Perspective on Engineering Transition-Metal Oxides - American

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A Perspective on Engineering Transition-Metal Oxides John B. Goodenough Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/cm402063u • Publication Date (Web): 20 Aug 2013 Downloaded from http://pubs.acs.org on August 27, 2013

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Chemistry of Materials

Perspective on Engineering Transition-Metal Oxides

John B. Goodenough*

*

Texas Materials Institute, The University of Texas at Austin, 204 E. Dean Keeton, C2200, Austin, Texas 78712,USA, e-mail: [email protected], phone: 512-471-1646

ACS Paragon Plus Environment


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Abstract: Engineering transition-metal oxides depends on understanding a few general concepts. Three of these are discussed: (1) orbital mixing and the roles of cation-d/O-2p covalent bonding as distinct from on-site cation-orbital hybridization; (2) cooperativity in ordering (a) localized orbitals to remove an orbital degeneracy, (b) ferroic atomic displacements and (c) bond lengths in a charge-density wave; (3) cation-site expansion at the crossover from itinerant to localized electron behavior. The latter can stabilize a first-order transition to a ferromagnetic metallic phase on the approach to crossover from the itinerant-electron side or, in a single-valent compound, an intermediate charge-density-wave phase on the approach to crossover from either the localized- or itinerant-electron side. In a mixed-valent compound, a two-phase segregation at a first-order cross-over may be static or mobile, and a mobile second phase may become ordered at low temperature to stabilize high-temperature superconductivity.

Key Words:

Orbital mixing, orbital ordering, charge-density waves, metallic ferromagnetism Introduction Transition-metal oxides exhibit a wide range of technically important electronic phenomena: these include magnetic, dielectric and ferroelectric, catalytic, superconductive, redox, thermoelectric, and multiferroic properties, insulator-metal transitions, charge-density waves, and mixed electronic/oxide-ion conduction. Engineering these oxides to provide a material that optimizes a given application or illustrates how these oxides are transformed under pressures that increase to that of Earth's lower mantle is a vast topic. All materials engineering depends on understanding a few principal concepts. In this brief perspective, three principal concepts are highlighted: the role of the covalent component of the d-orbital bonding, cooperative static and dynamic d-orbital ordering, and the character of the transition from localized to itinerant d-electron behavior.


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Atomic Orbital Mixing

(1) Types of Mixing. There are two types of orbital mixing, intraatomic hybridization and interatomic covalent bonding. Both increase the strength of bonding between atoms. Intraatomic hybridization is most commonly associated with mixing of outer s and p orbitals of the same principal quantum number n > 1 that are separated in energy ∆l by a different screening from the nuclear charge by core electrons; ∆l increases with the atomic number. However, d-orbital mixing with the s and p orbitals can also occur. Intraatomic orbital mixing increases the overlap of bonding orbitals on neighboring atoms. Interatomic covalent bonding involves an electron transfer between two atoms; the expectation of an electronic transfer is the integral bij ≡ , ij(ψi,ψj)

(1)

where εij is a one-electron energy and ' is the perturbation of the atomic potential at atomic position Rj by a neighboring atom (like or unlike) at position Ri; (ψi,ψj) is the overlap integral of the bonding orbitals on neighboring atoms. Two situations are distinguished: bonding between like atoms in like lattice potentials and bonding between unlike atoms or like atoms in unlike lattice potentials.

(2) Bonding between like atoms in like lattice potentials: Where (ψi,ψi) is large, the electron transfer between like atoms in like lattice potentials costs no energy and can therefore be treated in first-order perturbation theory. This condition is assumed to be the case in the molecular orbital theory (MO) of a homopolar bond and in independent-electron broad-band theories of bonding in solids. In this model, each electron belongs equally to all bonded atoms (i.e. are itinerant) and the mixed orbitals are linear combinations of atomic orbitals (LCAOs). In a diatomic homopolar bond, the bonding states are stabilized and the antibonding states are destabilized by b. The electron-electron coulomb interaction energies of a multielectron bond or of a broad band of one-electron states are incorporated into the atomic potentials; the width of a band of one-electron states is Wb ~ zb, where z is the number of nearest like neighbors with a bij = b. However, where the (ψi, ψj) are small, intraatomic electron energies may dominate the


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interatomic interactions to trap the electrons by a site expansion in localized-electron configurations. The 4fn configurations remain localized in an oxide whereas outer s and p electrons are normally itinerant. The transition-metal d electrons of a lattice of like transitionmetal cations in an oxide may be localized or itinerant, which opens the possibility of engineering transition-metal compounds at the crossover from localized to itinerant d-electron behavior. Localized-electron configurations are Mm+1/Mm+ and Mm+/M(m-1)+ redox centers separated from one another by a finite on-site electron-electron coulomb energy U. This MottHubbard energy separation needs to be distinguished from an energy gap Eg opened in a band of itinerant one-electron energy states by a change of the translational symmetry of like atoms providing a narrow metallic bond. (3) Covalent bonding is distinguishable from homopolar and itinerant-electron bonding by an energy gap ∆E between the donating and receiving orbitals of the two atoms of a bond. In oxides, the ionic component of the bonding introduces a large (ca. 5 eV) separation between O2p and M-s and p orbitals, but between O-2p and empty M-d orbitals, ∆E varies greatly and may vanish; moreover, a ∆Ε = U between localized dn and d(n+i) redox energies results in covalent d-d bonding with spin-spin superexchange interactions between like atoms in like lattice potentials. In a covalent bond, the occupied bonding orbitals are stabilized and the antibonding orbitals are destabilized by ∆ε = b2/∆E

(2)

and the orbital mixing is

ψ = N(ψo - λbϕ)

(3)

where ψo is the unperturbed wave function, N is the normalization constant, and λb ≡ b/∆E is the covalent-mixing parameter; ϕ is a symmetrized unperturbed wave function of all the z near neighbors that form bonding overlap integrals. The electron transfer in a covalent bond changes the charges on the atoms, but it does not introduce mobile charges as occurs where electrons are excited across an Eg. Therefore, covalent bonding is said to introduce a virtual charge transfer.


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Roles of O - 2p and M - d-Covalent Bonding (1) Effect of ionic bonding. Oxides contain primarily ionic bonds; electron transfer from a metal atom to a separated oxide ion costs an energy EI, but the electrostatic Madelung energy EM > EI between the charged ions of a solid creates an internal electrostatic field that stabilizes the O - 2p and raises all the M-atom orbitals to where the Fermi energy EF is above the top of the O-2p bands. Although virtual electron transfer back from filled oxide orbitals to the M atoms by covalent bonding reduces the ionic charges, which reduces EM, it increases the splitting of bonding oxide-ion and antibonding M-ion states to compensate for the loss of EM. In transitionmetal oxides, d orbitals as well as s and p orbitals may be raised above the top of the O-2p bands by EM. The covalent orbital mixing between the antibonding M-d and bonding oxide-ion orbitals, Eq.(3), has several important consequences for d-block transition-metal oxides: (2) Crystal-field splittings of the d orbitals are primarily due to differences in their overlap integrals with O - 2p orbitals[1a]. Note: Early electrostatic arguments with a point-charge model conserved the energy of only the d-state manifold, but symmetry arguments were correct so long as the crystal-field splittings were adjustable parameters. The five atomic d wave functions ψ = Rn,l (θ,ϕ) have the angular dependencies for quantum number l = 2: ψ0 ~ (3cos2θ - 1) = (3z2 - r2)/r2 ψ±1 ~ sin2θ exp(± iϕ) = 2(yz ± izx)/r2

(4)

ψ±2 ~ sin2θ exp (± i2ϕ) = [(x2 -y2) ± i2xy]/r2 In an oxide, the nearest-neighbor M-O interactions are introduced before the d - d interactions are considered. From Eqs. (2) and (3), M-O covalent bonding in an octahedral site raises the two antibonding σ orbitals of e symmetry, (3z2 - r2) = [(z2 - y2) + (z2 - x2)] and (x2 - y2), above the three antibonding π orbitals of t symmetry, xy and (yz ± izx), by ∆c = ∆M + ( - ) ∆Ep + ∆Εs

(5)

where ∆M is a small electrostatic term of uncertain sign, ∆Εp and ∆Εs are the energies separating the M - d and O-2p and 2s orbitals, and λσ, λπ are the covalent mixing parameters. Since the z


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component of the angular-momentum operator is Lz = -i/ ϕ, it follows that an angular momentum is only associated with orbitals having an imaginary component. Therefore, the two e orbitals ψe = Nσ(fc − λσϕσ) carry no angular momentum (ml = 0,0) whereas the three t orbitals ψt = Nπ(ft − λπϕπ) have an ml = 0, ± 1. In a tetrahedral site, ∆c is smaller and the t orbitals are raised above the e orbitals. With more than one d electron on a free ion, the Hund intraatomic exchange energy ∆ex responsible for the atomic highest spin-state rule stabilizes majority-spin states relative to minority-spin states. In an oxide, the transition-metal cations retain a high-spin state if ∆ex > ∆c, but they have a low-spin state if ∆ex < ∆c. The crystal-field splitting and the spin state determine the cation preference energy for an octahedral versus a tetrahedral site.

(3) Interatomic d - d bonding across an oxide ion is made possible by the mixing of O-2p orbitals into orbitals of d-electron symmetry, Eq.(3). M orbitals of d-electron symmetry on opposite sides of an oxide ion may share the same O-2p orbital to give an interatomic overlap of orbitals of d-wave symmetry and, therefore, a d-d charge-transfer integral b across the oxide ion. The fraction of O-2p character, and hence the magnitude of the d-d overlap integral across an oxide ion, increases with a decrease of the energy ∆E between the d-electron LUMO and the top of the O-2p bands. As the redox couple of a 3dn localized-electron LUMO approaches the top of the O-2p bands, the fraction of O-2p character in the states of d-electron symmetry increases to where the d-d overlap across an oxide ion can become strong enough to make 3d electrons itinerant. This situation is illustrated in the metallic perovskites and in the oxide cathodes of a Liion battery. Pinning of a redox energy at the top of the O-2p bands makes ∆E = 0, and the O-2p character of the d orbital of Eq.(3) becomes more and more dominant as the couple is oxidized. At a critical oxidation state, the holes in the narrow band of d-orbital symmetry may become trapped in O−O bonds to create surface peroxide ions (O2)2− with subsequent loss of O2. The approach to oxygen loss may prove to be an important contribution to the catalysis of the oxygen-reduction and/or oxygen-evolution reaction on an oxide. The larger radial extension of the 4d and 5d orbitals compared to the 3d orbitals can introduce strong enough O−2p mixing for M-O-O-M interactions to give itinerant electrons in bands of d-orbital symmetry where a 4d or 5d redox couple is still above the O−2p bands. Where a redox couple approaches or becomes pinned at the top of the O−2p bands, Zaanen, Sawatzky, and Allen[1b] have interpreted their spectroscopic data in terms of electron


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excitations from O−2p to d orbitals, which conforms to the ∆l = ± 1 selection rule, with an O−2p bandwidth Wo narrow enough for successive oxygen redox energies to be split by U > Wo. This important finding does not, however, alter the general discussion above on the origin of narrow bands of d symmetry where a redox couple approaches or becomes pinned at the top of an O−2p band. In the absence of a d-electron redox couple, oxidation of an oxide-ion ligand results in the formation of peroxides as in BaO2, not metallic O−2p bands.. (4) Site expansion on localization of a dn manifold. Where ∆E is larger, a smaller λb in Eq. (3) makes the fraction of O-2p character in the orbital of d-electron symmetry too small for electron transfer across an oxide ion to compete with the intraatomic energies that stabilize localized d-electron manifolds dn having a localized spin S; the dn manifolds are an energy U below the empty dn+1 redox manifold. Localization of the dn electrons also occurs in the absence of 180° M-O-M interactions where the direct d-d overlap is too small. Localization of electrons to an atomic site is accompanied by an expansion of the equilibrium M-O bond length; the transition to itinerant electrons by a vanishing of ∆E is accompanied by a shrinkage of the equilibrium M-O bond length. The observation of an equilibrium

(M-O)loc > (M-O)itin

(6)

can be understood from the Virial Theorem for particles in a central-force field

2 + = 0

(7)

where is the mean kinetic energy of the particles and is their mean potential energy. The for localized electrons in stationary states increases as their mean distance from the nucleus decreases. Since the atomic potential is negative, a 2 = || means that localizing electrons to an atomic site requires increasing ||. Since the d states are antibonding and the O-2p are bonding, increasing || requires decreasing the M:d -O:2p covalent contribution to the M-O bond by expanding the equilibrium M-O bond length. (5) Covalent bonding between localized dn manifolds. Covalent-bonding, both direct and across an oxide ion between localized dn manifolds having a localized spin transforms the bij of Eq. (2) to a tij = bij f( θij/2), where θij is the angle between the interacting spins and the energy ∆E


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= U[2]. For ferromagnetic coupling, tij = bijcos (θij/2) and for antiferromagnetic coupling, tij = bijsin(θij/2). Since the spin angular momentum is conserved in an electron transfer, this superexchange spin-spin coupling then has the form of the Heisenberg exchange energy ∆Espin = −2JijSiSj

(8)

Coupling between half-filled orbitals gives a Jij < 0 (antiferromagnetic) since the Pauli Exclusion Principle restricts transfer of an electron to one that is antiparallel to the spin on the receiving atom. Coupling between a half-filled and an empty or full orbital on an atom that retains a localized spin in other orbitals gives a Jij > 0 (ferromagnetic) since the intraatomic ferromagnetic Hund coupling Jintra favors, by a factor Jintra/U, transfer of an electron having a spin parallel to the spin on the receiving atom. These rules for the sign of superexchange coupling have been proven to hold over a wide range of compounds. (Note: On the other hand, the double exchange mechanism of Zener[3] and as amended by De Gennes[4] is invalid; where it has been applied[3,4], the predicted[4,5] paramagnetic Weiss constant θ 1) with the same [Mn2]O4 framework raises the Mn(IV)/Mn(III) redox couple by 1 eV[8].


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(7) Ionic radii tabulated by Shannon and Prewitt[9] were obtained by measuring M-O bond lengths, by assuming ionic M-O bonding, and by taking the radius of the O2- ion to be 1.40 Å. In fact, the equilibrium M-O bond lengths depend on the localization of the d electrons as well as on the fraction of the bonding that is covalent, and this fraction varies with λb of Eq. (3). The tabulated ionic radii are a useful guide, but a quantitative equilibrium M-O bond length may not be the sum of the tabulated radii. (8) Ferroic displacements change a symmetric O-M-O bond to an asymmetric bond in order to increase the net energy gain by covalent bonding, particularly of O-2p orbitals to empty M-d orbitals or to an empty hybrid 6s-6p orbital on Pb(II) or Bi(III) ions.

Cooperative Static and Dynamic d-orbital Ordering

Where the d-orbital crystal-field splitting by O-2p covalent bonding to M-d orbitals leaves a d-orbital degeneracy, the degeneracy may be removed by a local Jahn-Teller site deformation; the site distortion increases with the strength of the M:d-O:2p covalent bonding and the cooperativity between site distortions. In transition-metal oxides, static local distortions occur at very low temperatures unless they are cooperative. Cooperative local site distortions lower the crystal symmetry below an orbital-ordering temperature Tt that may be well-above room temperature for degenerate σ-bonding orbitals: Tt is lower for degenerate π-bonding orbitals. Cooperativity requires a coupling between locally distorted sites. Where any orbital angular momentum is quenched by the site symmetry, an elastic coupling is needed; where the orbital angular momentum is not completely quenched by the site symmetry, spin-orbit coupling can give a cooperative distortion where atomic spins are ordered collinearly below a magneticordering temperature. In a paramagnetic phase, cooperative elastic coupling requires a site distortion that quenches any orbital angular momentum. The spinels Zn[Mn2]O4 and Li[Mn2]O4 both contain high-spin Mn(III) : t3e1 configurations in octahedral sites, but the concentration of Mn(III) in Zn[Mn2]O4 is twice that in [Mn2]O4 where the Mn(IV): t3e0 have no orbital degeneracy. The twofold-degenerate σantibonding e orbitals at Mn(III) bond strongly with oxygen (∆Εp of Eq. (8) is relatively small) and the M: e-O:2pσ overlap integral is relatively large, which makes the Jahn-Teller site distortion strong, and the elastic coupling between distorted sites makes the site distortion


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strongly cooperative in Zn[Mn2]O4; below a Tt = 1025 °C, the spinel is tetragonal with c/a = 1.14 at room temperature[10]. With more dilute Mn(IV), the elastic coupling in Li[Mn2]O4 is weaker, and Tt is lowered to 280 K[11]. On the other hand, the spinel Fe[Ni1-xCoxFe]O4 has an orbital degeneracy only on the high-spin octahedral-site Co(II): t5 e2 with a threefold-degenerate minority-spin . In this spinel system, a cooperative site distortion to c/a < 1 appears below the ferrimagnetic ordering temperature, Tc = Tt, even for a very small x. Here the cooperativity is through Co(II) spin-orbit coupling[12] and the long-range ordering of collinear octahedral-site spins is a result of the strong antiferromagnetic superexchange between half-filled tetrahedral-site t orbitals and octahedral-site e orbitals. In this situation, the sign of the Co(II) site distortion stabilizes the xy orbital to leave a minority-spin electron in a twofold-degenerate orbital (yz ± izx)1β orbital. This twofold degeneracy is removed by spin-orbit coupling, which orients the spins along the [001] axis to give a magnetocrystalline anisotropy that increases in strength with x. Note: the z axis may be any of the axes.) This crystalline distortion is called magnetostriction since it reflects the spin orientation; it is to be distinguished from exchange striction, which is independent of the spin orientation. In the spinel Ni[Cr2]O4, the strong preference of Cr(III) for octahedral coordination (t3e0) versus tetrahedral coordination (e2 t1) forces the Ni(II) into tetrahedral sites where it has a threefold minority-spin degeneracy on its e4t4 manifold. This spinel has a weak Ni-Cr superexchange interaction, so below a Tt > Tc, the cooperative distortion is to tetragonal c/a > 1. Here stabilization of the xy orbital quenches the orbital angular momentum by leaving the (yz ± izx)β orbitals empty. Quenching the orbital angular momentum removes coupling of the orbital ordering to the disordered spins. However, substitution of Fe(III) for Cr(III) in Ni1-xFex[Cr2xNix]O4

allows the Ni(II) to occupy its preferred octahedral sites where Ni(II) : t6e2 has no orbital

degeneracy. As a result, Tt decreases and Tc increases until they cross; a Tt ≤ Tc makes the tetragonal distortion c/a < 1 in order to optimize the spin-orbit coupling on the tetrahedral-site Ni(II), see Fig. 1[13]. These static crystalline distortions resulting from cooperative localized-orbital ordering are well-known. Less widely appreciated is the fact that dynamic Jahn-Teller distortions above the long-range orbital ordering temperature Tt can induce chemical inhomogeneities where the site distortions fluctuate slowly enough to stabilize short-range like-atom clustering through local


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cooperativity at temperatures where the Jahn-Teller ions are mobile. Advantage of this phenomenon was taken in the 1950s in order to develop a squarish B-H hysteresis loop in the ferromagnetic memory elements of the first random-access memory (RAM) of the digital computer[14]. Static orbital order determines whether a particular d-d bond couples two half-filled or a half-filled and an empty or full orbital, thereby determining the magnetic order below the magnetic-ordering temperature. This situation is illustrated in the perovskite system La1-xYxTiO3. In an AMO3 oxide with the perovskite structure and a geometric tolerance factor t ≡ (A-O)/√2 (M-O)] < 1, a cooperative rotation of the corner-shared MO6/2 octahedral sites occurs. A rotation about the cubic [1-10] axis produces an orthorhombic structure; the symmetry of the octahedral site is distorted in the process[15], and this distortion biases orbital ordering to remove an orbital degeneracy. The site distortion of an orthorhombic (s.g. Pbnm) perovskite consists of two components: (1) an orthorhombic component with two long and two short M-O bonds alternating in the (001) plane and a medium M-O bond length in the site [001] direction and (2) a rhombohedral component with an

Perspective on Engineering Transition-Metal Oxides - American

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