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Normal modes of the LaMnO3 Pnma phase: comparison with La2CuO4 Cmca phase
Physica B 262 (1999) 247—261
Normal modes of the LaMnO Pnma phase: comparison with La CuO Cmca phase I.S. Smirnova* Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow district, 142432, Russia Received 22 July 1998; accepted 12 October 1998
Abstract A theoretical analysis of the phonon frequencies and polarization vectors at the ! point of the Brillouin zone for LaMnO Pnma phase necessary for the interpretation of Raman and infrared spectra has been carried out. The analysis is based on group theoretical analysis as well as on a lattice dynamical calculation. The calculation has been made in the framework of the effective-charge- rigid-ion model by using the short-range Born-Mayer and long-range Coulomb potentials. A comparative study of phonon modes of LaMnO (Pnma phase) and La CuO (Cmca phase) has also been performed. 1999 Elsevier Science B.V. All rights reserved. PACS: 31.15#q; 61.50Em; 63.20Dj; 78.30Hv Keywords: Lattice dynamics; Normal modes; Space group; Selection rules
1. Introduction A variety of structural, magnetic, and transport properties of transition metal oxides REMO (RE is a rare-earth element, M is a 3d transition metal) have been the subject of numerous studies for a long time. Jonker and Santen [1,45] observed the correlation between the magnetic order and conduction in the system La A MnO , where \V V A"Ca, Sr, Ba. Wollan and Koehler [2] proposed the scheme of magnetic structures of a series of perovskite-like compounds La Ca MnO (0) \V V
* Fax:#7-096-576-4111; e-mail: [email protected].
x)1). Goodenough [3] gave the interpretation of the magnetic lattice, the electrical resistivity of this system in the framework of the semicovalent-exchange model. Changes were also observed in the electric and magnetic properties on substituting rare-earth elements in RETiO and RENiO [4,5] which were discussed in terms of a change of the charge transfer energy [6,7]. Rare-earth orthoferrites REFeO demonstrate considerable variations in the magnetic properties depending on the RE number. Orthoferrites, where rare-earth ions possess magnetic moment exhibit spin-reorientation transitions (SR) [8,9]. The recently discovered colossal magnetoresistance effect in rare-earth manganates [10,46,47] has renewed interest in the system La A MnO \V V
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 1 5 4 - 5
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(A"Ca, Sr, Ba, Pb). A search for new compounds having large magnetoresistance has led, in particular, to such a compound as Bi Ca MnO [11,12] \V V where Bi is substituted for RE. Optical spectroscopy is a powerful method to investigate electron structure. The results of systematic experimental studies of optical properties of the titanites [13,14] the orthoferrites [15—17] and the manganates [18,19] were reported. The aim of this paper is to present lattice-dynamical calculations of the phonon frequencies and polarization vectors at the ! point of the Brillouin zone for LaMnO Pnma phase which can help in the assignment of the phonon spectra. The crystal structure with Pnma space group symmetry is characteristic for the large number of oxides mentioned at room temperature: RENiO [5,6,20]; RETiO [7]; REFeO [21,48]; LaMnO [22—26] La Ca MnO [27]; La Sr MnO \V V [28]; Bi Ca MnO [12]. The following features \V V of Pnma orthorhombic phase should be emphasized. 1. The structure has two oxygen O1 and O2 ions with different site symmetry. The O1 ion lies in the
symmetry plane (Fig. 1). The O2 ion has no symmetry of its own. Since these ions have different crystal environments and different hybridization of their electronic states with those of neighbouring ions they can have different charges. For example, in La CuO oxygen O1 and O2 have different site symmetry (see Table 1 ) and the tight binding band structure calculation of La CuO suggests that the O1 ion charge is 1.5e and that of O2 is 1.0e [29]. This result is in agreement with anisotropy of the effective oxygen charge found experimentally: 1.1e for O2 (ENc) and 1.4e for O1 (E#c) [30]. The LaMnO Pnma phase has a lower symmetry than the La CuO Cmca phase. The modes in LaMnO are more complex and the effective charges of the two oxygen ions can only be defined by fitting calculated oscillator strengths to experimental values. The attempts to estimate the effective charges of symmetrically nonequivalent oxygen ions were recently made by Henn and co-workers for YBa Cu O [31], and La CuO [32]. 2. The Pnma structure has a simple orthorhombic Bravais lattice, its unit cell has four Mn ions and four oxygen octahedra equivalent with respect to
Fig. 1. Structure of LaMnO Pnma orthorhombic phase. (a) the origin at Mn 4a (0, 0, 0), (b) the origin at Mn 4b (1/2, 0, 0).
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Table 1 Symmetry of normal modes for space group D (62) proposed for LaMnO in two coordinate system orientations: Pnma (Interna tional Table orientation) and Pbnm. Symmetry of normal modes for space group Cmca D (64) proposed for La CuO , and for space group R3 c D (167) proposed for LaMnO in International Table orientation. IR and RS denote infrared active modes and modes active in Raman scattering Atom position
Normal modes
Normal modes grouped according to activity Pnma D16 2h (62)
La 4c CVX O1 4c CVX Mn 4a C O2 8d C
2A #A #B #2B #2B #B #B #2B 2A #A #B #2B #2B #B #B #2B 3A #3B #3B #3B 3A #3A #3B #3B #3B #3B #3B #3B !"7A #8A #5B #10B #7B #8B #5B #10B Pbnm D16 2h (62)
Acoustic B #B #B IR 9B #7B #9B RS 7A #5B #7B #5B Silent 8A
La 4c CVW O1 4c C VW Mn 4a C O2 8d C
2A #A #2B #B #B #2B #B #2B 2A #A #2B #B #B #2B #B #2B 3A #3B #3B #3B 3A #3A #3B #3B #3B #3B #3B #3B !"7A #8A #7B #8B #5B #10B #5B #10B Cmca D18 2h (64) A #2B #2B #B A #A #2B #2B #B #B #2B #2B 2A #A #B #2B #B #2B #2B #B 2A #A #B #2B #B #2B #2B #B !"5A #4A #4B #8B #3B #7B #6B #5B R3 cD63d (167)
Acoustic B #B #B IR 7B #9B #9B RS 7A #7B #5B #5B Silent 8A
A #A #E #E A #A #2E A #A #2A #2A #3E #3E !"A #2A #3A #4A #4E #6E
Acoustic A #E IR 3A #5E RS A #3A #4E Silent 2A
Cu O2 O1 La
4a C V 8e CW 8f C WX 8f C WX
La 2a D Mn 2b S O 6e C
symmetry operations. Note, that the displacement vector directions of these four Mn ions are different (see Section 4.2). 3. Because all four Mn ions are symmetrically equivalent, their charges have to be equal. Therefore, an introduction of Mn>, Mn> ions which are arranged in an ordered manner is incompatible with the Pnma symmetry. The same is true for R3 c. In these space groups Mn>, Mn> can be arranged only in a disordered manner. The charge ordering (Mn>, Mn>) is possible only at symmetry lowering, for example in the P2 /m space group proposed by Radaelli et al. [27] for a description of the La Ca MnO structure at tem peratures below 100 K. Symmetry lowering was observed by means of electron diffraction in La
Acoustic B #B #B IR 7B #6B #4B RS 5A #4B #3B #6B Silent 4A
Ca MnO [33] (commensurate charge-ordered state is observed below 130 K), La Sr MnO \V V [34] (¹"110 K), and Bi Ca MnO [35] (¹"130 K). From the electron microscopy data, it follows that the low symmetry phase has a modulated structure, the basic structure being Pnma. The transition from the Pnma phase to the modulated structure should give rise to many new modes in phonon spectra. However, if the modulation wave amplitude is small, the occurrence of new modes should lead to broadening of the lines; if the modulation wave amplitude is large, then new lines will possibly appear at the background of the broadened lines. 4. Huang et al. [26], Hauback et al. [24], and To¨pfer and Goodenough [36] distinguish two
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orthorhombic phases. Both phases are paramagnetic insulators at room temperature and have the Pnma symmetry structure. One of the phases with a strongly distorted octahedron (O ) acquired a long-range antiferromagnetic order as the temperature was decreased, another (O), with a less distorted octahedron, has a simple ferromagnetic structure. Since Mn—O2 and La—O bond lenghts in these two orthorhombic phases are considerably different the frequencies of some modes can differ noticeably also. In addition to Pnma and R3 c phases, the monoclinic structure was also observed at room temperature. For example, in the paper by Wollan and Koehler [2] all the samples of undoped LaMnO and doped La Ca MnO (x"0.15, 0.5) had \V V a monoclinic structure at room temperature (see Table III of Ref. [2]). To¨pfer and Goodenough [36] reported that at 0)d)0.06 the O orthorhombic phase of LaMnO is stable, whereas in the region >B of 0.1)d)0.18 the R3 c trigonal phase is stable that transits to the O orthorhombic phase as the temperature is decreased. No monoclinic phase was observed at room temperature [36]. Mitchell et al. [28] and Huang et al. [26] have shown that LaMnO can have a monoclinic structure at room temperature under certain conditions of synthesis. For interpretation of the phonon spectra and comparison of the experimental and calculated modes, one has to know the phase of the crystal under study since the number of modes, frequencies, and polarization vectors can be different for these phases. Keeping in mind application first to Raman and IR spectra the calculation presented here was performed for phonons of small wave vectors. Unfortunately, experimental data available at the time were very limited. Since phonon modes of La CuO are well-known, a part of information needed can be obtained by means of a comparative analysis of modes of LaMnO and La CuO . The published results of group theoretical analysis of the space group Pnma are controversial [13,15—17], therefore, Section 2 is concerned with reasons for these controversies which may lead to errors in the interpretation of phonon spectra. Section 3 gives the description of the
model and its parameters. Section 4 gives a discussion of the features of modes in the two perovskite-type structures whose symmetry is, however, described by different nonsymmorphic space groups.
2. Group theoretical analysis of normal modes of Pnma phase A group theoretical analysis of phonon spectra of space groups, particularly of orthorhombic syngony, necessitates a correct choice of the coordinate system. The structural works [23,24,27] have been performed in the Pnma coordinate axes orientation (International Tables orientation). Here a" 5.74 As , b"7.69 As , c"5.54 As , that is, the largest lattice constant is the o½-axis. Gilleo [22], Lacorre et al. [5] and Mitchell et al. [28] describe this structure in the Pbnm orientation a"5.54 As , b" 5.74 As , c"7.69 As , that is, the largest lattice constant is the oZ-axis. (The lattice constant values are taken from Ref. [23] as an example, they are different for different oxides and, moreover, they can slightly vary for the same oxide with different authors.) Fig. 1 shows the structure of LaMnO , in one case the coordinate origin is taken for Mn 4a (0, 0, 0) (Fig. 1a), in the other case for Mn 4b (1/2, 0, 0) (Fig. 1b). The axis orientation for these two cases is given in Fig. 1a and b. In Pnma space group the La and O1 atoms have the site symmetry C . This point group consists of two elements, E and m (m is the reflection plane). In Pnma orientation m is the p plane, and the site W symmetry is described by the CVX point group, in the Pbnm m is p , and the site symmetry is CVW. The X point group C has two irreducible representations, namely, A and A with characters 1, 1 and 1, !1, respectively. Three degrees of freedom of each atom occurring in this position can be represented as two displacements in the plane that is transformed by the 2A irreducible representation (the x, z components in the Pnma orientation, the x, y in the Pbnm) and the displacement in the direction perpendicular to this plane that is transformed by A (the y-component in the Pnma and the z in Pbnm). The correlations between the site symmetry and the
I.S. Smirnova / Physica B 262 (1999) 247—261
crystallographic symmetry D for different ori entations yield Pnma:
2A N2A , 2B , 2B , 2B ; A NA , B , B , B , Pbnm: 2A N2A , 2B , 2B , 2B ; A NA , B , B , B . The Mn atom in Pnma group occupies Wyckoff ’s position 4a or 4b. The site symmetry of this ion is C . This point group has two irreducible representations, A and A with characters 1, 1 and 1,!1 respectively. All three displacement components are transformed by the 3A , therefore the Mn ion does not participate in the Raman active representations whereas in all the IR active representations the Mn ion can have all the three displacement components. Since for the considered space group, D, the coordinate origin is in the inversion center, then for both the Pnma and Pbnm, the correlation theorem yields the same result. The O2 ion has no symmetry of its own, its site symmetry group is C . This group has only one element — identical transformation, therefore for the Pnma and Pbnm in all the D irreducible representations the O2 ion can have three displacement components. Table 1 gives the results of the group theoretical analysis of the space group D for the Pnma and Pbnm orientations. For comparison, the results of the group theoretical analysis of La CuO Cmca and LaMnO R3 c phases are given. When investigating optical phonon spectra, experimentalists often use Pbnm orientation (for example in Ref. [13], where a"5.4 As , b"5.72 As , c"7.67 As ), but they perform the decomposition of phonon modes into irreducible representations based on the work of Rousseau et al. [37]. The authors [37] consider all the space groups in the International Tables axes orientation, for the structure in question Pnma orientation. Note that in two works of Venugopalan et al. [15,16] the correct decompositions of phonon modes are given but the confusion arises due to the fact that in Ref. [15] the Pnma orientation is declared but the decomposition is given for the Pbnm. In Ref. [16] the decomposition is given for the Pnma, but the axes are
251
drawn so that the largest lattice constant is the oZ-axis, that is characteristic of the Pbnm or Pnam orientations. The definition of the activity of modes is not correct in works [15,16]. In the monograph [38] are given the selection rules both for the IR absorption and the Raman scattering, the orientation of the system of coordinates being clearly shown for each point group. With other coordinate orientations one has, using the transition matrix, to perform appropriate transformations of both the electron polarizability tensors and the electric dipole moment vector (M). The M vector directions for two crystal axes orientations are given in Fig. 1. The RS tensors for Pnma and Pbnm orientations are given in Table 2.
3. Model The unit cell of nonsymmorphic Pnma phase of LaMnO contains 20 ions which have low site symmetries. Usually, one considers the number of “chemical” types of atoms (for example for LaMnO : La, Mn and O) but not the “structural” types (La, Mn, O1, O2) in order to decrease the number of parameters. However, in complex HTSC structures, Kulkarni et al. [40] and Chaplot et al. [41] were forced to introduce “anisotropic” potential for the Tl—O interplanar and Tl—O intraplanar interactions [40], and for the in-plane and out-ofplane Cu—O interactions [41], i.e. they were forced to introduce additional parameters. In the case of (La, Mn, O1, O2) the number of parameters increases especially for a shell model, but the introduction of four ionic types enable a more correct account of not only “anisotropy” of interactions, but also, Coulomb interactions since symmetrically nonequivalent ions have different charges. The calculations were performed in the rigid-ion model with effective charges using the structure data of Elemans et al. [23]. The interionic interaction between the atoms k and k is described by the potential: » (r)"»! (r)#» +(r), where »! (r)" IIY IIY IIY IIY Z Z /(4pe r) is the long-range Coulomb potential, I IY and » +(r)"a exp(!b r) is the short-range IIY IIY IIY Born—Mayer potential. This model requires the information on effective charges of ions that can be used as starting parameters to calculate the
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Table 2 Raman tensors for space group D (62) in two coordinate system orientations: Pnma and Pbnm. The lattice constant values are taken from Ref. [23] as an example Lattice constants
RS tensors and their symmetry Pnma D16 2h (62)
A"5.74 As , B"7.69 As , C"5.44 As Oz#CX , Ox#p W
a 0 0 0 b 0 0 0 c A
0 d 0
d
0 0
0
0 0
B
0 0 e
0
0 0
e
0 0
B
0 0 0
0
0 f
0
f
0
B
16 2h
Pbnm D (62) A"5.54 As , B"5.74 As , C"7.69 As Oz#CX , Ox#p W
b 0 0 0 c
0
0 0 a A
Coulomb part of the dynamic matrix. An estimation of the effective charges based on depolarized IR spectra was made for LaMO (M is transition metal) by Arima and Tokura [18] at a fixed charge of La #3, that can strongly misrepresent the charge values. Crandles et al. [4] report more real values of effective charges for perovskite-like oxides RETiO (RE"La, Ce, Pr, Nd, Sm, Gd). The in formation on effective charges of simple perovskites can be also found in the papers by Gervais [42] and Choen [43]. The calculation of the Coulomb part of the dynamic matrix necessitates the information on electric field gradients (EFG) on each ion. The second rank tensor of EFG has to satisfy three requirements: (1) the values of the tensor components do not have to change (within the present accuracy) as the number of the coordination spheres increases. In our case the number of coordination spheres NMAX"200; (2) the form of the tensor for each ion has to comply with the site symmetry of this ion; (3) the trace of the tensor for each ion has to be zero which reflects the fact that at the summation using the Ewald method the parameters are chosen such that the electric neutrality condition, div E"0, be fulfilled. Knowledge of the ion site symmetries and the number of the independent components of the force defined by the first potential derivatives for given site symmetry groups allows one to
0 0 d
0
0 0
d
0 0
B
0 0 0
0
0 e
0
e
B
0
0 f
0
f
0 0
0
0 0
B
determine the number of force components which are not zero: Pnma La CVX f V , 0, f X * * O1 CVX f V , 0, f X - - Mn C 0, 0, 0 O2 C fV ,fW ,fX - - - I4/mmm La CuO La C 0, 0, f X * O1 C 0, 0, f X - Cu D 0, 0, 0 O2 D 0, 0, 0 Due to symmetry, the force which acts on Mn is zero at any value of the parameters. Analogously, one component of the force acting on the La and O1 ions (y-component for the Pnma or z-component for the Pbnm) is zero as these ions lie in the symmetry plane. Note, that if ions have high site symmetry the potential energy minimum condition is automatically fulfilled due to symmetry at any parameter value. For comparison, the tetragonal I4/mmm phase of La CuO where ions have high site symmetry is given. In the case of complex crystals with the low site symmetry ions the question of the potential energy minimum condition becomes actual.
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Table 3 Model parameters: A , b : Born—Mayer constants; ¸ — longitudinal force constants between the ions i and j; ¹ — transverse force GH GH GH GH constants between the ions i and j. R: length of respective bound; Z: ionic charge b (As \) GH
¸ (N/m) GH
6050
4.80 4.784
3200
3.675
6050
4.10
3200
4.81
3200
3.845
179.31 249.71 62.83 56.86 4.46 85.0 6.96 71.9 25.56 31.25 1.35 2.58 2.13 0.89 0.82 11.54 10.37 2.86 1.76
Interaction
A (ev) GH
Mn—O1 Mn—O2 Mn—O2 La—O1 La—O1 La—O1 La—O1 La—O2 La—O2 La—O2 La—O2 O1—O2 O1—O2 O1—O2 O1—O2 O2—O2 O2—O2 O2—O2 O2—O2 Atom type Z[e]
Mn 1.52
O2 !1.04
Table 3 gives the parameters at which the results were obtained. The longitudinal ¸ and transverse GH ¹ force constants are also given. GH 4. Results and discussion The structures of LaMnO Pnma and La CuO Cmca, although different, have in their base a distorted perovskite cube and octahedral oxygen environment for Cu and Mn ions. The site symmetry of all ions is different in these structures, which can be seen in Table 1. We shall consider the differences of normal modes of these structures from the standpoint of symmetry. 4.1. Raman active modes Fig. 2 illustrates Raman active modes related to stretching vibrations of oxygen ions in La CuO and LaMnO . The La and O1 ions in both the space groups have the site symmetry C (E, m), but
O1 !1.18
¹ (N/m) GH
R (As )
!19.01 !14.0 !3.06 !6.05 !0.26 !8.2 !0.44 !6.2 !2.0 !2.5 !0.08 !0.21 !0.18 !0.07 !0.07 !0.65 !0.56 !0.13 !0.08
1.964 1.905 2.188 2.559 3.252 2.450 3.131 2.446 2.698 2.649 3.416 2.715 2.750 2.932 2.949 2.88 2.91 3.24 3.37
La 1.74
in Cmca m is p , and in Pnma m is p (axes orientaV W tion is the same in both the phases). Consider A modes. There are five A modes in La CuO . Fig. 2 depicts two of these. The 417 cm\ mode [29] (experimental 426 cm\ [44]) is governed preferentially by the stretching vibration of O1 ion. For the A symmetry the vibrations lying in the symmetry plane are allowed for the O1 ion, therefore the mode associated with the stretching vibration can be assigned to the A symmetry if the Cu—O1 bond (or Mn—O1 in LaMnO ) lies in the symmetry plane. Since in La CuO (Cmca) the symmetry plane is normal to the x-axis, the O1 stretching vibration lies in this plane and is symmetry allowed. In LaMnO there are seven A modes. In this crystal the symmetry plane is normal to the y-axis. The vibration of the O1 ion along the Mn—O1 bond is normal to this plane, and for A symmetry such vibration is forbidden. So, in the Pnma no A mode will exist associated with the O1 stretching vibrations. The O1 vibration along the Mn—O1 bond is symmetry allowed only in the
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Fig. 2. The comparison of the stretching vibrations of oxygen ions in Raman active modes of LaMnO and La CuO . The Mode analogous to the 217 cm\ mode related to stretching La ion vibrations is imposible in LaMnO .
B and B modes. In fact, the B 573 cm\ mode (Fig. 2) has eigenvectors which are analogous to the eigenvectors of the 417 cm\ mode in La CuO . The quite large difference in the frequencies is
accounted for by the fact that the Cu—O1 distance amounts to 2.43 As [39], and Mn—O1 is 1.96 As . The A 217 cm\ mode (Fig. 2) (experimental 226 cm\ [44]) is observable in La CuO , but in LaMnO
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such a vibration is impossible. Due to the occurrence of two La—O layers in La CuO there are La—O1 bonds both in La—O layer and normal to it. The 217 cm\ mode is associated with the La vibration along the normal bonds. There are no double La—O layers in LaMnO , therefore there are bonds only in the La—O plane and, so, no mode can exist analogous to 217 cm\ in La CuO . Consider B modes for both crystals (Fig. 2). Three such modes can exist in La CuO since only one component is symmetry allowed for each ion: u for O2 (u , u ,0) and u for O1 and La W V X V (u , u ,0). Since for O2 u , u ,0, the vibration of W X V X this ion along the Cu—O2 bond is impossible. Due to the low site symmetry of the O2 ion in LaMnO this ion has three displacement components, therefore the 603 cm\ B mode associated with the stretching vibration of the O2 ion is possible. In this mode the u O0, but it is very small. The W 405 cm\ B mode has, on the contrary, a large component u of the ion O2 and small components W u , u , therefore the eigenvectors of this mode are V X analogous to the eigenvectors of the 410 cm\ mode in La CuO . Note, that the O2 vibrations of the 603 cm\ B modes in two octahedra inclined to one side are not identical, they are equal in modulus but opposite in direction. Fig. 3 illustrates all 24 Raman active modes in LaMnO . The figure of 406 cm\ mode shows vi brations of 20 ions marked by the digits in Fig. 1a. 4.2. IR active modes Fig. 4 shows IR active modes related to the stretching vibrations of oxygen ions in La CuO and LaMnO . In La CuO the highest-frequency B and B modes are not exactly O2 stretching vibrations (Fig. 4, right side) since in these representations the displacement component u ,0 due W to symmetry, but the tilt angle of octahedron being small, these vibrations are very close to the O2 stretching vibrations. In LaMnO the highest-frequency B and B modes are also not exactly stretching vibra tions (Fig. 4, left side). The same as in La CuO , these modes are close in frequency but have a lower frequency 558 cm\, as suggested by experiment [18], than 687 cm\ [30] in La CuO . This fre
255
quency difference is likely to be caused first by the difference in the mean Cu—O2 distance (1.94 As ) and Mn—O2 (2.04 As ) and then by the fact that in LaMnO the vibrations are “less stretching” due to a greater tilt of the octahedron. The situation is different with the B modes. In La CuO , oxygen O2 has only the u component in W these modes, and u , u ,0, i.e. there is no stretching V X mode associated with O2. In LaMnO , B stretch ing mode is possible due to a lower site symmetry of O2. At first glance the summary dipole moment of this mode (562 cm\, Fig. 4) is zero but u O0, it is W small and not shown in figure. However, precisely this component produces the summary dipole moment. It is small in magnitude, therefore the LO—TO splitting and the oscillator strength of this mode are small. So, in LaMnO there are two stretching B modes: one is due to the O2 vibra tions (with a small LO—TO splitting), the other is due to the O1 vibrations (with a large LO—TO splitting). Fig. 5 illustrates all 25 IR active modes of LaMn O . In every orthorhombic crystal all irreducible representations (IRR) are one dimensional, therefore the dipole moment vector has one component in each IR active IRR. This means that every IR active mode in an orthorhombic crystal is such that for all symmetrically equivalent ions, for example for eight O2 ions, one of the components ought to have like signs (z- for B , y- for B , x- for B in Pnma orientation). The other displacement components of the symmetrically equivalent ions (O2 for example) must have such signs that the corresponding component of the summary dipole moment be zero. Some modes in IR active IRR have a small component giving the summary dipole moment. 562 and 207 cm\ B modes are the examples of such modes. So, the group theoretical analysis suggests that there has to be seven modes of B sym metry, but experimentally only five modes will possibly be observed. Fig. 6 shows silent modes of LaMnO : 217, 171 and 562 cm\ clearly show the vibrations of four symmetrically equivalent Mn ions. Such ions have the displacement vectors being equal in modulus but its displacement vector directions are different. The following difference in phonon modes of two structures in question is worth mentioning. In La
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Fig. 3. Normal frequencies and eigenvectors of A , B , B , and B modes of Pnma orthorhombic phase of LaMnO . In the Fig. of 406 cm\ A mode, the digits designate all 20 atoms of the unit cell as in Fig. 1a.
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MnO vibrations of oxygen ions in two octahedra tilted to one side can have different displacement directions for example 582 cm\ B mode and 562 cm\ B mode (Fig. 4) or 603 cm\ B mode and 573 cm\ B mode (Fig. 2). Such a situation is impossible in La CuO since in this more symmet rical structure all octahedra tilted to one side are translationally equivalent, and in each mode the displacements of all the octahedron’s ions of the same type (for example O2) are equal in magnitude and direction.
5. Conclusions
Fig. 4. The comparison of the stretching vibrations of oxygen ions in IR active modes of LaMnO and La CuO .
The calculation was performed for stoichiometric LaMnO using the structure data of Elemans et al. [23]. Hauback et al. [24], Mitchell et al. [28], Huang et al. [26], and To¨pfer and Goodenough [36] have shown that both Mn—O and La—O bond lengths can change in orthorhombic structure, depending on the growth conditions and the cation stoichiometry. Therefore, when comparing the calculated and experimental frequencies one has to know how much the crystal composition is close to the stoichiometric one. The LaMnO modes have been calculated for Pnma orientation (International Table orientation). With some other coordinate axes, for instance Pbnm, one has, taking into account the transition matrix from one system of coordinates to another, to make corresponding conversions both at decomposition of normal modes into irreducible representations (IRR) and transformations of the dipole moment vector (selection rules for IR spectra, see Fig. 1) and the electron polarizability tensor (selection rules for Raman spectra, see Table 2). The decompositions of normal modes into IRR in ! point have been given as an example for two systems: Pnma and Pbnm (see Table 1). It is well-known that the agreement between the theory and the experiment in the high-frequency region is crucial for the shell model and the rigideffective charge-ion model. Unfortunately, the information on higher phonon frequencies of oxides under consideration is incomplete and controversial.
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Fig. 5. Normal frequencies and eigenvectors of B , B , B modes of the Pnma orthorhombic phase of LaMnO . The respective LO frequencies are enclosed in parentheses. In the Fig. of 76 cm\ B mode, the digits designate all 20 atoms of the unit cell.
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259
Fig. 6. “Silent” A modes of the Pnma orthorhombic phase of LaMnO .
First at all consider experimental data of Raman spectra. Reedyk et al. [13] have reported the highest frequency of 645 cm\ in Raman spectra of CeTiO . The authors assign this frequency to all Raman active symmetries. Our calculation shows that such high-frequency modes of B , B , B symmetries are possible, however its frequencies should be different. This difference is a direct consequence of symmetry of the orthorhombic crystal rather than any model. The fact that splitting was not revealed in the experiment [13], probably points to an imperfection of samples under investigation or the values of splitting are )10 cm\. The values of splitting of B modes (i"1, 2, 3) are very useful for the fitting of model parameters. Iliev et al. [19] observed 611 cm\ B mode in LaMnO . For each of B , B symmetries authors have found only one mode instead of five modes. Venugopalan and Becker [16] and Koshizuka and Ushioda [17] have reported frequencies )500 cm\ of B (i"1, 2, 3) symmetries in or thoferrites REFeO , although the number of
B , B and B observed modes is less than that predicted. Let us consider the A mode with highest frequen cy. One of A modes has the frequency of 645 cm\ as Reedyk et al. [13] have reported. Venugopalan and Becker [16] and Koshizuka and Ushioda [17] investigated A spectra in three polarizations (xx, yy, zz) and observed more modes than it follows from the group theoretical analysis. As a result, the 645 cm\ mode was assigned as a twophonon mode in the work [16]. Koshizuka and Ushioda [17] attribute this mode to an impurityrelated phonon. Venugopalan and Becker [16] and Koshizuka and Ushioda [17] excluded the 645 cm\ mode from a one-phonon spectrum due to the eighth added mode of A symmetry. If this mode has A symmetry as Reedyk et al. [13] sug gest, then another mode should be excepted from a one-phonon spectrum due to the eighth added mode. Iliev et al. [19] have not detected the 645 cm\ mode of A symmetry, but they have observed only five instead of 7 A modes.
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Thus, the experimental data were interpreted from two viewpoints: 1. 645 cm\ mode is not a one-phonon mode [16,17] therefore the Raman spectra do not contain one-phonon modes with frequencies more than 500 cm\ 2. 645 cm\ mode is of a one-phonon nature [13] and moreover it is observed in all Raman-active symmetries. Then it is a highest frequency of the Raman spectra. The calculations presented show that the crystal can well have one-phonon modes of B symmetry (i"1, 2, 3) in the frequency region of &580— 650 cm\. However, the crystal cannot have A mode with such a high frequency, since in the considered model such a high (*600 cm\) frequency of A symmetry cannot be obtained taking into account that the highest frequency of IR-active spectra is 540—570 cm\ [18,14]. One can hope that these contradictions will be resolved in the course of further experimental studies. The information on IR-active modes available at that time are very limited, since all measurements have been fulfilled with depolarized light. However, these measurements give the approximate value of a highest frequency, for example, it is equal to 558 cm\ in LaMnO [18] and 545 cm\ in LaTiO [14]. According to the group theoretical analysis there has to be seven modes of the B symmetry. The calculation shows that two modes (one of them has a highest frequency) should have small oscillator strengths and they may not be observed in the experiment. However, the model parameters should be slightly adjusted if an experiment will show that all seven modes have rather large oscillator strengths. Note that in IR spectra highest frequencies of B , B , B symmetries should be different as well. The information about this splitting is absent, since measurements of IR spectra were carried out with depolarized light. The measurement with polarized light may give both the splitting of highest frequencies and LO—TO splitting. This information is required for a more exact calculation of this complex phase.
Finally it should be mentioned that Reedyk et al. [13] have found a very interesting experimental fact. At a complete substitution of rare-earth cations in ReTiO the authors observed a strong displacement of one of the A symmetry modes. According to the analysis of the Pnma phonon modes, one can suggest that this mode is, to a large extent, related to the vibration in the symmetry plane, i.e., to a change in the Re—O distance.
Acknowledgements The author thanks V.M. Edelstein, A.V. Bazhnov, and S.S. Nazin for stimulating discussions. This work was supported in part by VW Stiftung.
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Normal modes of the LaMnO Pnma phase: comparison with La CuO Cmca phase I.S. Smirnova* Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow district, 142432, Russia Received 22 July 1998; accepted 12 October 1998
Abstract A theoretical analysis of the phonon frequencies and polarization vectors at the ! point of the Brillouin zone for LaMnO Pnma phase necessary for the interpretation of Raman and infrared spectra has been carried out. The analysis is based on group theoretical analysis as well as on a lattice dynamical calculation. The calculation has been made in the framework of the effective-charge- rigid-ion model by using the short-range Born-Mayer and long-range Coulomb potentials. A comparative study of phonon modes of LaMnO (Pnma phase) and La CuO (Cmca phase) has also been performed. 1999 Elsevier Science B.V. All rights reserved. PACS: 31.15#q; 61.50Em; 63.20Dj; 78.30Hv Keywords: Lattice dynamics; Normal modes; Space group; Selection rules
1. Introduction A variety of structural, magnetic, and transport properties of transition metal oxides REMO (RE is a rare-earth element, M is a 3d transition metal) have been the subject of numerous studies for a long time. Jonker and Santen [1,45] observed the correlation between the magnetic order and conduction in the system La A MnO , where \V V A"Ca, Sr, Ba. Wollan and Koehler [2] proposed the scheme of magnetic structures of a series of perovskite-like compounds La Ca MnO (0) \V V
* Fax:#7-096-576-4111; e-mail: [email protected].
x)1). Goodenough [3] gave the interpretation of the magnetic lattice, the electrical resistivity of this system in the framework of the semicovalent-exchange model. Changes were also observed in the electric and magnetic properties on substituting rare-earth elements in RETiO and RENiO [4,5] which were discussed in terms of a change of the charge transfer energy [6,7]. Rare-earth orthoferrites REFeO demonstrate considerable variations in the magnetic properties depending on the RE number. Orthoferrites, where rare-earth ions possess magnetic moment exhibit spin-reorientation transitions (SR) [8,9]. The recently discovered colossal magnetoresistance effect in rare-earth manganates [10,46,47] has renewed interest in the system La A MnO \V V
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 1 5 4 - 5
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(A"Ca, Sr, Ba, Pb). A search for new compounds having large magnetoresistance has led, in particular, to such a compound as Bi Ca MnO [11,12] \V V where Bi is substituted for RE. Optical spectroscopy is a powerful method to investigate electron structure. The results of systematic experimental studies of optical properties of the titanites [13,14] the orthoferrites [15—17] and the manganates [18,19] were reported. The aim of this paper is to present lattice-dynamical calculations of the phonon frequencies and polarization vectors at the ! point of the Brillouin zone for LaMnO Pnma phase which can help in the assignment of the phonon spectra. The crystal structure with Pnma space group symmetry is characteristic for the large number of oxides mentioned at room temperature: RENiO [5,6,20]; RETiO [7]; REFeO [21,48]; LaMnO [22—26] La Ca MnO [27]; La Sr MnO \V V [28]; Bi Ca MnO [12]. The following features \V V of Pnma orthorhombic phase should be emphasized. 1. The structure has two oxygen O1 and O2 ions with different site symmetry. The O1 ion lies in the
symmetry plane (Fig. 1). The O2 ion has no symmetry of its own. Since these ions have different crystal environments and different hybridization of their electronic states with those of neighbouring ions they can have different charges. For example, in La CuO oxygen O1 and O2 have different site symmetry (see Table 1 ) and the tight binding band structure calculation of La CuO suggests that the O1 ion charge is 1.5e and that of O2 is 1.0e [29]. This result is in agreement with anisotropy of the effective oxygen charge found experimentally: 1.1e for O2 (ENc) and 1.4e for O1 (E#c) [30]. The LaMnO Pnma phase has a lower symmetry than the La CuO Cmca phase. The modes in LaMnO are more complex and the effective charges of the two oxygen ions can only be defined by fitting calculated oscillator strengths to experimental values. The attempts to estimate the effective charges of symmetrically nonequivalent oxygen ions were recently made by Henn and co-workers for YBa Cu O [31], and La CuO [32]. 2. The Pnma structure has a simple orthorhombic Bravais lattice, its unit cell has four Mn ions and four oxygen octahedra equivalent with respect to
Fig. 1. Structure of LaMnO Pnma orthorhombic phase. (a) the origin at Mn 4a (0, 0, 0), (b) the origin at Mn 4b (1/2, 0, 0).
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249
Table 1 Symmetry of normal modes for space group D (62) proposed for LaMnO in two coordinate system orientations: Pnma (Interna tional Table orientation) and Pbnm. Symmetry of normal modes for space group Cmca D (64) proposed for La CuO , and for space group R3 c D (167) proposed for LaMnO in International Table orientation. IR and RS denote infrared active modes and modes active in Raman scattering Atom position
Normal modes
Normal modes grouped according to activity Pnma D16 2h (62)
La 4c CVX O1 4c CVX Mn 4a C O2 8d C
2A #A #B #2B #2B #B #B #2B 2A #A #B #2B #2B #B #B #2B 3A #3B #3B #3B 3A #3A #3B #3B #3B #3B #3B #3B !"7A #8A #5B #10B #7B #8B #5B #10B Pbnm D16 2h (62)
Acoustic B #B #B IR 9B #7B #9B RS 7A #5B #7B #5B Silent 8A
La 4c CVW O1 4c C VW Mn 4a C O2 8d C
2A #A #2B #B #B #2B #B #2B 2A #A #2B #B #B #2B #B #2B 3A #3B #3B #3B 3A #3A #3B #3B #3B #3B #3B #3B !"7A #8A #7B #8B #5B #10B #5B #10B Cmca D18 2h (64) A #2B #2B #B A #A #2B #2B #B #B #2B #2B 2A #A #B #2B #B #2B #2B #B 2A #A #B #2B #B #2B #2B #B !"5A #4A #4B #8B #3B #7B #6B #5B R3 cD63d (167)
Acoustic B #B #B IR 7B #9B #9B RS 7A #7B #5B #5B Silent 8A
A #A #E #E A #A #2E A #A #2A #2A #3E #3E !"A #2A #3A #4A #4E #6E
Acoustic A #E IR 3A #5E RS A #3A #4E Silent 2A
Cu O2 O1 La
4a C V 8e CW 8f C WX 8f C WX
La 2a D Mn 2b S O 6e C
symmetry operations. Note, that the displacement vector directions of these four Mn ions are different (see Section 4.2). 3. Because all four Mn ions are symmetrically equivalent, their charges have to be equal. Therefore, an introduction of Mn>, Mn> ions which are arranged in an ordered manner is incompatible with the Pnma symmetry. The same is true for R3 c. In these space groups Mn>, Mn> can be arranged only in a disordered manner. The charge ordering (Mn>, Mn>) is possible only at symmetry lowering, for example in the P2 /m space group proposed by Radaelli et al. [27] for a description of the La Ca MnO structure at tem peratures below 100 K. Symmetry lowering was observed by means of electron diffraction in La
Acoustic B #B #B IR 7B #6B #4B RS 5A #4B #3B #6B Silent 4A
Ca MnO [33] (commensurate charge-ordered state is observed below 130 K), La Sr MnO \V V [34] (¹"110 K), and Bi Ca MnO [35] (¹"130 K). From the electron microscopy data, it follows that the low symmetry phase has a modulated structure, the basic structure being Pnma. The transition from the Pnma phase to the modulated structure should give rise to many new modes in phonon spectra. However, if the modulation wave amplitude is small, the occurrence of new modes should lead to broadening of the lines; if the modulation wave amplitude is large, then new lines will possibly appear at the background of the broadened lines. 4. Huang et al. [26], Hauback et al. [24], and To¨pfer and Goodenough [36] distinguish two
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orthorhombic phases. Both phases are paramagnetic insulators at room temperature and have the Pnma symmetry structure. One of the phases with a strongly distorted octahedron (O ) acquired a long-range antiferromagnetic order as the temperature was decreased, another (O), with a less distorted octahedron, has a simple ferromagnetic structure. Since Mn—O2 and La—O bond lenghts in these two orthorhombic phases are considerably different the frequencies of some modes can differ noticeably also. In addition to Pnma and R3 c phases, the monoclinic structure was also observed at room temperature. For example, in the paper by Wollan and Koehler [2] all the samples of undoped LaMnO and doped La Ca MnO (x"0.15, 0.5) had \V V a monoclinic structure at room temperature (see Table III of Ref. [2]). To¨pfer and Goodenough [36] reported that at 0)d)0.06 the O orthorhombic phase of LaMnO is stable, whereas in the region >B of 0.1)d)0.18 the R3 c trigonal phase is stable that transits to the O orthorhombic phase as the temperature is decreased. No monoclinic phase was observed at room temperature [36]. Mitchell et al. [28] and Huang et al. [26] have shown that LaMnO can have a monoclinic structure at room temperature under certain conditions of synthesis. For interpretation of the phonon spectra and comparison of the experimental and calculated modes, one has to know the phase of the crystal under study since the number of modes, frequencies, and polarization vectors can be different for these phases. Keeping in mind application first to Raman and IR spectra the calculation presented here was performed for phonons of small wave vectors. Unfortunately, experimental data available at the time were very limited. Since phonon modes of La CuO are well-known, a part of information needed can be obtained by means of a comparative analysis of modes of LaMnO and La CuO . The published results of group theoretical analysis of the space group Pnma are controversial [13,15—17], therefore, Section 2 is concerned with reasons for these controversies which may lead to errors in the interpretation of phonon spectra. Section 3 gives the description of the
model and its parameters. Section 4 gives a discussion of the features of modes in the two perovskite-type structures whose symmetry is, however, described by different nonsymmorphic space groups.
2. Group theoretical analysis of normal modes of Pnma phase A group theoretical analysis of phonon spectra of space groups, particularly of orthorhombic syngony, necessitates a correct choice of the coordinate system. The structural works [23,24,27] have been performed in the Pnma coordinate axes orientation (International Tables orientation). Here a" 5.74 As , b"7.69 As , c"5.54 As , that is, the largest lattice constant is the o½-axis. Gilleo [22], Lacorre et al. [5] and Mitchell et al. [28] describe this structure in the Pbnm orientation a"5.54 As , b" 5.74 As , c"7.69 As , that is, the largest lattice constant is the oZ-axis. (The lattice constant values are taken from Ref. [23] as an example, they are different for different oxides and, moreover, they can slightly vary for the same oxide with different authors.) Fig. 1 shows the structure of LaMnO , in one case the coordinate origin is taken for Mn 4a (0, 0, 0) (Fig. 1a), in the other case for Mn 4b (1/2, 0, 0) (Fig. 1b). The axis orientation for these two cases is given in Fig. 1a and b. In Pnma space group the La and O1 atoms have the site symmetry C . This point group consists of two elements, E and m (m is the reflection plane). In Pnma orientation m is the p plane, and the site W symmetry is described by the CVX point group, in the Pbnm m is p , and the site symmetry is CVW. The X point group C has two irreducible representations, namely, A and A with characters 1, 1 and 1, !1, respectively. Three degrees of freedom of each atom occurring in this position can be represented as two displacements in the plane that is transformed by the 2A irreducible representation (the x, z components in the Pnma orientation, the x, y in the Pbnm) and the displacement in the direction perpendicular to this plane that is transformed by A (the y-component in the Pnma and the z in Pbnm). The correlations between the site symmetry and the
I.S. Smirnova / Physica B 262 (1999) 247—261
crystallographic symmetry D for different ori entations yield Pnma:
2A N2A , 2B , 2B , 2B ; A NA , B , B , B , Pbnm: 2A N2A , 2B , 2B , 2B ; A NA , B , B , B . The Mn atom in Pnma group occupies Wyckoff ’s position 4a or 4b. The site symmetry of this ion is C . This point group has two irreducible representations, A and A with characters 1, 1 and 1,!1 respectively. All three displacement components are transformed by the 3A , therefore the Mn ion does not participate in the Raman active representations whereas in all the IR active representations the Mn ion can have all the three displacement components. Since for the considered space group, D, the coordinate origin is in the inversion center, then for both the Pnma and Pbnm, the correlation theorem yields the same result. The O2 ion has no symmetry of its own, its site symmetry group is C . This group has only one element — identical transformation, therefore for the Pnma and Pbnm in all the D irreducible representations the O2 ion can have three displacement components. Table 1 gives the results of the group theoretical analysis of the space group D for the Pnma and Pbnm orientations. For comparison, the results of the group theoretical analysis of La CuO Cmca and LaMnO R3 c phases are given. When investigating optical phonon spectra, experimentalists often use Pbnm orientation (for example in Ref. [13], where a"5.4 As , b"5.72 As , c"7.67 As ), but they perform the decomposition of phonon modes into irreducible representations based on the work of Rousseau et al. [37]. The authors [37] consider all the space groups in the International Tables axes orientation, for the structure in question Pnma orientation. Note that in two works of Venugopalan et al. [15,16] the correct decompositions of phonon modes are given but the confusion arises due to the fact that in Ref. [15] the Pnma orientation is declared but the decomposition is given for the Pbnm. In Ref. [16] the decomposition is given for the Pnma, but the axes are
251
drawn so that the largest lattice constant is the oZ-axis, that is characteristic of the Pbnm or Pnam orientations. The definition of the activity of modes is not correct in works [15,16]. In the monograph [38] are given the selection rules both for the IR absorption and the Raman scattering, the orientation of the system of coordinates being clearly shown for each point group. With other coordinate orientations one has, using the transition matrix, to perform appropriate transformations of both the electron polarizability tensors and the electric dipole moment vector (M). The M vector directions for two crystal axes orientations are given in Fig. 1. The RS tensors for Pnma and Pbnm orientations are given in Table 2.
3. Model The unit cell of nonsymmorphic Pnma phase of LaMnO contains 20 ions which have low site symmetries. Usually, one considers the number of “chemical” types of atoms (for example for LaMnO : La, Mn and O) but not the “structural” types (La, Mn, O1, O2) in order to decrease the number of parameters. However, in complex HTSC structures, Kulkarni et al. [40] and Chaplot et al. [41] were forced to introduce “anisotropic” potential for the Tl—O interplanar and Tl—O intraplanar interactions [40], and for the in-plane and out-ofplane Cu—O interactions [41], i.e. they were forced to introduce additional parameters. In the case of (La, Mn, O1, O2) the number of parameters increases especially for a shell model, but the introduction of four ionic types enable a more correct account of not only “anisotropy” of interactions, but also, Coulomb interactions since symmetrically nonequivalent ions have different charges. The calculations were performed in the rigid-ion model with effective charges using the structure data of Elemans et al. [23]. The interionic interaction between the atoms k and k is described by the potential: » (r)"»! (r)#» +(r), where »! (r)" IIY IIY IIY IIY Z Z /(4pe r) is the long-range Coulomb potential, I IY and » +(r)"a exp(!b r) is the short-range IIY IIY IIY Born—Mayer potential. This model requires the information on effective charges of ions that can be used as starting parameters to calculate the
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Table 2 Raman tensors for space group D (62) in two coordinate system orientations: Pnma and Pbnm. The lattice constant values are taken from Ref. [23] as an example Lattice constants
RS tensors and their symmetry Pnma D16 2h (62)
A"5.74 As , B"7.69 As , C"5.44 As Oz#CX , Ox#p W
a 0 0 0 b 0 0 0 c A
0 d 0
d
0 0
0
0 0
B
0 0 e
0
0 0
e
0 0
B
0 0 0
0
0 f
0
f
0
B
16 2h
Pbnm D (62) A"5.54 As , B"5.74 As , C"7.69 As Oz#CX , Ox#p W
b 0 0 0 c
0
0 0 a A
Coulomb part of the dynamic matrix. An estimation of the effective charges based on depolarized IR spectra was made for LaMO (M is transition metal) by Arima and Tokura [18] at a fixed charge of La #3, that can strongly misrepresent the charge values. Crandles et al. [4] report more real values of effective charges for perovskite-like oxides RETiO (RE"La, Ce, Pr, Nd, Sm, Gd). The in formation on effective charges of simple perovskites can be also found in the papers by Gervais [42] and Choen [43]. The calculation of the Coulomb part of the dynamic matrix necessitates the information on electric field gradients (EFG) on each ion. The second rank tensor of EFG has to satisfy three requirements: (1) the values of the tensor components do not have to change (within the present accuracy) as the number of the coordination spheres increases. In our case the number of coordination spheres NMAX"200; (2) the form of the tensor for each ion has to comply with the site symmetry of this ion; (3) the trace of the tensor for each ion has to be zero which reflects the fact that at the summation using the Ewald method the parameters are chosen such that the electric neutrality condition, div E"0, be fulfilled. Knowledge of the ion site symmetries and the number of the independent components of the force defined by the first potential derivatives for given site symmetry groups allows one to
0 0 d
0
0 0
d
0 0
B
0 0 0
0
0 e
0
e
B
0
0 f
0
f
0 0
0
0 0
B
determine the number of force components which are not zero: Pnma La CVX f V , 0, f X * * O1 CVX f V , 0, f X - - Mn C 0, 0, 0 O2 C fV ,fW ,fX - - - I4/mmm La CuO La C 0, 0, f X * O1 C 0, 0, f X - Cu D 0, 0, 0 O2 D 0, 0, 0 Due to symmetry, the force which acts on Mn is zero at any value of the parameters. Analogously, one component of the force acting on the La and O1 ions (y-component for the Pnma or z-component for the Pbnm) is zero as these ions lie in the symmetry plane. Note, that if ions have high site symmetry the potential energy minimum condition is automatically fulfilled due to symmetry at any parameter value. For comparison, the tetragonal I4/mmm phase of La CuO where ions have high site symmetry is given. In the case of complex crystals with the low site symmetry ions the question of the potential energy minimum condition becomes actual.
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253
Table 3 Model parameters: A , b : Born—Mayer constants; ¸ — longitudinal force constants between the ions i and j; ¹ — transverse force GH GH GH GH constants between the ions i and j. R: length of respective bound; Z: ionic charge b (As \) GH
¸ (N/m) GH
6050
4.80 4.784
3200
3.675
6050
4.10
3200
4.81
3200
3.845
179.31 249.71 62.83 56.86 4.46 85.0 6.96 71.9 25.56 31.25 1.35 2.58 2.13 0.89 0.82 11.54 10.37 2.86 1.76
Interaction
A (ev) GH
Mn—O1 Mn—O2 Mn—O2 La—O1 La—O1 La—O1 La—O1 La—O2 La—O2 La—O2 La—O2 O1—O2 O1—O2 O1—O2 O1—O2 O2—O2 O2—O2 O2—O2 O2—O2 Atom type Z[e]
Mn 1.52
O2 !1.04
Table 3 gives the parameters at which the results were obtained. The longitudinal ¸ and transverse GH ¹ force constants are also given. GH 4. Results and discussion The structures of LaMnO Pnma and La CuO Cmca, although different, have in their base a distorted perovskite cube and octahedral oxygen environment for Cu and Mn ions. The site symmetry of all ions is different in these structures, which can be seen in Table 1. We shall consider the differences of normal modes of these structures from the standpoint of symmetry. 4.1. Raman active modes Fig. 2 illustrates Raman active modes related to stretching vibrations of oxygen ions in La CuO and LaMnO . The La and O1 ions in both the space groups have the site symmetry C (E, m), but
O1 !1.18
¹ (N/m) GH
R (As )
!19.01 !14.0 !3.06 !6.05 !0.26 !8.2 !0.44 !6.2 !2.0 !2.5 !0.08 !0.21 !0.18 !0.07 !0.07 !0.65 !0.56 !0.13 !0.08
1.964 1.905 2.188 2.559 3.252 2.450 3.131 2.446 2.698 2.649 3.416 2.715 2.750 2.932 2.949 2.88 2.91 3.24 3.37
La 1.74
in Cmca m is p , and in Pnma m is p (axes orientaV W tion is the same in both the phases). Consider A modes. There are five A modes in La CuO . Fig. 2 depicts two of these. The 417 cm\ mode [29] (experimental 426 cm\ [44]) is governed preferentially by the stretching vibration of O1 ion. For the A symmetry the vibrations lying in the symmetry plane are allowed for the O1 ion, therefore the mode associated with the stretching vibration can be assigned to the A symmetry if the Cu—O1 bond (or Mn—O1 in LaMnO ) lies in the symmetry plane. Since in La CuO (Cmca) the symmetry plane is normal to the x-axis, the O1 stretching vibration lies in this plane and is symmetry allowed. In LaMnO there are seven A modes. In this crystal the symmetry plane is normal to the y-axis. The vibration of the O1 ion along the Mn—O1 bond is normal to this plane, and for A symmetry such vibration is forbidden. So, in the Pnma no A mode will exist associated with the O1 stretching vibrations. The O1 vibration along the Mn—O1 bond is symmetry allowed only in the
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Fig. 2. The comparison of the stretching vibrations of oxygen ions in Raman active modes of LaMnO and La CuO . The Mode analogous to the 217 cm\ mode related to stretching La ion vibrations is imposible in LaMnO .
B and B modes. In fact, the B 573 cm\ mode (Fig. 2) has eigenvectors which are analogous to the eigenvectors of the 417 cm\ mode in La CuO . The quite large difference in the frequencies is
accounted for by the fact that the Cu—O1 distance amounts to 2.43 As [39], and Mn—O1 is 1.96 As . The A 217 cm\ mode (Fig. 2) (experimental 226 cm\ [44]) is observable in La CuO , but in LaMnO
I.S. Smirnova / Physica B 262 (1999) 247—261
such a vibration is impossible. Due to the occurrence of two La—O layers in La CuO there are La—O1 bonds both in La—O layer and normal to it. The 217 cm\ mode is associated with the La vibration along the normal bonds. There are no double La—O layers in LaMnO , therefore there are bonds only in the La—O plane and, so, no mode can exist analogous to 217 cm\ in La CuO . Consider B modes for both crystals (Fig. 2). Three such modes can exist in La CuO since only one component is symmetry allowed for each ion: u for O2 (u , u ,0) and u for O1 and La W V X V (u , u ,0). Since for O2 u , u ,0, the vibration of W X V X this ion along the Cu—O2 bond is impossible. Due to the low site symmetry of the O2 ion in LaMnO this ion has three displacement components, therefore the 603 cm\ B mode associated with the stretching vibration of the O2 ion is possible. In this mode the u O0, but it is very small. The W 405 cm\ B mode has, on the contrary, a large component u of the ion O2 and small components W u , u , therefore the eigenvectors of this mode are V X analogous to the eigenvectors of the 410 cm\ mode in La CuO . Note, that the O2 vibrations of the 603 cm\ B modes in two octahedra inclined to one side are not identical, they are equal in modulus but opposite in direction. Fig. 3 illustrates all 24 Raman active modes in LaMnO . The figure of 406 cm\ mode shows vi brations of 20 ions marked by the digits in Fig. 1a. 4.2. IR active modes Fig. 4 shows IR active modes related to the stretching vibrations of oxygen ions in La CuO and LaMnO . In La CuO the highest-frequency B and B modes are not exactly O2 stretching vibrations (Fig. 4, right side) since in these representations the displacement component u ,0 due W to symmetry, but the tilt angle of octahedron being small, these vibrations are very close to the O2 stretching vibrations. In LaMnO the highest-frequency B and B modes are also not exactly stretching vibra tions (Fig. 4, left side). The same as in La CuO , these modes are close in frequency but have a lower frequency 558 cm\, as suggested by experiment [18], than 687 cm\ [30] in La CuO . This fre
255
quency difference is likely to be caused first by the difference in the mean Cu—O2 distance (1.94 As ) and Mn—O2 (2.04 As ) and then by the fact that in LaMnO the vibrations are “less stretching” due to a greater tilt of the octahedron. The situation is different with the B modes. In La CuO , oxygen O2 has only the u component in W these modes, and u , u ,0, i.e. there is no stretching V X mode associated with O2. In LaMnO , B stretch ing mode is possible due to a lower site symmetry of O2. At first glance the summary dipole moment of this mode (562 cm\, Fig. 4) is zero but u O0, it is W small and not shown in figure. However, precisely this component produces the summary dipole moment. It is small in magnitude, therefore the LO—TO splitting and the oscillator strength of this mode are small. So, in LaMnO there are two stretching B modes: one is due to the O2 vibra tions (with a small LO—TO splitting), the other is due to the O1 vibrations (with a large LO—TO splitting). Fig. 5 illustrates all 25 IR active modes of LaMn O . In every orthorhombic crystal all irreducible representations (IRR) are one dimensional, therefore the dipole moment vector has one component in each IR active IRR. This means that every IR active mode in an orthorhombic crystal is such that for all symmetrically equivalent ions, for example for eight O2 ions, one of the components ought to have like signs (z- for B , y- for B , x- for B in Pnma orientation). The other displacement components of the symmetrically equivalent ions (O2 for example) must have such signs that the corresponding component of the summary dipole moment be zero. Some modes in IR active IRR have a small component giving the summary dipole moment. 562 and 207 cm\ B modes are the examples of such modes. So, the group theoretical analysis suggests that there has to be seven modes of B sym metry, but experimentally only five modes will possibly be observed. Fig. 6 shows silent modes of LaMnO : 217, 171 and 562 cm\ clearly show the vibrations of four symmetrically equivalent Mn ions. Such ions have the displacement vectors being equal in modulus but its displacement vector directions are different. The following difference in phonon modes of two structures in question is worth mentioning. In La
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Fig. 3. Normal frequencies and eigenvectors of A , B , B , and B modes of Pnma orthorhombic phase of LaMnO . In the Fig. of 406 cm\ A mode, the digits designate all 20 atoms of the unit cell as in Fig. 1a.
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MnO vibrations of oxygen ions in two octahedra tilted to one side can have different displacement directions for example 582 cm\ B mode and 562 cm\ B mode (Fig. 4) or 603 cm\ B mode and 573 cm\ B mode (Fig. 2). Such a situation is impossible in La CuO since in this more symmet rical structure all octahedra tilted to one side are translationally equivalent, and in each mode the displacements of all the octahedron’s ions of the same type (for example O2) are equal in magnitude and direction.
5. Conclusions
Fig. 4. The comparison of the stretching vibrations of oxygen ions in IR active modes of LaMnO and La CuO .
The calculation was performed for stoichiometric LaMnO using the structure data of Elemans et al. [23]. Hauback et al. [24], Mitchell et al. [28], Huang et al. [26], and To¨pfer and Goodenough [36] have shown that both Mn—O and La—O bond lengths can change in orthorhombic structure, depending on the growth conditions and the cation stoichiometry. Therefore, when comparing the calculated and experimental frequencies one has to know how much the crystal composition is close to the stoichiometric one. The LaMnO modes have been calculated for Pnma orientation (International Table orientation). With some other coordinate axes, for instance Pbnm, one has, taking into account the transition matrix from one system of coordinates to another, to make corresponding conversions both at decomposition of normal modes into irreducible representations (IRR) and transformations of the dipole moment vector (selection rules for IR spectra, see Fig. 1) and the electron polarizability tensor (selection rules for Raman spectra, see Table 2). The decompositions of normal modes into IRR in ! point have been given as an example for two systems: Pnma and Pbnm (see Table 1). It is well-known that the agreement between the theory and the experiment in the high-frequency region is crucial for the shell model and the rigideffective charge-ion model. Unfortunately, the information on higher phonon frequencies of oxides under consideration is incomplete and controversial.
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Fig. 5. Normal frequencies and eigenvectors of B , B , B modes of the Pnma orthorhombic phase of LaMnO . The respective LO frequencies are enclosed in parentheses. In the Fig. of 76 cm\ B mode, the digits designate all 20 atoms of the unit cell.
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259
Fig. 6. “Silent” A modes of the Pnma orthorhombic phase of LaMnO .
First at all consider experimental data of Raman spectra. Reedyk et al. [13] have reported the highest frequency of 645 cm\ in Raman spectra of CeTiO . The authors assign this frequency to all Raman active symmetries. Our calculation shows that such high-frequency modes of B , B , B symmetries are possible, however its frequencies should be different. This difference is a direct consequence of symmetry of the orthorhombic crystal rather than any model. The fact that splitting was not revealed in the experiment [13], probably points to an imperfection of samples under investigation or the values of splitting are )10 cm\. The values of splitting of B modes (i"1, 2, 3) are very useful for the fitting of model parameters. Iliev et al. [19] observed 611 cm\ B mode in LaMnO . For each of B , B symmetries authors have found only one mode instead of five modes. Venugopalan and Becker [16] and Koshizuka and Ushioda [17] have reported frequencies )500 cm\ of B (i"1, 2, 3) symmetries in or thoferrites REFeO , although the number of
B , B and B observed modes is less than that predicted. Let us consider the A mode with highest frequen cy. One of A modes has the frequency of 645 cm\ as Reedyk et al. [13] have reported. Venugopalan and Becker [16] and Koshizuka and Ushioda [17] investigated A spectra in three polarizations (xx, yy, zz) and observed more modes than it follows from the group theoretical analysis. As a result, the 645 cm\ mode was assigned as a twophonon mode in the work [16]. Koshizuka and Ushioda [17] attribute this mode to an impurityrelated phonon. Venugopalan and Becker [16] and Koshizuka and Ushioda [17] excluded the 645 cm\ mode from a one-phonon spectrum due to the eighth added mode of A symmetry. If this mode has A symmetry as Reedyk et al. [13] sug gest, then another mode should be excepted from a one-phonon spectrum due to the eighth added mode. Iliev et al. [19] have not detected the 645 cm\ mode of A symmetry, but they have observed only five instead of 7 A modes.
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Thus, the experimental data were interpreted from two viewpoints: 1. 645 cm\ mode is not a one-phonon mode [16,17] therefore the Raman spectra do not contain one-phonon modes with frequencies more than 500 cm\ 2. 645 cm\ mode is of a one-phonon nature [13] and moreover it is observed in all Raman-active symmetries. Then it is a highest frequency of the Raman spectra. The calculations presented show that the crystal can well have one-phonon modes of B symmetry (i"1, 2, 3) in the frequency region of &580— 650 cm\. However, the crystal cannot have A mode with such a high frequency, since in the considered model such a high (*600 cm\) frequency of A symmetry cannot be obtained taking into account that the highest frequency of IR-active spectra is 540—570 cm\ [18,14]. One can hope that these contradictions will be resolved in the course of further experimental studies. The information on IR-active modes available at that time are very limited, since all measurements have been fulfilled with depolarized light. However, these measurements give the approximate value of a highest frequency, for example, it is equal to 558 cm\ in LaMnO [18] and 545 cm\ in LaTiO [14]. According to the group theoretical analysis there has to be seven modes of the B symmetry. The calculation shows that two modes (one of them has a highest frequency) should have small oscillator strengths and they may not be observed in the experiment. However, the model parameters should be slightly adjusted if an experiment will show that all seven modes have rather large oscillator strengths. Note that in IR spectra highest frequencies of B , B , B symmetries should be different as well. The information about this splitting is absent, since measurements of IR spectra were carried out with depolarized light. The measurement with polarized light may give both the splitting of highest frequencies and LO—TO splitting. This information is required for a more exact calculation of this complex phase.
Finally it should be mentioned that Reedyk et al. [13] have found a very interesting experimental fact. At a complete substitution of rare-earth cations in ReTiO the authors observed a strong displacement of one of the A symmetry modes. According to the analysis of the Pnma phonon modes, one can suggest that this mode is, to a large extent, related to the vibration in the symmetry plane, i.e., to a change in the Re—O distance.
Acknowledgements The author thanks V.M. Edelstein, A.V. Bazhnov, and S.S. Nazin for stimulating discussions. This work was supported in part by VW Stiftung.
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