Fingerprints of Spin-Orbital Physics in Crystalline O$_2$

The alkali hyperoxide KO$_2$ is a molecular analog of strongly-correlated systems, comprising of orbitally degenerate magnetic O$_2^-$ ions. Using first-principles electronic structure calculations, we set up an effective spin-orbital model for the low-energy \textit{molecular} orbitals and argue that many anomalous properties of KO$_2$ replicate the status of its orbital system in various temperature regimes.

Since O 2 is a good oxidizer and can easily take an additional electron when it brought in contact with alkali elements, there is another way of making the crystalline O 2 , in a form of ionic crystals (called "alkali hyperoxides"), which are similar to NaCl, but distorted due to elastic interactions involving O 2 molecules. One typical example is KO 2 , which around the room temperature forms the body-centered tetragonal (bct) lattice ( Fig. 1) [8,9]. The uniqueness of this situation is that O 2 has an extra electron residing in the doubly-degenerate π g shell, and KO 2 is the first molecular crystal, where the magnetism not only exists, but interplays with the orbital degrees of freedom, like in famous KCuF 3 [10] and LaMnO 3 [10,11]. Typically, such materials are known for their rich magnetic properties, which depend on the orbital state. This is clearly the case also for KO 2 . The Curie-Weiss temperature (T CW ), extracted from the behavior of magnetic susceptibility (χ m ) in two paramagnetic regions changes the sign at around 200 K [8], indicating at the change of character of intermolecular interactions, from antiferromagnetic (AFM) to ferromagnetic (FM) with decrease of the temperature (T ). The role of the crystal distortion in this region is somewhat controversial. On the one hand, there are some indications that the actual symmetry is lower than bct [8,9]. On the other hand, no profound change of the magnetic behavior has been observed. For example, a sizable orbital contribution to the magnetization persists down to 13 K. Therefore, even if exists, the crystal distortion does not seem to fully control the magnetic behavior of KO 2 in the paramagnetic region. If so, what causes the change of T CW ? Another interesting point is that below 13 K the situation is completely different and the behavior of KO 2 is strongly affected by frozen-in rotations of O 2 molecules, which quenche the orbital magnetization and stabilize an AFM order. Nevertheless, the transition temperature is surprisingly low. Moreover, the AFM order can be broken up by a magnetic field. The transition is accompanied by structural changes, and the effect is called "magnetogyration" [12].
The purpose of this work is to establish a theoretical basis for understanding the fascinating magnetic properties of KO 2 and other alkali hyperoxides. Using firstprinciples electronic structure calculations, we set up an effective spin-orbital model for the low-energy molecular states. Then, we show that the magnetic phase diagram of KO 2 replicates the status of its orbital system, which evolves (under heating) from a quenched state, to a region of relativistic spin-orbital correlations, and finally to the region of independent spin and orbital disorder.
The O 2 molecule appears to be the building block not only of the crystal, but also of the electronic structure of KO 2 . The hybridization within a single molecule is so strong that it leads to the splitting and formation of quite distinguishable molecular levels. The interaction between molecules is considerably weaker, so that the molecular orbitals form a group of narrow nonoverlapping bands (Fig. 1, we use the experimental lattice parame-ters reported in Ref. [9]). Thus, there is a clear analogy with atomic limit in the theory of strongly-correlated systems [10], except that now the molecular orbitals play the same role as atomic orbitals in a conventional case.
According to electronic structure calculations in the local-(spin)-density approximation [L(S)DA], the doublydegenerate π g band located near the Fermi level is formed by antibonding molecular p x and p y orbitals ( Fig. 1) [13]. The π g bandwidth is comparable with the exchange spinsplitting (∆ ex ∼1 eV). Hence, the system is half-metallic, and its magnetic properties can be understood in terms of FM double exchange (D) and AFM superexchange (S) interactions, which can be expressed through the first and second moments of occupied density of states for the ↓-spin π g band as J D =−m (1) /2z and J S =−m (2) /z∆ ex , where z is the coordination number [11]. Then, the Curie temperature will also consist of two parts, T C =T D C +T S C , which can be evaluated in the mean-field approximation [14]. This yields Thus, T C ≈ 324 K, and KO 2 is expected to be a good ferromagnet, being in straight contrast with the experimental data. The root of the problem is the Coulomb correlations in the narrow π g band, which are greatly oversimplified in LSDA.
In order to treat these effects rigorously, we derive an effective Hubbard-type model for the π g band, starting from the LDA band structure: The method has been explained in details in Ref. [15]. The basic difference here is that each lattice-point (i or j) corresponds to an O 2 molecule (rather than to a single atomic site). Each Greek symbol stands for a combination (m, s) of orbital (m= p x or p y ) and spin (s= ↑ or ↓) indices. Each orbital index refers to the antibonding molecular p-orbital of either x (m=1) or y (m=2) symmetry. The off-diagonal (with respect to i and j) elements of h αβ ij stand for transfer integrals, which do not depend on spin indices: i.e.,ĥ ij ≡ h αβ ij = t mm ′ ij δ ss ′ for any combination of α=(m, s) and β=(m ′ , s ′ ). They are derived from the LDA band structure using an exact version of the downfolding method [15,16]. The results are summarized in Table I, up to fourth nearest neighbor. Other parameters are considerably smaller. The diagonal matrix elements, h αβ ii = α|ξL ·Ŝ|β , stand for the spin-orbit interaction (SOI) is a single molecule (ξ ≈ 34 meV) and the crystal field, if the lattice distortion is applied.
The matrix of screened Coulomb interactions in the π g band has been computed in the two steps. First, we derive the interaction parameters between atomic porbitals, using the constraint-LDA method. It enable us to obtain the following parameters of intraatomic Coulomb interaction u ≈ 11.37 eV, interatomic intramolecular Coulomb interaction v ≈ 2.52 eV, and intraatomic exchange interaction j ≈ 2.30 eV. Then, we  , 0), and b4=(0, 0, c). Other matrix elements t mm ′ ij are obtained using symmetry operations and permutability of orbital indices, t 12 take into account the screening of Coulomb interactions between p-orbitals in the π g band by other bands, which are constructed from the same p-orbitals (i.e., the π u band, which is the bonding combination of p x -and p yorbitals, as well as the σ * g and σ * u bands, which have a strong weight of p z -orbitals). This part is done in the random-phase approximation, which starts with interaction parameters obtained in constraint-LDA. Then, we are able to derive the interaction parameters between molecular p-orbitals in the π g band: the intraorbital Coulomb interaction U ≈ 3.66 eV and the exchange interaction J ≈ 0.62 eV. The interorbital Coulomb interaction U ′ is related with U and J by the identity U ′ =U −2J. These parameters define the whole matrixÛ = U αβγδ of screened Coulomb interactions in the π g band.
Since any of t mm ′ ij is smaller than U by at least one order of magnitude, all transfer integrals can be treated as a perturbation, starting with isolated molecular orbitals. Then, it is convenient to use the hole representation. The advantage is that there is only one hole state associated with each molecular site. Therefore, if molecules were fully isolated, the holes would not interact with each other, and the ground state would be a single Slater determinant. By denoting the hole-orbitals associated with i and j as α i and α j , and constructing the two-hole determinant G(α i , α j )= 1 √ 2 {α i (1)α j (2)−α j (1)α i (2)}, one can find the energy gain caused by virtual hoppings in each bond [17,18]: where E jM and |jM stand for eigenvalues and eigenstates of excited two-hole configurations at the site j, andP j is a projector operator, which enforces the Pauli principle and prevent any hoppings onto α j [17]. Eq. (2) incorporates some intramolecular correlations beyond the Hartree-Fock approximation, which are included into the multiplet structure of the excited states. First, we discuss the properties of the bct phase of KO 2 in an intermediate-temperature region, where the crystal distortion is small and k B T ≪ξ (note that ξ/k B ≈ 395 K), so that the splitting of the molecular π g levels is largely controlled by the SOI, which provides the natural basis for the hole orbitals {α i }. In this case, each |α i can be presented as a linear combination of only two orbitals, |p + ↑ and |p − ↓ , which are split off by the SOI, and where |p ± =∓ 1 √ 2 (|p x ±i|p y ) are the complex harmonics. Now, the spin and orbital variables are rigidly bound, and the problem can be reformulated in terms of the pseudospin states: , |τ + z =|p + ↑ , and |τ − z =|p − ↓ , which allow us to make a formal mapping of energies (2) onto the anisotropic Heisenberg model for the bct lattice and the pseudospin 1/2: whereτ x ,τ y , andτ z are the 2×2 Pauli matrices in the basis of |p + ↑ and |p − ↓ orbitals.
The parameters, calculated as 2J ⊥ ij = T (τ + ix , τ − jx )−T (τ + ix , τ + jx )=T (τ + iy , τ − jy )−T (τ + iy , τ + jy ) and 2J ij = T (τ + iz , τ − jz )−T (τ + iz , τ + jz ), are summarized in Table I. The direct exchange interactions between O 2 molecules are considerably weaker and can be neglected [5]. The interlayer coupling J operating between neighboring molecules separated by b 2 stabilizes the FM ground state. Hence, the pseudospin moments are parallel to the z axis. Other interactions are antiferromagnetic and frustrated on the bct lattice. The expectation values of the magnetic moments are given by the following identities: τ ± x |L x +2Ŝ x |τ ± x = τ ± y |L y +2Ŝ y |τ ± y =0 and τ ± z |L z +2Ŝ z |τ ± z =±2: i.e., the finite magnetization is allowed only along the z-axis, while it is totally quenched in the perpendicular (x and y) directions. The same matrix elements define the g-tensor (e.g., in χ m ).
Then, the magnetic properties can be easily calculated using renormalized spin-wave (SW) theory [19]. The idea is to extend the conventional formalism to finite temperatures by replacing in the expression for the SW frequency the value of local magnetic moment (in our case, the pseudospin moment) by its thermal average τ z : On the other hand, τ z is expressed through the averaged number of excited spinwaves, yielding the equation τ z k coth ω k /2k B T = 1, which is solved self-consistently for each T and the external field b, acting onτ z . The expression for the Curie temperature, (k B T C ) −1 = q (J 0 −J ⊥ q ) −1 , can be derived by assuming b= τ z =0 or from the divergence of χ m =4µ 2 B τ z /b. The spectrum of SW excitations itself is rather remarkable (Fig. 2). All excitations are gapped. The gap in the Brillouin zone center is proportional to ξ, which is large on the temperature scale. However, all low-energy excitations correspond to the zone boundary, where ω q is largely reduced by AFM interactions J ⊥ ij . Thus, the FM order is expected only at T C ≈ 66 K, which is largely reduced in comparison with ξ/k B (and, apparently, can be further reduced by small lattice distortions, which are seen in KO 2 [8,9]). The renormalizes SW theory also predicts some small deviation from the Curie-Weiss law in the paramagnetic region, and the effective moment (µ eff ), derived from the slope of χ −1 m , will generally depend on T . Indeed, µ eff varies from 2 µ B at large T till 2.17 µ B near T C . This may help to explain some deviations of µ eff from its nominal value [8]. The unrenormalized spin-wave dispersion of anisotropic Heisenberg model: ω 0 q =2(J 0 −J ⊥ q ). The inset shows the inverse magnetic susceptibility in the paramagnetic region, as obtained in the renormalized spin-wave theory.
In the high-temperature region, k B T > ∼ ξ, the thermal fluctuations will eventually destroy the coupling between spin and orbital degrees of freedom, which now can be treated as independent variables. Then, the spin Hamiltonian can be derived by averaging the pair interactions (2) over orbital variables [10]. Practically, for each projection of spin, one can consider the whole family of possible hole-orbitals, |α ↑,↓ i =cos θ i |p x ↑, ↓ +sin θ i e iφi |p y ↑, ↓ (where 0≤θ i ≤π and 0≤φ i ≤2π), and to average (numerically) each T (α ↑,↓ i , α ↑,↓ j ) over all combinations of (θ i , φ i ) and (θ j , φ j ). The simplest approximation, which is justified for large T , is to assume that all combinations are equivalent and to neglect the spacial correlations between orbitals. Obviously, the spin system will be isotropic, and the parameters of Heisenberg model can be found as 2J ij =T ↑↓ ij −T ↑↑ ij , whereT ↑↓ ij andT ↑↑ ij are the averaged values of T (α ↑ i , α ↓ j ) and T (α ↑ i , α ↑ j ), respectively. All param-eters are antiferromagnetic (Table I), which naturally explain the sign of T CW in the high-temperature region [8].
The absolute value of T CW can be estimated from the behavior of spin susceptibility (χ s ), in the framework of renormalized SW theory. Note that χ s alone should provide a good estimate for the total χ m in the perpendicular xy-plane, where the orbital magnetization is quenched. This yields T CW ≈ −100 K, which was derived from the linear interpolation of χ −1 s at large T . Thus, the existence of several paramagnetic segments with different T CW 's can be related with a gradual increase of on-site correlations between spin and orbital degrees of freedom, driven by the SOI in the bct phase.
In the high-temperature region, the spin and orbital variable are totally decoupled, that explains the AFM character of intermolecular interactions. On the other hand, the strong spin-orbital correlations in the intermediatetemperature regime would lead to a ferromagnetism.
The crystal distortion plays a sizable role only at very low T , as it is clearly manifested in the quenching of orbital moments and the observed AFM order [8]. Although exact details of the low-temperature structure are unknown, a number of experimental data speak in favor of uniform rotations of O 2 molecules around one of tetragonal axes as the main distortion. Certainly, such a distortion will lift the degeneracy of molecular p x and p y levels, and lead to a FM orbital order, which typically coexists with AFM correlations between the spins [10]. However, the situation is not so simple. Let us consider a distorted structure, which was obtained by the rigid clockwise rotation of O 2 molecules around the y-axis by 30 • (the experimental value [8,9]) in the fixed tetragonal frame, and derive parameters of the model Hamiltonian (1) for the (deformed) π g band. The obtained "crystal-field splitting" of molecular π g levels is about 290 meV, which exceeds ξ by nearly one order of magnitude and stabilizes the p y -orbital (opposite to the assumption made in the phenomenological theory of magnetogyration [12]). Generally, the transfer integrals are specified by all three parameters (t 11 , t 12 , t 22 ) [20].  (0, a, 0) Thus, the transfer integrals are strongly anisotropic, both in and between the tetragonal xyplanes. Since the SOI is no longer important, the energies (2) can be mapped onto rotationally invariant (J ⊥ ij =J ij ) Heisenberg model. The natural choice of the hole orbitals, |p y ↑ or |p y ↓ , is dictated by the crystal-field splitting. This yields the following interaction parameters J ij = −0.77, 0, −0.59, 4, −0.09, and −0.01 meV between O 2 molecules separated by the vectors b 1 , b * 1 , b 2 , b * 2 , b 3 , and b 4 , respectively. They form quasi-twodimensional networks, where interactions in the planes perpendicular to [b 1 ×b 2 ] are much stronger than those between the planes. This suppresses any long-range mag-netic order [21]. Moreover, the AFM interactions in each plane are frustrated. Therefore, the Néel temperature (T Q ) is expected to be small. Indeed, from the mean-field analysis of the Heisenberg model we conclude that the ground state is a (nearly AFM) spin-spiral with Q≈ (0, 3.77 a , 0.02 c ). Then, the renormalized SW theory yields T Q ≈ 11 K, being in good agreement with the experiment [8]. The first-order transition to the FM bct phase in a magnetic field is also anticipated.
In summary, KO 2 is the new molecular analog of correlated spin-orbital systems. The geometry of molecular orbitals adds new functionalities into this classical problem. There is also a number of open questions. Particularly, any quantitative theory of magnetogyration is missing. Why do the O 2 molecules tend to rotate at low T ? Why do their orientation is different in KO 2 and NaO 2 ? These problems prompt further research. I wish to thank P. Mahadevan for drawing my attention to KO 2 , as well as M. Boero and Y. Motome for useful discussions. This work is partly supported by Grant-in-Aids for Scientific Research in Priority Area "Anomalous Quantum Materials" from the Ministry of Education, Culture, Sports, Science and Technology of Japan.