Charge/orbital disordered states with smaller volume and higher entropy in transition-metal oxides

INTRODUCTION

Transition-metal oxides exhibit surprisingly rich electrical, magnetic, and structural properties due to the correlated d electrons^[1,2]. The d orbitals with five-fold degeneracy in the atomic limit are split into three-fold degenerate t_2g (xy, yz, and zx) and two-fold degenerate e_g (3z²-r² and x²-y²) orbitals under the cubic ligand field. When the number of d electrons per transition-metal site is integer, the d electrons can be localized with the strong on-site electron-electron interaction (Mott insulators). A typical phase diagram of such Mott insulators is illustrated in Figure 1A. For example, pyrite-type NiS₂ exhibits a pressure-induced phase transition from a Mott insulator to a paramagnetic metal^[1,2]. The paramagnetic metallic phase with itinerant d electrons has smaller volume than the Mott insulating phase with localized d electrons. In the Mott insulating phase, the localized spins tend to order antiferromagnetically (sometimes ferromagnetically) unless strong frustration effect sets in due to lattice geometry or other degrees of freedom. In NiS₂ with a face-centered cubic lattice, the Ni²⁺ ion (d⁸ configuration with S = 1) does not have orbital degrees of freedom, and the S = 1 spins are antiferromagnetically ordered at low temperatures. In the paramagnetic insulating phase, the disordered spins can provide entropy of k_Blog3 per Ni site. Since the symmetry of the paramagnetic insulating phase is the same as that of the paramagnetic metallic phase, their phase boundary is terminated at the critical point. In such a case without orbital degrees of freedom, the paramagnetic insulating phase with relatively large volume is stabilized at high temperatures due to the spin entropy k_Blog3 per Ni relative to the paramagnetic metallic phase without it. Therefore, it is rather difficult to realize negative thermal expansion (NTE). On the other hand, in Mott insulators with orbital degrees of freedom, magnetic ordering temperature is usually lower than the orbital ordering temperature^[1,2]. This suggests that the paramagnetic insulating phase can be accompanied by orbital order which is driven by electron-electron and/or electron-lattice interaction. If the orbital order survives up to the insulator-metal transition temperature, the symmetry of the paramagnetic insulating phase differs from that of the paramagnetic metallic phase which can be stabilized at high temperatures [Figure 1B]. A layered perovskite Ca₂RuO₄ is one of such examples in which orbital degrees of freedom play a vital role at the insulator-metal transition^[3], and, indeed NTE in the temperature range 150-700 K was reported in Sn-doped Ca₂RuO₄ by Takenaka et al.^[4]. When the charge-transfer energy between the transition-metal d orbitals and the oxygen 2p orbitals becomes close to zero or negative, charge degrees of freedom may play significant roles. Perovskite-type RNiO₃ (R = rare earth) indeed has almost zero or negative charge-transfer energy and the metal-insulator transition is assigned to 2Ni³⁺(d⁸L)→Ni²⁺(d⁸) + Ni⁴⁺(d⁸L²) charge disproportionation rather than Mott localization^[5]. Here, L represents a hole in the O 2p orbitals. In the case of BiNiO₃, the oxygen 2p hole is bound to the Bi site, and, consequently, the valence state of Bi³⁺_0.5Bi⁵⁺_0.5Ni²⁺O₃ is realized^[6]. The electronic configuration of Bi⁵⁺ is s²L² rather than s⁰. Such oxygen 2p charge degrees of freedom are involved in the insulator-to-metal transition of BiNiO₃ and its sister materials. The high-temperature metallic phase with Bi³⁺Ni³⁺O₃ has smaller volume than the insulating phase of Bi³⁺_0.5Bi⁵⁺_0.5Ni²⁺O₃, providing the NTE in La-doped BiNiO₃ with a temperature range of 320-380 K as discovered by Azuma et al.^[6]. More recently, Pachoud et al. have discovered NTE behaviors with the temperature range of 600-700 K in V₂OPO₄ where both the charge and orbital degrees freedom are important^[7]. The V²⁺/V³⁺ mixed valence and V 3d orbital polarization have been observed by x-ray absorption/photoemission spectroscopy^[8,9]. It has been established that the Bi-Ni charge transfer and the V-V charge transfer are responsible for the NTE of BiNiO₃^[6] and V₂OPO₄^[7].

Figure 1. (A) Schematic phase diagram of NiS₂ as functions of temperature T and pressure P. (B) Schematic phase diagram of Ca₂RuO₄. PI, AFI, PM, and AFM represent paramagnetic insulating, antiferromagnetic insulating, paramagnetic metallic, and antiferromagnetic metallic phases. (C) Energy per unit cell calculated by mean-filed approximation for the layered perovskite-type d-p model with d⁴ as a function of Jahn-Teller distortion (the ratio between the compressed/elongated apical Ru-O bond length and the in-plane Ru-O bond length. F and AF represent ferromagnetic and antiferromagnetic states. With the compression (elongation), the xy (yz or zx) orbitals are doubly occupied.

When the charge/orbital orderings are associated with magnetic ordering and electron-electron interaction, the charge/orbital ordered states at low temperatures tend to have relatively large volume due to the localized d electrons with weak hybridization between neighboring d electrons. When the charge/orbital ordering is driven by formation of spin singlet bonds due to electron-lattice interaction (such as CuIr₂S₄ and MgTi₂O₄), the charge/orbital ordered state at low temperature has smaller volume. In the former case, since the exchange interaction between localized spins depends on the charge and orbital arrangement, the magnetic ordering temperature is usually lower than the charge and/or orbital ordering temperature. There are several types of lattice distortions to stabilize their charge/orbital order: Jahn-Teller distortion for Ca₂RuO₄/V₂OPO₄ and breathing distortion for BiNiO₃. The lattice distortions are removed in the paramagnetic metallic phase with smaller volume, and the additional entropy by charge/orbital fluctuations helps the NTE of Ca₂RuO₄ and BiNiO₃. V₂OPO₄ is unique in that the high-temperature phase can be stabilized by a different kind of lattice distortion: V-V dimerization. In the present work, charge/orbital ordered/disordered states in Ca₂RuO₄, BiNiO₃ and V₂OPO₄ are analyzed based on mean-field calculations on multiband d-p models. Based on the calculated results, the mechanism of NTE of the three systems is discussed.

METHODS

The spin-charge-orbital order/disorder transitions are analyzed by mean-field calculations on d-p models in which the d-d Coulomb interaction is expressed by Kanamori parameters u, u′ and j with u′ = u-2j^[10,11]. The transfer integrals between the transition-metal d and oxygen 2p orbitals are given by Slater-Koster parameters (pds) and (pdp) with the ratio (pdp)/(pds) = -0.45. Following the previous studies on the d-p model for 3d and 4d transition-metal oxides^[11,12], u′ and j are set to 3.0 and 0.5 eV for Ca₂RuO₄, 7.0 and 0.8 eV for RNiO₃, and 4.0 and 0.7 eV for V₂OPO₄. (pds) is set to -2.4, -1.8, and -2.0 eV, and charge transfer energy Δ from the oxygen 2p to transition-metal d orbitals is 2.0, 0.0, and 6.0 eV for Ca₂RuO₄, RNiO₃, and V₂OPO₄, respectively. The transfer integrals are scaled with the bond length by using Harrison’s rule with the exponent of -3.5. The interplay between the orbital ordering and the distortion of the RuO₆ octahedron is studied for Ca₂RuO₄^[12]. As for RNiO₃, following the mean field calculation on the negative U Hubbard model for BiNiO₃^[13], the previous d-p model work for the ground state^[14] is extended to the finite temperature. The interplay between the V-V dimer and the orbital order in V₂OPO₄ is examined based on the previous study on BaV₁₀O₁₅^[15].

RESULTS AND DISCUSSION

Layered perovskite-type Ca₂RuO₄ exhibits an insulator-metal transition around 350 K and an antiferromagnetic transition at 110 K [Figure 1B]^[16-19]. RuO₆ octahedra share their corners forming a square lattice of Ru in the ab plane of the layered perovskite. The Ru 4d t_2g orbitals accommodate four electrons (two holes) in the low spin configuration. In the low (high) temperature phase of Ca₂RuO₄, the RuO₆ octahedron is compressed (elongated) along the c-axis or the z-axis and the xy (yz/zx) orbital is stabilized relative to the yz/zx (xy) orbital among the three t_2g orbitals. As shown in Figure 1C, the orbital occupancy change due to the compression and elongation of the octahedron can be described by mean-filed calculations although the spin and orbital fluctuations are not considered. The compressed case has no orbital degrees of freedom. In the elongated case, the doubly degenerate yz/zx orbitals accommodate one hole per Ru site which may provide orbital entropy of k_Blog2 and spin entropy of k_Blog3 in the localized limit of S = 1. If the Ru 4d yz/zx electrons are fully itinerant, the spin and orbital entropy should be reduced. However, the strong electronic correlation in Ca₂RuO₄ suggests that the orbital contribution may remain even in the metallic phase.

The spin entropy can be reduced even in the paramagnetic insulating phase since the space-time fluctuations of spin and orbital are restricted by Ru 4d spin-orbit interaction^[3]. While the RuO₆ octahedron is strongly compressed in the antiferromagnetic insulating phase, it is almost regular in the paramagnetic insulating phase. When the energy splitting between the xy and yz/zx orbitals becomes comparable or smaller than the spin-orbit interaction, the three t_2g orbitals are mixed to form the complex orbitals of

and

where ↑ and ↓ indicate spin up and down. Under the strong spin-orbit interaction, $$\alpha=\sqrt{2 / 3},~\beta=\sqrt{1 / 3}$$. When the two holes occupy these two orbitals in this limit, the spin and orbital entropy can be quenched. The effect of the spin-orbit interaction is weakened in the metallic phase and the spin and orbital fluctuations can revive. Therefore, the metallic phase with elongated octahedron would have substantial electronic entropy higher than that of the paramagnetic insulating phase. The spin and orbital entropy released by the orbital transition may help the NTE discovered in Ca₂RuO₄^[4].

Perovskite-type RNiO₃ (R = rare earth) becomes insulating at low temperatures due to charge disproportionation of 2Ni³⁺→Ni²⁺+Ni⁴⁺. The metallic phase has smaller volume and is thus stabilized by pressure, as schematically shown in Figure 2A^[20-22]. The charge disproportionation is stabilized by the breathing type distortion of the NiO₆ octahedra, as illustrated in Figure 2B. The effect of the breathing type distortion can be studied by mean-filed calculations on the perovskite-type d-p model. The results [Figure 2C] indicate that, without the distortion, the various magnetic states with orbital orderings are stable. Even the small breathing type distortion limits the orbital state, and the G-type and C-type antiferromagnetic solutions are excluded. With further increase in the distortion, the compressed site becomes nonmagnetic, and the ferromagnetic and A-type antiferromagnetic solutions degenerate in energy. The O 2p-to Ni 3d charge-transfer energy is almost zero or even negative in RNiO₃^[23-25]. As a result, the electronic configuration of Ni³⁺ is given by d⁸L hybridized with d⁷. For simplicity, the Ni³⁺ state with d⁸L-d⁷ hybridization is described as d⁸L in this article. Here, it should be noted that the oxygen hole states represented by L are constructed from the O 2p orbitals in the NiO₆ octahedron and have the same orbital symmetry as the Ni 3d orbitals. Therefore, the d⁸L state (hybridized with d⁷) cannot avoid the Jahn-Teller instability for the low spin d⁷ configuration where one of the doubly degenerate e_g orbitals is occupied by one electron. In the case of the low spin d⁸L state, one of the doubly degenerate L states with e_g symmetry is occupied by one hole [Figure 2D]. The charge disproportionation corresponds to oxygen hole transfer to the Ni⁴⁺ site where the actual electronic configuration is nonmagnetic d⁸L² [Figure 2E]. With the charge disproportionated state of d⁸ and d⁸L², the NiO₆ octahedron undergoes breathing distortion rather than Jahn-Teller distortion. The d⁸L² state (hybridized with d⁷L and d⁶) of Ni⁴⁺ is also identified in SrNiO₃/LaFeO₃ heterostructure^[26].

Figure 2. (A) Schematic phase diagram for RNiO₃ as functions of temperature T and pressure P. PI, AFI, PM, and CD represent paramagnetic insulating, antiferromagnetic insulating, paramagnetic metallic, and charge disproportionated phases. (B) Distortions of the NiO₆ octahedron and the electronic configurations for the charge disproportionated Ni²⁺ and Ni⁴⁺ sites without oxygen 2p holes. The yellow and red circles indicate oxygen and transition metal ions, respectively. (C) Energy per unit cell for ferromagnetic and antiferromagnetic (A, G, and C type) states calculated by mean-filed approximation for the perovskite-type d-p model with d⁷ as a function of breathing distortion (the ratio between the long and average Ni-O bond length). Without the distortion, each site takes the low spin d⁷ state with S = 1/2. With the distortion, the expanded site becomes d⁸ (S = 1) and the compressed site becomes d⁶ (S = 0). (D) Electronic configuration of d⁸L. (E) Electronic configuration of d⁸L².

Since the phase diagram of RNiO₃ resembles that of Ca₂RuO₄, RNiO₃ itself may exhibit NTE. However, the Ni-O bond length is considerably shortened in the Ni⁴⁺O₆ octahedron due to the low-spin d⁸L² configuration which is hybridized with the low-spin d⁷L and d⁶ configurations. The volume of the charge-disproportionated insulating state is only slightly larger than that of the metallic state, and the NTE effect would be limited. Instead, Azuma et al. have established that a sister material BiNiO₃ exhibits one of the best NTE performances^[6]. In the insulating phase of BiNiO₃, the oxygen hole of Ni³⁺(d⁸L) is taken by Bi and the Ni valence becomes +2^[18]. As a result, the valence state of Bi³⁺_0.5Bi⁵⁺_0.5Ni²⁺O₃ is realized where the configuration of Bi⁵⁺ is s²L² rather than s⁰. Since the Ni²⁺-O bond length is longer than the Ni⁴⁺-O one, the volume of the insulating BiNiO₃ is larger than that of the charge disproportionated RNiO₃. Also, the Bi 6s² lone pair introduces additional lattice distortions to BiNiO₃. As expected, pressure or chemical substitution (such as Bi_1-xLa_xNiO₃) induces a valence transition from Bi³⁺_0.5Bi⁵⁺_0.5Ni²⁺O₃ to Bi³⁺Ni³⁺O₃ with substantial volume collapse^[6]. This transition can be viewed as oxygen hole transfer from the Bi⁵⁺ site to the Ni³⁺ site, and this picture is indeed supported by the recent ab initio calculations^[27].

In the NTE BiNiO₃, the Ni²⁺-O-Ni²⁺ superexchange interaction is enhanced due to the relatively small but positive charge-transfer energy (about 4 eV), and the magnetic order (or short-range order) can survive near the Ni-to-Bi charge transfer transition under pressure or with chemical substitution. Therefore, the spin entropy of the insulating Bi³⁺_0.5Bi⁵⁺_0.5Ni²⁺O₃ tends to be suppressed. The metallic Bi³⁺Ni³⁺O₃ can be viewed as a charge-disordered metallic state of the charge-disproportionated RNiO₃. Theoretically, the phase diagram has been studied using mean field calculations on a negative U Hubbard model^[13]. In the charge-disordered metallic state, oxygen holes are dissociated from the Ni and Bi sites. Most probably, in the metallic phase of Bi³⁺Ni³⁺O₃ and Bi³⁺_1-xLa³⁺_xNi³⁺O₃, the d⁸L state of Ni³⁺ is disassembled into d⁸ at the Ni site and L. While the oxygen hole L is responsible for the metallicity, the localized d⁸ state can provide spin entropy of k_Blog3 per Ni site. This picture is too simplified because the hybridization between the d⁸L and d⁷ configurations and the band formation are neglected. In the metallic phase, the spin entropy of the localized limit should be reduced due to the d⁸L-d⁷ hybridization and the band formation. However, substantial spin and charge entropy can remain in the metallic state where the bandwidth is reduced to the order of 0.1 eV by the strong electron-electron and electron-lattice interaction. In this sense, the spin degrees of freedom of Ni 3d⁸ state and the charge degrees of freedom of oxygen hole are correlated in BiNiO₃ providing the NTE^[6,28].

V₂OPO₄ consists of face-sharing and edge-sharing VO₆ octahedra and is supposed to have a body-centered tetragonal unit cell^[7,29]. However, it undergoes V²⁺/V³⁺ charge ordering at 605 K with monoclinic lattice distortion. The face-sharing V²⁺ and V³⁺ sites form chains along the [110] direction of the monoclinic lattice which corresponds to the a-axis of the tetragonal lattice of the high-temperature phase. The corner-sharing V³⁺ sites are connected approximately along the [001] direction of the monoclinic lattice which is inclined by ~30 degrees relative to the c-axis of the tetragonal lattice. The face-sharing V²⁺ and V³⁺ chain along the [110] direction is illustrated in Figure 3A. The x, y, and z axes are along the V-O bonds. In the present work, a d-p model with a face-sharing octahedron chain is employed. The ferrimagnetic charge-ordered (CO) state is stable only when the V-V bond is elongated or the VO₆ octahedron is elongated along the chain. The spin difference between the neighboring V sites is about 0.6. When the octahedron is close to the regular shape, there is no stable solution. Interestingly, another mean-field solution appears when the V-V bond is substantially compressed. In this solution, the spin difference between the neighboring V sites is as small as 0.1, indicating formation of the molecular orbital (MO) as illustrated in Figure 3B. The calculated energies are plotted as a function of compression/elongation along the V-V bond [Figure 3C].

Figure 3. (A) Ferrimagnetic CO state viewed along the face-sharing VO₆ chain and from the side of the chain. The yellow circles indicate oxygen ions. The x, y, and z axes are along the V-O bonds. (B) Ferromagnetic MO state viewed from the side of the chain. The a_1g orbitals form the MO. (C) Energy per unit cell (open triangles) calculated by mean-filed approximation for the face-sharing octahedron d-p model with d^2.5 as a function of compression/elongation along the V-V bond (ratio between the compressed/elongated V-V bond length and the original V-V bond length).

In the real system, the VO₆ octahedron of V³⁺ is compressed along the z-axis in the CO phase. Therefore, the xy orbital is lower in energy than the yz/zx orbitals. Considering the Hund coupling, one of the yz/zx orbitals is occupied at the V³⁺ site as shown in Figure 3A. Such orbital order is consistent with polarization dependence of V 2p X-ray absorption spectrum^[9]; consequently, the orbital fluctuation is quenched. V₂OPO₄ exhibits a ferrimagnetic transition at 165 K. In the ferrimagnetic CO phase, V²⁺ and V³⁺ spins are antiferromagnetically arranged along the face-sharing bond. The antiferromagnetic coupling is consistent with the orbital state in Figure 3A since all the xy, yz, zx electrons are equally transferred to the neighboring site via the V-V and V-O-V pathways in the face-sharing V₂O₉ cluster. Above 165 K, the magnetic susceptibility shows a paramagnetic moment of 1.61 μ_B per V₂OPO₄ unit. The reduction from the ionic values for V²⁺ and V³⁺ spin suggests the effect of spin-orbit interaction or partial singlet bond formation. In the high-temperature phase, the charge order is destroyed and the compression of the V³⁺O₆ octahedron disappears. With the face-sharing geometry, the trigonal ligand field becomes important and the t_2g orbitals are split into a_1g and e_g^π orbitals. In such a situation, the a_1g orbitals can form bonding and antibonding orbitals [Figure 3B]. This situation is similar to those of Ti₂O₃^[30,31] and BaV₁₀O₁₅^[15]. Since the bonding orbital is occupied by one electron (two electrons) in the ferromagnetic (ferrimagnetic) state, the V-V bond length is shortened which is consistent with the reduction of the c-axis lattice constant above the transition temperature. On the other hand, the e_g^π electrons remain localized with spin, charge, and orbital fluctuations. In the ferrimagnetic arrangement, the four e_g^π orbitals of the face-sharing V₂O₉ cluster accommodate three electrons providing entropy of k_Blog4 per cluster. In addition, the spin 1/2 and the spin 1 have k_Blog2 and k_Blog3, respectively. Since the tetragonal phase above 605 K with average V^2.5+ valence has a smaller volume than the monoclinic phase with V²⁺ and V³⁺ charge order, NTE is realized around 605 K, as reported by Pachoud et al.^[7,29]. The NTE in V₂OPO₄ is driven by the charge transfer between V²⁺ and V³⁺ sites which is coupled with spin and orbital degrees of freedom and the orbital dependent bond formation. The a_1g-a_1g bonds keep the smaller volume while the spin, charge, and orbital fluctuations of the e_g^π electrons provide entropy higher than the charge/orbital ordered phase.

CONCLUSIONS

Based on the mean-field calculations on the d-p models, the relationship between the charge/orbital states and the lattice distortion has been discussed for Ca₂RuO₄, RNiO₃, and V₂OPO₄. The changes of the charge/orbital states by the lattice distortion suggest their disordered states harbor relatively high entropy due to charge/orbital fluctuations. Across the insulator-metal transition in Ca₂RuO₄, the RuO₆ octahedron is elongated and the yz/zx orbital fluctuation becomes relevant in the metallic phase. The yz/zx orbital fluctuation in the metallic phase is intimately related to the orbital-dependent band renormalization in the metallic phase of Ca_2-xSr_xRuO₄^[32]. Above the transition temperature of Ca₂RuO₄, the strongly renormalized Ru 4d electrons (bandwidth ~0.1 eV) are almost incoherent and exhibit localized character. The insulator-metal transition in BiNiO₃ is accompanied by the oxygen hole transfer from Bi to Ni. In V₂OPO₄, the charge/orbital disordered state has relatively small volume due to the a_1g-a_1g bond in the face-sharing octahedra while the spin, charge, and orbital fluctuations of the e_g^π electrons provide higher entropy than the charge/orbital ordered state. In Ca₂RuO₄ and V₂OPO₄, the orbital-dependent fluctuations of the t_2g electrons play key roles in realizing the NTE behaviors. While the orbital order/disorder is coupled with the Jahn-Teller distortion in the corner-sharing octahedra of Ca₂RuO₄, it is governed by the metal-metal dimerization in the face-sharing octahedra of V₂OPO₄. Such t_2g electron systems with orbital ordering can be candidates for new NTE materials. Orbital and/or charge states are rearranged across their insulator-metal transitions of Ca₂RuO₄ and BiNiO₃, and the metallic phases with orbital and/or charge fluctuations can be stabilized at high temperatures relative to the insulating phases without them. The negative charge-transfer energy and the oxygen 2p holes are responsible for the charge-transfer mechanism and provide the charge fluctuations in the high-temperature phase of BiNiO₃. In Ca₂RuO₄, the Jahn-Teller distortion gradually develops below the transition temperature due to the spin-orbit coupling^[3]. The spin-orbit interaction would be important in the extremely wide temperature range of NTE in Ca₂RuO₄.