#
Coupling between Spin and Charge Order Driven by Magnetic Field in Triangular Ising System LuFe_{2}O_{4+δ}

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}O

_{4+δ}through single crystal neutron diffraction. In the absence of a magnetic field, the strong diffuse neutron scattering observed below the Neel temperature (T

_{N}= 240 K) indicates that LuFe

_{2}O

_{4+δ}shows short-range, two-dimensional (2D) correlations in the FeO

_{5}triangular layers, characterized by the development of a magnetic scattering rod along the 1/3 1/3 L direction, persisting down to 5 K. We also found that on top of the 2D correlations, a long range ferromagnetic component associated with the propagation vector

**k**= 0 sets in at around 240 K. On the other hand, an external magnetic field applied along the c-axis effectively favours a three-dimensional (3D) spin correlation between the FeO

_{1}_{5}bilayers evidenced by the increase of the intensity of satellite reflections with propagation vector

**k**= (1/3, 1/3, 3/2). This magnetic modulation is identical to the charge ordered superstructure, highlighting the field-promoted coupling between the spin and charge degrees of freedom. Formation of the 3D spin correlations suppresses both the rod-type diffuse scattering and the

_{2}**k**component. Simple symmetry-based arguments provide a natural explanation of the observed phenomenon and put forward a possible charge redistribution in the applied magnetic field.

_{1}## 1. Introduction

_{2}O

_{4}, bearing significant spin and charge frustration in a triangular lattice, has attracted wide interest thanks to reported high-temperature multiferroic properties, pivotal for technological applications in spintronics [1,2,3,4,5,6]. Such properties have been related to the charge ordering of Fe

^{2+}and Fe

^{3+}cations, which appears as a three-dimensional (3D) long range ordering below 320 K [2]. An apparent coupling between polarization and magnetism has also been previously proposed based on the observation of significant changes in the electric polarization when spin ordering sets in at T

_{N}= 240 K [3]. Although the polar nature of the charge ordering was not confirmed by the later structural study [7], LuFe

_{2}O

_{4}remains a focus of intensive investigations as a model frustrated system.

_{2}O

_{4}(space group R-3m) (Figure 1) is characterized by the stacking of three FeO

_{5}face-sharing bipyramidal layers along the c-axis separated by LuO

_{2}rock salt layers [1,2]. The resulting iron sublattice arrangement consists of three stacked triangular bilayers with an equal amount of Fe

^{2+}and Fe

^{3+}in the unit cell, and this makes the system geometrically frustrated in both the charge and spin channels. Even though intensive experimental and theoretical efforts to clarify the underlying pattern of spin and charge order have been made, its exact nature is still controversial [8].

^{3+}cations surrounded by a honeycomb lattice of Fe

^{2+}cations and a lower B layer has the opposite Fe

^{2+}/Fe

^{3+}arrangement (see Figure 1c). Whereas the AB model has been used to account for the ferroelectricity observed in LuFe

_{2}O

_{4}due to the net electric dipole moment along the c-axis [2,3], X-ray scattering experiments have suggested an antiferroelectric AB–BA bilayer stacking [6,9]. Later, several investigations have found that ferroelectric order can be induced based on this antiferroelectric model when an external electric field is applied [10,11,12]. A charge order model with AA–BB stacking has also been recently demonstrated through X-ray diffraction study, inconsistent with charge-ordering-based ferroelectricity [7].

_{2}O

_{4}are governed by the Ising character of the iron spin in the triangular lattice essentially due to spin-orbit coupling [8,13]. Both two-dimensional (2D) [1] and 3D magnetic structures [12,14] have been previously found in different samples, reflecting the important role played by oxygen stoichiometry [15,16,17,18]. In addition, two magnetic phase transitions have been observed by Christianson et al. via single crystal neutron diffraction study. The first transition, observed below T

_{N}= 240 K, is characterized by finite magnetic correlations within ferrimagnetic (FIM) monolayers stacked ferromagnetically along the c direction; while at the second transition (T

_{L}= 175 K) a significant broadening of specific magnetic reflections is observed, indicative of an additional decrease in the magnetic correlation length [14,19]. Moreover, a metamagnetic state in the vicinity of T

_{N}, consisting of nearly degenerate FIM and antiferromagnetic (AFM) spin order has been proposed [20]. A distinct FIM spin configuration on the bilayers based on the AB charge order model, where all Fe

^{2+}spins in the adjacent monolayers are FM, while Fe

^{3+}spins are antiparallel to Fe

^{2+}rich layers and nearest neighbour Fe

^{3+}spins in the Fe

^{3+}rich layer are antiparallel, has been put forward through X-ray magnetic circular dichroism [10,12,21], and has been supported by theoretical studies using both density functional theory and Monte Carlo simulations as well as inelastic scattering experiments [21,22,23]. Apart from this 2D spin arrangement in the bilayer, Mulders et al. have suggested a long range AFM order along the c direction that facilitates the long range order of electric dipole moment, suggesting the presence of spin-charge coupling [12]. A magnetic field vs. temperature phase diagram above 180 K has been recently proposed, which reveals that magnetic field promotes the FIM order from the degenerate AFM–FIM phases [20].

_{2}O

_{4+δ}under an applied magnetic field along the Ising direction in the temperature range 5–275 K. The striking effect of the magnetic field on the magnetic structure from our experiment together with previous works brings new insights into the complex magnetic field versus temperature phase diagram and reveals clear evidence for a field-induced coupling between the spin and charge orderings.

## 2. Results

_{2}O

_{4+δ}under zero field cooled (ZFC) and field cooled (FC) conditions measured by applying a magnetic field of 1 T parallel to the c-axis. In accordance with previous results, the ferrimagnetic order appears below T

_{N}= 240 K [1]. A significant irreversibility between the ZFC and FC data can be seen at around 185 K. However, the second magnetic phase transition at 175 K reported in Reference [14] is absent in this sample. The magnetization measurements are consistent with a sample containing a slight oxygen excess [1,3]. As shown in the inset of Figure 2a, the magnetic hysteresis loop at 200 K is indicative of a net spontaneous moment.

_{5}triangular bilayers. As shown in Figure 3a, the integrated intensity of the reciprocal space cut along the 1/3 1/3 L direction becomes more intense below 240 K, in good agreement with the magnetization measurement. Remarkably, a long range spin order associated with a propagation vector

**k**= 0 and coupled to the 2D spin correlations is unveiled below 240 K, and is characterized by sharp magnetic contribution to the 110, 011 and −221 nuclear reflections, as shown in Figure 3b. This

_{1}**k**= 0 component is related to a ferromagnetic contribution with the R-3m’ symmetry as can be appreciated in Figure A1 (an AFM arrangement with the R-3’m’ symmetry would result in null magnetic intensity in the HK0 plane reflections). A net moment for a single triangular unit of Ising spins is naturally associated with up–up–down (down–down–up) spin configurations. The presence of the coherent scattering with

_{1}**k**= 0 implies, in the simplest case, an averaging between up–up–down, up–down–up and down–up–up configurations for each triangle. Practically, it means that the numbers of spins with up and down polarization are not equal in each triangular layer, resulting in a net ferrimagnetic moment. An illustrative example of such a ferrimagnetic layer is shown in Figure 3c. The layers are then stacked along the c-axis with a random shift within the (ab) plane. The strong 2D magnetic scattering rod together with the 3D ferromagnetic component lead to a physical picture of spin ordering in LuFe

_{1}_{2}O

_{4+δ}from 5 K to 270 K in which the lack of spin correlations along the c direction coexists with the long range Ising ferrimagnetic order in each triangular monolayer.

_{5}triangular bilayers with a propagation vector

**k**= (1/3, 1/3, 3/2) with respect to the parent R-3m unit cell. It should be pointed out that this propagation vector is identical to the structural modulation associated with the charge ordering reported in Reference [6] and confirmed in the present study, as can be appreciated by the observation of charge satellite reflections in the high Q region in Figure A2. As presented in Figure 4d,e, the temperature dependence of integrated intensities of 1/3 1/3 0 and 1/3 1/3 1/2 satellite reflections under 5 T shows the onset of spin ordering at 260 K, slightly higher than that under 0 T.

_{2}**k**-related component. It is also important to stress that the magnetic scattering corresponding to the

_{1}**k**propagation vector cannot simply be ascribed to a redistribution of the diffuse scattering intensity but instead includes an extra component, likely coming from the magnetic scattering related to the

_{2}**k**propagation vector in zero field.

_{1}## 3. Discussion

_{2}

^{+}(H

_{−}) of the parent R-3m space group. The symmetry of the charge order cannot be uniquely identified from the available diffraction data and the only solid experimental evidence is that the propagation vector for this type of distortion is (1/3, 1/3, 3/2), in agreement with other experimental results [6]. There are two six-dimensional irreducible representations associated with this propagation vector whose matrix operators for the generators of the R-3m space group are specified in Table 1. Whichever of them is responsible for the symmetry breaking related to the charge ordering, there is always a way to form a tri-linear free-energy coupling term with a time-odd quantity (magnetic order) with the same periodicity as the charge ordering. Specifying the relevant order parameters as (η

_{+}, η*

_{+}, ξ

_{+}, ξ*

_{+}, ρ

_{+}, ρ*

_{+}) for the charge ordering and (η

_{−}, η*

_{−}, ξ

_{−}, ξ*

_{−}, ρ

_{−}, ρ*

_{−}) for the field-induced magnetic order, the free-energy invariant reads:

_{−}(η

_{−}ξ*

_{+}+ ξ

_{−}ρ*

_{+}+ ρ

_{−}η*

_{+}+ η

_{+}ξ*

_{−}+ ξ

_{+}ρ*

_{−}+ ρ

_{+}η*

_{−})

## 4. Experiments and Methods

## 5. Conclusions

_{2}O

_{4+δ}through magnetization measurement and neutron diffraction. In the absence of a magnetic field, the spin and charge subsystems are effectively decoupled. The spin forms 2D correlations featured by the magnetic diffuse scattering rod in the 1/3 1/3 L direction and a 3D ferromagnetic component below T

_{N}= 240 K. An external magnetic field suppresses the diffuse scattering and promotes 3D spin correlations with the propagation vector

**k**= (1/3, 1/3, 3/2). This periodicity is common for both the magnetic and charge-ordered sublattices and a combination of these experimental results with symmetry consideration provides evidence of a magnetic field imposed coupling between the spin and charge degrees of freedom.

_{2}## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Figure A1.**Simulated magnetic reflections in HK0 (

**Top**) and HKL (

**Bottom**) plane for a ferromagnetic (

**Left**) and antiferromagnetic order (

**Right**) along the c direction, the radius and the brightness of the reflections are an indication of the magnetic structure factor. The simulations are performed with the Jana2006 software.

**Figure A2.**Neutron scattering in the high Q region at the (

**a**) 1/3 1/3 L and (

**b**) 2/3 2/3 L positions showing the presence of charge ordering satellite reflections ascribable to the

**q**= (1/3, 1/3, 3/2) modulation vector.

## References

- Iida, J.; Tanaka, M.; Nakagawa, Y.; Funahashi, S.; Kimizuka, N.; Takekawa, S. Magnetization and spin correlation of two-dimensional triangular antiferromagnet LuFe
_{2}O_{4}. J. Phys. Soc. Jpn.**1993**, 62, 1723–1735. [Google Scholar] [CrossRef] - Yamada, Y.; Kitsuda, K.; Nohdo, S.; Ikeda, N. Charge and spin ordering process in the mixed-valence system LuFe
_{2}O_{4}: Charge ordering. Phys. Rev. B**2000**, 62, 12167–12174. [Google Scholar] [CrossRef] - Ikeda, N.; Ohsumi, H.; Ohwada, K.; Ishii, K.; Inami, T.; Kakurai, K.; Murakami, Y.; Yoshii, K.; Mori, S.; Horibe, Y.; et al. Ferroelectricity from iron valence ordering in the charge-frustrated system LuFe
_{2}O_{4}. Nature**2005**, 436, 1136–1138. [Google Scholar] [CrossRef] [PubMed] - Nagano, A.; Naka, M.; Nasu, J.; Ishihara, S. Electric polarization, magnetoelectric effect, and orbital state of a layered iron oxide with frustrated geometry. Phys. Rev. Lett.
**2007**, 99, 217202. [Google Scholar] [CrossRef] [PubMed] - Xiang, H.J.; Whangbo, M.H. Charge order and the origin of giant magnetocapacitance in LuFe
_{2}O_{4}. Phys. Rev. Lett.**2007**, 98, 246403. [Google Scholar] [CrossRef] [PubMed] - Angst, M.; Hermann, R.P.; Christianson, A.D.; Lumsden, M.D.; Lee, C.; Whangbo, M.H.; Kim, J.W.; Ryan, P.J.; Nagler, S.E.; Tian, W.; et al. Charge order in LuFe
_{2}O_{4}: Antiferroelectric ground state and coupling to magnetism. Phys. Rev. Lett.**2008**, 101, 227601. [Google Scholar] [CrossRef] [PubMed] - De Groot, J.; Mueller, T.; Rosenberg, R.A.; Keavney, D.J.; Islam, Z.; Kim, J.W.; Angst, M. Charge order in LuFe
_{2}O_{4}: An unlikely route to ferroelectricity. Phys. Rev. Lett.**2012**, 108, 187601. [Google Scholar] [CrossRef] [PubMed] - Ikeda, N.; Nagata, T.; Kano, J.; Mori, S. Present status of the experimental aspect of RFe
_{2}O_{4}study. J. Phys. Condens. Matter**2015**, 27, 053201. [Google Scholar] [CrossRef] [PubMed] - Xu, X.S.; De Groot, J.; Sun, Q.C.; Sales, B.C.; Mandrus, D.; Angst, M.; Litvinchuk, A.P.; Musfeldt, J.L. Lattice dynamical probe of charge order and antipolar bilayer stacking in LuFe
_{2}O_{4}. Phys. Rev. B**2010**, 82, 014304. [Google Scholar] [CrossRef] - Ko, K.T.; Noh, H.J.; Kim, J.Y.; Park, B.G.; Park, J.H.; Tanaka, A.; Kim, S.B.; Zhang, C.L.; Cheong, S.W. Electronic origin of giant magnetic anisotropy in multiferroic LuFe
_{2}O_{4}. Phys. Rev. Lett.**2009**, 103, 207202. [Google Scholar] [CrossRef] [PubMed] - Wen, J.S.; Xu, G.Y.; Gu, G.D.; Shapiro, S.M. Magnetic-field control of charge structures in the magnetically disordered phase of multiferroic LuFe
_{2}O_{4}. Phys. Rev. B**2009**, 80, 020403. [Google Scholar] [CrossRef] - Mulders, A.M.; Bartkowiak, M.; Hester, J.R.; Pomjakushina, E.; Conder, K. Ferroelectric charge order stabilized by antiferromagnetism in multiferroic LuFe
_{2}O_{4}. Phys. Rev. B**2011**, 84, 140403(R). [Google Scholar] [CrossRef] - Harris, A.B.; Yildirim, T. Charge and spin ordering in the mixed-valence compound LuFe
_{2}O_{4}. Phys. Rev. B**2010**, 81, 134417. [Google Scholar] [CrossRef] - Christianson, A.D.; Lumsden, M.D.; Angst, M.; Yamani, Z.; Tian, W.; Jin, R.; Payzant, E.A.; Nagler, S.E.; Sales, B.C.; Mandrus, D. Three-dimensional magnetic correlations in multiferroic LuFe
_{2}O_{4}. Phys. Rev. Lett.**2008**, 100, 107601. [Google Scholar] [CrossRef] [PubMed] - Wang, F.; Kim, J.; Kim, Y.; Gu, G.D. Spin-glass behaviour in in LuFe
_{2}O_{4+δ}. Phys. Rev. B**2009**, 80, 024419. [Google Scholar] [CrossRef] - Bourgeois, J.; Andre, G.; Petit, S.; Robert, J.; Poienar, M.; Rouquette, J.; Elkaim, E.; Hervieu, M.; Maignan, A.; Martin, C.; et al. Evidence of magnetic phase separation in LuFe
_{2}O_{4}. Phys. Rev. B**2012**, 86, 024413. [Google Scholar] [CrossRef] - Hervieu, M.; Guesdon, A.; Bourgeois, J.; Elkaim, E.; Poienar, M.; Damay, F.; Rouquette, J.; Maignan, A.; Martin, C. Oxygen storage capacity and structural flexibility of LuFe
_{2}O_{4+x}(0 ≤ x ≤ 0.5). Nat. Mater.**2013**, 13, 74–80. [Google Scholar] [CrossRef] [PubMed] - Bourgeois, J.; Hervieu, M.; Poienar, M.; Abakumov, A.M.; Elkaim, E.; Sougrati, M.T.; Porcher, F.; Damay, F.; Rouquette, J.; Van Tendeloo, G.; et al. Evidence of oxygen-dependent modulation in LuFe
_{2}O_{4}. Phys. Rev. B**2012**, 85, 064102. [Google Scholar] [CrossRef] - Xu, X.S.; Angst, M.; Brinzari, T.V.; Hermann, R.P.; Musfeldt, J.L.; Christianson, A.D.; Mandrus, D.; Sales, B.C.; McGill, S.; Kim, J.W.; et al. Charge order, dynamics, and magnetostructural transition in multiferroic LuFe
_{2}O_{4}. Phys. Rev. Lett.**2008**, 101, 227602. [Google Scholar] [CrossRef] [PubMed] - De Groot, J.; Marty, K.; Lumsden, M.D.; Christianson, A.D.; Nagler, S.E.; Adiga, S.; Borghols, W.J.H.; Schmalzl, K.; Yamani, Z.; Bland, S.R.; et al. Competing Ferri- and Antiferromagnetic phases in geometrically frustrated LuFe
_{2}O_{4}. Phys. Rev. Lett.**2012**, 108, 037206. [Google Scholar] [CrossRef] [PubMed] - Kuepper, K.; Raekers, M.; Taubitz, C.; Prinz, M.; Derks, C.; Neumann, M.; Postnikov, A.V.; De Groot, F.M.F.; Piamonteze, C.; Prabhakaran, D.; et al. Charge order, enhanced orbital moment, and absence of magnetic frustration in layered multiferroic LuFe
_{2}O_{4}. Phys. Rev. Lett.**2009**, 80, 220409(R). [Google Scholar] - Gaw, S.M.; Lewtas, H.J.; McMorrow, D.F.; Kulda, J.; Ewings, R.A.; Perring, T.G.; McKinnon, R.A.; Balakrishnan, G.; Prabhakaran, D.; Boothroyd, A.T. Magnetic excitation spectrum of LuFe
_{2}O_{4}measured with inelastic neutron scattering. Phys. Rev. B**2015**, 91, 035103. [Google Scholar] [CrossRef] - Xiang, H.J.; Kan, E.J.; Wei, S.H.; Whangbo, M.H.; Yang, J.L. Origin of the Ising ferrimagnetism and spin-charge coupling in LuFe
_{2}O_{4}. Phys. Rev. B**2009**, 80, 132408. [Google Scholar] [CrossRef] - Chapon, L.; Manuel, P.; Radaelli, P.G.; Benson, C.; Perrott, L.; Ansell, S.; Rhodes, N.J.; Raspino, D.; Duxbury, D.; Spill, E.; et al. Wish: The new powder and single crystal magnetic diffractometer on the second target station. Neutron News
**2011**, 22, 22–25. [Google Scholar] [CrossRef] - Campbell, B.J.; Stokes, H.T.; Tanner, D.E.; Hatch, D.M. ISODISPLACE: A web-based tool for exploring structural distortions. J. Appl. Cryst.
**2006**, 39, 607–614. [Google Scholar] [CrossRef] - Perez-Mato, J.M.; Gallego, S.V.; Tasci, E.S.; Elcoro, L.; de la Flor, G.; Aroyo, M.I. Symmetry-based computational tools for magnetic crystallography. Annu. Rev. Mater. Res.
**2015**, 45, 217–248. [Google Scholar] [CrossRef] - Petricek, V.; Dusek, M.; Palatinus, L. Crystallographic computing system JANA2006: General features. Z. Kristallogr. Cryst. Mater.
**2014**, 229, 345–352. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The crystal structure of LuFe

_{2}O

_{4}. (

**b**) Two adjacent triangular FeO

_{5}layers shifted by $\sqrt{3}/3$ a within the (ab) plane. (

**c**) The Fe

^{2+}-(A) and Fe

^{3+}-rich (B) monolayer charge models. Oxygen ions are omitted for clarity.

**Figure 2.**(

**a**) Magnetic susceptibility of LuFe

_{2}O

_{4+δ}measured with an applied field of 1 T parallel to the c-axis. Insets show the magnetic hysteresis loops at 200 and 300 K. (

**b**) The strong spin correlations along 1/3 1/3 L rod at 5 K and 0 T. The absence of intensity when L < −0.6 and >1 is an artefact of the magnet coverage.

**Figure 3.**Temperature dependence of the integrated intensities of (

**a**) the diffuse scattering at 1/3 1/3 L and (

**b**) the 110 reflection at 150 K. The inset shows the 110 reflection at 165, 175 and 220 K. (

**c**) Schematic drawing of a spin configuration with a net ferrimagnetic moment within a layer (upper panel) and a stacking of two adjacent monolayers into a bilayer (lower panel). Red (resp. blue) spheres represent up (resp. down) spins for the lower diagram.

**Figure 4.**(

**a**,

**b**) Neutron scattering rod along 1/3 1/3 L direction under 0 and 5 T at 150 K. The inset shows the incommensurate character of 1/3 1/3 0 reflection when magnetic field of 5 T is applied along the c-axis. (

**c**) Magnetic field dependence of neutron scattering along 1/3 1/3 L direction at 150 K. The inset demonstrates the magnetic field dependence of the integrated intensities around the 1/3 1/3 1/2 position and the 110 reflections at 150 K. (

**d**–

**f**) Temperature dependencies of the integrated intensities of the 1/3 1/3 0, 1/3 1/3 1/2 and 110 reflections at 150 K under 5 T.

**Table 1.**Irreducible representations for generating symmetry elements of the R-3m space group, associated with the propagation vector

**k**= (1/3, 1/3, 3/2).

_{2}Irrep | {3+|0,0,0} | {2_{xx}|0,0,0} | {−1|0,0,0} | |
---|---|---|---|---|

Γ_{2}^{+} | 1 | −1 | 1 | |

Y1 | $\left(\begin{array}{ccc}\begin{array}{cc}0& 0\\ 1& 0\end{array}& \begin{array}{cc}1& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 1\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}\end{array}\right)$ | $\left(\begin{array}{ccc}\begin{array}{cc}1& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 1& 0\end{array}\end{array}\right)$ | $\left(\begin{array}{ccc}\begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}\\ \begin{array}{cc}0& 0\\ 1& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 0& 0\end{array}\\ \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\end{array}\right)$ | |

Y2 | $\left(\begin{array}{ccc}\begin{array}{cc}0& 0\\ 1& 0\end{array}& \begin{array}{cc}1& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 1\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}\end{array}\right)$ | $\left(\begin{array}{ccc}\begin{array}{cc}-1& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ -1& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& -1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& -1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& -1\\ -1& 0\end{array}\end{array}\right)$ | $\left(\begin{array}{ccc}\begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}\\ \begin{array}{cc}0& 0\\ 1& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 0& 0\end{array}\\ \begin{array}{cc}0& 1\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\end{array}\right)$ | |

Irrep | {1|1,0,0} | {1|0,0,1} | ||

Γ_{2}^{+} | 1 | 1 | ||

Y1 | $\left(\begin{array}{ccc}\begin{array}{cc}{e}^{-\frac{2}{3}\pi i}& 0\\ 0& {e}^{\frac{2}{3}\pi i}\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}{e}^{-\frac{2}{3}\pi i}& 0\\ 0& {e}^{\frac{2}{3}\pi i}\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}{e}^{-\frac{2}{3}\pi i}& 0\\ 0& {e}^{\frac{2}{3}\pi i}\end{array}\end{array}\right)$ | $\left(\begin{array}{ccc}\begin{array}{cc}-1& 0\\ 0& -1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}-1& 0\\ 0& -1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}-1& 0\\ 0& -1\end{array}\end{array}\right)$ | ||

Y2 | $\left(\begin{array}{ccc}\begin{array}{cc}{e}^{-\frac{2}{3}\pi i}& 0\\ 0& {e}^{\frac{2}{3}\pi i}\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}{e}^{-\frac{2}{3}\pi i}& 0\\ 0& {e}^{\frac{2}{3}\pi i}\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}{e}^{-\frac{2}{3}\pi i}& 0\\ 0& {e}^{\frac{2}{3}\pi i}\end{array}\end{array}\right)$ | $\left(\begin{array}{ccc}\begin{array}{cc}-1& 0\\ 0& -1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}-1& 0\\ 0& -1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}-1& 0\\ 0& -1\end{array}\end{array}\right)$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ding, L.; Orlandi, F.; Khalyavin, D.D.; Boothroyd, A.T.; Prabhakaran, D.; Balakrishnan, G.; Manuel, P.
Coupling between Spin and Charge Order Driven by Magnetic Field in Triangular Ising System LuFe_{2}O_{4+δ}. *Crystals* **2018**, *8*, 88.
https://doi.org/10.3390/cryst8020088

**AMA Style**

Ding L, Orlandi F, Khalyavin DD, Boothroyd AT, Prabhakaran D, Balakrishnan G, Manuel P.
Coupling between Spin and Charge Order Driven by Magnetic Field in Triangular Ising System LuFe_{2}O_{4+δ}. *Crystals*. 2018; 8(2):88.
https://doi.org/10.3390/cryst8020088

**Chicago/Turabian Style**

Ding, Lei, Fabio Orlandi, Dmitry D. Khalyavin, Andrew T. Boothroyd, Dharmalingam Prabhakaran, Geetha Balakrishnan, and Pascal Manuel.
2018. "Coupling between Spin and Charge Order Driven by Magnetic Field in Triangular Ising System LuFe_{2}O_{4+δ}" *Crystals* 8, no. 2: 88.
https://doi.org/10.3390/cryst8020088