Orbital structure and magnetic ordering in stoichiometric and doped crednerite CuMnO2
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Orbital structure and magnetic ordering in stoichiometric and doped credneriteCuMnO A. V. Ushakov, S. V. Streltsov,
1, 2 and D. I. Khomskii Institute of metal physics, Russian academy of science,S. Kovalevskaya str. 18, 620041 Ekaterinburg, Russia Ural Federal University, Mira St. 19, 620002 Ekaterinburg, Russia II. Physikalisches Institut, Universit¨at zu K¨oln, Z¨ulpicherstraße 77, D-50937 K¨oln, Germany
The exchange interactions and magnetic structure in layered system CuMnO (mineral crednerite)and in nonstoichiometric system Cu . Mn . O ,with triangular layers distorted due to orbital or-dering of the Mn ions, are studied by ab-initio band-structure calculations, which were performedwithin the GGA+U approximation. The exchange interaction parameters for the Heisenberg modelwithin the Mn-planes and between the Mn-planes are estimated. We explain the observed in-planemagnetic structure by the dominant mechanism of the direct d − d exchange between neighboringMn ions. The superexchange via O ions, with 90 ◦ Mn-O-Mn bonds, plays less important role forthe in-plane exchange. The interlayer coupling is largely dominated by one exchange path betweenthe half-filled 3 z − r orbitals of Mn . The change of interlayer coupling from antiferromagneticin pure CuMnO to ferromagnetic in doped material is also explained by our calculations. PACS numbers: 75.50.Cc, 71.20.Lp, 71.15.Rf
I. INTRODUCTION
Magnetic systems with geometric frustrations attractnowadays considerable attention. Due to competition ofdifferent exchange paths often rather complicated mag-netic structures are realized, which are very sensitive toexternal influences and can be changed e.g. by smallmodifications of composition.There exist many magnetic materials containing thetriangular layers as a main building block. Among themthere are well-known systems like NaCoO , delafossites A MeO ( A = Cu, Ag, Pd, ...; Me – transition metals),such as multiferroic CuFeO or AgCrO , compoundswith unusual charge state like Ag NiO , ferroaxial mul-tiferroic FeRb(MoO ) and even some organic systemswith spin-liquid ground states. A regular triangularlattice with equilateral triangles is frustrated, which canlead to complicated ground states, especially in the pres-ence of strong magnetic anisotropy. Such frustrationscould in principle be lifted due to lattice distortion, forexample caused by orbital ordering (which, on theother hand, can itself lead to frustration in some, evennot geometrically frustrated lattices ). However incertain cases frustrations can remain even after such or-bital ordering.Apparently an example of this type is delafos-site CuMnO (mineral crednerite). The crystal struc-ture of CuMnO is shown in Fig. 1. Two features,disclosed by the experimental studies of stoichiometricCuMnO and similar system with the excess of Cu,Cu . Mn . O , are quite nontrivial and require ex-planation. First, the presence of the Jahn-Teller Mn ( t g e g ) ions in this system leads to change of the crys-tal structure (with corresponding ferro–orbital ordering)from the usual for delafossites rhombohedral R m to amonoclinic C /m structure already at room tempera-ture. In this structure different directions in the triangu- lar ab -plane become inequivalent: there exist for each Mntwo short and four long Mn-Mn distances in the plane.This in principle could lift frustration, if the antiferro-magnetic (AFM) exchange J on the long Mn-Mn bondswould be stronger than the exchange J on the shortones: then the topology of the plane would essentiallybecome that of a square lattice Fig. 2(a).However experimentally it turned out that this is notthe case: apparently the exchange at short bonds isstronger, which, with the uniaxial magnetic anisotropyof Mn (spins are oriented predominantly along thelong Mn-O bonds) still leaves the system frustrated [the FIG. 1. (Color online) The crystal structure of CuMnO inthe low temperature phase. Mn ions are shown as violet, Cuas blue, and O as red balls. Two different intralayer exchangeconstants via the long ( J ) and short ( J ) Mn-Mn bonds areshown. The MnO layers alternating in c direction are equiva-lent, but presented in slightly different ways to show commonedges of the MnO octahedra (upper layer) and the Mn-Obonds (lower layer). The image was generated using VESTAsoftware. stacking of the antiferromagnetic chains along the shortMn-Mn direction is frustrated for equivalent long bondswhich couple such chains, see Fig. 2(b)]. According toRef. 12 this degeneracy is lifted below magnetic transi-tion: at T < T N = 65 K the structure changes frommonoclinic to triclinic C d -orbitals ofMn, usually also the superexchange via oxygen plays animportant role, especially for the heavier 3 d elements.And one could expect that for the ferro-orbital orderinglike that found in CuMnO this contribution could bestronger for longer Mn-(O)-Mn bonds (see in more detailsbelow). Why is this not the case, is a priori not clear.Another surprising phenomenon was found in non-stoichiometric crednerite with small excess of copper,Cu . Mn . O , with part of the in-plane Mn substi-tuted by Cu. It was found in Ref. 14 that, whereas thein-plane magnetic ordering (and magnetostrictive distor-tions) remain practically the same as for pure CuMnO ,the interlayer exchange coupling changes to the opposite:if it was antiferromagnetic in CuMnO , it becomes fer-romagnetic in Cu . Mn . O . Interlayer coupling isnot often treated seriously in the study of such layeredfrustrated systems; it is usually considered rather as anuisance (although of course everyone understands thatsome such interlayer coupling is required to make a real3 D long-range magnetic ordering). Here, however, wehave to seriously address the question of such interlayercoupling and its dependence on the exact composition ofthe material.To explain the observed features, one has to know thevalues of exchange interaction between different Mn-Mnpairs, both in-plane and out-of-plane. For the lattice ge-ometry at hand, with its low symmetry and with manydifferent exchange paths (exchange due to direct Mn-Mnoverlap, different superexchange processes such as t g - t g , t g - e g , e g - e g , see e.g. Ref. 17), especially in the presenceof orbital ordering existing in CuMnO , it becomes aformidable task, difficult to solve “by hand”. Thereforewe decided to address this problem by ab initio band-structure calculations, which automatically take into ac-count all exchange mechanisms, different bond lengthsand angles present in this system. This allows us to com-pare relative stability of different types of magnetic or-dering, for both stoichiometric and for doped CuMnO .To avoid symmetry changing of the system we did notuse the supercell calculations with some particular Mnions substituted by Cu to model Cu . Mn . O , butmodeled the doped situation by changing the electronconcentration, which is equivalent to virtual crystal ap-proximation (VCA).The results of our calculations, first of all, confirm thatthe experimentally observed magnetic structure is thetheoretical ground state and also confirm the results of FIG. 2. (Color online) Possible magnetic structures for thedistorted triangular lattice. a) Exchange parameters J onthe long bonds are larger than that on short bonds J ; b)Exchange parameters on long bonds smaller that on shortones. This structure corresponds to the observed in CuMnO experimentally. For both cases the strongest exchange pathsare shown in red. the previous first principle calculations for the stoichio-metric case. The analysis shows, that the hopping be-tween the 3 z − r and xy orbitals and the direct hoppingbetween xy orbitals should give the main contribution tothe J exchange, and this AFM exchange turns out to bemuch stronger than J , leading to the magnetic structureobserved experimentally. The interlayer ordering fromthe calculations also coincides with that observed exper-imentally. For the nonstoichiometric CuMnO the in-plane ordering remains the same, whereas the interlayerordering changes to the opposite, in agreement with thefindings of Ref. 14. We discuss the physical mechanismof this change. II. CALCULATION DETAILS
The calculations were carried out by pseudopotentialmethod in the PWscf code in the framework of thedensity functional theory (DFT). We used the Vanderbiltultrasoft pseudopotentials for all ions. 3 d , 4 s , 4 p for Cuand Mn, 2 s and 2 p for O were considered as valence. Theplane-wave cutoff energy for these pseudopotentials wastaken be 40 Ry. The strong electronic correlations wereincluded on Mn sites via the GGA+ U approximation .Effective U eff = U − J H for the Mn 3 d states was chosento be 4.1 eV. We checked that main conclusions doesn’tdepend on the choice of U eff and also present the resultsfor U eff =3.6 eV.The integration in the course of self-consistency iter-ations was performed over a mesh of 256 k -points in awhole part of the Brillouin zone with switching off thesymmetry of the space group to allow any possible or-bital order. We used the Perdew-Burke-Ernzerhof (PBE)exchange-correlation functional. In order to estimate the exchange interaction parame-ters within the Mn triangular layers and the interlayer ex-change we calculated the total energies of different mag-netic configurations and used the Heisenberg model inthe form H = X ij J ij S i S j , (1)where summation runs twice over each pair. The in-planeexchange constants J and J (see Fig. 2) were recal-culated from the total energies of three magnetic struc-tures: (1) fully ferromagnetic, (2) Mn ions along the short(long) Mn–Mn bond are ordered ferromagnetically (anti-ferromagnetically), (3) Mn ions along the short Mn–Mnbonds are antiferromagnetically coupled. The unit cellcontains 4 f.u. In the LT phase the exchange parameters J along two long Mn–Mn bonds were assumed to be thesame, since the difference of the bond lengths is quitesmall, ∼ . A . For the calculation of the exchange con-stants along the b − axis the unit cell was doubled in thisdirection, the magnetic ordering in plane was chosen asin the experiment (ferromagnetic chains are going alongthe longest Mn–Mn bond).The lattice parameters (unit cell; atomic coordinates)used in the calculations were taken from the experimentaldata of Ref. 12–14 for 300 K (HT phase) and for 10 K (LTphase). III. RESULTS AND DISCUSSION
The first principles calculations of the stoichiometricCuMnO were performed for the high-temperature (HT)and low-temperature (LT) phases. The magnetic mo-ment on Mn ( d ) is close 3 . µ B in both phases (3.68 µ B in LT and 3.74 µ B in HT phase). The occupancy of the3 d shell for the interlayer linearly-coordinated Cu wasfound to be 9.73 in LT and 9.71 in HT phase. This is closeto the d configuration expected from the simple ionicconsideration. The Mn ion has 5 .
58 (5.54) electronsin the LT (HT) phases. The deviation from the nomi-nal filling d is related with the hybridization effects asdiscussed below.According to the GGA+U calculations CuMnO is ametal at the HT phase in the ferromagnetic configura-tion, while at lower temperatures in the experimental AFM magnetic structure the energy gap opens, E g = 0 . b -axis [Fig. 2(b)] is the ground state in the LT phase withthe lowest total energy.The partial Cu 3 d , Mn 3 d and O 2 p densities of statesfor the LT phase are presented in Fig. 3. One can see,that the Mn d states are lower and broader than the Cu3 d ones, which are closer to the Fermi level. A substan-tial hybridization between the Mn 3 d and O 2 p statesis noticeable. Because of the hybridization the occupa-tion of the 3 d shell of Mn is larger than the nominal, but -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 Energy (eV) -4-3-2-101234 DO S ( s t a t e s / [ e V * a t o m * s p i n ]) O 2p Mn 3dCu 3d E F FIG. 3. (Color online) The partial densities of states(DOS) obtained in the GGA+ U calculation for stoichiomet-ric CuMnO at LT phase in the AFM configuration, shownin Fig. 2(b). Positive (negative) values correspond to spin up(down). The Fermi energy is in zero. the spin moment is consistent with the d configuration(3 . µ B ).The structural distortions caused by the Jahn-Tellercharacter of the Mn and orbital ordering results in thestructure with two types of inequivalent Mn-Mn pairs inthe MnO plane: those along the a -direction with theshort Mn-Mn distance of 2 .
88 ˚A, and the long bondsin two other directions, with the Mn-Mn distance of ∼ .
14 ˚A(they become inequivalent in the magnetically-ordered phase), see Fig. 2. In effect one may expect thatthe exchange coupling in the ac plane can be character-ized by two exchange constants: J (short Mn-Mn bonds)and J (long Mn-Mn bonds). The direct calculation inthe GGA+U approximation shows that J = − . K (FM) and J = 16 . K (AFM).The MnO octahedra in the basal triangular layer havecommon edge, so that the Mn-O-Mn angle is close to90 ◦ . There exist in this geometry several contributionsto the nearest neighbor magnetic coupling. First, thereis a direct hopping between the Mn orbitals, especially t g ones, with the lobes directed toward one another, seeFig. 4(a). This contribution must be antiferromagnetic.The direct exchange is known to be quite efficient for firstfew 3 d transition metal ions, but it gradually decreasesgoing from left to the right in the periodic table, andalready for Cr and Mn the superexchange may becomelarger. Let us consider contributions to the superexchangevia O ions, some of which are illustrated in Fig. 4(b,c)(see more details e.g. in Ref. 17). For a such Mn-O-Mn 90 ◦ bond, according to Goodenough-Kanamori-Anderson rules (GKA), the superexchange between FIG. 4. (Color online) Different processes for the exchangecoupling of the neighboring Mn ions in the basal plane forthe MnO octahedra with common edge. (a) Direct overlap ofthe t g orbitals; (b) antiferromagnetic t g – t g superexchangevia oxygen; (c) antiferromagnetic t g – e g superexchange viaoxygen; the Jahn-Teller distortion makes the distance be-tween these two Mn ions long. Common edge of the neigh-boring MnO octahedra is marked by wavy line. Oxygen ionsare shown by circles. the half-filled t g orbitals via oxygens will be also antifer-romagnetic ∼ t pdπ ∆ ( U + U pp ), where ∆ is the charge-transfer energy (energy of the excitation, in our case,Mn ( d )O − (2 p ) → Mn ( d ) O − (2 p )), and U pp isthe repulsion of oxygen p -electrons.But potentially the most important contribution to theMn-Mn exchange via oxygens could be the t g - e g ex-change, which, for the hopping to the half-filled e g or-bital, is antiferromagnetic, and could be quite strong, ∼ t pdπ t pdσ ∆ ( U + U pp ). In pair of the MnO octahedraforming long Mn-Mn bond the single half-filled e g orbital(2 z − x − y ) will lie in the plane defined by commonedge and two Mn ions and will take part in such an ex-change process as shown in Fig. 4(c). Thus this contri-bution will be stronger for longer Mn-Mn bond. This iswhat we had in mind in the Sec. I, where mentioned thatsuperexchange may leads to the situation when J wouldbe stronger than J . But this not the case and directexchange obviously dominates.It has to be mentioned again that actually, it is a pri-ori not clear whether – the direct d − d exchange on ashort Mn-Mn bond or the superexchange via oxygens onthe long bonds would be stronger. In the second case wewould have expected that the magnetic ordering wouldbe simple two-sublattice antiferromagnetism in an effec-tive square lattice formed by the long Mn-Mn bonds,Fig. 2(a) (it can also be viewed as the stripe ordering in the original triangular lattice, but with stripes of parallelspins running along short bonds, not along long ones, asin Fig. 2(b)). Note that such a situation is indeed realizedon some triangular latties, e.g. in NaVO (in this casethe orbital ordering and the direct d – d exchange relievegeometric frustration).There exist also other contributions to the nearestneighbor exchange, e.g. those involving one occupied andone empty orbital; according to the GKA rules, these pro-cesses would give ferromagnetic contribution, but weakerby the factor of J H U or J H . Still, there are many such ex-change channels, so that the total contribution of theseprocesses can be significant.The results of our ab initio calculations give an answerto the question raised above; it turns out that the an-tiferromagnetic exchange J on the short Mn–Mn bondis stronger than the superexchange contributions on thelong bonds. Apparently the rather short Mn–Mn dis-tance (2 .
88 ˚A) on a short bond makes direct t g – t g ex-change dominant, but the superexchange between e g and t g orbitals via oxygen should also be important.In order to estimate the interlayer exchange interac-tion parameters and also to simulate the doped systemCu x Mn − x O we used the supercell with 8 inequiva-lent Mn ions. Two different magnetic configurations werecalculated. In both structures Mn ions constituting shortin-plane Mn–Mn bonds were antiferromagnetically cou-pled, while the interlayer spin order was taken different,ferro- or antiferromagnetic. The calculated exchangeconstant J inter = 0 . K (AFM) reproduced the observedinterlayer ordering for undoped CuMnO . the maincontribution to the interlayer exchange is given by theprocess illustrated in Fig. 5.The strongest exchange path goes from the occupied(half-filled) 3 z − r orbital of Mn ion in one layer, wherethe local z -axis is directed along the long Mn-O bond,via corresponding oxygen 2 p orbitals and eventually dia-magnetic Cu sitting in between layers, and then to thesimilar 3 z − r orbital of one particular Mn in the nextlayer (see Fig. 5(a)). This exchange coupling is antiferro-magnetic, which provides the antiferromagnetic couplingbetween layers observed for CuMnO . Note that theMn ions connected by this exchange path do not belongto one unit cell, i.e. one must be careful in compar-ing the obtained magnetic ordering with the experimen-tal one (which is defined in Ref. 12 in terms of relativeorientation of crystallographically equivalent Mn ions inneighboring layers).As it was mentioned above the same (as for pureCuMnO ) supercell consisting of the 8 Mn ions was usedto simulate the magnetic properties of the nonstoichio-metric Cu x Mn − x O with x=0.04. Since the exactpositions of the doped Cu in the Mn -plane is un-known, the virtual crystal approximation was used: anextra 0 .
32 holes were added in the calculations of theaforementioned superell. This corresponds to the uni-form distribution of this hole over a whole cell.The occupations of the 3 d shell of the Cu and Mn ions FIG. 5. (Color online) a) The strongest exchange path for in-terlayer antiferromagnetic coupling (involving the half-filled e g -orbitals) in the stoichiometric CuMnO . Oxygens areshown by circles, Cu ( d ) ion is shown by square. b) Thesame exchange path, but for the upper Mn (for the non-stoichiometric Cu x Mn − x O ); the hopping would be to theempty e g -orbital which is not shaded. According to the GKArules this exchange would be ferromagnetic. in doped system are slightly different from the stoichio-metric case, being 9.71 and 5.55 respectively. The mag-netic moment on Mn in triangular plane is 3 . µ B . Theoccupations for different 3 d orbitals are the same as inpure CuMnO . The interlayer exchange for x = 0 .
04 wasfound to be J inter = − . K , ferromagnetic, instead of an-tiferromagnetic interlayer coupling of +0 . K for undopedcompound. The intralayer exchanges are J = − . J = − . This can be explained inthe following way: when we substitute some Mn ions byCu (formally trivalent Cu is rather difficult to ob-tain, and it cannot be formed at these conditions, whichis confirmed by our calculations), we induce two changes.One is that Cu itself is magnetic and has different or-bital occupation, so that for some ions the same exchangepath from the 3 z − r orbital of the Mn in the lowerplane would connect to Cu in the neighboring plane,for which the 3 z − r orbital will be completely filled.The d -hole will be, as always, on x − y orbital. Dueto different orbital occupation this exchange would beferromagnetic, in accordance with the GKA rules.Another consequence of the substitution of Mn byCu is that to guarantee electroneutrality one Mn ionper each Cu should become Mn . These Mn ions,e.g. in the upper layer, would couple to Mn in thelower layer also ferromagnetically, see Fig. 5(b). Besides,as just explained, each Cu in triangular layer, as well as each Mn , would couple ferromagnetically to boththe layer above and layer below. Thus effectively eachextra Cu would make four interlayer bonds ferromag-netic instead of antiferromagnetic. Apparently all thesefactor combine to lead to the inversion of the interlayermagnetic ordering in nonstoichiometric crednerite withthe excess of Cu. And the fact that already very smallamount of excess copper, only 4% in Cu . Mn . O studied in Ref. 14, is sufficient to lead to this inver-sion of interlayer ordering, is probably connected withthe factors discussed above: that effectively every extraCu changes to the opposite the exchange of four inter-layer bonds, so that the effective doping is not 4%, butis rather ∼
16% (i.e. 16% of strongest interlayer bondschange sign).There may be also other factors contributing to thesame effect. Notably, it is known, e.g. on the exam-ple of CMR manganites, that the substitution of Mn byother ions with different valence can modify the in-planemagnetic ordering in a rather large region around thedopant. It is not excluded that also here the in-planeordering could be modified close to Cu and especiallyto Mn which should be created simultaneously. Theinteraction of this distorted region in a given plane withthe next plane with its, say, original ordering could alsohave opposite sign from the interaction of two “virgin”planes. The eventual presence of magnetic distortionsdue to doping in CuMnO should be checked by specialexperiments (the simplest indication of that would becertain broadening of magnetic reflexes in nonstoichio-metric crednerite as compared to those in pure CuMnO or ESR measurements).It’s worthwhile mentioning that the results obtainedin the present investigation does not strongly dependon the value on site Hubbard repulsion U eff . In or-der to check U dependence we performed additional cal-culations with smaller U eff = 3 . U eff leads to increase of the AFM contributions tothe intralayer exchange coupling, which are known to be ∼ t /U . As a result both in-plane exchanges becomemore AFM: J equals 21.0 K and 17.2 K, while J is − . − . and nonstoichio-metric case respectively. For U eff = 3 . ( J inter = 0 . . Mn . O ( J inter = − . U eff =4.1 eV.The results obtained in the present paper agree withthe conclusions of Ref. 27, where it was shown thatthe magnetic properties of CuMnO strongly depends ofwhat kind of dopant is used: magnetic Cu with notcompletely filled 3 d shell or non-magnetic Ga . In thefirst case the interlayer coupling changes to the opposite,while in the second in remains the same (antiferromag-netic). This demonstrates that this change is not dueto a modification of the crystal structure, but is con-nected with the substitution of Mn by magnetic Cu ,with the generation of another magnetic ion Mn , asexplained above. IV. CONCLUSION
In conclusion, on the basis of ab initio band structurecalculations we obtained the physical picture, which ex-plains experimentally observed stripy antiferromagneticorder in stoichiometric crednerite CuMnO , as well asin the system with the excess of Cu, Cu x Mn − x O .Ferro-orbital ordering present in this system plays veryimportant role in determining the exchange constantsand finally the magnetic structure. We argue that thein-plane magnetic ordering is mainly provided by the di-rect exchange interaction between the t g , while superex-change between e g and t g orbitals on different sites is ex-pected to be smaller. The interlayer exchange is mainlygiven by the exchange path involving the half-filled e g or-bitals of Mn . Substitution of a part of Mn by Cu ,with corresponding creation of compensating Mn ions,leads to the inversion of the interlayer coupling alreadyfor rather small doping, which explains the puzzling ex-perimental observation of Poienar et al. in Ref. 14 The obtained results once again demonstrate the im-portance of orbital ordering for magnetic structures, thistime in a frustrated system. It also shows that such sys-tems are very sensitive to even small variations of con-ditions, e.g. small doping, and their properties can beeffectively modified even by small perturbations. V. ACKNOWLEDGMENTS
The authors are very grateful to Christine Martin forattracting our attention to this problem and for numer-ous discussions. This work was supported by the Ger-man projects FOR 1346, by the University of Colognevia German Excellence Initiative and by the Europeanproject SOPRANO, Samsung by GRO program, and bythe Russian Foundation for Basic Research via RFFI-13-02-00374 of the Ministry of education and science ofRussia (grant MK-3443 . .
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