Spatial inhomogeneity and the metal-insulator transition in Ca$_3$(Ru$_{1-x}$Ti$_x$)$_2$O$_7$

Turning a pristine Mott insulator into a correlated metal by chemical doping is a common procedure in strongly correlated materials physics, e.g. underlying the phenomenology of high-$T_c$ cuprates. The ruthenate bilayer compound Ca$_3$Ru$_2$O$_7$ is a prominent example of a reversed case, namely a correlated metal at stoichiometry that realizes a transition into an insulating state via Ti doping. We here investigate this puzzling metal-insulator transition (MIT) by first-principles many-body theory and elucidate a challenging interplay between electronic correlations and symmetry breakings on the Ru sublattice. While average effects on the Ca$_3$Ru$_2$O$_7$ crystal structure are still relevant, key to the MIT is the cooperation of electronic correlations with the spatial inhomogeneity in the defect regime. Together they give rise to the emergence of site-selective Mott criticality and competing orbital-ordering tendencies.

Introduction.-Theinterplay of various degrees of freedom, e.g. of charge, orbital, spin or structural kind, is key to an understanding of many realistic metalinsulator transitions (MITs) in nature 1 .In this respect, the Ruddlesden-Popper series of n-layered calcium ruthenates Ca n+1 Ru n O 3n+1 poses a particularly challenging problem.It is agreed that electronic correlations arise from lowspin Ru 4+ (4d 4 ) sites in these compounds, formally locating them in a Hund-metal [2][3][4] regime.While the n → ∞ perovskite CaRuO 3 is metallic 5 with competing magnetic interactions 6 down to lowest temperatures, the distorted perovskite Ca 2 RuO 4 (n=1) undergoes a paramagnetic MIT at T MIT = 357 K and displays antiferromagnetic (AFM) order below T N = 110 K 7 .From these limiting cases, Mott criticality is expectedly intricate in the bilayer system.And indeed, though Ca 3 Ru 2 O 7 shows several electronic and magnetic transitions for T < 100 K [8][9][10][11][12] , a robust insulating state is not reached at stoichiometry.With its non-centrosymmetric B b2 1 m space group, the bilayer ruthenate marks the case of a 'polar metal' [13][14][15] , and has gained recent strong interest due to a complex fermiology 10,[16][17][18] .Doping of about 5% titanium gives rise to a MIT in the bilayer compound 12,19 at T MIT ∼ 80 K.The magnetic order switches from the stoichiometric A-type ordering of AFMcoupled ferromagnetic bilayers to G-type AFM ordering.There is apparently no paramagnetic Mott-insulating phase in Ca 3 (Ru 1−x Ti x ) 2 O 7 .Weakly-localized states are already observed for very small Ti doping 20 , and perculative behavior is also detected 12 .According to Ke et al. 19 , the substitutional dopants enter as Ti 4+ (3d 0 ) impurities, therefore do not provide any significant charge doping.Hence the doping-induced MIT has originally been associated with the blocking of hopping paths 19 .Furthermore, the averaged crystal structure of the Ti-doped bilayer 12 displays also an enhanced two-dimensionality of the bilayers as well as a somewhat increased tilting of the RuO 6 octahedra.These global structural changes lead to a larger averaged crystalfield (CF) splitting ∆ between the three partially occupied Ru(4d) states m = x y, xz/y z of t 2g character.In fact, a large ∆ is the major driving force for the MIT in Ca 2 RuO 4 21 .
Since the energy scales in Ca 3 Ru 2 O 7 are generally smaller than in the latter single-layer ruthenate, the competition between the various degrees of freedom is much more subtle.In this work, the goal is to unveil the detailed cooperation of defect physics and electronic correlations that drive the MIT in Ti-doped Ca 3 Ru 2 O 7 .By means of the realspace combination of density functional theory (DFT) and dynamical mean-field theory (DMFT) applied to a defect supercell with 6.25% Ti concentration in a fully charge selfconsistent manner, we profoundly account for average and local effects on an equal footing.While average effects from doping are notable, the spatial inhomogeneity introduced by Ti defects and cooperating with electron correlations is the crucial driving force towards the insulating phase.
Theoretical approach.-Chargeself-consistent DFT+DMFT [22][23][24] is used to access the correlated electronic structure.For the DFT part, a mixed-basis pseudopotential method 25,26 , employing the generalized-gradient approximation in the Perdew-Burke-Ernzerhof form 27 , is put into practise.Spin-orbit coupling is neglected in this work.The multi-single-site DMFT impurity problems encountered in the basic unit cell as well as in the defect supercell are solved by the hybridization-expansion continuous-time quantum Monte Carlo scheme 28 , as implemented in the TRIQS code 29,30 .The correlated subspace consists of the effective transition-metal (TM) t 2g Wannier-like functions w m (t 2g ), i.e. is locally threefold.These functions are obtained from the projected-local-orbital formalism 31,32 , using as projection functions the linear combinations of atomic-like t 2g orbitals that diagonalize the TM local w m (t 2g )-orbital density matrix on each site.
Local Coulomb interactions in Slater-Kanamori form are parametrized by a Hubbard U and a Hund's exchange J H .The intraorbital interaction U on the Ru sites is treated as a parameter, ranging at most from 1.5 eV to 7.5 eV.The Hund's exchange is chosen J H = 0.4 eV for U < 2 eV and J H = 0.7 eV for U > 2 eV, in order to allow for comparison with previous theory work on ruthenates 21 .For the Ti sites, a value U = 5 eV is chosen to account for the fact that interactions in the Ti(3d) shell are larger than in the Ru(4d) shell.To obtain the spectral information, analytical continuation from Matsubara space by the maximum-entropy method and the Padé method is performed.
-To set the stage, we start with the correlated electronic structure of stoichiometric Ca 3 Ru 2 O 7 .Figure 1a displays the experimental unit cell 13   Note that the given CF splitting is about half the value of ∆ ∼ 320 meV in single-layer Ca 2 RuO 4 21 .
The correlated spectral data at T = 100 K is depicted in Figs.1b-d.The compound is metallic, but compared to DFT, strong renormalization and loss of coherence is revealed at low energy, in accordance with angle-resolved photoemission 10,16,17 .Significant inplane anisotropy in the correlated fermiology takes place between M a and M b , in line with experiment 10 .A more thorough investigation of further lowertemperature dispersions asks for an inclusion of spin-orbit coupling, and is not an objective of the present study.Finally, the local spectral functions show that the xz/y z states are stronger correlated with a lower Hubbard band of enhanced weight and deeper energy location of ∼ −2.9 eV than the x y states.
Local electronic structure of Ca 3 (Ru 0.9375 Ti 0.0625 ) 2 O 7 .-Inorder to describe the bilayer ruthenate with finite Ti doping, we start from the experimentally-averaged (EA) struc- ture at 10 K with 5% Ti doping 19 .The corresponding experimental system is insulating below T MIT ∼ 80 K.As an averaged structure, the atom-number size of the primitive cell is identical to the one at stoichiometry, but with different lattice parameters a = 5.38 Å, b = 5.56 Å and c = 19.37Å, as well as modified atomic positions.Thus the effect of Ti impurities is only taken into account implicitly via an averaged structure modification.The effective CF splitting in this structure amounts to ∆ = −171 eV in DFT, hence is about 20 meV larger than in the stoichiometric case.This is mainly attributed to an enhanced two-dimensionality and increaced tilting of the RuO 6 octahedra.It can be seen from the dashed lines in Fig. 2b that this treatment of Ti doping is not sufficient to render the system insulating for a reasonable value of U = 4.25 eV, but electronic correlations are somewhat enhanced compared to the stoichiometric case.
Let us turn to the supercell description of Ti-doped Ca 3 Ru 2 O 7 .A 192-atom-site cell (cf.Fig. 2a) is constructed, starting from the EA structure and introducing two Ti impurities in adjacent bilayers, i.e. each bilayer carries one substitutional Ti defect.There are 32 TM sites in the defect supercell, 30 of Ru and 2 of Ti kind.Fixing the scaled EA lattice parameters, we structurally relax this supercell within DFT+U assuming G-type AFM order.No symmetry constraints are enforced in the structural relaxation.This leads to site-symmetry breakings, not unexpected in this puzzling polar-metal system 33 .For the DFT+DMFT investigation at a system temperature of T = 40 K, we identify 14 Ru-site classes as symmetry inequivalent.Together with both symmetry-equivalent Ti sites, there are hence 15 coupled impurity problems solved at each self-consistency step.As in previous DFT calculations 19 , the Ti impurities are indeed of Ti 4+ (3d 0 ) kind, and the t 2g electronic spectrum is accordingly located high in energy within the unoccupied region (see green part in Fig. 2b).Note that local structural relaxation shifts the Ti spectrum to somewhat smaller energies.The resulting CF splitting on the Ru sites is however distributed over a surprisingly large energy window [−99, −283] eV.In order to simplify notation, we will in the following address the different Ru-site classes as 'Ru∆'.Interestingly, the Ru sites just below the Ti impurities have the largest CF splitting.These Ru-283 sites show a comparatively large relaxation away from Ti.The latter may be explained by the fact that due to the 3d 0 character of titanium, the effective xz/y z hopping perpendicular to the plane is strongly weakened, resulting also in an overall reduced Ru-Ti bonding.Hence, the original effect of Ti sites blocking hopping paths is affirmed, however this single-particle-based mechanism alone cannot be sufficient to drive the MIT at the given comparatively small amounts of doping.
Notably, the average DFT crystal-field splitting in the supercell amounts to ∆ av = −172 eV and is thus indeed identical to the one in the starting EA structure.But for equal local Couloumb interactions, the system is Mott insulating with strong orbital polarization as shown in Fig. 2b.On average, the x y state becomes fully occupied and the xz/y z states each host one electron, such that the four-electron occupation of Ru(4d)-t 2g is realized.This orbital-polarization scenario is reminiscent of the one in single-layer Cu 2 RuO 4 21 .But since the average CF splittings of the EA structure and the aligned supercell correspond to each other, the spatial inhomogeneity has to play a key role in the bilayer MIT.The site-selective data shown in Fig. 3 and corresponding electronic spectra in Fig. 4 render indeed obvious, that the various Ru sites behave quite differently, in connection with their respective ∆ value.In order to not only rely on analytical continuation, Ru local spectral weight at lowest energy, i.e.A loc (ω) at zero frequency ω = 0, is plotted in the top of Fig. 3

in its approximate form
where G m loc (τ) is the local one-particle Green's function for orbital m and imaginary times τ at inverse temperature β = 1/T .The Ru-283 sites are much stronger correlated than e.g. the Ru-99 or Ru-114 sites with small ∆.In fact, for U = 3.6 eV the former sites have already gapped Ru-t 2g states, while on average, the Ru sublattice still shows metallic response (see inset in top of Fig. 4).The wide spectrum of CF values based on the significant spatial inhomogeneity introduced by Ti doping therefore gives rise to a site-selective Mott scenario 34,35 in Ca 3 (Ru 0.9375 Ti 0.0625 ) 2 O 7 .It occurs on the Ru sites perpendicular-adjacent to the Ti impurities and is precursory to the MIT of the complete system at U ∼ 4 eV.
Nonsurprisingly, the orbital polarization, here simply defined as p = n x y − (n xz + n y z )/2, is much weaker for small-∆ sites.Those sites also cause site-selective physics, namely an oscillatory-in-U revival of metallicity for U > 4 eV.Only for U = 6 eV all sites behave insulating 'in line'.The given intermediate large U -regime can be traced back to strong orbital(-order) competition originating from the small-∆ Ru sites.For instance, the Ru-99 sites mark a dominant xz orbital filling for U = 4.75 eV (see bottom of Fig. 4).
In order to render the comparison between the spatial homogeneous and -inhomogeneous case more explicit, Fig. 5 displays the −∆ vs. U phase diagram of effective Ti-doped Ca 3 Ru 2 O 7 .It is constructed by utilizing the EA structure and gradually fixing ∆ within the DFT+DMFT calculations.
Note that charge self-consistency is an important ingredient, since orbital polarization and MIT tendencies depend thereon 36 .Since the true experimental system is AFM ordered in the insulating phase, results for G-AFM order of the EA structure are included.Those were obtained by enabling a spin-polarized self-energy in the DFT+DMFT cycle.As expected, the MIT occurs at somewhat smaller U with G-AFM order, yet the net effect of magnetic ordering on the Mott criticality is not decisive.Increasing the absolute value of ∆ fosters the driving toward the MIT, but the slope remains steep for −∆ < 200 meV.The MIT occurs for moderate U values only for rather large CF splittings.Note in this respect that a paramagnetic MIT was realized for single-layer Ca 2 RuO 4 with ∆ ∼ −320 meV for U = 3.1 eV in a previous one-shot DFT+DMFT calculation 21 .But such one-shot approaches are known for tending to overestimate orbital polarizations 36 .Furthermore, the MIT value of U = 4.25 eV for the defect supercell (see green pentagon in Fig. 5) is still well in the metallic region of the EA-based phase diagram.Yet remember that the 'orbital-competing regime' up to a robust spatially-coherent insulating state extends also up to U = 6 eV in the supercell calculations.Still, it may be inferred that spatial inhomogeneity, which is not included in the EA structure, is proactive in driving the MIT toward smaller U values.
We moreover performed DFT+DMFT calculations with allowing for AFM ordering in the Ti-doped supercell.Also there, competing orderings, now between different AFM tendencies occur: while or moderate 3 < U < 4 eV the system tends to A-type like ordering, reminiscent of the stoichiometric magnetic order, for U > 4 eV indeed G-type like ordering becomes more favorable.Above U = 6 eV a very robust G-AFM ordering pattern is established. Summary.
-By means of large-scale first-principles manybody calculations, we showed that the MIT in Ti-doped Ca 3 Ru 2 O 7 is mainly driven by the interplay of spatialinhomogeneity features and strong electronic correlations.Introduction of Ti impurities leads to a substantial energy spread of the CF splitting ∆ across the Ru sublattice, resulting in site-selective Mott transitions for large-∆ sites and competitions in the orbital-ordering tendencies on small-∆ sites.This explains not only the occurrence of a dopinginduced metal-insulator transition for reasonable interaction strengths, but accounts furthermore for the findings of perculative behavior.Other effects such as hopping-blocking or average structural modification via a change of lattice parameters furthermore support the insulating tendencies with doping, but appear not decisive.The here reported siteselective physics may also be relevant for recently discovered non-equilibrium features in Ti-doped Ca 3 Ru 2 O 7 37 .
Let us finally emphasize that sole averaged-structure investigations of defect problems in correlated materials may have their shortcomings.In a recent realistic DMFT study on impurities in V 2 O 3 36 , it was shown that local point-group symmetry breaking from trigonal to monoclinic is essential to understand the Cr-induced paramagnetic MIT.And here, Ti-induced site-symmetry breakings are key to the MIT in Ca 3 Ru 2 O 7 .Both results corroborate that the explicit and detailed cooperation of defect chemistry and many-body physics is at the heart of various doping problems in correlated matter.

FIG. 1 .
FIG. 1. (color online) Characterization of stoichiometric Ca 3 Ru 2 O 7 .(a) Unit cell with two bilayers; Ca (large grey), Ru (blue) and O (small red).(b-d) Paramagnetic DFT+DMFT results for U = 4.25 eV at T = 100 K. (b) Spectral function A(k, ω) along high-symmetry lines, compared to DFT bands (red).(c) Fermi surface in k z = 0 plane, gray rectangular marks the Brillouin zone.(d) Local Ru-t 2g spectral function discriminating the x y and xz/y z orbitals and comparing to DFT spectrum.
FIG. 2. (color online) Structure and global spectrum of Ca 3 (Ru 0.9375 Ti 0.0625 ) 2 O 7 .(a) Structurally relaxed supercell showing Ru (blue) and Ti (green) ions, indicating the crystal-field splitting ∆ via the diameter of the Ru-site spheres (Ca and O sites are not shown).Numbers indicate the actual ∆ value of a given Ru-site symmetry class (in meV).The diameter of the Ti spheres is chosen as the average ∆ of the Ru sites.(b) Average Ru(4d)t 2g and Ti(3d)-t 2g paramagnetic local spectrum with 6.25% Ti for U = 4.25 eV (T = 40 K).Dashed lines show the corresponding Rut 2g spectrum of the EA structure for comparison.Inset: blow up of the low-energy region.

FIG. 3 .
FIG. 3. (color online)The Ru(4d)-t 2g spectral weight A 0 at the Fermi level (top) and the orbital polarization p (bottom) with increasing U , for selected Ru-site classes in the Ti-doped supercell, respectively, at T = 40 K.

FIG. 5 .
FIG.5.Crystal-field vs. U phase diagram for Ti-doped Ca 3 Ru 2 O 7 based on the EA structure at T = 100 K (see text for further details).Blue color scale amounts to A 0 of Ru(4d)-t 2g in the PM regime.Full(Dashed) line marks the interpolated metal-insulator phase boundary, here roughly defined by A 0 = 0.2, in the PM(G-AFM) regime.Green pentagon marks the position of the onset of the insulating phase in the supercell calculations (here located via its average ∆ value).