Evolution of the Kondo lattice electronic structure above the transport coherence temperature

TL;DR

This study combines ARPES experiments and DMFT calculations to reveal that in the Kondo lattice system CeCoIn$_5$, $f$ electrons participate in the Fermi surface at temperatures well above the coherence temperature, challenging previous assumptions.

Contribution

It provides a microscopic understanding of the temperature evolution of $f$-$d$ hybridization and Fermi surface participation in CeCoIn$_5$, linking experimental ARPES data with realistic DMFT calculations.

Findings

$f$ electrons participate in the Fermi surface above $T^*$

Identification of a crystalline electric field degeneracy crossover below $T^*$

ARPEs and DMFT results show temperature-dependent hybridization behavior

Abstract

The temperature-dependent evolution of the Kondo lattice is a long-standing topic of theoretical and experimental investigation and yet it lacks a truly microscopic description of the relation of the basic $f$-$d$ hybridization processes to the fundamental temperature scales of Kondo screening and Fermi-liquid lattice coherence. Here, the temperature-dependence of $f$-$d$ hybridized band dispersions and Fermi-energy $f$ spectral weight in the Kondo lattice system CeCoIn$_5$ is investigated using $f$-resonant angle-resolved photoemission (ARPES) with sufficient detail to allow direct comparison to first principles dynamical mean field theory (DMFT) calculations containing full realism of crystalline electric field states. The ARPES results, for two orthogonal (001) and (100) cleaved surfaces and three different $f$-$d$ hybridization scenarios, with additional microscopic insight provided by DMFT, reveal $f$ participation in the Fermi surface at temperatures much higher than the lattice coherence temperature, $T^*\approx$ 45 K, commonly believed to be the onset for such behavior. The identification of a $T$-dependent crystalline electric field degeneracy crossover in the DMFT theory $below$ $T^*$ is specifically highlighted.

Full text

I Introduction

The class of heavy fermion materials exhibits at low temperature ($T$) an itinerant Fermi-liquid due to Kondo screening that arises from hybridization of $f$ and conduction band ($d$) states, and emerges from an $f$ local moment regime at high $T$ Varma (1976); Stewart (1984); Coleman (2007). The $f$-electrons are predicted Coleman (2007); Choi et al. (2012) to be included in a “large” Fermi surface (FS) for the former and excluded from a “small” FS for the latter. One of the key unsolved problems is finding what determines the $T$-scale(s) for the evolution of this behavior for the dense periodic “Kondo lattice” of $f$-moments. Theoretical models of the Kondo lattice involve two basic temperature-scales, the single-impurity Kondo temperature ($T_{\textrm{K}}$) and the lattice coherence temperature ($T^{*}$). There has been debate as to the relative magnitudes of the two $T$ scales Burdin and Zlatić (2009) and whether only one $T$-scale is relevant to the lattice problem Nakatsuji et al. (2004); Grenzebach et al. (2006). Although there have been intensive studies on the Kondo breakdown of $f$-$d$ hybridization and consequent abrupt changes of the FS size at low $T$ or with variation of a tuning parameter near a quantum critical point Gegenwart et al. (2008); Si and Steglich (2010), there have been few studies on the microscopic understanding of $f$-$d$ hybridization $T$-scales extending to high $T$ Burdin and Zlatić (2009). A commonly held belief that $f$-$d$ hybridization occurs only below $T^{*}$ is not citable in any microscopic description, but is codified in a universal scaling formula of the 4$f$ density of states (DOS) proposed for the phenomenological two-fluid model Yang and Pines (2008).

The issue of the FS size has recently been highlighted in angle-resolved photoelectron spectroscopy (ARPES) measurements of YbRh2Si2, a hole analog to Ce heavy fermion materials, in which the “large” (hole) FS is observed to much higher $T$ than expected, and the transition to the “small” FS is yet to be observed Kummer et al. (2015). In this work, we use ARPES and dynamical mean field theory (DMFT) calculations, including both spin-orbit and crystalline electric field (CEF) splittings of the $f$-states, to investigate the $T$-dependent electronic structure of the Kondo lattice system CeCoIn5. Resonant enhancement of the ARPES Ce 4$f$ spectral weight is used to highlight Fermi-level ($E_{\textrm{F}}$) participation of $f$-electrons in the 3D Fermi surface, whose detailed topology is revealed using measurement from two orthogonal (001) and (100) surfaces. The DMFT calculations confirm the ARPES result of $f$-electron participation in the FS to temperatures much higher than $T^{*}$, far into the logarithmic $T$-regime of “incoherent” Kondo spin-flip scattering, and provide insight into the role(s) of CEF $f$-states in the high $T$ behavior. Specifically, the DMFT spectral functions explicitly show and confirm the concept Cornut and Coqblin (1972) of a $T$-dependent crossover of the Kondo resonance effective degeneracy of the two lowest CEF $f$-states.

The $T$ scales specific to CeCoIn5 are illustrated in relation to its resistivity profile in Fig. 1(a) with schematic images of the Kondo lattice screening $T$ regimes in Fig. 1(b). First, the single impurity Kondo temperature $T_{\textrm{K}}$ corresponds to the crossover from a logarithmic regime (extending far above $T_{\textrm{K}}$) of incoherent spin-flip scattering with anti-ferromagnetic Kondo coupling (described by perturbation theory) to a nonperturbative strong Kondo coupling regime that ultimately leads to a fully screened Kondo singlet ground state (well below $T_{\textrm{K}}$). The high-temperature $onset$ appearance of the -ln($T$) Kondo scattering regime, for which we introduce the label $T_{\textrm{K}}^{\prime}$, is approximately signified by the resistivity minimum crossover ($\sim$200 K) from $T^{5}$ lattice phonon scattering Kondo (1964). In dilute $f$ moment systems, the resistivity profile eventually plateaus to a constant value below $T_{\textrm{K}}$. Resistivity profile scaling behavior in a La dilution study of CeCoIn5 has determined a very small value of $T_{\textrm{K}}$ $\approx$ 1.7 K Nakatsuji et al. (2002), or $T_{\textrm{K}}$ $\approx$ 5 K estimated from the temperature at which the entropy obtained from specific heat measurements Zapf et al. (2001) reaches a value of \frac{1}{2}$$Rln2.

For a dense periodic array of $f$ magnetic moments, intersite coupling between $f$-electrons (schematically represented by overlapping Kondo screening clouds in Fig. 1(b)) leads to coherence of the $f$-$d$ scattering and a downturn in the resistivity. Hence the $transport$ lattice coherence temperature $T^{*}$  is identified experimentally as the resistivity maximum, $\sim$45 K in CeCoIn5. $Partial$ screening of the $f$-moments in the two intermediate $T$-regimes in the Fig. 1(b) and $partial$ coherence below $T^{*}$ are important concepts for our understanding, which naturally allow for T_{\textrm{K}}$$< T^{*}$$< $T_{\textrm{K}}^{\prime}$. The broad crossover behavior and $T$ scale definitions are further discussed in the Supplementary sup .

II 3D Fermi Surface $k$-locations

The temperature dependence of three different $f$-$d$ hybridization scenarios, schematically shown in Fig. 1(c), are studied. Their $k$-locations relative to the experimentally determined FS topology Dudy et al. (2013); Denlinger are indicated in Fig. 1(d). The $E_{\textrm{F}}$ crossing of the quasi-two-dimensional (2D) $\alpha$-band corresponds to the scenario #1, where the low $T$ $f$-$d$ hybridization causes a heavy mass dispersion of the $d$-band near $E_{\textrm{F}}$ with enhanced $f$-weight in the dispersion kink. In scenario #2, two $d$-band dispersions form a narrow hole pocket at high $T$ and induce even greater $f$-weight below $E_{\textrm{F}}$ at low $T$ with possible removal of the $E_{\textrm{F}}$-crossings. This scenario arises at the tip of the diamond-shaped $\gamma$ sheet near the zone center. In scenario #3, an unoccupied electron-like $d$-band minimum exists above $E_{\textrm{F}}$, invisible to ARPES at high $T$. At low $T$ its hybridization with the flat $f$-band just above $E_{\textrm{F}}$  creates a strong $f$-weight “hotspot” corresponding to a very shallow electron $E_{\textrm{F}}$-crossing. This scenario #3 occurs at multiple points in the Brillouin zone (BZ), including the edge of $Z$-point holelike $\gamma$-band FS indicated in Fig. 1(d).

The ARPES study of the $f$-weight $T$-dependence is assisted by the 4$d$-4$f$ resonant enhancement of the 4$f$ photoionization cross section at 122 eV photon energy. Fig. 1(e) shows a 2D schematic of the cross section of the bulk tetragonal BZ with Fermi-energy $k$-space arcs for photon energies in the range of 90-140 eV. Two different orthogonal cleave surfaces of CeCoIn5 were measured with ARPES. For the (001) cleave surface, the photon energy range spans $\sim$1.5 BZs along the $c$-axis with high symmetry $\Gamma$- and $Z$-points occuring at $\approx$ 90, 105, 122, and 140 eV. Thus the resonant energy of 122 eV cuts very close the high symmetry $\Gamma$ plane. In contrast, for the orthogonal (100) cleave surface, the same photon energy range covers less than a full BZ along the $x$-axis with high symmetry $\Gamma$- and $X$-planes at $\approx$110 and 140 eV, respectively, and the resonant energy of 122 eV cuts midway between $\Gamma$ and $X$.

Earlier ARPES studies of CeIrIn5 Fujimori et al. (2003, 2006) and CeCoIn5 Koitzsch et al. (2008, 2009) have essentially concluded that those systems are “nearly localized” even down to low $T$ from comparison to density functional (DFT) band calculations Fujimori et al. (2003); Koitzsch et al. (2009), but yet have a “small itinerant” low energy scale component as revealed also by $f$-resonant ARPES along $\Gamma$-$X$ Fujimori et al. (2006) or along $X$-$M$ Koitzsch et al. (2008). Such seemingly contradictory localized yet itinerant character in the ARPES measurement is a natural consequence of (i) the weak $f$-$d$ hybridization for a low $T_{\textrm{K}}$ system, (ii) experimental ARPES resolution limitations, and (iii) the well-known $f$-bandwidth deficiencies of DFT. More sophisticated theoretical treatments such as “renormalized band theory” Zwicknagl (1992, 2011) or DFT+DMFT Shim et al. (2007); Choi et al. (2012) include the necessary ingredients of electron correlation and dynamical screening for proper description of the low energy scale physics. DMFT additionally provides single-particle excitation spectral function results that are directly comparable to ARPES spectra.

Theoretical and experimental slices of the 3D Fermi surface, presented in Fig. 2, highlight the three specific $k$-locations for subsequent $T$-dependent study. Figure 2(a) shows the DMFT spectrum of the Fermi surface in the high symmetry $\Gamma$-plane with labeling of $M$-centered $\alpha$ and $\beta$ electron sheet contours and also holelike $\gamma$ tube FS along $\Gamma$-$X$ that connects to a diamond-shaped FS centered on the $\Gamma$-point. In Fig. 2(b), we show a resonant energy Fermi-edge intensity map from the (001) cleave surface, symmetrized about $k_{x}$=0, showing agreement with the DMFT results. More detailed $\Gamma$- and $Z$-plane comparisons are given elsewhere Dudy et al. (2013); Denlinger . The FS map in Fig. 2(b), using $s$-polarization of the incident light, also exhibits distinct enhancement of $f$-weight at the edges of the $\alpha$-band along $X$-$M$ (cut #1) and at the tip of the $\Gamma$-centered diamond-shaped FS (cut #2). Line cuts #1 and #2 through these two $k$-points are used for the $T$-dependent measurements presented later.

A DMFT Fermi-energy spectral image of the orthogonal $\Gamma$-$X$-$Z$-$R$ plane in Fig. 2(c) highlights the existence of a $Z$-centered hole FS (labeled $\gamma_{Z}$) and its relation to the tubular $\gamma$ sheet along $\Gamma$-$X$. Diagonal features along $\Gamma$-$R$ are associated with the $\beta$’ FS represented in Fig. 1(d). The “X”-shape of the $Z$-centered hole FS is nicely observed in the high symmetry 110 eV angle-dependent map of an orthogonally cleaved (100) surface shown in Fig. 2(d). An angle-dependent map at the $f$-resonance energy of 122 eV is then observed in Fig. 2(e) to be dominated by two bright $f$-hotspots at the BZ boundaries where the $k_{x}$ location is at the edge of the $Z$-centered hole FS where the curvature becomes concave (electronlike). A line cut (#3) through these two hotspots is used for $T$-dependent measurements in Fig. 4.

The complex shape of the $Z$-centered hole-FS with strong $f$ participation, identified here in DMFT and ARPES, and further detailed in the Supplementary sup , is notable in that it exists in localized ($f$-core) DFT calculations, inherently without any $f$-contribution, and yet is completely $absent$ in itinerant DFT calculations. This highlights the artificial and sometimes misleading conclusions derived from the standard “itinerant-versus-localized” DFT theory comparison. While simple post-facto energy-scale renormalization of itinerant DFT can be a sufficient correction in cases of isolated $\alpha$ or $\beta$ Fermi surface band crossings, the low energy scale complexity of the non-$f$ $\gamma$-band structure along $\Gamma$-$Z$ in CeCoIn5 is a prime example where the itinerant DFT large $f$-bandwidth disruption is too great. High symmetry valence band dispersions of the full occupied bandwidth of the $\alpha$, $\beta$ and $\gamma$ $d$-bands for the (001) and (100) cleave surfaces are also provided in the Supplementary sup .

III Large-to-small Fermi surface $T$-dependence

In this section we focus attention on the low energy scale $f$-$d$ hybridization modification of the $\alpha$ and $\beta$-band along $X$-$M$ (line cut #1) to highlight the large-to-small Fermi surface size change with temperature. The DMFT spectral function at 10 K for this $k$-cut, shown in Fig. 3(a), shows the $f$-$d$ hybridization interaction between two $d$-bands and three crystalline electric field (CEF) split $f$-levels, with a rich complexity of connectivity and selectivity that arises from the close proximity of the $d$-bands and the relative symmetries of the $f$ and $d$ states. The three 4$f_{5/2}$ CEF doublets, labeled $f_{0}$, $f_{1}$, and $f_{2}$, correspond to $\Gamma_{7}^{(1)}$, $\Gamma_{7}^{(2)}$ and $\Gamma_{6}$ orbitals, respectively, and their relative energies of \sim$$E_{\textrm{F}}, +8 meV and +40 meV, are in good agreement with neutron scattering measurements of 1st and 2nd excited states at +8 meV and +25 meV Bauer (2004); Willers et al. (2010). Whereas the hybridized outer $\beta$-band connects with very heavy mass dispersion to the $f_{0}$ ground state level, the close proximity of the $\alpha$-band requires its hybridized dispersion to immediately connect to the 1st excited $f_{1}$ level, thereby giving it an order-of-magnitude larger Fermi velocity (v_{\textrm{F}}$$\approx0.2 eV-Å) compared to the $\beta$-band ($\approx$0.02 eV-Å). This close proximity effect, specific for this $k$-region, contributes to the much smaller average effective mass ($m^{*}<$18) for the $\alpha$-sheet orbits in dHvA as compared to the $\beta$-sheet orbits ($m^{*}>$48) Settai et al. (2001).

The differences in the occupied $\alpha$ and $\beta$ dispersions are also visualized in Fig. 3(b) where the DMFT spectral function has been multiplied by the 10 K Fermi-Dirac cutoff. In addition to the very different \Delta$$k_{\textrm{F}} shifts relative to the extrapolated $d$-band dispersion Fermi wave-vector ($k_{\textrm{F}}$) values, the relatively weak $f$ weight at $k_{\textrm{F}}$ in the $\alpha$-band is further diminished for the even heavier $\beta$-band dispersion. An experimental 122 eV resonance energy cut through the $\alpha$- and $\beta$-bands, slightly displaced from the $X$-$M$ line, is shown in Fig. 3(c) with overplotted DMFT dispersions. While quantification of the $\beta$-band is limited by the resolution of $\sim$15 meV, a relatively stronger $f$-weight in the $\alpha$-band, similar to that of the theory calculation, is present for both $s$- and $p$-polarization of the incident light.

Figure 3(d) shows $\alpha$-band energy dispersion images for the line cut #1 for four temperatures selected out of a $T$-series measured from 8 K up to 86 K. The enhanced $f$-weight near $E_{\textrm{F}}$ is observed to diminish simultaneously with the low energy kink becoming less visible. The overplotted DMFT dispersion at 86 K shows still a small dispersion kink at this temperature. The weak $f$-weight enhancement and $T$ dependence is also shown in the $k_{\textrm{F}}$ line spectra in Fig. 3(e). A previous resonant ARPES analysis of $k$-integrated windows just inside and outside the $\alpha$-band dispersion at three temperatures has also reported a weakened but still discernible low energy scale $f$-peak at 105 K in comparison to 20 K and 180 K Koitzsch et al. (2008).

Confident that the signatures of the large-to-small FS change have been observed experimentally with basic agreement to theory, we go beyond the ARPES resolution and $T$-range limitations and additionally analyze the DMFT spectral functions to extract the peak dispersion, Fermi velocity and Fermi momentum of the $\alpha$-band for many intermediate and high temperatures in Figs. 3(f), 3(g) and 3(h). Upon cooling from 1000 K to 200 K, Fig. 3(f) shows a gradual $k_{\textrm{F}}$ shift resulting from a near linear band velocity change extending to 100 meV below $E_{\textrm{F}}$. Then below 200 K (\approx$$T_{\textrm{K}}^{\prime}) a weak kink in the dispersion develops around -20 meV and the Fermi velocity begins to more rapidly decrease. Upon further cooling, the rapid $v_{\textrm{F}}$ change is observed to slow down around 50 K (near $T^{*}$) and then becomes constant below 30 K where $k_{\textrm{F}}$ also stops changing.

Figures 3(g) and 3(h) also show comparison of the theory $T$-dependence to experimental $v_{\textrm{F}}$ and $k_{\textrm{F}}$ values at high and low $T$. The high binding energy ARPES band velocity of 2.5 eV-Å is compared to the DMFT at high $T$, whereas at low $T$, the ARPES $v_{\textrm{F}}$ and $k_{\textrm{F}}$ values are estimated from a visual triangular fit of the dispersion kink near $E_{\textrm{F}}$. We do not claim to experimentally verify the detailed DMFT $v_{\textrm{F}}$ and $k_{\textrm{F}}$ $T$-dependent profile(s), but we note that the 33 K spectrum bears greater resemblance to the 8 K spectrum than to the 86 K spectrum, indicating that the FS evolution is not just beginning at $T^{*}$.

Thus a central finding, so far, is that the resistivity downturn temperature $T^{*}$, associated with lattice coherence, does not signify the $onset$ of the heavy effective mass $f$-$d$ hybridization dispersion curvature, nor the $onset$ of FS size changes indicated by $k_{\textrm{F}}$ changes. Previous DMFT calculations, without the inclusion of CEF states have similarly predicted dHvA orbit FS size changes occuring as high as 130 K ($>$2.5 $T^{*}$) for CeIrIn5  Choi et al. (2012), and near vertical kinks in the non-$f$ dispersive states in CeCoGe2 as high as $\sim$200 K $prior$ to the formation of heavy mass band dispersion below $\sim$90 K Choi et al. (2013). The additional presence of the CEF states in the DMFT calculations here may be responsible for the even higher onsets of the heavy mass $v_{\textrm{F}}$ changes and FS size $k_{\textrm{F}}$-shifts (up to 1000K) theoretically observed in Fig. 3. Similar analysis of the DMFT $\beta$-band dispersion, shown in the Supplementary sup , indicate that the transport coherence temperature in CeCoIn5 is more closely associated with the most rapid $T$-dependent changes of $v_{\textrm{F}}$ and $k_{\textrm{F}}$.

Scanning tunneling microscopy (STM) quasiparticle interference (QPI) on the Ce-In terminated surface has also observed a heavy mass band dispersion kink along the (100) direction of CeCoIn5 at 20 K with a scattering $q$-vector of 0.2\cdot$$(2\pi/a) that is consistent with $\alpha$-sheet FS nesting in the $Z$-plane Aynajian et al. (2012). Hence those results can be directly compared to the $\alpha$-band ARPES measurements in Fig. 3. At 70K, the STM-QPI observes the linear light mass dispersion above $E_{\textrm{F}}$, without the heavy mass dispersion kink, but with still a weak hybridization intensity dip.

IV Hotspot $f$-weight $T$-dependence

Next we turn our attention from $T$-dependent $dispersion$ analysis to the $T$-dependent $f$-$weight$ signatures of $f$ participation in the Fermi surface for Fig. 1(c) $f$-$d$ hybridization scenarios #2 and #3, where the stronger $f$-weight allows experimental analysis up to temperatures as high as 200 K. Figures 4(a) and 4(b) show 122 eV energy dispersion images for selected temperatures for (001) and (100) cleave surface line cuts #2 and #3 in Fig. 2, respectively. Overplotted DMFT bands (white lines) at low and high $T$ illustrate the $f$-$d$ hybridization scenarios of $f$-weight being pulled below $E_{\textrm{F}}$ at the center of a narrow hole-like $d$-band dispersion (Fig. 4(a)) and an electronlike $f$-dispersion at the zone boundaries being induced by a non-$f$ band above $E_{\textrm{F}}$ (Fig. 4(b)).

The $T$-dependent line spectra at the (001) $f$-hotspot and at one of the two (100) hotspots is shown in Fig. 4(c) and Fig. 4(d), respectively. The hotspot $f$-bandwidth below and above $E_{\textrm{F}}$ is notably larger for the (100) surface ($\pm$100 meV) as compared to the (001) surface ($\pm$50 meV), consistent with larger $f$-band dispersions in the overplotted DMFT theory. In both cases, while the $f$ peak appears to be suppressed in the energy dispersion images at 180 - 200 K, the line spectra reveal a continuous decrease of the peak amplitude all the way up to the highest measured temperature. A $T$-dependent Fermi-Dirac distribution (FDD) function convolved with a Gaussian instrumental broadening of 15 meV, illustrated for low and high $T$ in Figs. 4(c) and 4(d), is used for background subtraction Reinert et al. (2001) for the extraction of the normalized 4$f$ $amplitude$ $T$-profiles in Fig. 5. Note that this implies a finite 4$f$ DOS even for a flat line spectrum that does not exhibit a visual peak. The sensitivity to different background subtraction methods is discussed in the Supplementary sup .

For theoretical comparison, DMFT $k$-$integrated$ $f$ density of states (DOS) spectral weights were calculated from 10 K to 1000 K, shown in Fig. 5(a) for only the Kondo resonance and CEF-split states. The DMFT spectra are then multiplied by a Fermi-Dirac distribution (FDD) function and convolved with a Gaussian instrumental broadening of 15 meV, to simulate the photoemission measurement, and plotted in Fig. 5(b).

Using the 750 K spectrum for background subtraction, the $T$-dependent Fermi-edge weight is plotted in Fig. 5(c) with comparison to the ARPES $f$-peak $amplitudes$. The experimental results for the two $f$-hotspots are very comparable with each other, despite the different $f$-$d$ hybridization scenarios and $f$-peak widths. A reasonable agreement of the approximate logarithmic $T$-dependence between 40K and 200K is found for normalization of the DMFT $T$-profile to 1.2 at low $T$. Both experiment and theory agree as to the existence of a long $f$-weight tail extending to high $T$ far above $T^{*}$. A reason for the low $T$ discrepancy and saturation of the ARPES $T$-profile is the instrumental resolution suppression of the low $T$ peak amplitude.

V CEF Effects

It is important to carefully delineate the various effects that contribute to the high $T$ extension of the $f$-weight in Fig. 5(c). First, analysis of the full-range $k$-integrated DMFT spectra, presented and discussed in the Supplementary sup . reveal that the large residual high $T$ $f$-DOS in the Kondo resonance (KR) region (including CEF and spin-orbit excitations) does not result from a strong spectral weight transfer (it is $<$10%) from the KR region to the higher $f^{2}$ energy range above 2 eV. Rather, the primary $T$-dependence in the KR region is that of peak broadening of spectral weight from the peak center(s) to the peak tails. This leads to a number of generic effects which contribute to the appearance of relatively enhanced $E_{\textrm{F}}$ $f$-weight at higher temperatures, even with background subtraction of the high $T$ $f$-weight. (i) The key notable effect is that of broadening of CEF weight into the KR peak and $E_{\textrm{F}}$ energy windows. Their $T$-dependent profiles will then include the $T$-dependence of the tail of the CEF peak. Such CEF broadening origin of enhanced $E_{\textrm{F}}$ $f$-weight at high $T$ has been noted before Reinert et al. (2001); Ehm et al. (2007) within single-impurity NCA calculations Bickers et al. (1987). (ii) Similarly the $E_{\textrm{F}}$ weight will inherently have a weaker $T$-profile than the KR peak because it exists in the KR tail with a smaller low $T$ amplitude and a similar high $T$ amplitude. (iii) Finally large energy-window $area$ analyses will generally have weaker $T$-dependences than narrow energy-window $amplitudes$.

Another important CEF effect concerns the $T$-dependent effective $f$-degeneracy as discussed in the early theoretical perturbation theory calculations of resistivity profiles of Ce heavy fermion compounds Cornut and Coqblin (1972). The effect leads to a larger effective Kondo temperature and an extended high $T$ logarithmic resistivity regime relative to the N=2 doublet groundstate Kondo effect. The degeneracy crossover effect was initially predicted to manifest as thermal depopulation hump(s) in the resistivity profile around the CEF energy-level splitting temperatures Cornut and Coqblin (1972), e.g. 100 K and 300K for CeCoIn5. However, a theoretical treatment going beyond third order perturbation theory and including Kondo broadening of the CEF levels predicts a modification to the resistivity profile at much lower $T$’s than the nominal CEF splittings i. Kashiba et al. (1986). The KR and CEF peak broadening within single-impurity NCA calculations of the $f$-DOS, discussed above, does in general lead to indistinguishability of CEF peaks at temperatures below the actual CEF splitting energies.

CeCoIn5 experimentally exhibits only a single resistivity maximum (at 45 K) whose downturn is firmly associated with lattice coherence from La-dilution studies Nakatsuji et al. (2002), with no apparent additional secondary humps that one might associate with the CEF merging effect. An enhanced N=6 Kondo temperature due to both CEF excitations, was estimated to be $T_{\textrm{K}}$(6) $\approx$ 35 K Nakatsuji et al. (2002); Hanzawa et al. (1985) relative to $T_{\textrm{K}}$=1.7 K for the dilute system, and so the coherence downturn below 45 K could possibly be masking the CEF degeneracy crossover(s).

The manifestation of such a CEF $f$-degeneracy crossover of the two lowest CEF levels is observed in the $k$-integrated DFT+DMFT spectral function $T$-dependence shown in Fig. 5(d) and 5(e). While single-impurity NCA calculations exhibit broadening and monotonic decline of fixed energy KR and CEF $f$-peaks, here both the ground state $\Gamma_{7}^{(1)}$ and first-excited $\Gamma_{7}^{(2)}$ CEF peaks are observed to shift $towards$ each other and merge into a single enhanced-amplitude peak as early as 30 K, before the subsequent monotonic broadening decline to higher $T$. This is suggestive of the crossover from a narrow $N$=2 degeneracy KR peak ($\approx$1 meV center with 3 meV width) at 10 K to a new $N$=4 degeneracy KR peak ($\approx$4 meV center with 9 meV width) in which the $\Gamma_{7}^{(1)}$ and $\Gamma_{7}^{(2)}$ states are indistinguishable. The higher peak energy above $E_{\textrm{F}}$ plus the broader width and enhanced amplitude of the new KR peak are all consistent with the crossover to a larger Kondo temperature. A $k$-resolved DFT+DMFT band image view of this CEF merging effect for the $\alpha$ and $\beta$-band crossings is shown in the Supplementary sup .

The closeness of this CEF degeneracy crossover temperature in the DFT+DMFT theory to the resistivity maximum $T^{*}$ suggests that lattice coherence may be intimately involved in the more rapid CEF merging effect, and that a clean separation of the $T^{*}$ and CEF $T$ scales is not realized in CeCoIn5, resulting in only a single resistivity peak and downturn. A related $T$-dependent DFT+DMFT study of Ce2IrIn8 uses tuning of the CEF splittings to reveal CEF effects on the $f$-orbital anisotropy, the relative Fermi-edge scattering coherence, and the appearance of secondary humps in calculated resistivity profiles Kim et al. .

VI Discussion

The ARPES and DMFT demonstrations of $f$-$d$ hybridization effects well above $T^{*}$ are supported by several other experimental spectroscopy and scattering measurements of CeCoIn5, including the signature of a hybridization gap in early optical spectroscopy Singley et al. (2002) and in scanning tunneling microscope spectroscopy Aynajian et al. (2012), which decreases for increasing $T$, but are still very much present at 70-100 K ($\approx$2$T^{*}$). Similarly, resonant elastic x-ray scattering at the Ce 3$d$-4$f$ $M_{5}$ edge has recently demonstrated a $q$-dependent sensitivity to bulk $f$-states along the (110) direction Gyenis et al. (2016), with the $f$-resonant peak intensity exhibiting a logarithmic dependence persisting up to at least 150 K ($\approx$3$T^{*}$). Further discussion and comparison of this $T$-profile are provided in the Supplementary sup .

The “small” to “large” Fermi surface transition can be additionally visualized from the DMFT calculation by the total FS $volume$ as a function of temperature, as plotted in Fig. 5(f) with conversion to electron occupation, $n_{FS}$, e.g. 2 electrons per Brillouin Zone volume. It is observed that $n_{FS}\sim$3 at high $T$ increases to $n_{FS}=$4 at 15 K where one $f$ electron fully participates in the FS. In this case, $T^{*}$ appears to be associated with the onset of a more rapid change in FS size, even though $\sim$0.5 electron has been gradually incorporated from high $T$ down to $T^{*}$. Similarly, the ‘small” to “large” FS transition of the $\alpha$ and $\beta$ sheets can be analyzed by the cross sectional FS $area$, provided in the Supplementary sup , with comparison to Ce versus La dHvA orbits Shishido et al. (2002). Note that the large FS volume $f$-occupation $increase$ to low $T$ should be distinguished from the few percent $decrease$ of the local $f$-occupation from n_{f}$$\sim1, also plotted in Fig. 5(f), which is consistent with recent analysis of Ce 3$d$ hard x-ray core-level photoemission obtaining $n_{f}$=0.97 at 20 K Sundermann et al. (2016).

Theoretical efforts to compute the lattice coherence scale Pruschke et al. (2000); Burdin et al. (2000); Assaad (2004); Burdin and Zlatić (2009); Choi et al. (2012, 2013) rely on a variety of definitions including onset of Fermi-liquid transport coefficients, and effective mass scaling Choi et al. (2012, 2013), and others, with unclear relation to the experimental CeCoIn5 resistivity maximum $T^{*}$. In addition, while the proposed two-fluid model universal scaling formula for the $f$-DOS Yang and Pines (2008) contains a logarithmic term that extends out to $\approx$2.7 $T^{*}$, a multiplying “order parameter” term defines a sharp termination of the $f$-DOS at $T^{*}$, as illustrated in the Supplementary sup . CEF effects are not discussed within the two-fluid model.

In summary, we have presented a detailed view of the experimental 3D Fermi surface of CeCoIn5, including the complex-shaped holelike $Z$-sheet, using ARPES measurements from two orthogonal (001) and (100) cleave surfaces. We have used an $f$-resonant photon energy to highlight the $k$-locations of enhanced $f$-weight corresponding to three different $f$-$d$ hybridization scenarios, including the well-known $\alpha$-band crossing, and have measured the $T$-dependence of these $f$-weights. We find declining, but finite $f$-weight extending up to $\sim$200 K, surprisingly far above the transport coherence temperature of $\sim$45 K.

Theoretical $k$-resolved DFT+DMFT calculations provide confirmation of the experimental 3D FS topology, and provide $T$-dependent $f$-spectral functions that predict both dispersion and $f$-weight changes that extend even higher than the ARPES measurements. The inclusion of CEF states in the theory provides a glimpse of the complexity of $k$-resolved $f$-$d$ hybridization interactions above $E_{\textrm{F}}$, origins of the disparity in effective masses of $\alpha$- and $\beta$-band crossings, and a first time explicit spectral function view of a $T$-dependent crossover of the Kondo resonance effective degeneracy involving the two lowest CEF $f$-states. These CEF effects may explain much of the long high $T$ tail of KR $f$-weight, but the role of CEF states in the observed DMFT high $T$ onset of the effective mass and large-to-small FS size $T$-evolution still needs to be elucidated. The observed mismatch between the transport-defined coherence temperature and the higher $T$ $onset$ of signatures of $f$-$d$ hybridization and coherence-related effective mass changes in CeCoIn5 may have relevance to the missing high $T$ “small” Fermi surface in YbRh2Si2 Kummer et al. (2015).

In the preparation of this manuscript, we became aware of a purely experimental ARPES result on CeCoIn5 by Chen $et\ al.$ Chen et al. (2016) with similar $T$-dependent resonant-enhanced $f$-weight and dispersion results, combined with bulk-sensitive soft x-ray 3D Fermi surface measurements. Differences in the results and interpretations here compared to Chen $et\ al.$ are discussed in the Supplementary sup .

VII Methods

Experimental. Single crystal samples of CeCoIn5 were grown with a molten indium flux technique Zapf et al. (2001). The (001) surface was cleaved in vacuum using the top-post method. Orthogonal (100) surfaces were similarly fractured using side mounting of platelet samples with narrow $c$-axis thickness.

Temperature-dependent ARPES measurements were performed at the MERLIN beamline 4.0.3 of the Advanced Light Source (ALS) employing both linear horizontal and linear vertical polarization from an elliptically polarized undulator. A Scienta R8000 electron spectrometer with 2D parallel detection of electron kinetic energy and angle in combination with a six-axis helium cryostat goniometer with 6 K base temperature and $<5\times 10^{-11}$ torr base pressure. Total energy resolution of approximately 15 meV was used for measurements at h$\nu$ = 122 eV corresponding to the Ce 4$d$-4$f$ resonant enhancement of the $f$ photoionization cross section.

Theory. The correlation effect of the Ce 4$f$ orbital is treated by the charge self-consistent DFT+DMFT method Haule et al. (2010); Kotliar et al. (2006). The WIEN2k package was used for the DFT part, which is based on the full potential linearized augmented plane-wave+local orbitals (FP-LAPW+lo) method Blaha et al. (2001). The generalized gradient approximation (PBE-GGA) is employed for exchange-correlation potential Perdew et al. (1996). Ce 4$f$ electrons are treated dynamically with the DMFT local self-energy, where the full atomic interaction matrix is taken into account to describe the crystal field splitting at low temperature. We used previously determined Coulomb and exchange interaction parameters of 5.0 eV and 0.68 eV, respectively Shim et al. (2007), in which the occupation of Ce 4$f$ orbital was estimated to be 0.96 at 20 K. For the impurity solver in the DMFT step, the non-crossing approximation (NCA) is used.

In the description of the spin orbit interaction (SO) and CEF splittings of the Ce 4$f$ states under the tetragonal symmetry, both the diagonal basis and the simple atomic $jj$ basis of the interaction matrix were tested, and give similar results. Because the atomic $jj$ basis has quite small off-diagonal components, it was used inside the impurity solver. There are three doubly degenerate CEF states with ground state $\Gamma_{7}^{(1)}$, first excited state $\Gamma_{7}^{(2)}$ and second excited state $\Gamma_{6}$. The CEF splittings are estimated to be $\sim$8 meV and $\sim$40 meV, which are consistent with other Ce$M$In5 ($M$ = Rh and Ir) compounds Willers et al. (2010).

Acknowledgements

We wish to acknowledge useful discussions with F. F. Assaad and C. M. Varma. Supported by U.S. DOE at the Advanced Light Source (DE-AC02-05CH11231), and at U. Michigan (DE-FG02-07ER46379). JHS was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2015R1A2A1A15051540).