Electronic structure and 4f-electron character in Ce2PdIn8 studied by angle-resolved photoemission spectroscopy

TL;DR

This study investigates the electronic structure of Ce2PdIn8, revealing a high-temperature localized-to-itinerant transition of Ce 4f electrons around 120K using advanced photoemission techniques.

Contribution

First direct observation of the localized-to-itinerant transition of Ce 4f electrons in Ce2PdIn8 over a wide temperature range.

Findings

Hybridization forms a Kondo resonance near the Fermi level at low temperatures.

The transition temperature is around 120K, higher than the coherence temperature.

Electronic structure characterized by high-resolution photoemission spectroscopy.

Abstract

The localized-to-itinerant transition of f electrons lies at the heart of heavy-fermion physics, but has only been directly observed in single-layer Ce-based materials. Here, we report a comprehensive study on the electronic structure and nature of the Ce 4f electrons in the heavy-fermion superconductor Ce2PdIn8, a typical n=2 CenMmIn3n+2m compound, using high-resolution and 4d-4f resonance photoemission spectroscopies. The electronic structure of this material has been studied over a wide temperature range, and hybridization between f and conduction electrons can be clearly observed to form a Kondo resonance near the Fermi level at low temperatures. The characteristic temperature of the localized-to-itinerant transition is around 120K, which is much higher than its coherence temperature Tcoh~30K.

Full text

In heavy fermion materials, $f$ electrons generally exhibit both itinerant and localized feature as the temperature changes. According to the standard Kondo lattice model, $f$ electrons are localized at high temperatures, while conduction electrons would screen the local moments of $f$ electrons to form Kondo singlets upon cooling, eventually resulting in an itinerant heavy Fermi-liquid ground state Coleman and Nevidomskyy (2010). On the other hand, the recent two-fluid theory suggests that part of an atom’s $f$ electrons give rise to the itinerant properties at low temperatures, while others account for the localization behavior at higher temperatures Yang et al. (2008); Yang and Pines (2008, 2012); Yang et al. (2009). Experimentally, detailed evolution of the localized-to-itinerant transition has also been hotly debated Shishido et al. (2002); Haule et al. (2010); Koitzsch et al. (2008, 2009); Chen et al. (2017). Recently, the hybridization between $f$ and conduction electrons were extensively studied, and direct evidence for such localized-to-itinerant transition has been observed in the Ce$M$In5 family ($M$=Co, Rh, Ir) Fujimori et al. (2003); Chen et al. (2017); Fujimori et al. (2006); Chen et al. (2018a, b). However, in other Ce-based heavy-fermion compounds, temperature evolution of the localized-to-itinerant transition of the $f$ electrons has never been reported to the best of our knowledge (Raj et al., 2005; Jiang et al., 2015).

The Cen$M$In3n+2 ($M$=Co, Rh or Ir, with $n$=1, 2, $\infty$) heavy-fermion compounds consist of $n$ layers of CeIn3, stacked sequentially along the $c$ axis with one intervening layer of $M$In2 Bao et al. (2001). The electronic structure of the $n$=$\infty$ and 1 compounds, CeIn3 and Ce$M$In5, has been extensively studied by angle-resolved photoemission spectroscopy (ARPES) Zhang et al. (2018, 2016); Fujimori et al. (2003); Chen et al. (2017); Fujimori et al. (2006), which has been proved to be a powerful tool to directly observe the behavior of $f$ electrons. For the $n$=2 compounds, Ce2CoIn8 and Ce2RhIn8 have been investigated through off-resonant ARPES, and Ce 4$f$ electrons were found to be predominantly localized in both compounds Raj et al. (2005); Jiang et al. (2015). However, to date there have been no reports on these $n$=2 compounds by means of on-resonance ARPES, in which the use of the 4$d$-4$f$ resonance would largely enhance the $f$-electron photoemission matrix element. Moreover, for another important $n$=2 member, Ce2PdIn8 with the unique paramagnetic ground state, de Haas-van Alphen (dHvA) measurements Götze et al. (2015) and X-ray photoelectron spectroscopy (XPS) Werwi$\acute{n}$ski et al. (2015) both suggest delocalized character of the Ce 4$f$ states therein, yet a direct observation of the $f$-electron behavior, and particularly the itinerant-to-localized transition, in this compound is still lacking.

In this Rapid Communication, we present a systematic ARPES study on Ce2PdIn8. Its electronic structure is comprehensively revealed and directly compared with first-principles calculations. In particular, the character of the $f$ electrons is studied by resonant ARPES, and the localized-to-itinerant transition is directly found to set in at a surprisingly high temperature.

High-quality single crystals of Ce2PdIn8 were synthesized by a self-flux method as described in Ref. Kaczorowski et al. (2009a). ARPES measurements were conducted with photons from Beamline I05 of the Diamond light source, which is equipped with a Scienta R4000 electron analyzer. The overall energy resolution is 15$\sim$25 meV depending on the photon energy, and the angular resolution was set to 0.3 ∘. Samples were cleaved at 10 K and then measured between 10 and 190 K under a vacuum better than 5 $\times$10*-11* Torr. Electronic band structure calculations of Ce2PdIn8 were performed with the all-electron general potential linearized augmented plane-wave (FL-APW) method as implemented in the Wien2k code K. S. Blaha et al. (2001). The exchange and correlation effects were treated using generalized gradient approximation (GGA) in the form proposed by Perdew, Wang and Ernzerhof Perdew et al. (1996). The Ce 2$f$ electrons were removed from valence band as it was previously done in DFT calculations for Ce2RhIn8 Jiang et al. (2015); Fujimori et al. (2006). Spin-orbit coupling (SOC) was included as a second variational step with a basis of scalar-relativistic eigenfunctions, after the initial calculation was converged to self-consistency. The Monkhorst–Pack Monkhorst and Pack (1976) special $k$-point scheme with $23\times 23\times 9$ mesh was used resulting in 390 $k$ points in the irreducible part of the Brillouin zone. The wave functions in the interstitial region were expanded using plane waves with a cutoff of ($R_{mt}K_{max}$) = 8, where $K_{max}$ is the plane wave cut-off, and $R_{MT}$ is the smallest of all atomic sphere radii. Experimental lattice parameters have been used with atoms fixed in their bulk positions.

Fig. 1(a) illustrates the crystal structure of Ce2PdIn8 and the corresponding three-dimensional Brillouin zone (BZ). This compound forms in the tetragonal Ho2CoIn8-type crystal structure ($P4/mmm$) with lattice parameters $a$=$b$=4.69 Å and $c$=12.185 Å, and consists of a sequence of two CeIn3 layers alternating with a PdIn2 layer along the $c$ axis Tursina et al. (2013). To investigate its three-dimensional electronic structure, we have performed detailed $k_{z}$-dependent measurements by varying the photon energy between 75 and 95 eV. As shown in Fig. 1(b), there exists weak photon-energy dependence of peak positions in the momentum distribution curves (MDCs) taken along the high symmetry $\Gamma$-$M$ direction, indicating the quasi-two-dimensional character. An inner potential of 13 eV, comparable to that of CeIn3 Zhang et al. (2016), has been obtained through the best fit to the periodic variation of the MDCs’ peak positions according to the free-electron final-state model Hüfner (2003), allowing us to determine the photon energies corresponding to the high-symmetry planes.

Figs. 1(c) and 1(d) compare photoemission intensity maps for the \varGamma$$X$$M and Z$$A$$R planes, and their detailed photoemission intensity plots along high-symmetry directions are illustrated in Figs. 1(e-j) using the same colors. The Fermi surfaces in the \varGamma$$X$$M plane consist of a square Fermi pocket around $\varGamma$ and a racetrack pocket centered at $X$ (both formed by the holelike band $\gamma$), and four pockets around $M$ resulting from the Fermi crossings of the electronlike bands $\alpha$, nearly-degenerate $\varepsilon$/$\delta$, and $\beta$. In the Z$$A$$R plane, both the shape of most Fermi pockets and the corresponding band dispersions exhibit only minor variations, confirming the quasi-two-dimensional character of these bands. Still, differences can be resolved between the Fermi surface topologies of these two $k_{z}$ planes. Changes in both the shape and size of the $\gamma$ pocket in the Z$$A$$R plane imply a three-dimensional character of this band. The main bands in the vicinity of the Fermi level ($E_{F}$) have similar orbital character to those in the $n$=1 compound CeCoIn5, while band splitting due to interlayer coupling makes the band structure of Ce2PdIn8 more complex. Overall, our experimental data show a qualitative agreement with first-principles calculations, as displayed in Figs. 1(k) and 1(l).

In order to highlight the $f$-electron behavior, we chose 121 eV photons (the energy of the Ce 4$d$-4$f$ transition) to realize resonant enhancement of the Ce 4$f$ photoionization cross section. Figs. 2(b) and 2(c) show the off-resonance and on-resonance photoemission intensity plots along $\varGamma$-$M$, taken with 114 eV and 121 eV photons, respectively, using $s$-polarized light. The corresponding on-resonance photoemission intensity plot taken with $p$-polarized light is displayed in Fig. 2(d). Here, $p$- and $s$-polarized light is defined to be from electric fields in and out of the emission plane, which is determined by the analyzer slit and the sample surface normal, as illustrated in Fig. 2(a).

As shown in Fig. 2(b), Pd 4$d$ and Ce 5$d$ states dominate the off-resonance spectra, consistent with the data shown in Fig. 1(e). The Ce 4$f$ character is strongly enhanced in the on-resonance cases, as demonstrated by the flat bands in the photoemission intensity plots [Figs. 2(c) and 2(d)] and the apparent peaks in the corresponding integrated spectra [Fig. 2(e)]. The three evident features located around $E_{F}$, -0.25 eV and -2.5 eV below $E_{F}$ can be well understood by the single impurity Anderson model (SIAM) and were thus assigned as 4$f^{1}_{5/2}$, 4$f^{1}_{7/2}$ and 4$f^{0}$, respectively Gunnarsson and Schönhammer (1983); Chen et al. (2017). Moreover, we note that the 4$f$ state near $E_{F}$ shows a sharp peak under $s$-polarized light but a broad hump in $p$-polarized light, and the overall $f$ electron intensity observed with $p$-polarized light is weaker than with $s$-polarized light. Therefore, further investigations of the Ce 4$f$ states in this compound were performed with $s$-polarized light.

Next we performed temperature-dependent ARPES measurements on this compound to investigate the evolution of the itinerant/localized nature of the $f$ electrons with temperature. At 5 K, a flat $f$-electron feature near $E_{F}$ can be clearly observed in the intensity plot, as demonstrated in Fig. 3(a). At this temperature, large $f$ spectral weight can be clearly observed near $E_{F}$, suggesting a rather itinerant nature of the $f$ electrons. Upon increasing the temperature, the spectral weight of the $f$ states gradually diminishes while that of the $\beta$ band is still preserved even to rather high temperature (180 K). This temperature dependence is also clearly visible when directly comparing the EDCs at $\Gamma$ taken at different temperatures [Fig. 3(b)]. We observe a suppression of the $f$ spectral weight upon increasing temperature, and the sharp $f$-electron quasi-particle peak near $E_{F}$ is no longer visible at 150 K. Fig. 3(c) presents the temperature dependence of $f$-electron spectral weight in the vicinity of $\Gamma$ in Ce2PdIn8, which is obtained by integrating the EDCs over [$E_{F}$ - 100 meV, $E_{F}$ + 10 meV]. Here, the spectral weight has been normalized at 200K and is found to keep arising with the decreasing temperature, showing the same trend as Fig. 3(b).

For comparison, Fig. 3(d) and Fig. 3(e) show the normalized photoemission intensity maps taken at 5 K and 190 K, respectively. In the 5 K Fermi surface map, the normalized intensity around the Brillouin zone center is strongly enhanced, originating from the itinerant $f$ states. In contrast, the $f$ electron spectral weight around the same momentum is greatly suppressed for the map contour taken at 190 K. Note that we have presented both maps using the same color scale, so the much higher spectral weight around the zone center at 5 K should not be artificial. Moreover, as shown in the supplementary materialSM , direct comparison of MDCs taken at 5 K and 190 K can further confirm the spreading out of the $f$-band spectral weight from the zone center to somewhere with the increasing temperature. All the above data explicitly show the characteristic localized-to-itinerant crossover of $f$ electrons, which has never been reported in the prototypical Ce 218 heavy fermion systems before. Remarkably, the occurrence of hybridization starts from about 120 K in Ce2PdIn8, which is much higher than $T_{coh}\sim 30$ K Kaczorowski et al. (2009a, 2010a, b, 2010b).

In Fig. 4 we present the zoomed-in photoemission intensity plot near $E_{F}$ to concentrate on the $f$-electron character and quasiparticle bands. Compared with spectra taken at high temperatures, when the $\beta$ band shows linear dispersion, evident bending of this band around $E_{F}$ can be observed at the low temperature [Fig. 4(a)]. This is a direct evidence of the hybridization between the $f$ and conduction electrons. According to the standard view on the heavy fermion ground state based on periodic Anderson Model (PAM), the hybridization between the $f$ and conduction electrons would create two separate bands with band bending near $E_{F}$. Meanwhile, the $f$ spectral weight should be redistributed, with significant enhancement to the “inside” of the hole-like bands and the “outside” of the electron-like bands. In Fig. 4(b), we indeed observe the enhancement of the $f$ spectral weight “inside” of the hole-like $\beta$ band, which is one more evidence for the hybridization between the $f$ and conduction electrons in Ce2PdIn8. Such behavior is similar to that observed in the heavy-fermion compound CeIrIn5 Chen et al. (2018b).

To further quantitatively investigate the hybridization of conduction bands and $f$ electrons, we zoom in the spectral feature around $\varGamma$ in the vicinity of $E_{F}$ and then performed the PAM fitting, which has been proved an effective way to study the $f$ electron behaviors Chen et al. (2017, 2018a). Details of the band dispersion extraction and the PAM fitting can be found in the supplementary material SM . Here, the resolution-convoluted Fermi-Dirac distribution has been divided in order to probe the band structure slightly above $E_{F}$, as shown in Fig. 5(a).

Quantitatively, a fit to PAM picture gives a renormalized hybridization $V_{k}$=15$\pm$5 meV for both conduction bands, which implies the substantial hybridization strength between $f$ electrons and conduction $d$ bands. Moreover, as shown in Fig. 5(b), a heavy quasiparticle band is induced with an energy dispersion of around 6 meV, which is larger than that in CeIn3 Zhang et al. (2018); Fujimori (2016), indicating a stronger hybridization in CeIn3.

To summarize, we have characterized the electronic structure and $f$ electron behavior of Ce2PdIn8 using high-resolution resonant and non-resonant ARPES. Besides presenting the low-lying electronic structure of this material, we have shown how localized $f$ electrons in Ce2PdIn8 become partially itinerant and evolve into the heavy-fermion state from a much higher temperature than the coherence temperature $T_{coh}$. These findings provide a comprehensive experimental picture of the character of $f$ electrons in Ce2PdIn8, complementing our understanding of $f$ electrons’ itinerant-to-localized evolution in the CenMmIn3n+2m heavy fermion family.

We thank Diamond Light Source for the beam time at beamline I05 (Proposal No. SI16345) that contributed to the results presented here. We grateful acknowledge helpful discussions with Dr. D. C. Peets. The work was supported by the National Science Foundation of China (Grants No. 11874330, 11274332, 11574337, 11704073, 11504342, and U1332209), the National Key R&D Program of the MOST of China (Grants No. 2016YFA0300200, 2017YFA0303104). Also, this work is supported by the National Science Centre (Poland) under research grant No. 2015/19/B/ST3/03158 and U1332209), and the Science Challenge Project (No. TZ2016004). P.S.’s work at Ames Laboratory (electronic structure calculations) was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering. Ames Laboratory is operated for the US Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. D.W.S is also supported by “Award for Outstanding Member in Youth Innovation Promotion Association CAS”.