Spin and orbital ordering in bilayer SrCrO
Using maximally localized Wannier functions obtained from DFT calculations, we derive an effective Hubbard Hamiltonian for a bilayer of SrCrO, the member of the Ruddlesden-Popper SrCrO system. The model consists of effective orbitals of Cr in two square lattices, one above the other. The model is further reduced at low energies and two electrons per site, to an effective Kugel-Khomskii Hamiltonian that describes interacting spins 1 and pseudospins 1/2 at each site describing spin and orbitals degrees of freedom respectively. We solve this Hamiltonian at zero temperature using pseudospin bond operators and spin waves. Our results confirm a previous experimental and theoretical study that proposes spin ordering antiferromagnetic in the planes and ferromagnetic between planes, while pseudospins form vertical singlets, although the interplane separation is larger than the nearest-neighbor distance in the plane. We explain the physics behind this rather unexpected behavior.
Some decades ago, Kugel and Khomskii studied theoretically the interplay between orbital and spin degrees of freedom in compounds like KCuF and KCuF.kugel () They showed that in these compounds the orbital degrees of freedom (leaving the hole in the 3d configuration of Cu in the orbital with symmetry either or ) can be described by a pseudospin, and these pseudospins interact between them and with the spins of the Cu ions, in such a way that the preferred ordering is antiferromagnetic for the pseudospins and the spin ordering is ferromagnetic in the plane and antiferromagnetic in the direction. The staggering ordering of the orbitals leads to a staggering of quadrupolar distorted CuF units in the planes, as expected from any electron-phonon interaction.liech ()
The interest in systems with spin and orbitals degrees of freedom was rising over the years. See for example Refs. liech, ; mizo, ; Feiner, ; ling, ; Ulrich, ; Horsch, ; Fang, ; ruo, ; Sugai, ; Pavarini, ; Manaka, ; bruce, ; Stingl, ; Oles, ; kov, ; fepc, ; fepc2, ; srcro, . A rich physics has been observed in the members of the Ruddlesden-Popper series of the form , where denote transition-metal atoms that form two square lattices in the plane, one displaced with respect to the other in the direction.ling (); Manaka (); Stingl (); srcro () In the layered colossal magnetoresistance manganite LaSrMnO, different spin and orbital orderings are observed as is varied indicating that orbital polarization is the driving force behind spin ordering.ling () Particularly interesting is KCuO, where distortions reveal antiferromagnetic orbital ordering, while the system presents a spin gap due to spin dimers in the direction.Manaka () This is an exotic ordering involving spins and pseudospins both of magnitude 1/2. In SrRuO, an applied magnetic field induces domains with distorted lattice parameters likely related to orbital ordering.Stingl ()
Recently the compound SrCrO has been studied by several experimental and theoretical techniques.srcro () The resistivity and specific heat measurements indicate that the system is insulating. The calculations using density functional theory (DFT) with LDA+U approximation indicate that occupancy of Cr is consistent with an oxidation state Cr (two electrons in the 3d shell) and there is an orbital degeneracy between and . Hund rules favor a total spin . There is a magnetic transition at 210 K with a huge total entropy change near indicating a simultaneous spin and orbital ordering. The magnetic structure observed by neutron diffraction is consistent with the DFT results and correspond to antiferromagnetic alignment between nearest-neighbors in the plane and ferromagnetic between planes. Orbital ordering is usually not detected in DFT with LDA and derived potentials due to the difficulties of these techniques to obtain orbital polarization.liech (); note () Nevertheless, the absence of observable distortions that accompany the orbital ordering (as in KCuF and KCuF mentioned above) is consistent with the formation of vertical singlet orbital dimers, so that quantum fluctuations destroy long-range pseudospin ordering. This spin/pseudospin state has some analogy to the case of KCuO (with both spin and pseudospin ) mentioned in the previous paragraph with spin and pseudospin (orbital) degrees of freedom interchanged.
The authors have also derived a Kugel-Khomskii Hamiltonian from a multiband Hubbard model containing the relevant orbitals of Cr and O.srcro (). Solving this Hamiltonian by Lanczos in a cluster containing eight sites, they found that effectively the ground state corresponds to the observed magnetic ordering and vertical pseudospin singlets if , where () is the hopping between nearest-neighbor Cr and O orbitals perpendicular to (in) the CrO planes. Since is expected to be smaller than because of the larger distances in the plane, and the size of the cluster is very small, further theoretical work is necessary to confirm that the proposed exotic spin and orbital ordering is in fact the ground state. Note that for a spin 1/2 Heisenberg model in a bilayer system (our bilayer system for pseudospins only, i.e. ) with interplane interaction and vertical one , the quantum phase transition from a Neel ordered phase to the quantum disordered dimer-singlet phase takes place for ,sand (); wang () and for the three-dimensional extension with two cubic sublattices the transition moves to .qin () Therefore naively one would expect a phase with long range pseudospin ordering in SrCrO for .
In this work we first construct a tight-binding model for effective orbitals at the Cr sites, using maximally localized Wannier functions (MLWF) and add to it the on-site interactions.ruo (); spli () This leads to a three-band Hubbard model. This starting approach is similar to that followed by Ogura et al. to study the possible occurrence of superconductivity in hole doped SrCrO and SrMoO.superc () These effective orbitals are not pure Cr 3d orbitals of , , and symmetry but contain an important admixture with O orbitals. Next we derive a Kugel-Khomskii Hamiltonian for the low-energy subspace of two electrons per site, by degenerate perturbation theory in the hopping terms. We explain the meaning of the different terms and the expected physics. Finally this Hamiltonian is solved for the infinite system, using a combination of bond operators and spin waves. For the resulting parameters we obtain that the state of singlet dimers and the spin ordering proposed in Ref. srcro, is in fact the ground state. To destabilize it, one would need to reduce the hopping mentioned above by a factor near 1/2.
The paper is organized as follows. In Section II we describe the atomic and electronic structures. Section III describes the ab initio method used to obtain the effective hoppings and on-site energies used in Section IV to derive an effective multiband model for the system. In Section V we use this model to derive an effective Kugel-Khomskii Hamiltonian to describe the spin and orbital degrees of freedom of the model. This model is solved in Section VI using a generalized spin-wave theory. Section VII contains a summary and discussion.
Ii The System
As a starting point we calculate the electronic structure of the SrCrO system within the framework of DFT. The structure has I4/mmm symmetry, which is tetragonal with lattice parameters Å and Å srcro (). We distinguish three types of O atoms according to its bonding role. As shown in Fig. 1, the structure is a stacking of CrO layers and SrO layers. We label as O the O atoms inside the CrO layers, O denote the O atoms in the SrO layers between two CrO layers, and O refer to the O atoms of the SrO layers that lie between another Sr-O layer and a CrO one. Two consecutive Sr-O layers are displaced by a vector , where is the interlayer Sr-O distance, as can be seen in the small scheme at the right of Fig. 1. In the CrO layers, both, O and Cr atoms form a square lattice. Each Cr atom has four nearest neighbors that are O atoms in the plane and two more O neighbors in the direction, the one between CrO layers is type O and the other is type O.
Since we are looking for the hopping parameters, the DFT calculations are performed as spin unpolarized, but using both, cell parameters and atomic coordinates obtained from the spin-polarized case. The band structure calculations are done using Wien2k codewien2k (), with precision parameters , which reads as the product between smallest muffin tin sphere radius with the plane-wave cutoff (). The Brillouin zone was sampled with a regular mesh containing 800 irreducible points and we use the GGA approximation for the potential of exchange-correlation.
The band structure along with atom-projected density of states are shown in Fig. 2, where band character is emphasized by color, red means strong Cr component and blue stands for O component. The electronic structure close to the Fermi energy is dominated by Cr-d states, which are split by the tetragonal component of the crystal field into and states. The Cr- states are the partially filled states and share a peak with O at -0.7 eV, also they hybridize with in plane O atoms. Above the Fermi level, the states of Cr share a peak with O at 0.09 eV. These characteristics in the DOS expose the bridging character of the O between next CrO layers. The band structure presents a region around the Fermi level, where the states of the Cr prevail. The -type orbitals are located at 0.9 eV above the Fermi level. Thus, they have no influence on the orbital ordering.
Iii Hopping parameters from MLWF
The MLWF approach provides a physically intuitive and also rigorous representation of the electronic band structure of a system in an energy region of interest, which defines a Hilbert subspace. Then, the Hamiltonian expressed in the base of MLWF can be mapped to a tight binding-based model which describes the system in the target Hilbert subspace. These MLWF can be derived from Bloch states of a DFT calculation by the so called wannierization process as implemented in the Wannier90 codewannier90 (). However, the input of Wannier90 requires the overlap matrices and projections to the Hilbert subspace, so we use the Wien2Wannierwien2wannier () routine as interface among them.
The target Hilbert subspace is chosen in order to describe the Cr -orbitals near the Fermi energy, which are responsible of the magnetic ordering in the system. In Fig. 2, the horizontal dashed line at -1.5 eV and 1 eV wrap the chosen energy window where the wannierization process takes place. Within the selected energy window, the number of desired Wannier functions is equal to the number of -orbitals multiplied by the number Cr of atoms in the unit cell, which in our case is 12 (3 -orbitals times 4 Cr atoms per unit cell). Also, we have to take into account that there are 14 bands lying inside the chosen energy window, meaning that the disentanglement procedure must be used before the wannierization procedure takes place. Nevertheless, fast and accurate wannierization can be done using as initial guess the expected -like wavefunctions.
The obtained MLWF corroborate the -type symmetry and are centered at Cr atoms as shown in Fig. 3. Nevertheless, the isosurface plot in real space of the MLWF reveals the strong hybridization between Cr and O’s. In particular, the point group at the Cr atoms does not contain the reflection through the CrO layers, a fact that is evident in the different content of O and O orbitals in the Wannier functions with approximate and symmetry. This fact allows us to conclude that the hoppings obtained from MLWF-base Hamiltonian includes the effect of O atoms in the effective Cr-Cr hopping processes. This suggests that a simplified multiband model can be proposed (as we do in the following Section) without the need to include crossed terms between and orbitals.
Iv The multiband Hubbard model
The multiband model is constructed from effective orbitals at each Cr site, the difference between on-site energies and the effective hopping between orbitals at different sites calculated with DFT+MLWF, and the interactions between electrons at the same site. The Hamiltonian is
Here indicates the upper or lower square sublattice at positions Å/2, and , with Å and integers, denotes the position of a Cr atom within the plane. The creation operators of effective orbitals at site with spin are denoted by , where , , or . As discussed in the previous Section, these effective orbitals contain some admixture with O orbitals, and do not have a definite symmetry under the reflection . However, for simplicity we keep the notation corresponding to orbitals in cubic symmetry (, , or ).
The first term in Eq. (1) corresponds to the tetragonal crystal field, which raises the energy of an electron in the orbital with respect to the other two.
In the following we will take that corresponds to spherical symmetry spli () (as for the free atom).
The hopping between orbitals is mediated by Cr-O hopping through O 2p orbitals and the symmetry of the orbitals imposes restrictions on the allowed processes. As a consequence, the orbitals cannot hop in the direction. Similarly the () orbitals cannot hop in the () direction. Then, the hopping term of the multiband model has the form
where is a vector connecting two nearest Cr atoms in the or direction.
The crystal-field splitting and the hopping parameters , and were determined from the MLWF, as described in Section III.
The resulting crystal-field splitting is eV. This is likely an underestimation, because orbital polarization, which is not properly taken into account by DFT leads to larger splittings.spli () In any case, its detailed value affects very little our results and does not change our conclusions as long as . A negative would lead to a trivial pseudospin singlet configuration at each site and is inconsistent with a change of entropy near observed in the magnetic transition.srcro ()
The resulting hopping parameters are eV, eV and eV. They are of the same order of magnitude. The fact that is expected in perturbation theory in the Cr-O hopping because in the expression for , the denominator involves the energy necessary to take an electron from the O atom and put it in an orbital, and this is smaller than the corresponding denominator for . Instead, the fact that is rather unexpected, because the interlayer distance is almost 2% larger than , the shortest distance between Cr atoms in the plane. However, different on-site energies of the O orbitals lying in between the Cr atoms and the fact that the effective orbitals are deformed with respect to the ideal shape (as evidenced in Fig. 3) can modify the result. The deformation of the orbitals can also explain an effective hopping between () orbitals in the () direction of magnitude 0.035 eV absent in perturbation theory in the Cr-O hopping because of symmetry. We neglect this contribution.
Concerning the values of the interactions and , a fit of the lowest atomic-energy levels along the 3d series gives eV.kroll () We take this value as a basis for our study. Since does not involve charge transfer, it is usually not screened in the solids in contrast to . However, in our case since the effective orbitals contain some orbital admixture, can be smaller. A reasonable value for for early transition metals, already used to study the orbital Kondo effect in V-doped 1T-CrSe,kov () is eV. We shall analyze the dependence of the results with .
V The Kugel-Khomskii Hamiltonian
For two electrons per site and large enough the system described by the multiband Hamiltonian Eq. (1) leads to an insulating ground state, in agreement with the experimental evidence in SrCrO.srcro () In this case, the hopping term can be eliminated from by means of a canonical transformation (similarly to the derivation of the Heisenberg model from the Hubbard one heis ()) leading to an effective Kugel-Khomskii Hamiltonian for the spin 1 and pseudospin 1/2 (orbital) degrees of freedom. The eigenstates and corresponding energies of that we need [see Eqs. (1), (2) and (3)] have been calculated if Ref. ruo, . We restrict to second order in and denote the spin at each site by and the pseudospin by , with (1/2) corresponding to the () configuration. The resulting effective Hamiltonian can be written as
where contains the vertical interactions (in the direction) for each two-dimensional position in the plane, and describes the interactions in the plane , 2. Dropping irrelevant constants, one has
where the factor 1/4 in the first term is introduced to compensate for factors that come from in classical orderings and render easier the qualitative discussion below. The coefficients are
is the energy necessary to take a () electron from the ground state of the () configuration and add it to the () configuration of a neighboring site to build the ground state of the configuration.
Similarly for the interactions in each plane
At this point we discuss qualitatively the meaning of and the expected physics. We begin discussing the two-site vertical interactions [Eq. (6)]. For , all interactions are equal [see Eq. (7)]: . This means that without the spin-pseudospin interaction both spins and pseudospin minimize the energy for an antiferromagmetic (AF) alignment, but the term in is minimized for one ferromagnetic (FM) and the other AF alignment. As a consequence from the four classical possibilities of orienting the spin and pseudospin FM or AF, all of them are part of the degenerate ground state with energy - except the FM-FM one. This result is easy to understand: the second order correction to the energy of these states contains virtual processes in which one electron in the (pseudospin ) or (pseudospin ) orbital and spin or jumps to the other site and comes back. The corresponding gain in energy is the same for any alignment of spin and pseudospin except in the case in which the same orbital with the same spin is occupied at both sites because of Pauli principle. If the orbitals were absent, leaving spins 1/2, this picture would not be modified by quantum fluctuations. Actually in this case the model would have SU(4) symmetry with spin and pseudospin playing a similar role.fepc2 () In our actual case with , the pseudospins 1/2 are more quantum than the spins 1 and the ground state of the dimer is a pseudospin singlet and spin triplet with energy . The first excited state is a pseudospin triplet and spin singlet with energy .
When (the interaction responsible for the Hund rules) is increased, as expected the ferromagnetic spin interactions are favored. From Eqs. (7) it is apparent that decreases more strongly than the other two, clearly favoring the pseudospin singlet and spin triplet. A disadvantage of the pseudospin singlet is that it cannot take advantage of the pseudospin interactions in the plane (except for some fluctuations).
Leaving aside for the moment the contribution due to the hopping of the orbitals, the interactions in the plane , , are exactly half of the corresponding ones in the vertical direction if . This factor is due to the fact that for a given direction in the plane, only one of the degenerate , orbitals can hop. The anisotropy in direction is reflected by the term proportional to . Another consequence of the fact that () orbitals can only hop in the () direction in the plane is the absence of pseudospin flip terms in [see Eq. (9)].
For one has . From the influence of on the parameters (similar to the case of the vertical interaction) one would expect AF pseudospin ordering and FM spin ordering favored. However the contribution due to the hopping of the orbitals dominates the spin ordering. For , . The prefactor 4 with respect to the other interactions in the plane is due to a factor 2 because the orbitals an hop in both the and directions, and another factor 2 because the FM spin alignment cannot gain energy even for AF pseudospin ordering. For hoppings of the same order of magnitude clearly dominates over the other interactions and one expects AF spin ordering within the planes. From the argument given above, one expects in addition FM spin ordering between planes, in agreement with the spin ordering observed by neutron scattering and calculated with ab initio methods.srcro ()
Concerning pseudospin ordering, from the values of and given above and results in the literature sand (); wang () for (neglecting spins) one would expect the quantum phase transition between an AF Neel ordered phase and the phase with vertical singlet dimers to take place for or . However for the actual spin of the system, the observed spin ordering and the effect of the interactions between spins and pseudospins and , favor singlet ordering between planes and weakens the effective intraplane AF pseudospin interaction.
Taking eV eV, which implies eV, leads to the values tabulated in Table 1 for the parameters of .
From the results (rather expected from the above discussion), it is clear that the dominant vertical interaction is which favors pseudospin singlets, or possibly AF vertical order. This together with the effect of , which is about ten times larger than , overcomes the weak antiferromagnetic interaction and FM vertical spin alignment is clearly favored. Instead, in the planes the dominant interaction is which favors spin AF order. In addition the interaction between spins and pseudospins is smaller than and therefore AF orbital order in the plane is also expected. Finally the anisotropic interaction which favors FM orbital order is clearly smaller and has no relevant effect.
A detailed study of the competition between vertical pseudospin singlets and long-range pseudospin AF ordering is the subject of the following Section.
Vi The spin and pseudospin ordering
In this Section, we report on our study of the stability of two phases I and II, which are the most likely according to the analysis of the previous section and a numerical study on a small cluster.srcro () In both of them, the spin ordering corresponds to the experimentally observed one: antiferromagnetic in the planes and ferromagnetic between planes. In phase I, the pseudospins form vertical dimer singlets, and in phase II they order antiferromagnetically in both directions.
To calculate the energy and stability of phase I, we used the idea of the bond-operator formalism,chubu (); sach (); gopa (); matsu (); bond () but in the form of a generalized spin-wave theory,muniz () which allows us to avoid the use of Lagrange multipliers. For phase II we use ordinary spin-wave theory.
The vertical pseudospin singlet , where the first arrow denotes can be represented by a using a boson as . The interplane term in the Hamiltonian mixes this state with the triplet with projection 0: , because , , and the same interchanging and . Following Ref. muniz, we assume for phase I that the number of triplet excitations is small, and ”condense” the singlets using
For the spins we use the usual Holstein–Primakoff bosons proceeding in a similar way.muniz () Performing a rotation of the spins in half of the sites by around the axis to convert the AF order in the plane in a translationally invariant FM order mart () and retaining as usual terms up to quadratic in the bosonic operators, the Hamiltonian [see Eqs. (5), (6) and (9)] for phase I becomes
where is the number of sites in a plane and creates a spin excitation at two-dimensional position of plane .
Diagonalizing the Hamiltonian by means of a standard Bogoliubov transformation, The ground-state energy becomes
Note that when , the system becomes unstable against creation of triplet excitations of long wavelength and Eq. (13) becomes meaningless. In general if for some parameters the assumed pseudospin or spin arrangements become unstable, the situation is detected in the numerical algorithm used to calculate the two-dimensional integral over by the non-analyticity of some expression for small . In fact, as we show below, phase I becomes unstable near the transition to phase II (as it might be expected).
For phase II with long range spin and pseudospin ordering, a similar treatment as above using Holstein–Primakoff bosons leads to the following energy
As a test of our procedure we have compared the energy of the two phases when all interactions involving spin are zero (this is equivalent to take ) leaving only and . We obtain a transition between the long-range ordered phase II for small to the phase of vertical dimers I for large at , 17% larger than the value near 2.522 obtained by Monte Carlo calculations.sand (); wang () Thus, our approach underestimates the stability of phase I.
For the parameters listed in Table 1, we obtain that the energy of the dimerized phase I is lower than the long-range ordered II by 19.8 meV. These agrees with the structural measurements, which do not detect any distortion of the lattice, or displacement of the O atoms expected for long-range orbital ordering. As a test of the stability of this phase, we have lowered the vertical hopping by a factor and searched for the value of that leads to the equality of both energies () for different values of . The results are shown by the full line in Fig. 4. For the expected value of eV ( eV), the resulting value of is slightly larger than the value of that corresponds to the instability of the dimerized phase against the formation of triplet excitations [given by , see the discussion after Eq. (13)] corresponding to the dashed line in the figure. Both values of are of the order of 0.5 reflecting the fact that the real system is far from the boundary of the phase diagram. For larger values of , the dimerized phase is stabilized further. For eV (), the dimerized phase I becomes unstable at a point at which its energy is still lower than that of the long-range-ordered one II. This is probably a shortcoming of the approximations. From the physics of the case of spin ,sand (); wang () and its extension to two cubic sublattices,bruce () one would expect a second-order transition between both phases and a coincidence of both transitions (full and dashed lines).
In Fig. 5 we show how the previous results change when is reduced from 0.7 to 0.4 eV. We consider that this value is a lower bound of the interaction responsible of the Hund rules due to the fact that the effective orbitals are not pure Cr ones, but have some admixture of neighboring O atoms with smaller interactions. As one can see, the changes with respect to the previous figure are minor. We conclude that for the calculated values of the hopping terms obtained as described in Section III, and reasonable values of the interactions, the dimerized phase I is the stable one.
Vii Summary and discussion
Using maximally localized Wannier functions in bilayer SrCrO, we have derived a six-band Hubbard model for effective Cr orbitals (three per Cr site) which contain some admixture with neighboring O atoms. Using the resulting hopping and on-site energy parameters, and reasonable values of the Coulomb interaction and interaction responsible for the Hund rules , we have derived an effective Kugel-Khomskii Hamiltonian using a procedure equivalent to degenerate second-order perturbation theory in the hopping terms.
A similar has been derived in Ref. srcro, using forth-order perturbation theory in the Cr-O hopping, but the parameters were not determined and it was not clear where the system lies in the phase diagram, although the absence of observable distortions is consistent with same phase that we obtain here.
Our analysis of and calculations based on bond-order operators and spin waves, show that the ground state of the system has long-range spin order, with antiferromagmetic order in the layers and ferromagnetic order between layers, in agreement with experiment and ab initio calculations.srcro () Instead the orbital degrees of freedom form singlet dimers perpendicular to the planes. This rather exotic arrangement of the orbital degrees of freedom is rare. For interplane and intraplane interactions of the same order of magnitude, one expects long-range antiferromagmetic ordering of the pseudospin (orbital) degrees of freedom. Although the methods used to solve are semiquantitative, with errors of the order of 17% for a known case, the obtained ground state is rather far from the phase boundary to the phase of long-range antiferromagmetic pseudospin ordering.
The reason for the stability of the dimerized phase is twofold. On one-hand, the intraplane pseudospin interactions are smaller due to restrictions of the and orbitals to hop in certain directions of the plane. On the other hand, the interactions between spins and pseudospins strengthen (weaken) the antiferromagmetic pseudospin ordering normal to (in the) layers.
A. A. A. (C. H.) thanks B. Normand and C. Batista (Rubén Weht) for helpful discussions. A. A. A. (C. H.) is sponsored by PIP 112-201501-00506 of CONICET and PICT 2013-1045 of the ANPCyT (PICT 2014-1555 of the ANPCyT).
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