz stays at a zero-field qz in B∥½110 value of ∼0.45 up to 2 T, and decreases gradually down to 0.41 on approaching the P1-P2 boundary above 3 T. We found that qz discontinuously jumps to the value 1=3 at ∼3.7 T, close to the P1-P2 phase boundary (vertical solid line) determined from the magnetization measurements in Fig. S1(c) of the Supplemental Material [28]. On the other hand, no anomaly is observed at the P2-P3 phase boundary near 4.5 T (vertical dotted line). These results provide clear evidence that the qz jump occurs at the P1-P2 phase boundary.
147202-2
PRL 113, 147202 (2014)
12
0.40 2K B || [1-10]
1.3 10 0.30 0.33 0.36 0.30 0.33 0.36
(hh0) [r. l. u.]
qz [r. l. u.]
20
0.35 1/3 0.30
4
6
3
0
30
30 1.4
8
4
2 para.
2
0 P1 P2 P3
1
2 3 B [T]
4
5
FIG. 2 (color online). Contour plot of the magnetic Bragg intensity in the (h, h, l) scattering plane for (a) P1 phase (B ¼ 0) ¯ and (b) P2 phase at B ¼ 3.8 T (B∥½110). (c) Field variation of qz (magnetic periodicity along c) in the ordering vector (1=3, 1=3, qz ). Vertical solid line (dotted line) indicates the P1-P2 (P2-P3) phase boundary. All the data were taken at T ¼ 2 K.
In Fig. 3(a), Pc and M at 1.5 K are given as functions of B applied along the a axis. The data confirm that P1 and P2 are ferroelectric, whereas the polarization disappears in P3, in agreement with previous results [3]. We observe that P4 and P5 are nonferroelectric, in line with the presumption that P4 and P5 are noncollinear phases with no chirality [25]. Note that a clear 1=3-magnetization plateau is observed in P3, where the spins are expected to align in a collinear up-up-down state. We now examine the P1-P2 transition in detail. In Fig. 3(b), we show Pc ðBÞ and its field derivative dPc =dB in the low-field region (< 6 T). The data show that Pc ðBÞ is quasicontinuous across the transition. A tiny anomaly is observed in dPc =dB near 3.8 T, confirming the occurrence of the P1-P2 transition. However, this anomaly is significantly smaller than the huge negative peak observed at the P2-P3 transition at ∼4.8 T. In view of the fact that a significant jump has been observed in the qz value from 0.41 to 1=3 across the P1-P2 transition, our experimental data strongly suggest that the qz value is irrelevant to MFE in this system. In order to show this point more quantitatively, we estimate the field variation of Pc by using simple models. First, we assume that σ t , which is independent of qz , is the predominant factor contributing to MFE ½Pc ðBÞ ∝ σ t . The field variation of σ t can be roughly estimated by assuming a three-sublattice structure with a symmetrical spin arrangement with respect to B∥a [see Fig. 3(a), inset], and the resultant Pc ðBÞ is shown by a dot-dashed line in Fig. 3(b) (chirality model). The result well reproduces the experimentally obtained Pc ðBÞ. Next, we consider the case where σ h is the predominant factor contributing to MFE ½Pc ðBÞ ∝ σ h . It is difficult to evaluate precisely the field variation of σ h in the P1 phase except for B ¼ 0 because of incommensurability (qz ≈ 0.45). On the other hand, the field variation of the
20
helicity model
-20
1
0
0
-10 1.5 K E || c, B || a
P4
P5
-2
-1
-4
-2 20
0
5
10 B [T]
15
Pc [µC/m2]
1.5
0.45
(b) 40
5
10
dPc/dB [µC/m2T]
40
10
(c)
6
(a)
1.5 K E || c, B || a
M [µB/Fe]
(00l) [r. l. u.]
50 1.6
Intensity [arb. units]
1.7
0.50
Pc [µC/m2]
(b)
-30 chirality model
0 0
P1 (IC) 1 2
3 B [T]
-40 P2 P3 (C) (C) 4 5 6
FIG. 3 (color online). (a) Magnetic field variation of the magnetization M and the c component of the electric polarization Pc of RFMO for B∥ [110] (a axis). Insets indicate the expected three-sublattice spin structures in the basal plane. (b) Pc (open circles) and its field derivative dPc =dB (open triangles) in the low-field region measured under a pulsed magnetic field. Only the field-increasing data are shown for clarity. The upward arrow for dPc =dB near 3.8 T indicates a tiny anomaly observed at the P1-P2 transition. The dot-dashed line indicates the expected variation of Pc ðBÞ assuming Pc ∝ σ t (chirality model), whereas the dashed line represents the case of Pc ∝ σ h (helicity model). The latter exhibits a large jump across the P1-P2 transition where a jump in qz from ∼0.41 to 1=3 is observed. The zero-field extrapolation of the helicity with qz fixed to 1=3 is 2.8 times larger than that for qz ≈ 0.45 (P1), as indicated by the dotted line.
P2 phase (qz ¼ 1=3) is easily estimated by assuming a three-sublattice structure along the c axis, similar to the one assumed for the in-plane structure in P1 and P2. Note that σ h in P1 should be much smaller than in P2 owing to the nearly 180° rotations of spins along the c axis [Fig. 4(b)]. Thus, the ratio of the zero-field-extrapolated value of σ h in the P2 phase (qz fixed to 1=3) to that in the P1 10
(a)
(b) B
B || a
c axis
P4
8
P3
6
B [T]
(a)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
P2 4
P2
2
P1 para.
0
0
1
2
3
4
T [K]
5
6
P1 7
8
FIG. 4 (color online). (a) Phase diagram of RFMO for B∥ [110] (a axis) determined from Pc ðBÞ, ϵðT; BÞ, MðT; BÞ, and specific heat CðT; BÞ measurements. The highlighted regions indicate the ferroelectric phases (P1 and P2). Insets depict the expected inplane spin structures. (b) Spin arrangements along the c axis for P1 and P2 phases.
147202-3
PRL 113, 147202 (2014)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
TABLE I. Symmetry relations relevant for the electrical polarization parallel to the c direction in TLAFMs. The state (lattice and spin) is invariant (○) or noninvariant (×) for each operation. All three lattice structures listed here are nonferroelectric in the paramagnetic state because there is at least one symmetry operation that makes the þz direction equivalent to −z. When the magnetic order sets in, chirality or helicity completely breaks the c-axis inversion symmetry, thereby realizing MFE. For ¯ the space groups P3m1 and P321, only chirality has this property (see column C2x ). mz
Symmetry operations S3 C2y i S6
C2x
Lattice
P3¯ ¯ P3m1 P321
× × ×
× × ×
× × ×
○ ○ ×
○ ○ ×
× ○ ○
Spin
Chirality Helicity
○ ×
○ ×
○ ○
× ×
× ×
× ○
phase (B ¼ 0) is large, namely, ∼2.8 ½≈ sinð2π=3Þ= sinð2π × 0.45Þ. Accordingly, if the MFE arises solely from the helicity, then a large jump in Pc ðBÞ must be observed at the P1-P2 transition, as shown by a dashed line in Fig. 3(b) (helicity model). The predicted discontinuity in Pc is 13 μC=m2 , considerably larger than the observed value of −0.2 μC=m2 . Therefore, we may rule out the possibility that MFE is driven by the helicity along the c axis. Figure 4(a) shows the phase diagram of RFMO for B∥a determined from the present MðT; BÞ, Pc ðT; BÞ, ϵðT; BÞ, and specific heat measurements (see Supplemental Material [28]). The highlighted phases (P1 and P2) are ferroelectric. The phase diagram is essentially the same as that previously reported [3,22], although the P1-P2 boundary is missing in Fig. 3 of Ref. [3]. Comparing this phase diagram with that for classical Heisenberg spin TLAFMs [25,29], one finds that all the ferroelectric (nonferroelectric) phases correspond to the states with (without) in-plane chirality. These findings, along with the independence of Pc from the c-axis helicity, provide strong evidence that the MFE in this compound is induced by chirality, and, to the best of our knowledge, constitute the first experimental proof for the field variation of spin chirality in classical spin TLAFMs [25]. Table I summarizes the symmetry relations of MFE in triangular lattices. There are up to six symmetry operations relevant to ferroelectricity along the c axis, namely, mz , reflection with respect to the (001) plane; S3 , threefold ¯ rotatory reflection; C2y , twofold rotation around ½110 (y axis); i, inversion with respect to the Fe site; S6 , sixfold rotatory reflection; and C2x , twofold rotation around ½110 ¯ (x axis) [9]. When the lattice symmetry is lowered to P3, two symmetry operations i and S6 ensure that the system remains nonferroelectric. In the RFMO crystal, a MoO2− 4 cluster is situated asymmetrically on either side of a unit triangle. Accordingly, the local mz symmetry is broken, and
(a)
(b)
FIG. 5 (color online). Local electrical polarization in RFMO. (a) Paramagnetic state. Thick vertical arrows indicate local electrical polarizations due to the asymmetric arrangement of the MoO4 pyramids with respect to the triangular plane. As i or S6 lattice symmetry ensures equal magnitude of up- and downpointing polarizations, no macroscopic polarization appears. (b) Positive (left triangle) and negative (right triangle) spin chirality may induce a change in the magnitude of the local electrical polarization, leading to a macroscopic polarization.
one may assume that a finite local electrical polarization exists perpendicular to the plane [Fig. 5(a)]. Because of the i or S6 symmetry of the lattice, the upward-pointing local polarizations exactly cancel out the downward-pointing ones; hence, there is no macroscopic polarization in the paramagnetic state. Chirality breaks the remaining i and S6 symmetry, thus allowing ferroelectricity. Microscopically, the appearance of Pc suggests that chirality induces a difference in the amplitude of the local electrical polarization on the unit triangle [Fig. 5(b)]. It should be noted that P3¯ is not a unique structure for the occurrence of MFE in TLAFMs. Symmetry considerations ¯ imply that the space groups P3m1 and P321 can also yield MFE (Table I). In these two space groups, the spin helicity along the c axis does not induce MFE. Thus, it would be very easy to probe the spin-chirality-driven MFE in these systems. We also note a possible MFE in kagome-lattice antiferromagnets. Amongpthe ffiffiffi various pffiffiffi allowed ground-state spin configurations, the 3 × 3 spin structure [30] may exhibit MFE if the crystal lattice belongs to the space group ¯ P3m1, ¯ P3, or P321. In summary, we have demonstrated that an electrical polarization perpendicular to the triangular plane appears in RbFeðMoO4 Þ2 in those phases for which in-plane triangular-spin chirality is predicted. On the other hand, no strong anomaly is observed in polarization at the field where the c axis modulation of the spin structure qz exhibit a jump. Thus, we conclude that magnetoelectricity in this compound is driven predominantly by triangular-spin chirality. Similar chirality-driven ferroelectricity can be expected for ¯ the structures P3m1 and P321. We thank T. Inami and T. Waki for critical discussions and useful technical advice. The work at ORNL was supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. WAND is operated jointly by ORNL and Japan Atomic Energy Agency under US-Japan Cooperative Program on Neutron Scattering. The present study was supported by a Grant-in-Aid for Scientific Research C (No. 26400329).
147202-4
PRL 113, 147202 (2014) *
PHYSICAL REVIEW LETTERS
[email protected]‑tokyo.ac.jp [1] S. Miyashita and H. Shiba, J. Phys. Soc. Jpn. 53, 1145 (1984). [2] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature (London) 426, 55 (2003). [3] M. Kenzelmann, G. Lawes, A. B. Harris, G. Gasparovic, C. Broholm, A. P. Ramirez, G. A. Jorge, M. Jaime, S. Park, Q. Huang, A. Y. Shapiro, and L. A. Demianets, Phys. Rev. Lett. 98, 267205 (2007). [4] A. B. Harris, Phys. Rev. B 76, 054447 (2007). [5] S. Seki, Y. Onose, and Y. Tokura, Phys. Rev. Lett. 101, 067204 (2008). [6] K. Kimura, H. Nakamura, K. Ohgushi, and T. Kimura, Phys. Rev. B 78, 140401(R) (2008). [7] A. B. Harris, A. Aharony, and O. Entin-Wohlman, J. Phys. Condens. Matter 20, 434202 (2008). [8] A. J. Hearmon, F. Fabrizi, L. C. Chapon, R. D. Johnson, D. Prabhakaran, S. V. Streltsov, P. J. Brown, and P. G. Radaelli, Phys. Rev. Lett. 108, 237201 (2012). [9] H. Mitamura, R. Watanuki, N. Onozaki, Y. Shimura, S. Kittaka, T. Sakakibara, and K. Suzuki, J. Phys. Conf. Ser. 391, 012099 (2012). [10] K. Kimura, T. Otani, H. Nakamura, Y. Wakabayashi, and T. Kimura, J. Phys. Soc. Jpn. 78, 113710 (2009). [11] H. Kawamura, Prog. Theor. Phys. Suppl. 101, 545 (1990). [12] M. Soda, K. Kimura, T. Kimura, M. Matsuura, and K. Hirota, J. Phys. Soc. Jpn. 78, 124703 (2009). [13] T. Arima, J. Phys. Soc. Jpn. 76, 073702 (2007). [14] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005). [15] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006). [16] I. A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006).
week ending 3 OCTOBER 2014
[17] C. Jia, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B 74, 224444 (2006). [18] T. Inami, Y. Ajiro, and T. Goto, J. Phys. Soc. Jpn. 65, 2374 (1996). [19] T. Inami, J. Solid State Chem. 180, 2075 (2007). [20] L. E. Svistov, A. I. Smirnov, L. A. Prozorova, O. A. Petrenko, L. N. Demianets, and A. Y. Shapiro, Phys. Rev. B 67, 094434 (2003). [21] L. E. Svistov, A. I. Smirnov, L. A. Prozorova, O. A. Petrenko, A. Micheler, N. Büttgen, A. Y. Shapiro, and L. N. Demianets, Phys. Rev. B 74, 024412 (2006). [22] A. I. Smirnov, H. Yashiro, S. Kimura, M. Hagiwara, Y. Narumi, K. Kindo, A. Kikkawa, K. Katsumata, A. Y. Shapiro, and L. N. Demianets, Phys. Rev. B 75, 134412 (2007). [23] J. S. White, C. Niedermayer, G. Gasparovic, C. Broholm, J. M. S. Park, A. Y. Shapiro, L. A. Demianets, and M. Kenzelmann, Phys. Rev. B 88, 060409(R) (2013). [24] L. E. Svistov, L. A. Prozorova, N. Büttgen, A. Y. Shapiro, and L. N. Dem’yanets, JETP Lett. 81, 102 (2005). [25] H. Kawamura and S. Miyashita, J. Phys. Soc. Jpn. 54, 4530 (1985). [26] H. Mitamura, S. Mitsuda, S. Kanetsuki, H. A. Katori, T. Sakakibara, and K. Kindo, J. Phys. Soc. Jpn. 76, 094709 (2007). [27] T. Sakakibara, H. Mitamura, T. Tayama, and H. Amitsuka, Jpn. J. Appl. Phys. 33, 5067 (1994). [28] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.113.147202 for the results of magnetization, dielectric constant, specific heat, and electric polarization measurements. [29] S. Watarai, S. Miyashita, and H. Shiba, J. Phys. Soc. Jpn. 70, 532 (2001). [30] A. Chubukov, Phys. Rev. Lett. 69, 832 (1992).
147202-5
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
Spin-Chirality-Driven Ferroelectricity on a Perfect Triangular Lattice Antiferromagnet H. Mitamura,1,* R. Watanuki,2 K. Kaneko,3 N. Onozaki,2 Y. Amou,2 S. Kittaka,1 R. Kobayashi,1,4 Y. Shimura,1 I. Yamamoto,2 K. Suzuki,2 S. Chi,4 and T. Sakakibara1 1
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan 2 Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan 3 Quantum Beam Science Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan 4 Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Received 24 February 2014; revised manuscript received 20 June 2014; published 1 October 2014) Magnetic field (B) variation of the electrical polarization Pc (∥c) of the perfect triangular lattice antiferromagnet RbFeðMoO4 Þ2 is examined up to the saturation point of the magnetization for B⊥c. Pc is observed only in phases for which chirality is predicted in the in-plane magnetic structures. No strong anomaly is observed in Pc at the field at which the spin modulation along the c axis, and hence the spin helicity, exhibits a discontinuity to the commensurate state. These results indicate that the ferroelectricity in this compound originates predominantly from the spin chirality, the explanation of which would require a new mechanism for magnetoferroelectricity. The obtained field-temperature phase diagram of ferroelectricity agree well with those theoretically predicted for the spin chirality of a Heisenberg spin triangular lattice antiferromagnet. DOI: 10.1103/PhysRevLett.113.147202
PACS numbers: 75.85.+t, 75.50.Ee, 77.80.-e
The ground state of a triangular lattice antiferromagnet (TLAFM) is known to be a “120° structure”. This spin configuration can be viewed as a staggered order of triangular-spin chirality (hereafter “chirality”) σ t ¼ n123 · ðS1 × S2 þ S2 × S3 þ S3 × S1 Þ=S2 , where the unit vector n123 normal to the triangle plane is defined according to the right-hand screw rule. In particular, the in-plane 120° structure of a planar TLAFM is known to be doubly degenerate with respect to the chirality [1]. Whereas the chirality is microscopically distinguishable by a polarized neutron scattering experiment, whether or not the chirality controls the macroscopic physical properties of TLAFMs remains unanswered. In this Letter, we provide clear evidence that the chirality can indeed induce ferroelectricity. Recently, magnetoferroelectricity (MFE), which deals with the phenomenon of the simultaneous appearance of electrical polarization and magnetic ordering [2], has been observed in some TLAFMs with an electrical polarization P perpendicular to the spin rotational plane and discussed in the context of inversion symmetry breaking by chirality [3–9]. In CuCrO2 [5], P is induced parallel to the triangular plane in the out-of-plane 120° ordering state below T N ¼ 23.6 K [6]. However, in this compound, magnetoelastic coupling induces a lattice distortion [10], and the threefold lattice symmetry is broken in the ordered state. Thus, the spin structure becomes incommensurate [11,12], and MFE can probably be explained in the same framework as proper helical multiferroics [12,13]. On the other hand, RbFeðMoO4 Þ2 (RFMO) is extremely interesting because MFE appears in an in-plane 120° spin structure [3], for which none of the existing microscopic models [13–17] predict the appearance of electrical polarization. 0031-9007=14=113(14)=147202(5)
In RFMO, Fe3þ ions (spin 5=2 with zero orbital angular moment) form equilateral triangular lattice planes, each separated by a MoO4 -Rb-MoO4 layer [Fig. 1(a)] [18]. The crystal structure at room temperature belongs to space ¯ group P3m1 [Fig. 1(b)] [19]. Below 190 K (T s ), the symmetry is lowered to P3¯ by a lattice distortion (rotation of the MoO4 pyramids) that preserves the threefold axis [Fig. 1(c)] [19]. This compound orders antiferromagnetically below T N ¼ 3.9 K, becoming ferroelectric at the same temperature [3]. According to neutron diffraction measurements under zero magnetic field, the ordered moments are confined to the c plane and have the wave vector q ¼ ð1=3; 1=3; 0.458Þ [3]; the magnetic structure is an in-plane 120° structure with helicity along the c axis. Since no evidence of threefold symmetry breaking has been observed [19], the electrical polarization (Pc ) must be parallel to the c axis. In previous pyroelectric measurements [3], it has been reported that Pc ðTÞ is proportional to (a)
(b)
(c)
Fe Rb O4 MoO
FIG. 1 (color online). (a) Crystal structure of RbFeðMoO4 Þ2 (RFMO). Fe sites form simple-stacking perfect triangular lattice layers. Each Fe-Fe nearest-neighbor bond is equivalent. (b) Top ¯ view above T s (space group P3m1). (c) Top view below T s (space ¯ where threefold symmetry remains. Arrows indicate group P3), the 120° spin structure below T N with positive (þ) and negative (−) chirality.
147202-1
© 2014 American Physical Society
PRL 113, 147202 (2014)
PHYSICAL REVIEW LETTERS
the zero-field temperature variation of the magnetic Bragg intensity at (1=3, 1=3, 0.458). When an external magnetic field B is applied along the a axis at 1.5 K, five phases P1–P5 successively appear with increasing B [Fig. 3(a)], before reaching a paramagnetic saturation near 20 T [20–22]. All these phases are characterized by the wave vector (1=3, 1=3, qz ), where qz ≈ 0.458 for P1, 1=3 for P3-P4, and qz ≈ 0.44 for P5 [3,23]. Phase P3 was found to be paraelectric; however, P2, occurring in a narrow field region below P3 [20–22], was not resolved in previous neutron and pyroelectric measurements [3]. On the other hand, a NMR experiment [24] has reported P2 to be a commensurate phase. Accordingly, it has remained controversial whether the switch of qz from 0.458 to 1=3 occurs at the P1-P2 transition or the incommensurate state continues until the P3 phase appears. Owing to weak interlayer magnetic interactions 100 times smaller than the in-plane nearest-neighbor interaction [22], RFMO is considered to be a rather good example of a two-dimensional TLAFM. Hence, the magnetic phases of RFMO can be intimately related to those of the 2D classical Heisenberg spin TLAFM model [25]. P1 and P2 correspond to the low-field noncollinear state (phase I). P3 can be assigned to a collinear phase II with a 1=3-magnetization plateau, while P4 and P5 correspond to a high-field noncollinear state (phase III). According to theoretical predictions, only P1 and P2 have finite chirality. In previous work, MFE was discussed on the basis of a phenomenological model according to which the chirality in RFMO itself breaks the inversion symmetry [3,4]. Later on, however, it was pointed out that the spin helicity along the c axis may also contribute to MFE [8]. The magnetoelectric free energy takes the form Pz ðc1 vσ t þ c2 Aσ h Þ [8], where σ h ¼ rab · ðSa × Sb Þ denotes the helicity (rab is a unit vector connecting spins Sa and Sb along the c axis), and v and A are the z components of the polar and axial vectors v ¼ ð0; 0; vÞ and A ¼ ð0; 0; AÞ, respectively, arising from the crystal structure. Note that σ t in Ref. [8] is equal to vσ t in the present definition. Indeed, σ t and σ h are coupled together in this lattice structure, and it was demonstrated that both switch simultaneously when Pc is reversed by an electric field [8]. As pointed out [8], whether σ t or σ h contributes to MFE more remains undetermined. In order to find out, we examine the field variation of qz and Pc of RFMO in more detail. In particular, we focus on the field-induced P1-P2 transition. If these two phases differ from each other only in their z component of the wave vector qz as suggested by the NMR experiment [24], then only σ h exhibits a large discontinuity across the transition while σ t remains unchanged. In that case, knowing whether Pc ðBÞ exhibits a strong anomaly across the P1-P2 transition is crucial in determining whether σ t or σ h is more important for MFE in this system. Single crystals of RFMO were grown in a platinum crucible. The as-grown crystals had a thin hexagonal shape
week ending 3 OCTOBER 2014
with flat surfaces in the c plane. The typical dimensions were ∼7.5 × 6 × 0.3 mm3 for neutron diffraction and ∼3 × 3 × 0.1 mm3 for other measurements. Elastic neutron scattering measurements were performed in vertical fields ¯ up to 5 T using the wide-angle neutron along ½110 diffractometer (WAND) installed at the high-flux isotope reactor at Oak Ridge National Laboratory (ORNL). Pc and the magnetization M were measured by a nondestructive pulsed magnet system in the Faculty of Engineering, Yokohama National University (YNU). The pulse width was approximately 15 ms. We employed an inductive and a pyroelectric method to measure MðBÞ and Pc ðBÞ [26], respectively. The electrodes for measuring the pyroelectric currents I pyro were made from silver paste on either side of the largest flat surfaces of the sample. Since RFMO has relatively high electrical conductivity, it is very difficult to measure I pyro under a quasistatic magnetic field owing to a leakage current I leak across the sample. Nevertheless, the short measurement time of the pulsed magnetic field makes it possible to measure I pyro . In the pyroelectric method, the total current measured (I total ) is the sum of I pyro and I leak . Under a time-varying magnetic field, I pyro can be expressed as I pyro ¼ dPc =dB⋅dB=dt, which can be much larger than I leak because of a large dB=dt, 106 times as large as the typical value for an ordinary superconducting magnet. Temperature variation of the dielectric constant ϵ was measured using a digital capacitance bridge (AndeenHaagerling 2500 A) in a steady magnetic field up to 14 T produced by a superconducting magnet. dc magnetization was measured by a force magnetometer at temperatures down to 0.25 K [27] and a SQUID magnetometer (S700X-R, Cryogenic Ltd., installed in YNU). Specific heat was measured by a semiadiabatic method in magnetic fields of up to 8.5 T. Neutron scattering intensity maps in the (h, h, l) scattering plane, measured at T ¼ 2 K, are shown in Fig. 2 for [2(a)] zero magnetic field and [2(b)] B ¼ ¯ direction. For B ¼ 0, strong 3.8 T applied along the ½110 Bragg peaks were observed at (1=3, 1=3, l), with l ¼ 1.45 and 1.55, in agreement with the reported ordering vector of q ¼ ð1=3; 1=3; qz Þ with qz ¼ 0.458 [3] or 0.448 [19]. At B ¼ 3.8 T, the peak position moves along l to be l ≈ 1.33 and 1.66, indicating that the ordering vector of the P2 phase is (1=3, 1=3, 1=3). Figure 2(c) shows the field variations of ¯ obtained at T ¼ 2 K. qz stays at a zero-field qz in B∥½110 value of ∼0.45 up to 2 T, and decreases gradually down to 0.41 on approaching the P1-P2 boundary above 3 T. We found that qz discontinuously jumps to the value 1=3 at ∼3.7 T, close to the P1-P2 phase boundary (vertical solid line) determined from the magnetization measurements in Fig. S1(c) of the Supplemental Material [28]. On the other hand, no anomaly is observed at the P2-P3 phase boundary near 4.5 T (vertical dotted line). These results provide clear evidence that the qz jump occurs at the P1-P2 phase boundary.
147202-2
PRL 113, 147202 (2014)
12
0.40 2K B || [1-10]
1.3 10 0.30 0.33 0.36 0.30 0.33 0.36
(hh0) [r. l. u.]
qz [r. l. u.]
20
0.35 1/3 0.30
4
6
3
0
30
30 1.4
8
4
2 para.
2
0 P1 P2 P3
1
2 3 B [T]
4
5
FIG. 2 (color online). Contour plot of the magnetic Bragg intensity in the (h, h, l) scattering plane for (a) P1 phase (B ¼ 0) ¯ and (b) P2 phase at B ¼ 3.8 T (B∥½110). (c) Field variation of qz (magnetic periodicity along c) in the ordering vector (1=3, 1=3, qz ). Vertical solid line (dotted line) indicates the P1-P2 (P2-P3) phase boundary. All the data were taken at T ¼ 2 K.
In Fig. 3(a), Pc and M at 1.5 K are given as functions of B applied along the a axis. The data confirm that P1 and P2 are ferroelectric, whereas the polarization disappears in P3, in agreement with previous results [3]. We observe that P4 and P5 are nonferroelectric, in line with the presumption that P4 and P5 are noncollinear phases with no chirality [25]. Note that a clear 1=3-magnetization plateau is observed in P3, where the spins are expected to align in a collinear up-up-down state. We now examine the P1-P2 transition in detail. In Fig. 3(b), we show Pc ðBÞ and its field derivative dPc =dB in the low-field region (< 6 T). The data show that Pc ðBÞ is quasicontinuous across the transition. A tiny anomaly is observed in dPc =dB near 3.8 T, confirming the occurrence of the P1-P2 transition. However, this anomaly is significantly smaller than the huge negative peak observed at the P2-P3 transition at ∼4.8 T. In view of the fact that a significant jump has been observed in the qz value from 0.41 to 1=3 across the P1-P2 transition, our experimental data strongly suggest that the qz value is irrelevant to MFE in this system. In order to show this point more quantitatively, we estimate the field variation of Pc by using simple models. First, we assume that σ t , which is independent of qz , is the predominant factor contributing to MFE ½Pc ðBÞ ∝ σ t . The field variation of σ t can be roughly estimated by assuming a three-sublattice structure with a symmetrical spin arrangement with respect to B∥a [see Fig. 3(a), inset], and the resultant Pc ðBÞ is shown by a dot-dashed line in Fig. 3(b) (chirality model). The result well reproduces the experimentally obtained Pc ðBÞ. Next, we consider the case where σ h is the predominant factor contributing to MFE ½Pc ðBÞ ∝ σ h . It is difficult to evaluate precisely the field variation of σ h in the P1 phase except for B ¼ 0 because of incommensurability (qz ≈ 0.45). On the other hand, the field variation of the
20
helicity model
-20
1
0
0
-10 1.5 K E || c, B || a
P4
P5
-2
-1
-4
-2 20
0
5
10 B [T]
15
Pc [µC/m2]
1.5
0.45
(b) 40
5
10
dPc/dB [µC/m2T]
40
10
(c)
6
(a)
1.5 K E || c, B || a
M [µB/Fe]
(00l) [r. l. u.]
50 1.6
Intensity [arb. units]
1.7
0.50
Pc [µC/m2]
(b)
-30 chirality model
0 0
P1 (IC) 1 2
3 B [T]
-40 P2 P3 (C) (C) 4 5 6
FIG. 3 (color online). (a) Magnetic field variation of the magnetization M and the c component of the electric polarization Pc of RFMO for B∥ [110] (a axis). Insets indicate the expected three-sublattice spin structures in the basal plane. (b) Pc (open circles) and its field derivative dPc =dB (open triangles) in the low-field region measured under a pulsed magnetic field. Only the field-increasing data are shown for clarity. The upward arrow for dPc =dB near 3.8 T indicates a tiny anomaly observed at the P1-P2 transition. The dot-dashed line indicates the expected variation of Pc ðBÞ assuming Pc ∝ σ t (chirality model), whereas the dashed line represents the case of Pc ∝ σ h (helicity model). The latter exhibits a large jump across the P1-P2 transition where a jump in qz from ∼0.41 to 1=3 is observed. The zero-field extrapolation of the helicity with qz fixed to 1=3 is 2.8 times larger than that for qz ≈ 0.45 (P1), as indicated by the dotted line.
P2 phase (qz ¼ 1=3) is easily estimated by assuming a three-sublattice structure along the c axis, similar to the one assumed for the in-plane structure in P1 and P2. Note that σ h in P1 should be much smaller than in P2 owing to the nearly 180° rotations of spins along the c axis [Fig. 4(b)]. Thus, the ratio of the zero-field-extrapolated value of σ h in the P2 phase (qz fixed to 1=3) to that in the P1 10
(a)
(b) B
B || a
c axis
P4
8
P3
6
B [T]
(a)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
P2 4
P2
2
P1 para.
0
0
1
2
3
4
T [K]
5
6
P1 7
8
FIG. 4 (color online). (a) Phase diagram of RFMO for B∥ [110] (a axis) determined from Pc ðBÞ, ϵðT; BÞ, MðT; BÞ, and specific heat CðT; BÞ measurements. The highlighted regions indicate the ferroelectric phases (P1 and P2). Insets depict the expected inplane spin structures. (b) Spin arrangements along the c axis for P1 and P2 phases.
147202-3
PRL 113, 147202 (2014)
week ending 3 OCTOBER 2014
PHYSICAL REVIEW LETTERS
TABLE I. Symmetry relations relevant for the electrical polarization parallel to the c direction in TLAFMs. The state (lattice and spin) is invariant (○) or noninvariant (×) for each operation. All three lattice structures listed here are nonferroelectric in the paramagnetic state because there is at least one symmetry operation that makes the þz direction equivalent to −z. When the magnetic order sets in, chirality or helicity completely breaks the c-axis inversion symmetry, thereby realizing MFE. For ¯ the space groups P3m1 and P321, only chirality has this property (see column C2x ). mz
Symmetry operations S3 C2y i S6
C2x
Lattice
P3¯ ¯ P3m1 P321
× × ×
× × ×
× × ×
○ ○ ×
○ ○ ×
× ○ ○
Spin
Chirality Helicity
○ ×
○ ×
○ ○
× ×
× ×
× ○
phase (B ¼ 0) is large, namely, ∼2.8 ½≈ sinð2π=3Þ= sinð2π × 0.45Þ. Accordingly, if the MFE arises solely from the helicity, then a large jump in Pc ðBÞ must be observed at the P1-P2 transition, as shown by a dashed line in Fig. 3(b) (helicity model). The predicted discontinuity in Pc is 13 μC=m2 , considerably larger than the observed value of −0.2 μC=m2 . Therefore, we may rule out the possibility that MFE is driven by the helicity along the c axis. Figure 4(a) shows the phase diagram of RFMO for B∥a determined from the present MðT; BÞ, Pc ðT; BÞ, ϵðT; BÞ, and specific heat measurements (see Supplemental Material [28]). The highlighted phases (P1 and P2) are ferroelectric. The phase diagram is essentially the same as that previously reported [3,22], although the P1-P2 boundary is missing in Fig. 3 of Ref. [3]. Comparing this phase diagram with that for classical Heisenberg spin TLAFMs [25,29], one finds that all the ferroelectric (nonferroelectric) phases correspond to the states with (without) in-plane chirality. These findings, along with the independence of Pc from the c-axis helicity, provide strong evidence that the MFE in this compound is induced by chirality, and, to the best of our knowledge, constitute the first experimental proof for the field variation of spin chirality in classical spin TLAFMs [25]. Table I summarizes the symmetry relations of MFE in triangular lattices. There are up to six symmetry operations relevant to ferroelectricity along the c axis, namely, mz , reflection with respect to the (001) plane; S3 , threefold ¯ rotatory reflection; C2y , twofold rotation around ½110 (y axis); i, inversion with respect to the Fe site; S6 , sixfold rotatory reflection; and C2x , twofold rotation around ½110 ¯ (x axis) [9]. When the lattice symmetry is lowered to P3, two symmetry operations i and S6 ensure that the system remains nonferroelectric. In the RFMO crystal, a MoO2− 4 cluster is situated asymmetrically on either side of a unit triangle. Accordingly, the local mz symmetry is broken, and
(a)
(b)
FIG. 5 (color online). Local electrical polarization in RFMO. (a) Paramagnetic state. Thick vertical arrows indicate local electrical polarizations due to the asymmetric arrangement of the MoO4 pyramids with respect to the triangular plane. As i or S6 lattice symmetry ensures equal magnitude of up- and downpointing polarizations, no macroscopic polarization appears. (b) Positive (left triangle) and negative (right triangle) spin chirality may induce a change in the magnitude of the local electrical polarization, leading to a macroscopic polarization.
one may assume that a finite local electrical polarization exists perpendicular to the plane [Fig. 5(a)]. Because of the i or S6 symmetry of the lattice, the upward-pointing local polarizations exactly cancel out the downward-pointing ones; hence, there is no macroscopic polarization in the paramagnetic state. Chirality breaks the remaining i and S6 symmetry, thus allowing ferroelectricity. Microscopically, the appearance of Pc suggests that chirality induces a difference in the amplitude of the local electrical polarization on the unit triangle [Fig. 5(b)]. It should be noted that P3¯ is not a unique structure for the occurrence of MFE in TLAFMs. Symmetry considerations ¯ imply that the space groups P3m1 and P321 can also yield MFE (Table I). In these two space groups, the spin helicity along the c axis does not induce MFE. Thus, it would be very easy to probe the spin-chirality-driven MFE in these systems. We also note a possible MFE in kagome-lattice antiferromagnets. Amongpthe ffiffiffi various pffiffiffi allowed ground-state spin configurations, the 3 × 3 spin structure [30] may exhibit MFE if the crystal lattice belongs to the space group ¯ P3m1, ¯ P3, or P321. In summary, we have demonstrated that an electrical polarization perpendicular to the triangular plane appears in RbFeðMoO4 Þ2 in those phases for which in-plane triangular-spin chirality is predicted. On the other hand, no strong anomaly is observed in polarization at the field where the c axis modulation of the spin structure qz exhibit a jump. Thus, we conclude that magnetoelectricity in this compound is driven predominantly by triangular-spin chirality. Similar chirality-driven ferroelectricity can be expected for ¯ the structures P3m1 and P321. We thank T. Inami and T. Waki for critical discussions and useful technical advice. The work at ORNL was supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. WAND is operated jointly by ORNL and Japan Atomic Energy Agency under US-Japan Cooperative Program on Neutron Scattering. The present study was supported by a Grant-in-Aid for Scientific Research C (No. 26400329).
147202-4
PRL 113, 147202 (2014) *
PHYSICAL REVIEW LETTERS
[email protected]‑tokyo.ac.jp [1] S. Miyashita and H. Shiba, J. Phys. Soc. Jpn. 53, 1145 (1984). [2] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature (London) 426, 55 (2003). [3] M. Kenzelmann, G. Lawes, A. B. Harris, G. Gasparovic, C. Broholm, A. P. Ramirez, G. A. Jorge, M. Jaime, S. Park, Q. Huang, A. Y. Shapiro, and L. A. Demianets, Phys. Rev. Lett. 98, 267205 (2007). [4] A. B. Harris, Phys. Rev. B 76, 054447 (2007). [5] S. Seki, Y. Onose, and Y. Tokura, Phys. Rev. Lett. 101, 067204 (2008). [6] K. Kimura, H. Nakamura, K. Ohgushi, and T. Kimura, Phys. Rev. B 78, 140401(R) (2008). [7] A. B. Harris, A. Aharony, and O. Entin-Wohlman, J. Phys. Condens. Matter 20, 434202 (2008). [8] A. J. Hearmon, F. Fabrizi, L. C. Chapon, R. D. Johnson, D. Prabhakaran, S. V. Streltsov, P. J. Brown, and P. G. Radaelli, Phys. Rev. Lett. 108, 237201 (2012). [9] H. Mitamura, R. Watanuki, N. Onozaki, Y. Shimura, S. Kittaka, T. Sakakibara, and K. Suzuki, J. Phys. Conf. Ser. 391, 012099 (2012). [10] K. Kimura, T. Otani, H. Nakamura, Y. Wakabayashi, and T. Kimura, J. Phys. Soc. Jpn. 78, 113710 (2009). [11] H. Kawamura, Prog. Theor. Phys. Suppl. 101, 545 (1990). [12] M. Soda, K. Kimura, T. Kimura, M. Matsuura, and K. Hirota, J. Phys. Soc. Jpn. 78, 124703 (2009). [13] T. Arima, J. Phys. Soc. Jpn. 76, 073702 (2007). [14] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005). [15] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006). [16] I. A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006).
week ending 3 OCTOBER 2014
[17] C. Jia, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B 74, 224444 (2006). [18] T. Inami, Y. Ajiro, and T. Goto, J. Phys. Soc. Jpn. 65, 2374 (1996). [19] T. Inami, J. Solid State Chem. 180, 2075 (2007). [20] L. E. Svistov, A. I. Smirnov, L. A. Prozorova, O. A. Petrenko, L. N. Demianets, and A. Y. Shapiro, Phys. Rev. B 67, 094434 (2003). [21] L. E. Svistov, A. I. Smirnov, L. A. Prozorova, O. A. Petrenko, A. Micheler, N. Büttgen, A. Y. Shapiro, and L. N. Demianets, Phys. Rev. B 74, 024412 (2006). [22] A. I. Smirnov, H. Yashiro, S. Kimura, M. Hagiwara, Y. Narumi, K. Kindo, A. Kikkawa, K. Katsumata, A. Y. Shapiro, and L. N. Demianets, Phys. Rev. B 75, 134412 (2007). [23] J. S. White, C. Niedermayer, G. Gasparovic, C. Broholm, J. M. S. Park, A. Y. Shapiro, L. A. Demianets, and M. Kenzelmann, Phys. Rev. B 88, 060409(R) (2013). [24] L. E. Svistov, L. A. Prozorova, N. Büttgen, A. Y. Shapiro, and L. N. Dem’yanets, JETP Lett. 81, 102 (2005). [25] H. Kawamura and S. Miyashita, J. Phys. Soc. Jpn. 54, 4530 (1985). [26] H. Mitamura, S. Mitsuda, S. Kanetsuki, H. A. Katori, T. Sakakibara, and K. Kindo, J. Phys. Soc. Jpn. 76, 094709 (2007). [27] T. Sakakibara, H. Mitamura, T. Tayama, and H. Amitsuka, Jpn. J. Appl. Phys. 33, 5067 (1994). [28] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.113.147202 for the results of magnetization, dielectric constant, specific heat, and electric polarization measurements. [29] S. Watarai, S. Miyashita, and H. Shiba, J. Phys. Soc. Jpn. 70, 532 (2001). [30] A. Chubukov, Phys. Rev. Lett. 69, 832 (1992).
147202-5
