Orientation of ground-state orbital in CeCoIn$_5$ and CeRhIn$_5$

TL;DR

This study uses advanced x-ray scattering techniques to determine the orientation of the ground-state orbital in heavy fermion compounds CeCoIn$_5$ and CeRhIn$_5$, revealing the specific orbital configuration and hybridization effects.

Contribution

It provides the first direct experimental determination of the ground-state orbital orientation in CeCoIn$_5$ and CeRhIn$_5$ using NIXS, clarifying the orbital symmetry and hybridization effects.

Findings

The ground state is the $ ext{Γ}_7^-$ doublet with lobes along (110).

In CeCoIn$_5$, the ground state includes contributions from the first excited state.

The hybridization affects the orbital composition, reducing the expected $ ext{α}^2$ value.

Abstract

We present core level non-resonant inelastic x-ray scattering (NIXS) data of the heavy fermion compounds CeCoIn$_5$ and CeRhIn$_5$ measured at the Ce $N_{4,5}$-edges. The higher than dipole transitions in NIXS allow determining the orientation of the $\Gamma_7$ crystal-field ground-state orbital within the unit cell. The crystal-field parameters of the Ce$M$In$_5$ compounds and related substitution phase diagrams have been investigated in great detail in the past; however, whether the ground-state wavefunction is the $\Gamma_7^+$ ($x^2\,-\,y^2$) or $\Gamma_7^-$ ($xy$ orientation) remained undetermined. We show that the $\Gamma_7^-$ doublet with lobes along the (110) direction forms the ground state in CeCoIn$_5$ and CeRhIn$_5$. For CeCoIn$_5$, however, we find also some contribution of the first excited state crystal-field state in the ground state due to the stronger hybridization of 4$f$ and conduction electrons, suggesting a smaller $\alpha^2$ value than originally anticipated from x-ray absorption. A comparison is made to the results of existing density functional theory plus dynamical mean-field theory calculations.

Full text

I Introduction

At high temperature, heavy-fermion materials are described by decoupled localized $f$ electrons and conduction electron bands. Upon cooling, the localized $f$ electrons start to interact with the conduction electrons ($cf$-hybridization) and become partially delocalized. The resulting entangled fluid consists of heavy quasiparticles with masses up to three orders of magnitude larger than the free electron mass. These quasiparticles may undergo magnetic or superconducting transitions. In the Doniach phase diagram temperature $T$ versus exchange interaction $\cal{J}$, magnetic order prevails for small $\cal{J}$ whereas a non-magnetic Kondo singlet state forms for large $\cal{J}$. Between these two regimes quantum critical behaviour occurs which is often accompanied by a superconducting dome that hides a quantum critical point (QCP).Löhneysen et al. (2007); Wirth and Steglich (2016) Understanding how these quasiparticles, that have atomic-like as well as itinerant character, give rise to these ground states is a challenging question in condensed-matter physics, and the answer to this question will provide predictive understanding of these quantum states of matter.Coleman (2007)

The tetragonal compounds Ce$M$In5 ($M$ = Co, Rh, Ir) are heavy fermion compounds that display different ground states for different transition metal ions; for $M$ = Co and Ir the ground state is superconducting ($T_{c}$ = 2.3 and 0.4 K) and for $M$ = Rh it is antiferromagnetic ($T_{N}$ = 3.8 K).Thompson and Fisk (2012) High-quality Ce$M$In5 crystals can be grown, making this family suitable for determining the parameter that drives the different ground states. Within the above mentioned extended Doniach phase diagram, CeRhIn5 is on the weak side of hybridization, CeCoIn5 close to the QCP and CeIrIn5 is on the side of stronger $cf$-hybridization, i.e. superconductivity goes along with stronger $cf$-hybridization. Although there are strong indications for localization (Rh) and delocalization (Co,Ir) in, e.g., the size of the Fermi surface,Haga et al. (2001); Fujimori et al. (2003); Harrison et al. (2004); Shishido et al. (2005); Settai et al. (2007); Shishido et al. (2007); Goh et al. (2008); Allen et al. (2013) it is not possible to detect the differences in $f$ occupations. They are so subtle that they are below the detection limit.Sundermann et al. (2016)

A light-polarization analysis of soft X-ray absorption spectroscopy (XAS) spectra shows that the crystal-field wavefunction of the ground state correlates with the ground-state properties in the temperature - transition metal (substitution) phase diagram of CeCoIn5 - CeRhIn5 - CeRh1-δIrδIn5 - CeIrIn5; orbitals more compressed in the tetragonal $ab$-plane favor an antiferromagnetic ground state as for CeRhIn5 and the Rh rich compounds with $\delta$ $\leq$ 0.2. The compounds with more elongated orbitals along the $c$ axis, however, have superconducting ground states (CeCoIn5, CeIrIn5 and also the Ir rich compounds with $\delta$ $\geq$ 0.7).Pagliuso et al. (2002); Willers et al. (2015) The obvious conclusion is that the more pronounced extension of the ground state orbitals in the direction of quantization (crystallographic $c$ direction) promotes stronger hybridization in the $z$ direction and hence superconductivity. This is supported by combined local density approximation plus dynamical mean field theory (LDA+DMFT) calculations by Shim et al. Shim et al. (2007) that find for CeIrIn5 the strongest hybridization with the out-of-plane In(2) ions (see unit cell in Fig. 1 (a)). It was also shown that the suppression of superconductivity in CeCo(In1-ySny)5 by about 3% of Sn is due to a homogeneous increase of hybridization in the tetragonal $ab$ plane since the Sn ions go preferably to the In(1) sites.Sakai et al. (2015) Accordingly, we found that here the hybridization with In(1) ions plays a decisive role; the 4$f$ ground state orbital extends increasingly in the plane as the Sn content is increased. Chen et al. (2018)

Hence, the ground state wavefunction is a very sensitive probe for quantifying hybridization. Haule et al. obtained a 4$f$ Weiss field hybridization function for Ce$M$In5 based on realistic lattice parameters using density functional theory plus dynamical mean field (DFT+DMFT) calculations which they have decomposed into crystal-field components (see Fig. 1 (b)).Haule et al. (2010) Here our goal is to verify that the crystal-field components that were extracted in these calculations are in agreement with reality.

The tetragonal point symmetry of Ce in Ce$M$In5 splits the Ce Hund’s rule ground state into three Kramers doublets, two $\Gamma_{7}$ doublets $\Gamma_{7}^{+/-}$ = $\left|\alpha\right||\pm 5/2\rangle$ $+/-$ $\sqrt{1-\alpha^{2}}|\mp 3/2\rangle$ and $\Gamma_{7}^{-/+}$ = $\sqrt{1-\alpha^{2}}|\pm 5/2\rangle$ $-/+$ $\left|\alpha\right||\mp 3/2\rangle$, and one $\Gamma_{6}$ = $|\mp 1/2\rangle$. We write $+/-$ or $-/+$ because the sign has not yet been determined, and this is the scope of the present manuscript. $\Gamma_{6}$ as a pure $J_{z}$ state has full rotational symmetry around the quantization axis $c$ but the mixed states have lobes with fourfold rotational symmetry. The magnitude of $\alpha$ describes the shape and aspect ratio of the $\Gamma_{7}^{+/-}$ orbitals whereas the sign in the wavefunction determines how the orbitals are oriented within the unit cell; with the lobes along [100] ($\Gamma_{7}^{+}$: $x^{2}\,-\,y^{2}$) or with the lobes along [110] ($\Gamma_{7}^{-}$: $xy$).

The crystal-field potential of the Ce$M$In5 has been determined with inelastic neutron scattering (INS) Christianson et al. (2002, 2004) and the ground state wavefunctions were studied in greater detail with linear polarized soft XAS. Willers et al. (2010, 2015); Chen et al. (2018) Hence, the crystal-field energy splittings, the sequence of states ($\Gamma_{7}^{+/-}$, $\Gamma_{7}^{-/+}$, $\Gamma_{6}$) and also the magnitude of the $\alpha^{2}$-values are known (0.13, 0.38, 0.25 for Co, Rh and Ir). Only the sign of the wavefunction remains unknown because it cannot be determined with any of these dipole-selection-rule based spectroscopies. We, therefore, set up an experiment to determine the sign of the ground-state wavefunction in the Ce$M$In5 compounds in order to find out which one of the two scenarios in Fig. 1 (a) applies.

II Method

We performed a core level non-resonant inelastic x-ray scattering (NIXS) experiment at the Ce $N_{4,5}$-edges (4$d$ $\rightarrow$ 4$f$). It has been shown previously that this method is able to detect anisotropies with higher than twofold rotational symmetry. Gordon et al. (2009); Willers et al. (2012); Rueff et al. (2015); Sundermann et al. (2017, 2018a, 2018b) In the following, we briefly recap the principles of NIXS, a photon-in photon-out technique with hard x-rays (E${}_{\text{in}}$ $\approx$ 10 keV). Because of the high incident energies, NIXS is bulk sensitive and allows one to reach large momentum transfers $\left|\vec{q}\right|$ of the order of 10 Å*-1* when measuring in back scattering geometry. At such large momentum transfers, the transition operator in the scattering function S($\vec{q}$,$\omega$) can no longer be truncated after the dipole term. As a result, higher order scattering terms contribute to the scattering intensity. Haverkort et al. (2007); Gordon et al. (2008, 2009); Bradley et al. (2010, 2011); Caciuffo et al. (2010); Willers et al. (2012) For a Ce $4d$ $\rightarrow$ $4f$ transition at about 10 Å*-1*, octupole (rank $k$ =3) and triacontadipole ($k$ =5) terms dominate the scattering intensity whereas the dipole part ($k$ = 1) is less prominent. Accordingly, the directional dependence of the scattering function in a single crystal experiments follows multipole selection rules, in analogy to the dipole selection rules in linearly polarized XAS. Thus single crystal NIXS yields information not only about the orbital occupation but also the sign of the wavefunction that distinguishes the $xy$ and $x^{2}-y^{2}$ orientations of a $\Gamma_{7}$ when comparing two directions within the xy-plane; here [100] and [110].

III Experiment

CeCoIn5 and CeRhIn5 single crystals were gown using the standard In-flux technique. CeCoIn5 crystals are plate-like with the [001] direction perpendicular to the plate, whereas CeRhIn5 crystals are more three-dimensional. A very detailed structural investigation on the 115 compounds shows that more than 98 % of the crystal volumes form in the HoCoGa5 structure.Wirth et al. (2014) All samples were aligned by Laue diffraction before the experiment. For each compound two samples were cut, one with a (100) and a second one with a (110) surface so that specular geometry could be realized in the experiment.

The experiments were performed at the Max-Planck NIXS end station P01 at PETRA III/DESY in Hamburg, Germany. P01 has a vertical scattering geometry and the incident energy was selected with a Si(311) double monochromator and twelve Si(660) 1 m radius spherically bent crystal analyzers were arranged in 3 x 4 array as shown in Fig. 2 of Ref. Sundermann et al., 2017 so that the fixed final energy was E${}_{\text{final}}$ = 9690 eV. The analyzers were positioned at scattering angles of 2 $\theta$ $\approx$ 150∘, 155∘, and 160∘ which provide an averaged momentum transfer of $|\vec{q}|$ = 9.6 $\pm$ 0.1 Å*-1*. The scattered beam was detected by a position sensitive custom-made detector (LAMBDA), based on a Medipix3 chip detector. The elastic line was consistently measured and a pixel-wise calibration yields instrumental energy resolutions of $\approx$ 0.7 eV full width at half maximum (FWHM). For both samples the $N_{4,5}$ edges were measured with the momentum transfer $\vec{q}$ parallel to [001] and parallel to [110] ($\vec{q}$ $\|$ [001] and $\vec{q}$ $\|$ [110]).

We used the full multiplet code $Quanty$ Haverkort (2016) for simulating the NIXS data. A Gaussian broadening of 0.7 eV accounts for the instrumental resolution and an additional Lorentzian broadening of 0.4 eV FWHM accounts for life-time effects. The atomic parameters were taken from the Cowan code,Cowan (1981) whereby the Hartree-Fock values of the Slater integrals were reduced to about 60 % for the 4$f$-4$f$ and to about 80 % for the 4$d$-4$f$ Coulomb interactions to reproduce the energy distribution of the multiplet excitations of the Ce $N_{4,5}$-edges. This reduction accounts for configuration interaction processes not included in the Hartree-Fock scheme. Tanaka and Jo (1994)

IV Results

Figure 2 shows NIXS data (circles) at the Ce $N_{4,5}$-edges of CeRhIn5 (a) and CeCoIn5 (b) plus simulations (lines) for two scattering directions, $\vec{q}$ $\|$ [100] (blue) and $\vec{q}$ $\|$ [110] (green). The overall shape of the spectra looks very similar and represents the multipole scattering expected for the Ce $N_{4,5}$-edges. Gordon et al. (2008, 2009); Willers et al. (2012); Rueff et al. (2015); Sundermann et al. (2017) Figure 2 (c) and (d) show the directional dependencies I${}_{\vec{q}\|[110]}$ - I${}_{\vec{q}\|[100]}$ (dichroism), the experimental data as circles and simulations for the $\Gamma_{7}^{-}$ and $\Gamma_{7}^{+}$ as orange and gray lines, respectively. The expected dichroisms for a $\Gamma_{7}^{-}$ and $\Gamma_{7}^{+}$ ground state are opposite in sign so that the present experiment provides an either-or result which makes the interpretation of the data straight forward.

For CeRhIn5 the $N_{4,5}$ edges in Fig. 2 (a) as well as the dichroism in Fig. 2 (c) are fairly well reproduced by the simulation with a $\Gamma_{7}^{-}$ ground state (orange line). Here we used the $\alpha^{2}$ value of 0.38 as determined with XAS.Willers et al. (2010) The same simulation with a $\Gamma_{7}^{+}$ ground state is clearly in contradiction to the observation (gray line). For CeCoIn5 the agreement between simulated and experimental directional dependence in Fig. 2 (d) is not as good, and we will discuss below the possible reasons for this. Also here the simulation was performed with the corresponding $\alpha^{2}$ value from XAS, $\alpha^{2}$ = 0.13.Willers et al. (2010) Most importantly, however, we conclude that the ground state of CeCoIn5 must be also predominantly of $\Gamma_{7}^{-}$ character because the size of the scaling between experiment and calculation is clearly positive (0.63 $\pm$ 0.19) com and because the $\Gamma_{7}^{+}$ is, as for CeRhIn5, in clear contradiction to the observation.

V Discussion

In Figure 3, we compare the directional dependence I${}_{\vec{q}\|[110]}$ - I${}_{\vec{q}\|[100]}$ for several $\alpha^{2}$ values. For $\alpha^{2}$ = 0 (or 1) the dichroism is zero because in this case the $\Gamma_{7}$ state is a pure $J_{z}$ state and rotational invariant; for $\alpha^{2}$ = 0.5 the dichroism is largest. For CeRhIn5 ($\alpha^{2}$ = 0.38) the mixing factor $\alpha^{2}$ is closer to 0.5 than for CeCoIn5 ($\alpha^{2}$ = 0.13) so that the expected dichroism for CeCoIn5 is smaller than for CeRhIn5. But, the expected reduction due to the different $\alpha^{2}$ values still does not account for the strongly reduced directional effect in CeCoIn5. A natural explanation could be the stronger $cf$-hybridization in CeCoIn5 with respect to CeRhIn5: in CeCoIn5 the coherence temperature $T^{*}$ is of the order of 50 KPetrovic et al. (2001); Willers et al. (2010) which is comparable to the energy splitting of the two lowest crystal-field states (6.8 meV or $\approx$75 K) (see Fig. 4) so that the first excited crystal-field state will contribute to the ground state via hybridization. The first excited crystal-field state is the $\Gamma_{7}^{+}$ which has the opposite dichroism of the crystal-field ground state $\Gamma_{7}^{-}$ so that the net dichroism of the hybridized ground state will be reduced. In short, the strongly reduced directional effect in CeCoIn5 is due to the presence of strong hybridization. Assuming the first excited crystal-field state $\Gamma_{7}^{+}$ contributes 19 % to the ground state of CeCoIn5 yields a very good agreement of measured dichrosim and simulation (see dark red line in Fig. 2 (d)).com However, 19 % of $\Gamma_{7}^{+}$ mixed into the ground state by hybridization must be an overestimation because the $\Gamma_{7}^{+}$ admixture was not accounted for when describing the linear dichroism in XAS.Willers et al. (2010) It turns out that both data sets, the directional dependence in NIXS and the linear dichroism in XAS, can be analyzed consistently and are well described with $\alpha^{2}$ = 0.10 and 13 % of $\Gamma_{7}^{+}$.

Figure 4 summarizes what we know now about the crystal-field splittings of the $J$ = 5/2 multiplet of Ce$M$In5. The splittings and $\alpha^{2}$ values are taken from inelastic neutron scattering and XAS as published in Ref. Christianson et al., 2002, 2004; Willers et al., 2010. The present NIXS experiments on CeCoIn5 and CeRhIn5 add the missing information that the $\Gamma_{7}^{-}$ dominates the ground state for both compounds, i.e. the lobes of the crystal-field ground state are along the crystallographic (110) direction. These $\Gamma_{7}^{-}$ ground state orbitals extend more in the $z$-direction than the $\Gamma_{7}^{+}$ at about 6-7 meV so that the scenario as shown in Fig.1 (c) applies to CeCoIn5 and CeRhIn5, whereas the wavefunctions projected out by DFT+DMFT calculationsHaule et al. (2010) have opposite signs (see Fig.1 (b)).

The tips of the lobes of the $\Gamma_{7}^{-}$ ground state wavefunctions of CeCoIn5 and CeRhIn5 are pointing towards the triangle In2-In1-In2 (see Fig. 1 or Fig.1 (c)). It is therefore reasonable to conclude that the impact of the hybridization with the out-of-plane In2 atoms is more important than the hybridization with the in-plane In1 atoms. This is in agreement with the results of the CeRh1-δIrδIn5 substitution series Willers et al. (2015) where the orbitals that are more extended along the $c$-axis tend to hybridize more strongly. Nevertheless, the present results also show that hybridization with the In1 atoms is important and this supports the results of the CeCo(In1-ySny)5 substitution series;Chen et al. (2018) the Sn atoms go preferably to the In1 sites leading to a stronger hybridization in the plane.Sakai et al. (2015) We would like to note that the revised value of $\alpha^{2}$ for CeCoIn5 that is obtained when taking into account the first excited crystal-field state leads to a crystal-field ground state orbital that is even more extended in the $z$-direction than for the originally anticipated value. The same should apply to CeIrIn5 when allowing a hybridization induced contribution of the first excited crystal-field state. Hence, the correlation of stronger hybridization with the In2 atoms due to ground state orbitals that are more extended in $z$-direction and superconductivity still holds.Willers et al. (2015)

A modest increase in the contribution of $J_{z}$ = $|\pm 5/2\rangle$ to the $\Gamma_{7}^{-}$-state of CeRhIn5 will promote overlap of $f$-orbitals with $p$-states of In(1) at the expense $f$-In(2) hybridization. Mixing of Zeeman-split ground and $\Gamma_{7}^{+}$ first excited crystal-field levels, in principle, could produce such a modest increase in the $J_{z}$ = $|\pm 5/2\rangle$ contribution. Indeed, recent high-field magnetostriction Rosa et al. (2019) and nuclear magnetic resonance Lesseux et al. (2019) measurements on CeRhIn5 are consistent with this possibility that appears to be a significant contributing factor to field-induced Fermi-surface reconstruction in CeRhIn5 subject to a magnetic field near 30 T.

In the limit of strong intra-atomic Coulomb interactions, which is typical of strongly correlated metals like CeCoIn5 and CeRhIn5, the magnetic exchange $\cal{J}$ is proportional to the square of the matrix element $\left\langle V_{kf}\right\rangle$ that mixes conduction and $f$-wavefunctions. Schrieffer and Wolff (1966) Both Kondo and long-range Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions depend on the magnitude of $\cal{J}$ that is set by $\left\langle V_{kf}\right\rangle$2 and, consequently, by the $f$-orbital configuration. These interactions are fundamental for a description of Kondo-lattice systems and their relative balance can be tuned by non-thermal control parameters, such as magnetic field and pressure. Modest pressure applied to CeRhIn5 tunes its antiferromagnetic transition temperature toward zero temperature where a dome of unconventional superconductivity emerges with a maximum transition temperature very close to that of CeCoIn5 Thompson and Fisk (2012) and also changes the Fermi surface from small to large as in CeCoIn5.Shishido et al. (2005) We do not know if the $f$-orbital configuration of CeRhIn5 at these pressures is the same as that of CeCoIn5, but this is an interesting possibility that merits study.

VI Summary

In $f$-based materials, the shape of the crystal-field wavefunctions ultimately determines the origin of anisotropic hybridization in these materials and their ground state. Here, we show that the ground state of Ce$M$In5 ($M$ = Co,Rh) is a $\Gamma_{7}^{-}$ = $\left|\alpha\right||\pm 5/2\rangle$ $-$ $\sqrt{1-\alpha^{2}}|\mp 3/2\rangle$ doublet with lobes $\Gamma_{7}^{-}$ pointing toward the 110 direction, i.e., the lobes have $xy$ character. Though careful DFT+DMFT calculations shed light on these materials, the crystal-field scheme obtained is different from our experimental one. Our work settles the question on the orientation of $f$-orbitals in the ground state of Ce$M$In5 and will stimulate theoretical developments that take into account the actual wavefunctions.

VII Acknowledgment

We thank Peter Thalmeier for fruitful discussions. Parts of this research were carried out at PETRA III/DESY, a member of the Helmholtz Association HGF, and we would like to thank Christoph Becker, Manuel Harder and Frank-Uwe Dill for skillful technical assistance. Work at Los Alamos was performed under the auspices of the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering. A.A., A.S., and M.S. gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft under projects SE 1441-4-1.