Magnetostriction and ferroelectric state in AgCrS 2
Sergey V. Streltsov, Alexander I. Poteryaev, Alexey N. Rubtsov
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Magnetostriction and ferroelectric state in AgCrS Sergey V. Streltsov,
1, 2
Alexander I. Poteryaev, and Alexey N. Rubtsov
4, 5 M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620137, Ekaterinburg, Russia Ural Federal University, Mira St. 19, 620002 Ekaterinburg, Russia ∗ Institute of Quantum Materials Science, Bazhova St. 51, Ekaterinburg 620075, Russia Department of Physics, Moscow State University, Moscow 119991, Russia Russian Quantum Center, Moscow 143025, Russia (Dated: July 30, 2018)The band structure calculations in the GGA+U approximation show the presence of additionallattice distortions in the magnetically ordered phase of AgCrS . The magnetostriction leads toformation of the long and short Cr-Cr bonds in the case when respective Cr ions have the same oropposite spin projections. These changes of the Cr lattice are accompanied by distortions of theCrS octahedra, which in its turn leads to development of the spontaneous electric polarization. PACS numbers: 75.25.-j, 75.30.Kz, 71.27.+a
I. INTRODUCTION
Transition metal oxides and sulfides with delafossitestructure are actively investigated nowadays due to di-verse physical properties observed in these materials. Forexample, CuMnO was found to show a quite strong de-pendence of magnetic properties on doping [1, 2], CuAlO is one of rare p − type transparent semiconductors [3, 4],CuFeO was intensively studied over past years due toits multiferroicity [5]. Another system with crystal struc-ture closely related to delafossites, AgCrS , was recentlyfound to be multiferroic [6]. However the mechanism ofthe coupling between electric and magnetic characteris-tics in this material is still unknown.The crystal structure of AgCrS is shown in Fig. 1.Triangular planes of Cr ions are stacked along c direc-tion. At very high temperatures it is characterized bythe space group R ¯3 m , which is centro-symmetric. Withdecrease of the temperature to T c =670 K it transformsto R m , which is still of high symmetry but with lack-ing inversion and thus noncentro-symmetric. However,spontaneous electric polarization is not observed downto N´eel temperature, T N ≈
41 K, where magnetically or-dered phase develops [6]. This low temperature phaseis characterized by monoclinic noncentro-symmetric Cm space group and doubling of unit cell [7]. Neverthelessthe general triangular plane geometry is preserved and itcan be regarded as distorted high symmetry. Magneticstructure for T < T N consists of the double ferromagnetic(FM) stripes coupled antiferromagnetically (see Fig. 4 inRef. [7]). This magnetic order is developed due to strongantiferromagnetic (AFM) exchange interaction betweenthird nearest neighbors [8].The absence of the electric polarization in R m phaseis ascribed to disorder of Ag + ions [6], that addition-ally complicates analysis of interplay between electric andmagnetic characteristics in AgCrS . Different scenarios ∗ Electronic address: [email protected] were proposed to get an insight about mechanism of mul-tiferroicity, [7, 9] but this riddle is far from being solved.In the present paper we performed the optimization ofthe crystal structure obtained previously in the experi-ments on powder samples of AgCrS and found the pres-ence of additional lattice distortions in the magneticallyordered phase. In the direction perpendicular to the mag-netic stripe (in the Cr planes) the chromium ions withthe opposite spins become closer to each other and theyare moved apart for the same spin projections. These dis-tortions result in the development of spontaneous electric FIG. 1: (color online) Crystal structure of AgCrS . Cr ionsforming triangular planes are shown in blue, while S and Agions are in yellow and grey colors, respectively. It is importantthat there are two crystallographically different sulfur ions:each Ag is connected with one S , but with three S , whichresults in different Cr-S bond lengths. polarization, which has a magnetostrictive origin there-fore. II. CALCULATION DETAILS
We used pseudo-potential Vienna ab initio simulationpackage (VASP) for the calculation of electronic andmagnetic properties of AgCrS [10]. The Perdew-Burke-Ernzerhof [11] version of the exchange-correlation poten-tials was utilized. The strong Coulomb correlations weretaken into account via the GGA+U method [12]. Theon-cite Coulomb interaction ( U ) and Hund’s rule cou-pling ( J H ) parameters were taken to be U =3.7 eV and J H =0.8 eV [13, 14]. The integration in the course ofthe self-consistency was performed over a mesh of 175 k -points in the irreducible part of the Brillouin-zone.The electric polarization was calculated using theBerry phase formalism [15, 16]. The crystal structuresof AgCrS were taken from Ref. 7. The results presentedin Sec. III were obtained for the data corresponding to T =10 K ( Cm ), while in Sec. IV both T =10 K ( Cm ) and T =300 K ( R m ) structures were used. III. ELECTRONIC AND MAGNETICPROPERTIES AND LATTICE DISTORTIONS
It is interesting to note that oxides based on Cr ionsare usually Mott insulators [13, 17, 18], where top ofthe valence band and bottom of the conduction band areformed by the Cr 3 d states, and increase of the Cr oxida-tion state up to 4+ is needed to move them on the vergebetween Mott and charge-transfer regimes [14, 19–21].The density of states calculated within GGA+U approx-imation for experimentally observed magnetic structureare presented in Fig. 2. An analysis of the GGA+U re-sults shows that Cr 3 d ↑ states are split and occupied partlies about -4...-2 eV while empty states that form bottomof conduction band are at 1.5...2 eV. The minority spinstates, Cr 3 d ↓ , are totally empty and located at 2...3 eV.Magnetic moment on Cr ions is 2.9 µ B which is in goodaccordance with the experimental value of 2.7 µ B [7].The top of valence band is formed predominantly by theS 3 p states, and thus, AgCrS is the charge-transfer in-sulator. This is related to much larger spatial extensionof the S 3 p orbitals compared with O 2 p . As a resultthe charge-transfer energy defined as energy costs for the d n p → d n +1 p transition is drastically decreased in sul-fides [22]. The effect of the on-site Coulomb repulsionis also important since it splits partially occupied Cr 3 d states and moves them away from the Fermi level. Theband gap in AgCrS for values of U used was found tobe ∼ ions (e.g. in Cr pyroxenes the band gapis ∼ Cm crystal structure each -6 -5 -4 -3 -2 -1 0 1 2 3 Energy (eV) -6-4-2024681012 D O S [ s t a t e s / ( e V f. u . ) ] Total DOSCr 3d E F FIG. 2: (color online) Total and partial density of states(DOS) for Cr 3 d obtained in the GGA+U calculation for theexperimental double stripe magnetic structure. Positive andnegative values of partial DOS correspond to spin majorityand minority. The Fermi energy is set to zero. chromium atom has six in-plane nearest neighbors: twoof them lie along b direction on 3.5 ˚A distance and fourCr are at 3.48 ˚A distance [7]. The long bonded along b di-rection Cr-Cr ions form ferromagnetically coupled chain,while these chains are magnetically ordered as ↑↑↓↓ (seeFig. 4 in Ref. [7]). It is important to note that accordingto experiment the inter-chain distance remains unalteredregardless the ↑↑ or ↑↓ magnetic ordering of differentchains. Naively thinking one may expect that magne-tostriction differentiates the distances between FM ( ↑↑ )and AFM ( ↑↓ ) coupled chains and therefore the crystalstructure could be somewhat different from the one re-ported previously.In order to clarify the discrepancy between experimentand theoretical considerations we carried out the struc-tural optimization of the low temperature phase usingGGA+U method. The total energy calculations for ex-perimental parameters display that double stripe AFMorder does not correspond to the ground state, being1.7 meV/f.u. higher than FM state. The relaxation ofthe atomic positions and lattice parameters dramaticallychanges this situation stabilizing the double stripe mag-netic structure and making this magnetic order the lowestin total energy.An analysis of the relaxed atomic positions for thedouble stripe AFM structure shows that the Cr-Cr dis-tance along stripe stays the same, d || =3.5 ˚A, while inperpendicular directions they change substantially: Cr-Cr bonds with the same spin projection are stretched, d ↑↑⊥ = d ↓↓ =3.52 ˚A, and with opposite spins are shrunk, d ↑↓⊥ =3.44 ˚A (see Fig. 3). However the relaxation does notlead to corrugation of the Cr planes leaving average Cr-Crdistance the same (3.48 ˚A) in the directions perpendic-ular to the stripe. Thus the effect of the magnetostric-tion for Cr-Cr bond lengths exceeds δd ⊥ = d ↑↑⊥ − d ↑↓⊥ ∼ .
04 ˚A for given U . For the FM order relaxation leadsto the minor changes of the crystal structure.It has to be noted that the effect of magnetostrictiondepends strongly on the value of the on-site Coulomb re-pulsion parameter U . For example, the decrease of U on 1 eV results in increase of δd ⊥ (up to 0 .
05 ˚A). Oneof the possible explanations could be that we gain mag-netic energy moving some of the Cr nearest neighborscloser together. For short Cr-Cr pairs corresponding ex-change interaction has to be AFM, since the direct ex-change should dominate over other contributions in theedge sharing geometry [8]. This is exactly observed inthe present calculations: Cr ions having opposite spinprojections are getting closer. Moreover, a decrease ofthe Coulomb interaction parameter increases this effectsince direct exchange is inverse proportional to it, ∼ /U .The gain in magnetic energy due to direct exchange in ↑↓ pairs is compensated by the growth of the elastic energyand decrease of the magnetic energy gain in ↑↑ ( ↓↓ ) Crpairs, so that an exact value of δd ⊥ depends on detailson different internal parameters of the system.It was found in Ref. [8] that in the LSDA (local spindensity approximation) the double stripe AFM struc-ture in AgCrS is stabilized due to strong AFM ex-change coupling between the third nearest neighbours.While the magnetostriction, as it was explained above,will certainly modify nearest neighbour interaction theexchange to the third nearest neighbours will not bechanged drastically because of unaltered average dis-tances in chromium plane. IV. ELECTRIC POLARIZATION
For the calculation of the electric polarization ( P )one needs to choose a reference structure and find thedifference ( δP ) in polarizations for given and referencestructures [24]. One usually takes the high temperaturecentro-symmetric lattice as the reference. However, asit was explained above in AgCrS the polarization ap-pears at the transition between two noncentro-symmetricstructures, R m and Cm . Therefore in spite of the factthat real reason of the absence of electric polarization inintermediate temperature R m structure is unknown [6],we are forced to consider this structure as the reference.The transition from paraelectric to ferroelectric stateis accompanied by the transition from paramagnet toAFM with formation of the long-range magnetic order.The paramagnetic state having local spins and short-range magnetic correlations can hardly be simulated inthe GGA or GGA+U calculations. In principle one maymodel this state with such a technique as averaging overdifferent spin configurations [25], but even in this casethis state is useless since for Cr with 3 d electronic con-figuration LSDA/GGA is expected to give a metal andhence one cannot calculate P . The nonmagnetic state FIG. 3: (color online) a) Distortions dS , dS , dS , dS (grey dashed arrows) which appear in experimentally ob-served double stripe AFM order due to magnetostriction.The direction perpendicular to double stipes ( ↑↑↓↓ ) is shown.Black arrows show the spin order. dS = dS = dS = dS . b) Distorted structure and corresponding electric po-larization. Note that since the distortions in two neighboringCrS octahedra are different, ~P and ~P ′ do not compensateeach other. could neither be used, since the electronic configurationof each Cr ion would be not ( d ↑ ) , but ( d ↑ ) . ( d ↓ ) . ,which is quite unnatural because of the absence of the lo-cal spin. The isotropic AFM state (all Cr neighbours areantiferromagnetically coupled) which would be the bestto simulate paramagnet is also impossible due to frus-trated triangular lattice. Therefore we simulate referenceparaelectric state by the R m crystal structure with theFM order.On the first stage we calculate δP due to appearanceof the double stripe AFM order in the R m structure,relaxing atomic positions both for the double stripe (2S)AFM and FM orders. This results in | δ ~P S → F M | ∼ µC/m and polarization was found to be directedperpendicular to the Cr triangular planes (i.e. alongrhombohedral c axis). We also checked that the use ofthe antiferromagnetically coupled AFM ( ↑↓↑↓ ) chains asthe reference state does not change the result.Since for the R m structure the lattice parameters areknown only for quite high temperature (300 K, muchabove transition to ferroelectric state) and the unit cellvolumes V KR m and V KCm are different, we repeated thesecalculations for the value corresponding to the lowestin temperature structure and found that | δ ~P S → F M | ∼ µC/m , i.e. practically does not change. Moreover,additional calculation for the R m structure, but withfixed ionic positions (i.e. there is only electronic contribu-tion to the polarization) gives | δ ~P elec S → F M | ∼ µC/m .I.e. it is a redistribution of the electronic charge densitydue to the double stripe AFM structure that mainly leadsto the development of the spontaneous electric polariza-tion in the low temperature phase of AgCrS . The latticefollows this tendency and changes the absolute value ofthe polarization.The double stripe magnetic structure was shown toinduce ferroelectric state due to specific lattice distor-tions in CdV O [26], which seems to be related withthe orbital-selective behavior [27, 28]. While the crystalstructure is quite different in case of AgCrS (delafossite-like instead of spinel), the microscopic mechanism beyondthe ferroelectric state is similar.In the R m crystal structure there are two inequiv-alent sulfur atoms (S and S in Fig. 1) with differentchromium-sulfur bond lengths [7]. In effect already inthe experimental R m structure the Cr-S -Cr and Cr-S -Cr triangles (formed by two neighboring Cr and oneof the common S) are also different in terms of angles andbond lengths Cr-S, as shown in Fig. 3. The magnetostric-tion results in the formation of short ( d ↑↓⊥ ) and long ( d ↑↑⊥ )metal-metal bonds (see Fig. 3). As a result two neigh-boring Cr ↑ -S -Cr ↑ and Cr ↑ -S -Cr ↓ triangles turn out tobe differently distorted (the same is valid for two adja-cent triangles with S ions). These dS , dS , dS , dS distortions do not compensate each other, which leads tonon-zero electric polarization. It has to be noted, thatwhile this mechanism is based on specific atomic displace-ments, the distortions by themselves are triggered by dif-ferent charge-density distributions for the ↑↑ ( ↓↓ ) and ↑↓ bonds.Albeit the magnetostriction strongly modifies all bondlengths, both Cr-S -Cr and Cr-S -Cr triangles stayisosceles, as shown in Fig. 3b. Therefore there is no com-ponent of the electric polarization along Cr-Cr bonds. Incontrast since all the displacements, dS ij , are different intwo neighboring plaquettes (Cr ↑ -S -Cr ↑ -S and Cr ↑ -S -Cr ↓ -S ), there are two resulting polarizations ~P and ~P ′ ,which lie in the planes of plaquettes, perpendicular to Cr-Cr bonds. Each CrS octahedron from the double chain(AFM double chain) share its edges with two neighbor-ing octahedra and the net polarization is directed nearlyperpendicular to the CrS plane as shown in Fig. 4. V. CONCLUSIONS
The total energy calculations in the GGA+U approxi-mation for the low temperature Cm crystal structure ofAgCrS compound show an importance of atomic relax-ation to set the experimentally observed magnetic orderas ground state. The relaxed atomic positions for Crions display the changes of distances between FM chainsthat coupled ferro- and antiferromagnetically. Thereforeon the base of the first principles calculations we predictthe existence of magnetostriction effect in this materialwhich was not observed in experiment [7]. The evaluatedelectric polarization is mostly due to electronic rather structural origin and it is much larger than 20 µC/m measured experimentally [6]. This discrepancy can be FIG. 4: (color online) Two net electric dipole moments (shownin blue) in two Cr S plaquettes form net polarization (shownin red) which is directed nearly perpendicular to the CrS plane. The plaquettes do not lie in the same plane (in contrastto Fig. 3). Blue and green balls correspond to Cr ions withdifferent spin projections. partially explained by the improper ferromagnetic (notparamagnetic) reference point (which we are forced touse) to calculate initial polarization. The calculationsthat treat properly the insulating nature of material at alltemperatures and paramagnetism with local moments athigh temperatures and long range magnetic order at lowtemperatures can be done within DFT+DMFT method.In this case one would expect that the calculated valueof electric polarization will be reduced. At the sametime the experimental estimation of the electric polar-ization was obtained on the polycrystalline samples [6],which are known to provide substantially smaller valuesthan in single crystals, due to features of the pyroelec-tric measurements, see e.g. Refs. [29–32]. The use ofpolycrystalline samples could also explain the absenceof the magnetostriction effect in experiment. Therefore,the DFT+DMFT calculations as well as refined measure-ments on single crystal samples are required for furtherinvestigation of the electronic and magnetic properties ofAgCrS . VI. ACKNOWLEDGMENTS
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