Abstract
Interactiondriven metal–insulator transitions or Mott transitions are widely observed in condensed matter systems. In multiorbital systems, manybody physics is richer in which an orbitalselective metal–insulator transition is an intriguing and unique phenomenon. Here we use firstprinciples calculations to show that a magnetic transition (from paramagnetic to longrange magnetically ordered) can simultaneously induce an orbitalselective insulator–metal transition in rocksalt ordered double perovskite oxides A_{2}BB′O_{6}, where B is a nonmagnetic ion (Y^{3+} and Sc^{3+}) and B′ a magnetic ion with a d^{3} electronic configuration (Ru^{5+} and Os^{5+}). The orbitalselectivity originates from geometrical frustration of a facecenteredcubic lattice on which the magnetic ions B′ reside. Including realistic structural distortions and spinorbit interaction do not affect the transition. The predicted orbitalselective transition naturally explains the anomaly observed in the electric resistivity of Sr_{2}YRuO_{6}. Implications of other available experimental data are also discussed. This work shows that by exploiting geometrical frustration on nonbipartite lattices, new electronic/magnetic/orbitalcoupled phase transitions can occur in correlated materials that are in the vicinity of metal–insulator phase boundary.
Introduction
Interactiondriven metal–insulator transition (socalled Mott transition) is one of the most striking phenomena in condensed matter systems.^{1} With the development of manybody methods such as dynamical mean field theory, we can coherently describe the Mott transition using a singleorbital Hubbard model.^{2,3}
In multiorbital systems, more complicated Mott physics emerges and the orbitalselective Mott transition (OSMT) is a most intriguing phenomenon.^{4} OSMT refers to the phenomenon in which as the transition occurs, conduction electrons become localized on some orbitals and remain itinerant on other orbitals. The idea, which was first introduced to explain the transport properties of Ca_{2−x}Sr_{x}RuO_{4},^{4,5,6,7} has stimulated many theoretical investigations^{8,9,10,11,12,13,14} and different mechanisms underlying this phenomenon have been proposed: for example different orbitals have different intrinisic band widths,^{4} different onsite energies,^{15} different pd hybridization^{16}, and/or different band degeneracies.^{17}
In this work, we use firstprinciples calculations to introduce a new approach to induce orbitalselective insulator–metal transition in multiorbital systems. We show that in a multiorbital Mott insulator with its magnetic ions residing on a nonbipartite lattice, the occurrence of longrange magnetic ordering can drive electrons on one orbital into a metallic state while leaving electrons on other orbitals insulating. The orbitalselectivity originates from ‘geometrical frustration’ of nonbipartite lattices, which enforces some magnetic moments to be ferromagnetically coupled in an antiferromagnetic ordering.
Figure 1a shows the crystal structure of a rocksalt ordered double perovskite oxide A_{2}BB′O_{6}. Blue and brown oxygen octahedra enclose two different types of transition metal ions B and B′. Green balls are A ions and red balls are oxygen ions. In panels b and c of Fig. 1, we show a simplified structure of an ordered double perovskite oxide in which only transition metal ions B and B′ are shown. The small blue balls are nonmagnetic transition metal ions B and the large black balls are magnetic transition metal ions B′. The red arrows denote magnetic moments of B′ ions. The magnetic ions B′ reside on a facecenteredcubic (fcc) lattice. Panel b shows a schematic of a paramagnetic state in which magnetic moments on B′ ions have random orientations and fluctuate in time. Panel c shows a schematic of an antiferromagnetic state. We note that if nearestneighbor exchange is antiferromagnetic in nature, it is impossible to have a ‘complete’ antiferromagnetic ordering on a fcc lattice in which each pair of nearestneighbor magnetic moments is antiferromagnetically coupled because fcc lattice has ‘geometrical frustration’.^{18,19} Instead a socalled typeI antiferromagnetic ordering is widely observed in ordered double perovskite oxides.^{20,21,22,23,24,25,26,27} This ordering is shown in panel c, in which magnetic moments alternate their directions between adjacent atomic planes along the zaxis. Mathematically the magnetic moment configuration is characterized by an ordering wave vector \({\mathbf{Q}} = \frac{{2\pi }}{a}(001)\), where a is the lattice constant. Our firstprinciples calculations show that ordered double perovskite oxides which contain magnetic Ru^{5+} and Os^{5+} ions are promising candidate materials which are Mott insulators in hightemperature paramagnetic state but undergo the aforementioned orbitalselective insulator–metal transition as the typeI antiferromagnetic ordering occurs at low temperatures. Experimental evidence for this transition and implications of other available experiment data will be discussed.
The computational details of our firstprinciples calculations are found in Methods Section.
Results
Table 1 lists five candidate materials in this study. In those ordered double perovskite oxides, Ru^{5+} and Os^{5+} are magnetic, and Y^{3+} and Sc^{3+} are nonmagnetic. Both Ru^{5+} and Os^{5+} have a d^{3} configuration in which, due to Hund’s rule, three d electrons fill three t_{2g} orbitals and form a spin \(S = \frac{3}{2}\).^{28} All those four ordered double perovskite oxides exhibit typeI antiferromagnetic ordering below Néel temperature T_{N}.^{20,21,22,23,24,25,26,27} For clarity, we first study Ba_{2}YRuO_{6} as a representative material. We discuss other four materials in section Discussion. Ref. ^{20} shows that Ba_{2}YRuO_{6} crystallizes in a cubic Fm3m structure (space group No. 225) and retains Fm3m symmetry from room temperature down to 2.8 K (below T_{N}). The change in lattice constant due to thermal expansion is very small (<0.15%). Experimentally, it is found that Y^{3+} and Ru^{5+} site mixing is negligible or at most very low (about 1%)^{20} because the size difference between Y^{3+} and Ru^{5+} is significant (0.260 Å), which stabilizes the ordered structure.^{29} Our calculations use its experimental lowtemperature ordered structure (the details are shown in Supplementary Materials).
Spectral functions
We show in Fig. 2 spectral functions of Ba_{2}YRuO_{6} in both paramagnetic state (panel a) and typeI antiferromagnetic state (panel b) (A spinresolved spectral function of a single Ru atom in the typeI antiferromagnetic state is shown in Supplementary Materials). The blue curves are total spectral functions and the red curves are Ru t_{2g} projected spectral functions. The paramagnetic state is insulating with a Mott gap of about 0.2 eV. However, the typeI antiferromagnetic state shows interesting properties: the lower and upper Hubbard bands of Ru t_{2g} states exhibit sharper peaks, compared to those in the paramagnetic state, but the Mott gap is closed and the state is metallic.
We first note that the transition shown in Fig. 2 is opposite to Slater transition.^{30,31} Whereas both transitions are driven by antiferromagnetic ordering, in Slater transition a gap is opened in a paramagnetic metal with the occurrence of antiferromagnetic ordering, whereas Fig. 2 shows that the appearance of antiferromagnetic ordering closes the gap of a paramagnetic insulator and induces a metallic state.
Second, we show that the gap closing has nothing to do with charge transfer between Ru^{5+} and Y^{3+} ions.^{32,33} In Fig. 3, we show the spectral functions of Ba_{2}YRuO_{6} in a larger energy window. In addition to total and Ru t_{2g} projected spectral functions, we also show Ru e_{g} projected spectral function (green), Y t_{2g} projected spectral function (purple) and O p projected spectral function (orange). We find that Ru e_{g} states have higher energy than Ru t_{2g} states due to crystal field splitting, and Y t_{2g} state have even higher energy than Ru e_{g} states. This is consistent with the nominally empty d configuration of Y^{3+}. We note that even in plain DFTPBE calculations (without Hubbard U), Y t_{2g} states have higher energy than Ru t_{2g} and e_{g} states (see Fig. 1 in Supplementary Materials). This indicates that there is no charge transfer between Y^{3+} and Ru^{5+} ions in both paramagnetic and typeI antiferromagnetic states of Ba_{2}YRuO_{6}.
Orbitalselective transition
In this section, we show that the gap closing in Ba_{2}YRuO_{6} is driven by the orbitalselective insulator–metal transition as we mentioned in the Introduction. Figure 4 is the key result, in which we decompose the spectral function of Ba_{2}YRuO_{6} into three Ru t_{2g} orbital projections, in the paramagnetic state and in the typeI antiferromagnetic state (the ordering wave vector \({\mathbf{Q}} = \frac{{2\pi }}{a}(001)\)). We can see that in the paramagnetic state, three Ru t_{2g} orbitals have identical projected spectral functions due to cubic symmetry. A small Mott gap is opened up in the paramagnetic state. However, in the typeI antiferromagnetic state, three Ru t_{2g} orbitals have different projected spectral functions. Ru d_{xy} orbital exhibits metallic property with the gap closed, in contrast to Ru d_{xy} orbital in the paramagnetic state (column 1 of Fig. 4). On the other hand, Ru d_{xz} and Ru d_{yz} orbitals show stronger insulating property with the gap size increased and the peaks of lower/upper Hubbard bands becoming sharper (columns 2 and 3 of Fig. 4).
The orbitalselectivity, i.e. which Ru t_{2g} orbital undergoes the insulator–metal transition with the occurrence of typeI antiferromagnetic ordering is related to the Ru magnetic moment configuration, which is characterized by the ordering wave vector Q. For typeI antiferromagnetic ordering, there are three ordering wave vectors: \({\mathbf{Q}} = \frac{{2\pi }}{a}(001)\), \(\frac{{2\pi }}{a}(010)\) or \(\frac{{2\pi }}{a}(100)\), where a is the lattice constant. They correspond to different axes along which Ru magnetic moments alternate their directions between adjacent atomic planes. As is shown in Fig. 5, for each ordering wave vector Q, Ru magnetic moments are parallel in \(\frac{1}{3}\) of nearestneighbor Ru pairs and are antiparallel in the other \(\frac{2}{3}\) of nearestneighbor Ru pairs. The Ru magnetic moments that are parallel single out a plane and the Ru t_{2g} orbital that lies in the plane (rather than out of the plane) undergoes an insulator–metal transition. For example, in Fig. 5a, the ordering wave vector \({\mathbf{Q}} = \frac{{2\pi }}{a}(001)\) and the parallel Ru magnetic moments single out xy plane. Together, we show an isovalue surface, which is the spinresolved (orange and green) integrated local spectral function around the Fermi level (Supplementary Materials for details). The shape of the isovalue surface clearly indicates that the manybody density of states close to the Fermi level has a d_{xy} character, which is consistent with Fig. 5. In Fig. 5b, c, we repeat the calculations with different ordering wave vectors \({\mathbf{Q}} = \frac{{2\pi }}{a}(010)\) and \(\frac{{2\pi }}{a}(100)\). As we change Q, the states at the Fermi surface show d_{xz} and d_{yz} orbital character, respectively. This partial ‘ferromagnetic coupling’ in the typeI antiferromagnetic ordering is the key to explain the orbitalselective insulator–metal transition. In Fig. 5a, the Ru magnetic moments are ferromagnetically coupled in the xy plane and antiferromagnetically coupled in the xz and yz planes. The largest hopping matrix element for Ru d_{xy} orbital is the one in the xy plane between the Ru nearestneighbors. In the xy plane, the parallel Ru magnetic moments facilitate scattering upon excitation and thus increase coherence and band width for Ru d_{xy} orbital^{34}. If the band width is large enough, the Mott gap is closed for the Ru d_{xy} orbital, which is exactly what Fig. 4b1 shows. Similarly, for Ru d_{xz} (d_{yz}) orbital, the largest hopping matrix element is the one in xz (yz) plane, but in that plane the Ru t_{2g} magnetic moments are antiparallel, which hinders scattering upon excitation and thus decreases band width and further increases band gap^{34} (A simple material with a halffilled t_{2g} shell is studied in Supplemenatray Materials to demonstrate the correlation between longrange magnetic ordering and band widths). We note in Fig. 4 that compared to the paramagnetic state, in the typeI antiferromagnetic state, the gaps of Ru d_{xz} and d_{yz} orbitals are indeed larger and the peaks of lower/upper Hubbard bands of Ru d_{xz} and d_{yz} orbitals become sharper. Applying the same analysis to different magnetic configurations in Fig. 5b, c shows that Ru d_{xz} (d_{yz}) undergoes the insulator–metal transition with the occurrence of typeI antiferromagnetic ordering of \({\mathbf{Q}} = \frac{{2\pi }}{a}(010)\) \(\left( {{\mathbf{Q}} = \frac{{2\pi }}{a}(100)} \right)\). We emphasize here that because both paramagnetic state and antiferromagnetic state in Fig. 4 are calculated at the same low temperature, it indicates that the occurrence of typeI antiferromagnetic ordering is the driving force to induce the orbitalselective insulator–metal transition.
Electric conductivity
A direct consequence of the electronic structure shown in Fig. 4 is anisotropic transport properties of Ba_{2}YRuO_{6} in a typeI antiferromagnetic state. We calculate electric conductivity using DFT + U method within linear response theory and semiclassical approximation framework.^{35,36} We explain that for longrange magnetically ordered states, because the selfenergy is small and its frequency dependence is weak, DFT + DMFT and DFT + U methods yield very similar results.
In DFT + U method, electric conductivity origins from intraband transitions, which can be calculated from band structure. Using linear response theory and semiclassical approximation, we have:^{35,36}
where f(ε) is the FermiDirac distribution, α, β = x, y, z, and τ is the relaxation time. Note that τ is not directly calculated by DFT + U method, but is treated as a parameter. Our calculations find that the offdiagonal components of electric conductivity vanish due to crystal symmetry (Ba_{2}YRuO_{6} has a Fm3m structure). The diagonal components of electric conductivity have two independent values: σ_{xx} = σ_{yy} and σ_{zz}. This is because typeI antiferromagnetic ordering breaks cubic symmetry (given \(\frac{{2\pi }}{a}\) (001) ordering wave vector). Anisotropy in electric conductivity arises from the fact that in a typeI antiferromagnetic state (given \(\frac{{2\pi }}{a}\) (001) ordering wave vector), Ru d_{xy} orbital is metallic while Ru d_{xz} and d_{yz} orbitals are insulating. This means that intraband transitions contribute to σ_{xx} and σ_{yy}, but not to σ_{zz}. Our calculations find a finite electric conductivity σ_{xx} = σ_{yy} (Fig. 6) and a vanishing electric conductivity σ_{zz} = 0.
Magnetic energetics
We have shown that an orbitalselective insulator–metal transition can occur in ordered double perovskite Ba_{2}YRuO_{6} as the material transitions from the paramagnetic state into the typeI antiferromagnetic (AFM) state with decreasing temperatures. Although typeI AFM ordering has been observed in experiment (Table 1), as a selfconsistent check, we calculate other types of longrange magnetic orderings: ferromagnetic ordering (FM) and antiferromagnetic ordering with magnetic moments alternating directions along (111) axis (the ordering wave vector \({\mathbf{Q}} = \frac{{2\pi }}{a}\left( {\frac{1}{2}\frac{1}{2}\frac{1}{2}} \right)\) and we refer to it as typeII AFM) (Supplementary Materials for details). We use DFT + U method (with the same U_{Ru} and J_{Ru}) to calculate the energy difference between these magnetic orderings because technically (i) DFT + U method can calculate larger systems than DFT + DMFT method (we need an 80atom cell to calculate typeII antiferromagnetic ordering^{37}); (ii) DFT + U method can achieve much higher accuracy than CTQMCbased DFT + DMFT method.^{38} Due to the quantum Monte Carlo nature of CTQMC algorithm, the accuracy we can obtain from DFT + DMFT method is on the order of 10 meV per cell. DFT + U method can converge a total energy of 1 meV per cell accuracy or even higher. In addition, as we have mentioned in the previous section, DFT + DMFT and DFT + U methods produce consistent results for longrange ordered states. That is the physical reason why we may alternatively use DFT + U method to calculate the total energy for magnetically ordered states.
Using typeI AFM state as the reference, we find FM and typeII AFM are higher in energy than typeI AFM by 110 meV/f.u. and 37 meV/f.u., respectively. The result that FM has higher energy than typeI AFM shows that the nearestneighbor exchange coupling is indeed antiferromagnetic in nature. The reason that typeI AFM is more stable than typeII AFM is because in typeI AFM state, for each Ru magnetic moment, \(\frac{2}{3}\) of its nearestneighbor magnetic moments are antiparallel and the other \(\frac{1}{3}\) of its nearestneighbor magnetic moments are parallel; in typeII AFM state, for each Ru magnetic moment, half of its nearestneighbor magnetic moments are antiparallel and the other half are parallel. As the nearestneighbor Ru exchange coupling is intrinsically antiferromagnetic, and typeI AFM ordering has more antiferromagnetic coupled nearestneighbor pairs of Ru magnetic moments than typeII AFM ordering, this explains why typeI AFM ordering is more stable. Our results are consistent with the experimental measurements.^{20,21,22}
We note that the fcc lattice on which the magnetic ion Ru resides has ‘geometrical frustration’, therefore complicated magnetic orderings (noncollinear and/or noncoplanar etc.) are possible in the ground state.^{28,39,40} However, at finite temperatures, by the mechanism of ‘order by disorder’, collinear magnetic orderings are favored by thermal fluctuations^{18,19} and collinear typeI AFM ordering is indeed observed in experiments.^{20,21,22} In our current study, it is the first longrange magnetic ordering which emerges from a paramagnetic state that is relevant to the orbitalselective insulator–metal transition.
Spinorbit interaction
We notice that Ru has 4d orbitals and spinorbit (SO) interaction plays a more pronounced role in 4d and 5d magnetic ions than 3d magnetic ions. In this section, we discuss whether spinorbit interaction may affect the magnetically driven orbitalselective insulator–metal transition.
We note that currently DFT + DMFT + SO method is not feasible in multiorbital systems because spinorbit interaction induces an intrinsic sign problem in the CTQMC algorithm.^{38} But we find that in the antiferromagnetic (AFM) ordered state, the frequency dependence in the selfenergy is much weaker than that in the paramagnetic state (see the Supplementary Materials for details). This indicates that HartreeFock approximation is as good as DMFT to describe the AFM ordered state. Therefore we compare DFT + U and DFT + U + SO methods.
In the presence of spinorbit interaction, spin is directly coupled to crystal lattice. In typeI AFM state with \({\mathbf{Q}} = \frac{{2\pi }}{a}(001)\), we globally rotate all Ru magnetic moments in real space and find that they are stabilized along the zaxis.
Figure 7 shows the spectral functions for typeI AFM state of Ba_{2}YRuO_{6} (with an ordering wave vector \({\mathbf{Q}} = \frac{{2\pi }}{a}(001)\)), calculated using DFT + U method (panel a) and DFT + U + SO method (panel b). The red, blue, and green lines are the spectral functions projected onto Ru d_{xy}, Ru d_{xz}, and Ru d_{yz} orbitals, respectively. Using both methods, we find that with the ordering wave vector \({\mathbf{Q}} = \frac{{2\pi }}{a}(001)\), Ru d_{xz} and Ru d_{yz} orbitals are insulating while Ru d_{xy} orbital is metallic. This orbitaldependent feature is also consistent with the spectral function calculated by DFT + DMFT method (Fig. 4).
This result is in fact not surprising because in the current study, the magnetic ions of double perovskite oxides have a d^{3} configuration. Due to Hund’s rule, the three electrons fill three t_{2g} orbitals and form a spin \(S = \frac{3}{2}\) state. The orbital degree of freedom is completely quenched and the system is presumably well described by a spinonly Hamiltonian.^{28,40} Therefore including spinorbit interaction does not significantly change the electronic structure, as is shown in Fig. 7.
Phase diagram with Hubbard U
In the previous sections, we use a single value of Hubbard U_{Ru} to perform all the calculations. Now, we discuss the phase diagram as a function of Hubbard U_{Ru} with J_{Ru} fixed at 0.3 eV, calculated by DFT + DMFT method. We find that there are two critical values of Hubbard U (Fig. 8): (i) as U > U_{c1}, both the hightemperature paramagnetic state (PM) and the lowtemperature typeI antiferromagnetic state (AFM) are insulating; (ii) as U < U_{c2}, both the hightemperature PM and lowtemperature AFM states are metallic and (iii) as U_{c2} < U < U_{c1}, the hightemperature PM state is insulating and the lowtemperature AFM state is metallic. It is precisely in the region of U_{c2} < U < U_{c1} that the magnetically driven orbitalselective insulator–metal transition can occur at the magnetic critical temperature T_{N}. For Ba_{2}YRuO_{6}, we find U_{c1} = 3.2 eV and U_{c2} = 1.5 eV. Although the accurate value of Hubbard U for Ru is yet to be determined, the range set by U_{c1} and U_{c2} is achievable for a 4d transition metal ion. We also note that Fig. 8 shows two types of phase transition. One is the AFMmetallic to AFMinsulating transition on the Hubbard U axis (at low temperatures). The other is the PMinsulator to AFMmetal state transition as temperature decreases. Both types of phase transition are continuous. The Udriven phase transition is continuous because increasing U gradually separates the majority and minority spins of Ru d_{xy} orbital (given a \(\frac{{2\pi }}{a}\) (001) ordering wave vector) and eventually opens a gap. The PMinsulator to AFMmetal transition is continuous too, because the gap closing of Ru d_{xy} orbital (given a \(\frac{{2\pi }}{a}\) (001) ordering wave vector) is achieved by gradually aligning the Ru d_{xy} moments and increasing the band width of Ru d_{xy} orbital till the majority and minority spins of Ru d_{xy} orbital overlap in energy.
Discussion
We have provided a comprehensive study on the magnetically driven orbitalselective insulator–metal transition in Ba_{2}YRuO_{6} in section Results. However, the transition is not unique to Ba_{2}YRuO_{6}; it is general to ordered double perovskite oxides with one d^{3} magnetic ion and one nonmagnetic ion as long as the material is a Mott insulator that lies close to the metalinsulator phase boundary in the paramagnetic state.
In this section, we study other four ordered double perovskite oxides that are listed in Table 1 and discuss the connection of our theoretical results to the available experimental data. Ba_{2}ScRuO_{6}, Sr_{2}YRuO_{6}, Sr_{2}ScOsO_{6}, and Sr_{2}YOsO_{6} have been synthesized and their experimental structures (used in the calculations) are shown in Supplementary Materials. We use DFT + DMFT method to calculate the U phase diagram for these four double perovskite oxides (the Hund’s J_{Ru} and J_{Os} are fixed at 0.3 eV^{41,42,43}). The results are shown in Fig. 8. Like Ba_{2}YRuO_{6}, Ba_{2}ScRuO_{6} also crystallizes in the cubic Fm3m structure (space group No. 225 Fm3m).^{23} However, the lattice constant of Ba_{2}ScRuO_{6} is smaller than that of Ba_{2}YRuO_{6} by about 2%,^{20,23} which leads to larger hopping matrix elements. Therefore the critical Hubbard U_{c1} and U_{c2} for Ba_{2}ScRuO_{6} are both larger than those for Ba_{2}YRuO_{6}. On the other hand, Sr_{2}YRuO_{6}, Sr_{2}ScOsO_{6} and Sr_{2}YOsO_{6} all crystallize in a distorted structure (space group No. 14 P2_{1}/n).^{24,25,26,27} Due to rotations and tilts of RuO_{6} and OsO_{6} oxygen octahedra, metaloxygenmetal bond angle is smaller than that in a cubic structure (In Sr_{2}YRuO_{6}, Sr_{2}ScOsO_{6}, and Sr_{2}YOsO_{6}, the average metaloxygenmetal bond angle is about 160°, while in Ba_{2}YruO_{6} and Ba_{2}ScRuO_{6}, the metaloxygenmetal bond angle is 180°). This results in reduced hopping and therefore the critical Hubbard U_{c1} and U_{c2} for all three double perovskite oxides are smaller than those for Ba_{2}YRuO_{6}. We note that while in our calculations there is uncertainty about the accurate value of Hubbard U on transition metal ions (Ru^{5+} and Os^{5+}), different ‘isoelectronic’ materials (Table 1) provide a fairly large range of U in which the predicted transition can occur (shown in Fig. 8).
Next we turn to available experimental data. Magnetic properties of the five materials listed in Table 1 have been carefully studied.^{21,22,23,24,25,26,27} TypeI antiferromagnetic ordering has been observed in all these double perovskite oxides. Remarkably, Cao et al. observes a sharp anomaly in the electric resistivity ρ(T) of Sr_{2}YRuO_{6} at the magnetic ordering temperature T_{N}.^{26} Ref. ^{26} measures two types of resistivity: ρ_{ab}(T) for the basal plane and ρ_{c}(T) for the outofplane caxis. As the temperature T is above the Néel temperature T_{N}, both ρ_{ab}(T) and ρ_{c}(T) exhibit insulating properties: they rapidly increase as the temperature decreases. However, just below T_{N}, ρ_{ab}(T) exhibits a clear anomaly: it changes the sign of its slope and slowly decreases with lowering temperatures (a metalliclike behavior). Interestingly, this anomaly is only evident in ρ_{ab}(T) but is absent in ρ_{c}(T). ρ_{c}(T) exhibits insulating property both above and below T_{N} with a weak “kink” feature at T_{N}. Just below T_{N}, ρ_{c}(T) increases slightly faster with decreasing temperatures than it does just above T_{N}. Our predicted phase transition provides an explanation for the anomaly observed in the resistivity of Sr_{2}YRuO_{6} at T_{N}. Considering that the magnetic ordering wave vector is along the caxis,^{26} the anomaly in ρ_{ab}(T) shows that Ru d_{xy} orbital (which lies in the ab plane) undergoes an insulator–metal transition at T_{N} (see panel 1 of Fig. 4). On the other hand, Ru d_{xz} and d_{yz} orbitals remain insulating at T_{N} and therefore the anomaly is not observed in ρ_{c}(T). The gap size associated with Ru d_{xz} and d_{yz} orbitals increases at T_{N} (see panels 2 and 3 of Fig. 4), which explains the “kink” behavior at T_{N}.
However, as the temperature further decreases, ρ_{ab} of Sr_{2}YRuO_{6} undergoes a second phase transition from an antiferromagnetic metal to an antiferromagnetic insulator.^{26} According to the authors of ref. ^{26} the second phase transition arises from the fact that Dzyaloshinskii–Moriya interaction (DMinteraction) cants Ru spins and induces weak ferromagnetism, which eventually reopens the gap.
The second phase transition is interesting by itself and deserves further investigation, but is outside the scope of our current study. In our calculations, we consider typeI antiferromagnetic state (no weak ferromagnetism) in all material candidates.
Because, Sr_{2}YRuO_{6} has a distorted structure and the presence of DMinteraction complicates the analysis of transport measurements, we suggest that a very similar compound Ba_{2}YRuO_{6} is a cleaner system to observe our predicted phase transition. Ba_{2}YRuO_{6} has a cubic structure (space group Fm3m) and inversion symmetry of Fm3m space group forbids DMinteraction. Without the second phase transition, ρ_{ab} should show a turningpoint at T_{N} (this has already been observed in Sr_{2}YRuO_{6}) and then monotonically decrease with decreasing temperatures.
Another cleaner material candidate is probably Ba_{2}ScRuO_{6}, which also crystallizes in a Fm3m structure. Ref. ^{23} shows that in double perovskite Ba_{2}ScRuO_{6}, a doublekink feature is observed in its magnetic susceptibility, which indicates two ordering temperatures (T_{N} = 31 and 44 K). However, only one peak is observed in its heat capacity, which corresponds to the higher ordering temperature. The origin of the transition at the lower ordering temperature is not clear. A measurement of lowtemperature electric resistivity for Ba_{2}ScRuO_{6} is desirable, which will probe the predicted orbitalselective transition and may help unlock the puzzle of two ordering temperatures.
Finally, we mention that in order to observe the transition, we need the material to lie close to the metalinsulator phase boundary in the paramagnetic state (but still on the insulating side). Therefore, for 3d transition metal ions such as Mn^{4+} (d shell configuration 3d^{3}), because a typical U is about 4–5 eV (larger than all the U_{c1} calculated), we do not expect to observe the orbitalselective insulator–metal transition in 3d double perovskite oxides, such as Sr_{2}TiMnO_{6}. For 4d, 5d transition metal ions such as Ru^{5+} and Os^{5+}, because the Hubbard U gets smaller and the metal d band width gets larger, complex oxides that contain 4d and 5d transition metal ions are much closer to metalinsulator phase boundary in paramagnetic state and therefore they are more promising candidate materials to observe the transition we predict here.
In conclusion, we use firstprinciples calculations to demonstrate a magnetically driven orbitalselective insulator–metal transition in ordered double perovskite oxides A_{2}BB′O_{6} with a nonmagnetic ion B (Y^{3+} and Sc^{3+}) and a d^{3} magnetic ion B′ (Ru^{5+} and Os^{5+}). With decreasing temperatures, as the material transitions from paramagnetic insulating (Mott) state to typeI antiferromagnetic (AFM) state, one t_{2g} orbital of the magnetic ion becomes metallic while the other two t_{2g} orbitals of the magnetic ion become more insulating. The origin of the transition arises from ‘geometric frustration’ of a fcc lattice, which enforces some magnetic moments to be ferromagnetically coupled in an antiferromagnetic ordering. The orbitalselectivity is associated with the ordering wave vector Q of typeI AFM state. We hope our study can stimulate further experiments to provide more compelling evidence for the predicted electronic phase transition in ordered double perovskite oxides that contain 4d and 5d transition metal ions.
Methods
We perform firstprinciples calculations by using density functional theory (DFT)^{44,45} plus Hubbard U correction (DFT + U),^{46} DFT plus Hubbard U correction and spinorbit interaction (DFT + U + SO)^{47} and DFT plus dynamical mean field theory (DFT + DMFT).^{3} Both DFT + U and DFT + U + SO methods are implemented in the Vienna Ab initio Simulation Package (VASP).^{48,49} In DMFT method, a continuoustime quantum Monte Carlo algorithm (CTQMC)^{50} is used to solve the impurity problem.^{38} The impurity solver was developed by K. Haule’s group at Rutgers University.^{51} In DMFT calculations, both paramagnetic and antiferromagnetic states for all material candidates are computed at a temperature of 116 K. Convergence of key results is checked at 58 K and no significant changes are observed in electronic structure.
For longrange magnetically ordered calculations using DFT + U, DFT + U + SO and DFT + DMFT as well as paramagnetic calculations using DFT + DMFT, a nonspinpolarized exchange correlation functional is used in the DFT component.^{52,53} The spin symmetry is broken by the Hubbard U and Hund’s J interactions.
Electronic structures are calculated using DFT + DMFT method. Magnetic energy differences are calculated using DFT + U method and effects of spinorbit (SO) coupling are studied by using DFT + U + SO method.
In the DFT part, we use generalized gradient approximation with Perdew–Burke–Ernzerhof (PBE) parametrization^{54} for the exchange correlation functional. For DFT + DMFT calculations, the correlated metal d orbitals and the oxygen p orbitals are constructed using maximally localized Wannier functions.^{55} As for the interaction strengths, we first use one set of interaction parameters U_{Ru} = 2.3 eV and J_{Ru} = 0.3 eV to show the representative electronic structure and then study Hubbard U dependence. We show that the transition we predict can occur in a range of interaction strength for all candidate materials. We note that recent calculations of SrRu_{2}O_{6}^{56,57,58} show that for a t_{2g}p model, U_{Ru} is about 5 eV from constrained randomphaseapproximation (cRPA),^{57,58} which is larger than the upper limit U_{c1} below which our predicted transition can be observed. However, the “kink” observed in the resistivity of Sr_{2}YRuO_{6} indicates that the antiferromagnetic ordered state of Sr_{2}YRuO_{6} exhibits metallike behavior around T_{N}, implying that the interaction strength U_{Ru} in Sr_{2}YRuO_{6} might be smaller than that in SrRu_{2}O_{6} probably due to different crystal structure, or singlesite DMFT method with a cRPA value of interaction strength may favor the insulating phase. This deserves further study in future work.
In DFT + U, DFT + U + SOC and DFT + DMFT calculations, we use a chargeonly exchange correlation functional (i.e. not depending on spin density) in the DFT component. A chargeonly double counting is also used in all methods. Previous works show that this choice can avoid an unphysically large exchangesplitting in spindependent exchange correlation functionals.^{52,53,59}
More computational details are found in Supplementary Materials.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
 1.
Imada, M., Fujimori, A. & Tokura, Y. Metalinsulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
 2.
Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996).
 3.
Kotliar, G. et al. Electronic structure calculations with dynamical meanfield theory. Rev. Mod. Phys. 78, 865–951 (2006).
 4.
Anisimov, V. I., Nekrasov, I. A., Kondakov, D. E., Rice, T. M. & Sigrist, M. Orbitalselective Mottinsulator transition in Ca_{2−x}Sr_{x}RuO_{4}. Eur. Phys. J. B 25, 191–201 (2002).
 5.
Nakatsuji, S. & Maeno, Y. Quasitwodimensional Mott transition system. Phys. Rev. Lett. 84, 2666–2669 (2000).
 6.
Fang, Z., Nagaosa, N. & Terakura, K. Orbital dependent phase control in Ca_{2−x}Sr_{x}RuO_{4}. Phys. Rev. B 69, 045116 (2004).
 7.
Ko, E., Kim, B. J., Kim, C. & Choi, H. J. Strong orbitaldependent dband hybridization and fermisurface reconstruction in metallic Ca_{2−x}Sr_{x}RuO_{4}. Phys. Rev. Lett. 98, 226401 (2007).
 8.
Koga, A., Kawakami, N., Rice, T. M. & Sigrist, M. Orbitalselective Mott transitions in the degenerate Hubbard model. Phys. Rev. Lett. 92, 216402 (2004).
 9.
De’Medici, L., Georges, A. & Biermann, S. Orbitalselective Mott transition in multiband systems: slavespin representation and dynamical meanfield theory. Phys. Rev. B 72, 205124 (2005).
 10.
Ferrero, M., Becca, F., Fabrizio, M. & Capone, M. Dynamical behavior across the Mott transition of two bands with different bandwidths. Phys. Rev. B 72, 205126 (2005).
 11.
Liebsch, A. Novel Mott transitions in a nonisotropic twoband Hubbard model. Phys. Rev. Lett. 95, 116402 (2005).
 12.
Biermann, S., de’ Medici, L. & Georges, A. Nonfermiliquid behavior and doubleexchange physics in orbitalselective Mott systems. Phys. Rev. Lett. 95, 206401 (2005).
 13.
Liebsch, A. & Ishida, H. Subband Filling and Mott transition in Ca_{2−x}Sr_{x}RuO_{4}. Phys. Rev. Lett. 98, 216403 (2007).
 14.
Hoshino, S. & Werner, P. Spontaneous orbitalselective Mott transitions and the jahnteller metal of A _{3}C_{60}. Phys. Rev. Lett. 118, 177002 (2017).
 15.
Werner, P. & Millis, A. J. Highspin to lowspin and orbital polarization transitions in multiorbital Mott systems. Phys. Rev. Lett. 99, 126405 (2007).
 16.
Wu, J., Phillips, P. & Castro Neto, A. H. Theory of the magnetic moment in iron pnictides. Phys. Rev. Lett. 101, 126401 (2008).
 17.
De’Medici, L., Hassan, S. R., Capone, M. & Dai, X. Orbitalselective Mott transition out of band degeneracy lifting. Phys. Rev. Lett. 102, 126401 (2009).
 18.
Henley, C. L. Ordering by disorder: groundstate selection in fcc vector antiferromagnets. J. Appl. Phys. 61, 3962–3964 (1987).
 19.
Henley, C. L. Ordering due to disorder in a frustrated vector antiferromagnet. Phys. Rev. Lett. 62, 2056–2059 (1989).
 20.
Aharen, T. et al. Magnetic properties of the S = 3/2 geometrically frustrated double perovskites La_{2}LiRuO_{6} and Ba_{2}YRuO_{6}. Phys. Rev. B 80, 134423 (2009).
 21.
Battle, P. & Jones, C. The crystal and magnetic structures of Sr_{2}LuRuO_{6}, Ba_{2}YRuO_{6}, and Ba_{2}LuRuO_{6}. J. Solid State Chem. 78, 108–116 (1989).
 22.
Carlo, J. P. et al. Spin gap and the nature of the 4d ^{3} magnetic ground state in the frustrated fcc antiferromagnet Ba_{2}YRuO_{6}. Phys. Rev. B 88, 024418 (2013).
 23.
Kayser, P. et al. Magnetic and structural studies of Sc containing ruthenate double perovskites A_{2}ScRuO_{6} (A = Ba, Sr). Inorg. Chem. 56, 9009–9018 (2017).
 24.
Taylor, A. E. et al. Magnetic order and electronic structure of the 5d ^{3} double perovskite Sr_{2}ScOsO_{6}. Phys. Rev. B 91 100406(R) (2015).
 25.
Paul, A. K. et al. Magnetically frustrated double perovskites: synthesis, structural properties, and magnetic order of Sr_{2} BOsO_{6} (b = Y, In, Sc). Z. für Anorg. und Allg. Chem. 641, 197–205 (2015).
 26.
Cao, G., Xin, Y., Alexander, C. S. & Crow, J. E. Weak ferromagnetism and spincharge coupling in singlecrystal Sr_{2}YRuO_{6}. Phys. Rev. B 63, 184432 (2001).
 27.
Battle, P. & Macklin, W. The crystal and magnetic structures of Sr_{2}YRuO_{6}. J. Solid State Chem. 52, 138–145 (1984).
 28.
Chen, G. & Balents, L. Spinorbit coupling in d ^{2} ordered double perovskites. Phys. Rev. B 84, 094402 (2011).
 29.
Barnes, P. W., Lufaso, M. W. & Woodward, P. M. Structure determination of A _{2} M ^{3+}TaO_{6} and A_{2}M^{3+}NbO_{6} ordered perovskites: octahedral tilting and pseudosymmetry. Acta Crystallogr. Sect. B 62, 384–396 (2006).
 30.
Slater, J. C. Magnetic effects and the HartreeFock equation. Phys. Rev. 82, 538–541 (1951).
 31.
Calder, S. et al. Magnetically driven metalinsulator transition in NaOsO_{3}. Phys. Rev. Lett. 108, 257209 (2012).
 32.
Chen, H., Millis, A. & Marianetti, C. Engineering correlation effects via artificially designed oxide superlattices. Phys. Rev. Lett. 111, 116403 (2013).
 33.
Kleibeuker, J. E. et al. Electronic reconstruction at the isopolar LaTiO^{3}/LaFeO_{3} interface: an Xray photoemission and densityfunctional theory study. Phys. Rev. Lett. 113, 237402 (2014).
 34.
Salamon, M. B. & Jaime, M. The physics of manganites: structure and transport. Rev. Mod. Phys. 73, 583–628 (2001).
 35.
Gajdoš, M., Hummer, K., Kresse, G., Furthmüller, J. & Bechstedt, F. Linear optical properties in the projectoraugmented wave methodology. Phys. Rev. B 73, 045112 (2006).
 36.
Judith Harl’s thesis (http://othes.univie.ac.at/2622/).
 37.
Vasala, S. & Karppinen, M. A _{2} B′B″O_{6} perovskites: a review. Prog. Solid State Chem. 43, 1–36 (2015).
 38.
Gull, E. et al. Continuoustime Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).
 39.
Tiwari, R. & Majumdar, P. Noncollinear magnetic order in the double perovskites: double exchange on a geometrically frustrated lattice. Int. J. Mod. Phys. B 27, 1350018 (2013).
 40.
Chen, G., Pereira, R. & Balents, L. Exotic phases induced by strong spinorbit coupling in ordered double perovskites. Phys. Rev. B 82, 174440 (2010).
 41.
Dang, H. T., Mravlje, J., Georges, A. & Millis, A. J. Electronic correlations, magnetism, and Hund’s rule coupling in the ruthenium perovskites SrRuO_{3} and CaRuO_{3}. Phys. Rev. B 91, 195149 (2015).
 42.
Han, Q., Dang, H. T. & Millis, A. J. Ferromagnetism and correlation strength in cubic barium ruthenate in comparison to strontium and calcium ruthenate: a dynamical meanfield study. Phys. Rev. B 93, 155103 (2016).
 43.
Lo Vecchio, I. et al. Electronic correlations in the ferroelectric metallic state of LiOsO_{3}. Phys. Rev. B 93, 161113 (2016).
 44.
Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964).
 45.
Kohn, W. & Sham, L. J. Selfconsistent equations including exchange and correlation effects. Phys. Rev. 140, A1133 (1965).
 46.
Liechtenstein, A. I., Anisimov, V. I. & Zaanen, J. Densityfunctional theory and strong interactions: orbital ordering in MottHubbard insulators. Phys. Rev. B 52, R5467 (1995).
 47.
Takeda, T. The scalar relativistic approximation. Z. für Phys. B Condens. Matter Quanta 32, 43–48 (1978).
 48.
Kresse, G. & Furthmüller, J. Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
 49.
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
 50.
Werner, P., Comanac, A., De’ Medici, L., Troyer, M. & Millis, A. J. Continuoustime solver for quantum impurity models. Phys. Rev. Lett. 97, 076405 (2006).
 51.
Haule, K. Quantum Monte Carlo impurity solver for cluster dynamical meanfield theory and electronic structure calculations with adjustable cluster base. Phys. Rev. B 75, 155113 (2007).
 52.
Park, H., Millis, A. J. & Marianetti, C. A. Density functional versus spindensity functional and the choice of correlated subspace in multivariable effective action theories of electronic structure. Phys. Rev. B 92, 035146 (2015).
 53.
Chen, H. & Millis, A. J. Spindensity functional theories and their +U and +J extensions: a comparative study of transition metals and transition metal oxides. Phys. Rev. B 93, 045133 (2016).
 54.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
 55.
Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).
 56.
Streltsov, S., Mazin, I. I. & Foyevtsova, K. Localized itinerant electrons and unique magnetic properties of SrRu_{2}O_{6}. Phys. Rev. B 92, 134408 (2015).
 57.
Hariki, A., Hausoel, A., Sangiovanni, G. & Kuneš, J. DFT + DMFT study on soft moment magnetism and covalent bonding in SrRu_{2}O_{6}. Phys. Rev. B 96, 155135 (2017).
 58.
Okamoto, S., Ochi, M., Arita, R., Yan, J. & Trivedi, N. Localizeditinerant dichotomy and unconventional magnetism in SrRu_{2}O_{6}. Sci. Rep. 7, 11742 (2017).
 59.
Chen, J., Millis, A. J. & Marianetti, C. A. Density functional plus dynamical meanfield theory of the spincrossover molecule Fe(phen)_{2}(NCS)_{2}. Phys. Rev. B 91, 241111 (2015).
Acknowledgements
We are grateful to useful discussion with Andrew J. Millis, Jernej Mravlje, Sohrab IsmailBeigi, Gang Chen, Yuan Li, and Hongjun Xiang. The work is funded by the National Science Foundation of China under the grant No. 11774236. Computational facilities are provided via Extreme Science and Engineering Discovery Environment (XSEDE).
Author information
Affiliations
Contributions
H.C. conceived the project, performed firstprinciples calculations, analyzed data, and wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, H. Magnetically driven orbitalselective insulator–metal transition in double perovskite oxides. npj Quant Mater 3, 57 (2018). https://doi.org/10.1038/s4153501801312
Received:
Accepted:
Published:
Further reading

Linear magnetoresistance with a universal energy scale in the strongcoupling superconductor Mo8Ga41 without quantum criticality
Physical Review B (2020)

Muon spin rotation and neutron scattering investigations of the B site ordered double perovskite Sr2DyRuO6
Physical Review B (2020)

Realization of the orbitalselective Mott state at the molecular level in Ba3LaRu2O9
Physical Review Materials (2020)

Experimental and theoretical determination of physical properties of Sm2Bi2Fe4O12 ferromagnetic semiconductors
Journal of Materials Chemistry C (2020)

Study of the microstructure and the optical, electrical, and magnetic feature of the Dy2Bi2Fe4O12 ferromagnetic semiconductor
Journal of Materials Research and Technology (2020)