Optics of Chromites and Charge-Transfer Transitions

Abstract

Specific features of the charge-transfer (CT) states and O_2p→Cr_3d transitions in the octahedral (CrO₆)⁹⁻ complex are considered in the cluster approach. The reduced matrix elements of the electric-dipole transition operator are calculated on many-electron wave functions of the complex corresponding to the initial and final states of a CT transition. Modeling the optic spectrum of chromites has yielded a complicated CT band. The model spectrum is in satisfactory agreement with experimental data which demonstrates the limited validity of the generally accepted concept of a simple structure of CT spectra.

1. Introduction

Since discovering the high-𝑇_𝑐 superconductivity in oxides with Cu²⁺ ions and the colossal magnetoresistance in Mn³⁺ systems, a considerable interest has been renewed in transition-metal oxide systems. An important problem of the physics of these strongly correlated systems concerns the nature of their electronic structure and energy spectrum. Optic and magneto-optic techniques are well suitable for studying them. However, the origin of low-energy electron-hole excitations in 3d-oxides is still under discussion.

First proposed for iron garnets [1] and developed further for a wider class of ferric oxide compounds [2], the idea that their optic and magneto-optic properties in the near UV range and the fundamental optic absorption edge are determined by the oxygen-to-3d metal ion charge-transfer (CT) transitions has become widespread [3, 4]. However, there is no consistent theory of CT states and transitions in 3d oxides. In most studies, the idea of CT transitions has not changed significantly since its appearance. One tries to explain the peculiarities of optic spectra of these compounds by some “2p-3d” CT transitions or, at best, keeping in mind the 𝑡_2𝑔-, 𝑒_𝑔-classification of d-electrons in the crystal field, one reasons in terms of “2p→3d𝑡_2𝑔,” “2p→3d𝑒_𝑔” CT transitions [5]. But such one-electron approach to CT states and transitions is insufficient even for qualitatively explaining the observed optic properties of 3d oxides. The one-electron band models, including LDA+𝑈, do not solve the problem, since they fail to reproduce the important intra-atomic correlation effects which form the term structure of the ground and excited CT states.

We consider here the CT states and transitions in chromites and their optic spectra. The analysis is based on the cluster model of one-center CT transitions which has been successfully used to explain the optics and magneto-optics of orthoferrites, iron garnets, and a number of other 3d oxides [6–12]. The advantages of this model are its physical clarity and the possibility of taking detailed account of the electron-electron correlations and the crystal field effects.

2. Ct States and Transitions in Chromites

Slightly distorted octahedral CrO₆ complexes are the main optic centers of oxide compounds such as perovskite-like 𝑅CrO₃ (𝑅 is a trivalent rare-earth or yttrium cation). The ground state configuration of the (CrO₆)⁹⁻ complex includes filled molecular orbitals (MOs) O_1s-2p predominantly anionic in nature, filled “cationic” MOs Cr_1s-3p and 3d-type MOs. The ground state of the complex is the orbital singlet |𝛾⁶²_p𝑡^342𝑔𝐴_2𝑔⟩. The excited CT configuration of the complex arises at the transition of an electron from the “anionic” 𝛾_2p MO (𝛾=𝑡_2𝑢(𝜋),𝑡_1𝑢(𝜋),𝑡_1𝑢(𝜎)) into the 3d-type MO (𝑡_2𝑔, 𝑒_𝑔) of the central Cr³⁺ ion of the complex. The CT configuration can be written as 𝛾̃¹²_p3d⁴ where 𝛾̃¹²_p denotes the hole in the anion MO core. The transition between the ground and excited configurations is the CT transition 𝛾_2p→3d.

The electric-dipole CT transitions from the ground ⁴𝐴_2𝑔 state of the complex are only allowed to the ⁴𝑇_2𝑢 terms of 𝛾̃¹²_p𝑡^42𝑔 and 𝛾̃¹²_p𝑡^32𝑔𝑒_𝑔 configurations (electron transfer to 𝑡_2𝑔 and 𝑒_𝑔 shell, resp.). Among the highest-energy 2p-type MOs, there are two MOs with the 𝑡_1𝑢 symmetry (𝑡_1𝑢(𝜎),𝑡_1𝑢(𝜋)) and one 𝑡_2𝑢(𝜋)-type MO. Thus, there arise six low-energy one-electron CT transitions, namely, 𝑡_2𝑢,𝑡_1𝑢(𝜋),𝑡_1𝑢(𝜎)→𝑡_2𝑔, and 𝑡_2𝑢,𝑡_1𝑢(𝜋),𝑡_1𝑢(𝜎)→𝑒_𝑔. These one-electron transitions give rise to the following electric-dipole allowed many-electron CT transitions of ⁴𝐴_2𝑔→⁴𝑇_2𝑢 type: 𝛾⁶²_p¹𝐴_1𝑔; 𝑡^342𝑔𝐴_2𝑔: ⁴𝐴_2𝑔→𝛾⁵²_p²𝛾_𝑢; 𝑡^432𝑔𝑇_1𝑔: ⁴𝑇_2𝑢 (one-electron 𝛾_2p→𝑡_2𝑔 transition) and 𝛾⁶²_p¹𝐴_1𝑔; 𝑡^342𝑔𝐴_2𝑔: ⁴𝐴_2𝑔→𝛾⁵²_p²𝛾_𝑢; 𝑡^342𝑔𝐴_2𝑔; 𝑒^1𝑔2𝐸_𝑔(^2𝜎+1𝐸_𝑔): ⁴𝑇_2𝑢 (one-electron 𝛾_2p→𝑒_𝑔 transition). Here, 𝜎 takes the values 1 and 2.

The transition intensity measure is the line strength: the squared modulus of the reduced matrix element (RME) of the operator of a transition between states. The RME of the electric dipole moment on many-electron wave functions of the final and initial configurations corresponding to one-electron 𝛾_2p→𝑡_2𝑔 CT transition has the form 𝛾^52p2𝛾_𝑢;𝑡^432𝑔𝑇_1𝑔∶⁴𝑇_2𝑢‖‖𝐝‖‖̂𝛾⁶²_p¹𝐴_1𝑔;𝑡^342𝑔𝐴_2𝑔∶⁴𝐴_2𝑔=3(−1)_𝑗(𝛾){𝑇¹𝑇₁𝑇₂𝛾𝑇₂𝐴₂}𝑡_2𝑔‖‖𝐝‖‖̂𝛾_𝑢.(1) Here, 𝑗(𝛾) is the phase factor which is even for representations 𝛾=𝐴₁,𝐸,𝑇₂ of the cubic group, and odd for 𝛾=𝐴₂,𝑇₁; {^⋯⋯} is the 6Γ symbol. According to (1), three one-electron 𝛾_2p→𝑡_2𝑔 CT transitions beget three many-electron ⁴𝐴_2𝑔→⁴𝑇_2𝑢 CT transitions, the RME on many-electron wave functions being equal to (−⟨𝑡_2𝑔‖𝐝‖̂𝜸_𝑢⟩) both for 𝛾=𝑡₁ and 𝛾=𝑡₂.

In the case of the one-electron 𝛾_2p→𝑒_𝑔 transition, the RME is 𝛾^52p2𝜸_𝑢;𝑡^342𝑔𝐴_2𝑔;𝑒^1𝑔2𝐸_𝑔^2𝜎+1𝐸_𝑔∶⁴𝑇_2𝑢𝑒_0𝑔1‖‖𝐝‖‖̂𝛾⁶²_p¹𝐴_1𝑔;𝑡^342𝑔𝐴_2𝑔;=𝐴^1𝑔−⁴1𝐴)^2𝑔∶⁴_{𝑗(𝛾)+𝜎+1}𝐴_2𝑔√3(2𝜎+1){𝐸𝐸𝐴²𝑇₂𝑇₁𝛾}𝑒^𝑔‖‖𝐝‖‖̂𝜸_𝑢.(2) Three one-electron 𝛾_2p→𝑒_𝑔 transitions give rise to six many-electron ⁴𝐴_2𝑔→⁴𝑇_2𝑢 CT transitions; the RMEs are gathered in Table 1.

Table 1 

Many-electron CT transitions generated by one-electron 𝑡_2𝑢,𝑡_1𝑢(𝜋),𝑡_1𝑢(𝜎)→3d𝑒_𝑔 CT transitions.

These results can be parameterized in the local approximation [13] consisting in the full neglect of many-center integrals when calculating the RME of the dipole moment. All such RMEs can be expressed in terms of 𝜆_𝜎, 𝜆_𝜋; the effective covalency parameters for 𝜎 and 𝜋 bonds. Since the RME ⟨𝑡_2𝑢(𝜋)‖𝐝‖𝑒_𝑔⟩, ⟨𝑡_1𝑢(𝜋)‖𝐝‖𝑒_𝑔⟩, ⟨𝑡_1𝑢(𝜎)‖𝐝‖𝑡_2𝑔⟩ go to zero in the local approximation, it reveals itself as a supplementary selection rule which prohibits the 𝜎→𝜋(𝑡_1𝑢(𝜎)→𝑡_2𝑔) and 𝜋→𝜎(𝑡_1,2𝑢(𝜋)→𝑒_𝑔) transitions even if they are electric-dipole allowed. Only 𝜎→𝜎 type CT transitions (𝑡_1𝑢(𝜎)→𝑒_𝑔) and 𝜋→𝜋 type ones (𝑡_1,2𝑢(𝜋)→𝑡_2𝑔) are allowed in this approximation (‘‘strong’’ transitions, in contrast with ‘‘weak’’ 𝜎→𝜋, 𝜋→𝜎 ones).

3. Modeling of the Ct Transition Band in Chromites

We apply here the expounded theoretical notions to model the optic properties of chromites like 𝑅CrO₃. They are usually regarded as Mott-Hubbard insulators, the Coulomb energy being reputedly small in comparison with the CT energy [3], but this attribution seems to be not universally justified [7]. We will treat the chromites from the CT point of view using the following plausible assumptions for modeling the optic spectra.

(1) Since the 3d⁴ configuration of the central atom formally corresponding to Mn³⁺ ion arises at the CT transition, to compute the Tanabe-Sugano diagram (see below) for the excited 3d⁴ CT configuration we use the Racah parameters 𝐵 and 𝐶 of the Mn³⁺ ion ground state [14] in the octahedral oxygen surroundings.

(2) The corrections to the level energies owing to 2p-hole-3d-shell interaction are neglected. Our estimations [12] show that these corrections may take on the value of several tenths of an eV.

(3) The energies of the nonbonding oxygen states in the (CrO₆)⁹⁻ complex are supposed to coincide with those in the (FeO₆)⁹⁻ complex [15]. This assumption seems justified since the energies of purely oxygen states mainly depend on crystalline environment.

Thus, the energy spectrum of the one-electron CT transitions depends on the energies of 3d 𝑡_2𝑔, 𝑒_𝑔 orbitals and rigidly shifts if the latter are changed. It remains true for many-electron CT transitions as well. The referencing of the computed theoretical spectrum to the experimental one has been performed according to the fundamental optic absorption edge; thereby the positions of all peaks of the computed spectrum become defined.

To find the energies of many-electron CT transitions and the transition intensities under the mixing of CT states in (1) and (2), we have computed the Tanabe-Sugano diagram for d⁴ configuration with parameters 𝐵=0.12 eV and 𝛾=𝐶/𝐵=4.5 [14]. At the crystal field parameter 𝐷𝑞=0.23 eV [14] we obtain an intensive CT band with the total extent of about 8 eV, from ≈2 to ≈10 eV: each of the one-electron CT transitions 𝑡_2𝑢(𝜋),𝑡_1𝑢(𝜋),𝑡_1𝑢(𝜎)→𝑡_2𝑔 gives rise to seven many-electron CT transitions, and each of the one-electron CT transitions 𝑡_2𝑢(𝜋),𝑡_1𝑢(𝜋),𝑡_1𝑢(𝜎)→𝑒_𝑔 generates four many-electron CT transitions, altogether 33 lines!

We have performed a modeling of the band of electric-dipole allowed CT transitions in the optic spectrum of orthochromite YCrO₃ (note that in our assumptions the specific compound affects the reference energy only). Since in the local approximation the contribution of transitions 𝑡_1,2𝑢(𝜋)→𝑒_𝑔 and 𝑡_1𝑢(𝜎)→𝑡_2𝑔 cannot be taken into account, we have introduced ‘‘by hand’’ the corresponding covalency parameter 𝜆_𝜋𝜎≡𝜆_𝜎𝜋. Based on reasonable physical notions and the concept of strong and weak transitions, we assumed the following model relationship of covalency parameters: 𝜆_𝜋=(1/5)𝜆_𝜎, 𝜆_𝜋𝜎=(1/10)𝜆_𝜎. The line half-width Γ is taken to be equal to 0.4 eV for many-electron CT transitions due to weak transitions, 0.8 eV for those due to strong 𝑡_1,2𝑢(𝜋)→𝑡_2𝑔 transitions, and 1.2 eV for those due to strong 𝑡_1𝑢(𝜎)→𝑒_𝑔 transitions. The model Lorentzian line shape used is4𝜔𝜔₀Γ𝜔²−𝜔²⁰−Γ²²+4𝜔²Γ²,(3)𝜔₀ being the transition frequency.

The computation results are shown in Figure 1. The computed total contribution of all electric-dipole-allowed strong and weak CT transitions to the imaginary part 𝜀₂ of the diagonal permittivity tensor component 𝜀=𝜀₁+𝑖𝜀₂ is presented in the main panel. The experimental spectra for Lu_0.85Y_0.15CrO₃ [16], YCrO₃ [17], and LaCrO₃ [17] converted by us from optic conductivity to permittivity are shown in the inset (lines 1, 2, and 3, resp.). The computation results reasonably agree with them. The 3D picture of 𝜀₂ spectra versus the crystal field parameter 𝐷𝑞 is shown in Figure 2. Variation of 𝐷𝑞 causes not merely the relative change of peaks height, but also the restructuring of the whole spectrum.

Figure 1 

Modeling of O2p-Cr3d CT band in chromite. The computed total contribution of electric-dipole allowed CT transitions to Im𝜀 and (inset) the experimental spectra of Im𝜀 in Lu0.85Y0.15CrO3 [16], YCrO3 [17], and LaCrO3 [17] (lines 1, 2, and 3, resp.) are shown.

Figure 2 

Theoretical modeling of O2p-Cr3d CT band in chromite. The spectral dependence of all electric-dipole allowed CT transitions contribution to Im𝜀 versus the crystal field strength.

4. Conclusion

In the cluster approach which combines the crystal field and MO notions, we have considered the peculiarities of CT states and O_2p→Cr_3d CT transitions in the octahedral (CrO₆)⁹⁻ complex. Using the Racah algebra for the group of rotations and the cubic point group, we have calculated the reduced matrix elements of the electric dipole operator on many-electron wave functions being the initial and final states of a CT transition. Based on the parameterization of the results obtained, we have calculated the relative intensities of the allowed CT transitions. Within the crystal field theory, we have computed the energies of many-electron CT transitions.

Modeling the optic spectrum of chromites yielded an intricate CT transition band consisting of 33 lines, with the main maxima at about 6.8 and 9.1 eV and satellites in the 2–10 eV range. The total extent of the CT band is about 8 eV. The influence of crystal field magnitude on the spectrum structure is examined.

For all the roughness of the model assumptions, the computed spectrum is in satisfactory agreement with the experimental ones. Hence it appears that the rooted opinion of the CT transitions as having a simple, one- or two-peaked structure is groundless. There is a need for refining upon the customary theoretical treatment of peculiarities in the optic spectra of 3d metal-based oxide compounds.

Acknowledgment

This work is supported by the Russian Foundation of Basic Research, Grant no. 07-02-96036.