Absence of orbital-dependent Mott transition in Ca 2 x Sr x RuO 4

Transcription

1 EUROPHYSICS LETTERS 1 July 23 Europhys. Lett., 63 (1), pp (23) Absence of orbital-dependent Mott transition in Ca 2 x Sr x RuO 4 A. Liebsch Institut für Festkörperforschung, Forschungszentrum Jülich Jülich, Germany (received 31 January 23; accepted in final form 6May 23) PACS Be Transition metals and alloys. PACS a Strongly correlated electron systems; heavy fermions. PACS Bm Clean metal, semiconductor, and insulator surfaces. Abstract. The Mott transition between the metallic and insulating paramagnetic phases of the layer perovskite Ca 2 xsr xruo 4 is analyzed within the Dynamical Mean Field Theory. To simulate the band narrowing caused by finite Ca doping concentrations, quasi-particle spectra appropriate for Sr 2RuO 4 are evaluated for increasing values of the on-site Coulomb energy U. At small U the planar geometry splits the t 2g bands near E F into a wide, two-dimensional d xy band and two narrow, nearly one-dimensional d xz,yz bands. At larger U, however, the spectral distribution of these states exhibits similar correlation features, suggesting a common metal-insulator transition for all t 2g bands at the same critical U. The layer perovskite Ca 2 x Sr x RuO 4 has attracted considerable interest during the recent years because of a variety of fascinating properties. Sr 2 RuO 4 (x = 2) exhibits unconventional p-wave superconductivity [1,2] and is in fact the only known layered perovskite without copper that becomes superconducting in the absence of doping. The ground state is non-magnetic whereas the volume perovskite SrRuO 3 is a strong ferromagnet. Substitution of Sr with Ca makes Sr 2 RuO 4 undergo a transition to an antiferromagnetic Mott insulator [3,4]. In contrast, doping with La leads to a ferromagnetic instability accompanied by a breakdown of Fermiliquid behavior [5]. Spin fluctuations in Sr 2 RuO 4 are believed to have both ferromagnetic and antiferromagnetic components [6]. Moreover, inelastic neutron scattering data indicate a softphonon mode associated with in-plane rotations of oxygen octahedra [7]. Clearly, Sr 2 RuO 4 is close to structural, electronic and magnetic instabilities. Substitution of Sr by the smaller Ca ions also causes a crystallographic distortion consisting of tilting and rotation of oxygen octahedra. These modifications reduce the effective hopping between Ru 4d orbitals via O 2p levels. The resulting narrowing of the 4d bands is believed to be the driving mechanism for the metal-insulator transition observed in Ca 2 x Sr x RuO 4 for x =.2 [3, 4, 8 1]. The layer perovskite Sr 2 RuO 4 can be regarded as a prototype of a strongly correlated, highly anisotropic multi-orbital transition metal oxide with coexisting wide and narrow bands near the Fermi level. Due to the planar crystal structure the partially filled t 2g bands separate into a two-dimensional d xy band with a van Hove singularity (vhs) just above E F [11], and two nearly one-dimensional d xz,yz bands. In the single-particle picture these bands do not hybridize for symmetry reasons. Local electron-electron interactions in the Ru 4d shell, however, are strong. According to angle-resolved photoemission spectra [12], the on-site Coulomb energy lies between the single-particle band widths of the d xz,yz and d xy bands: W xz,yz <U< W xy [13]. Thus, although correlations primarily affect the narrow d xz,yz bands, multi-orbital interactions also influence the wider d xy band, in particular, its singularity above E F. Recent c EDP Sciences

2 98 EUROPHYSICS LETTERS quasi-particle calculations [13] based on the Quantum Monte Carlo (QMC) Dynamical Mean Field Theory (DMFT) [14 16] showed that on-site correlations give rise to a charge transfer from the narrow bands to the wider d xy band and to a red-shift of the van Hove singularity to within approximately 1 mev of E F. Thus, doping with a few % La moves the Fermi level through the d xy singularity, thereby causing deviations from Fermi-liquid behavior [5]. According to the phase diagram of Ca 2 x Sr x RuO 4 [3], Ca 2 RuO 4 (x = ) is an antiferromagnetic Mott insulator below T < T N with T N = 113 K and a paramagnetic insulator up to 36 K. At even higher temperatures it becomes a bad metal. Fang and Terakura [8] have carried out extensive electronic-structure calculations within the local density approximation (LDA) and showed that the rotation, tilting and flattening of RuO 6 octahedra caused by Ca doping are key factors for the electronic and magnetic properties. The flattening is particularly important since it gives rise to increased O 2p Ru4d bond lengths within the Ru plane and significantly reduces the width of Ru t 2g bands. Rotation and tilting enhances the hybridization between t 2g orbitals and leads to redistribution of spectral weight. A full treatment of these structural modifications in the presence of local Coulomb interactions is difficult and has not been attempted yet. To qualitatively investigate the metal-insulator transition induced by iso-electronic substitution of Sr with Ca, Anisimov et al. [1] recently performed DMFT calculations for undistorted Sr 2 RuO 4 for increasing values of the local Coulomb repulsion U. The key assumption in their work is that the Mott transition caused by Ca doping is driven by the enhancement of U/W, where W is the t 2g band width, and that additional spectral changes due to rotation and tilting of RuO 6 octahedra play a minor role. This picture is very interesting from a conceptual point of view, since in the anisotropic t 2g configuration of undistorted Sr 2 RuO 4 U is comparable at first to W xz,yz 1.3 ev and subsequently to W xy 3.5 ev. The remarkable result of these calculations were two successive Mott transitions for the d xz,yz and d xy bands, implying an intermediate region of U or, equivalently, of Ca concentration, where the narrow bands exhibit an excitation gap while the wide band is still metallic. Only at even larger U a gap appeared also in the wide d xy band. The possible coexistence of metallic and insulating phases within the same d electron shell induced by orbital-dependent Mott transitions is clearly a problem of fundamental interest. Analogous problems arise in a variety of other contexts, such as the coexistence of insulating surface layers with a metallic bulk and of paramagnetic surface layers with a ferromagnetic bulk. Also, orbital-dependent ferromagnetism in rare-earth metals, orbital-dependent superconductivity in the anisotropic layer systems Sr 2 RuO 4 and MgBr 2, and pseudogaps in specific areas of the Fermi surface of high-t c materials coexisting with robust metallic regions are conceptually closely related. Remarkably, in all of these cases a true separation of phases does not seem possible unless there exist mechanisms for a severe decoupling of subsystems. For instance, although the magnetic moments in the surface layers of a ferromagnet might be reduced, bulk and surface have the same Curie temperature [17]. The same applies to the orbital-dependent moments in ferromagnetic rare-earth metals [18]. Similarly, even though the order parameters in anisotropic superconductors might be orbital dependent, they evidently share a common transition temperature [2, 19]. In high-t c systems strong scattering processes involving specific wave vectors can lead to a low but finite density of states in certain areas of the Fermi surface [2]. Theoretical predictions also exclude the coexistence of insulating surface layers and a metallic bulk [21]. Although previous photoemission data on Ca 1 x La x VO 3 [22] seem to be in conflict with this prediction, their analysis is difficult. Sample preparation might also play a role. Analogous surface/bulk photoemission measurements are currently being carried out on VO 2 [23] to search for insulating surface phases. To study the possibility of orbital-dependent Mott transitions caused by the highly aniso-

3 A. Liebsch: Absence of orbital-dependent Mott transition in Ca 2 x Sr x RuO (a).6 (a) xy -1-2 N xy (ω) (1/eV).4.2 Density of states (1/eV) -3 Γ X M Γ 1.5 (b) 1 xz,yz.5 xy N xz (ω) (1/eV) (b) xz Fig. 1 Fig. 2 Fig. 1 (a) Tight-binding fit to the two-dimensional t 2g bands of Sr 2RuO 4 calculated within the LDA (E F = ); (b) density of states ρ i(ω) ofd xy bands (solid curve) and of d xz,yz bands (dashed curve) [13]. Fig. 2 Quasi-particle density of states N i(ω) ofsr 2RuO 4 derived from DMFT for T =145K, assuming independent threefold degenerate (a) d xy and (b) d xz,yz bands (solid curves; see text). U =3.eV, J =.2eV; dashed curves: bare densities. tropic electronic structure in layer perovskite systems, we have extended our previous DMFT calculations for Sr 2 RuO 4 [13] to larger values of the on-site Coulomb energy. Qualitatively, this approach might serve to understand the transition between the metallic and insulating paramagnetic phases of Ca 2 x Sr x RuO 4 at temperatures above the Néel temperature of the end member Ca 2 RuO 4. Surprisingly, we find a behavior that differs fundamentally from the one reported in ref. [1]. For small U, thed xy and d xz,yz quasi-particle spectra indeed have quite different shapes owing to their predominantly two- vs. one-dimensional electronic structure, respectively. As U increases, however, these spectral distributions begin to resemble one another, with only minor differences in the weight and shape of the coherent peaks near E F and of the lower and upper Hubbard bands. Our results therefore suggest that for sufficiently large U the correlated local electronic structure within the t 2g shell dominates the one-electron hopping. As a consequence, one common metal-insulator transition exists at an intermediate critical value of U between those of the isolated d xz,yz and d xy bands. The reason for the discrepancy between the two approaches is not clear at present, but could be related to the fact that in ref. [1] the quantum impurity problem was treated within the non-crossing approximation (NCA) [24, 25], whereas we use the Quantum Monte Carlo method [15]. Since impurity calculations are carried out at imaginary times, uncertainties could also be introduced by the maximum entropy reconstruction [26] of the quasi-particle spectra at real frequencies. Figure 1 shows the tight-binding fit [13] to the LDA band structure of Sr 2 RuO 4 [11] in the vicinity of E F and the corresponding partial densities of t 2g states. In the singleparticle picture the d xy van Hove singularity lies about 5 mev above E F. All three bands are approximately 2/3 filled. The non-symmetric shape of the two-dimensional ρ xy (ω) iscaused by second-neighbor Ru-Ru hopping terms. The d xz,yz densities are dominated by quasi onedimensional hopping along Ru rows. To take account of local Coulomb interactions we calculate the quasi-particle spectra within

4 1 EUROPHYSICS LETTERS the multi-orbital QMC-DMFT approach [13, 27]. Since the d xy and d xz,yz bands do not hybridize, the self-energy is diagonal in orbital space. In the DMFT the elements Σ i (i = xy, xz, yz) are functionals of the bath Green s functions G 1 i = G 1 i +Σ i, where the local G i is given by ρ i (ω) G i (iω n )= dω (1) iω n + µ Σ i (iω n ) ω and µ is the chemical potential. Most QMC calculations were done for β =8(T = 145 K) with 64 time slices and several runs using up to sweeps. In the low-temperature calculations for β =4(T = 3 K) 128 time slices were used, with up to sweeps. The quasi-particle density of states N i (ω) = Im G i (ω)/π is obtained via maximum entropy reconstruction [26]. As shown in ref. [13], the angle-resolved photoemission spectra of Sr 2 RuO 4 can qualitatively be represented by quasi-particle distributions for local Coulomb and exchange energies U =1.2eV, J =.2eV. The d xy and d xz,yz spectra then look like moderately deformed versions of their respective single-particle densities of states. For instance, the lower van Hove singularity of the d xz,yz bands near 1 ev is broadened and shifted to about.5ev. Larger Coulomb energies would yield too small binding energies for this spectral feature. This result is consistent with the fact that on-site Coulomb energies for 4d transition metals are smaller than for 3d metals. The d xy vhs is also shifted and lies only about 1 mev above E F. Both bands show very low spectral weight in the energy region of the Hubbard peaks. The effective masses calculated within the QMC-DMFT for these Coulomb energies also agree with experiment. We now explore the paramagnetic metal-insulator transition for this anisotropic multiband system by increasing the on-site Coulomb energy instead of reducing the band width. To illustrate the effect of the different widths of the t 2g subbands on the quasi-particle spectra we show first in fig. 2 the results for hypothetical threefold degenerate bands consisting purely either of d xy or d xz,yz character. The total filling is 4 as in actual Sr 2 RuO 4.ForU =3.eVthe wide d xy band is still dominated by the strong quasi-particle peak at E F. Since U<W xy,itis moderately affected by correlations and shows only weak shoulders in the range of the Hubbard bands. Considerably larger values of U are required to drive the pure d xy band towards a Mott transition. On the other hand, since U W xz,yz the narrow band derived from the d xz,yz density of states exhibits clear signs of a Mott transition already for U 3 ev. The upper and lower Hubbard peaks are the dominant spectral features. Because of the finite temperature used in the QMC-DMFT calculation (effective broadening 125 mev), the remaining quasiparticle peak at E F obscures the gap between the Hubbard bands. Nevertheless, according to the temperature dependence of similar features found in one-band systems [15, 28] the value of U used in the spectra shown in fig. 2 should be very close to the critical value for a metalinsulator transition within the d xz,yz bands. Clearly, therefore, as long as the d xy and d xz,yz orbitals are treated as independent without Coulomb coupling between them, the different widths and shapes of their respective densities of states imply very different local Coulomb energies U crit to enforce a Mott transition. The quasi-particle spectra for the actual t 2g bands of Sr 2 RuO 4 are shown in fig. 3(a) for the same on-site energies as in fig. 2. Evidently, as a result of the strong inter-orbital Coulomb and exchange interactions both d xy and d xz,yz spectral distributions have nearly lost their original density-of-states character and look remarkably similar: with coherent peaks at E F and upper and lower Hubbard bands of about the same intensity, position and shape. These spectra suggest a common Mott transition for the t 2g bands at a value of U only slightly larger than 3 ev. The local correlations for U = 3 ev induce a charge transfer of about.5 electrons from the narrow d xz,yz bands to the d xy band. This would imply that the d xy van Hove singularity

5 A. Liebsch: Absence of orbital-dependent Mott transition in Ca 2 x Sr x RuO 4 11 N xy,xz (ω) (1/eV) (a) xy,xz N xy (ω) (1/eV) (a) xy N xy,xz (ω) (1/eV) (b).4.2 xz N xz (ω) (1/eV) (b) xz xy Fig. 3 Fig. 4 Fig. 3 Quasi-particle density of states N i(ω) ofsr 2RuO 4 derived from DMFT for T =145K. (a) U =3.eV, J =.2eV; (b) U =4.eV, J =.7eV. Solid curves: d xy states; dashed curves: d xz,yz states; dotted and dash-dotted curves: corresponding bare densities. Fig. 4 Quasi-particle density of states N i(ω) ofsr 2RuO 4 derived from DMFT for T = 3K, U =3.eV, J =.2eV. Solid curves: (a) d xy states; (b) d xz,yz states; dashed curves: bare densities. has shifted below E F and that the γ sheet of the Fermi surface is hole-like rather than electronlike as observed in de Haas-van Alphen measurements [29]. This supports our previous choice of a smaller U for the actual Sr 2 RuO 4 t 2g bands [13]. Figure 3(b) illustrates the effect of increasing the exchange energy to J =.7 ev. In order to keep the average Coulomb energy Ū unchanged, U is increased to 4 ev. (For a t 2g complex Ū coincides with the inter-orbital Coulomb energy U = U 2J [1]. Thus, Ū =2.6eV in figs. 3(a) and (b).) Although the d xy and d xz,yz spectra are not quite as similar as in fig. 3(a), they have coherent and incoherent peaks of about the same intensity. Only the positions of these features are shifted. Extrapolation to lower temperatures does not seem to be quite as straightforward as in the previous case. Nevertheless, qualitatively both d xy and d xz,yz spectra are equally correlated and therefore also suggest a common Mott transition. The comparison of figs. 3(a) and (b) shows that the d xy, d xz,yz transition energies between lower and upper Hubbard bands depend on the local Coulomb interactions and do not necessarily coincide. This picture is analogous to orbital-dependent superconductivity and to ferromagnetism in rare-earth metals [18, 19]. Although these systems exhibit different order parameters and orbital-dependent magnetic moments, respectively, they have only one critical temperature. Since the incoherent peaks in fig. 3(a) and (b) are not quite as pronounced as in fig. 2(b) for the pure d xz,yz bands, we conclude that Uxz,yz crit <Ut2g crit <Uxy crit, i.e., the Mott transition for the actual t 2g complex requires a critical U between those of the independent d xz,yz and d xy bands. The calculations discussed above are performed at rather high temperatures (145 K). It is therefore important to check to what extent the results remain applicable at lower temperatures. As mentioned above, Ca 2 RuO 4 becomes a paramagnetic insulator at about 36 K and is antiferromagnetic below 113 K. To test the idea of a common Mott transition for the t 2g complex we have performed multi-band QMC-DMFT calculations at about 3 K which are computationally much more costly. Figure 4 shows the d xy and d xz,yz quasi-particle distribu-

6 12 EUROPHYSICS LETTERS tions for U =3.eV, J =.2eV. Qualitatively similar results are obtained for U =4.eV, J =.7eV. Near E F the d xz,yz spectrum exhibits a double-peak structure which is the remnant of the upper and lower edge singularities of the d xz,yz bands. The d xy spectrum also reveals a fine structure which might be caused by strong orbital interactions. The important point is that both d xy and d xz,yz spectra are similarly correlated: the coherent peak has about the same weight and the upper Hubbard bands are almost identical. The main difference is that the lower Hubbard d xz,yz satellite is split off from the main peak by a gap while the d xy satellite lies in the lower tail of the d xy density of states and therefore forms a continuum with the main peak. Leaving aside the uncertainties caused by the maximum entropy reconstruction, such a gap does not imply that the spectral weight at the Fermi level vanishes at a different U crit or T crit than in the case of the d xy states. On the contrary, the comparable strength of the coherent peaks indicates that the d xy and d xz,yz bands undergo a common Mott transition. We emphasize here that the finer details of the spectral distributions shown in figs. 2-4 are prone to vary between different QMC runs and for different parameters used in the maximum entropy reconstruction. For instance the precise positions and weights of the Hubbard peaks of the t 2g subbands are difficult to converge. Nevertheless the general trend emerging from many repeated calculations, in particular the qualitative difference obtained for the independent t 2g bands shown in fig. 2 and the actual fully interacting bands in fig. 3 were found to be reliable. The above results show that the layer perovskite structure, in spite of its highly anisotropic electronic properties, exhibits only one metal-insulator transition. This finding differs qualitatively from the successive orbital-dependent Mott transitions obtained at U = 1.5eV and U =2.5eV (J =.7eV) in the work by Anisimov et al. [1, 3]. The NCA used in their DMFT approach should in principle also be applicable to multi-band materials. On the other hand, even in one-band cases it is known to be unreliable in some situations and to violate Fermi-liquid behavior [15,16,27]. Some uncertainty might also stem from different parameters employed in the maximum entropy procedures. It would be desirable to perform numerical renormalization group calculations for this system at low temperatures. Unfortunately, multiband calculations within this scheme are computationally not yet within reach. In summary, we have used the QMC-DMFT to calculate quasi-particle spectra of Sr 2 RuO 4 in the range of on-site electron-electron interactions appropriate for a possible paramagnetic metal-insulator transition. Although these calculations may primarily be regarded as an illustrative model study of a highly anisotropic multi-orbital system, increasing the value of U serves to simulate the reduced t 2g band width caused by distortions of O octahedra when Sr is replaced by Ca. At low Coulomb energies, the spectral distributions of the d xy and d xz,yz bands differ strongly because of the planar geometry of this perovskite material. Despite this anisotropy, at sufficiently large U these spectra resemble one another in the sense that they are equally strongly affected by local correlations. In particular, they exhibit similar upper and lower Hubbard bands and a coherent peak of similar strength. Thus, in the critical region of U local Coulomb interactions dominate the one-electron hopping, giving rise to a common Mott transition for the anisotropic t 2g complex. This result is consistent with other multi-band phenomena, such as orbital-dependent superconductivity and orbital-dependent ferromagnetism. I would like to thank A. Bringer, R. Bulla, O. Gunnarsson, A. Kampf, J. Keller, N. Nagaosa, Th. Pruschke, G. A. Sawatzky, D. Vollhardt and K. Yamada for useful discussions. I also thank A. I. Lichtenstein for the QMC-DMFT code. REFERENCES [1] Maeno Y. et al., Nature (London), 372 (1994) 532.

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