Lattice dynamics and electronic transitions in a structurally-complex layered copper borate Cu 3 (BO 3 ) 2
A. D. Molchanova, M. A. Prosnikov, R. M. Dubrovin, V. Yu. Davydov, A. N. Smirnov, R. V. Pisarev, K. N. Boldyrev, M. N. Popova
LLattice dynamics and electronic transitions in a structurally-complexlayered copper borate Cu (BO ) A. D. Molchanova, ∗ M. A. Prosnikov, R. M. Dubrovin, V. Yu. Davydov, A. N. Smirnov, and R. V. Pisarev
Ioffe Physical Technical Institute, Russian Academy of Sciences, 194021 St.-Petersburg, Russia
K. N. Boldyrev and M. N. Popova
Institute of Spectroscopy, Russian Academy of Sciences, 108840 Moscow, Troitsk, Russia (Dated: November 28, 2017)Copper borate Cu (BO ) is a complex compound with a layered crystallographic structure inwhich the Jahn – Teller active and magnetic copper Cu ions occupy sixteen nonequivalent posi-tions in the unit cell displaying controversial magnetic behavior. In this paper, we report on theinfrared and Raman spectroscopic studies of the lattice dynamics and the electronic structure of3 d copper states below the fundamental absorption band. The lattice dynamics is characterizedby a large number of phonons due to a low P Z = 10. Unusually rich set of phonons was found in the low-energy part of the infrared and Ramanspectra below 100 cm − , which we tentatively assign to interlayer vibrations activated by a crystalsuperstructure and/or to weak force constants for modes related to some structural groups. Severalphonons show anomalous behavior in the vicinity of the magnetic phase transition at T N =10 K thusevidencing pronounced magnetoelastic interaction. No new phonons were found below T N , whichexcludes the spin – Peierls type of the magnetic transition. In the region of electronic transitions,a strong broad absorption band centered at ∼ d states of Cu ions split by the crystal field in nonequivalent positions.The fundamental charge-transfer absorption band edge has a complex structure and is positionedaround ∼ PACS numbers: 63.20.-e, 78.30.-j, 63.20.Ls, 78.40.-q
I. INTRODUCTION
Numerous copper oxide compounds display a wide va-riety of unique chemical and physical properties and findimportant applications. Many of these properties are re-lated, first of all, to an intrinsic ability of copper ionsto adopt different valence states, typically Cu , Cu ,Cu , and others. Another intrinsic property of cop-per ions is their ability to occupy crystallographic posi-tions with different coordination that results, as a rule,in markedly-distorted geometries. There are many com-pounds in which copper ions occupy positions with 2,3, 5, and 6 nearest-neighbor anions. In some cases,these specific properties of copper ions are consequencesof the electronic Jahn – Teller effect [1]. For example,Cu ions with the 3 d ionic state are susceptible todistortions in octahedral sixfold-coordinated sites. Cop-per ions play a decisive role in many materials by creat-ing their specific physical properties and some examplescan be cited. First of all, that is the simplest, from thepoint of view of chemical composition, antiferromagnetCuO which demonstrates multiferroic properties [2]. An-other simple compound is a diamagnetic Cu O in whichthe existence of giant Rydberg excitons with principalquantum numbers as large as n = 25 has recently beendemonstrated [3]. A singlet magnetic ground state is re-alized due to a spatial spin-dimerization of spins S = 1/2at the spin – Peierls phase transition in a chemically-simple quasi-one-dimensional compound CuGeO [4].SrCu O is an example of a two-leg spin-ladder sys- tem [5]. Strontium-boron cuprate SrCu B O is a rareexample of a quasi-two-dimensional crystal with a singletmagnetic ground state [6–8]. A complex mineral her-bertsmithite Cu Zn(OH) Cl is a spin S = 1/2 kagome-type spin-liquid system [9]. Multiferroic properties [10]and anomalous optical properties [11] were reported in amixed-valence antiferromagnet LiCu O in which Cu and Cu ions coexist. Another complex-structure exam-ple is trigonal green dioptase Cu [Si O ] · O. Neutronscattering experiments were done on natural crystals, re-vealing complex magnetic ground state and excitationspectra [12]. Other noticeable and well-studied chiralmagnetic cuprate is Cu OSeO , known as a skyrmionand half-skyrmion host [13]. Besides that, there is a lotof examples of molecular magnets based on Cu [14].Many other examples can be given. And last but notleast, high- T C copper-oxide superconductors are, with-out any doubt, among the most interesting, important,and actively studied materials [15].Among numerous oxide cuprates, there are severalcompounds in the binary system CuO-B O . The sim-plest compound in this group is a very interesting copper-boron oxide CuB O , which crystallizes in the spacegroup I d ( Z = 12) [16]. This crystal possessesa very intricate magnetic phase diagram with commen-surate and incommensurate magnetic structures, multi-ple magnetic phase transitions, and a separate order-ing of the two magnetic Cu -subsystems [17]. Lin-ear and nonlinear optical studies in the region of tran-sitions within the 3 d configuration split by the crys-tal field made it possible to observe a unique of set all a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov zero-phonon (ZP) lines accompanied by a rich picture ofphonon sidebands [18–20]. Among other results, this ma-terial provided a rare, and probably a unique, possibilityto evaluate the genuine crystal-field parameters using theexact positions of the ZP lines for the both Cu subsys-tems [19]. Moreover, ZP lines allowed one to observe theantiferromagnetic dichroism and revealed new details inthe magnetic phase diagram [21].These observations of intriguing magnetic propertiesand unique manifestations of the fine structure in theelectronic spectra of CuB O raise the question to whatdegree they are related either to its particular crystalstructure or to the chemical composition of an oxide com-pound which contains only Cu and B cations [22].This question remains unanswered so far.Also other copper borates belong to a binary family3CuO · B O . Unlike CuB O with a unique structure,they can crystallize in a triclinic [23], a monoclinic [24],and an orthorhombic [25] structures. To the best of ourknowledge, there are no ab initio calculations which couldexplain structural stability or metastability of these threephases. We may suppose that in the case when chemi-cally the same compound can adopt different crystallo-graphic structures is a good example of a large versatilityof copper ions for occupying crystallographically differ-ent positions. In this group of three materials, the tri-clinic Cu (BO ) compound is the most actively studiedone [26–32], practically nothing is known about physicalproperties of the two other Cu (BO ) compounds.However, up to now, no data are available in literatureon the lattice dynamics and electronic structure of thiscrystal. In the present paper, we report on a detailedstudy of the lattice dynamics, electronic d-d transitions,and optical properties of this triclinic crystal, which,from the chemical point of view, is similar to CuB O .Both these compounds belong to the binary system CuO-B O , but the former has the composition 3CuO · B O whereas the latter CuO · B O . Obviously, these two com-pounds possess different crystallographic structures, withthe space groups P Z = 10) and I d ( Z = 12), respec-tively. The unit cell volume and the number of inequiv-alent Cu positions in Cu (BO ) is much larger thanthose in CuB O . Both Cu (BO ) and CuB O werereported to be good photocatalysts [33], but no opticaldata on the former compound are available so far. Wewill show that 3 d electronic spectra of these two com-pounds strongly differ but the main transitions are ob-served in the same spectral range. Possible reasons ofsimilarities and differences between optical properties ofthese crystals will be discussed.The paper is organized as follows. In Sec. II, the crystalstructure and other properties of Cu (BO ) are brieflyreviewed. In Sec. III, experimental details of the in-frared, Raman, and optical absorption measurements aredescribed. In Sec. IV, we analyze the phonon modes anddiscuss experimental results of infrared and Raman stud-ies. In Sec. V, we analyze the local positions of Cu ionsand the relevant electronic structure, describe the optical properties and discuss them in comparison with literaturedata on some other cuprate oxides. Sec. VI deals withconclusions. II. CRYSTAL STRUCTURE AND MAGNETICPROPERTIESA. Crystal structure
The triclinic crystal structure of Cu (BO ) withthe lattice parameters a = 3.353(2), b = 19.665(7), c = 19.627(8) ˚A, α = 88.77 ◦ , β = 69.71 ◦ , and γ = 69.24 ◦ was solved in Ref. [23]. Two bc and ac projections ofthe triclinic unit cell of Cu (BO ) are shown in Fig. 1using the vesta software [34]. For a more convenientdescription of our experimental data, we introduce anadditional set of axes, namely, a *, b *, and c *, which arenormals to the bc -, ac -, and ab -planes, respectively. Itshould be noted that a * is the normal to the crystalcleavage planes. All the atoms build a pseudo-tetragonalnearly-planar layer structure. All B ions have trigo-nal nearly-planar coordination by O − ions within thesame layer forming [BO ] groups. A part of these tri-angles are interconnected through the common O − ionsforming [B O ] groups. The Cu ions within each layerhave nearly planar quadratic coordination by four O − ions. Some of the Cu ions in the layer are additionallycoordinated by one or two O − ions in the neighboringlayers.Examination of the relevant Cu–O bond lengths (notlonger than 3 ˚A) within the 16 different positions of Cu ions allows us to distinguish three different types of po-sitions, which are shown in Fig. 1 by different colors.These types of copper positions, according to Ref. [23],correspond to different Cu–O polyhedra, namely, i) aplanar square (coordination number 4); ii) a distortedsquare pyramid (4 + 1); and iii) a strongly distorted oc-tahedral coordination (4 + 2). We will analyze thesepositions in more detail in Section V where we dis-cuss the effective coordination numbers and electronicstructure of the 3 d states of Cu ions in the crystalfield. We remind that the i) and iii) types of the Cu–O polyhedra are structurally close to two basic groupsof Cu ions in CuB O [16]. It is also worth notingthat in Cu (BO ) different kinds of interconnection be-tween groups are realized, for example, vertex-connected[CuO ]–[CuO ], [CuO ]–[CuO ], [CuO ]–[CuO ], and[CuO ]–[BO ] groups, edge-connected [CuO ]–[CuO ],and [CuO ]–[BO ] groups, and others. B. Magnetic properties
According to the structural data, the average Cu–Obond length in Cu (BO ) within the layers is 1.95 ˚A,whereas the bond length between the layers is 2.90 ˚A. a a*b c b) a bc [B O ] a) [BO ][CuO ][CuO ][CuO ] CuBO FIG. 1. (Color Online) a) Polyhedral representation of thecrystal structure projected along the [100] direction. Num-bers correspond to nonequivalent crystallographic positionsof the Cu ions. Distorted [CuO ] groups are shown in blue,near-square coordinated groups by dark-blue, and pyramidalgroups by light-blue color. b) Projection along the [010] di-rection; set of the crystal layers in the bc plane with perfectcleavage. Note that the a * axis is normal to the cleavagelayers. Single-crystal unit cell is marked by thin solid lines. This large difference defines a two-dimensional charac-ter of the crystal structure and, consequently, implies atwo-dimensional character of the magnetic structure [26].Measurements of the magnetic susceptibility [26; 27], spe-cific heat [27; 29], and inelastic neutron scattering [27]revealed the formation of a gap of ∆ ∼ . T N = 10 K, which pointsto a singlet ground state. However, magnetic susceptibil-ity measurements along the a * axis [29], observation ofa spin-flop transition at H c = 9.5 T [31], and NMR mea-surements on B nuclei [31] point to a 3D antiferromag-netic (AFM) ordering. The authors of Ref. [31] arguedthat the ground magnetic state is an AF state with a spin-amplitude modulation similar to a spin-density wave. Acomplex magnetic ordering was suggested in [27] as beinga combination of a 3D-AFM ordering and a spin-singletstate.
III. EXPERIMENTAL DETAILS
In our experiments we used single crystals ofCu (BO ) grown by L. N. Bezmaternych (Institute ofPhysics, Krasnoyarsk, Russia) using a spontaneous crys-tallization method [26]. Single crystals as large as ∼ were obtained. The lattice parameters of the grown crys-tals were measured with the use of x rays and were inaccordance with the literature data for the triclinic struc-ture. A perfect cleavage along the bc planes as well as thepresence of well-defined { } , { } , { } , { } andother forms could be also identified on the single crystals.In thin layers, the studied crystals have a deep-greencolor and noticeable pleochroism (see inset to Fig. 7),while orthorhombic polymorphs were reported to havea blue color [25]. All these observations allowed us tounambiguously assign the grown single crystals to the P (BO ) possessed aflaky structure most possibly related to the layered crys-talline structure of the compound and weak interactionsbetween the individual structural elements. Such crys-tals are difficult to analyze by using conventional infrared(IR) spectroscopy because of scattering of the incidentlight on their rough surfaces resulting in poor quality ofthe spectra. A much better method for examining suchcrystals is the so-called attenuated total reflection (ATR)spectroscopy [35; 36], when almost complete reflectionof the light beam from the sample surface is observed. Infact, the radiation slightly penetrates inside the samplewhere it is partially absorbed. Depending on the angleof incidence, the radiation beam can be reflected fromthe sample surface one or several times. As a result,a signal resembling that in the absorption spectrum isrecorded, and the frequencies of the absorbed radiationcoincide with the frequencies obtained in the IR trans-mission spectroscopy. In our experiment, a special SingleReflection ATR Microsampler MVP-Pro with a diamondATR crystal was used for registering the ATR spectra.Infrared transmission and ATR spectra were registeredin a broad spectral range using a high-resolution Fourier-transform IR spectrometer Bruker IFS 125HR. Helium-cooled bolometer for the far IR region (10–500 cm − ),and a liquid-nitrogen cooled MCT detector for the mid-dle IR region (400–3000 cm − ) were used. Samples werecooled in a closed-cycle cryostat Cryomech ST403.Raman spectra were measured in the range 25–2000 cm − with the use of a T64000 (Jobin-Yvon)spectrometer equipped with a cooled charge-couple de-vice (CCD) camera. The line λ = 532 nm (2.33 eV) of aNd:YAG laser (Laser Quantum) was used as the excita-tion source with the incident laser power limited to 3 mWin order to avoid the sample overheating. All measure-ments were done in the back-scattering geometry withthe use of a 50 × objective. Cooling and stabilizationof the sample’s temperature between 10 and 300 K weredone with the use of a closed-cycle helium cryostat. Mul-tiple polarization settings were applied to register thescattering spectra.For measuring absorption and reflection spectra in thenear infrared, visible, and ultraviolet regions, a BrukerIFS 125HS and a Shimadzu UV-3600 Plus spectropho-tometers were used. IV. LATTICE DYNAMICSA. Symmetry analysis
The primitive unit cell of triclinic Cu (BO ) with Z = 10 contains 110 atoms which results in total330 phonon modes. The group-theoretical analysis givesthe following set of the lattice modes in the center of theBrillouin zone:330Γ = 162 A g ( xx, yy, zz, xy, xz, yz ) + 168 A u ( x, y, z )(1)Among them, there are 3 A u acoustic modes; 165 A u odd optical modes are IR-active, and 162 A g even modesare Raman-active. Low crystal symmetry of Cu (BO ) allows one to observe Raman-active modes in all diagonaland off-diagonal polarization settings. The main types oflattice vibrations in the high-symmetry [BO ], [CuO ],[CuO ], and [CuO ] groups can be found in [37]. Becauseof a low symmetry of the crystal as a whole and low sitesymmetries, all the phonons are nondegenerate. B. Infrared spectroscopy
Fig. 2 shows the ATR spectrum from a cleavage plateof Cu (BO ) measured at room temperature. The spec-trum demonstrates a very rich phonon structure due tothe presence of a large number of atoms in the primi-tive cell. One can distinguish several frequency regionsrelated to the quasi-molecular internal vibrations ν , ν , ν , and ν of triangular [BO ] − groups [37; 38], namely,550–700 cm − , 700–850 cm − , around 1000 cm − , and1200–1550 cm − , respectively. Vibrations at lower fre-quencies are connected with translational and librationalmovements of [BO ] − groups (molecular weight M = 59)and movements of copper atoms ( M = 63). The observedvibrational spectrum of Cu (BO ) occupies the fre-quency range 100–1550 cm − , which is somewhat broaderthat the range 100–1200 cm − previously observed inCuB O [20], the crystal with a similar chemical com-position. The strongest bonds in both structures areB–O bonds but whereas in Cu (BO ) these bonds arewithin BO planar triangles, in CuB O they form BO tetrahedrons. The frequencies of eigenvibrations of a freeBO molecule are higher than those of a BO tetrahedralmolecule [37], and this can explain why higher frequenciesare observed in Cu (BO ) . It is worth noting, however,that in the compounds R Fe (BO ) ( R is a rare earth n n n Attenuated total reflection (a.u.)
W a v e n u m b e r ( c m - 1 ) T = 3 0 0 K C u ( B O ) k | | a * n FIG. 2. (Color Online) Attenuated total reflection (ATR)spectrum of Cu (BO ) in the range 40-1800 cm − at roomtemperature. Frequency regions corresponding to internal vi-brations of the [BO ] groups are indicated. ion), which also contain planar triangular BO groups,the highest observed vibrational frequency was of about1440 cm − . Even higher frequencies of about 1550 cm − in Cu (BO ) can be tentatively assigned to internal vi-brations of tightly bound B O units.The main difference between the infrared spectra ofCu (BO ) and CuB O is that for the former one thetransmission spectrum contains a large number of lineswith frequencies below 100 cm − . There are two phononlines with frequencies around 30 cm − (see Fig. 3), andabout ten lines in the range 55–97 cm − , (see Fig. 4).Low-frequency vibrations are characteristic for layeredcompounds (to which Cu (BO ) belongs), because ofweak couplings between the layers. The number of inter-layer optical vibrations (induced by acoustic vibrationsof layers) is N = 3( N L -1) [39], where N L is the number oflayers in the primitive unit cell. According to the struc-tural data [23], N L = 1 for Cu (BO ) , and, thus, N = 0.However, we may assume that the crystal structure ofCu (BO ) is even more complex and a superstructurealong the a axis exists resulting in a corresponding fold-ing of the Brillouin zone. Acoustic phonons from the zoneboundary are thus transferred to the zone center wherethey become optically active. The presence of such ex-tra folded modes was observed in the IR spectra of thecopper oxide CuO [40]. As the most probable mecha-nism for the formation of a superstructure in CuO, theauthors of [40] considered the formation of polar Jahn –Teller centers [CuO ] − and [CuO ] − , according to themodel developed earlier for explaining some experimentalresults on CuO and high- T c cuprates [41; 42]. One can-not exclude such a mechanism in the case of Cu (BO ) ,but, obviously, the problem requires a further study.Assumption about the superstructure in Cu (BO ) allows us to explain in a natural way low-frequency “ex- FIG. 3. (Color Online) Low-frequency transmission spec-trum of Cu (BO ) as a function of temperature. The valuesof transmittance are indicated by the color scale in the rightupper corner. Inset shows the temperature dependence of thefrequency of the ν = 32.6 cm − mode. tra” modes in the IR spectra. Two lowest modes atabout 30 cm − originate from relative shifts of the twoneighboring layers in the bc plane. The third inter-layermode corresponding to the relative movement of two lay-ers along the a * direction cannot be excited by the lightpolarized in the bc plane. Modes with frequencies in therange 55–97 cm − , most probably, correspond to intra-layer vibrations at specific points of the Brillouin zone,activated by the zone folding. Extremely small width(of about 0.3 cm − ) of these low-frequency modes andtheir low intensity (they are not observed in reflectionbut only in transmission) supports their interpretationas folded modes [43].We consider now the temperature behavior of the ob-served infrared-active vibrational modes. Two modesshown in Fig. 3 harden and narrow with decreas-ing the temperature. At T = 3.5 K, ν = 30.2 cm − ,∆ ν = 0.3 cm − , and ν = 32.6 cm − , ∆ ν = 0.5 cm − .Such behavior is typical for the lattice phonons. How-ever, in the low-temperature region different behavior isobserved for the 32.6 cm − line, namely, in the vicinityof the magnetic transition at T N = 10 K the trend forits frequency change becomes negative, as it is shown inFig. 3, Inset. Such behavior evidences pronounced mag-netoelastic interaction, which leads to a softening of theelastic constants for this particular mode and a softeningof the relevant frequency. The 2D magnetic correlationstake place at temperatures well above T N [44] and thisexplains why the frequency changes caused by magne-toelastic interactions are noticable above T N . Superpo- FIG. 4. (Color Online) Transmission spectrum of Cu (BO ) in the spectral region 50-107 cm − as a function of tempera-ture. The values of transmittance are indicated by the colorscale in the right upper corner. Inset shows the intensity ofthe ν = 57.5 cm − line as a function of temperature. sition of the two contributions with opposite tempera-ture trends results in the dependence with a maximumat 13 K.Similar behavior is observed for some phonons athigher frequencies. However, several phonons, for exam-ple at ν = 80 cm − in Fig. 4, demonstrate a slight increaseof the frequency in the vicinity of the magnetic phasetransition. The matter is that the magnetoelastic inter-action may result in both softening and hardening of theelastic constants for a particular mode [40]. The mag-netoelastic interaction also introduces an additional con-tribution to anharmonic interactions in the crystal [45],which leads, in particular, to a redistribution of intensi-ties between the modes. As an example, Inset in Fig. 4shows the temperature dependence of the intensity of thephonon at ν = 57.5 cm − , demonstrating a peculiarity at13 K. It is important to note that no new phonon modeswere found below T N , and this observation excludes thespin – Peierls character of the magnetic transition.Table I presents the frequencies of all IR-active modesobserved in the IR transmission and ATR spectra. C. Raman spectroscopy
Raman studies of low-frequency excitations are alwaysquite a challenging task because of strict requirements tostray-light rejection. The use of a triple monochroma-tor in dispersion subtraction mode allowed us to registerthe Raman scattering spectra of Cu (BO ) beginning
50 100 150 0 200 400 600 800 1000 1200 1400 1600
Raman shift (cm -1 ) . . . . . . . . . . . . . . . . . . a) I n t e n s it y ( a . u . ) . T = 10 KCu (BO ) b) c(a * b)cc(bb)ca * (cc)a * a * (cb)a * a * (bb)a * FIG. 5. (Color Online) Raman scattering spectra of Cu (BO ) for five different polarization settings at T = 10 K. a) Lowenergy part of the spectrum below <
200 cm − . Numbers represent Raman shift of several selected intense modes. b) Spectrain the whole energy range of 20–1630 cm − vertically shifted for clarity.TABLE I. Frequencies (cm − ) of the IR-active A u modesobserved on the IR transmission and ATR spectraIR transmission spectra at T = 3.5 K ( T = 235 K)30.2 (29.2) 59.3 (58.9) 78.5 (78.7) 90.8 (90.5)32.6 (31.3) 66.2 (64.7) 81.0 (83.0) 97.4 (96.3)55.3 (55.1) 72.2 (72.4) 81.7 (-) 104.6 (103.3)57.5 (55.9) 73.8 (72.8) 86.8 (86.3)ATR spectrum at T = 300 K111.2 224.4 328.3 511.9 685.7 1150.6 1420.1116.3 230.6 381.3 552.7 722.3 1170.1 1450.7138.6 240.7 403.0 591.6 738.9 1195.7 1481.7181.2 249.3 436.9 609.3 860.0 1232.2 1503.7190.9 255.8 448.4 638.5 972.1 1267.2 1527.5200.3 288.3 474.1 653.3 1038.1 1344.7 1539.1210.7 299.1 499.0 672.8 1105.9 1373.5 1585.6 from 25 cm − . They are shown for different polarizationsettings at T = 10 K in Fig. 5. The highest density ofthe Raman lines is observed in the range 30–800 cm − ,and several more lines are detected at higher energies,with the highest frequency at 1566 cm − . Parametersof the all observed phonons were extracted through fit-ting spectra to Voigt profiles with the use of fityk code[46] and summarized in Table II. It is interesting to com-pare the observed spectra with those of the chemically-related compound CuB O where the lattice excitationspectra lie in the range 110–1230 cm − . An importantdifference between CuB O and Cu (BO ) , besides ob-vious differences in lattice symmetry, is the coordinationof boron, namely, in the former there are [BO ] tetrahe-dral groups while in the latter one finds either isolated[BO ] groups or [B O ] groups with shorter B–O bonds. Thus, it is quite reasonable to expect higher-frequencymodes in the lattice excitation spectra of Cu (BO ) .Some other examples of Raman spectra of characteris-tic [BO ] and [B O ] groups in other borate crystals andmelts with phonon modes up to 1500 cm − can be foundin [47]. Obviously, this assignment of the high-frequencymodes to [BO ] or [B O ] groups does not exclude a pos-sibility that some of them could be related to two-phononexcitations. Some similarities between Cu (BO ) andCuB O spectra can be noticed, e.g., characteristic andintense “triplet” at 727-733-757 cm − similar to a tripletat 695-704-725 cm − in CuB O but shifted to a lowerfrequency by 30 cm − .It is also worth noting, that, in contrast to CuB O with T N =21 K in which a strong two-magnon scatteringcentered at about 80 cm − was observed [48], no mag-netic scattering was detected in Cu (BO ) at tempera-tures down to 10 K. First of all, that might be due to amuch lower T N = 10 K in this compound. Secondly, themagnetic structure of Cu (BO ) is completely different,much more complex, and only partially ordered. Thus,we expect that a two-magnon scattering in Cu (BO ) should be positioned at much lower energy and smearedover a broader frequency range due to a greater number ofspin-wave branches and magnon-magnon interaction. Wesuppose that all these factors make two-magnon scatter-ing in Cu (BO ) indistinguishable from the background.All nine Raman-tensor elements are non-zero for tri-clinic crystals, and therefore all phonons should be ob-served at any polarization; nonequivalence of these ele-ments leads to a variation of intensities in the spectra indifferent polarization settings. For example, one of thelowest-frequency phonon at 41.1 cm − is the most pro-nounced in the a *( cc ) a * polarization. TABLE II. Frequencies (cm − ) of the experimentally ob-served Raman active A g modes extracted from the spectrawith different polarizations at T = 10 K.36.4 118.9 215.3 350.6 525.1 691.0 1228.541.1 125.2 222.6 364.4 531.5 733.1 1269.754.2 139.3 240.8 370.2 546.0 743.4 1272.064.3 145.8 246.0 376.3 566.9 757.6 1287.871.5 147.2 250.5 384.7 580.1 850.5 1362.874.0 152.6 265.7 403.2 601.2 869.1 1369.381.9 159.8 288.9 412.3 612.1 966.8 1396.885.6 168.5 296.2 439.6 636.1 985.2 1399.692.5 175.2 305.5 456.5 644.4 1043.3 1462.496.5 183.8 316.3 470.5 650.8 1147.5 1531.6106.5 193.5 329.2 491.4 661.9 1158.7 1536.1111.5 203.0 336.1 504.5 668.6 1166.1 1539.6115.9 211.7 341.4 518.9 685.8 1181.2 1565.9 The rich set of low-frequency phonons in Cu (BO ) described above deserves a further discussion. Typically,low-frequency phonons can be observed in systems con-taining heavy ions, e.g. ZrW O [49], or as soft modesin materials with structural and/or ferroelectric transi-tions. An example of a material containing both heavyions and soft modes is the lead zirconate PbZrO withseveral modes below 100 cm − [50]. Another possibil-ity is often found in organic systems with phonon modescharacterized by extremely low force constants. The cys-tine is a good example in which a bunch of phonon linesbelow 100 cm − is observed corresponding mainly to tor-sion vibrations of large organic groups [51]. This com-pound is even used as a calibrant for ultra-low frequencyRaman scattering measurements. Another example isCu Mo O in which phonons with frequencies as low as31 cm − were observed and tentatively assigned to rigid-chain modes when each of the structure-forming chainsmoves as a rigid unit [52]. Since Cu (BO ) is com-posed of relatively light ions and cannot be consideredas a molecular or soft-mode crystal, alternatively to thefolding mechanism discussed in the previous subsection,the observed low-frequency modes could be assigned toexternal vibrations of individual or coupled groups dueto the low symmetry and large unit cell. Evidently, theseinteresting observations require further experimental andtheoretical studies. It should be noted that computationof lattice dynamics properties of such a low-symmetrycompound with a very big unit cell is an extremely chal-lenging task not only for ab initio methods but also for(semi-)classical force-field methods due to rich variety ofdifferent coordination polyhedra. That is a challengingtask but it could be of a great help in understanding thenature of low-frequency part of the lattice dynamics inCu (BO ) .The above-discussed low-frequency IR and Ramanphonons in Cu (BO ) lie in the terahertz frequencyrange (1–5 THz) and thus can provide a playgroundfor either ultrafast optical excitation of such modes byfemtosecond laser pulses or for selective THz excitation of particular modes. Such methods were applied forphonon-assisted phase transitions or even for a modu-lation of the exchange interaction [53]. V. ELECTRONIC STRUCTURE AND OPTICALPROPERTIESA. Analysis of crystallographic positions of Cu ions in Cu (BO ) In Cu (BO ) , the Cu ions with the 3 d orbitalsoccupy sixteen nonequivalent crystallographic positionsof three types [23] as shown in Fig. 1. Among them,four are planar-square positions with the four nearestoxygens O − ; two are distorted square pyramids with(4 + 1) oxygens; and ten are strongly distorted octahe-dral positions with (4 + 2) oxygens. However, such con-clusions about the coordination of Cu ions were basedon a somewhat arbitrary basis. While the well-knowncoordination-number scheme works well in the case ofregular polyhedra it becomes not quite applicable in somecases. In fact, in Cu (BO ) there is a large number of in-equivalent positions with irregular and strongly distortedCu–O polyhedra and therefore it becomes more prefer-able to use the so-called effective coordination number (ECoN) approach [54; 55]. This approach is based onadding weighting scheme for selected bonds, where thefractions between 0 and 1 are assigned depending on itslength. For calculating N ECoN one needs to know onlybond lengths of the polyhedra under analysis [54].As the first step, the weighted average bond lengthshould be calculated according to the equation l av = (cid:80) i l i exp (cid:2) − ( l i /l min ) (cid:3)(cid:80) i exp (cid:2) − ( l i /l min ) (cid:3) , (2)where l min is the length of the shortest bond in the poly-hedron. The subsequent steps are aimed on calculatingthe relevant weights for each bond and the sum of theseweighted bonds gives the N ECoN as N ECoN = (cid:88) i exp (cid:104) − (cid:16) l i l av (cid:17) (cid:105) . (3)Such analysis was performed for the all sixteen posi-tions of Cu ions in Cu (BO ) presented in Fig. 1.Calculated values of N ECoN along with average bondlengths l av are shown in Fig. 6. The analysis showsthat the average bond lengths for all positions are closeto l av = 2.116 ˚A while the average effective coordina-tion number N avECoN = 3.94. Thus, this analysis allowsus to describe all the Cu-containing groups as fourfold-coordinated nearly square-planar groups. We performedsuch analysis for some other cuprates. For example, N ECoN = 4.0 in the monoclinic CuO [56]. Analyzing thestructural data of several other cuprates we have foundthat the value of N ECoN = 4.0 and nearby is typical formany other Cu complex oxides. But there are some FIG. 6. Analysis of the sixteen crystallographic positions ofCu ions in Cu (BO ) (see Fig. 1) for the nearest Cu–Obonds with the length up to 3 ˚A according to the crystallo-graphic data from [23]. Filled purple squares show averagedbond lengths for all polyhedra. Empty diamonds representthe length of all bonds. Red filled circles show calculated val-ues of effective coordination numbers. Dashed line shows thevalue of N ECoN = 4 for a perfectly square-planar coordina-tion. Green symbols show the N ECoN values for two differentcrystallographic positions, 4 b and 8 d , of CuB O [58]. interesting exceptions. For copper ions in the above-mentioned orthorhombic Cu (BO ) [25] which is struc-turally equivalent to other orthorhombic borates [57], N ECoN is very close to six for both types (2 b and 4 d ) ofthe [Cu–O] groups. We note that the triclinic Cu (BO ) where N ECoN ≈ N ECoN ≈ ρ = 4.54 and 5.12 g/cm , respectively, while the density ofmonoclinic phase is 4.434 g/cm . It should be noted thatsuch difference in densities of chemically complex com-pounds is quite a rare phenomenon. The difference in theeffective coordination of copper ions leads to a markedlymore friable structure of the triclinic Cu (BO ) . Thiscomparison of the two materials with the same chemicalcomposition but crystallizing in different crystal struc-tures and characterized by different N ECoN values em-phasizes once more the remarkable ability of Cu ionsfor adopting different positions in the unit cell, prefer-ably, with a lower local symmetry as a result of the Jahn –Teller effect. B. Optical absorption in the region of d-d electronic transitions
The crystal-field analysis of the 3 d orbitals (3 d in thehole representation) for the all types of positions showsthat the ground state is x - y . Obviously, the relative en-ergy positions of the excited states vary from one positionto another because they are defined by such factors as i) the symmetry of the position and ii) specific values of thecrystal-field parameters. Electronic d-d transitions fromthe ground state to excited states are forbidden by theparity selection rule, in particular, in the square-planarposition possessing the inversion center 1 (position num-bers 1 and 16 in Fig. 1). However, this rule is broken insquare pyramids and distorted octahedra, due to admix-ture of wave functions with different parity (from excitedconfigurations of copper or from charge-transfer bands)to the d wave functions by the odd crystal-field parame-ters. Another important contribution to the d-d absorp-tion intensity may come from vibronic transitions causedby the interaction between d electrons of Cu andodd-parity vibrations of the nearest surrounding. Thistemperature-dependent mechanism dominates in the op-tical absorption of CuGeO [59; 60]. For Cu (BO ) ,however, the intensity of the optical d-d absorption bands(situated in the spectral region 1.0–2.5 eV) practicallydoes not depend on the temperature (see Fig. 7), whichconfirms a dominant role of a mixing between the 3 d states of Cu and the states of a different parity in theCu centers without the inversion symmetry.Only the bc -plane-cleaved samples of Cu (BO ) wereused in absorption and reflection measurements. Thinsamples were obtained due to the perfect cleavage of thesingle crystals on the planes shown in Fig. 1. They wereof a dark-green color with a noticeable pleochroism inthe visible spectral range when they had the thickness t ≤ µ m (see the photo at the left side of Fig. 7).The room-temperature spectra shown in Fig. 7 comparethe optical absorption of Cu (BO ) ( k (cid:107) a *, unpolar-ized), CuB O ( k (cid:107) c , unpolarized), and CuGeO ( k (cid:107) a , E (cid:107) b , and E (cid:107) c ) in the regions of both intraconfigurational d-d transitions and the fundamental absorption edge.We concentrate first on the spectra of the d-d tran-sitions displayed in more detail in Fig. 8, upper panel.For Cu (BO ) (and also CuGeO ) they remain broadwhen the temperature decreases but peculiarities of thespectral shape become more pronounced.Absolute values of the absorption coefficient were cal-culated from measurements on several samples with thethickness of 5–6 µ m, and we estimate the accuracy ofthe measured absorption coefficient as not better than ∼
10 %. At the maximum (1.82 eV), the absorption co-efficient reaches the value of α = 5800 cm − . Stronganisotropy of the optical absorption and index of refrac-tion is expected but this is a subject of future studies.For comparison, the lower panel in Fig. 8 shows the low-temperature absorption spectrum of CuB O [18], whichradically differs from that of Cu (BO ) . First, charac-teristic rich fine structure of the electronic spectrum ofCuB O with well-defined ZP lines and sharp phononsidebands at the both crystallographic positions com-pletely disappears in Cu (BO ) . We note that no ZPlines or any fine structure were observed in a structurallymore simple CuGeO , the spectrum of which [59–61] isshown in Fig. 7. Second, the absorption coefficient atthe maximum is roughly an order of magnitude larger FIG. 7. (Color online) Absorption spectra of Cu (BO ) at T = 7 and 300 K ( k (cid:107) a *, unpolarized), of CuB O at 40 K( k (cid:107) c , unpolarized), and of CuGeO at 300 K for two polar-izations [60]. The photo at the left-hand side shows a thinsample of Cu (BO ) cleaved perpendicular to the a * axisand placed between the crossed polarizers. Green and bluearrows represent the angles of a polarizer and analyzer, re-spectively. Inset: Fringes in the reflected light due to theinterference of the beams reflected from the crystal faces inthe transparency window (upper panel) and the refractive in-dex calculated from these fringes (lower panel). for Cu (BO ) than for CuB O and CuGeO . On theother hand, the value α = 5800 cm − falls in the range ofabsorption coefficients found in several other Cu -basedoxide minerals, see, for example, Table 5.20 in Ref. [62].A question arises why there are such drastic differencesbetween the absorption spectra of chemically similar bo-rates Cu (BO ) and CuB O . Several factors can beassumed. First of all, CuB O should be considered as amore ionic compound whereas Cu (BO ) is more cova-lent, due to relatively larger concentration of Cu–O bondsby a factor of three to one. We suggest that this factoralong with the presence of a larger number of nonequiva-lent positions lead to a broadening of narrow ZP lines andvibronic transitions in Cu (BO ) in which only broadoverlapping bands are observed. A low crystal-field sym-metry of all Cu positions in this compound, on the onehand, leads to an appreciable admixture of excited elec-tronic configurations of the opposite parity to the 3 d configuration, thus enhancing the intensity of the d-d transitions. On the other hand, the low local symmetrycompletely removes the degeneracy of the 3 d states ofCu ions. In the upper panel of Fig. 8, we show resultsof the fitting of the experimental spectrum by four or-bital subbands related to the electronic transitions fromthe ground x - y state to the xy , z , xz ( yz ), and yz ( xz )excited states (see inset in upper panel). Broken linesconnecting the zero-phonon lines observed in CuB O and the broad spectrum of Cu (BO ) show that themaxima of partial bands in this compound are shifted x - y x y z x y ( y z ) 2 . 0 E x p e r i m e n t T = 7 K z C u ( B O ) x y x z ( y z ) ( x z ) y z y z ( x z ) 2 . 2 e V 3 d D q z x y E x p e r i m e n t (cid:1) s p e c t r u m T = 9 K Absorption coefficient (cid:1) (cm-1)
E n e r g y ( e V )C u B O x z ( y z ) FIG. 8. (Color online) Comparison of electronic absorp-tion spectra of Cu (BO ) (upper panel) and CuB O (lowerpanel) at 7 and 9 K, respectively. Inset shows a schematicrepresentation of the nondegenerate energy levels of the Cu ions in Cu (BO ) . Comparison of the two spectra shows acomplete absence of any fine structure in Cu (BO ) . On theother hand, the positions of maxima of the absorption bandsin Cu (BO ) (upper panel) lie very closely to the positions ofthe relevant bands in CuB O (lower panel). We note that the xz and yz states in CuB O are degenerate because of a highsymmetry of the both Cu positions whereas in Cu (BO ) these states are split. from the relevant zero-phonon lines in CuB O by thevalue of 0.08–0.12 eV lying in the energy range of thewhole phonon spectrum. Position of the maximum ofthe first partial band defines the cubic crystal-field pa-rameter 10 Dq = 1.5 eV averaged over all 16 inequivalentpositions of Cu ions. This 10 Dq value is very close tothe cubic crystal-field parameter in several Cu -basedminerals given in Table 5.20 in Ref. [62]. For comparison,the first ZP lines in CuB O in the 4 b and 8 d positionsdefine the genuine cubic parameter 10 Dq = 1.403 eV and1.577 eV, respectively.Figure 7 demonstrates another important difference inthe optical properties of Cu (BO ) and CuB O . A sig-nificant red shift of the fundamental absorption edge toa value of about 2.8 eV takes place in Cu (BO ) in com-parison to about 4 eV in CuB O . We can explain suchshift by a larger relative concentration of Cu–O bonds(approximate ratio is three to one) in Cu (BO ) . In fact,in pure copper oxide CuO, which is usually characterizedas a semiconductor with a direct fundamental band gap,the fundamental absorption edge is positioned at about1.9 eV [63–65]. This comparison allows us to make a con-0clusion that, in Cu -containing compounds, there is asystematic shift of the absorption edge due to charge-transfer transitions, from the lowest value of about 1.4 eVto more than 4.0 eV as the concentration of Cu–O bondsdecreases. This strong red shift of the absorption edgeis due to an increased hybridization between the 3 d evenstates of the copper ions and 4 p odd states of the oxygenions. The same hybridization contributes to an enhance-ment of the d-d absorption intensity, which explains inpart a strong increase of the electronic absorption upto α = 5800 cm − . Polarization measurements on the bc -plane-cleaved sample depicted at the left side of Fig. 7,show that extinction angle is close to the angle betweenthe b and c axis, 89 ◦ and 88.77 ◦ , respectively. VI. CONCLUSIONS
The present study of the triclinic copper borateCu (BO ) has shown that this crystal possesses severalunusual properties which, to a large extent, are related toits complex crystallographic layered structure and a largeunit cell with Z = 10. A bright feature of this structure isa perfect cleavage on the bc -type crystallographic planes.We suppose that a decisive role in forming the struc-ture itself and the properties of the compound belongsto the Jahn – Teller magnetic copper Cu ions ( S = 1/2)occupying sixteen strongly distorted nonequivalent po-sitions. In the original publication [23], these positionswere divided into the three main types, on the basis of thenumber of their nearest neighbors. However, our analy-sis of all types of the distorted polyhedra around Cu ions in Cu (BO ) , on the basis of an effective coordi-nation number (ECoN) approach [54], has shown that N ECoN is close to 4 for all the positions, confirming thatall of the Cu–O groups can be treated as nearly planar-square [CuO ] groups. The large difference, more than10%, in densities of triclinic, monoclinic, and orthorhom-bic phases of Cu (BO ) may point to an important roleof Jahn – Teller distortions in stabilizing each particularphase.The lattice dynamics of Cu (BO ) was studied withthe use of the infrared absorption and attenuated totalreflection, and Raman scattering spectroscopies. Thesestudies confirmed the presence of a very complex struc-ture of lattice excitations in both infrared and Ramanspectra. In particular, unusual low-frequency phononswere found below 100 cm − , which was not observed pre-viously in the chemically similar crystal CuB O [18]. Atthe present state of our research we can only tentativelyassign these low-frequency phonons either to interlayervibrations activated by the presence of the crystal super-structure or to phonons responsible for the movement oflarge groups of ions. Perhaps, a more definite explanation of this unusual low-frequency spectrum could be givenas a result of theoretical calculations of phonons in thiscomplex and low-symmetry structure with a very largeunit cell containing 110 atoms. Some of phonons showan anomalous behavior in the vicinity of the magneticphase transition at T N = 10 K evidencing a pronouncedmagnetoelastic interaction. We should emphasize thatno new phonons were found below T N , which excludesthe spin S = 1/2 dimerization through a magnetostruc-tural phase transition of the spin – Peierls type observed,e.g., in CuGeO [66–68].Absorption and reflection measurements in the range0.35–6.7 eV allowed us to characterize the electronicstructure formed by the 3 d states of Cu ions in thelow-symmetry crystal fields. A large number of nonequiv-alent positions of the Cu ions results in a broadeningof the range of the crystal-field parameters and a corre-sponding broadening and averaging of the whole 3 d ab-sorption spectrum originating from optical transitions indifferent ion positions.In a striking contrast to CuB O , no zero-phonon linesand fine phonon-sideband structure was found in the op-tical absorption spectra of Cu (BO ) characterized byan intense band positioned at about 1.8 eV, with themaximum absorption coefficient α = 5800 cm − . Thus,our study confirmed once more that CuB O remains,to the best of our knowledge, the only known compoundin which all zero-phonon lines at the both types of crys-tallographic positions can be identified, allowing calcu-lations of the genuine crystal field parameters [19]. Thestrong absorption band centered at 1.8 eV in Cu (BO ) was assigned to overlapping electronic transitions in allnonequivalent positions from the ground state x - y ofthe Cu ions to the xy , z , xz ( yz ), yz ( xz ) excited states.The first transition defines the cubic crystal-field param-eter 10 Dq = 1.5 eV (see inset to the upper panel, Fig. 8).We suppose that the strong increase of absorption re-lated to these electronic transitions below the fundamen-tal band edge by about an order of magnitude in compar-ison to CuB O is due to a larger relative concentrationof strongly-covalent Cu–O bonds in Cu (BO ) , in an ap-proximate ratio three to one. The stronger covalency alsoexplains a noticeable red shift of the fundamental absorp-tion edge to a value of about ∼ (BO ) ,in comparison with a value of about ∼ O . VII. ACKNOWLEDGEMENTS
This work was supported by the Russian Science Foun-dation under the project No. 16-12-10456. K.N.B. andM.N.P. acknowledge a support of the Russian Academyof Sciences under the Programs for Basic Research “Top-ical problems of the low-temperature physics”. ∗ [email protected]; Also at the Institute of Spec- troscopy, Russian Academy of Sciences, 108840 Moscow, Troitsk, Russia I. B. Bersuker,
The Jahn – Teller effect and vibronic inter-actions in modern chemistry (Springer Science & BusinessMedia, 2013). T. Kimura, Y. Sekio, H. Nakamura, T. Siegrist, and A. P.Ramirez, Nat Mater , 291 (2008). T. Kazimierczuk, D. Fr¨ohlich, S. Scheel, H. Stolz, andM. Bayer, Nature , 343 (2014). M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. , 3651 (1993). M. Azuma, Z. Hiroi, M. Takano, K. Ishida, and Y. Ki-taoka, Phys. Rev. Lett. , 3463 (1994). H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov,K. Onizuka, M. Kato, K. Kosuge, C. P. Slichter, T. Goto,and Y. Ueda, Phys. Rev. Lett. , 3168 (1999). H. Kageyama, M. Nishi, N. Aso, K. Onizuka, T. Yosihama,K. Nukui, K. Kodama, K. Kakurai, and Y. Ueda, Phys.Rev. Lett. , 5876 (2000). K. Kodama, M. Takigawa, M. Horvati´c, C. Berthier,H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, andF. Mila, Science , 395 (2002). F. Bert, S. Nakamae, F. Ladieu, D. L’Hˆote, P. Bonville,F. Duc, J.-C. Trombe, and P. Mendels, Phys. Rev. B ,132411 (2007). S. Park, Y. J. Choi, C. L. Zhang, and S.-W. Cheong, Phys.Rev. Lett. , 057601 (2007). R. V. Pisarev, A. S. Moskvin, A. M. Kalashnikova, A. A.Bush, and T. Rasing, Phys. Rev. B , 132509 (2006). A. Podlesnyak, L. M. Anovitz, A. I. Kolesnikov, M. Mat-suda, T. R. Prisk, S. Toth, and G. Ehlers, Phys. Rev. B , 064426 (2016). O. Janson, I. Rousochatzakis, A. A. Tsirlin, M. Belesi,A. A. Leonov, U. K. R¨oßler, J. van den Brink, and H. Ros-ner, Nature Communications , 5376 (2014). C. P. Landee and M. M. Turnbull, European Journal ofInorganic Chemistry , 2266 (2013). B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, andJ. Zaanen, Nature , 179 (2015). M. Martinez-Ripoll, S. Mart´ınez-Carrera, and S. Garc´ıa-Blanco, Acta Crystallographica Section B , 677 (1971). M. Boehm, B. Roessli, J. Schefer, A. S. Wills, B. Oulad-diaf, E. Leli`evre-Berna, U. Staub, and G. A. Petrakovskii,Phys. Rev. B , 024405 (2003). R. V. Pisarev, I. S¨anger, G. A. Petrakovskii, andM. Fiebig, Phys. Rev. Lett. , 037204 (2004). R. V. Pisarev, A. M. Kalashnikova, O. Sch¨ops, and L. N.Bezmaternykh, Phys. Rev. B , 075160 (2011). R. V. Pisarev, K. N. Boldyrev, M. N. Popova, A. N.Smirnov, V. Y. Davydov, L. N. Bezmaternykh, M. B.Smirnov, and V. Y. Kazimirov, Phys. Rev. B , 024301(2013). K. N. Boldyrev, R. V. Pisarev, L. N. Bezmaternykh, andM. N. Popova, Phys. Rev. Lett. , 247210 (2015). It is interesting to note that Pd(4 d )B O , according toW. Depmeier and H. Schmid, Acta Crystallographica Sec-tion B , 605 (1982) is the only other known crystal thatcrystallizes in the same I d group. However, to the bestof our knowledge no reports on its magnetic and opticalproperties is available in literature. H. Behm, Acta Crystallographica Section B , 2781(1982)Structural data (.cif) can be found in CrystallographyOpen Database with ID: 2105418. N. V. Kuratieva, D. Mikhailova, and H. Ehrenberg, ActaCrystallographica Section C , i85 (2009). R.-H. Zhang, D. Zhao, L. Zhang, F.-X. Ma, X. Xin, and F.-F. Li, Inorganic and Nano-Metal Chemistry , 521 (2017). G. A. Petrakovskii, K. A. Sablina, A. M. Vorotynov, O. A.Bayukov, A. F. Bovina, G. V. Bondarenko, R. Szymczak,M. Baran, and H. Szymczak, Physics of the Solid State , 610 (1999). G. A. Petrakovski˘ı, L. N. Bezmaternykh, O. A. Bayukov,M. A. Popov, J. Schefer, C. Neidermayer, P. Aleshkevich,and R. Szymczak, Physics of the Solid State , 1315(2007). K. Kudo, T. Noji, Y. Koike, T. Sakon, M. Motokawa,T. Nishizaki, and N. Kobayashi, Journal of the PhysicalSociety of Japan , 569 (2003). K. Kudo, T. Noji, and Y. Koike, Journal of the PhysicalSociety of Japan , 935 (2001). D. A. Balaev, K. A. Sablina, A. L. Freydman, A. A.Krasikov, and A. F. Bovina, Physics of the Solid State , 284 (2016). H. Sakurai, N. Tsuboi, M. Kato, K. Yoshimura, K. Kosuge,A. Mitsuda, H. Mitamura, and T. Goto, Phys. Rev. B ,024428 (2002). A. Fukaya, I. Watanabe, and K. Nagamine, Journal of thePhysical Society of Japan , 2868 (2001). J. Liu, S. Wen, X. Zou, F. Zuo, G. J. O. Beran, andP. Feng, J. Mater. Chem. A , 1553 (2013). K. Momma and F. Izumi, Journal of Applied Crystallog-raphy , 1272 (2011). N. J. Harrick, J. Phys. Chem , 1110 (1960). J. Fahrenfort, Spectrochimica Acta , 698 (1961). K. Nakamoto, “Applications in inorganic chemistry,” in
Infrared and Raman Spectra of Inorganic and CoordinationCompounds (John Wiley & Sons, Inc., 2008) pp. 149–354. D. Fausti, A. A. Nugroho, P. H. M. van Loosdrecht, S. A.Klimin, M. N. Popova, and L. N. Bezmaternykh, Phys.Rev. B , 024403 (2006). Y. E. Kitaev, M. F. Limonov, A. G. Panfilov, A. P. Mir-gorodskij, and R. A. Evarestov, Fizika Tverdogo Tela ,865 (1994). A. B. Kuz’menko, D. van der Marel, P. J. M. van Bentum,E. A. Tishchenko, C. Presura, and A. A. Bush, Phys. Rev.B , 094303 (2001). A. S. Moskvin, JETP Lett. , 345 (1993). A. S. Moskvin, N. N. Loshkareva, Y. P. Sukhorukov, M. A.Sidorov, and A. A. Samokhvalov, JETP , 967 (1994). M. N. Popova, A. B. Sushkov, A. N. Vasil’ev, M. Isobe,and Y. Ueda, JETP Letters , 743 (1997). L. J. de Jongh and A. R. Miedema, Advances in Physics , 1 (1974). S. A. Klimin, A. B. Kuzmenko, M. A. Kashchenko, andM. N. Popova, Phys. Rev. B , 054304 (2016). M. Wojdyr, Journal of Applied Crystallography , 1126(2010). W. Bues, G. F¨orster, and R. Schmitt, Zeitschrift f¨ur anor-ganische und allgemeine Chemie , 148 (1966). V. G. Ivanov, M. V. Abrashev, N. D. Todorov, V. Tomov,R. P. Nikolova, A. P. Litvinchuk, and M. N. Iliev, Phys.Rev. B , 094301 (2013). G. Ernst, C. Broholm, G. R. Kowach, and A. P. Ramirez,Nature , 147 (1998). J. Hlinka, T. Ostapchuk, E. Buixaderas, C. Kadlec,P. Kuzel, I. Gregora, J. Kroupa, M. Savinov, A. Klic,J. Drahokoupil, I. Etxebarria, and J. Dec, Phys. Rev. Lett. , 197601 (2014). N. N. Brandt, A. Y. Chikishev, A. V. Kargovsky, M. M.Nazarov, O. Parashchuk, D. A. Sapozhnikov, I. N.Smirnova, A. P. Shkurinov, and N. V. Sumbatyan, Vi-brational Spectroscopy , 53 (2008). T. Sato, K. Aoki, R. Kino, H. Kuroe, T. Sekine, M. Hase,K. Oka, T. Ito, and H. Eisaki, in
Proceedings of the Inter-national Conference on Strongly Correlated Electron Sys-tems (SCES2013) (2014) p. 014035. R. V. Mikhaylovskiy, E. Hendry, A. Secchi, J. H. Mentink,M. Eckstein, A. Wu, R. V. Pisarev, V. V. Kruglyak, M. I.Katsnelson, T. Rasing, and A. V. Kimel, Nature commu-nications , 8190 (2015). C. Giacovazzo,
Fundamentals of crystallography , Vol. 7(Oxford university press, USA, 2002). R. Hoppe, S. Voigt, H. Glaum, J. Kissel, H. P. M¨uller, andK. Bernet, Journal of the Less Common Metals , 105(1989). G. Tunell, E. Posnjak, and C. J. Ksanda, Zeitschrift f¨urKristallographie-Crystalline Materials , 120 (1935). R. V. Pisarev, M. A. Prosnikov, V. Y. Davydov, A. N.Smirnov, E. M. Roginskii, K. N. Boldyrev, A. D.Molchanova, M. N. Popova, M. B. Smirnov, and V. Y.Kazimirov, Phys. Rev. B , 134306 (2016). G. K. Abdullaev and K. S. Mamedov, Journal of StructuralChemistry , 637 (1981). M. Bassi, P. Camagni, R. Rolli, G. Samoggia, F. Parmi-giani, G. Dhalenne, and A. Revcolevschi, Phys. Rev. B , R11030 (1996). M. N. Popova, A. B. Sushkov, S. A. Golubchik, A. N.Vasil’ev, and L. I. Leonyuk, JETP , 1227 (1996). P. H. M. van Loosdrecht, in
Contemporary Studies in Con-densed Matter Physics , Solid State Phenomena, Vol. 61(Trans Tech Publications, 1998) pp. 19–26. R. G. Burns,
Mineralogical applications of crystal field the-ory , Vol. 5 (Cambridge University Press, 1993). S. C. Ray, Solar energy materials and solar cells , 307(2001). D. Tahir and S. Tougaard, Journal of Physics: CondensedMatter , 175002 (2012). W. Y. Ching, Y.-N. Xu, and K. W. Wong, Phys. Rev. B , 7684 (1989). H. Kuroe, T. Sekine, M. Hase, Y. Sasago, K. Uchinokura,H. Kojima, I. Tanaka, and Y. Shibuya, Phys. Rev. B ,16468 (1994). A. Damascelli, D. van der Marel, F. Parmigiani,G. Dhalenne, and A. Revcolevschi, Phys. Rev. B ,R11373 (1997). M. N. Popova, A. B. Sushkov, S. A. Golubchik, A. N.Vasil’ev, and L. I. Leonyuk, Phys. Rev. B57