Magnetic and crystal structures of the magnetoelectric pyroxene LiCrSi2O6

HAL Id: hal-00967503

https://hal.archives-ouvertes.fr/hal-00967503

Submitted on 31 Mar 2014

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Magnetic and crystal structures of the magnetoelectric

pyroxene LiCrSi2O6

Gwilherm Nenert, M. Isobe, Clemens Ritter, Olivier Isnard, A.-N. Vasiliev, Y.

Ueda

To cite this version:

Gwilherm Nenert, M. Isobe, Clemens Ritter, Olivier Isnard, A.-N. Vasiliev, et al.. Magnetic and

crystal structures of the magnetoelectric pyroxene LiCrSi2O6. Physical Review B: Condensed Matter

and Materials Physics (1998-2015), American Physical Society, 2009, 79, pp.064416.

�10.1103/Phys-RevB.79.064416�. �hal-00967503�

Magnetic and crystal structures of the magnetoelectric pyroxene LiCrSi

₂

O

₆

G. Nénert,1,

_*

_{M. Isobe,}2_{C. Ritter,}3_{O. Isnard,}4_{A. N. Vasiliev,}5_{and Y. Ueda}2 1_{CEA-Grenoble INAC/SPSMS/MDN, 17 rue des martyrs, 38054 Grenoble, Cedex 9, France} 2_{Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwa, Chiba 277-8581, Japan}

3_{Institut Laue-Langevin, Boîte Postale 156, 38042 Grenoble, Cedex 9, France} 4_{Institut Néel, CNRS, Université Joseph Fourier, Boîte Postale 166, 38042 Grenoble, France}

5_{Low Temperature Physics Department, Moscow State University, Moscow 119991, Russia}

共Received 6 October 2008; revised manuscript received 29 January 2009; published 24 February 2009兲

We investigated the magnetic and crystal structures of the recent reported magnetoelectric system LiCrSi2O6

by powder neutron diffraction. Below T_N= 11.5 K, an antiferromagnetic order appears. It is characterized by an antiferromagnetic coupling within the CrO₆octahedra chains and a ferromagnetic coupling between the chains. The magnetic order is commensurate with the lattice with k = 0. The associated magnetic space group is P2₁⬘/c. This symmetry is in agreement with the reported magnetoelectric effect. We show that the magnetic frustration in this system is small. Finally, we discuss our results using a Landau phenomenological model and in the light of the literature.

DOI:10.1103/PhysRevB.79.064416 PACS number共s兲: 75.25.⫹z, 91.60.Pn, 75.50.Ee

I. INTRODUCTION

In recent years, the coupling between magnetic and di-electric properties in transition-metal oxides gave rise to a significant research effort.1–3 _{This effort is governed by the} emergence of new fundamental physics and potential techno-logical applications.2–4 _{Multiferroic materials exhibit} simul-taneously 共ferro兲magnetic, pyroelectric, and ferroelastic properties. Contrary to multiferroic materials, magnetoelec-tric materials show an induced elecmagnetoelec-trical polarization by a magnetic field. A proper understanding of the interplay be-tween the various physical properties of these two types of materials relies heavily on the knowledge of the detailed crystal and magnetic structures.

Recently pyroxene materials with the general formula

AMSi₂O₆共A=Li,Na;M=Fe,Cr兲 have been reported as mul-tiferroic materials.5This family of materials provides a large playground for physicists in condensed matter where A may be alkali or alkaline-earth elements, M being various metals with valence state 2+ or 3+ and Si4+ _{can be replaced by}

Ge4+_{. Due to the presence of chains of octahedra which can}

be magnetic, this family of materials has attracted much at-tention. It exhibits interesting properties such as spin gap system in NaTiSi2O6共Ref.6兲 and low dimensional magnetic

system in LiVGe₂O₆.7

While these pyroxenes have been the subject of various studies,5–12_{the complete magnetic structures are known only} for few members of this class.7,11–13 _{In order to understand} and interpret properly the interplay between the dielectric and magnetic properties of this new multiferroic family, the determination of the magnetic structure of their members is necessary. Jodlauk et al.5_{suggested that several members of} the pyroxene family among which LiCrSi2O6exhibit an

in-commensurate magnetic structure. This incommensurability of the magnetic structure would be the result of the geometri-cal magnetic frustration present in this family and would explain the reported multiferroic properties.

In this contribution, we have investigated the crystal and magnetic structures of LiCrSi₂O₆ using powder neutron

dif-fraction as function of temperature and magnetic field. We show that LiCrSi2O6 exhibits a rather unexpectedly simple

magnetic structure characterized by k = 0 with a magnetic symmetry compatible with a linear magnetoelectric effect. We discuss this linear magnetoelectric effect using a Landau phenomenological model. Finally we discuss the literature. We illustrate through the known reported magnetic structure of several pyroxenes that all of them exhibit a simple mag-netic structure with k = 0. In particular, we show that the reported magnetic structure of LiFeSi₂O₆ is incompatible with the reported magnetoelectric effect.

II. EXPERIMENT

Polycrystalline samples of LiCrSi₂O₆were prepared by a solid-state reaction with an appropriate molar ratio of Li2CO3, Cr2O3, and SiO2. The weighted mixtures were

pressed into pellets and heated at 1273 K in air for several days with one intermediate grinding.

Neutron-diffraction measurements were carried out on powder LiCrSi2O6. The data were collected with the

double-axis multicounter high-flux diffractometer D1B at the Institut Laue-Langevin ILL Grenoble using 2.52 Å wavelength se-lected by a pyrolytic graphite monochromator. In the con-figuration used, the resolution of D1B was about full width at

half maximum 共FWHM兲 ⬃0.3°. The multicounter is

com-posed of 400 cells covering a total angular domain of 80°共in 2␪兲. The data collection was made in the temperature range 1.8⬍T⬍15 K. The stoichiometry of the compound as well as the precise crystal structure was investigated using high-resolution powder data at room temperature, 15 and 1.8 K from D2B diffractometer at the ILL. The measurements were carried out at a wavelength of 1.594 Å corresponding to the 共335兲 Bragg reflexion of a germanium monochromator. The neutron detection is performed with 3He counting tubes spaced at 1.25° intervals for D2B. A complete diffraction pattern is obtained after about 25 steps of 0.05° in 2␪. D1B and D2B are powder diffractometers operating with the take-off angle of the monochromator at 44° and 135° 共in 2␪兲,

respectively. Neutron-diffraction measurements have also been recorded on LiCrSi2O6samples in magnetic fields of up

to 6 T at 2 K using a cryomagnet. The powder was pressed in tablets in order to preclude the grains’ movement in the mag-netic field. Diffraction data analysis was done using the

FULLPROFrefinement package.14

III. RESULTS A. Crystal structure

The pattern was refined in the space group P121/c1,

tak-ing as starttak-ing structural model the structure reported by Redhammer and Roth.9

The refined lattice parameters are a = 9.5355共3兲 Å, b = 8.5809共3兲 Å, c=5.24898共16兲 Å, and ␤= 109.9303共5兲° at room temperature. Within the error bars, our refinement at room temperature is similar to the single crystal reported previously.9_{Since the crystal structure of the pyroxene} fam-ily has been investigated in details already in the literature, we shall present here only our results at 15 K. The good agreement between the calculated and observed patterns at 15 K is presented in Fig. 1.

The most characteristic parameters after the refinement are listed in TableI. A selection of the most important atomic distances and bonding angles are included in Tables II and

III. Attempts of refinements of Cr and Li occupancies did not give evidence for the formation of vacancies neither for an-tisite disorder.

We recall here for the general reader the main structural features of the pyroxene family. This system exhibits a chain of edge-sharing CrO₆octahedra running along the crystallo-graphic axis c. These quasi-one-dimensional chains are con-nected by chains of SiO4tetrahedra. The packing of the CrO6

octahedra chains linked by SiO4tetrahedra chains gives rise

to a triangular magnetic lattice.

The SiO₄ tetrahedra form infinite chains running parallel to the c axis and are interconnected by the O_3a 共or O_3b兲 oxygen atoms via shared corners. One chain is made of Si1,

O1a, O2a, and O3a while the other one is made of Si2, O1b,

O2b, and O3b. They form two distinct chains in the P21/c

symmetry and one single chain in the high temperature form with the C2/c symmetry. At 15 K, the S-rotated tetrahedra chains made from Si1 are characterized by the angle

O3a-O3a-O3a= and the angle O3b-O3b-O3b for the O-rotated

chains made from Si2. The O3a-O3a-O3aangle is of 194.7共1兲°

and the O_3b-O_3b-O_3bangle is of 156.3共1兲° at 15 K.

Within the CrO6chains, there are two different magnetic

interaction paths: Cr-O1a-Cr and Cr-O1b-Cr. The respective

angles are 98.4共1兲° and 97.8共1兲° at room temperature. Within the error bars, these angles remain unchanged while lowering the temperature down to 1.8 K.

B. Determination of the magnetic structure

The temperature dependence of the neutron powder-diffraction patterns collected as function of temperature on

the D1B diffractometer is shown in Fig. 2. Below T

⯝11.5 K, new diffraction peaks appear while some Bragg peaks increase in intensity. This confirms the appearance of a magnetic ordering below T = 11.5 K, in good agreement with

TABLE II. Selected bond distances at room temperature in Å. Cr3+_O 6 Si14+O4 Si24+O4 Cr-O_1a 2.038共5兲 Si₁-O_1a 1.647共4兲 Si₂-O_1b 1.644共4兲 Cr-O1a 2.013共4兲 Si1-O2a 1.582共4兲 Si2-O2b 1.597共4兲 Cr-O_1b 2.053共5兲 Si₁-O_3a 1.641共4兲 Si₂-O_3b 1.630共4兲 Cr-O1b 1.980共4兲 Si1-O3a 1.613共4兲 Si2-O3b 1.638共5兲 Cr-O_2a 1.919共5兲 Cr-O2b 1.920共5兲

FIG. 1. 共Color online兲 The neutron pattern 共␭=1.594 Å兲 of LiCrSi2O6sample collected at 15 K using the D2B diffractometer.

The refinement has been done in the P2₁/c space group with

a = 9.5121共5兲 Å, b = 8.5728共4兲 Å, c = 5.2241共3兲 Å and ␤ = 109.7657共6兲° with the following statistics 共corrected from back-ground兲: Rp= 4.25% and Rwp= 5.50%.

TABLE I. Crystallographic coordinates extracted from the Rietveld refinement carried out on powder neutron diffraction 共D2B兲 using the space group P121/c1 at 15 K. The associated cell

parameters are a = 9.5121共5兲 Å, b=8.5728共4兲 Å, c=5.2241共3兲 Å, and␤=109.7657共6兲°. Atom Wyckoff x y z Li 4e 0.2511共10兲 0.0109共7兲 0.2294共16兲 Cr 4e 0.2515共5兲 0.6582共4兲 0.2348共8兲 Si1 4e 0.0499共3兲 0.3412共4兲 0.2742共5兲 Si₂ 4e 0.5505共4兲 0.8409共4兲 0.2479共6兲 O1a 4e 0.8661共3兲 0.3327共3兲 0.1658共5兲 O1b 4e 0.3670共3兲 0.8348共3兲 0.1289共5兲 O_2a 4e 0.1172共3兲 0.5118共3兲 0.3079共5兲 O2b 4e 0.6249共3兲 0.0059共3兲 0.3558共4兲 O_3a 4e 0.1099共3兲 0.2696共3兲 0.5838共5兲 O3b 4e 0.6068共2兲 0.7180共3兲 0.5003共5兲

NÉNERT et al. PHYSICAL REVIEW B 79, 064416共2009兲

the magnetic data.15_{The magnetic reflections can be indexed} by the propagation vector k = 0.

The possible magnetic structures compatible with the symmetry of LiCrSi₂O₆are determined by following the rep-resentation analysis technique described by Bertaut.16_{For the} propagation vector k = 0, the small group Gk, formed by

those elements of the space group that leave k invariant, coincides with the space group P21/c. For k=0, the

irreduc-ible representations of the group Gkare those shown in Table

IV.18

A representation ⌫ is constructed with the Fourier com-ponents mk _{corresponding to the Cr atoms of the Wyckoff}

position 4e. The Cr atoms at the site 4e are denoted as 共1兲 共x,y,z兲, 共2兲 共x¯,y+1

2, z¯ + 1

2兲, 共3兲 共x¯,y¯,z¯兲, and 共4兲 共x,y¯+ 1 2, z +1₂兲. The decomposition of the representation ⌫ in terms of

the irreducible representations⌫kis for the 4e site,

⌫共4e兲 = 3⌫1+ 3⌫2+ 3⌫3+ 3⌫4. 共1兲

We have four Cr3+ ions per unit cell. Consequently, there are four different possible couplings between them: one fer-romagnetic configuration and three antiferfer-romagnetic ones. Equation共2兲 shows these four possibilities. The different

ba-sis vectors associated with each irreducible representation and calculated by using the projection operator technique implemented in BASIREPS.17 _{They are presented in Table V.} There are four possible magnetic structures. We have carried out refinements at the lowest measured temperature on D2B 共T=1.8 K兲 with the four possible magnetic structures. We present in Fig. 3 the results of the refinements. The associ-ated Rmag factors associated to the models ⌫j are 24.8%,

10.7%, 53.4%, and 7.7% for⌫₁,⌫₂,⌫₃, and⌫₄, respectively M = S₁+ S₂+ S₃+ S₄,

L1= S1− S2+ S3− S4,

L₂= S₁+ S₂− S₃− S₄,

L3= S1− S2− S3+ S4. 共2兲

According to the experimental results the magnetic struc-ture is given by the basis vectors of the irreducible represen-tation ⌫4. This corresponds to a magnetic structure with the

moments oriented mostly in the ac plane with a small com-ponent along the b axis, as shown in Fig. 4. The chains of CrO6octahedra are ferromagnetically coupled while the

cou-pling within the chains is antiferromagnetic. The thermal evolution of the magnetic moments is displayed in Fig. 5. The magnetic components of the Cr3+_{ion determined using}

the D2B data at 1.8 K are L_2x= 2.12共4兲␮B; L3y= 0.29共8兲␮B, and L2z= 1.15共8兲␮B. The values of the magnetic moments

reach saturation below 5 K and the orientation of the mo-ments with respect to the a axis remains nearly constant below TN. The magnetic moment at T = 1.8 K is 储␮共Cr3+_兲储

= 2.06共4兲␮B. This is about 16% of reduction compared to the saturation of the magnetic form factor of Cr3+ determined experimentally which is 2.5␮B.19_{This reduced magnetic} mo-ment could be interpreted as a result from the geometrical magnetic frustration presents in this system.

In Fig.6, we show the temperature dependence of the cell parameters obtained from refinements of the D1B data. We observe no significant changes on passing through the Néel temperature. This observation suggests that the magnetostric-tion is extremely small in this material. This is confirmed by the refinements of the high-resolution data obtained at 1.8 and 15 K on the diffractometer D2B.

TABLE III. Selected bond angles at room temperature in degrees. Cr-O1a-Cr 97.7共2兲 Cr-O_1b-Cr 98.2共2兲 O1b-Cr-O1a 176.0共2兲 O_1b-Cr-O_1b 95.6共2兲 O1b-Cr-O2a 89.2共2兲 O_1b-Cr-O_2b 96.1共2兲

TABLE IV. Irreducible representations of the space group P21/c

for k = 0. The symmetry elements are written according to Kova-lev’s notation共Ref.18兲,␶=共0,1₂,1₂兲.

h1 h3/共␶兲 h25/共␶兲 h27/共␶兲

⌫1 1 1 1 1

⌫2 1 1 −1 −1

⌫3 1 −1 1 −1

⌫4 1 −1 −1 1

TABLE V. Basis vectors for the atoms of the 4e site. Basis vectors x y z

⌫1 L1x My L1z

⌫2 L3x L2y L3z

⌫3 Mx L1y Mz

⌫4 L2x L3y L2z

FIG. 2. Neutron diffraction patterns collected on the D1B dif-fractometer in the temperature range 1.8–15 K.

C. Study of the magnetic field dependence

In the work of Jodlauk et al.,5 the authors claimed the existence of a structural phase transition induced by the mag-netic field giving rise to a ferroelectric phase. Although these authors did not give any evidence for this statement共no hys-teresis polarization loop was recorded兲, we have investigated the magnetic and crystal structures as function of magnetic field. We have measured LiCrSi2O6at T = 2 K on the

diffrac-tometer D2B under magnetic field using a cryomagnet up to

6 T. The powder was pressed in tablets in order to preclude the grains’ movement in the magnetic field. We present a selected area of the recorded patterns where the changes as function of the magnetic field are the most important in Fig.

7.

The effect of the magnetic field is to reduce the magnetic moment from 2.06共4兲␮B at zero field to 1.93共4兲␮B at 6 T. This tendency is clearly confirmed by the progressive

reduc-FIG. 3. 共Color online兲 Fragment of the D2B neutron-diffraction pattern of LiCrSi2O6at 1.8 K and refined by using the models with

different orientation of the Cr magnetic moment.

FIG. 4. 共Color online兲 Magnetic structure of LiCrSi₂O₆. The magnetic coupling between the CrO6chains is ferromagnetic while

the magnetic coupling within the chains is antiferromagnetic.

FIG. 5. Temperature dependence of the magnetic moment of Cr3+_{ions obtained from the refinement results of the D1B neutron}

diffractometer共␭=2.52 Å兲.

NÉNERT et al. PHYSICAL REVIEW B 79, 064416共2009兲

tion in the magnetic intensity upon increasing magnetic field 共see inset Fig. 7兲. Within the resolution of our data, we did

not find any evidence for a magnetically induced ferroelec-tric phase. Our results suggest that LiCrSi₂O₆ is a linear magnetoelectric material in agreement with the magnetic structure at zero magnetic field.

IV. PHENOMENOLOGICAL MODEL

In this section we describe a phenomenological model describing the magnetic ordering displayed by LiCrSi₂O₆ be-low its Néel temperature. In addition, we investigate the cou-pling terms giving rise to the linear magnetoelectric effect in this material.

We use here the notations for collinear structures de-scribed in Eq.共2兲 and the magnetic modes defined in Table V. The magnetic structure below TNis described by the irre-ducible representation ⌫4. This irreducible representation is

characterized by the magnetic modes L2x, L3y, and L2z.

Ex-perimentally, we have determined that the contribution L3yis

smaller by a factor 4 to 7 compared to L_2x or L_2z. Conse-quently in our phenomenological treatment, we will consider that only L2 has nonzero values below TN. We can

decom-pose the expression of the free energy⌽ in several contribu-tions: ⌽=⌽₀+⌽_ex+⌽_rel+⌽_me+⌽H.20,21 _{The various}

cou-pling terms between the various order parameters are derived from symmetry considerations. Every product of the axial-vector components belonging to the same irreducible repre-sentation is invariant by time reversal and the crystal symmetry.20 _{The minimization of the exchange part ⌽}

ex of

the free energy determines only the relative orientation of the magnetic moments Sj with respect to each other

⌽ex= ␨ 2L2 2 +␰ 4L2 4 +

兺

i ␹i 2Mi 2 . 共3兲

The variable ␨ is the only parameter depending of the temperature such that ␨=␨0共T−TN兲. The

␹i

2 terms are the

components of the inverse magnetic susceptibility. The mini-mization of the relativistic part ⌽_rel of the free energy will determine the orientation of the Sjwith respect to the crystal axes20 ⌽rel= 1 2

兺

i ␯iL2i 2 . 共4兲

The reported magnetoelectric effect in LiCrSi₂O₆requires the introduction in the thermodynamic potential of two addi-tional contributions. One contribution must contain the pos-sible magnetoelectric coupling terms which are of the type

LiMjPk. These terms are the couplings between the sublattice magnetization Li, the total magnetization Mj, and the electri-cal polarization Pk. The other contribution is the magnetic energy21 ⌽me=␭1共L2x+ L2z兲PxMy+␭2共L2x+ L2z兲PzMy +␭₃共L_2x+ L_2z兲PyMx+␭4共L2x+ L2z兲PyMz+

兺

i,j ␬i,j 2 Pj 2_, 共5兲

FIG. 6. Temperature dependence of the cell parameters as func-tion of temperature in the range 1.8–18 K from the D1B data re-finements共␭=2.52 Å兲. In 共a兲, we show the cell parameters a and b and in共b兲 the cell parameter c and the angle␤.

FIG. 7.共Color online兲 Neutron-diffraction patterns recorded at 2 K on pressed pellets of LiCrSi₂O₆in magnetic field up to 6 T. In black, measurement was done with H = 0 T, H = 4 T, and H = 6 T. The inset shows a zoom of the reflection共110兲 on which the varia-tion of intensity is the most important.

⌽H= − M · H. 共6兲 The ␭i terms in ⌽_me, which express the coupling of the order parameter共the antiferromagnetic order induced by L2兲

with the electrical polarization P and the magnetization M, are of relativistic origin. The terms ␬i,j

2 are the tensor

compo-nents of the inverse dielectric susceptibility. In⌽me, we have

also included quadratic terms in P which correspond to the dielectric energy.⌽Hwill be taken into account for discuss-ing the effect of the applied magnetic field.

Minimization of ⌽ with respect to the components of P and M yields the following relations:

Px= − ␭1LxLzMy ␬x , Py= −␭3LxLzMx−␭4LxLzMz ␬y , Pz= − ␭2LxLzMx ␬z , 共7兲 Mx= Hx−␭3LxLzPy ␹x , My= Hy−␭1LxLzPx−␭2LxLzPz ␹y , Mz= Hz−␭4LxLzPy ␹z . 共8兲

As a consequence of relations given in Eqs. 共7兲 and 共8兲,

one can work out the magnetic field induced polarization along the following three axes:

Px= −␭1␬zL2xL2z ␹y␬x␬z−共␭21␬z+␭22␬x兲L2x 2 L_2z2 Hy, Py= −␭3␹zL2xL2z ␹x␹z␬y−共␭3 2_␹ z+␭4 2_␹x兲L 2x 2 L_2z2 Hx + −␭4␹xL2xL2z ␹x␹z␬y−共␭32␹z+␭42␹x兲L2x2L2z2 Hz, Pz= −␭2␬xL2xL2z ␹y␬x␬z−共␭1 2_␬ z+␭2 2_␬x兲L 2x 2 L_2z2 Hy. 共9兲

Using the results of Eq.共9兲, we can write down the linear

magnetoelectric tensor Pi=␣ijHj corresponding to the irre-ducible representation⌫₄ 共magnetic space group P2₁

⬘

/c兲,

␣ij=

冢

0 ␣12 0

␣21 0 ␣23

0 ␣₃₂ 0

冣

.

We note that this magnetoelectric tensor is in agreement with the linear magnetoelectric effect reported in LiCrSi2O6.5

Indeed Jodlauk et al. reported an induced electrical

polariza-tion along the b axis while applying a magnetic field along the c axis. This corresponds to the linear magnetoelectric tensor term␣₂₃. This observation confirms that the magnetic structure that we have determined is in agreement with the reported linear magnetoelectric effect. Our phenomenologi-cal model is in agreement with the absence of field induced structural transition. The temperature dependence of␣ijcan be modeled using Eq.共9兲. Knowing that L2_{is proportional to}

␨, we are able to find the temperature dependence of␣ij. This is given in Fig.8.

Our phenomenological model also gives information about the two magnetic components of L2becoming nonzero

below TN. If one assumes that␭i2are small, one can work a simple expression for L2x and L2z,

L_2x2 =−␨共␰+␥兲 +␥␯z+␰␯x ␰2_−␥2 , L_2z2 =−␨共␰+␥兲 +␥␯x+␰␯z ␰2_−␥2 , 共10兲 where ␥=

冢

Hy 2

冉

␭12 ␬x +␭2 2 ␬z

冊

␹y 2 + 共␹zHx␭3+␹xHz␭4兲2 ␹x 2_␹ z 2_␬ y

冣

.

If there is no applied magnetic field, we find the usual expression for a magnetic component below its Néel temperature,20

L_2x2 =−␨+␯x

␰ ,

L_2z2 =−␨+␯z

␰ . 共11兲

Looking at Eq. 共11兲, one can notice that the relativistic

contribution is important 共terms in ␯x and ␯z兲 in order to describe properly the magnetic order below TN共L2x⫽L2z兲.

V. DISCUSSION

The crystal structure solution can be confirmed by calcu-lation of the bond valence sum共BVS兲 for the various ions.22

0 2 4 6 8 10 0.00 0.01 0.02 0.03 0.04 0.05 Temperature
K Magnetoelectric Tensor Αij
arb.units 

FIG. 8. Temperature dependence of the magnetoelectric coeffi-cients␣ijgiven by Eq.共9兲 for TN= 11.5 K.

NÉNERT et al. PHYSICAL REVIEW B 79, 064416共2009兲

The Cr3+ _{ion is almost saturated 共bond valence sum}

关⌺s=2.95共2兲兴 and the Si4+ _{ions are also saturated 关⌺s}

= 4.04共2兲兴. The Li cation in pyroxenes generally is under-bonded and possess valence sums of ⌺s=0.79–0.82.10 _This is in agreement with our refinements at room temperature with⌺s=0.801共8兲. It is assumed that the long Li-O3bond is

responsible for the low valence sum 共coordination number = 5, Li-O1a= 2.071 Å; Li-O1b= 2.083 Å; Li-O2a= 2.183 Å;

Li-O2b= 2.115 Å; Li-O3a= 2.318 Å兲.10 The O1a, O1b, O3a,

and O3b oxygen atoms also are saturated in charge 关⌺s

= 2.046共13兲, 2.021共12兲, 2.077共13兲, and 1.990共13兲, respec-tively兴. However, the O2a and O2b oxygen atoms possess a

distinct deficit in negative charge 关underbonded, ⌺s = 1.849共12兲 and 1.851共13兲 respectively兴. The apparent deficit in negative charge is compensated for by the short Sij-O2j

and Li-O_2jbond lengths共see also TableII兲.

Neutron powder-diffraction experiments confirm that LiCrSi₂O₆ exhibits an antiferromagnetic ordering below TN = 11.5 K. The magnetic structure is commensurate with the chemical unit cell with k = 0 with a magnetic moment of Cr= 2.06共4兲␮B. The magnetic space group displayed by LiCrSi₂O₆is P2₁

⬘

/c allowing a linear magnetoelectric effect in agreement with the reported induced electrical polariza-tion along the b axis when a magnetic field is applied along the c axis.5,23 _{Jodlauk et al.}15 _{suggested that this material} exhibits a spiral magnetic structure resulting from the mag-netic frustration. Our results show that this is not the case and that the magnetic frustration is small共magnetic moment reduced only by 16%兲 in agreement with the ratio 兩T␪N兩=2.6.

This reduced magnetic moment is similar to what has been reported for LiFeSi2O6.11While magnetic frustration is

ob-vious from considerations on the crystal structure of these pyroxenes, this family is not a good example of a spatially anisotropic triangular lattice24_{contrary to the expectations of} Jodlauk et al.5_{This discrepancy can be explained by the fact} that in the model proposed by Zhang et al., there are only two antiferromagnetic coupling constants J and not three like in LiCrSi2O6as illustrated in Fig. 1 of Ref.5. This probably

explains the absence of spiral magnetic structure.

We notice also that our refinements suggest a magnetic symmetry which do not allow any ferromagnetic moment 共no magnetic Mj components belong to the irreducible rep-resentation ⌫4, see Table V兲. This is in perfect agreement

with the reported magnetoelectric effect. However, this is in opposition with the reported small ferromagnetic moment of 0.005␮B by Jodlauk et al.5 We believe that this moment is most likely the result of a small ferromagnetic impurity in their sample.

The linear magnetoelectric effect is strongly dependent on the magnetic symmetry of the system.23 _{Jodlauk et al. have} reported that they observe a magnetically induced electrical polarization not only in LiCrSi2O6 but also in LiFeSi2O6.

Several studies using powder and single crystal report a mag-netic structure described by the magmag-netic space group

P21/c.11,12This magnetic symmetry forbids any linear

mag-netoelectric effect since it contains the inversion symmetry.23 Consequently there is a discrepancy between the reported induced electrical polarization by Jodlauk et al. and the re-ported magnetic symmetry. Further investigations of the di-electric properties and reinvestigation of the magnetic struc-ture of LiFeSi2O6 are necessary in order to resolve this

discrepancy. Additionally, we notice that LiVGe2O6 has the

same magnetic structure as LiCrSi₂O₆.7 _{Consequently, the} magnetic symmetry of LiVGe2O6is compatible with the

ex-istence of a linear magnetoelectric effect. VI. CONCLUSION

We have investigated the magnetic and crystal structures of LiCrSi₂O₆as function of temperature and magnetic field using powder neutron diffraction. Below TN= 11.5 K, LiCrSi2O6 exhibits a long-range antiferromagnetic order

commensurate with the lattice with k = 0. It is characterized by antiferromagnetic coupling within the CrO₆chains and a ferromagnetic coupling between the chains. The associated magnetic symmetry is P2₁

⬘

/c in agreement with the reported magnetoelectric effect. The magnetic field has for conse-quence to reduce the magnetic moment from 2.06共4兲 at zero field to 1.93共4兲␮Bat H = 6 T. We propose a Landau phenom-enological model describing the linear magnetoelectric effect in this material. We show that further investigations are re-quired on the dielectric and magnetic properties of LiFeSi₂O₆ in order to understand the present discrepancies. Additionally, we notice that LiVGe2O6 has the same

mag-netic symmetry of LiCrSi2O6and thus should exhibit a linear

magnetoelectric effect.

ACKNOWLEDGMENTS

We acknowledge Jacques Schweizer for critical reading of the manuscript. This study is partly supported by a Grant-in Aid for Scientific Research No. 19052008 and RFBR Grant No. 07-02-91201 from the Ministry of Education, Culture, Sports, Science and Technology of Japan and for JSPS 共Japan兲-RFBR 共Russia兲 Joint Research Project.

*Corresponding author: Gwilherm Nénert. nenert@ill.eu. Present address: Institut Laue-Langevin, Boîte Postale 156, 6 rue Jules Horowitz, 38042 Grenoble Cedex 9, France.

1_{M. Fiebig, J. Phys. D 38, R123共2005兲.}

2_{W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 共London兲}

442, 759共2006兲.

3_{S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13共2007兲.}

4_{A. Pimenov, A. A. Mukhin, V. Yu. Ivanov, V. D. Travkin, A. M.}

Balbashov, and A. Loidl, Nat. Phys. 2, 97共2006兲; A. B. Sush-kov, R. V. Aguilar, S. Park, S. W. Cheong, and H. D. Drew, Phys. Rev. Lett. 98, 027202共2007兲.

5_{S. Jodlauk, P. Becker, J. A. Mydosh, D. I. Khomskii, T. Lorenz,}

S. V. Streltsov, D. C. Hezel, and L. Bohatý, J. Phys.: Condens. Matter 19, 432201共2007兲.

6_{M. Isobe, E. Ninomiya, A. N. Vasiliev, and Y. Ueda, J. Phys.}

Soc. Jpn. 71, 1423共2002兲; S. V. Streltsov, O. A. Popova, and D. I. Khomskii, Phys. Rev. Lett. 96, 249701共2006兲.

7_{M. D. Lumsden, G. E. Granroth, D. Mandrus, S. E. Nagler, J. R.}

Thompson, J. P. Castellan, and B. D. Gaulin, Phys. Rev. B 62, R9244共2000兲.

8_{M. Isobe and Y. Ueda, J. Magn. Magn. Mater. 272-276, 948}

共2004兲; S. V. Streltsov and D. I. Khomskii, Phys. Rev. B 77, 064405共2008兲; G. J. Redhammer, H. Ohashi, and G. Roth, Acta Crystallogr., Sect. B: Struct. Sci. 59, 730 共2003兲; C. Satto, P. Millet, and J. Galy, Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 53, 1727共1997兲; P. Millet, F. Mila, F. C. Zhang, M. Mambrini, A. B. Van Oosten, V. A. Pashchenko, A. Sulpice, and A. Stepanov, Phys. Rev. Lett. 83, 4176 共1999兲; E. Baum, W. Treutmann, M. Behruzi, W. Lottermoser, and G. Amthauer, Z. Kristallogr. 183, 273共1988兲; B. Pedrini, S. Wessel, J. L. Gavil-ano, H. R. Ott, S. M. Kazakov, and J. Karpinski, Eur. Phys. J. B

55, 219共2007兲; M. Isobe, Y. Ueda, A. N. Vasiliev, T. N.

Vo-loshok, and O. L. Ignatchik, J. Magn. Magn. Mater. 258-259, 125共2003兲; A. N. Vasiliev, O. L. Ignatchik, A. N. Sokolov, Z. Hiroi, M. Isobe, and Y. Ueda, JETP Lett. 78, 551共2003兲.

9_{G. J. Redhammer, and G. Roth, Z. Kristallogr. 219, 585共2004兲.}

Please note that the coordinates given in the paper contain an error. The x coordinate of Si₁is 0.0487 and not 0.487.

10_{G. J. Redhammer and G. Roth, Z. Kristallogr. 219, 278共2004兲.} 11_{G. J. Redhammer, G. Roth, W. Paulus, G. André, W.}

Lotter-moser, G. Amthauer, W. Treutmann, and B. Koppelhuber-Bitschnau, Phys. Chem. Miner. 28, 337共2001兲.

12_{W. Lottermoser, G. J. Redhammer, K. Forcher, G. Amthauer, W.}

Paulus, G. André, and W. Treutmann, Z. Kristallogr. 213, 101

共1998兲.

13_{O. Ballet, J. M. D. Coey, G. Fillion, A. Ghose, A. Hewat, and J.}

R. Regnard, Phys. Chem. Miner. 16, 672共1989兲.

14_{J. Rodriguez-Carvajal, Physica B 192, 55共1993兲.}

15_{A. N. Vasiliev, O. L. Ignatchik, A. N. Sokolov, Z. Hiroi, M.}

Isobe, and Y. Ueda, Phys. Rev. B 72, 012412共2005兲.

16_{E. F. Bertaut, in Magnetism, edited by G. T. Rado and H. Shul}

共Academic, New York, 1963兲, Vol. III, Chap. 4.

17_{J. Rodríguez-Carvajal, BasIreps: A program for calculating}

irre-ducible representations of space groups and basis functions for axial and polar vector properties 共see http://wwwold.ill.fr/dif/ Soft/fp/php/downloads.html兲.

18_{O. V. Kovalev, in Representations of the Crystallographic Space}

Groups: Irreducible Representations, Induced Representations and Corepresentations, edited by H. T. Stokes and D. M. Hatch

共Gordon and Breach, Amsterdam, 1993兲.

19_{P. J. Brown, J. B. Forsyth, and F. Tasset, Solid State Sci. 7, 682}

共2005兲.

20_{J.-C. Tolédano and P. Tolédano, The Landau Theory of Phase}

Transitions共World Scientific, Singapore, 1987兲.

21_{P. Toledano, Ferroelectrics 161, 257共1994兲.}

22_{N. E. Brese and M. O’Keefee, Acta Crystallogr. B 47, 192}

共1991兲; I. D. Brown, ibid. 48, 553 共1992兲.

23_{International Tables for Crystallography in Physical Properties}

of Crystals, edited by A. Authier 共Kluwer, Dordrecht, 2003兲,

Vol. D.

24_{W. M. Zhang, W. M. Saslow, and M. Gabay, Phys. Rev. B 44,}

5129共1991兲.

NÉNERT et al. PHYSICAL REVIEW B 79, 064416共2009兲