Lonkai, T.; Tomuta, D.G.; Amann, U.; Ihringer, J.; Hendrikx, R.W.A.; Többens, D.M.; Mydosh,
J.A.
Citation
Lonkai, T., Tomuta, D. G., Amann, U., Ihringer, J., Hendrikx, R. W. A., Többens, D. M., &
Mydosh, J. A. (2004). Development of the high-temperature phase of hexagonal manganites.
Physical Review B, 69(13), 134108. doi:10.1103/PhysRevB.69.134108
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Development of the high-temperature phase of hexagonal manganites
Th. Lonkai,1,*D. G. Tomuta,2,†U. Amann,1J. Ihringer,1R. W. A. Hendrikx,2D. M. To¨bbens,3and J. A. Mydosh2,4
1
Institut fu¨r Angewandte Physik, Universita¨t Tu¨bingen, Auf der Morgenstelle 10, 72076 Tu¨bingen, Germany
2Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands 3BENSC, HMI Berlin, Glienicker Straße 100, 14109 Berlin, Germany
4Max-Planck Institut fu¨r Chemische Physik Fester Stoffe, No¨thnitzer Straße 40, 01187 Dresden, Germany
共Received 14 October 2003; published 19 April 2004兲
Reports about the ferroelectric ordering temperatures in the multiferroic hexagonal RMnO3 system are
controversial: transition temperatures varying between ⬇900 K and ⬇1300 K are reported for the same material. To elucidate the structural changes leading to ferroelectric distortions in hexagonal manganites, we calculate the irreducible representations of the distortions from the possible high-temperature symmetry P63/mmc to the low-temperature symmetry P63cm. There are four different orthogonal modes, of which
only one allows a spontaneous electric polarization. Structure refinements and an accurate statistical analysis of neutron powder-diffraction data of TmMnO3, based on this group-theoretical analysis, reveal two phase
tran-sitions: We extrapolate a polar to nonpolar transition temperature of Tn pt⫽1433(27) K, where the hexagonal
bitetrahedra start to tilt, while the ferroelectric distortion appears at TFE⫽1050(50) K. For R⫽Lu, Yb the tilt
of the bitetrahedra and the buckling of the R layers as well as the ferroelectric distortion were extrapolated to comparable temperatures.
DOI: 10.1103/PhysRevB.69.134108 PACS number共s兲: 77.80.Bh, 61.12.Ld, 61.10.Nz, 02.20.⫺a
I. INTRODUCTION
Multiferroic compounds of hexagonal manganites
RMnO3(R⫽Er,Ho,In,Lu,Sc,Tm,Y,Yb兲 represent a challeng-ing and fascinatchalleng-ing field in solid-state physics. The simulta-neous presence of electric and magnetic ordering (TFE ⬇1000 K, TN⬇100 K),1 the famous problem with two
nearly homometric magnetic lattices2and the unusual strong magnetic two-dimensional共2D兲 short-range order,3the exis-tence of ferroelectromagnetic domains,4and the coupling of electric and magnetic order parameters5 attracted interest of experimentalists, engineers and theoreticians. Possible appli-cations as random access memory devices are proposed ac-cording to the ferromagnetic and ferroelectric behaviors of epitaxial thin films.6,7 Though systematic studies on this class started in the mid 1960s,2,8and recent publications are numerous, e.g., Refs. 9–12, the magnetic structure remained controversial until optical measurements were applied13 and a novel statistical method of data analysis for diffraction data was developed.14
The analysis of the electric ordering process is likewise problematic. Hexagonal manganites order in the space group
共SG兲 P63cm. 8
The absence of the inversion center in
P63cm allows off-center displacements along the polar axis and by this a spontaneous electric polarization. Furthermore, local-spin-density approximations 共LSDA兲 based on the paraelectric space group P63/mmc showed that ferroelectric displacements in hexagonal manganites will only be favored along the polar axis.15LSDA were able to achieve even more remarkable results, e.g., YMnO3 is correctly described as an insulator,16,17thus LSDA studies of the ferroelectric displace-ments are now possible.
In recent publications the origin of the ferroelectric dis-tortion in hexagonal manganites is suggested to be generated by a temperature-dependent tilt of the MnO5 coordination polyhedra and a buckling of the R layers,18 indicating a
structural phase transition from the nonpolar SG P63/mmc
共Ref. 19兲 (1¯苸P63/mmc) to the polar SG P63cm (1¯ 苸P63cm). This agrees with early conjectures by Lukaszewicz and Karat-Kalicinska.20
However, macroscopic measurements seem to contradict the results from diffraction data: for YMnO3, TFE⫽920 K detected by direct measurement of the pyroelectric current21 is several hundred kelvins lower than the reported structural phase transition at about 1270 K,22 obtained via x-ray dif-fraction. Table I lists the ferroelectric to nonferroelectric transition temperatures (TFE) as collected by Smolenskii and Chupis22 in comparison to the structural transition tempera-tures, estimated by Abrahams19(Tn pt). Note here the
signifi-cant differences. Despite the successes of experimental and computational physics, the electric ordering process is still not understood—the polyhedral tilting and the ferroelectric distortion occur at different temperatures.
In this manuscript we present a detailed analysis of the development of the high-temperature structure of hexagonal manganites. Calculations of the irreducible presentation of the high-temperature structure explain the necessity of two different transition temperatures TFE and Tn pt, as a
ferro-electric phase transition with a nonzero propagation vector contradicts Landau’s theory of structural phase transitions. Rietveld refinements and a careful statistical analysis of x-ray- and neutron-diffraction data of RMnO3-powdered samples, R⫽Lu, Tm, Yb, reveal indeed two distinct transi-tions TFEand Tn pt. We explain the tripling of the unit cell at Tn pt by a tilt of the fivefold coordination polyhedron of Mn
and a corrugation of the R layers, while the ferroelectric phase transition at TFE is generated without change of sym-metry by a displacement of the oxygen and manganese ions within the fivefold coordination polyhedron of Mn.
The present work represents the results of a cooperation between the Kamerlingh-Onnes Laboratory, Leiden
Univer-PHYSICAL REVIEW B 69, 134108 共2004兲
sity 共sample preparation, x-ray powder diffraction, neutron powder diffraction, Rietveld refinements兲 and the Institut fu¨r Angewandte Physik, Universita¨t Tu¨bingen 共Rietveld refine-ments, statistical analysis, irreducible representations兲. The neutron powder diffraction was performed at the E9 spec-trometer, Hahn-Meitner Institut, Berlin.
II. NUCLEAR STRUCTURE
The nuclear structure of hexagonal manganites at room temperature, SG P63cm, is well known,8,23–26see Table II. Hexagonal manganites are formed from layers of 2D con-nected distorted and tilted MnO5 polyhedra in an unusual fivefold coordination, which are separated by corrugated R layers, see Fig. 1.
To gain better insight in the evolution of the nuclear phases, it is useful to begin the symmetry analysis with the highest possible symmetry, the so-called aristotype.28 Re-lated structures of lower symmetry are called hettotypes. The obvious aristotype of hexagonal manganites, as suggested by Abrahmas19 as a possible nonpolar high-temperature SG of hexagonal manganites, is P63/mmc 共Table III兲. The next step is a comparison to its hettotype P63cm, the low-temperature structure of hexagonal manganites.
We indicate the crystal axes of the high- and
low-temperature phases as aជ, bជ, cជ, and Aជ, Bជ, Cជ, respectively. With
冑
3a⬇A,冑
3b⬇B, c⬇C, 共1兲 the hexagonal axes aជ and Aជ of both structures include an angle of⫽30° as shown in Fig. 1, (AB) plane. The atomic positions of each SG as well as the relations between them are indicated in the fifth columns of Tables II and III.III. IRREDUCIBLE REPRESENTATIONS
According to Landau’s theory of structural phase transi-tions, any symmetry reduction is a linear combination of atomic displacements that transform as irreducible represen-tations of the high-symmetry group.
As the unit cell is tripled, we look for a displacement for each atom Dជ(rជ0)苸C3,
Dជ共rជ兲⫽Re关Dជ共rជ0兲exp共2ikជ⌬ជr兲兴, ⌬ជr⫽rជ⫺rជ0 共2兲
TABLE I. Transition temperatures for the hexagonal RMnO3
compounds. TFEdenotes the transition temperature from
ferroelec-tric to nonferroelecferroelec-tric state共Ref. 22兲, and Tn ptdenotes the
struc-tural transition temperature from a high symmetry, e.g., P63/mmc
to the low symmetry P63cm as estimated by Abrahams共Ref. 19兲.
Compound TFE (K) Tn pt(K) YMnO3 920 ⬇1220 LuMnO3 ⬎750 ⬇1290 YbMnO3 993 ⬇1270 ScMnO3 ⬇1220 TmMnO3 ⬎573 ErMnO3 833 ⬇1310 HoMnO3 873
TABLE II. Low-temperature SG P63cm of hexagonal manganites, atom type, Wyckoff notation, site
symmetry, and atomic position. In the fifth column the corresponding atomic positions of the possible high-temperature phase共Ref. 19兲 are given. Values not given in decimal notation are fixed by symmetry. As the structure is polar, the z position of one atom, usually Mn, is fixed to zero. Approximations for the atomic positions共Refs. 8, 14, and 23–26兲 or for isostructural YGaO3共Ref. 27兲 are given, but values may vary for
different R and different temperatures.
Mn 6(c) ..m (0.335,0,0.000) (1/3,0,0) R(1) 2(a) 3.m (0,0,0.27) (0,0,1/4) R(2) 4(b) 3.. (1/3,2/3,0.235) (1/3,2/3,1/4) O共1兲 6(c) ..m (0.31,0,0.167) (1/3,0,0.167) O共2兲 6(c) ..m (0.36,0,⫺0.167) (1/3,0,⫺0.167) O共3兲 2(a) 3.m (0,0,⫺0.02) (0,0,0) O共4兲 4(b) 3.. (1/3,2/3,0.015) (1/3,2/3,0)
FIG. 1. Hexagonal RMnO3 at room temperature, SG P63cm,
(AB) plane共left兲, (BC) plane 共right兲, big spheres R, small spheres O, Mn lies in the middle of the coordination polyhedra. In addition the hexagonal axes of the low-temperature phase P63cm Aជ , Bជ and
leading from the high-temperature symmetry P63/mmc to the low-temperature symmetry P63cm.
Here, Re关Dជ(rជ0)兴 denotes the distortion of the original unit cell in P63/mmc. The propagation vector kជ苸R*3, kជ
⫽(1/3,1/3,0), as can be calculated easily, modulates this
dis-tortion, tripling the unit cell. There are four subgroups of
P63/mmc, which are supergroups of P63cm:
兵P63/mmc, P63mc, P63/mcm, P63cm其. 共3兲 Using the programISOTROPY,29based on Ref. 30, we derive two one-dimensional modes ⌫1⫹: P63/mmc⇒P63/mmc and ⌫2⫺: P63/mmc⇒P63mc and two two-dimensional modes K1: P63/mmc⇒P63/mcm and K3:
P63/mmc⇒P63cm, describing the distortions ⌬ជ(X) of atom X共Ref. 31兲, see Table IV.
共1兲 The full symmetric or so-called breathing mode ⌫1⫹:
P63/mmc⇒P63/mmc, with the refineable order parameter
1(Oa p),
⌬ជ„O共1兲…⫽⫺⌬ជ„O共2兲…⫽1共Oa p兲Cជ. 共4兲 共2兲 The proper ferroelectric mode ⌫2⫺:
P63/mmc⇒P63mc, with the refineable order parameters
2(Mn), 2(R), 2(Oeq), and2(Oa p),
⌬ជ共Mn兲⫽2共Mn兲Cជ, 共5兲
⌬ជ„R共1兲…⫽⌬ជ„R共2兲…⫽2共R兲Cជ, 共6兲
⌬ជ„O共1兲…⫽⌬ជ„O共2兲…⫽2共Oa p兲Cជ, 共7兲 ⌬ជ„O共3兲…⫽⌬ជ„O共4兲…⫽2共Oeq兲Cជ 共8兲
leading to a ferroelectric distortion.32
共3兲 The nonferroic mode K1: P63/mmc⇒P63/mcm, with refineable order parameters3(Mn) and3(Oa p),
⌬ជ共Mn兲⫽3共Mn兲Aជ, 共9兲
⌬ជ„O共1兲…⫽⌬ជ„O共2兲…⫽3共Oa p兲Aជ 共10兲
leading to a displacement of the O共1兲-Mn-O共2兲 axis of the MnO5 bitetrahedra parallel to Aជ.
共4兲 Finally, the improper ferroelectric mode K3:
P63/mmc⇒P63cm, with refineable order parameters
4(R), 4(Oeq), and4(Oa p),
⌬ជ„R共1兲…⫽⫺⌬ជ„R共2兲…⫽4共R兲Cជ, 共11兲
⌬ជ„O共1兲…⫽⫺⌬ជ„O共2兲…⫽4共Oa p兲Aជ, 共12兲 ⌬ជ„O共3兲…⫽⫺⌬ជ„O共4兲…⫽4共Oeq兲Cជ. 共13兲
leading to a tilt of the MnO5 bitetrahedra and a corrugation of the R layers.
It is important to realize that though K3 leads directly to the nonpolar SG P63cm and allows displacements parallel to
Cជ, a spontaneous electric polarization can only be activated by a second mode: Let Eជ be the sum of the displacements of an atom at the position rជ0 due to the K3:
P63/mmc⇒P63cm mode, summarized over the tripled unit cell of the SG P63cm. Then Eជ can be written as
Eជ⫽Re
冋
兺
j⫽02
Dជ共rជ0兲exp
冉
2i3 j
冊
册
⫽共0,0,0兲 共14兲 as exp(2/3i) denotes the third complex root of⫺1. In factany phase transition from a nonpolar SG to a polar SG with a propagation vector kជ⫽(0,0,0) cannot lead to a spontaneous electric polarization.
Thus the discrepancy between the transition temperatures obtained by measurements of the pyroelectric current and x-ray powder diffraction were to be expected. While cooling, a peak in the pyroelectric current indicates the freezing-out of the proper ferroelectric mode, while the appearance of new peaks in x-ray powder-diffraction patterns indicates the tripling of the unit cell. The latter contradicts a proper ferro-electric mode, and different modes are usually correlated to different energies, and, thus, transition temperatures.
Figure 2 is a diagram of the group-subgroup relations be-tween the nuclear SG P63/mmc and P63cm in terms of
irreducible representations. The diagram is depicted as a two-step phase transition, because two modes are always re-quired to explain a distortion of a nonferroic crystal obeying
TABLE III. Possible high-temperature SG P63/mmc of
hex-agonal manganites 共Ref. 19兲, atom type, Wyckoff notation, site symmetry, and atomic position. In the fifth column the correspond-ing atoms of the low-temperature phase are given. Values not given in decimal notation are fixed by the symmetry. The value for z(Oa p)
is taken from Abrahams共Ref. 19兲.
Mn 2(d) ¯ m26 共1/3, 2/3, 3/4兲 Mn R 2(a) 3¯ m 共0, 0, 0兲 R(1), R(2)
Oa p 4( f ) 3m 共1/3, 2/3, 0.917兲 O共1兲,O共2兲
Oeq 2(b) ¯ m26 共0, 0, 1/4兲 O共3兲,O共4兲
TABLE IV. The possible high-temperature phase transition P63/mmc to P63cm in terms of irreducible representations共Refs.
29 and 30兲.
Mode Volume SG Type kជ
⌫1⫹ 1 P63/mmc Full 共0,0,0兲 共194兲 D6h 4 symmetric ⌫2⫺ 1 P63mc Proper 共0,0,0兲 共186兲 C6v 4 ferroelectric K1 3 P63/mcm Non 共1/3,1/3,0兲 共193兲 D6h 3 ferroic K3 3 P63cm Improper 共1/3,1/3,0兲 共185兲 C6v 3 ferroelectric
DEVELOPMENT OF THE HIGH-TEMPERATURE PHASE . . . PHYSICAL REVIEW B 69, 134108 共2004兲
the symmetry P63/mmc to a ferroelectric crystal obeying the symmetry P63cm. The three routes are as follows.
共1兲 The K1: P63/mmc⇒P63/mcm mode displaces the Oa p-Mn-Oa p axis of the MnO5 bitetrahedron from its high-symmetric position without tilting it and triples the unit cell, is followed by the ⌫2⫺: P63/mcm⇒P63cm mode, which
tilts the bitetrahedra, corrugates the R layers, and generates a ferroelectric distortion.
共2兲 The ⌫2⫺: P63/mmc⇒P63mc mode, which generates a ferroelectric distortion, is followed by the K1:
P63mc⇒P63cm mode, tilting the bitetrahedra, moving them from their high-symmetric positions, corrugates the R layers, and triples the unit cell.
共3兲 The K3: P63/mmc⇒P63cm mode tilts the MnO5 bitetrahedra, without displacing them, corrugates the R layers and triples the unit cell, is followed by the ⌫1:
P63cm⇒P63cm mode, which displaces the O(1)-Mn-O(2) axis of the MnO5bitetrahedra and generates the ferroelectric distortion.33
We expect order parametersi(X)(T) of the same mode j
to have the same temperature dependence ⌳j(T),
indepen-dent of atom X. Thus, measuring the temperature dependence of the order parameters will indicate the correct sequence of phase transitions:
᭙X᭚ ⌳1共T兲᭚ ⌳2共T兲:
3共X兲共T兲⬀⌳1共T兲,
2共X兲共T兲⬀4共T兲⬀⌳2共T兲 共15兲
supports the first scenario in Fig. 2,
᭙X᭚ ⌳1共T兲᭚ ⌳2共T兲:
2共X兲共T兲⬀⌳1共T兲,
3共X兲共T兲⬀4共X兲共T兲⬀⌳2共T兲 共16兲 supports the second scenario, and
᭙X᭚ ⌳1共T兲᭚ ⌳2共T兲:
4共X兲共T兲⬀⌳1共T兲,
2共X兲共T兲⬀3共X兲共T兲⬀⌳2共T兲 共17兲 supports the third scenario.
The second scenario is ruled out immediately, as TFE
⬍Tn pt in Table I for the entire class of hexagonal
mangan-ites. An order parameter 3(X) of ⬃0.01Aជ 共as calculated from the low-temperature structure23–26兲 generates only tiny additional peaks to the diffractogram indexed by P63/mmc and would be nearly impossible to detect at T⬎1000 K. This rules out the first scenario and leaves us with the third sce-nario as the only possibility to explain the known data of the transition temperatures as well as the low-temperature struc-ture.
Thus, a comparison of the results from group theory, Fig. 2, and the known experimental data, Table I, suggests the third scenario depicted in Fig. 2:
K3: P63/mmc⇒P63cm
a tilt of the bitetrahedra and a corrugation of the R layers at
Tn pt, followed by
⌫1: P63cm⇒P63cm
a ferroelectric distortion of the crystal at TFE for the entire
class of hexagonal manganites.
In summary, P63/mmc is indeed the high-temperature phase.
IV. SAMPLE PREPARATION AND EXPERIMENTAL DETAILS
The neutron powder method is ideally suited to probe hexagonal manganites, since Mn has a fairly large negative scattering length (b⫽⫺3.37 fm) and Tm, Yb, Lu, and O have large positive scattering lengths (7⭐b⭐13 fm and 5.8 fm兲. Thus, there is a high level of contrast for both cations and oxygen displacements over the 2⌰ range. On the other hand, though contrast of light atoms, e.g. oxygen, is minimal, x-ray diffraction is a fast and readily available method to detect the tripling of the unit cell connected with the sym-metry reduction from P63/mmc to P63cm.
For neutron-diffraction experiments large amounts of powder sample were required. The ceramic samples investi-gated (TmMnO3, YbMnO3, and LuMnO3) were prepared by the solid-state reaction technique at ambient pressure. Cation oxides of R2O3 共99.99%兲 and MnO2 共99.99%兲,
ob-FIG. 2. There are two modes required to explain a distortion of the possible high-temperature phase of hexagonal manganites P63/mmc 共Ref. 19兲 to the known low-temperature structure of
hexagonal manganites共Refs. 23–26兲, a crystal obeying the symme-try P63cm with a tilted bitetrahedron with its axis displaced from
tained from Alpha Aesar,34were mixed in a 1:2 molar ratio to achieve the stoichiometry of hexagonal RMnO3. The mix-ture was well pulverized and calcinated in air at 1100°C for 24 h. To ensure better homogeneity, the mixtures were ground again and reheated in air at 1000°C for 48 h for all compounds.
After preparation, the samples were checked by x-ray powder diffraction at room temperature to verify the correct hexagonal structure of SG P63cm. Small amounts of Mn3O4 (⬇3 –6 %) were found for LuMnO3 after preparation.
Neutron powder-diffraction data were collected on the E9 beam line at the Hahn-Meitner-Institut, Berlin, Germany. Diffractograms were recorded at various temperatures using 10 g samples mounted in a vanadium can. For high-temperature investigation we have used the high-high-temperature furnace共HTF兲, which allowed us to carry out experiments up to 1273 K and 1373 K for TmMnO3. Details on the E9 beam line and spectrometer are given in Ref. 35.
The samples were measured in a 4°⭐2⌰⭐155° range with a step size of 0.08°. Effectively, runs lasted around 2–3 h each for every temperature set point. Temperatures were stabilized within⫾5 K. The vanadium can is mounted in the HTF sample chamber which is continuously pumped to avoid oxidation while warming above room temperature. The pressure during all runs was ⬇10⫺5 mbar. We note that changing the pressure and temperature might induce decom-position or loss of oxygen stoichiometry, e.g., LuMnO3 showed small extra reflections after heating to 1273 K共not shown here兲, which were associated later with Lu2O3 by x-ray diffraction at room temperature and electron probe mi-croanalysis.
Furthermore, additional x-ray-diffraction data of TmMnO3 and YbMnO3 were taken at the HTF of the Siemens Bragg-Brentano diffractometer in Leiden,
⫽1.54 Å 共Cu K␣), step size ⫽0.01°, in a 10°⬍2⌰⬍80° range up to temperatures of 1273 K.
Figure 3 shows x-ray-diffraction data of YbMnO3at 1273 K, 50°⬍2⌰⬍80°. As can be seen clearly, the high-temperature structure P63/mmc can be ruled out; the struc-ture has to be indexed using a lower symmetry. While a drop of the integrated intensity can be observed with increasing temperature 关Fig. 3, bottom, I(026)/I(032)], the large error bars prevent an extrapolation to the transition temperature. For LuMnO3 and TmMnO3, a decrease of intensity of共026兲 with increasing temperature was also observed, but the peak remained clearly visible for the full temperature range, indi-cating Tn pt⬎1300 K.
V. STATISTICAL EVALUATION OF THE RIETVELD REFINEMENTS
To obtain the different temperature-dependent order pa-rameters connected to the different modes we refined the neutron powder data using the programSIMREF2.6 共Ref. 36兲
and the statistical qualifier N 共Ref. 37兲, taking into account the standard deviation of 2⌰ 共Ref. 38兲, an absorption correction,39,40and an asymmetry correction due to the finite height of the detector.41
N is defined by NªM⫺共n⫺P兲
冑
2共n⫺P兲 , Mªi兺
⫽1 n 共Yi obs⫺Y i calc兲2 i 2 , 共18兲with n being the number of data points, P the number of refined parameters, Yiobs the observed data points关cts.兴, i the standard deviation of observed data points 关cts.兴, and
Yicalc the calculated data points关cts.兴. N itself is obviously random; in an ideal experiment without systematic errors, the expected value of N and its standard deviation can be esti-mated by
᭙i苸关1,n兴:
具
Yiobs典
⫽Yicalc,具
i典
⫽冑
Yicalc⇒
具
N典⬇0,
具
共N兲典
⬇1. 共19兲Thus N indicates the quality of a fit independent of the number of degrees of freedom (n⫺P).
As large error bars easily hide a small ferroelectric distor-tion, we lowered the number of refined parameters by con-straining the order parameters and the anisotropic tempera-ture factorsi j 共Ref. 42兲 by
3共Mn兲⫽3共Oa p兲, 共20兲
FIG. 3. X-ray powder-diffraction data of YbMnO3 at 1273 K
共top兲 and integrated intensity 共026兲 normalized by 共032兲 as a
func-tion of temperature. The high-temperature structure P63/mmc has
to be ruled out, a distortion obeying the K3 mode is evident and
starts to vanish with increasing temperature共see text兲.
DEVELOPMENT OF THE HIGH-TEMPERATURE PHASE . . . PHYSICAL REVIEW B 69, 134108 共2004兲
2共R兲⫽2共Oeq兲, 共21兲 2共Oa p兲⫽2共Mn兲⫽0, 共22兲
i j„R共1兲…⫽i j„R共2兲…, 共23兲
i j„O共1兲…⫽i j„O共2兲…, 共24兲
i j„O共3兲…⫽i j„O共4兲…. 共25兲
By this, we explain the spontaneous electric polarization by an in-phase displacement of the O共1兲-Mn-O共2兲 axis parallel to the polar axis Cជ: Mn is displaced from the basal plane of the MnO5 bitetrahedron. Indeed the atomic distances Mn-Oeq and Mn-Oa p compared to R-Oa p in the
high-temperature phase19give further support to the approach of a displacement of the O共1兲-Mn-O共2兲 axis: the distances be-tween Mn and the equatorial oxygen positions tend to be far larger (⬇2.1 Å) than between Mn and the apical positions (⬇1.85 Å). The latter are comparable to the sum of the added ionic diameters,43 as well as the atomic distances be-tween R and the apical oxygen positions R-Oa p. We propose
that a thermal stabilization of Mn in the basal plane of the MnO5 bitetrahedron is removed in a soft-mode transition by lowering the temperature, leading to a spontaneous electric polarization. The displacement of the apical oxygen positions accompanying the Mn displacement is directly explained by the small Mn-Oa pdistance.
Figure 4 shows the statistical qualifiers of Rietveld refine-ments of RMnO3 powder-diffraction data, R⫽Lu, Tm, and Yb. Obviously the effect of a refinement of 2 is minimal: refinements using the constraints共20兲–共25兲 result in qualifi-ers comparable to those of the approach without constraints in the full temperature region. For R⫽Lu and Tm qualifiers
of refinements without 2 but with the constraints 共20兲 and
共23兲–共25兲 tend to increase 共lower right兲. For R⫽Yb, a
com-parison of the qualifiers remains ambiguous. Based on the comparison of the statistical qualifiers of the refinements of the different models, we propose transition temperatures of
TFE⫽1050(50) K for R⫽Tm and of TFE⬍1100 K for R ⫽Lu. The transition temperature TFE for R⫽Yb is taken
from Table I. Nonferroelectric regions are shown shaded in Figs. 4 – 6.
Figure 5 displays the results of our refinements of the TmMnO3powder-diffraction data using the constraints共20兲–
共25兲: the order parameters 2, 3, and4, the crystal axes and the anisotropic temperature factors of Mn, the apical oxygen positions, Tm and the equatorial oxygen positions. The order parameters 4(X) are normalized to ⇒1 for
T⇒0 K. For clarity, the tilting anglesa pandeq,
tan共a p兲⫽ 4共Oa p兲A z„O共1兲…C, 共26兲 tan共eq兲⫽ 3 2 4共Oeq兲C 1 2A ⫽34共Oeq兲C A , 共27兲
of the O共1兲-Mn-O共2兲 axis and of the basal plane of the MnO5 bitetrahedron are depicted additionally in Fig. 6, instead of the order parameters of4(Oa p) and 4(Oeq). As the z po-sition of Oeqstays constant within the error bars (1⬇0, not shown here兲 and changes in the ratio of the axes are minimal, Fig. 6 is in fact equivalent to the depiction of4(X) in Fig. 5. Anisotropic temperature factors, which are not shown here, are either constrained by the symmetry of the SG or zero within the error bars of our Rietveld refinements.
Though the order parameter3remains ambiguous within the error bars, we propose tentatively 2⬀3, with a
transi-FIG. 4. Statistical qualifiers
N of Rietveld refinements of
RMnO3 neutron powder data, R
⫽Lu, Tm, Yb. The influence of
the proper ferroelectric mode ⌫1
is minimal. Nevertheless a transi-tion temperature of TFE⫽1050 K can be estimated for R⫽Tm, and
TFE⬍1100 K for R⫽Lu 共lower right兲. The transition temperature for R⫽Yb is taken from Table I.
Nonferroelectric regions are
FIG. 5. Temperature dependence of order parameters, C axis, and anisotropic temperature factors of TmMnO3. The order parameter
4(X) is normalized to 4⇒1 for T⇒0 K. A fit to an empirical formula 共Ref. 44兲 gives clear evidence for the common origin of the
displacement of oxygen and R due to the K3mode. A kink in the C axis and kinks in the anisotropic temperature factors of Mn and the apical
oxygen positions give evidence for the onset of a spontaneous electric polarization at T⬇1050(50) K, due to a soft-mode transition, while the development of the anisotropic temperature factors of Tm and the equatorial oxygen positions remain nearly unaffected. This agrees with
2vanishing in the error bars.3remains ambiguous due to low statistics, but is connected to 2by group theory. The nonferroelectric
regions are shown shaded.
DEVELOPMENT OF THE HIGH-TEMPERATURE PHASE . . . PHYSICAL REVIEW B 69, 134108 共2004兲
tion temperature of TFE, as group theory suggests only one
transition temperature共Fig. 2兲.
While obviously the K3mode is active and the unit cell is tripled in the full temperature range, the values of the three order parameters of the K3 mode decrease clearly with in-creasing temperature, keeping a constant ratio, as can be seen in Fig. 5. Using a fit with the empirical formula44
4共X兲⫽4共X兲共T⫽0兲
冋
1⫺冉
T Tn pt冊
␣册
 , 共28兲with⌳2(X)(T⫽0) and␣, free parameters, the three order parameters4(X) can be normalized to the order parameter
⌳2共T兲⫽ 1
4共X兲共T⫽0兲
4共X兲共T兲, 共29兲 fit parameters are shown in Table V. Obviously the order parameter 4(X) obeys the same temperature dependence, giving clear evidence for the common origin of the displace-ment of oxygen and R due to the K3mode, while a common temperature dependence of 2(X) and3(X) cannot be ex-cluded.
In contrast, the anisotropic temperature factors of the api-cal oxygen atoms and Mn show clearly a symmetry change of the temperature movement at ⬇1050(50) K, e.g.,
com-pare 11 and22, while the polar C axis shows a kink and the ferroelectric distortion becomes nonsignificant within statistics at the same temperature. Again, this gives further evidence for the activation of the proper ferroelectric ⌫1 mode at ⬇1050(50) K and indicates a soft-mode transition. The anisotropic temperature factors of Oeq and Tm remain
nearly unaffected; the ferroelectric mode ⌫1:
P63cm⇒P63cm displaces mainly the O共1兲-Mn-O共2兲 axis of
the MnO5 bitetrahedron.
As can be seen in Fig. 7, the temperature-dependent order parameter4(R), R⫽Lu, Tm, Yb shows comparable behav-ior for the three measured RMnO3 compounds. As the dis-tortions are clearly visible, but cannot lead to a spontaneous electric polarization below T⫽TFE共14兲, hexagonal
mangan-ites must be described as triangular antiferroelectric for
TFE⬍T⬍Tn pt and as triangular ferroelectric, or in analogy to magnetism, canted antiferroelectric, for T⬍TFE. With the
knowledge of the magnetic ordering process,3,13,14 the tran-sition temperatures of Table I and our results, we propose a schematic phase diagram for the entire class of hexagonal manganites 共Fig. 8兲. Figure 9 shows the three phases, the modes, and the displacements of the atomic positions.
VI. SUMMARY
We presented calculations of the irreducible representa-tions for the low-temperature phase of hexagonal mangan-ites, resulting in four different orthogonal modes and order parameters. A comparison of our calculations to published experimental data19–26 indicates a two-step phase transition
P63/mmc⇒P63cm⇒P63cm at temperature Tn pt⬎TFE, in
particular, the existence of the aristotype at high tempera-tures is shown. By x-ray and neutron powder diffraction the low-temperature symmetry of P63cm was verified for R
⫽Lu, Tm, Yb over the full temperature range RT⬍T ⬍1273 K and up to 1373 K for TmMnO3. Rietveld refine-ments of the order parameters of neutron powder data
indi-TABLE V. Fit parameters of the order parameter⌳2.
4(R)T⫽0(C) 4(a p)T⫽0(deg) 4(eq)T⫽0 (deg) ␣
0.02123共26兲 4.3共11兲 8.8共22兲 1.35共37兲
 Tn pt(K) 2
0.168共32兲 1433共27兲 0.41
FIG. 6. Two order parameters of the improper ferroelectric mode K3depicted as tilting angles: with increasing temperature the
tilt gets smaller whilea p⬇1/2eq. The nonferroelectric region is shown shaded.
FIG. 7. Temperature dependence of the order parameters4(R),
R⫽Lu, Tm, Yb. A fit of 4(Tm) to Eq.共28兲 for R⫽Tm is added as
cated a soft-mode transition at a temperature of TFE ⫽1050(50) K for R⫽Lu, Tm, as the O共1兲-Mn-O共2兲 axis is
displaced from the basal plane of the MnO5 bitetrahedron. This generates a spontaneous electric polarization. A nuclear transition temperature at Tn pt⫽1433(27) K, connected with
a tripling of the unit cell by a tilt of the MnO5 bitetrahedra and a buckling of the R layers, was extrapolated by an em-piric formula for R⫽Tm. For R⫽Lu, Yb similar transition temperatures can be estimated. This allows us to present a schematic phase diagram of electric and magnetic ordering processes in hexagonal manganites. Our results are in full
agreement with both our group-theoretical calculations and published experimental data19–26of hexagonal manganites.
ACKNOWLEDGMENTS
The authors acknowledge the financial support of the Ger-man BMBF共Grant No. 03PRE7TU兲 and the Dutch founda-tion FOM.
*Electronic address: thomas.lonkai@uni-tuebingen.de
†Electronic address: tomuta@phys.leidenuniv.nl
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FIG. 9. Local coordination of the MnO5bitetrahedron in the AC
plane. Hexagonal manganites order in three nuclear phases: paraelectric P63/mmc for T⬎Tn pt 共left兲,
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DEVELOPMENT OF THE HIGH-TEMPERATURE PHASE . . . PHYSICAL REVIEW B 69, 134108 共2004兲
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