Symmetry disquisition on theTiOXphase diagram(X=Br,Cl)

University of Groningen Symmetry disquisition on the TiOX phase diagram (X=Br,Cl) Fausti, Daniele; Lummen, Tom T. A.; Angelescu, Cosmina; Macovez, Roberto; Luzon, Javier; Broer, Ria; Rudolf, Petra; van Loosdrecht, Paul H. M.; Tristan, Natalia; Buechner, Bernd Published in: Physical Review. B: Condensed Matter and Materials Physics DOI: 10.1103/PhysRevB.75.245114 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2007 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Fausti, D., Lummen, T. T. A., Angelescu, C., Macovez, R., Luzon, J., Broer, R., ... Büchner, B. (2007). Symmetry disquisition on the TiOX phase diagram (X=Br,Cl). Physical Review. B: Condensed Matter and Materials Physics, 75(24), [245114]. https://doi.org/10.1103/PhysRevB.75.245114 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 30-05-2020 PHYSICAL REVIEW B 75, 245114 共2007兲 Symmetry disquisition on the TiOX phase diagram „X = Br, Cl… Daniele Fausti,* Tom T. A. Lummen, Cosmina Angelescu, Roberto Macovez, Javier Luzon, Ria Broer, Petra Rudolf, and Paul H. M. van Loosdrecht† Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands Natalia Tristan and Bernd Büchner IFW Dresden, D-01171 Dresden, Germany Sander van Smaalen Laboratory of Crystallography, University of Bayreuth, 95440 Bayreuth, Germany Angela Möller, Gerd Meyer, and Timo Taetz Institut für Anorganische Chemie, Universität zu Köln, 50937 Köln, Germany 共Received 1 March 2007; published 18 June 2007兲 The sequence of phase transitions and the symmetry of, in particular, the low temperature incommensurate and spin-Peierls phases of the quasi-one-dimensional inorganic spin-Peierls system TiOX 共X = Br and Cl兲 have been studied using inelastic light scattering experiments. The anomalous first-order character of the transition to the spin-Peierls phase is found to be a consequence of the different symmetries of the incommensurate and spin-Peierls 共P21 / m兲 phases. The pressure dependence of the lowest transition temperature strongly suggests that magnetic interchain interactions play an important role in the formation of the spin-Peierls and the incommensurate phases. Finally, a comparison of Raman data on VOCl to the TiOX spectra shows that the high energy scattering previously observed has a phononic origin. DOI: 10.1103/PhysRevB.75.245114 PACS number共s兲: 68.18.Jk, 63.20.⫺e, 75.30.Kz, 78.30.⫺j I. INTRODUCTION The properties of low-dimensional spin systems are one of the key topics of contemporary condensed matter physics. Above all, the transition metal oxides with highly anisotropic interactions and low-dimensional structural elements provide a fascinating playground for studying novel phenomena, arising from their low-dimensional nature and from the interplay between lattice, orbital, spin, and charge degrees of freedom. In particular, low-dimensional quantum spin 共S = 1 / 2兲 systems have been widely discussed in recent years. Among them, layered systems based on a 3d9 electronic configuration were extensively studied in view of the possible relevance of quantum magnetism to high temperature superconductivity.1,2 Though they received less attention, also spin= 1 / 2 systems based on early transition metal oxides with electronic configuration 3d1, such as titanium oxyhalides 共TiOX, with X = Br or Cl兲, exhibit a variety of interesting properties.3,4 The attention originally devoted to the layered quasi-two-dimensional 3d1 antiferromagnets arose from considering them as the electron analog to the high-Tc cuprates.5 Only recently TiOX emerged in a totally new light, namely, as a one-dimensional antiferromagnet and as the second example of an inorganic spin-Peierls compound 共the first being CuGeO3兲.6,7 The TiO bilayers constituting the TiOX lattice are candidates for various exotic electronic configurations, such as orbital ordered,3 spin-Peierls,6 and resonating-valence-bond states.8 In the case of the TiOX family, the degeneracy of the d orbitals is completely removed by the crystal field splitting, so that the only d electron present, mainly localized on the Ti site, occupies a nondegenerate energy orbital.3 As a conse1098-0121/2007/75共24兲/245114共9兲 quence of the shape of the occupied orbital 共which has lobes oriented in the b and c directions, where c is perpendicular to the layers兲, the exchange interaction between the spins on different Ti ions arises mainly from direct exchange within the TiO bilayers, along the b crystallographic direction.3 This, in spite of the two-dimensional structural character, gives the magnetic system of the TiOX family its peculiar quasi-one-dimensional properties.6 Magnetic susceptibility6 and electron spin resonance3 measurements at high temperature are in reasonably good agreement with an antiferromagnetic, one-dimensional spin-1 / 2 Heisenberg chain model. At low temperature 共Tc1兲, TiOX shows a first-order phase transition to a dimerized nonmagnetic state, discussed in terms of a spin-Peierls state.6,9,10 Between this low temperature spin-Peierls phase 共SP兲 and the one-dimensional antiferromagnet in the high temperature 共HT兲 phase, various experimental evidences4,11–13 showed the existence of an intermediate phase, whose nature and origin is still debated. The temperature region of the intermediate phase is different for the two compounds considered in this work: for TiOBr, Tc1 = 28 K and Tc2 = 48 K, while for TiOCl, Tc1 = 67 K and Tc2 = 91 K. To summarize the properties reported so far, the intermediate phase 共Tc1 ⬍ Tc2兲 exhibits a gapped magnetic excitation spectrum,4 anomalous broadening of the phonon modes in Raman and IR spectra,9,13 and features of a periodicity incommensurate with the lattice.14–17 Moreover, the presence of a pressure induced metal to insulator transition has been recently suggested for TiOCl.18 Due to this complex phase behavior, both TiOCl and TiOBr have been extensively discussed in recent literature, and various questions still remain open: there is no agreement on the crystal symmetry of the spin Peierls phase, the nature and symmetry of the incommensurate phase are not clear, and the anomalous first- 245114-1 ©2007 The American Physical Society PHYSICAL REVIEW B 75, 245114 共2007兲 FAUSTI et al. order character of the transition to the spin-Peierls state is not explained. Optical methods such as Raman spectroscopy are powerful experimental tools for revealing the characteristic energy scales associated with the development of broken symmetry ground states, driven by magnetic and structural phase transitions. Indeed, information on the nature of the magnetic ground state, lattice distortion, and interplay of magnetic and lattice degrees of freedom can be obtained by studying in detail the magnetic excitations and the phonon spectrum as a function of temperature. The present paper reports on a vibrational Raman study of TiOCl and TiOBr, a study of the symmetry properties of the three phases and gives a coherent view of the anomalous first-order character of the transition to the spin-Peierls phase. Through pressure-dependence measurements of the magnetic susceptibility, the role of magnonphonon coupling in determining the complex phase diagram of TiOX is discussed. Finally, via a comparison with the isostructural compound VOCl, the previously reported13,19 high energy scattering is revisited, ruling out a possible interpretation in terms of magnon excitations. II. EXPERIMENT Single crystals of TiOCl, TiOBr, and VOCl have been grown by a chemical vapor transport technique. The crystallinity was checked by x-ray diffraction.12 Typical crystal dimensions are a few mm2 in the ab plane and 10– 100 ␮m along the c axis, the stacking direction.15 The sample was mounted in an optical flow cryostat, with a temperature stabilization better than 0.1 K in the range from 2.6 to 300 K. The Raman measurements were performed using a triple grating micro-Raman spectrometer 共Jobin Yvon, T64000兲, equipped with a liquid nitrogen cooled charge coupled device detector 共resolution of 2 cm−1 for the considered frequency interval兲. The experiments were performed with a 532 nm Nd: YVO4 laser. The power density on the sample was kept below 500 W / cm2 to avoid sample degradation and to minimize heating effects. The polarization was controlled on both the incoming and outgoing beams, giving access to all the polarizations schemes allowed by the backscattering configuration. Due to the macroscopic morphology of the samples 共thin sheets with natural surfaces parallel to the ab planes兲, the polarization analysis was performed mainly with the incoming beam parallel to the c axis 关c共aa兲c̄, c共ab兲c̄, and c共bb兲c̄, in Porto notation兴. Some measurements were performed with the incoming light polarized along the c axis, where the k vector of the light was parallel to the ab plane and the polarization of the outgoing light was not controlled. These measurements will be labeled as x共c 쐓 兲x̄. The magnetization measurements were performed in a Quantum Design magnetic property measurement system. The pressure cell used is specifically designed for measurement of the dc magnetization in order to minimize the cell’s magnetic response. The cell was calibrated using the lead superconducting transition as a reference, and the cell’s signal 共measured at atmospheric pressure兲 was subtracted from the data. FIG. 1. 共Color online兲 Polarized Raman spectra 共Ag兲 of TiOCl and TiOBr in the high temperature phase, showing the three Ag modes. Left panel: 共bb兲 polarization; right panel: 共aa兲 polarization. III. RESULTS AND DISCUSSION The discussion will start with a comparison of Raman experiments on TiOCl and TiOBr in the high temperature phase, showing the consistency with the reported structure. Afterward, through the analysis of Raman spectra, the crystal symmetry in the low temperature phases will be discussed, and in the final part, a comparison with the isostructural VOCl will be helpful to shed some light on the origin of the anomalous high energy scattering reported for TiOCl and TiOBr.13,19 A. High temperature phase The crystal structure of TiOX in the high temperature 共HT兲 phase consists of buckled Ti-O bilayers separated by layers of X ions. The HT structure is orthorhombic with space group Pmmn. The full representation20 of the vibrational modes in this space group is ⌫tot = 3Ag + 2B1u + 3B2g + 2B2u + 3B3g + 2B3u . 共1兲 Among these, the modes with symmetries B1u, B2u, and B3u are infrared active in the polarizations along the c, b, and a crystallographic axes,9 respectively. The modes with symmetries Ag, B2g, and B3g are expected to be Raman active: the Ag modes in the polarization 共aa兲, 共bb兲, and 共cc兲 and the B2g modes in 共ac兲 and the B3g ones in 共bc兲. Figure 1 shows the room temperature Raman measurements in different polarizations for TiOCl and TiOBr, and Fig. 2 displays the characteristic Raman spectra for the three different phases of TiOBr; the spectra are taken at 共a兲 100, 共b兲 30, and 共c兲 3 K. At room temperature, three Raman active modes are clearly observed in both compounds for the c共aa兲c̄ and c共bb兲c̄ polarizations 共Fig. 1兲, while none are observed in the c共ab兲c̄ polarization. These results are in good agreement with the group theoretical analysis. The additional weakly active modes observed at 219 cm−1 for TiOCl and at 217 cm−1 for TiOBr are ascribed to a leak from a different polarization. This is confirmed by the measurements with the optical axis parallel to the ab planes 关x共c 쐓 兲x̄兴 on TiOBr, where an intense mode is observed at the same frequency 关as shown in 245114-2 PHYSICAL REVIEW B 75, 245114 共2007兲 SYMMETRY DISQUISITION ON THE TiOX PHASE… TABLE I. 共a兲Vibrational modes for the high temperature phase in TiOCl, TiOBr, and VOCl. The calculated values are obtained with a spring model. The mode reported in italics are measured in the x共c 쐓 兲x̄ polarization; they could therefore have either B2g or B3g symmetry 共see experimental details兲. TiOBr Ag共␴aa , ␴bb , ␴cc兲 TiOCl VOCl Expt. Calc. Expt. Calc. Expt. Calc. 142.7 329.8 389.9 141 328.2 403.8 203 364.8 430.9 209.1 331.2 405.2 201 384.9 408.9 208.8 321.5 405.2 B2g共␴ac兲 105.5 328.5 478.2 157.1 330.5 478.2 156.7 320.5 478.2 B3u共IR , a兲 77a 417a 75.7 428.5 104b 438b 94.4 428.5 93.7 425.2 B3g共␴bc兲 60 216 598 86.4 336.8 586.3 c 219 129.4 336.8 586.3 129.4 327.2 585.6 131a 275a 129.1 271.8 176b 294b 160.8 272.1 159.5 269.8 194.1 301.1 192.4 303.5 B2u共IR , b兲 B1u共IR , c兲 155.7 304.8 a Value taken from Ref. 7. Value taken from Ref. 9. c Value obtained considering the leakage in the ␴yy polarization. b the inset of Fig. 2共a兲兴. In addition to these expected modes, TiOCl displays a broad peak in the c共bb兲c̄ polarization, centered at around 160 cm−1 at 300 K; a similar feature is observed in TiOBr as a broad background in the low frequency region at 100 K. As discussed for TiOCl,13 these modes are thought to be due to pretransitional fluctuations. Upon decreasing the temperature, this “peaked” background first softens, resulting in a broad mode at Tc2 关see Fig. 2共b兲兴, and then locks at Tc1 into an intense sharp mode at 94.5 cm−1 for TiOBr 关Fig. 2共c兲兴 and at 131.5 cm−1 for TiOCl. The frequencies of all the vibrational modes observed for TiOCl and TiOBr in their high temperature phase are summarized in Table I. Here, the infrared active modes are taken from the literature,7,9 and for the Raman modes, the temperatures chosen for the two compounds are 300 K for TiOCl and 100 K for TiOBr. The observed Raman frequencies agree well with previous reports.13 The calculated values re- ported in Table I are obtained with a spring-model calculation based on phenomenological longitudinal and transversal spring constants 共see Appendix兲. The spring constants used were optimized using the TiOBr experimental frequencies 共except for the ones of the B3g modes due to their uncertain symmetry兲 and kept constant for the other compounds. The frequencies for the other two compounds are obtained by merely changing the appropriate atomic masses and are in good agreement with the experimental values. The relative atomic displacements for each mode of Ag symmetry are shown in Table II. The scaling ratio for the lowest frequency mode 共mode 1兲 between the two compounds is in good agreement with the calculation of the atomic displacements. The low frequency mode is mostly related to Br/ Cl movement and, indeed, the ratio ␯TiOCl / ␯TiOBr = 1.42 is similar to the mass ratio 冑M Br / 冑M Cl. The other modes 共2 and 3兲 in- TABLE II. The ratio between the frequencies of the Ag Raman active modes measured in TiOBr and TiOCl is related to the atomic displacements of the different modes as calculated for TiOBr 共all the eigenvectors are fully c polarized, the values are normalized to the largest displacement兲. Mode ␯共TiOBr兲 ␯Cl / ␯Br Ti O Br 1 2 3 142.7 329.8 389.9 1.42 1.11 1.11 0.107 1 0.04 0.068 0.003 1 1 0.107 0.071 245114-3 PHYSICAL REVIEW B 75, 245114 共2007兲 FAUSTI et al. volve mainly Ti or O displacements, and their frequencies scale with a lower ratio, as can be expected. (a) P2/c (b) P21/m (c) Pmm2 B. Low temperature phases Although the symmetry of the low temperature phases has been studied by x-ray crystallography, there is no agreement concerning the symmetry of the SP phase; different works proposed two different space groups, P21 / m 共Refs. 14–16兲 and Pmm2.21 The possible symmetry changes that a dimerization of Ti ions in the b direction can cause are considered in order to track down the space group of the TiOX crystals in the low temperature phases. Assuming that the low temperature phases belong to a subgroup of the high temperature orthorhombic space group Pmmn, there are different candidate space groups for the low temperature phases. Note that the assumption is certainly correct for the intermediate phase, because the transition at Tc2 is of second order implying a symmetry reduction, while it is not necessarily correct for the low temperature phase, being the transition at Tc1 is of first order. Figure 3 shows a sketch of the three possible low temperature symmetries considered, and Table III reports a summary of the characteristic of the unit cell together with the number of phonons expected to be active for the different space groups. Depending on the relative position of the neighboring dimerized Ti pairs, the symmetry elements lost FIG. 2. 共Color online兲 Polarization analysis of the Raman spectra in the three phases of TiOBr, taken at 共a兲 3, 共b兲 30, and 共c兲 100 K. The spectra of TiOCl show the same main features and closely resemble those of TiOBr. Table IV reports the frequencies of the TiOCl modes. The inset shows the TiOBr spectrum in the x共c ⴱ 兲x̄ polarization 共see text兲. FIG. 3. 共Color online兲 Comparison of the possible low temperature symmetries. The low temperature structures reported are discussed, considering a dimerization of the unit cell due to Ti-Ti coupling and assuming a reduction of the crystal symmetry. The red rectangle denotes the unit cell of the orthorhombic HT structure. Structure 共a兲 is monoclinic with its unique axis parallel to the orthorhombic c axis 共space group P2 / c兲, 共b兲 shows the suggested monoclinic structure for the SP phase 共P21 / m兲, and 共c兲 depicts the alternative orthorhombic symmetry proposed for the low T phase Pmm2. in the dimerization are different and the possible space groups in the SP phase are P2 / c 关Table III共a兲兴, P21 / m 关Table III共b兲兴, or Pmm2 关Table III共c兲兴. The first two are monoclinic groups with their unique axis perpendicular to the TiO plane 共along the c axis of the orthorhombic phase兲 and lying in the TiO plane 共储 to the a axis of the orthorhombic phase兲, respectively. The third candidate 关Fig. 3共c兲兴 has orthorhombic symmetry. The group theory analysis based on the two space groups suggested for the SP phase 关P21 / m 共Ref. 14兲 and Pmm2 共Ref. 21兲兴 shows that the number of modes expected to be Raman active is different in the two cases 关Tables III共b兲 and III共c兲兴. In particular, the 12 fully symmetric vibrational modes 共Ag兲, in the P21 / m space group, are expected to be active in the ␴xx, ␴yy, ␴zz, and ␴xy polarizations, and 6Bg modes are expected to be active in the cross polarizations 共␴xz and ␴yz兲. Note that in this notation, z refers to the unique axis of the monoclinic cell, so ␴yz corresponds to c共ab兲c for the HT orthorhombic phase. For Pmm2, the 11 A1 vibrational modes are expected to be active in the ␴xx, ␴yy, and ␴zz polarizations, and only one mode of symmetry A2 is expected to be active in the cross polarization 关␴xy or c共ab兲c兴. The experiments, reported in Table IV for both compounds and in Fig. 2 for TiOBr only, show that ten modes are active in the 245114-4 PHYSICAL REVIEW B 75, 245114 共2007兲 SYMMETRY DISQUISITION ON THE TiOX PHASE… TABLE III. Comparison between the possible low temperature space group. 共a兲 Space group P2 / c Unique axis ⬜ to TiO plane, C42h Four TiOBr per unit cell ⌫ = 7Ag + 6Au + 9Bg + 11Bu 7Ag Raman active ␴xx, ␴yy, ␴zz, ␴xy 11Bg Raman active ␴xz, ␴yz 6Au and 9Bu IR active 共b兲 Space group P21 / m Unique axis in the TiO plane, C22h Four TiOBr per unit cell ⌫ = 12Ag + 5Au + 6Bg + 10Bu 12Ag Raman active ␴xx, ␴yy, ␴zz, ␴xy 6Bg Raman active ␴xz, ␴yz 5Au and 10Bu IR active 共c兲 Space group Pmm2 Four TiOBr per unit cell ⌫ = 11A1 + A2 + 4B1 + 5B2 11A1 Raman active ␴xx, ␴yy, ␴zz A2 Raman active ␴xy 4B1 and 5B2 Raman active in ␴xz and ␴yz c共aa兲c and c共bb兲c in the SP phase 关Fig. 2共c兲兴, and, more importantly, two modes are active in the cross polarization c共ab兲c. This is not compatible with the expectation for Pmm2. Hence, the comparison between the experiments and the group theoretical analysis clearly shows that of the two low temperature structures reported in x-ray crystallography,15,21 only the P21 / m is compatible with the present results. As discussed in the Introduction, the presence of three phases in different temperature intervals for TiOX is now well established even though the nature of the intermediate phase is still largely debated.7,12,15 The temperature depen- FIG. 4. 共Color online兲 The temperature dependence of the Raman spectrum of TiOBr is depicted 共an offset is added for clarity兲. The three modes present at all temperatures are denoted by the label RT. The modes characteristic of the low temperature phase 共disappearing at Tc1 = 28 K兲 are labeled LT, and the anomalous modes observed in both the low temperature and the intermediate phase are labeled IT. The right panel 共b兲 shows the behavior of the frequency of IT modes, plotted renormalized to their frequency at 45 K. It is clear that the low-frequency modes shift to higher energy while the high-frequency modes shift to lower frequency. dence of the Raman active modes for TiOBr between 3 and 50 K is depicted in Fig. 4. In the spin-Peierls phase, as discussed above, the reduction of the crystal symmetry16 increases the number of Raman active modes. Increasing the temperature above Tc1, a different behavior for the various low temperature phonons is observed. As shown in Fig. 4, some of the modes disappear suddenly at Tc1 共labeled LT兲, some stay invariant up to the HT phase 共RT兲, and some others undergo a sudden broadening at Tc1 and slowly disappear upon approaching Tc2 共IT兲. The polarization analysis of the Raman modes in the temperature region Tc1 ⬍ T ⬍ Tc2 shows TABLE IV. Vibrational modes of the low temperature phases. TiOBr Ag共␴xx , ␴yy兲 Ag共␴xy兲 TiOCl Ag共␴xx , ␴yy兲 Ag共␴xy兲 TiOBr 共30 K兲 Ag共␴xx , ␴yy兲 TiOCl 共75 K兲 Ag共␴xx , ␴yy兲 a 共a兲 Spin-Peierls phase 94.5 102.7 276.5 330 175,6 506.5 131.5 305.3 178.5 145.8 322.6 524.3 共b兲 Intermediate phase 94.5 142 344.5 390.4 132.8 380 206.2 420.6 142.4 351 167 392 219 411a 203.5 365.1 211.5 387.5 296.5 431a 221.5 277 328.5 302 317.2 364.8 The broad line shape of this feature suggests that it may originate from a two-phonon process. 245114-5 PHYSICAL REVIEW B 75, 245114 共2007兲 FAUSTI et al. Pmmn Hypothetical “standard” SP P21/m Symmetry Reduction: 2nd Order transition Unique axis Changed: 1st Order transition P2/c FIG. 5. 共Color online兲 The average crystal symmetry of the intermediate phase is proposed to be monoclinic with the unique axis parallel to the c axis of the orthorhombic phase. Hence, the low temperature space group is not a subgroup of the intermediate phase, and the transition to the spin-Peierls phase is consequently of first order. that the number of active modes in the intermediate phase is different from that in both the HT and the SP phases. The fact that at T = Tc1 some of the modes disappear suddenly while some others do not disappear strongly suggests that the crystal symmetry in the intermediate phase is different from both other phases and indeed confirms the first-order nature of the transition at Tc1. In the x-ray structure determination,15 the intermediate incommensurate phase is discussed in two ways. First, starting from the HT orthorhombic 共Pmmn兲 and the SP monoclinic space group 共P21 / m—unique axis in the TiO planes, 储 to a兲, the modulation vector required to explain the observed incommensurate peaks is two dimensional for both space groups. Second, starting from another monoclinic space group, with unique axis perpendicular to the TiO bilayers 共P2 / c兲, the modulation vector required is one dimensional. The latter average symmetry is considered 共in the commensurate variety兲 in Fig. 3共a兲 and Table III共a兲. In the intermediate phase, seven modes are observed in the ␴xx, ␴yy, and ␴zz geometry on both compounds 关see Table IV共b兲兴 and none in the ␴xy geometry. This appears to be compatible with all the space groups considered, and also with the monoclinic group with unique axis perpendicular to the TiO planes 关Table III共a兲兴. Even though from the evidence it is not possible to rule out any of the other symmetries discussed, the conjecture that in the intermediate incommensurate phase the average crystal symmetry is already reduced supports the description of the intermediate phase as a monoclinic group with a one-dimensional modulation,15 and, moreover, it explains the anomalous first-order character of the spin-Peierls transition at Tc1. The diagram shown in Fig. 5 aims to visualize that the space group in the spin-Peierls state 共P21 / m兲 is a subgroup of the high temperature Pmmn group, but not a subgroup of any of the possible intermediate phase space groups suggested 共possible P2 / c兲. This requires the phase transition at Tc1 to be of first order, instead of having the conventional spin-Peierls second-order character. Let us return to Fig. 4共b兲 to discuss another intriguing vibrational feature of the intermediate phase. Among the modes characterizing the intermediate phase 共IT兲, the ones at low frequency shift to higher energy approaching Tc2, while the ones at high frequency move to lower energy, seemingly converging to a central frequency 共⯝300 cm−1 for both TiOCl and TiOBr兲. This seems to indicate an interaction of the phonons with some excitation around 300 cm−1. Most likely J1 J2 J3 FIG. 6. 共Color online兲 共a兲 Magnetization as a function of temperature measured with fields of 1 and 5 T 共the magnetization measured at 1 T is multiplied by a factor of 5 to evidence the linearity兲. The inset shows the main magnetic interactions 共see text兲. 共b兲 Pressure dependence of Tc1. The transition temperature for transition to the spin-Peierls phase increases with increasing pressure. The inset shows the magnetization versus the temperature after subtracting the background signal coming from the pressure cell. this is, in fact, arising from a strong, thermally activated coupling of the lattice with the magnetic excitations and is consistent with the pseudospin gap observed in NMR experiments4,22 of ⬇430 K 共⯝300 cm−1兲. C. Magnetic interactions As discussed in the Introduction, due to the shape of the singly occupied 3d orbital, the main magnetic exchange interaction between the spins on the Ti ions is along the crystallographic b direction. This, however, is not the only effective magnetic interaction. In fact, one also expects a superexchange interaction between nearest and next-nearest neighbor chains 关J2 and J3 in the inset of Fig. 6共a兲兴.23 The situation of TiOX is made more interesting by the frustrated geometry of the interchain interaction, where the magnetic coupling J2 between adjacent chains is frustrated and the exchange energies cannot be simultaneously minimized. Table V reports the exchange interaction values for the three possible magnetic interactions calculated for TiOBr. These magnetic interactions were computed with a density functional theory broken symmetry approach24 using an atom cluster including the two interacting atoms and all the surrounding ligand atoms; in addition, the first shell of Ti3+ ions was replaced by Al3+ ions and also included in the cluster. The calculations were performed with the GAUSSIAN03 package25 using the hybrid exchangecorrelation functional B3LYP26 and the 6-3111G* basis set. Although the computed value for the magnetic interaction along the b axis is half of the value obtained from the mag- 245114-6 TABLE V. Calculated exchange interactions in TiOBr. J1 = −250 K J2 = −46.99 K J3 = 11.96 K PHYSICAL REVIEW B 75, 245114 共2007兲 SYMMETRY DISQUISITION ON THE TiOX PHASE… netic susceptibility fitted with a Bonner-Fisher curve accounting for a one-dimensional Heisenberg chain, it is possible to extract some conclusions from the ab initio computations. The most interesting outcome of the results is that in addition to the magnetic interaction along the b axis, there is a relevant interchain interaction 共J1 / J2 = 5.3兲 in TiOBr. First, this explains the substantial deviation of the Bonner-Fisher fit from the magnetic susceptibility even at temperature higher than Tc2. Second, the presence of an interchain interaction, together with the inherent frustrated geometry of the bilayer structure, was already proposed in literature12 in order to explain the intermediate phase and its structural incommensurability. The two competing exchange interactions J1 and J2 have different origins: the first arises from direct exchange between Ti ions, while the second is mostly due to the superexchange interaction through the oxygen ions.23 Thus, the two exchange constants are expected to depend differently on the structural changes induced by hydrostatic pressure; J1 should increase with hydrostatic pressure 共increases strongly with decreasing the distance between the Ti ions兲, while J2 is presumably weakly affected due only to small changes in the Ti-O-Ti angle 共the compressibility estimated from the lattice dynamics simulation is similar along the a and b crystallographic directions兲. The stability of the fully dimerized state is reduced by the presence of an interchain coupling, so that Tc1 is expected to be correlated to J1 / J2. Pressure dependent magnetic experiments have been performed to monitor the change of Tc1 upon increasing hydrostatic pressure. The main results shown in Fig. 6 are indeed consistent with this expectation: Tc1 increases linearly with pressure; unfortunately, it is not possible to address the behavior of Tc2 from the present measurements. D. Electronic excitations and comparison with VOCl The nature of the complex phase diagram of TiOX was originally tentatively ascribed to the interplay of spin, lattice, and orbital degrees of freedom.7 Only recently, infrared spectroscopy supported by cluster calculations excluded a ground state degeneracy of the Ti d orbitals for TiOCl, hence suggesting that orbital fluctuations cannot play an important role in the formation of the anomalous incommensurate phase.27,28 Since the agreement between the previous cluster calculations and the experimental results is not quantitative, the energy of the lowest 3d excited level is not accurately known, not allowing us to discard the possibility of an almost degenerate ground state. For this reason, a more formal cluster calculation has been performed using an embedded cluster approach. In this approach, a TiO2Cl4 cluster was treated explicitly with a CASSCF/CASPT2 quantum chemistry calculation. This cluster was surrounded by eight Ti3+ total ion potentials in order to account for the electrostatic interaction of the cluster atoms with the shell of the first neighboring atoms. Finally, the cluster is embedded in a distribution of punctual charges fitting the Madelung’s potential produced by the rest of the crystal inside the cluster region. The calculations were performed using the MOLCAS quantum chemistry package29 with a triple quality basis set; for the Ti TABLE VI. Crystal field splitting of 3d1 Ti3+ in TiOCl and TiOBr 共eV兲. xy xz yz x2 − r2 TiOCl TiOBr 0.29–0.29 0.66–0.68 1.59–1.68 2.30–2.37 0.29–0.30 0.65–0.67 1.48–1.43 2.21–2.29 atom polarization functions were also included. The calculations reported in Table VI confirmed the previously reported result27 for both TiOCl and TiOBr. The first excited state dxy is at 0.29– 0.3 eV 共⬎3000 K兲 for both compounds; therefore, the orbital degrees of freedom are completely quenched at temperatures close to the phase transition. A comparison with the isostructural compound VOCl has been carried out to confirm that the phase transitions of the TiOX compounds are intimately related to the unpaired S = 1 / 2 spin of the Ti ions. The V3+ ions have a 3d2 electronic configuration. Each ion carries two unpaired electrons in the external d shell and has a total spin of 1. The crystal field environment of V3+ ions in VOCl is similar to that of Ti3+ in TiOX, suggesting that the splitting of the degenerate d orbital could be comparable. The electrons occupy the two lowest t2g orbitals, of dy2−z2 共responsible for the main exchange interaction in TiOX兲 and dxy symmetries, respectively, where the lobes of the latter point roughly toward the Ti3+ ions of the nearest chain 共Table VI兲. It is therefore reasonable to expect that the occupation of the dxy orbital in VOCl leads to a substantial direct exchange interaction between ions in different chains in VOCl and thus favors a two-dimensional antiferromagnetic order. Indeed, the magnetic susceptibility is isotropic at high temperatures and well described by a quadratic two-dimensional Heisenberg model, and at TN = 80 K VOCl undergoes a phase transition to a twodimensional antiferromagnet.30 The space group of VOCl at room temperature is the same as that of TiOX in the high temperature phase 共Pmmn兲, and, as discussed in the previous section, three Ag modes are expected to be Raman active. As shown in Fig. 7共b兲, three phonons are observed throughout the full temperature range 共3 − 300 K兲, and no changes are observed at TN. The modes observed are consistent with the prediction of lattice dynamics calculations 共Table I兲. In the energy region from 600 to 1500 cm−1, both TiOBr and TiOCl show a similar highly structured broad scattering continuum, as already reported in literature.13,19 The fact that the energy range of the anomalous feature is consistent with the magnetic exchange constant in TiOCl 共J = 660 K兲 suggested at first an interpretation in terms of two-magnon Raman scattering.13 Later, it was shown that the exchange constant estimated for TiOBr is considerably smaller 共J = 406 K兲 with respect to that of TiOCl, while the high energy scattering stays roughly at the same frequency. Even though the authors of Ref. 19 still assigned the scattering continuum to magnon processes, it seems clear that the considerably smaller exchange interaction in the Br compound 共J = 406 K兲 falsifies this interpretation and that magnon scat- 245114-7 PHYSICAL REVIEW B 75, 245114 共2007兲 FAUSTI et al. (a) (b) a b b a a b FIG. 8. 共Color online兲 Sketch of the bilayer structure 共b兲 and of the interactions introduced in the spring-model calculation 共a兲. tion. Looking back at Fig. 2, the inset shows the measurements performed with the optical axis parallel to the TiOX plane, where the expected mode is observed at 598 cm−1. The two-phonon process related to this last intense mode is in the energy range of the anomalous scattering feature and has symmetry Ag 共B3g 丢 B3g兲. The nature of the anomalies observed is therefore tentatively ascribed to a multiplephonon process. Further detailed investigations of lattice dynamics are needed to clarify this issue. FIG. 7. 共Color online兲 Raman scattering features of VOCl. 共a兲 High energy scattering of TiOCl/ Br and VOCl, and 共b兲 temperature dependence of the vibrational scattering features of VOCl. No symmetry changes are observed at TN = 80 K. tering is not at the origin of the high energy scattering of the two compounds. Furthermore, the cluster calculation 共Table VI兲 clearly shows that no excited crystal field state is present in the energy interval considered, ruling out a possible orbital origin for the continuum. These observations are further strengthened by the observation of a similar continuum scattering in VOCl 关see Fig. 7共a兲兴, which has a different magnetic and electronic nature. Therefore, the high energy scattering has most likely a vibrational origin. The lattice dynamics calculations, confirmed by the experiments, show that a “high” energy mode 共⯝600 cm−1兲 of symmetry B3g 共Table I兲 is expected to be Raman active in the ␴yz polarizaTABLE VII. Elastic constants used in the spring-model calculation. The label numbers refer to Fig. 8, while the letters refer to the different inequivalent positions of the ions in the crystal. Number 1 2 3 4 5 6 7 Ions Longitudinal 共L兲 共N/m兲 Transversal 共T兲 共N/m兲 Ti共a兲-Ti共b兲 Ti共a兲-O共a兲 Ti共a兲-O共b兲 Ti共a兲 – X共a兲 O共a兲-O共b兲 X共a兲 – O共a兲 X共a兲 – X共b兲 18.5 18.5 53.1 29.0 20.6 18.5 11.7 32.7 11.1 9.5 4.4 7.3 3.5 0.7 IV. CONCLUSION The symmetry of the different phases has been discussed on the basis of inelastic light scattering experiments. The high temperature Raman experiments are in good agreement with the prediction of the group theoretical analysis 共apart from one broad mode which is ascribed to pretransitional fluctuations兲. Comparing group theoretical analysis with the polarized Raman spectra clarifies the symmetry of the spinPeierls phase and shows that the average symmetry of the incommensurate phase is different from both the high temperature and the SP phases. The conjecture that the intermediate phase is compatible with a different monoclinic symmetry 共unique axis perpendicular to the TiO planes兲 could explain the anomalous first-order character of the transition to the spin-Peierls phase. Moreover, an anomalous behavior of the phonons characterizing the intermediate phase is interpreted as evidencing an important spin-lattice coupling. The susceptibility measurements of TiOBr show that Tc1 increases with pressure, which is ascribed to the different pressure dependence of intrachain and interchain interactions. Finally, we compared the TiOX compounds with the “isostructural” VOCl. The presence of the same anomalous high energy scattering feature in all the compounds suggests that this feature has a vibrational origin rather than a magnetic or electronic one. ACKNOWLEDGMENTS The authors are grateful to Maxim Mostovoy, Michiel van der Vegte, Paul de Boeij, Daniel Khomskii, Iberio Moreira, and Markus Grüninger for valuable and insightful discus245114-8 PHYSICAL REVIEW B 75, 245114 共2007兲 SYMMETRY DISQUISITION ON THE TiOX PHASE… sions. This work was partially supported by the Stichting voor Fundamenteel Onderzoek der Materie 关FOM, financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek 共NWO兲兴 and by the German Science Foundation 共DFG兲. APPENDIX: DETAILS OF THE SPRING-MODEL CALCULATION The spring-model calculation reported in the paper was carried out using the software for lattice-dynamical calculation UNISOFT31 共release 3.05兲. In the calculations, the Born– von Karman model was used; here, the force constants are treated as model parameters and they are not interpreted in terms of any special interatomic potentials. 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