Formation and consequences of heavy d-electron quasiparticles in Sr3Ru2O7

We report angle-resolved photoelectron spectroscopy measurements of the quantum critical metal Sr3Ru2O7 revealing itinerant Ru 4d-states confined over large parts of the Brillouin zone to an energy range of<6 meV, nearly three orders of magnitude lower than the bare band width. We show that this energy scale agrees quantitatively with a characteristic thermodynamic energy scale associated with quantum criticality and illustrate how it arises from the hybridization of light and strongly renormalized, heavy quasiparticle bands. For the largest Fermi surface sheet we find a marked k-dependence of the renormalization and show that it correlates with the Ru 4d - O 2p hybridization.

Abstract. The phase diagram of Sr 3 Ru 2 O 7 shows hallmarks of strong electron correlations despite the modest Coulomb interaction in the Ru 4d shell. We use angle-resolved photoelectron spectroscopy measurements to provide microscopic insight into the formation of the strongly renormalized heavy d-electron liquid that controls the physics of Sr 3 Ru 2 O 7 . Our data reveal itinerant Ru 4d-states confined over large parts of the Brillouin zone to an energy range of <6 meV, nearly three orders of magnitude lower than the bare band width. We show that this energy scale agrees quantitatively with a characteristic thermodynamic energy scale associated with quantum criticality and illustrate how it arises from a combination of back-folding due to a structural distortion and the hybridization of light and strongly renormalized, heavy quasiparticle bands. The resulting heavy Fermi liquid has a marked k-dependence of the renormalization which we relate to orbital mixing along individual Fermi surface sheets.
The unusual physics of the bilayer ruthenate Sr 3 Ru 2 O 7 exemplifies some of the unsolved problems in condensed matter physics. Its rich phase diagram includes quantum criticality, magnetism and an electron nematic phase [1][2][3][4][5] and has generated considerable theoretical interest [6][7][8][9][10][11][12][13][14][15][16][17]. While there is no consensus on the precise microscopic origin of these properties, they are often associated with an instability of a heavy Fermi liquid resulting from strong correlations and the existence of a low-energy scale of unknown origin. This is consistent with the high electronic specific heat coefficient of γ ∼110 mJ/mol RuK 2 of Sr 3 Ru 2 O 7 [18] and with recent angle-resolved photoemission spectroscopy (ARPES) [19], scanning tunneling microscopy (STM) [20,21] and quantum oscillation measurements [22] that all indicate the existence of itinerant electrons with high quasiparticle masses. However, little is known about the structure of the heavy Fermi liquid and the origin of the putative lowenergy scale in Sr 3 Ru 2 O 7 . Within density functional theory in the local density approximation (LDA), Sr 3 Ru 2 O 7 has a wide bare conduction band, formed by relatively extended Ru 4d states hybridizing with O 2p electrons and a similar density of states at the Fermi level as the single-layer compound Sr 2 RuO 4 . Yet, the latter has remarkably different physical properties and a much smaller electronic specific heat indicative of weaker correlations than in Sr 3 Ru 2 O 7 . Recent dynamical mean field theory studies of generic trends in transition metal oxides [23][24][25] show strong correlation effects at low frequencies in ruthenates but the difference between Sr 3 Ru 2 O 7 and Sr 2 RuO 4 remains largely unexplored.
The existence of the low-energy scale in Sr 3 Ru 2 O 7 mentioned above is supported by recent transport and entropy data [2,5]. Relevant in the context of this paper is the observation of a maximum in the zero-field electronic specific heat C el (T )/T near 8 K. The position of this maximum can be suppressed continuously in an external field, terminating in a logarithmic divergence at the putative quantum critical end point [5]. This suggests that criticality in Sr 3 Ru 2 O 7 is driven by the suppression of a single, low-energy scale that persists in zero field. However, despite recent progress in characterizing the low-energy electronic structure of Sr 3 Ru 2 O 7 [19][20][21], the microscopic origin of this energy scale remained elusive.
Here, we report detailed ARPES measurements of Sr 3 Ru 2 O 7 revealing flat Ru 4d bands that define an energy scale consistent with thermodynamic measurements. We show how this energy scale arises microscopically from the hybridization of strongly renormalized bands with dispersive states. The resulting Fermi liquid shows pronounced multi-band effects and has a For the experiments presented in this paper, we used crystals grown by the floating-zone method as described in [26] with residual resistivities as low as 0.4 µ cm. Our photoemission experiments were performed using 16-57 eV photons from SSRL's beamline V-4, BESSY-II's 1 3 beamline and the SIS beamline at SLS, as well as He Iα radiation from monochromatized discharge lamps. The measurements were taken at temperatures around 8 K (figures 1, 3(a), (c) and 4) and 1.1 K (figures 3(b) and (c)) and energy and angular resolutions of 3.5-5 meV and ≈0.3 • , respectively. Density functional calculations within the LDA were performed using the all-electron code Wien2k including spin-orbit coupling and the pseudo-potential code Quantum Espresso [27,28].
The low-energy electronic structure of Sr 3 Ru 2 O 7 is summarized in figure 1(a) where we show an experimental Fermi surface with a model of the low-energy quasiparticle dispersion that spans the entire Brillouin zone. The model is derived from a smoothing interpolation between ∼10 3 experimentally determined initial state energies and is fully consistent with the analysis of a smaller k-space range reported in our previous work [19]. While the gross band topography resembles the LDA band structure [19,29], the experimental quasiparticle velocities are markedly lower than the calculated bare velocities. This behavior is well known for correlated Fermi liquids and can be characterized by a renormalization constant (Fermi liquid residue) where v LDA and v exp are the group velocities of bare bands as calculated within LDA and the measured quasiparticle bands at the Fermi level, respectively. However, the magnitude of Z −1 FL in Sr 3 Ru 2 O 7 is highly unusual. Whereas most metallic 3d and 4d transition metal oxides have average mass enhancements γ /γ LDA ∼2 [25], in Sr 3 Ru 2 O 7 Z −1 FL reaches values up to 25 for some Fermi surface sheets as shown in figure 1. It therefore exceeds the largest values found in the single-layer compound Sr 2 RuO 4 , which is often described as strongly correlated, by approximately a factor of 4. We illustrate the marked influence of correlations on the quasiparticle band structure of Sr 3 Ru 2 O 7 in figures 1(b)-(e), where we compare the parabolically dispersing δ-pocket with the weakly interacting electron gas found at the Cu(111) surface [30,31]. Both of these states have similar bare band masses of ≈0.4m e . Yet, while the quasiparticle dispersion in Cu closely follows the bare band, the width of the δ-pocket in Sr 3 Ru 2 O 7 is reduced to ≈5 meV, corresponding to a renormalization constant of Z −1 FL ≈25. At the same time the spectral weight of the coherent quasiparticle peak is reduced and pushed to higher energy, resulting in the characteristic 'peak-dip-hump' line shape of strongly interacting systems ( figure 1(d)).
The same energy scale and equally strong band renormalization, combined with marked multi-band effects, are observed over a large k-space volume spanned by the hybridized α 2 -γ 2 sheet (figures 1 and 3). For these bands of mixed x z/yz, x y orbital character (see [13]), the low-energy scale is much more important, since they span a large area in k-space and thus dominate the low-energy density of states. Several authors proposed that many properties of quantum critical materials as they are seen in Sr 3 Ru 2 O 7 and some heavy fermion systems can be explained assuming a narrow peak in the density of states close to or locked to the chemical potential [3,5,32,33]. In order to test this idea for Sr 3 Ru 2 O 7 we numerically computed the quasiparticle density of states g(ε) from our model of the low-energy band dispersion shown in figure 1(a), and used this function to calculate the temperature dependence of the electronic specific heat given by The result, shown in figure 2(a) assuming constant density of states above the Fermi energy, correctly reproduces the magnitude and the gross shape of C el (T )/T including the temperature of the maximum. Intriguingly, our data even reproduce the strong enhancement of the specific heat in an applied field ( figure 2(b)). In this calculation, we use a simple, rigid band shift of the Zeeman split states. With this approach we find that the specific heat at zero-temperature peaks for a Zeeman splitting of 1.8 meV corresponding to a field of 15 T for g = 2. Given that our model neglects any temperature and field dependence of the many-body density of states, the agreement with direct measurements of the temperature and field dependence of the specific heat [5] is excellent. This strongly suggests that the energy scale defined by the hybridized α 2 -γ 2 band, which dominates g(ε), is intimately involved in quantum criticality and the formation of an electron nematic state in high field. The complex shape and orbital character of this sheet, evolving from the α 2 pocket that stems from the out-of-plane x z/yz orbital, to x y-dominated states near the X-point, highlights the need for realistic models of nematicity in Sr 3 Ru 2 O 7 to include all three t 2g orbitals as well as spin-orbit coupling [9,13,16].
For the remainder of this paper we focus on the formation of heavy d-electron quasiparticles in Sr 3 Ru 2 O 7 and their unusual Fermi surface sheet-and momentum-dependent mass enhancement. In figure 3 we show ARPES data along X for selected photon energies together with the quasiparticle dispersion extracted from several measurements with different photon energies and polarizations. Attempting to describe the experimental dispersion with a minimal model, we approximate the peak positions by eight cosine bands tracking the dispersions of the fundamental bilayer split x y, x z/yz orbitals and their back-folded copies x y , x z /yz . (The bilayer splitting in the x y sheet is not resolved experimentally and is for illustrative purpose only.) Intriguingly, these cosine bands have very different widths with Fermi velocities varying by more than an order of magnitude. We attribute this to a combination of band structure effects and strong, orbital-dependent correlations.
The backfolding arises from a rotation of the RuO 6 octahedra around the z-axis by ∼7 • [34] which doubles the in-plane unit cell and folds all bands along the (0, π)-(π, 0) line of the undistorted Brillouin zone. For the quasi-one-dimensional x z/yz orbitals, the structural distortion should merely double the number of bands and invert the dispersion of the backfolded x z /yz states. In contrast, backfolding of the nearly isotropic fundamental x y sheet will result in Fermi surface contours that are nearly parallel to the X line and thus disperse weakly along this direction. This occurs in the LDA calculated bands as well, but the effect is far more pronounced  /yz , x y ). The low-energy contour arising from the hybridization of these bands is indicated by a thick blue line.
in the experimental data, which we relate to many-body effects not contained in the LDA. Comparing the cosine bands of our simple model with an LDA calculation we find that the fundamental x y sheet is renormalized by a factor of ∼6, whereas the dispersion of the backfolded x y sheet is 20-30 times smaller than in LDA, indicative of strongly enhanced correlation effects for the backfolded states which have extended flat regions and a higher density of states at very low energies. In the presence of spin-orbit coupling [35,36] this situation naturally leads to the hybridization of very itinerant light bands and heavy states near the Fermi level, akin to the situation in f-electron heavy fermion materials. The resulting low-energy contour (figure 3(c)) remains confined to an energy range of <6 meV over an extended part of the Brillouin zone and has a complex shape with multiple saddle-point and band-edge singularities in the vicinity of the X point. This includes a band maximum at ≈−1 meV that might lead to the putative γ 2 pocket, as discussed in our previous work [19], and, more recently, in a LDA + slave boson approach [17].
We also observe pronounced many-body effects in the out-of-plane orbitals. The x z /yz bands hybridize weakly to form the α 1,2 Fermi surface contours as shown in figure 4(c). Within LDA these sheets have nearly isotropic bare Fermi velocities, consistent with naive expectations. However, the quasiparticle dispersion along the α 2 Fermi surface, which was also detected in spectroscopic imaging STM [21], is clearly anisotropic. This is evident from the dispersion plots shown in figures 4(a) and (b), where we overlay an LDA dispersion, globally compressed by a factor of 6, onto the data. This reproduces the Fermi velocity along M but overestimates it by a factor of ≈3 along X. The renormalization Z −1 FL along the entire α 1,2 Fermi surface sheets is visualized in panel (c) by red arrows with lengths proportional to v LDA /v exp , with v exp extracted from a large number of cuts normal to the Fermi surface. A marked variation of Z −1 FL corresponding to a strongly momentum-dependent self-energy along the α 2 Fermi surface is evident from the data. Such a pronounced anisotropy is unexpected for a Fermi liquid where local electron-electron correlations dominate the mass enhancement. In fact, in the absence of multi-band effects, purely local interactions lead to a self-energy that lacks any momentum dependence. Anisotropic renormalizations Z −1 FL , as they have been reported in cuprates or cobaltates, are thus commonly attributed to coupling to bosonic modes [37][38][39]. And indeed, neutron scattering detected strong antiferromagnetic spin fluctuations in Sr 3 Ru 2 O 7 at energies <5 meV and wave vectors |q 1 | = 0.18π/a t and |q 2 | = 0.5π/a t [40,41] that are a near perfect match to the nesting vectors connecting parallel sections of α 1,2 . However, our data show that the correlation between nesting and renormalization is incomplete: while v LDA /v exp is clearly enhanced for the parallel sections of α 2 , this is not the case for the nearly square α 1 sheet, which is as highly nested as α 2 and whose nesting vector matches the strongest antiferromagnetic fluctuations at q 2 seen in neutron scattering. Thus, while nesting might play an important role in the physics of Sr 3 Ru 2 O 7 , it cannot be the sole mechanism behind the renormalization we observe here.
On the other hand, we find a clear correlation between Z −1 FL and the character of the wave functions at the Fermi surface. To illustrate this we introduce a simple tight-binding model describing the x z/yz orbitals of an isolated bilayer using the in-plane hopping elements (t x , t y ) and the out-of-plane nearest (t z ) and next-nearest-neighbor (t z ) elements. The dispersion of this model along X is ε k± = 2(t x ∓ t z )cos k x ± t z and its Fermi surface consists of the two sets of slightly curved light blue and pink lines in figure 4(c) which correspond to the symmetric (|+ ) and antisymmetric (|− ) combinations of the x z/yz orbitals in the upper and lower RuO 2 plane; |± = |upper ± |lower . It is evident from this analysis that the strongly and weakly renormalized bands exactly follow the |+ and |− states, rather than the nested parallel sections of the α 1,2 Fermi surfaces. We use an LDA calculation of the valence charge distribution (figures 4(d) and (e)) to elucidate the particular differences in the wave functions of the |+ , |− states in real space. The antisymmetric combination, which has a node at the apical O site in the naive tight-binding picture, indeed has little weight at this site but hybridizes with in-plane O p z states, whereas the symmetric combination hybridizes strongly with apical O p x,y states but has little weight on the in-plane O. This has a moderate influence on the bare band width along X which is around 1.6 eV for the antisymmetric and 0.9 eV for the symmetric combination. Our photoemission results suggest that correlations markedly amplify this difference, possibly by reducing the effective in-plane hopping for the |+ states of the many-body system with little weight on in-plane oxygen. Within our empirical tight-binding picture, the relative width r = W − /W + of the |− and |+ bands is controlled by the ratio t z /t x = (r − 1)/(r + 1) between the diagonal inter-layer hopping and the dominant in-plane hopping. The LDA dispersion can be described with t z /t x ≈0. 28, similar to what is used for many theoretical models 13 . However, the fit shown in figures 4(a) and (b) as a red line suggests that t z /t x is as high as 0.85 in the many-body dispersion, which should have a profound effect on the behavior of the large α 2 Fermi surface sheet in external fields. We point out that strong variations of the orbital character are not restricted to α 2 but are observed for most Fermi surface sheets including γ 1 and the putative γ 2 pocket. The complexity added over Sr 2 RuO 4 by the RuO 2 bilayer in Sr 3 Ru 2 O 7 goes far beyond simple band structure effects and includes substantially altered effects of many-body interactions, which might hold the clue to their remarkably different thermodynamic properties.
In conclusion, we illustrated how orbital-dependent renormalization, backfolding and hybridization lead to the formation of heavy d-electron quasiparticles in Sr 3 Ru 2 O 7 with a strongly structured low-energy density of states. We further argued that, in multi-band systems with strong orbital mixing along the Fermi surface such as Sr 3 Ru 2 O 7 , local electron correlations can cause a pronounced anisotropy in the mass enhancement, which should be included in future more realistic models of the electron nematic phase. We expect that both of these effects are of general relevance to 4d transition metal oxides and other strongly correlated multi-band systems.