Lattice dynamics of manganites RMnO3 (R =Sm, Eu or Gd): instabilities and coexistence of orthorhombic and hexagonal phases

The lattice dynamical and allied properties of the multiferroic manganites SmMnO3, EuMnO3 and GdMnO3 were investigated in this work by means of a shell model with transferable pairwise interionic interaction potential. This shell-model potential is able to reproduce the available crystal structure and phonon frequencies. A zone center imaginary Au mode is observed in these lattice dynamics calculations that indicates metastability of the perovskite phase. Comparison of the Gibbs free energies in the orthorhombic and hexagonal phases points to the possible coexistence of the two phases of these manganites under ambient conditions.


Introduction
It is well known that multiferroic materials possess any two or all three states of ferroelectricity, ferromagnetism and ferroelasticity occurring in the same phase and, hence, they are of great technological and fundamental importance. The recent discovery of large magnetoelectric effects [1] in the rare-earth manganites (RMnO 3 ) has kindled interest among investigators in understanding the complex relationships between lattice distortion, magnetism, dielectric and transport properties of undoped RMnO 3 . With respect to structure, these manganites have been grouped into the hexagonal phase (P6 3 cm) with R (= Ho, Er, Tm, Yb, Lu or Y) [2] having smaller ionic radius (r R ) and the orthorhombic structure (Pnma) with R (= La, Pr, Nd, Sm, Eu, Gd, Tb or Dy) [3] having larger r R . However, it is well known that the compounds in the first group can also be formed in a metastable perovskite Pnma phase by high-pressure synthesis, by soft chemistry synthesis or by epitaxy. On the other hand, some compounds of the second group could be synthesized [4]- [6] in a metastable hexagonal phase by epitaxial stabilization. The present work is devoted to a theoretical study of these multiferroic manganites, and through a host of computations of the Gibbs free energies and thermodynamic properties for these manganites, we intend to provide a microscopic basis for the metastability and phase competition between these two structures.
The crystal structure of RMnO 3 (R = Sm, Eu and Gd) has been studied previously using x-ray and neutron diffraction techniques [7]- [9]. The perovskite structure has a lower unit cell volume compared to the hexagonal phase. For hexagonal RMnO 3 (R = Sm, Eu or Gd), only the lattice parameters obtained through x-ray diffraction experiments on thin films are known. All three orthomanganites exhibit a Jahn-Teller distorted orthorhombic structure and show sharp peaks in the specific heat due to various magnetic transitions. A wealth of experimental data is available on these manganites; therefore it seems essential to theoretically understand their physical properties, like crystal structure, Raman and infrared frequencies, specific heat, elastic constants, phonon density of states (PDOS) and phonon dispersion curves.
Raman techniques [10] have been successfully employed to investigate the phonon excitations in various orthorhombic RMnO 3 compounds (R = Sm, Eu or Gd) of the manganite family. Recently, the lattice dynamical studies of the parent member LaMnO 3 have been 3 reported [11] using a seven-parameter shell model [12]- [14] based on an interatomic potential that consists of the long-range Coulomb and the short-range interaction terms. Earlier calculations on other manganite materials have used more complex lattice dynamical models [15]- [19], whereas we have used a simpler shell model with a smaller number of parameters [11]. Previously, a modified rigid ion model has been used to elucidate the cohesive, thermal and thermodynamic properties of doped and undoped perovskite manganites [20,21].
We have computed the phonon dispersion curves (PDCs) for the orthorhombic RMnO 3 (R = Sm, Eu or Gd) along three major symmetry ( , and ) directions. The method of computation is presented in section 2.

Lattice dynamics computations
The interatomic potential of the shell model employed in the calculations is expressed as [11]- [14] V where a = 1822 eV and b = 12.364 eV. Z (k) and R(k) are the effective ionic charge and radius parameters associated with atom k. The effect of polarizability of the oxygen ions in this model has been included by considering a massless shell (charge Y (O) = −1.8e) and shell-core force For the orthomanganites, the shell-model parameters are evaluated in such a way that the forces on each atom and the internal stresses in the crystal vanish and they reproduce the crystal structure close to that observed by the diffraction experiments at zero pressure. The crystal structure of orthorhombic RMnO 3 (R = Sm, Eu or Gd) has been calculated by minimizing the Gibbs free energy with respect to the structural parameters. Group-theoretical analysis for the orthorhombic perovskite manganites gives the classification of the phonon modes at the zone center ( ) point and along the , and directions as [11] : 7A g + 7B 2g + 5B 1g + 5B 3g + 10B 1u + 10B 3u + 8A u + 8B 2u , The optimized potential yields the long-wavelength phonon modes close to their measured values. Out of the 60 -point phonon modes, 24 (7A g + 5B 1g + 7B 2g + 5B 3g ) are Raman active, 25 (9B 1u + 7B 2u + 9B 3u ) are infrared active, 8 (8A u ) are optically inactive and 3 (B 1u + B 2u + B 3u ) are acoustic modes. For the computation of the PDCs, the equilibrium structure and the interatomic potential expressed by equation (1) along with the symmetry vectors obtained through the group theoretical analysis for Pnma space group [11] have been employed to diagonalize the dynamical matrix along the three high-symmetry directions and to classify the phonon frequencies obtained into their irreducible representations. The procedures adopted to compute the PDOS and allied thermodynamic properties like specific heat are discussed elsewhere [11]- [14], [22,23] in detail. The results of the computations (performed using the DISPR [14] program, developed at BARC) are presented and discussed below.

Results and discussion
The optimized parameters for orthorhombic RMnO 3 (R = Sm, Eu or Gd) thus obtained are listed in table 1.

Crystal structure
The multiferroic perovskite manganites RMnO 3 (R = Sm, Eu or Gd) (with space group Pnma) have an orthorhombic structure with four formula units per unit cell [7,9]. The computed cell parameters, atomic coordinates and strain parameter (s = 2(a − c)/(a + c)) of orthorhombic RMnO 3 (R = Sm, Eu or Gd) are shown in table 2 and are also compared with available experimental data [8,9]. It is seen from table 2 that our calculated results are in fairly good agreement with the corresponding experimental data [8,9]. The calculated lattice parameters (a, b and c) differ by only 0.7% on average from the experimentally observed data [8,9] for RMnO 3 (R = Sm, Eu or Gd). The calculated values of unit cell volume (V ) have also shown good agreement with experimental data. The decreasing trend of V with decreasing order of ionic radius (r R ) exhibited by both the experimental and our calculated results is similar, as can be seen from table 2.
It is obvious from these satisfactory predictions that our shell-model calculations are capable of reproducing the crystal structures of orthorhombic RMnO 3 (R = Sm, Eu or Gd) manganites.

Elastic constants
The longitudinal acoustic wave velocities as computed from the slopes of the acoustic phonon branches for RMnO 3 (R = Sm, Eu or Gd) are shown in table 3. These compare well with the available experimental data [24] at 130 K. We have also calculated the elastic constants for these materials through a computation of the acoustic wave velocities along different symmetry directions and listed them in table 3. These calculated values could not be compared due to a lack of experimental data on them.

Phonon density of states
The total PDOS has been computed and the results are depicted in figure 2 for the orthomanganites RMnO 3 (R = Sm, Eu or Gd). The total PDOS spans the spectral range up to 80 meV. Also, it has shown broad structures in the 10-50 meV range with peaks around 15, 25, 32 and 38 meV. The partial density of states of individual atoms (Sm/Eu/Gd), Mn, O1 and O2 have also been computed and shown in figure 2. It is seen from these that (Eu/Sm/Gd) atoms generally contribute up to 30 meV. The Mn atoms too contribute mainly up to 30 meV with a few broad peaks. It is interesting to also note, in figure 2, that the peaks around 65 and 72 meV in the total PDOS can be attributed to the oxygen ions alone.

Specific heat
The calculated density of states has been used to evaluate the specific heat at constant volume (C V ) as a function of temperature. The values of the specific heat for orthorhombic RMnO 3 (R = Sm, Eu or Gd) up to the temperature range of 1000 K are displayed in figure 3. They are compared with the respective available experimental data [27] in the figure. Our results have   [27]).
Phonon frequency (cm  followed a trend more or less similar to that exhibited by the experimental curve. The peaks observed in the experimental specific heat curves are not found in our calculated results; this might be due to the exclusion of magnetic interactions from the specific heat in our model calculations.

Phonon dispersion curves
The PDCs for (Sm, Eu, Gd)MnO 3 along the three high-symmetry directions , and in the Brillouin zone have been computed using the present shell model and DISPR program [14]. The Phonon frequency (cm -1 ) Figure  On transferring the shell-model parameters to the known [4]- [6] hexagonal structure (for the atomic coordinates, we used those known [28] for hexagonal DyMnO 3 ), forces on the atoms are nearly zero, but dynamical instability exists. The volume change (∼9%) in going from the orthorhombic to the hexagonal phase is reproduced in the computations. Comparison of the Gibbs free energy for the two phases (figures 7 and 8) shows that the energy difference is of the order of less than 2 kJ mol −1 , under ambient conditions, and hence coexistence is very much possible.

Conclusion
In summary, we have reported here our investigation of the lattice dynamical and thermodynamic properties of RMnO 3 (R = Sm, Eu or Gd) perovskites and their hexagonal polymorphs employing a transferable shell-model potential. For the orthorhombic manganites, our calculated crystal structure parameters, Raman frequencies and infrared frequencies are found to be in good agreement with the available experimental data.  shell-model potential reproduces the change in volume when going from the perovskite to the hexagonal phase. The Gibbs free energies are within 2 kJ mole −1 , which is in accord with thermodynamic estimation [28]. The existence of imaginary modes points to inherent instability in both perovskite and hexagonal phases. On the basis of our results, it may be concluded that the present shell model is suitable and appropriate for the description of the lattice dynamical and related properties of RMnO 3 (R = Sm, Eu or Gd) manganites.