Orbitally selective breakdown of Fermi liquid quasiparticles in Ca$_{1.8}$Sr$_{0.2}$RuO$_4$

TL;DR

This study uses angle-resolved photoemission spectroscopy to reveal that in Ca$_{1.8}$Sr$_{0.2}$RuO$_4$, some orbitals exhibit Fermi liquid behavior while others show non-Fermi-liquid properties, indicating an orbitally selective breakdown of quasiparticles.

Contribution

It provides the first detailed orbital-resolved analysis showing orbitally selective Fermi liquid breakdown in Ca$_{1.8}$Sr$_{0.2}$RuO$_4$ using combined experimental and theoretical methods.

Findings

$d_{xz}, d_{yz}$ bands show Fermi liquid behavior with mass renormalization.

$d_{xy}$ band exhibits non-Fermi-liquid behavior with heavy quasiparticles.

Ca$_{1.8}$Sr$_{0.2}$RuO$_4$ is a hybrid metal with orbitally selective quasiparticle breakdown.

Abstract

We present a comprehensive angle-resolved photoemission spectroscopy study of Ca$_{1.8}$Sr$_{0.2}$RuO$_4$. Four distinct bands are revealed and along the Ru-O bond direction their orbital characters are identified through a light polarization analysis and comparison to dynamical mean-field theory calculations. Bands assigned to $d_{xz}, d_{yz}$ orbitals display Fermi liquid behavior with fourfold quasiparticle mass renormalization. Extremely heavy fermions - associated with a predominantly $d_{xy}$ band character - are shown to display non-Fermi-liquid behavior. We thus demonstrate that Ca$_{1.8}$Sr$_{0.2}$RuO$_4$ is a hybrid metal with an orbitally selective Fermi liquid quasiparticle breakdown.

Full text

Correlated metals are typically classified either as Fermi liquids or non-Fermi liquids depending on whether resistivity scales with temperature squared or not. There is, however, transport evidence suggesting that some materials are hybrids of these two metal classes Cooper et al. (2009). This mixed regime is of particular interest as it provides insight into how Fermi liquids break down and the nature of non-Fermi liquid quasiparticles. In this context, multi-orbital metallic systems in conjunction with strong Hund’s coupling and electron correlations are of great conceptual importance Georges et al. (2013). Such Hund’s metals are expected to display orbital differentiated quasiparticle (QP) renormalization effects along with magnetic correlations (Mravlje et al., 2011). In the strongly correlated limit, orbitally selective Mott physics (OSMP) has been explored theoretically (Anisimov et al., 2002; Koga et al., 2004; Vojta, 2010; de’Medici et al., 2005; Biermann et al., 2005; Ferrero et al., 2005). The concepts of Hund’s metals and OSMP have both been applied to describe band structure renormalization effects in pnictide superconductor compounds (Yin et al., 2011; Yi et al., 2015; Gerber et al., 2017; Sprau et al., 2017; Aichhorn et al., 2010; Lee, 2017). It remains, however, unclear whether these systems exhibit genuine heavy Fermion and Mott physics. In contrast, the oxide compounds LiV2O4 and Ca1.8Sr0.2RuO4 are multi-orbital systems where the existence of heavy Fermions are clearly demonstrated from specific heat measurements Nakatsuji et al. (2003); Kondo et al. (1997). Ca1.8Sr0.2RuO4 is furthermore in close proximity to a Mott-Hubbard metal-insulator transition Nakatsuji et al. (2004). Angle resolved-photoemission experiments (ARPES) on this system have been interpreted in terms of both the Hund’s metal and the OSMP scenario (Shimoyamada et al., 2009; Neupane et al., 2009). Resistivity and specific heat indicate that the ground state is a Fermi liquid (FL). However, a thermal excitation of just 1 K turns the system into a non-Fermi liquid (nFL) state Nakatsuji et al. (2003). Here we present a high-resolution ARPES study, demonstrating that Ca1.8Sr0.2RuO4 is neither a standard Hunds metal nor representing OSMP. In fact, the thermally excited state constitutes an example of a hybrid metal. Along the Ru-O bond direction, bands with $d_{xz}$, $d_{yz}$ orbital character display FL behavior whereas $d_{xy}$ dominated bands host nFL QPs. Breakdown of FL QPs are therefore orbitally selective. This physics might apply to other ruthenate systems such as for example Sr3Ru2O7.

Single crystals of Ca1.8Sr0.2RuO4 were grown by the flux-feeding floating-zone technique Fukazawa et al. (2000); Nakatsuji and Maeno (2001). ARPES experiments were carried out at I05, SIS, $1^{3}$ beamlines of Diamond Light Source (DLS) Hoesch et al. (2017), Swiss Light Source (SLS), and BESSY – respectively. All samples were cleaved in-situ under UHV conditions and measured at temperatures $T=1-30$ K. ARPES spectra were collected with different incident photon energies $h\nu$ and light polarisations using Scienta R4000 electron analyzers. Depending on $h\nu$ and $T$, the overall energy resolution was in the order of 10 meV. As Ca1.8Sr0.2RuO4 has low-temperature L-Pbca Friedt et al. (2001) crystal structure ($a=5.33$ Å, $b=5.32$ Å and $c=12.41$ Å), orthorhombic notation is used. The electronic structure is calculated within the DFT+DMFT (density functional theory + dynamical mean field theory) framework using Wien2k (Blaha et al., 2001) and the TRIQS library (Aichhorn et al., 2009, 2016; Parcollet et al., 2015), including a strong-coupling continuous-time Monte Carlo impurity solver (Gull et al., 2011; Seth et al., 2016).

Wannier-like $t_{2g}$ orbitals are constructed out of Kohn-Sham bands within the energy window $[-2,1]$ eV with respect to the Fermi energy $E_{\mathrm{F}}$. For the correct description of atomic multiplets, a rotationally invariant Kanamori interaction is used (Georges et al., 2013). Inclusion of charge-self-consistency in the DFT+DMFT loop does not change our results. This validates the correlation induced changes of orbital occupancy in the DFT+DMFT in comparison with the DFT result.

Bulk Sr2RuO4 hosts three Fermi surface (FS) sheets $\alpha$, $\beta$ ($d_{xz}$, $d_{yz}$) and $\gamma$ ($d_{xy}$) (Damascelli et al., 2000; Bergemann et al., 2003; Zabolotnyy et al., 2012; Haverkort et al., 2008; Zabolotnyy et al., 2013; Iwasawa et al., 2010, 2012; Kondo et al., 2016). Upon Ca for Sr substitution, the $\gamma$-band is undergoing a Liftshitz transition, changing it from electron- to hole-like Wang et al. (2004). Simultaneously, an electron pocket emerging around the zone center is predicted Ko et al. (2007). Orthorhombic folding of these bands (shown schematically in Fig. 1a) captures all the observed FS sheets of Ca1.8Sr0.2RuO4. In total four sheets are observed and labelled $\alpha$, $\beta$, $\gamma$ and $\delta$ (Fig. 1b–d). The weakest $\delta$-band is further documented in the Supplementary Material (SM) SFig. 1 See Supplementary Material for details concerning $\delta$-band identification and the band structure fitting procedure. and light polarisation dependence of the $\alpha$ and $\gamma$ bands is shown in Fig. 2. The $\alpha$-band, observed with C+ and $\bar{\pi}$-polarised light, is suppressed completely in the $\bar{\sigma}$-channel. For the $\gamma$-band in the zone corner, the opposite trend is observed although complete suppression is not found. Self-energy $\Sigma(\omega)$ versus temperature and binding energy $\omega$ is extracted through a combination of momentum and energy distribution curve (EDC) analysis. For example, the $\alpha$-band QP dispersion is analyzed by fitting momentum distribution curves (MDC). The resulting band dispersion $\varepsilon_{k}^{\alpha}$ and line-width $\Gamma(\omega)$ led us to $\Re\Sigma(\omega)=\varepsilon_{k}^{\alpha}-\varepsilon_{k}^{b}$ and $\Im\Sigma(\omega)=\Gamma(\omega)v_{\mathrm{F}}^{b}$ where $\varepsilon_{k}^{b}$ and $v_{\mathrm{F}}^{b}$ are the DFT bare band and associated Fermi velocity (Fig. 3). Temperature dependence of spectral intensity along the zone diagonal for both the $\alpha$- and $\gamma$- band are analyzed in Fig. 4. In contrast to the $\alpha$-sheet, the $\gamma$-band QP peak amplitude has significant $T$-dependence.

DFT calculations provide an excellent description of the experimental FS of Sr2RuO4 Haverkort et al. (2008). Already without spin-orbit coupling (SOC), our DFT calculation of Ca1.8Sr0.2RuO4 produces several of the experimentally observed FS sheets (Fig. 2e). SOC is known to improve the calculation along the $\Gamma$–Y direction Haverkort et al. (2008); Kim et al. (2018), but has no effect along the $\Gamma$–S direction. The absence of the heavy Fermi pocket around the S-point in the DFT calculation is therefore a significant discrepancy (compare Fig. 2a, e). This motivated our DMFT calculations, using the same parameters of Coulomb interaction $U=2.3$ eV and Hund’s coupling $J_{\mathrm{H}}=0.4$ eV that successfully described Ca2RuO4 Sutter et al. (2017) and other ruthenates Dang et al. (2015); Mravlje et al. (2011). DMFT predicts strong bandwidth renormalization effects, which is particular clear for the $\delta$-band (see Fig. 2g). Moreover, our DMFT calculation reproduces qualitatively the heavy Fermions states around the zone corner.

Next, we discuss the orbital character of the $\alpha$ and $\gamma$ bands along $\Gamma-$S. The incident light and centre of our analyser slit, define a mirror plane to which the electromagnetic field has odd (even) parity for $\bar{\sigma}$ ($\bar{\pi}$) polarisation (see Fig. 2d,h). For final states with even character, selection rules Damascelli et al. (2003) dictate that odd (even) band character is suppressed in the $\bar{\pi}$ ($\bar{\sigma}$) polarisation channel. The $\alpha$-band being suppressed completely (see Fig. 2) in the $\bar{\sigma}$-channel therefore has even character. Assuming approximately tetragonal crystal structure, ($d_{xy}$, $d_{xz}$, and $d_{yz}$) have (odd, even, and odd) character along the $a$-axis. As a result and consistent with the DFT and DMFT calculations, the $\alpha$-band along the Ru-O bond direction has pure $d_{xz}$ character. The $\gamma$-band is placed further away from the mirror-plane due to the perpendicular electron analyser-slit. Hence, less strict selection rules are expected. Nevertheless, our experimental results and DMFT calculations both assign predominately $d_{xy}$ character to the $\gamma$-band.

As Hund’s coupling quenches inter-orbital fluctuations, the orbitals can be viewed approximately as single bands de’ Medici et al. (2014, 2011); Georges et al. (2013). For the ruthenates, this is valid only along the $\Gamma$-S direction as spin-orbit interaction mixes orbital characters along $\Gamma$-Y Haverkort et al. (2008). Experimentally, it is thus only sensible to evaluate orbital differentiated properties along $\Gamma$-S. Comparison of DFT and DMFT suggests that the $d_{xy}$ dominated $\delta$- and $\gamma$-bands are most strongly affected by electron correlations. It suggests that electron correlations are orbitally differentiated. Orbital fillings provide some insight into this effect. As the RuO4 octahedron is almost cubic, DFT yields essentially degenerate $d_{xy}$, $d_{xz}$, and $d_{yz}$ orbital energies with equivalent $4/3$-filling (inset Fig. 2e). The DMFT calculations, by contrast, indicate that electron interactions favor a less populated $d_{xy}$ orbital with $(n_{xy},n_{xz},n_{yz})=(1.18,1.42,1.42)$ (inset Fig. 2f). Electron interactions thus push the $d_{xy}$ channel closer to half-filling and effectively into a more correlated regime.

To describe the orbitally differentiated self-energy of Ca1.8Sr0.2RuO4, we distinguish between saturated and unsaturated FLs. The latter refers to QPs for which $Z$ has $\omega$ or temperature dependence. This implies a nFL self-energy, i.e., non-linear $\Re\Sigma(\omega)$ for $\omega\rightarrow 0$. The saturated regime, by contrast, refers to a standard FL with self-energy $\Sigma(\omega,T)=\gamma_{0}\omega+i\alpha_{0}[\omega^{2}+(\pi k_{\mathrm{B}}T)^{2}]$ where $\gamma_{0}$ and $\alpha_{0}$ are constants Varma et al. (2002); Damascelli et al. (2003). Hence, the QP residue $Z\equiv[1-\partial_{\omega}\Re\Sigma(\omega)]^{-1}=(1-\gamma_{0})^{-1}$ is independent of $\omega$ and $T$. A FL is therefore expected to display (1) a linear QP dispersion, (2) a line width that scales as $\omega^{2}$, $Z|\Im\Sigma(\omega)|<\omega$ Deng et al. (2013) below a cut-off energy scale. and (4) a QP amplitude proportional to $Z$, independent of $\omega$ and $T$. Using $\Re\Sigma(\omega)=\omega\,(1-1/Z)$, the third criterion can be rewritten as $|\Im\Sigma(\omega)|<\Re\Sigma(\omega)/(1-Z)$. For criterion four, the Fermi-Dirac distribution combined with finite instrumental resolution may induce a weak temperature dependence on the effectively observed QP peak amplitude. This weak effect is discussed in the SM Note I See Supplementary Material for details concerning $\delta$-band identification and the band structure fitting procedure. . Examination of the $\alpha$-band, with pure $d_{xz}$, $d_{yz}$ character, reveals an almost $T$-independent QP amplitude (Fig. 4g). The QP dispersion is approximately linear $\varepsilon_{k}^{\alpha}\approx v_{\mathrm{F}}|k-k_{\mathrm{F}}|$, implying $\Re\Sigma(\omega)=(1-v_{\mathrm{F}}^{b}/v_{\mathrm{F}})\,\omega$, with $v_{\mathrm{F}}$ and $v_{\mathrm{F}}^{b}$ being dressed and bare Fermi velocities Fatuzzo et al. (2014). Assuming an isotropic FL and using $v_{\mathrm{F}}^{b}=2.34\,$eVÅ, the QP residue yields $Z=v_{\mathrm{F}}/v_{\mathrm{F}}^{b}=0.26(4)$, consistent with DMFT that finds $Z_{xz}=0.23$. Analysis of the MDC linewidth (HWHM) at $T=30$ K yields $\Gamma(\omega)=\Gamma_{0}+\eta\omega^{2}$ with $\Gamma_{0}=0.020(2)\,$Å*-1* and $\eta=10.6(6)\,$Å*-1eV-2* being constants. This is documented by plotting $\Im\Sigma(\omega)=(\Gamma(\omega)-\Gamma_{0})\,v_{\mathrm{F}}^{b}$ versus $\omega$ (Fig. 3). By comparing $\Re\Sigma(\omega)/(1-Z)$ and $\Im\Sigma(\omega)$, criterion three is obviously satisfied and as shown in Fig. 4g,h the quasiparticle amplitude is temperature independent. The QP excitations of the $\alpha$-band thus fulfil, in the most strict sense, all criteria of a FL (see also SM SFig. 5 See Supplementary Material for details concerning $\delta$-band identification and the band structure fitting procedure. ).

Resistivity and specific heat measurements, however, display FL behavior for $T<1$ K only and much heavier QP masses Nakatsuji et al. (2003). Reconciliation is reached by analysis of the extremely dressed $\gamma$-band QP states around the S-point. These QP amplitudes are roughly proportional to $Z$. In contrast to the $\alpha$-band, the QP peak amplitude of the $\gamma$-band exhibits a pronounced suppression with increased $T$ (Fig. 4e–g). To circumvent the effects of (i) the Fermi-Dirac distribution, (ii) impurity scattering and (iii) finite instrumental resolution (see SM SFig. 3 and 4 See Supplementary Material for details concerning $\delta$-band identification and the band structure fitting procedure. ), it is useful to perform a box integration of spectral weight around $k_{\mathrm{F}}$ (see Fig. 4a–d). Again, the $\gamma$-band displays a pronounce spectral weight temperature dependence whereas the $\alpha$-band remains approximately unchanged. As both the $\alpha$- and $\gamma$-bands are measured simultaneously, this effect is not a result of surface degradation. We are thus led to conclude that the $d_{xy}$ dominated $\gamma$-band states display non-saturated FL behavior. Furthermore, the ratio between coherent and incoherent spectral weight (see Fig. 4e) indicates that $Z\ll 1$ around the S-point, in accordance with the DMFT value $Z_{xy}\approx 0.05$. We have thus demonstrated that the QP mass renormalization and FL QP breakdown are orbitally selective along the $\Gamma$-S direction. It is also worth noticing that temperature dependent spectral weight has also been reported in CeCoIn5 Chen et al. (2017) and Ce2PdIn8 Yao et al. (2019). This effect may therefore be generic to heavy fermion quasiparticles.

In summary, we have presented a combined ARPES, DFT, and DMFT study of Ca1.8Sr0.2RuO4. Our results revealed the complete low-energy electronic structure. Through light polarisation analysis and band structure calculations, insight into the orbital band character was obtained. By studying self-energy effects, it was demonstrated that QP masses and the FL breakdown are orbitally selective. Ca1.8Sr0.2RuO4 thus constitutes a unique example of a hybrid metal hosting orbitally differentiated FL and nFL QPs. As an outlook, it is interesting to consider the idea that nFL behavior found in Ba2RuO4 Burganov et al. (2016) and Sr2RuO4 Barber et al. (2018) under strain has a similar underlying origin.

D.S., M.H., T.N, and J.C. acknowledge support by the Swiss National Science Foundation and Y.S. was supported by the Wenner-Gren foundation. Experiments were carried out on the I05, SIS, and $1^{3}$ endstations at the Diamond Light Source, Swiss Light Source and BESSY, respectively. We acknowledge Diamond Light Source for time on beamline I05 under proposal SI15296. A.G. and M.K. acknowledge the support of the European Research Council grant ERC-319286-QMAC and the Swiss National Science Foundation (NCCR MARVEL), as well as support from the CPHT computer team. The Flatiron Institute is supported by the Simons Foundation. We thank all beamline staff for technical support.