Magnetoelastic hybrid excitations in CeAuAl$_3$

TL;DR

This study uses neutron spectroscopy and ab-initio calculations to reveal complex hybrid excitations involving phonons and crystal electric field states in CeAuAl$_3$, highlighting their role in emergent properties and potential for new functionalities.

Contribution

First detailed quantification of three coupling mechanisms between phonons, CEF states, and conduction electrons in CeAuAl$_3$, revealing profound renormalization of quasiparticle excitations.

Findings

Identification of a CEF-phonon bound state with low phonon density

Observation of anti-crossing between CEF states and acoustic phonons

Detection of broadening of CEF states due to hybridization

Abstract

The interactions between elementary excitations such as phonons, plasmons, magnons, or particle-hole pairs, drive emergent functionalities and electronic instabilities such as multiferroic behaviour, anomalous thermoelectric properties, polar order, or superconductivity. Whereas various hybrid excitations have been studied extensively, the feed-back of prototypical elementary excitations on the crystal electric fields (CEF), defining the environment in which the elementary excitations arise, has been explored for very strong coupling only. We report high-resolution neutron spectroscopy and ab-initio phonon calculations of {\ceaual}, an archetypal fluctuating valence compound. The high resolution of our data allows us to quantify the energy scales of three coupling mechanisms between phonons, CEF-split localized 4f electron states, and conduction electrons. Although these interactions do not appear to be atypically strong for this class of materials, we resolve, for the first time, a profound renormalization of low-energy quasiparticle excitations on all levels. The key anomalies of the spectrum we observe comprise (1) the formation of a CEF-phonon bound state with a comparatively low density of acoustic phonons reminiscent of vibronic modes observed in other materials, where they require a pronounced abundance of optical phonons, (2) an anti-crossing of CEF states and acoustic phonons, and (3) a strong broadening of CEF states due to the hybridization with more itinerant excitations. The fact that all of these features are well resolved in CeAuAl$_3$ suggests that similar hybrid excitations should also be dominant in a large family of related materials. This promises a predictive understanding towards the discovery of new magneto-elastic functionalities and instabilities.

Figures

Tables

Table 1. Table 1: Coupling constants as compared with the literature: (i) vibronic bound state, g VBS subscript 𝑔 VBS g_{\rm VBS} , (ii) anticrossing, g AC subscript 𝑔 AC g_{\rm AC} , and (iii) damping through conduction electrons, g BFK subscript 𝑔 BFK g_{\rm BFK} .

compound	$g_{\mathrm{VBS}}$ [$\mu$eV]	$g_{\mathrm{AC}}$ [$\mu$eV]	$g_{\mathrm{BFK}}$ [unitless]	Ref.
CeAuAl3	$\approx$ 400	12.1(2)	0.022(0)	this study
CeAl2	540		0.06	Thalmeier (1984)
CeCuAl3	800			Adroja et al. (2012)
PrNi5		4 111This value was obtained by fitting the data reported in Ref. Aksenov et al. (1983). See supplement SOM (2018) and Fig. S6 for details.		Aksenov et al. (1983)

Equations

$$H=H_{\mathrm{L}}+H_{\mathrm{CF}}+H_{\mathrm{ME}},$$

$$H_{\mathrm{CF}}^0=B_2^0{O}_2^0+B_4^0{O}_4^0+B_4^4{O}_4^4,$$

$${\omega }_{q\pm }^2=\frac{E^2+{\omega }_0^2}{2}\pm \sqrt{{\left(\frac{E^2-{\omega }_0^2}{2}\right)}^2+16{\alpha }^{2}E{\omega }_0^2{g}_{\mathrm{AC}}}$$

Full text

magneto-elastic coupling $|$ f-electron materials $|$ neutron spectroscopy

††preprint: APS/PRB

As the elementary excitations in solids reflect different aspects of the entire system of electrons, a wide range of coupling phenomena may be expected. For instance, the phonon-polariton, plasmon-polariton, or electro-magnon change their character continuously as a function of wavevector and energy Ashcroft and Mermin (1976). In metallic systems well-defined fermion quasiparticle excitations have been reported, featuring electron-phonon interactions, as well as strongly dispersive paramagnon or polaronic dressing clouds Lonzarich (1988). The interest in such coupled elementary excitations has been driven by the search for and discovery of anomalous materials properties and electronic instabilities. These may either be a direct consequence of the hybrid character of the low-energy dynamics, or result from from changes in lifetime and dispersion of pure elementary excitation, which affect the nature and range of their interactions.

Crystal electric fields (CEFs) may be considered one of the most important aspects in local-electron many-body physics, as they determine the environment in which elementary excitations arise. In this capacity, the CEFs control the nature and coupling of spin and orbital degrees of freedom, as well as electronic and magnetic anisotropies. The conventional static properties of the CEF excitations are extremely well understood and documented. In comparison, long-standing open questions concern the feed-back of elementary excitations on the CEFs, which leads to the formation of additional excitations beyond the expectations on the single ion level, as well as finite lifetimes and anomalous temperature dependences. Two primary mechanisms have been considered as the origin of such CEF properties. First, phonons may create CEF transitions between neighboring ions Sinha (1978), representing an important example of so-called magnetoelastic (ME) coupling Vlasov and Ishmukhametov (1964); Fennell et al. (2014); Naji et al. (2016); Boldyrev et al. (2017). Second, in metallic systems a coupling exists with particle-hole excitations Becker et al. (1977).

While various facets of the CEFs have been studied for nearly five decades, experimental evidence reflecting different coupling strengths as well as the full range of properties of the CEFs is surprisingly limited Steglich et al. (1979); Chapon et al. (2006); Adroja et al. (2012); Ruminy et al. (2016); Loewenhaupt et al. (1988); Fournier et al. (1991). This situation may be traced to the underlying interplay of energy scales, i.e., CEF excitations, phonons, particle-hole pairs, spin-orbit coupling, and magnetic interactions, all of which tend to be of similar strength, representing therefore a veritable chicken-and-egg-type of problem.

Experimentally, the direct measurement of such complex low-energy dynamics requires a single crystal spectroscopic method with high resolution in both energy and momentum transfer. In contrast, as CEF excitations are traditionally assumed to be momentum-independent they have typically been determined in polycrystalline samples. Moreover, studies reporting momentum-resolved data on single crystals focus on a single facet of the hybrid character only, and lack the necessary comprehensive information. In turn, there are a number of long-standing issues that may be resolved with targeted spectroscopic measurements, such as the feed-back of the ME coupling on the CEFs and spectrum of phonon excitations Loewenhaupt et al. (1987) the unambiguous identification of genuine hybrid excitations such as anticrossings Aksenov et al. (1983), and the origin and nature of finite lifetimes and temperature dependences of the CEFs Blackburn et al. (2006); Bernhoeft and Lonzarich (1995).

Taken together, these represent general problems, which are potentially relevant in any correlated-electron system. However, rare-earth intermetallics with moderate ME coupling are particularly suited for their study, because the different aspects of the hybrid character are well-defined and tractable. Namely, as both spin and orbital angular contributions generate the ME coupling, the well defined multiplett structure of the f-shells in rare earth compounds makes the ME coupling particularly tractable Mentink et al. (2018).

The rich diversity of CEF-driven magnetoelastic phenomena in rare-earth based compounds has been recognized for a long time. Perhaps most prominent is the formation of a bound state between the CEF excitations and phonons in CeAl2, known as vibronic-bound state (VBS) Loewenhaupt et al. (1979); Loewenhaupt and Witte (2003); Thalmeier and Fulde (1982); Thalmeier (1984). The key requirements of the VBS assumed so far are rather stringent, comprising a pristine (non-hybridized) CEF level at the same energy as a large phonon density of states. Moreover, as the phonons must have the same symmetry as the CEF level to permit hybridization and the phonon density of states must be excessive, it has been assumed that the VBS may only be formed with weakly dispersive (optical) phonons. Despite its conceptual importance, numerous questions are still unresolved in CeAl2. These include the importance of phonons other than the optical modes, the observation of complex excitations at momentum transfers away from the $\Gamma$ point Marshall and Lovesey (1971), and the absence of Raman scattering of the highest level VBS Güntherodt et al. (1983, 1985).

Several studies have proposed the formation of a VBS in other materials, considering an interplay of optical phonons with CEF excitations as in CeAl2. For instance, consistent with this mechanism evidence for a VBS in polycrystalline PrNi2 vanishes as the optical phonon levels shift to lower energies under doping Mühle et al. (1989). Putative evidence for vibronic states was also reported in cubic Ce3Pt23Si11 Opagiste et al. (2011) as well as the tetragonal systems CePd2Al2 Chapon et al. (2006); Klicpera et al. (2017a) and CeCuAl3 Adroja et al. (2012). Moreover, vibronic excitations have even been proposed in rare-earth doped cuprates Ruf (1996), as well as geometrically frustrated oxides such as Tb2Ti2O7 Fennell et al. (2014); Ruminy et al. (2016) and Ho2Ti2O7 Gaudet et al. (2018), which underscores the wide-spread relevance.

A different interplay with dispersive phonon branches as a function of momentum has been proposed in the regime of a steep crossing with the CEF excitations. For the case of dipolar magnetic interactions the emergence of an anticrossing in the presence of an applied magnetic field or spontaneous magnetic order has been observed, e.g., in Pr, PrAl2, and TmVO4 Thalmeier and Lüthi (1991); Jensen (1976); Houmann et al. (1979); Purwins et al. (1976); Thalmeier and Fulde (1975); Kjems et al. (1975). On the other hand, an anticrossing in zero magnetic field may be expected theoretically between phonons and quadrupolar excitons, as putatively observed in PrAlO3 Birgeneau et al. (1974) and TbVO4 Hutchings et al. (1975). In intermetallic rare-earth compounds only indirect evidence for an anti-crossing has been reported for PrNi5 Aksenov et al. (1983), where a mere shifting of the exciton energies before and after the crossing could be detected, whereas the actual anticrossing could not be resolved. Interestingly, it has been speculated that the scattering of the phonons by magnetic excitons causes anomalous contributions to the thermal conductivity of PrNi5 Reiffers et al. (1988).

Finally, the lifetime and temperature dependence of CEF excitations have been known to display pronounced deviations from the expected thermal population of single-ion states Lawrence and Shapiro (1980); Loewenhaupt et al. (1985); Hense et al. (2004); Blackburn et al. (2006); Iwasa et al. (2009, 2010). In a seminal theoretical study Becker, Fulde and Keller (BFK) Becker et al. (1977) successfully explained the anomalous temperature dependences of the CEF occupation in terms of the temperature dependence of the interaction with particle-hole excitations in metallic systems. In turn, a large number of studies have attributed the broadening and temperature dependent effects observed experimentally to this mechanism. Of particular interest is the possibility of superconductive pairing due to CEF excitations and quadrupolar fluctuations as proposed in UPd2Al3 and PrOs4Sb12 Blackburn et al. (2006); Iwasa et al. (2009) Bauer et al. (2002). In this context, the broadening and thermal effects of the CEF are of central importance for the understanding of the metallic state and putative electronic instabilities.

In this paper we report a comprehensive inelastic neutron scattering study of the spectrum of low-lying excitations in CeAuAl3 and ab initio phonon calculations. This compound is a member of the CeTAl3 series (T=Ce, Au, Pd, Pt), which forms part of a wider family of BaAl4-type materials Franz et al. (2016). Our findings for CeAuAl3 are therefore directly relevant for a large number of systems featuring strong electronic correlations and magnetic order at the border of a quantum phase transition. Early measurements of the thermal, magnetic an charge transport properties of polycrystalline samples established that CeAuAl3 is a valence fluctuating compound with antiferromagnetic order below $T_{\mathrm{N}}=1.32\,$K Paschen et al. (1998).

The enhancement of the linear temperature dependence of the specific heat and quadratic temperature dependence of the resistivity ($\gamma=227$ mJ mol*-1* K*-2* and $A=5\,\mu\Omega$cmK*-2*, respectively) are characteristic of a heavy Fermi liquid state. The CEF lifts the degeneracy of the Ce3+ $J=5/2$ manifold, which has a characteristic impact on both the magnetic susceptibility and specific heat of the material. However, the first and second doublet at $T_{\rm I}=57\,{\rm K}$ and $T_{\rm II}=265\,{\rm K}$ are split from the ground state such that they have no bearing on the bulk properties and the enhancement of the Fermi liquid ground state. Overall, CeAuAl3 appears to be a typical Ce-intermetallic, which shares many of it physical properties with a large number of related materials. One unusual feature is its anisotropic reduced thermal conductivity (compared to non-magnetic LaAuAl3). This has been interpreted in terms of enhanced magnetoelastic phonon scattering on the Ce ions taking into account the CEF splitting Aoki et al. (2000).

The observation of a VBS state in CePd2Al2 Chapon et al. (2006), a related tetragonal compound, appears to be intimately related to a structural phase transition and underscores a strong interplay of the CEF excitations and phonons in this class of systems. Indeed, time of flight (TOF) neutron spectroscopy revealed also a VBS in polycrystalline CeCuAl3 Adroja et al. (2012), as confirmed recently in polarized single crystal neutron spectroscopy 201 (2018) and in slightly off-stoichiometric samples Klicpera et al. (2017b). Here, too, electronic excitations are assumed to hybridize with optical phonons, which results in four doublets $|\Gamma_{6},{0}\rangle$, $|\Gamma_{6},{1}\rangle$ and $|\Gamma_{7}^{1,2},{0}\rangle$. This suggests that the symmetry of the lattice fluctuations imparts a different character to the VBS in tetragonal, as compared to cubic, systems. Yet, systematic time-of-flight neutron spectroscopy in polycrystalline CeRhGe3 Hillier et al. (2012) and CeAuAl3 Adroja et al. (2015) failed to detect a VBS. Moreover, the search for magnetoelastic phonon softening by inelastic x-ray scattering in CeCuAl3 and CeAuAl3 has been inconclusive Tsutsui et al. (2017).

In contrast, revisiting the properties of single-crystal CeAuAl3 in careful triple-axis neutron spectroscopy we find clear evidence for strong feed-back of phonons and particle-hole excitations on the CEF excitations. In particular we find a VBS, a clear anticrossing and substantial broadening of the CEFs. The observed behaviour is rather subtle and below the detection limit of neutron TOF spectroscopy in polycrystals Adroja et al. (2015), but consistent with the reduced thermal conductivity of CeAuAl3 Aoki et al. (2000). The rather weak coupling between CEFs, phonons and conduction electrons suggests that similar hybrid excitations must be generic in a wide range of materials.

Results

The direct observation of the interplay between the lattice and electronic properties requires a spectroscopic probe which couples both to the lattice and electronic transitions with high resolution in both energy- and momentum transfer. Inelastic neutron scattering (INS) is uniquely suited to the task. Careful comparison of the INS signal in different Brillouin zones, and the temperature dependence of this response allows to distinguish nuclear and magnetic contributions to the spectral response, even without the need for polarization analysis. We have therefore used neutron triple axis spectroscopy at the spectrometers PANDA and PUMA at the Heinz-Maier Leibniz Zentrum at the Technical University of Munich. For technical details, we refer to the methods section and supplementary information SOM (2018).

A high-quality and high-purity single crystals of CeAuAl3 was grown by the optical floating zone method, using a ultra-high-vacuum compatible preparation chain Bauer et al. (2016). The unit cell of the material (corresponding to the BaNiSn3 structure type) is shown in Fig. 1 (a). As previously reported, x-ray and neutron diffraction experiments revealed a very small amount of antisite disorder in these samples Klicpera and Javorský (2014); Franz et al. (2016). This was also confirmed in recent NMR measurements in the CeTX3 systems, which found minimal antisite disorder in CeAuAl3, compared to other members of the series Chlan et al. (2018).

Shown in Fig. 1 (b) is the first Brillouin zone (BZ) of the body-centered tetragonal unit cell of CeAuAl3, where high symmetry directions are marked according to the Bilbao notation Aroyo et al. (2014). Points, lines and planes in the BZ are denoted by black, blue and white letters, respectively. The relevant (h,k,0) and (h,0,l)-planes, and the trajectory of our measurement in reciprocal space are shown in Fig. 1 (b) and (c) as indicated by a red line. Starting at the $\Gamma$ point in the centre of the Brillouin zone, this trajectory proceeded initially along the $c$-axis towards the zone boundary at the $M$ point. It then covered the properties in the $ab$-plane between the $M$ and the $S$ points. From here the trajectory returned back to the $\Gamma$ point. Since these directions of momentum transfer do not coincide with the main crystallographic directions of the primitive unit cell, neutrons couple to all polarizations of phonon modes. This proves to be very helpful in the discussion of our data presented below.

An overview of the excitation spectra of CeAuAl3 as a function of reduced scattering wave vector q we observe is presented in Fig. 2 (a). For any reduced scattering vector $q$, the spectra feature two flat excitations, marked by red and green shading. The flat excitations are crossed by strongly dispersive phonon modes branching out of the $\Gamma$ points. The interplay of these key characteristics results in two of the three main experimental observations of our study, notably (i) formation of a new bound state as marked by green shading, and (ii) distinct anti-crossing of acoustic phonons with the crystal field along the $\Gamma$-$M$ direction shown in Figs. 2 (c). In addition we observe, (iii), a strongly enhanced broadening of the crystal field levels with temperature ((shown in more detail in Fig. 3 and discussed below)).

We now turn to a more detailed discussion of these observations. The crystal field excitation at $E_{\rm CF}=4.9\,{\rm meV}$ (red shading) may be attributed to the transition from the $|\Gamma_{6}\rangle$ ground state to the first excited doublet $|\Gamma_{7}^{1}\rangle$, cf. Fig. 2 (b). The energy of this transition is in excellent agreement with previous time-of-flight neutron spectroscopy and bulk data in a polycrystalline sample, which, however, do not allow to search for a momentum dependence Adroja et al. (2015); Paschen et al. (1998). The weak non-dispersive excitation at $E_{\rm VBS}=7.9\,{\rm meV}$ (marked in greed shading) is an unexpected new finding. This feature was not observed in the previous time-of-flight INS studies Adroja et al. (2015), probably due to the loss in spectral weight in the polycrystalline average.

The strongly dispersive excitations at the $\Gamma$-points may be clearly attributed to acoustic phonons as they emanate from nuclear Bragg peaks. Taking into account the tetragonal crystal symmetry, a longitudinal and a transverse acoustic branch are observed along the $\Gamma$ to $M$ direction, labeled as $\Lambda_{\rm LA}$ and $\Lambda_{\rm TA}$, respectively. As illustrated in Fig. 2 (c), these phonons display compelling evidence of an anticrossing with the crystal field excitation at $E_{\rm CF}$ in the three independent Brillouin zones studied, namely $(101)$, $(202)$ and $(114)$ SOM (2018). In contrast, for the $\Gamma$ to $S$ direction (labelled $\Sigma$) the two non-degenerate transverse acoustic phonons and a longitudinal acoustic phonon, denoted $\Sigma_{\rm TA1}$, $\Sigma_{\rm TA2}$ and $\Sigma_{\rm LA}$, respectively, cross the non-dispersive excitations at $E_{\rm CF}$ and $E_{\rm VBS}$ without apparent interaction (Fig. 2 (b)). We did not detect any evidence of phonons in the vicinity of the zone boundary between the $M$ and $S$-points, probably because the intensity was too low or they coincided with the VBS. The intensity marked by the white arrow in Fig. 2 (c) is a so-called Currat-Axe spurion, which is a common effect in triple-axis spectrometers.

The momentum dependence of the intensity of the dispersion-less excitations at $E_{\rm CF}$ and $E_{\rm VBS}$ follows the from factor of the Ce3+ ion (see Fig. S6 in the Supplemental Material). This is strong evidence that these excitations are essentially of magnetic character. By contrast, the strongly dispersive excitations at the $\Gamma$-points are essentially due to nuclear scattering. Moreover, as a function of increasing temperature the intensity of both dispersionless excitations decreases strongly as shown in Fig. 3 (a-c). This is qualitatively consistent with the thermal population of the first excited crystal field level, which provides further evidence for the magnetic character of these excitations. However, closer inspection reveals that the intensity decreases much faster than would be expected of a simple three-level system. We return to an accurate account of the temperature dependence below, which reveals dominant hybridization with the conduction electrons.

As illustrated by the energy scans at the $\Gamma$ and M points shown in Fig. 3 (d), the dispersionless excitation at $E_{\rm VBS}=7.9\,{\rm meV}$ strongly varies in intensity throughout the Brillouin zone. The intensity is large at the zone boundary and becomes very weak and difficult to discern at the zone centre. This behaviour was observed in all Brillouin zones investigated, as listed in Fig. 2 (a). In stark contrast, no such variation was observed for the dispersionless excitation at $E_{\rm CF}=4.9\,{\rm meV}$, except near the $\Gamma$-point in the $(101)$ Brillouin zone (cf. Fig. S7 in the supplement). Taken together, this suggests that the formation of the dispersionless excitation at $E_{\rm VBS}=7.9\,{\rm meV}$, as well as the enhancement of the excitation at $E_{\rm CF}=4.9\,{\rm meV}$ at the $\Gamma$-point are driven by virtue of the coupling to the phonons.

Discussion

For the discussion of our experimental results we assume that the magnetoelastic properties of CeAuAl3 are dominated by the interactions of the 4$f$-electrons with the electrostatic crystal field. A Hamiltonian describing this situation is given by

[TABLE]

where $H_{\rm L}$ accounts for the kinetic and potential energy of the Bravais lattice, $H_{\rm CF}$ denotes the conventional crystal field Hamiltonian, and $H_{\rm ME}$ represents the single-ion magnetoelastic coupling of the spin to lattice strains. It is important to note that the phonons and crystal field excitations must share the same symmetry to couple directly.

Treating the contribution of the Bravais lattice, $H_{\rm L}$, in a harmonic approximation we calculated the structural properties and spectrum of phonon excitations Liu et al. (2018) using the Vienna Ab-initio Simulation Package (VASP) in the frozen-core projector augmented wave (PAW) method. The calculated lattice constants $a=4.335\,\text{\AA}$ and $c=10.844\,\text{\AA}$ were found to be in excellent agreement with experiment Franz et al. (2016). Further, the spectrum of phonon excitations was calculated using a finite difference method. The results are shown as faint red lines in Fig. 2 (a) for energies up to $12\,{\rm meV}$ and also as total density of phonon states in Fig. 2 (d). For the five atoms in the primitive unit cell of CeAuAl3 the full phonon dispersion consists of 15 branches, which comprise three acoustic and twelve optical modes. Using group theoretical techniques that take into account all representations of the $C_{4\nu}$ point group are one-dimensional, an analysis of the irreducible representations for the high symmetry points establishes that the two acoustic branches along $\Gamma$ to $Z$ are degenerate. This justifies the assignment of the phonon branches in the vicinity of the $\Gamma$-points. At low energies the phonon branches are in excellent agreement with calculation. In contrast, for energies in the range 4 meV to 9 meV, we observe systematic deviations highlighting the presence of magneto-elastic coupling and the need for an advanced treatment.

For the tetragonal symmetry of CeAuAl3, point group $C_{4v}$ (4mm), contributions of the Ce3+ ions in the presence of a crystal electric field may be expressed as

[TABLE]

where $B_{n}^{m}$ and $O_{n}^{m}$ are the crystal electric field parameters and Steven’s operators, respectively Balcar and Lovesey (1989); Hutchings (1964). Due to the symmetry constraints all parameters should be real numbers Bauer and Rotter (2010). In the paramagnetic phase the sixfold degenerate 4$f^{1}$ states of Ce3+ ($J=\frac{5}{2}$) splits into three doublets. For our neutron data we find $B_{2}^{0}=1.203\,{\rm meV}$, $B_{4}^{0}=-0.001\,{\rm meV}$, and $B_{4}^{4}=\pm 0.244\,{\rm meV}$. This corresponds to a $|\Gamma_{6}\rangle$ doublet ground state and two excited states, $|\Gamma_{7}^{1}\rangle$ and $|\Gamma_{7}^{2}\rangle$, at energies of 4.95 meV and 24.3 meV, respectively. The associated eigenvectors are $|\Gamma_{6}\rangle=|\pm\frac{1}{2}\rangle$, $|\Gamma_{7}^{1}\rangle=-\alpha|\mp\frac{3}{2}\rangle+\beta|\pm\frac{5}{2}\rangle$, and $|\Gamma_{7}^{2}\rangle=\alpha|\mp\frac{5}{2}\rangle+\beta|\pm\frac{3}{2}\rangle$, with $\alpha=0.931$, $\beta=0.364$. These results are in excellent agreement with neutron time of flight spectroscopy of a powder sample of CeAuAl3 Adroja et al. (2015) as well as with the bulk properties Paschen et al. (1998). Further details of the CEF analysis may be found in the Supplementary Information SOM (2018).

The calculations of the phonon spectrum and crystal field levels presented so far, clearly identify the dispersionless excitation at $E_{\rm VBS}=7.9\,{\rm meV}$ (green shading in Fig. 2 (a)) as an additional state. It is tempting to interpret this intensity in terms of a vibronic state as observed in CeAl2 and CeCuAl3. A comparison of the associated energy level scheme of CeAl2 and CeCuAl3 with CeAuAl3 is shown in Fig. 4. For the cubic crystal structure of CeAl2, the $|\Gamma_{7}\rangle$ doublet and $|\Gamma_{8}\rangle$ quartet of cerium in the CEF forms a set of new eigenstates. By virtue of hybridization with optical phonons, this yields one electronic $|\Gamma_{7},{0}\rangle$, one phononic $|\Gamma_{6},{1}\rangle$ doublet and two mixed $|\widetilde{\Gamma}_{8}^{1,2}\rangle$ quartets, which are linear combinations of purely electronic and single-phonon states, depicted in Fig. 4(a).

Similarly, the formation of the VBS in the tetragonal crystal structure of CeCuAl3, illustrated in Fig. 4 (b), involves a hybridization of the $|\Gamma_{7}^{2}\rangle$ doublet with a high density of optical phonons to form a phononic $|\Gamma_{6},{1}\rangle$ doublet and three mixed doublets, denoted $|\widetilde{\Gamma}_{6}\rangle$ and $|\widetilde{\Gamma}_{7}^{1,2}\rangle$. Thus, in both cases the formation of a Thalmeier-Fulde VBS presumes strong magnetoelastic interactions between weakly dispersive optical phonons and a nearby crystal field level. In contrast, in CeAuAl3 at the energy of the putative VBS the phonon calculations do not yield the required high density of states. A low phonon density of states at at $E=8\,\,{\rm meV}$, as well as $E\approx 6\,\,{\rm meV}$ and $E=11-13\,\,{\rm meV}$ is also supported by our calculations (see Fig. 2 (d) and neutron time-of-flight spectroscopy on polycrystalline samples of LaAuAl3 Adroja et al. (2015). This means that one important prerequisite for the formation of a VBS is apparently absent.

Nevertheless, our measurements suggest that a comparatively weak coupling between the excitation from the $|\Gamma_{6}\rangle$ to $|\Gamma_{7}^{1}\rangle$ state with acoustic phonons is sufficient for the formation of a VBS at $E_{\rm VBS}=7.9\,{\rm meV}$ as illustrated in Fig. 4 (c). This implies that CEF excitations may form bound states with phonons that (i) do not cover large portions of the Brillouin zone, (ii) are dispersive and, (iii) of much reduced intensity. Further, with increasing magneto-elastic coupling the enhancement of the $|\widetilde{\Gamma}_{7}^{1}\rangle$ level may be expected to shift down. In contrast, the newly created $|\Gamma_{6},{1}\rangle$ level is shifting up.

Using similar methods as reported in Refs. Adroja et al. (2012); SOM (2018), the coupling constant may be roughly determined to be $g_{\rm VBS}=0.4\,{\rm meV}$ (for details of the calculations we refer to the Supplementary Information SOM (2018)). The reduced intensity of the excitation at $E_{\rm VBS}=7.9\,{\rm meV}$ as compared with the excitation at $E_{\rm CF}=4.9\,{\rm meV}$ imply that the low phonon density of states is just sufficient to reach the threshold for the bound state to become measurable. In fact, it is interesting to speculate, whether the weak maximum of the calculated density of states at $E=11\,\,{\rm meV}$ is just below the threshold, entailing an incipient VBS.

The magneto-elastic interactions may also be expected to affect the spectrum of phonon and crystal field excitations where their interplay is strongest. This is indeed the case in the regime of the anticrossing SOM (2018); Liu et al. (2018). Indeed, the anticrossing also proves to be a direct consequence of the magnetoelastic coupling. We follow considerations first reported for PrNi5 Aksenov et al. (1983), where the Hamiltonian of the magnetoelastic coupling, $H_{\rm ME}^{I}$, describes direct coupling between the deformations of the lattice and the 4$f$ shell. In a group theoretical analysis of this expression Callen and Callen (1965) , the energy of the associated coupled quadrupole-phonon excitation can be stated as

[TABLE]

where $\omega_{q\pm}$ represent the energies of the two anticrossing excitations; $\omega_{0}$ is the phonon energy which depends on $\bm{k}$, $E$ is the non-dispersive energy of the crystal field level involved and $g_{\rm AC}$ is an effective coupling constant related to the renormalization of elastic constant, see supplement for details SOM (2018). A fit of our data yields $g_{\rm AC}$ = 12.1(2) $\mu$eV.

The magneto-elastic coupling is also reflected in the temperature dependence of the scattering intensity, shown in terms of energy scans at the $M$ point in Fig. 3 (a). It is instructive to consider the reduced scattering intensity, $I/I_{0}$, normalized to its value at low temperatures as shown in Fig. 2 (b) for $E_{\rm CF}=4.9\,{\rm meV}$ and $E_{\rm VBS}=7.9\,{\rm meV}$. At 200 K we observe a large reduction of the intensity of the crystal field level at $E_{\rm CF}$ by $\sim 80\,\%$, whereas the intensity of the excitation at $E_{\rm VBS}$ already vanishes above 100 K. In contrast, a reduction of only 50 % would be expected of the intensity at $E_{\rm CF}$ for 200 K, when thermally populating the three crystal field excitations determined in the standard analysis, which ignores the weak mode at $E_{\rm VBS}$. This situation improves slightly with a reduction of 60 % at 200 K for four crystal field levels when additionally taking into account the mode at $E_{\rm VBS}$. However, the agreement is still far from satisfactory.

When additionally considering the coupling to the conduction electrons following the suggestion of Ref. Tsutsui et al. (2017), a BFK model of crystal field line broadening Becker et al. (1977) provides excellent agreement with our data.The fitting procedure incorporates the code of Keller Rotter (2004), where technical details may be found in the Supplementary Information SOM (2018). As shown in Fig. 3 (c) the improved account of the peak intensity in the BFK model is also reflected in an improved account of the energy dependence. The associated unitless coupling constant, $g_{\rm BFK}=0.022(0)$, is remarkably small. While the BFK model already provides a satisfactory agreement with the broadening further improvements may be expected when taking into account the interactions with the spectrum of phonons. A full analysis of these contributions lies beyond the present capabilities of established computational techniques Becker (shed).

As summarized in Tab. 1, the coupling constants $g_{\rm BFK}$ and $g_{\rm VBS}$ in CeCuAl3 are smaller than in CeAl2. This highlights that the spectrum of low-lying CEF excitations may be modified profoundly, even for systems with rather weak magneto-elastic coupling.

Conclusions

In summary, we find a remarkable combination of different facets of the interactions of phonons and conduction electrons with the CEF excitations in CeAuAl3. The magneto-elastic hybrid character of the excitation comprises the formation of a bound state akin to the VBS in CeAl2, a well-resolved anticrossing, and strong damping of the CEF levels. The combined observation of all of these different effects in the same material, the importance of acoustic phonons in their formation, and the comparatively weak coupling imply that these effects must be present in many materials. This underscores, that moderately interacting rare-earth compounds offer tractable insights towards a predictive understanding of magneto-elastic functionalities and instabilities.

Methods

Materials preparation and quality

A high quality single crystal of CeAuAl3 was prepared by optical float-zoning under ultra-high vacuum compatible conditions Bauer et al. (2016). For the inelastic neutron scattering experiments a crystal with a mass of approximately 2 g was used. High sample purity was confirmed by means of resistivity, magnetisation and specific heat of small pieces cut from the same ingot Franz (2014); Franz et al. (2016). The correct BaNiSn3-type structure and high crystalline quality were confirmed by powder and Laue x-ray diffraction as well as neutron diffraction Franz et al. (2016). Great care was exercised to confirm the correct crystal structure Franz et al. (2016). Neutron diffraction established that anti-site disorder is negligible in the present samples Franz et al. (2016).

Neutron scattering

Inelastic neutron scattering measurements were carried out on the triple axis spectrometers PUMA and PANDA at MLZ, Garching Sobolev and Park (2015); Schneidewind and Čermák (2015). For details of the experimental setup and the momentum and energy ranges covered in our experiments we refer to the supplementary information SOM (2018). The sample was cooled with a pulse-tube cooler. Data were recorded at a temperature of 5 K unless stated otherwise, i.e., well above the antiferromagnetic transition temperature at 1.3 K.

Theoretical calculations

Ab initio calculations have been carried using VASP and the frozen-core projector augmented wave (PAW) method. Taking into account weak interactions of the rare-earth (RE) ions with phonons in terms of magnetoelastic effects, an analytical expression for the hybridization of quadrupole excitations and phonons from the poles of the one-phonon Green-function is derived. For details see SOM (2018) and Ref.Liu et al. (2018).

Acknowledgements.

We acknowledge discussions with P. Thalmeier, W. Wulfhekel, J. Kulda, M. Loewenhaupt, M. Janoschek, M. Wilde and M. Klicpera. BL is supported by the Nat. Nat. Science Found. of China (Grant No. 11305150), and Direct. Found. of China Acad. of Eng. Phys. (Grant No. YZ2015009). The work of PC was supported by the Czech Science Found. (Grant No. 17-04925J). CP and AS acknowledge support through DFG TRR80 and DFG grant WI3320-3. CP acknowledges support through ERC AdG ExQuiSid.