Crystal Field Effect

Atoms in a solid with a regular crystal structure are influenced by the electric field due to the neighboring atoms in the crystal. Average of such electric fields are called crystal field [46]. Crystal field depends upon the nature of the local environments such as nature of atomic co-ordination, etc. The magnetism in transition elements originates from their unfilled d-shell electrons. In general, the d orbital has five degenerate energy levels and is divided into two sub orbitals namely t_2g (d_xy, d_yz and d_zx) and e_g (dx2₋y2and 2

dZ ) as shown in Fig. 1.3.

But when ligands approach the d orbital from different directions, not all parts of d orbitals interact directly with the ligands and it results in unequal electrostatic repulsion between them. Such interaction creates a splitting of d orbital due to the electrostatic environment and the d orbital energy is no longer degenerate. For example, in octahedral environment, the crystal field arises mainly from the electrostatic repulsion from the negatively charged electrons in the surrounding oxygen 2p orbitals. The electronic configurations of 2p orbitals are oriented along x, y and z axes and are referred as px, py and pz,respectively. Therefore, in the octahedral environment the e_g orbitals (dx2₋y2and dZ2) which also point along x, y and z axes, overlap predominantly with neighboring p-orbitals of oxygen ion compared to those of t2g orbitals that points in between x, y and z axes. Consequently, the electrons in eg orbitals experience greater electrostatic repulsion and hence larger electrostatic energy. As a result eg

levels are lifted up compared to those of t2g levels. But in the case of tetrahedral environment, the eg orbitals now maximally avoid the electronic charge density associated with the oxygen

atoms and hence the eg levels are now lowered in energy [46]. The energy level diagrams of octahedral and tetrahedral environments along with the respective figure of octahedral and tetrahedral coordinations are shown in Fig. 1.4.

Figure 1.3: The electronic distribution of 3d orbitals. In the cubic crystal field, this fivefold degeneracy is lifted and separated into two eg levels (dx2₋y2and dZ2) and three t2g

levels (d_xy, d_yz and d_zx) (Reproduced from Tokura et al. [47]).

e

g

t

_2g

Figure 1.4: Energy level diagram in octahedral and tetrahedral environments.

1.3.1 Orbital Quenching

The crystal field effect is mainly observed in transition elements, where their 3d valence electrons are close to the outermost shell and thus they are exposed to the electronic configuration of neighboring ions. In general, the effective magnetic moment (µ_eff) of an ion can be calculated by using the relation µ_eff =g J J( +1) in the unit of µ_B but it does not match with experimental values for most of the transition elements. This is due to the large crystal field effect, which dominates over the Hund’s spin-orbit coupling energy in 3d transition elements and hence the orbital angular momentum is quenched (L = 0). Hence, µ_eff is generally calculated using the relationµ_eff(µ_B)=g S S( +1). Here J and S refer quantum numbers corresponding to total and spin angular momentum of electrons, respectively.

Another interesting series of elements having strong magnetic moments is rare earths, where the magnetism originates from 4f shell. Unlike transition elements, here the 4f levels are deep

inside from the outermost orbital, i.e. with negligible overlapping with the electronic configuration of neighboring ions and hence they do not show any crystal field effect. In this class of materials µeff can be calculated using the general relation, µ_eff(µ_B)=g J J( +1) [46].

1.3.2 Jahn-Teller Distortion

In some of the 3d transition elements for a specific valence state of an ion, electrons asymmetrically occupy the degenerate t_2g or e_g orbital leading to net larger electronic energy in the system. In order to reduce the overall energy, the system undergoes lattice distortion either by stretching or compressing their bonding such that degeneracy in the above orbitals is lifted. This distortion is known as the JTD [46]. The above process leads to overall reduction in the energy of the system.

In the octahedral environment, the most pronounced JTD is observed when an odd number of electrons occupy the eg orbitals, i.e. when the ion has an orbital degeneracy in the eg orbitals. However, ions with orbital degeneracy in the t2g orbitals show very weak JTD since these orbitals do not point directly towards the ligands (non bonding orbitals) and are lower in energy. High spin complexes with d⁴, d⁹ (Mn³⁺, Cu²⁺_, etc.) electrons and low spin complexes with d⁷ (Co²⁺) electrons in octahedral environment undergo strong JTD. For example in case of d⁴ configuration, three electrons occupy the t2g orbitals and the fourth electron has the orbital degereracy of eg (dx2₋y2and dZ2) orbitals. In order to lift the orbital degeneracy, the JTD stretch the octahedra along z-direction and this process leads to smaller energy of 2

dZ due to smaller overlapping with adjacent legands. So, the fourth electron occupies the dZ2 level and this process leads to overall reduction in energy. This process is shown in Fig. 1.5 [46]. Conversely in some system, the net energy is redudced by compressing the octahedra, where the dx2₋y2 orbital will be lowered and the fourth electron occupies this orbital rather than that of 2

dZ level [48].

In tetrahedral environment, the t_2g orbitals are closer to the distribution of electrons in the 2p orbitals of the ligands (oxygen) and directed to each other compared to the e_g orbitals

and hence t2g orbitals are at higher energy level. So, here strong JTD occurs only for the configurations which are degenerate with respect to t2g orbital. High spin complexes with d³, d⁴, d⁸, d⁹ electrons and low spin complexes with d⁵, d⁶, d⁸, d⁹ electrons in tetrahedral environment undergo strong JTD. Spinel compounds having Jahn-Teller active ions, such as Ni²⁺, Cu²⁺ and Fe²⁺ at the tetrahedral site cause a cubic to tetragonal distortion. For example, the spinel chromites NiCr₂O₄, FeCr₂O₄ and CuCr₂O₄ undergo a cubic to tetragonal distortion at 310 K, 140 K and 853 K, respectively as the temperature is decreased. In NiCr₂O₄ and FeCr₂O₄ the JTD is of elongation type while in CuCr₂O₄ it is compression type [11, 35, 48].

Figure 1.5: Further splitting of both the t2g and e_g states in ions with d⁴ configuration due to the JTD.