{'fraction_non_alphanumeric': 0.03976038443815417, 'fraction_numerical': 0.021460075044434205, 'mean_word_length': 4.504347826086956, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Spin density wave as a superposition of two magnetic states of opposite chirality and its implications

A magnetic solid with weak spin frustration tends to adopt a noncollinear magnetic structure such as cycloidal structure below a certain temperature and a spin density wave (SDW) slightly above this temperature. The causes for these observa-tions were explored by studying the magnetic structures of BaYFeO4, which undergoes an SDW and a cycloidal phase transi-tion below 48 and 36 K, respectively, in terms of density func-tional theory calculations. We show that an SDW structure aris-es from a superposition of two magnetic states of opposite chi-rality, an SDW state precedes a chiral magnetic state due to the lattice relaxation, and whether an SDW is transversal or longitudinal is governed by the magnetic anisotropy of magnetic ions.

ABSTRACT: A magnetic solid with weak spin frustration tends to adopt a noncollinear magnetic structure such as cycloidal structure below a certain temperature and a spin density wave (SDW) slightly above this temperature. The causes for these observations were explored by studying the magnetic structures of BaYFeO4, which undergoes an SDW and a cycloidal phase transition below 48 and 36 K, respectively, in terms of density functional theory calculations. We show that an SDW structure arises from a superposition of two magnetic states of opposite chirality, an SDW state precedes a chiral magnetic state due to the lattice relaxation, and whether an SDW is transversal or longitudinal is governed by the magnetic anisotropy of magnetic ions.
A magnetic system composed of identical magnetic ions tends to adopt a noncollinear magnetic superstructure such as a cycloid (Fig. 1a) below a certain temperature when its spin exchange interactions are weakly spin frustrated, 1-6 while a magnetic system with strong spin frustration does not undergo a log-range magnetic ordering. 7 It is often observed that a spin density wave (SDW) state (Fig. 1b,c) occurs slightly above the onset temperature of the cycloid. [8][9][10][11][12][13] In a cycloidal structure, all moments of the ions are the same in magnitude although they differ in orientation (Fig. 1a). In an SDW (Fig. 1b,c), the moments of the magnetic ions change their magnitudes depending on their positions in the crystal lattice, i.e., they vary sinusoidally along the propagation direction of the SDW and may even vanish. The latter is apparently unphysical for any magnetic system consisting of identical magnetic ions although, mathematically, they correctly describe the observed magnetic structure. So far, this apparently puzzling observation has not been explained to the best of our knowledge, nor has been why a cycloidal state is preceded by a SDW. Furthermore, a SDW can be transversal (Fig. 1b) as found for BaYFeO4 8 or longitudinal (Fig. 1c) as found for TbMnO3. 9,10 What controls whether a SDW is transversal or longitudinal is not well understood. In the present work, we explore the answers to these questions by studying the magnetic oxide BaYFeO4 (BYFO), which undergoes a SDW and cycloidal magnetic phase transitions below 48 and 36 K, respectively. 8,14 Figure 1. Magnetic superstructures typically observed for a magnetic system with weak spin frustration: (a) a cycloid. (b) a transverse SDW. (c) a longitudinal SDW. For the sake of simplicity, the SDWs and cycloids illustrated here are commensurate ones. Our discussion is equally valid for incommensurate SDWs and incommensurate cycloids.
BYFO is made up of corner-sharing FeO6 octahedra and FeO5 square pyramids, both containing high-spin Fe 3+ (d 5 , S = 5/2) ions. 8,14 These polyhedra share their corners to form Fe4O18 tetramer units (Fig. 2a), and these tetramers share their corners to form folded-ladder (FL) chains of formula (FeO4)4 running along the b-direction (Fig. 2b). Such FLchains are packed in BYFO with two types of orientations, A and B (Fig. 2c). The axial Fe-O bonds of the FeO5 square pyramids are aligned along the (a -c) and (a + c) directions in type A and B orientations, respectively. The magnetic susceptibility of BYFO hints two antiferromagnetic (AFM) transitions at 36 and 48 K. 14 The neutron powder diffraction study revealed that the two magnetic transitions at 36 and 48 K arise from a long-range magnetic order. 7 The magnetic structure below 48 K is best described by a transversal SDW ( Fig. 1b) with propagation vector k = (0, 0, 1/3) and moments along the b-axis, and the magnetic structure below 36 K by a cycloid (Fig. 1a) with propagation vector k = (0, 0, 0.358) and moments lying in the bc-plane. The cycloidal structure below 36 K lacks inversion symmetry, so BYFO exhibits ferroelectric polarization induced by a magnetic order. 15 TbMnO3 adopts a longitudinal SDW ( Fig.  1c) with moments in the b-direction below 42 K and a cycloidal (Fig. 1a) magnetic structure with moments in the bc-plane below 28 K. 9,10 In what follows, we examine the nature of spin frustration in BYFO leading to its SDW and cycloidal states as well as under what condition they occur by evaluating six intrachain and four interchain spin In each folded-ladder chain, the nearest-neighbor Fe 3+ -spins are antiferromagnetically coupled. The latter is indicated by using shaded and unshaded spheres. (e) A view of how the spin moment of each Fe 3+ ion rotates in the bc-plane. The rotational plane at each spin site is represented by an ellipse. For simplicity, only the Fe 3+ -spins forming the interchain exchange paths J9 are shown. The angle between two adjacent spin sites in each exchange path J9 along the c-direction is . exchanges (Fig. 2b,c) and determining the magnetic anisotropy of the Fe 3+ ions on the basis of density functional theory (DFT) calculations. Subsequently, we explore important implications of our finding.
We carry out spin-polarized DFT calculations using the Vienna ab Initio Simulation Package, 16 the projector augmented wave method, 17 the PBE exchange-correlation functionals, 18 the plane wave cutoff energy of 500 eV, a set of 221 k-points for sampling the irreducible Brillouin zone, and the threshold of 10 -6 eV for self-consistent-field energy convergence. The electron correlation associated with the Fe 3d states were taken into consideration by performing the DFT+U calculations 19 with the effective onsite repulsion U eff = U -J (= 4, 5 eV) on Fe. The preferred spin directions of the Fe 3+ ions were determined by DFT+U+SOC calculations. 20 To explore the nature of the noncollinear magnetic structures of BYFO, we examine the six intrachain spin exchanges J1, J2, J5, J6, J7 and J8 (Fig. 2b) as well as the four interchain spin exchanges J3, J4, J9 and J10 (Fig. 2c). The geometrical parameters associated with these exchange paths are summarized in Table S1. To extract the values of these spin exchanges, we construct 11 ordered spin states of BYFO (Fig. S1) using a (a, 2b, 2c) supercell, which has 32 formula units (FUs). The magnetic energy spectrum of BYFO can be described in terms of the Heisenberg spin Hamiltonian, where Jij = J1 -J10. The total spin exchange energies of these states per (a, 2b, 2c) supercell can be expressed as where S refers to the spin of the high-spin Fe 3+ ion (i.e., S = 5/2). The values of ni (i = 1 -10) found for the 11 ordered spin states are listed in Table S2. The relative energies of these states determined by DFT+U calculations are summarized in Table S3. We obtain the values of J1  J10 by energy-mapping analysis, 1 in which the relative energies from the DFT+U calculations are mapped onto the corresponding energies expected from the spin exchange energies (Eq. 2). The values of J1  J10 obtained for the 280, 38 and 6 K structures 7 of BYFO by DFT+U calculations with U eff = 4 and 5 eV are summarized in Table S4 and S5, respectively.
For the 280, 38 and 6 K crystal structures, all spin exchanges (except for J9 at 6 K) are AFM, and the intrachain spin exchanges J1 and J2 dominate over all other intrachain spin exchanges as well as all the interchain spin exchanges. Due to the strong spin exchanges J1 and J2, the NN spins in each FLchain are strongly coupled antiferromagnetically (Fig. 2d) and their collinear spin arrangement should be most stable. This explains the collinear AFM arrangement within each FL-chain observed for the cycloidal magnetic structure at 6 K. We now examine the magnetic anisotropy of the Fe 3+ (S = 5/2, L = 0) ions of BYFO. Formally, L = 0 for the high-spin Fe 3+ ions, so the effect of SOC is weak. Thus, in determining the magnetic anisotropy of the Fe 3+ ions at low temperature, we employ a commensurate, low-energy AFM state of BYFO (e.g., Fig. S2) predicted by the spin exchange constants obtained from the DFT+U calculations. Using this AFM state, we perform DFT+U+SOC calculations with three different spin orientations, i.e., parallel to the a-, b-and c-directions. These calculations show that the ||b spin orientation is slightly more stable than the ||c spin orientation, while these two orientations are substantially more stable than the ||a spin orientation (Table S6 and S7). Thus, the spins of the Fe 3+ ions prefer to lie in the bc-plane, which agrees with the observation that the Fe 3+ moments lie in the bc-plane in the cycloidal magnetic structure at 6 K. 8 The cycloidal magnetic structure is determined by the noncollinear spin arrangement in the exchange paths J9 (Fig. 2c), which run along the c-direction. As depicted in Fig.  2e, the spins in each zigzag chain of J9 paths rotate by the angle of  in the bc-plane as they move from one spin site to another along the c-direction. In the cycloidal structure of BYFO at 6 K with propagation vector k = (0, 0, 0.358), which is equivalent to 2 = 128.8 (i.e.,  = 64.4). 7 This successive spin-rotation is related to the spin frustration between FLchains, which is brought about by the interchain spin exchanges J3, J4, J9 and J10 (Fig. 2c) together with the intrachain exchanges J1 and J2. These exchanges form (J3, J4, J10), (J1, J3, J9) and (J2, J4, J9) triangles (see Fig. S3 for more details). According to the packing pattern of the FL-chains (Fig. 2c), one FL-chain of a particular arrangement, say, A, interacts with four FL-chains of arrangement B and with two FL-chains of arrangement A. Thus, the total spin exchange energy E() that one FL-chains generates (per four FUs) is written as The angle min minimizing this energy is obtained by requiring that dE()/d = 0, which leads to the expression Note that a cycloidal magnetic state (Fig. 1a) is chiral, and a cycloid of one chirality is identical in energy with that of its opposite chirality as long as the lattice of the magnetic ions remains the same. Then, the two cycloidal states of opposite chirality depicted in Fig 3a,b should be equally populated to give rise to a transversal SDW of Fig. 1b, because the components of the moments along the propagation direction are canceled out while those perpendicular to the propagation direction are reinforced with their magnitudes varying sinusoidally along the propagation direction. The opposite happens when the cycloidal sates of opposite chirality depicted in Fig 4a,b are equally populated, leading to a longitudinal SDW of Fig. 1c. In short, when equally populated, two cycloidal states of opposite chirality generate a SDW magnetic structure. The apparently puzzling picture of a SDW, namely, that the moments of the magnetic ions change their magnitudes depending upon their positions in the lattice, is a direct consequence of equally populating the two cycloidal states of opposite chirality. Helical magnetic states are also chiral, so it is possible to have a SDW as a superposition of two helical states of opposite chirality. FIGURE 3. Two cycloids of opposite chirality with moments in the bc-plane, leading to a transversal SDW (Fig. 1b) propagating along the c direction. . Two cycloids of opposite chirality with moments in the bc-plane, leading to a longitudinal SDW shown (Fig. 1c) propagating along the b direction.
On lowering the temperature, a magnetic solid with a SDW state will adopt a cycloidal magnetic structure because its crystal lattice will be relaxed to become energetically more favorable to one of the two cycloidal states so the two cycloidal states become nondegenerate. On lowering the temperature, therefore, a SDW state should be followed by a cycloidal state, as found experimentally. 8-13 In NaFeSi2O6, a SDW is followed by a helical magnetic state. 21 We note that, in reproducing the entire experimental phase diagram of RMnO3 (R = rare earth), it is necessary to include the spinphonon coupling, 22,23 which is essential for a crystal lattice to relax and energetically favor one of the two cycloidal states. In this phase diagram study based on a model Hamiltonian, neither the origin of the SDW nor its implications was examined.
Finally, we consider why the SDW of BYFO is transversal (Fig. 1b) while that of TbMnO3 is longitudinal (Fig. 1c). In BYFO, the moment orientation along the ||b direction is slightly more stable than that along the ||c direction (Table  S6, S7). In the SDW of BYFO, the two cycloids (Fig. 3a, b) give rise to the nonzero moments perpendicular to the SDW propagation direction (Fig. 1b). The preferred orientation of the Mn 3+ spin in TbMnO3 is contained in the ab-plane and is aligned approximately along the b-direction (Fig. S4). 2,12 Due to this magnetic anisotropy plus the ferromagnetic spin exchanges between the nearest-neighbor Mn 3+ spins, 2 the Mn 3+ spins prefer to align along the b-axis. In the SDW of TbMnO3, the two cycloids (Fig. 4a, b) lead to the nonzero moments along the SDW propagation direction (Fig. 1c).
In summary, a SDW state is a superposition of two cycloidal states of opposite chirality. A spin density wave state precedes a cycloidal magnetic state, and the magnetic anisotropy of magnetic ions determines whether a SDW is transversal or longitudinal.

SYNOPSIS TOC
A magnetic solid with weak spin frustration tends to adopt a cycloidal structure below a certain temperature and a spin density wave (SDW) slightly above this temperature. Our analysis shows that a SDW arises from a superposition of two cycloids of opposite chirality. Implications of this finding were explored.    Figure S1. The 11 ordered spin states used to calculate J1  J10. S10 Figure S2. The AFM state used to determine the magnetic anisotropy of the Fe 3+ ions in BaYFeO4. S11 Figure S3. A perspective view of the interchain spin exchange paths J3, J4, J9 and J10. The intrachain spin exchanges J1 and J2 are also shown. S12 Figure S4. A layer of corner-sharing MnO6 octahedra parallel to the ab-plane. The axiallyelongated Mn-O bonds (indicated by blue arrows) lie in the ab-plane and are aligned more along the b-direction than along the a-direction.
The spin exchanges Jaa and Jbb are antiferromagnetic, but the spin exchange Jab is ferromagnetic.
These spin exchanges, together with the preferred spin orientation of each Mn 3+ ion along the elongated Mn-O bond, makes the Mn 3+ spins lie in the bc-plane in the cycloidal magnetic structure of TbMnO3 below 28 K.