Complex ordering phenomena

1.1.1 Charge order

Charge ordering (CO) is often observed in strongly correlated materials such as transition metal oxides or organic conductors with mixed valence state. As the term CO says it is the ordering of valence state of ions in a crystal from high temperature homogeneous intermediate valence state to a low temperature ordered-mixed valence state. CO is associated with a structural phase transition with a lowering of the symmetry, because of the long range ordering in which sites the electrons localizes. Though the charge order was first proposed by Eugene Wigner 1930, this concept and the associated phenomena were first observed in magnetite (Fe₃O₄) by Verwey in 1934 (for details, see chapter 4). Later it has been observed in mixed-valence pervoskites, e.g.,doped manganites (A1−xB_xMnO₃: the ratio Mn³⁺/Mn⁴⁺ depends on the doping level) [11], rare earth nickelates(RNiO₃:Ni³⁺ →N i^3+δ/N i^3−δ) [12, 13], in self-doped NaV₂O₅ (V⁴⁺/V⁵⁺) [14], SrFeO₃ (Fe³⁺/Fe⁵⁺) and SrFeO3−δ (Fe³⁺/Fe⁴⁺) [15], RFe₂O₄ (Fe²⁺/Fe³⁺) [16], Fe₂OBO₃

1.1 Complex ordering phenomena

(Fe²⁺/Fe³⁺) [17] etc. Charge ordering induces an electric polarization whenever it breaks the spacial inversion symmetry. The types of charge ordering and its related novel phenomena, that are relevant to the present work are discussed in the section 1.2. CO can alter the lattice periodicity as it is always companied by a slight lattice distortion.

Hence the observation of the charge ordering is possible. There are different methods to detect the CO phenomena. Few of them are: An empirical method, Bond Valence Sum (BVS) calculation [18, 19] through properly determined crystallographic bond lengths.

The valence V can be calculated by the formula:

V =X

exp(R₀−R_i) b

Where R_i is the observed bond length, R₀ is a tabulated parameter expressing the (ideal) bond length of the cation-anion pair and b is an empirical constant (0.37 ˚A).

Another method which has been widely used to study the CO is Resonance x-ray scattering [20]. The energy value of the absorption edge, also known as chemical shift for the different valences are slight different. Hence by tuning the incident x-ray energies near the absorption edge, the contrast between the atomic scattering factors for the different valence states can be significantly enhanced. The technique of Mossbauer spectroscopy is widely used to distinguish ions with different valence states by measuring the isomer shift [21]. However, it cannot determine the charge ordering, i.e. the spatial arrangement of different valence states.

1.1.2 Orbital order

The orbital degrees of freedom often plays a crucial role in the physics of strongly correlated 3d transition metal oxides for e.g., metal-insulator transition, colossal magnetoresistance etc. In order to understand these phenomena first we discuss the basic interactions which are necessary to understand the orbital ordering.

1.1.2.1 Crystal field effect

Crystal field theory has the basis on the basic principle of breaking of degeneracy of the d- orbitals due to Stark effect. d- orbitals are having their lobes that is the regions of highest electron density either along the axis of the orbitals or at 45 degree angle to the

axis. In case of a octahedral complex, the 3d level split in to doubly degenerate upper level, eg (d_z², d_x²−y²) and triply degenerate lower level, t2g (dxy, dyz, dzx), see fig 1.1. The difference between the energy of e_g and t_2g for an octahedral crystal field is called ∆_o. But in case of a tetrahedral complex, the lower level is doubly degenerate and upper level is triply degenerate. The total crystal field splitting of tetrahedral crystals is called ∆_t. In weak crystal field the energy associated with the first Hunds rule leads to a high spin states and when crystal field is stronger and it prefers to stay in the splitted low energy orbitals and a low spin state is found. Crystal field splitting energy depends upon the strength of the approaching crystal field. Different shapes of the d-orbitals and the level schemes of the 3d orbitals in an octahedral and in tetrahedral arrangement is shown the figure 1.1.

Figure 1.1: (a) Shapes of the d-orbitals, taken from reference [22]. (b) Crystal field effect:

d-orbitals in the presence of octahedral and tetrahedral field

1.1.2.2 Jahn-Teller effect

The original statement of Jahn-Teller (JT) theorem [23] is as follows: Any non-linear molecular system in a degenerate electronic state will be unstable and will undergo distortion to form a system of lower symmetry and lower energy thereby removing the

1.1 Complex ordering phenomena

degeneracy. If the two orbital ofe_g level have unsymmetrical distribution of electrons, this will leads to either of shortening or elongation of the bonds. This breaks the degeneracy of the e_g orbitals i.e. the e_g orbitals will again split with different energies for e_g and t_2g orbitals. If the crystal field is same then the inversion center of the orbitals are retained.

Similar but small and more complex effect can be observed in case of unsymmetrical electron distributions in thet2g orbitals also.

For example in the case of LaMnO₃, Mn³⁺ has a d⁴ configuration. According to first Hund’s rule, all the spins are aligned parallel resulting in total spin of S = 1/2 and the configuration is t³_2ge¹_g. Mn³⁺ is Jahn-teller active. Hence the oxygen octahedron is distorted. As a result the degeneracy of the eg orbitals is removed, shown in the figure 1.2. This leads to a long range orbital ordering of e_g electrons. Charge order determines what orbital order is possible and this couples charge, orbital and spin order together.

Figure 1.2: Jahn-teller splitting: The oxygen ions surrounding the Mn³⁺ is slightly distorted and the degeneracy of the e_g is remove

1.1.3 Spin order

The strong exchange interaction between the spins of neighboring magnetic ions leads to magnetic or spin order in a system. Good description of the interaction between the neighboring spins S_i and S_j are given by Heisenberg, within framework of model

Hamiltonian [24]:

H =X

ij

−JijS~iS~j (1.1)

Where J_ij is the exchange constant between the i^th and j^th spins, which describe the nature of the spins. J > 0 favors parallel alignment of the neighboring spins, hence the system is ferromagnetic. J < 0 favors antiparallel alignment of the spins, hence the system is antiferromagnetic. In case of ferromagnetic order the periodicity is equal to the separation of the magnetic moments. But in case of antiferromagnetic order the repeat period is doubled, which can lead to different types of magnetic structures. For example, commensurate antiferromagnetic order, where the period of the magnetic order is equal to an integer number of lattice units (A-type, G-type, C-type and E-type [see fig 1.3]).

Figure 1.3: Different types of commensurate antiferromagnetic ordering: (a) A-type, (b) G-type, (c) C-type, (d) E-type.

Competitive neighbor and next nearest neighbor ferro and antiferromagnetic exchange or relatively more complex anisotropic exchange interaction can lead to incommensurate antiferromagnetic order where, the period of the magnetic order is not equal to an integer number number of lattice units (e.g., sinusoidal modulated spin density waves and spiral order). In the later case the spins will change their orientation by a fixed angle relative to their neighbors along the propagation direction. This can be determined macroscopically for e.g., by measuring the net magnetization in different crystallographic directions or microscopically by neutron polarization analysis in different orientations (hhl-plane,