Interplay of charge and orbital ordering in manganese perovskites

A model of localized classical electrons coupled to lattice degrees of freedom and, via the Coulomb interaction, to each other, has been studied to gain insight into the charge and orbital ordering observed in lightly doped manganese perovskites. Expressions are obtained for the minimum energy and ionic displacements caused by given hole and electron orbital configurations. The expressions are analyzed for several hole configurations, including that experimentally observed by Yamanda et al. in La7/8Sr1/8MnO3. We find that, although the preferred charge and orbital ordering depend sensitively on parameters, there are ranges of the parameters in which the experimentally observed hole configuration has the lowest energy. For these parameter values we also find that the energy differences between different hole configurations are on the order of the observed charge ordering transition temperature. The effects of additional strains are also studied. Some results for La1/2Ca1/2MnO3 are presented, although our model may not adequately describe this material because the high temperature phase is metallic. 71.38.+i, 71.45.Lr, 71.20.Be, 72.15.Gd Typeset using REVTEX 1 Over the last few years much attention has been focused on manganese perovskite-based oxides, most notably the pseudocubic materials Re1−xAkxMnO3. (Here Re is a rare earth element such as La, and Ak is a divalent alkali metal element such as Ca or Sr.) The initial motivation came from the observation that for some range of x, and temperature, T , resistance can be reduced by a factor of up to 10 in the presence of a magnetic field. Two other interesting physical phenomena occurring in this class of materials are charge ordering and orbital ordering. In this paper, we study the connection between the two. The important electrons in Re1−xAkxMnO3 are the Mn eg electrons; their concentration is 1 − x. For many choices of Re, Ak, and x, especially at commensurate x values, the eg charge distribution is not uniform and it indeed appears that a fraction x of Mn ions have no eg electron while 1 − x have a localized eg electron. A periodic pattern of filled and empty sites is said to exhibit charge ordering. There are two eg orbitals per Mn ion. A localized Mn eg electron will be in one linear combination of these; a periodic pattern of orbital occupancy is said to exhibit orbital ordering. In this paper we present an expression for the coupling between charge and orbital ordering, with different charge ordering patterns favoring different orbital orderings. We also argue that the orbital ordering energy differences determine the observed charge ordering in lightly doped manganites. Localized charges induce local lattice distortions, which must be accommodated into the global crystal structure; the energy cost of this accommodation is different for different charge ordering patterns. To model the charge and orbital ordering, we assume that the electrons are localized classical objects, so that each Mn site is occupied by zero or one eg electron, and each eg electron is in a definite orbital state. This assumption seems reasonable in the lightly doped materials such as La7/8Sr1/8MnO3, which are strongly insulating at all temperatures, 3 but may not be reasonable for the La1/2Ca1/2MnO3 composition, 2 where the charge ordered state emerges at a low temperature from a metallic state. For localized electrons there are two energy terms. One is the coupling to the lattice, which will be discussed at length below, the other is the Coulomb interaction, which will be discussed briefly first. We argue that the Coulomb energy cannot explain the observed ordering pattern or transition temperature. To demonstrate this, we study the three configurations of hole ratio 1/8 shown in Fig. 1. Figure 1(b) is the configuration proposed by Yamanda et al. to explain their experimental results for La7/8Sr1/8MnO3. The configurations in Figs. 1(a) and 1(c) have been chosen because of their structural similarity to the observed Fig. 1(b) configuration. We take as reference the state with one eg electron per Mn and denote by δqi the charge of a hole on a Mn site. From the classical Coulomb energy UCoulomb = 1 2ǫ0 ∑ i6=j δqiδqj rij , (1) one finds that the difference in energy between the configurations in Fig. 1 is ∆UCoulomb, per hole = 1 2ǫ0 ∑ i6=o ∆(δqi) rio , (2) where o is a site containing a hole and ∆(δq) is the difference in charge between the two configurations. We estimated the above infinite sum by repeated numerical calculations for 2 larger and larger volumes of the unit cells around the origin. We find that Fig. 1(c) has the lowest energy; 12 meV/ǫo lower than Fig. 1(b), and 27 meV/ǫo lower than Fig. 1(a). To estimate the magnitude of the Coulomb energy differences, we need an estimate for the dielectric constant ǫ0, which we obtain from the measured reflectivity for La0.9Sr0.1MnO3, 4 and the Lyddane-Sachs-Teller relation ω L = ω 2 T ǫ0/ǫ∞. At frequencies greater than the greatest phonon frequency the reflectivity is close to 0.1, implying ǫ∞ ≈ 3.4; the reflectivity is near unity between ωT = 0.020 eV, and ωL = 0.024 eV, implying ǫ0 ≈ 5.0. Because both La7/8Sr1/8MnO3 and La0.9Sr0.1MnO3 are insulating and have similar compositions, their static dielectric constants are expected to be similar. Using ǫ0 ≈ 5.0, the energy difference between different configurations of holes is only around 2.4 meV, or 30 K per hole, which is small compared to the observed charge ordering temperature of 150 K − 200 K of these materials. The inconsistency with the experimentally observed hole configuration and the smallness of the energy difference scale indicate that the electrostatic energy is not the main origin of charge ordering for this material. We now turn our attention to the lattice energy. A classical model for the lattice distortions of the insulating perovskite manganites has been derived in Ref. 6, and shown to be consistent with experimental results on LaMnO3. This model is adopted here with an additional term, an energy cost for shear strain. We now briefly outline the model, which is explained in more detail in Ref. 6 and the Appendix. The ionic displacements included are the vector displacement ~δi of the Mn ion on site i, and the â directional scalar displacement ui (a = x, y, and z) of the O ion which sits between the Mn ion on site i and the Mn ion on site i + â. For convenience, ~δi and u a i are defined to be dimensionless in the following way: the lattice constant of the ideal cubic perovskite is b, the Mn ion position in the ideal cubic perovskite is ~ Ri, the actual Mn ion position is ~ Ri + b~δi, and the actual O ion position is ~ Ri + (b/2 + bu a i )â. The lattice energy is taken to be harmonic and depends only on the nearest neighbor Mn-O distance and the first and second nearest neighbor Mn-Mn distances. The spring constants corresponding to these displacements are K1, K2, and K3 as shown in Fig. 2. Because K1 and K2 involve bond stretching, while K3 involves bond bending, K1 ≥ K2 ≫ K3 is expected. Thus, Elattice = EMn-O + EMn-Mn,first + EMn-Mn,second, where EMn-O = 1 2 K1 ∑ i,a (δ i − ui ) + (δ i − ui−a), (3) EMn-Mn,first = 1 2 K2 ∑ i,a (δ i − δ i−a), (4) EMn-Mn,second = 1 2 K3 ∑ i,(a,b) [( δ i+a+b + δ b i+a+b √ 2 ) − ( δ i + δ b i √ 2 2 + [( δ i+a−b − δ i+a−b √ 2 ) − ( δ i − δ i √ 2 )]2 . (5) In the above equations a denotes x, y, and z, and (a, b) represents (x, y), (y, z), and (z, x). EMn-Mn,second was not considered in Ref. 6. The shear modulus produced by this term is important, because without it, a Mn ion on site i+ x̂ can have arbitrary large y directional displacement relative to the Mn ion on site i at no cost of energy. For this reason, the model with K3 = 0 has singularities, whose proper treatment requires K3 6= 0 in our model. 3 However, still we expect K3 will be much smaller than K1 or K2. Therefore, in order to simplify the calculation, the K3/K1 → 0 limit has been taken after the expression of minimized energy and equilibrium ionic displacements have been obtained. Second, we consider the electronic degree of freedom. We parameterize the electron density by the variable hi. If an electron is present on site i, hi = 0; if no electron is present, hi = 1. If there is an electron on site i, the electron orbital state, which is a linear combination of the two eg orbitals, is parameterized by an angle θi as follows. |ψi(θi) >= cos θi|d3z2−r2 > +sin θi|dx2−y2 > (6) with 0 ≤ θi < π. The electron orbital state couples to the distortion of the surrounding oxygen octahedra through the Jahn-Teller distortion. The coupling is given by EJT = −λ ∑ i (1− hi)[cos 2θi{v i − 1 2 (v i + v y i )}+ sin 2θi √ 3 2 (v i − v i )] = −λ ∑ i,a (1− hi)v i cos 2(θi + ψa), (7) where v i = u a i − ui−a, (8) ψx = −π/3, ψy = π/3, ψz = 0. (9) If a hole is present on site i, it attracts the surrounding oxygens equally, giving rise to a breathing distortion energy given by Ehole = βλ ∑ i hi(v x i + v y i + v z i ). (10) The parameter β represents the strength of the breathing distortion relative to the JahnTeller distortion. Finally, following Kanamori, we include a phenomenological cubic anharmonicity term given by Eanharm = −A ∑ i (1− hi) cos 6θi. (11) The sign has been chosen so that the electron orbital states of |3x2 − r >, |3y2 − r >, or |3z2 − r >, with x̂, ŷ, and ẑ pointing toward nearest oxygen ions are favored when A is positive. The total energy, which is the sum of all the above energy terms, is given by Etot = EMn-O + EMn-Mn,first + EMn-Mn,second + EJT + Ehole + Eanharm. (12) We minimized Etot about δ a i ’s and u a i ’s for fixed hole and orbital configurations. These are conveniently expressed in terms of the variables δ ~k , u a ~k , h~k, and c a ~k defined in the following way.