Symmetry disquisition on the TiOX phase diagram
Symmetry disquisition on the TiOX phase diagram
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
(Dated: November 2, 2018)
Abstract
The sequence of phase transitions and the symmetry of in particular the low temperature incom-
mensurate and spin-Peierls phases of the quasi one-dimensional inorganic spin-Peierls system TiOX
(TiOBr and TiOCl) 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 (P21/m) 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 incom-
mensurate phases. Finally, a comparison of Raman data on VOCl to the TiOX spectra shows that
the high energy scattering observed previously has a phononic origin.
PACS numbers: 68.18.Jk Phase transitions
63.20.-e Phonons in crystal lattices
75.30.Et Exchange and superexchange interactions
75.30.Kz Magnetic phase boundaries (including magnetic transitions, metamagnetism, etc.)
78.30.-j Infrared and Raman spectra
http://arxiv.org/abs/0704.0170v2
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 to
study 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 (S=1/2) 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 superconductivity1,2.
Though they received less attention, also spin=1/2 systems based on early transition metal
oxides with electronic configuration 3d1, such as titanium oxyhalides (TiOX, with X=Br
or Cl), exhibit a variety of interesting properties3,4. 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 (the first being CuGeO3)
The TiO bilayers constituting the TiOX lattice are candidates for various exotic electronic
configurations, such as orbital ordered3, spin-Peierls6 and resonating-valence-bond states8.
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 orbital3. As a consequence of the shape of the occupied
orbital (which has lobes oriented in the b− and c−directions, where c is perpendicular to
the layers), the exchange interaction between the spins on different Ti ions arises mainly
from direct exchange within the TiO bilayers, along the b crystallographic direction3. This,
in spite of the two-dimensional structural character, gives the magnetic system of the TiOX
family its peculiar quasi one-dimensional properties6. Magnetic susceptibility6 and ESR3
measurements at high temperature are in reasonably good agreement with an antiferromag-
netic, one-dimensional spin-1/2 Heisenberg chain model. At low temperature (Tc1) TiOX
shows a first-order phase transition to a dimerised nonmagnetic state, discussed in terms
of a spin Peierls state6,9,10. Between this low temperature spin Peierls phase (SP) and the
one-dimensional antiferromagnet in the high temperature phase (HT), various experimental
evidence4,11,12,13 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 so far reported, the intermediate
phase (Tc1 < Tc2) exhibits a gapped magnetic excitation spectrum
4, anomalous broadening
of the phonon modes in Raman and IR spectra9,13, and features of a periodicity incom-
mensurate with the lattice14,15,16,17. Moreover, the presence of a pressure induced metal to
insulator transition has been recently suggested for TiOCl18. 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 is not clear and
the anomalous first-order character of the transition to the spin Peierls state is not explained.
Optical methods like 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 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 magnon-phonon 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 trans-
port technique. The crystallinity was checked by X-ray diffraction12. Typical crystal di-
mensions are a few mm2 in the ab−plane and 10-100 µm along the c−axis, the stacking
direction15. The sample was mounted in an optical flow cryostat, with a temperature sta-
bilization better than 0.1 K in the range from 2.6 K to 300 K. The Raman measurements
were performed using a triple grating micro-Raman spectrometer (Jobin Yvon, T64000),
equipped with a liquid nitrogen cooled CCD detector (resolution 2 cm−1 for the considered
frequency interval). 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 beam, giving access
to all the polarizations schemes allowed by the back-scattering configuration. Due to the
macroscopic morphology of the samples (thin sheets with natural surfaces parallel to the
ab−planes) the polarization analysis was performed mainly with the incoming beam parallel
to the c−axis (c(aa)c̄, c(ab)c̄ and c(bb)c̄, in Porto notation). Some measurements were per-
formed 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(c⋆)x̄.
The magnetization measurements were performed in a Quantum Design Magnetic Prop-
erty 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
(measured at atmospheric pressure) was subtracted from the data.
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. After-
wards, 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 TiOBr13,19.
A. High Temperature Phase
The crystal structure of TiOX in the high temperature (HT) 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:
Γtot = 3Ag + 2B1u + 3B2g + 2B2u + 3B3g + 2B3u. (1)
Among these, the modes with symmetry B1u, B2u, and B3u are infrared active in the polar-
izations along the c, b, and a crystallographic axes9, respectively. The modes with symmetry
Ag, B2g, and B3g are expected to be Raman active: The Ag modes in the polarization (aa),
(bb), and (cc); the B2g modes in (ac) and the B3g ones in (bc). Fig.1 shows the room tem-
100 200 300 400 100 200 300 400
Energy (cm-1)
c(bb)c
 TiOBr (T=100 K)
c(aa)c
 TiOCl (T=300 K)
FIG. 1: (Color online) Polarized Raman spectra (Ag) of TiOCl and TiOBr in the high temperature
phase, showing the three Ag modes. Left panel: (bb) polarization; right panel: (aa) polarization.
perature 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 spec-
tra are taken at 100 (a), 30 (b) and 3K (c). At room temperature three Raman active modes
are clearly observed in both compounds for the c(aa)c̄ and c(bb)c̄ polarizations (Fig.1), while
none are observed in the c(ab)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 (x(c⋆)x̄) on
TiOBr, where an intense mode is observed at the same frequency (as shown in the inset of
Fig.2(a)). In addition to these expected modes, TiOCl displays a broad peak in the c(bb)c̄
polarization, centered at around 160 cm−1 at 300K; a similar feature is observed in TiOBr
as a broad background in the low frequency region at 100K. As discussed for TiOCl13, these
modes are thought to be due to pre-transitional fluctuations. Upon decreasing the tempera-
ture, this ”peaked” background first softens, resulting in a broad mode at Tc2 (see Fig.2(b)),
and then locks at Tc1 into an intense sharp mode at 94.5 cm
−1 for TiOBr (Fig.2(c)) and at
131.5 cm−1 for TiOCl.
The frequency 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 literature7,9 and for the Raman modes the temperatures chosen for the two compounds
are 300K for TiOCl and 100K for TiOBr. The observed Raman frequencies agree well with
100 200 300 400 500 600
c(aa)c
c(ab)c
Energy (cm-1)
c(bb)c
T=3 K
T=30 K
c(aa)c
c(ab)c
c(bb)c
200 250 550 600
Energy (cm-1)
TiOBr - Pol x(c*)x
T=100 K
c(aa)c
c(ab)c
c(bb)c
FIG. 2: (Color online) Polarization analysis of the Raman spectra in the three phases of TiOBr,
taken at 3 (a), 30 (b) and 100K (c). 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(c∗)x̄ polarization (see text).
previous reports13. The calculated values reported in Table I are obtained with a spring-
model calculation based on phenomenological longitudinal and transversal spring constants
(see Appendix). The spring constants used were optimized using the TiOBr experimental
frequencies (except for the ones of the B3g modes due to their uncertain symmetry) 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 (mode 1) 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 νT iOCl/νT iOBr =
1.42 is similar to the mass ratio
MCl. The other modes (2 and 3) involve mainly
Ti or O displacements, and their frequencies scale with a lower ratio, as can be expected.
B. Low Temperature Phases
Although the symmetry of the low temperature phases has been studied by X-ray crys-
tallography, there is no agreement concerning the symmetry of the SP phase; different works
TABLE I: (a)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 in Table I are
measured in the x(c⋆)x̄ polarization they could therefore have either B2g or B3g symmetry (see
experimental details).
(a) TiOBr TiOCl VOCl
Exp. Cal. Exp. Cal. Exp. Cal.
Ag (σaa, σbb, σcc) 142.7 141 203 209.1 201 208.8
329.8 328.2 364.8 331.2 384.9 321.5
389.9 403.8 430.9 405.2 408.9 405.2
B2g(σac) 105.5 157.1 156.7
328.5 330.5 320.5
478.2 478.2 478.2
B3u(IR, a) 77
a 75.7 104b 94.4 93.7
417a 428.5 438b 428.5 425.2
B3g(σbc) 60 86.4 129.4 129.4
216 336.8 219 c 336.8 327.2
598 586.3 586.3 585.6
B2u(IR, b) 131
a 129.1 176b 160.8 159.5
275a 271.8 294b 272.1 269.8
B1u(IR, c) 155.7 194.1 192.4
304.8 301.1 303.5
aValue taken from Ref.7.
bValue taken from Ref.9.
cValue obtained considering the leakage in the σyy polarization.
TABLE II: The ratio between the frequency 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
(all the eigenvectors are fully c−polarized, the values are normalized to the largest displacement).
(b) Mode ν(TiOBr) νCl/νBr Ti O Br
1 142.7 1.42 0.107 0.068 1
2 329.8 1.11 1 0.003 0.107
3 389.9 1.11 0.04 1 0.071
proposed two different space groups, P21/m
14,15,16 and Pmm221.
The possible symmetry changes that a dimerisation 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 symme-
try reduction, while it is not necessarily correct for the low temperature phase, being the
transition at Tc1 is of first-order.
FIG. 3: (Color online) Comparison of the possible low temperature symmetries. The low temper-
ature structures reported are discussed, considering a dimerisation 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 (a) is monoclinic with its unique axis parallel to
the orthorhombic c−axis (space group P2/c), (b) shows the suggested monoclinic structure for the
SP phase (P21/m), and (c) depicts the alternative orthorhombic symmetry proposed for the low
T phase Pmm2.
Fig.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 po-
sition of the neighboring dimerised Ti pairs, the symmetry elements lost in the dimerisation
are different and the possible space groups in the SP phase are P2/c (Table III(a)), P21/m
(b) or Pmm2 (c). The first two are monoclinic groups with their unique axis perpendicular
to the TiO plane (along the c−axis of the orthorhombic phase), and lying in the TiO plane
(‖ to the a−axis of the orthorhombic phase), respectively. The third candidate (Fig.3(c))
has orthorhombic symmetry.
The group theory analysis based on the two space groups suggested for the SP phase
(P21/m
14 and Pmm221) shows that the number of modes expected to be Raman active is
TABLE III: Comparison between the possible low temperature space group.
(a) Space group P2/c
Unique axis ⊥ to TiO plane, C42h
4TiOBr per unit cell
Γ = 7Ag + 6Au + 9Bg + 11Bu
7Ag Raman active σxx, σyy, σzz, σxy
11Bg Raman active σxz, σyz
6Au and 9Bu IR active
(b) Space group P21/m
Unique axis in the TiO plane, C22h
4 TiOBr per unit cell
Γ = 12Ag + 5Au + 6Bg + 10Bu
12Ag Raman active σxx, σyy, σzz, σxy
6Bg Raman active σxz, σyz
5Au and 10Bu IR active
(c) Space group Pmm2
4 TiOBr per unit cell
Γ = 11A1 +A2 + 4B1 + 5B2
11A1 Raman active σxx, σyy, σzz
A2 Raman active σxy
4B1 and 5B2 Raman active in σxz and σyz
different in the two cases (Table III(b) and (c)). In particular, the 12 fully symmetric vibra-
tional modes (Ag), in the P21/m space group, are expected to be active in the σxx, σyy, σzz
and σxy polarizations, and 6Bg modes are expected to be active in the cross polarizations
(σxz and σyz). Note that in this notation, z refers to the unique axis of the monoclinic
cell, so σyz corresponds to c(ab)c for the HT orthorhombic phase. For Pmm2 the 11 A1
vibrational modes are expected to be active in the σxx, σyy, σzz polarizations, and only one
mode of symmetry A2 is expected to be active in the cross polarization (σxy or c(ab)c). The
experiments, reported in Table IV for both compounds and in Fig.2 for TiOBr only, show
that 10 modes are active in the c(aa)c and c(bb)c in the SP phase (Fig.2(c)), and, more
importantly, two modes are active in the cross polarization c(ab)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 crystallography15,21, 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 debated7,12,15. The temperature dependence 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
(labeled LT ), some stay invariant up to the HT phase (RT ) and some others undergo a
sudden broadening at Tc1 and slowly disappear upon approaching Tc2 (IT ). The polarization
analysis of the Raman modes in the temperature region Tc1 < T < Tc2 shows that the number
TABLE IV: Vibrational modes of the low temperature phases.
spin Peierls phase
(a) TiOBr Ag(σxx, σyy) 94.5 102.7 142.4 167 219
276.5 330 351 392 411∗
Ag(σxy) 175,6 506.5
TiOCl Ag(σxx, σyy) 131.5 145.8 203.5 211.5 296.5
305.3 322.6 365.1 387.5 431∗
Ag(σxy) 178.5 524.3
Intermediate phase
(b) TiOBr (30K) Ag(σxx, σyy) 94.5 142 221.5 277 328.5
344.5 390.4
TiOCl (75K) Ag(σxx, σyy) 132.8 206.2 302 317.2 364.8
380 420.6
∗ The broad line shape of this feature suggests it may originate from a two-phonon process.
100 150 200 250 300 350 400 450 10 20 30 40 50
TITRTITITRTLT
Energy (cm-1)
27.5 K
32.5 K
37.5 K
42.5 K
: broadens above T
: high-T mode
: dissappears above T
 351cm-1
 328cm-1
 275cm-1
 220cm-1
 94cm-1
 Temperature(K)
FIG. 4: (Color online) The temperature dependence of the Raman spectrum of TiOBr is depicted
(an offset is added for clarity). The 3 modes present at all temperatures are denoted by the label
RT . The modes characteristic of the low temperature phase (disappearing at Tc1 = 28 K) are
labelled LT , and the anomalous modes observed in both the low temperature and the intermediate
phase are labelled IT . The right panel (b) 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.
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 determination15, the intermediate incommensurate phase is dis-
cussed in two ways. Firstly, starting from the HT orthorhombic (Pmmn) and the SP mon-
oclinic space group (P21/m - unique axis in the TiO planes, ‖ to a), the modulation vector
required to explain the observed incommensurate peaks is two-dimensional for both space
groups. Secondly, starting from another monoclinic space group, with unique axis perpen-
dicular to the TiO bilayers (P2/c), the modulation vector required is one-dimensional. The
latter average symmetry is considered (in the commensurate variety) in Fig.3(a) and Table
III(a).
In the IP, seven modes are observed in the σxx, σyy and σzz geometry on both compounds
(see Table IV(b)), and none in the σxy 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 (Table III(a)). Even though from the evidence it is not possible to rule
out any of the other symmetries discussed, the conjecture that in the intermediate incom-
mensurate phase the average crystal symmetry is already reduced, supports the description
of the intermediate phase as a monoclinic group with a one-dimensional modulation15, 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
FIG. 5: (Color online) 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.
(P21/m) is a subgroup of the high temperature Pmmn group, but not a subgroup of any of
the possible intermediate phase space groups suggested (possible P2/c). 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(b) to discuss another intriguing vibrational feature of the interme-
diate phase. Among the modes characterizing the intermediate phase (IT ), 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 (≃300 cm−1 for both TiOCl and
TiOBr). This seems to indicate an interaction of the phonons with some excitation around
300 cm−1. Most likely this is in fact arising from a strong, thermally activated coupling
of the lattice with the magnetic excitations, and is consistent with the pseudo-spin gap
observed in NMR experiments4,22 of ≈430 K (≃300 cm−1).
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 crystallo-
graphic b−direction. This, however, is not the only effective magnetic interaction. In fact,
FIG. 6: (Color online) (a) Magnetization as a function of temperature measured with fields 1 T
and 5 T (the magnetization measured at 1 T is multiplied by a factor of 5 to evidence the linearity).
The inset shows the main magnetic interactions (see text). (b) 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.
one also expects a superexchange interaction between nearest and next-nearest neighbor
chains (J2 and J3 in the insert of Fig.6(a))
23. The situation of TiOX is made more interest-
ing by the frustrated geometry of the interchain interaction, where the magnetic coupling J2
between adjacent chains is frustrated and the exchange energies can not 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 DFT
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 exchange-correlation functional B3LYP26 and the
6-3111G* basisset.
TABLE V: Calculated Exchange interactions in TiOBr
TiOBr
J1 = −250 K
J2 = −46.99 K
J3 = 11.96 K
Although the computed value for the magnetic interaction along the b−axis is half of the
value obtained from the magnetic 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 (J1/J2 = 5.3)
in TiOBr. Firstly, this explains the substantial deviation of the Bonner-Fisher fit from
the magnetic susceptibility even at temperature higher than Tc2. Secondly, 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 ions23. Thus, the two exchange constants are expected to
depend differently on the structural changes induced by hydrostatic pressure, J1 should
increase with hydrostatic pressure (increases strongly with decreasing the distance between
the Ti ions), while J2 is presumably weakly affected due only to small changes in the Ti–
O–Ti angle (the compressibility estimated from the lattice dynamics simulation is similar
along the a and b crystallographic directions). 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, indeed
is 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 freedom7. Only recently, infrared spec-
troscopy supported by cluster calculations excluded a ground state degeneracy of the Ti d
orbitals for TiOCl, hence suggesting that orbital fluctuations can not play an important role
in the formation of the anomalous incommensurate phase27,28. 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 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 clus-
ter was surrounded by eight Ti3+ TIP 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 per-
formed using the MOLCAS quantum chemistry package29 with a triple quality basis set; for
the Ti 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 (> 3000 K) 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
TABLE VI: Crystal field splitting of 3d1 Ti3+ in TiOCl and TiOBr (eV).
TiOCl TiOBr
xy 0.29-0.29 0.29-0.30
xz 0.66-0.68 0.65-0.67
yz 1.59-1.68 1.48-1.43
x2 − r2 2.30-2.37 2.21-2.29
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 (responsible for the main exchange interaction in TiOX) and
dxy symmetry respectively. Where the lobes of the latter point roughly towards the Ti
ions of the nearest chain (Table VI). It is therefore reasonable to expect that the occupa-
tion 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 two-dimensional antiferromagnet30.
200 300 400
1000 1500
 20 K
 40 K
 60 K
 78 K
 98 K
 116 K
Frequency (cm-1)
 VOCl (3 K)
 TiOCl (3 K)
 TiOBr (10 K)
FIG. 7: (Color online) Raman scattering features of VOCl. (a) High energy scattering of TiOCl/Br
and VOCl, and (b) temperature dependence of the vibrational scattering features of VOCl. No
symmetry changes are observed at TN = 80 K.
The space group of VOCl at room temperature is the same as that of TiOX in the high
temperature phase (Pmmn), and, as discussed in the previous section, three Ag modes are
expected to be Raman active. As shown in Fig.7(b), three phonons are observed throughout
the full temperature range (3 − 300 K), and no changes are observed at TN . The modes
observed are consistent with the prediction of lattice dynamics calculations (Table I).
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 literature13,19. The
fact that the energy range of the anomalous feature is consistent with the magnetic exchange
constant in TiOCl (J=660 K) suggested at first an interpretation in terms of two-magnon
Raman scattering13. Later it was shown that the exchange constant estimated for TiOBr
is considerably smaller (J=406 K) 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 taht the considerably
smaller exchange interaction in the Br compound (J=406 K) falsifies this interpretation
and that magnon scattering is not at the origin of the high energy scattering of the two
compounds. Furthermore, the cluster calculation (Table VI) 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 (see fig. 7(a)) 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 (≃600 cm−1) of symmetry B3g (Table I) is expected to be Raman active in
the σyz polarization. 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 (B3g ⊗ B3g). The nature of the
anomalies observed is therefore tentatively ascribed to a multiple-phonon process. Further
detailed investigations of lattice dynamics are needed to clarify this issue.
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 (apart from one broad mode which is
ascribed to pre-transitional fluctuations). Comparing group theoretical analysis with the
polarized Raman spectra clarifies the symmetry of the spin-Peierls phase and shows that
the average symmetry of the incommensurate phase is different from both the high temper-
ature and the SP phases. The conjecture that the intermediate phase is compatible with a
different monoclinic symmetry (unique axis perpendicular to the TiO planes) 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 fea-
ture in all the compounds suggests that this feature has a vibrational origin rather than a
magnetic or electronic one.
Acknowledgements 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 discussions. This work was partially supported by the Stichting voor Fundamenteel
Onderzoek der Materie [FOM, financially supported by the Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO)], and by the German Science Foundation (DFG).
V. 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 (release 3.05). 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 a special interatomic potential. Only short range
interactions between nearest neighbor ions are taken into account. Considering the forces to
be central forces, the number of parameters is reduced to two for each atomic interaction:
the longitudinal and transversal forces respectively defined as L =
d2V (r̄i,j)
and T = 1
dV (r̄i,j)
A custom made program was interfaced with UNISOFT to optimize the elastic constants.
Our program proceeded scanning the n dimensional space (n = number of parameters) with
a discrete grid, to minimize the squared difference between the calculated phonon frequencies
and the measured experimental frequencies for TiOBr, taken from both Raman and infrared
spectroscopy. The phonon frequencies of TiOCl and VOCl were obtained using the elastic
constants optimized for TiOBr and substituting the appropriate ionic masses. The optimized
force constants between different atoms are reported in N/m in the following Table.
TABLE VII: Elastic constants used in the spring model calculation. The label numbers refer to
Fig. ??, while the letters refer to the different inequivalent positions of the ions in the crystal.
Number Ions Longitudinal (L) (N/m) Transversal (T) (N/m)
1 Ti(a)-Ti(b) 18.5 32.7
2 Ti(a)-O(a) 18.5 11.1
3 Ti(a)-O(b) 53.1 9.5
4 Ti(a)-X(a) 29.0 4.4
5 O(a)-O(b) 20.6 7.3
6 X(a)-O(a) 18.5 3.5
7 X(a)-X(b) 11.7 0.7
∗ Electronic address: d.fausti@rug.nl
† Electronic address: P.H.M.van.Loosdrecht@rug.nl
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	Introduction
	Experiment
	Results and Discussion
	High Temperature Phase
	Low Temperature Phases
	Magnetic Interactions
	Electronic Excitations and Comparison with VOCl
	Conclusion
	Appendix: Details of the spring model calculation
	References
