Revealing the Orbital Composition of Heavy Fermion Quasiparticles in CeRu₂Si₂

Kevin P. Kramer¹ , Rina Tazai² , Karin von Arx¹ , Masafumi Horio¹ , Julia Küspert¹ , Qisi Wang¹ , Yasmine Sassa³ , Timur K. Kim⁴ , Céphise Cacho⁴ , Julien E. Rault⁵ , Patrick Le Fèvre⁵ , François Bertran⁵ , Marc Janoschek^1,6 , Nicolas Gauthier^7,8 , Daniel Mazzone⁹ , Ramzy Daou¹⁰, and Johan Chang¹

This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the article, journal citation, and DOI.

3. Results

On- and off-resonance ARPES spectra collected along high symmetry direction on a Ce-terminated surface, shown in Figs. 1(a)–1(c) and 1(e)–1(g), reveal a rich band structure. Numerous dispersive bands are observed along with the non-dispersive Ce \(4f_{7/2}\)^11,22) state at a binding energy of roughly 0.3 eV [best seen in Fig. 1(g)]. The associated on- and off-resonance Fermi surfaces perpendicular to \(k_{z}\) are displayed in Figs. 1(d) and 1(h), while Fermi surfaces parallel to \(k_{z}\) along the high symmetry directions ΓX and \(\Gamma\Sigma\) are shown in Figs. 1(i) and 1(j), respectively. From panels (d) and (h) it is apparent that the X-point exhibits 4-fold symmetry, in contrast to previous reports where this point has 2-fold symmetry.^23,26) The dotted outline of the simple tetragonal Brillouin zone (BZ) boundaries are a better match for the measured data than the more appropriate body-centered tetragonal BZ boundaries. Further discussion on this observation can be found in the supplemental material.³²⁾ In the following, we conclude from this observation that the data obtained on- and off-resonance is directly comparable in terms of band structure.

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Figure 1. (Color online) High-symmetry band structure and Fermi surface from a Ce-terminated surface. The first and second rows are recorded using photons tuned to be off- or on-resonance, respectively. Under both conditions, the band structure along high symmetry directions is displayed in (a)–(c) and (e)–(g). On- and off-resonance Fermi surface maps were recorded at 13 K (d, h). Colored dashed lines indicate the high-symmetry trajectories along which the band structure is displayed in (a)–(c) and (e)–(g). Panels (i) and (j) show the Fermi surface along the \(k_{z}\) direction for cuts along ΓX and \(\Gamma\Sigma\), respectively. White lines in (d, h, i, f) indicate the body-centered tetragonal Brillouin zone boundaries. Simple tetragonal Brillouin zone boundaries are also shown with dotted white lines. A schematic of the body-centered tetragonal Brillouin zone with the locations of the special points is shown in panel (k). Panels displaying data recorded with same incident photon energy share a common color scale normalization.

Several Fermi surface sheets can be identified. A small electron pocket [labeled γ in Fig. 2(a)] is found around the Y₁-point. Around the zone center, Z, a small (α) and a slightly larger (β) Fermi surface sheet are observed. They form the inner part of a flower-like shape, indicated with a dotted line in Fig. 1(h). The smaller α Fermi surface sheet displays a pronounced enhancement under resonant illumination, compare Figs. 1(b) and 1(c) with Figs. 1(f) and 1(g). This resonance effect implies that f-electronic spectral weight is folded into the low-energy QPs. The orbital nature of these composed QPs is our central point of focus here.

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Figure 2. (Color online) Zone corner band structure from Si-terminated Fermi surface. (a, b), (e, f) Band structure along the Z-Y₁ directions recorded with linear horizontal (π) and vertical (σ) light polarization for temperatures as indicated. (c, d), (g, h) Intensity difference I(36 K)–I(9 K) for the two polarizations at the resonance and off-resonance. The color scale is organized so that white indicates no difference, red indicates intensity gain and blue intensity loss upon heating. The color scale of the off resonance difference spectra (d, h) has been enhanced by 500% in order to make features visible at all. On the resonance, loss and gain traces the band structure whereas the off-resonance differences are generally a factor of 5 weaker and appear rather random.

When rare earth ions are surrounded by Si layers the chemical surrounding resembles that of the bulk.^11,33) In principle, by addressing the Si-terminated surface we ensure that the observed f-electron physics represents the bulk properties, rather than those of surface Ce atoms. Surface effects are, however, commonly observed in related systems. In particular, the absent \(k_{z}\) dispersion of the α band [apparent from Fig. 1(i)] is reminiscent of previous reports,^34–36) where the \(k_{z}\) dispersion is affected by surface effects. We note that even if we are observing strong \(k_{z}\) broadening or a surface projection of the bulk electronic state, the agreement with soft x-ray ARPES data²³⁾ makes it clear that the observed bands emerge from the same orbitals as the corresponding bulk states [compare panels b, f, d, and i of Fig. 1 to the respective figures in the x-ray ARPES report:²³⁾ Figs. 2(b), 2(a), 4(a), and 4(b)].

Band structure along the Z-Y₁ direction is systematically collected with on- and off-resonance, σ and π light polarization, and as a function of temperature up to 36 K (Figs. 2 and 3). The on-resonance data displayed in Figs. 2 and 3 exhibits a clear temperature dependence around the Fermi level. Here band structure changes as a function of temperature seem to occur [Figs. 2(c), 2(d), 2(g), 2(h), and 3]. We also notice that the spectral weight of the conduction bands seems to change even far from the Fermi level [Figs. 2(c) and 2(d)]. The off-resonance data by contrast display little or no temperature dependence [Figs. 2(d) and 2(h)], as expected. Altogether, this suggests that the observed resonant temperature dependence is associated with the \(cf\) hybridization, which is known to have a thermal dependence due to the Kondo effect. It can however not be excluded that the temperature effect at large binding energy is linked to resonant photoemission effects.

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Figure 3. (Color online) Temperature dependence of quasiparticles. Energy distribution maps recorded on the Ce-resonance along the Z-Y₁ direction for different temperatures. Top (bottom) panels are recorded with π (σ) light polarization. Solid lines are momentum distribution curves, integrated over a binding energy (\(E_{\text{B}}\)) range from 30 to 0 meV. The intensity of these curves is normalized to their values at the Y₁-point.

The γ-band forms a Dirac-cone like structure around the Y₁-point. This band is only observed in the π channel and vanishes when σ polarization is used [see Figs. 2(a) and 2(e)]. The direct implication is that this band has even orbital character with respect to the photoemission mirror plane.^37,38) We also stress that this band does not display any significant resonance effects. It does therefore not seem to hybridize with the f-electronic states. Furthermore, it should be pointed out that we do not observe any \(k_{z}\) dispersion for the γ-band [Fig. 1(i)]. This, like the previously mentioned discrepancy in the symmetry of the X point, is again in contrast to previous reports,^23,26) where the corresponding band shows a small but appreciably dispersion along \(k_{z}\).

The band structure around the zone center (Γ- and Z-points) also displays a strong dependence on light polarization, on/off resonance condition and temperature. With π polarization, a hole-like band structure (labeled α) is found at the lowest measured temperature. By contrast in the σ channel an electron-like structure accompanied with two resonance structures are observed around the Z-point [Figs. 2(a) and 2(e)]. Both the hole-like structure and the resonances found in the π and σ channels, respectively, are weakened in spectral weight as temperature is increased [Figs. 2(c), 2(g), and 3]. These are the observations that we are going to use in the following to elucidate the orbital nature of the \(cf\)-composed QPs.

4. Analysis

We now turn to an analysis of the \(cf\) hybridization that is dictated by symmetry and extent of orbital overlap. Generally, the conduction electrons can hybridize with the \(f_{5/2}\) states. In a tetrahedral crystal field they split into three levels: \(\Gamma_{7}^{1}= a |J_{z}=\pm 5/2\rangle + b |J_{z} =\mp 3/2\rangle\), \(\Gamma_{6} = |J_{z}=\pm 1/2\rangle\), and \(\Gamma_{7}^{2} = a|J_{z}=\mp 3/2\rangle - b|J_{z}=\pm 5/2\rangle\) with \(a^{2} + b^{2} = 1\). X-ray absorption spectroscopy and inelastic neutron scattering studies yield \(|a|=0.8\) for CeRu₂Si₂.³⁹⁾ The sign of a has not been determined experimentally but is established to be negative in CeCu₂Si₂, with a similar absolute value.³⁹⁾ In what follows, we assume that CeCu₂Si₂ and CeRu₂Si₂ both have \(a<0\) even though \(a>0\) has previously been used in theoretical literature.⁴⁾ Since the conduction bands display strong light polarization dependence, they must have significant even and odd orbital character with respect to the photoemission mirror plane. The in-plane Si p orbitals do not have the respective symmetries, along the projected unit cell diagonal. Furthermore, in LaRu₂Si₂, the low-energy electronic structure predominately stems from the ruthenium d-orbitals.⁴⁰⁾ In what follows, we assume that this conclusion extends to CeRu₂Si₂. Consequently, we are considering hybridization integrals between Ru d orbitals and \(f_{5/2}\) states.

The tetrahedral crystal field environment splits the Ru d-orbitals into \(e_{g}\) (\(d_{x^{2}-y^{2}}\) and \(d_{3z^{2}-r^{2}}\)) and \(t_{2g}\) \((d_{xy}, d_{xz}+d_{yz}, d_{xz}-d_{yz})\) states with \(e_{g}\) being energetically favorable. The Ru atoms are found at a mixed valence of Ru²⁺ and Ru³⁺, which correspond to \(4d^{6}\) or \(4d^{5}\) electronic configurations. Combined, these two facts imply fully (partially) occupied \(e_{g}\) (\(t_{2g}\)) states. The \(e_{g}\) states \(d_{x^{2}-y^{2}}\) and \(d_{3z^{2}-r^{2}}\) have, respectively, odd and even parity along the diagonals Γ-Σ and Z-Y₁. In the \(t_{2g}\)-sector, \(d_{xy}\) is even whereas \(d_{xz}\) and \(d_{yz}\) display mixed character. However, the linear combination \(d_{xz}+d_{yz}\) is even while \(d_{xz}-d_{yz}\) is odd along Γ-Σ.

We are now ready to make our first conclusions. The γ band around the Y₁-point displays even character and no significant resonance effect. This band does therefore not hybridize with the localized f-states. By evaluating \(\langle d|\Gamma_{7}^{1}\rangle\) and \(\langle d|\Gamma_{7}^{2}\rangle\), we find that only \(d_{xy}\) is consistent with this observation within σ-coupling,^41,42) i.e., only \(\langle d_{xy}|\Gamma_{7}^{1}\rangle=\langle d_{xy}|\Gamma_{7}^{2}\rangle=0\) (see Table I). The absent hybridization of Ru \(d_{xy}\) with the Ce f states is intuitively sound from geometric considerations. The spatial extent of the \(d_{xy}\) orbitals at the Ru sites is directed towards the neighbouring Ru atoms, prohibiting hybridization with Ce f states for symmetry reasons.

»View table Table I. Symmetry and \(cf\) hybridization overlap integrals. (Top) The parity with respect to the diagonal mirror plane of the different d orbitals is given in the second column. The remaining columns show the values of k-dependent overlap integrals \(|\langle d|f\rangle|\), with \(f\in \{\Gamma_{6},\Gamma_{7}^{1},\Gamma_{7}^{2}\}\) at the Z point in units of the coupling parameter \(t_{df\sigma}\) for \(k_{z}=1\). (Bottom) Parity along the diagonal for the bands α, β, and γ. Observation of hybridization with localized f-electrons is indicated by yes or no. The last column indicates the intensity ratio between \(T_{0}=9\) K and \(T_{1}=36\) K.

The β band around the Z-point completely disappears under π illumination. We can therefore assign odd orbital character to this band and thus, the orbitals \(d_{x^{2}-y^{2}}\) or \(d_{xz}-d_{yz}\) are the only possibilities. Given the tetrahedral crystal field environment, partial filling of \(d_{xz}-d_{yz}\) is expected. Hence, the β band is most likely composed of \(d_{xz}-d_{yz}\) orbital character. The α band is hole-like and located around the zone center. It disappears completely in the σ channel, implying even mirror symmetry. Arguing again with partial occupation of the \(t_{2g}\) states, the α band therefore most conceivably has \(d_{xz}+d_{yz}\) character.

Figure 3 presents a more detailed view of the temperature effect shown in Figs. 2(c), 2(d), 2(g), and 2(h). Both from the spectra and the momentum distribution curves it becomes clear that the α and β features selectively lose spectral weight when the temperature is increased beyond this system's Kondo temperature \(T_{\text{K}}\) of around 20 K.^43,44) This is the expected behaviour for Kondo QPs as the coupling between localized and itinerant states becomes weak above \(T_{\text{K}}\), leading to their break up.

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