Isoelectronic tuning of heavy fermion systems: Proposal to synthesize Ce3Sb4Pd3

TL;DR

This paper proposes an orbital-based approach to tune the Kondo coupling in heavy fermion systems via isoelectronic substitutions, demonstrating that orbital extent significantly influences electronic properties beyond simple lattice effects.

Contribution

It introduces an orbital scenario for tuning Kondo interactions through isoelectronic substitutions, supported by band-theory calculations and experimental data, advancing understanding of microscopic control in heavy fermion compounds.

Findings

Kondo coupling varies with orbital radial extent of substituents.

Predicted effective mass for Ce3Sb4Pd3 is approximately ten times the band mass.

Excellent agreement between simulated and experimental Kondo temperatures.

Abstract

The study of (quantum) phase transitions in heavy-fermion compounds relies on a detailed understanding of the microscopic control parameters that induce them. While the influence of external pressure is rather straight forward, atomic substitutions are more involved. Nonetheless, replacing an elemental constituent of a compound with an isovalent atom is---effects of disorder aside---often viewed as merely affecting the lattice constant. Based on this picture of chemical pressure, the unit-cell volume is identified as an empirical proxy for the Kondo coupling. Here instead, we propose an "orbital scenario" in which the coupling in complex systems can be tuned by isoelectronic substitutions with little or no effect onto cohesive properties. Starting with the Kondo insulator Ce$_3$Bi$_4$Pt$_3$, we consider---within band-theory---isoelectronic substitutions of the pnictogen (Bi$\rightarrow$Sb) and/or the precious metal (Pt$\rightarrow$Pd). We show for the isovolume series Ce$_3$Bi$_4$(Pt$_{1-x}$Pd$_x$)$_3$ that the Kondo coupling is in fact substantially modified by the different radial extent of the $5d$ (Pt) and $4d$ (Pd) orbitals, while spin-orbit coupling mediated changes are minute. Combining experimental Kondo temperatures with simulated hybridization functions, we also predict effective masses $m^*$, finding excellent agreement with many-body results for Ce$_3$Bi$_4$Pt$_3$. Our analysis motivates studying the so-far unknown Kondo insulator Ce$_3$Sb$_4$Pd$_3$, for which we predict $m^*/m_{band}=\mathcal{O}(10)$.

Tables

Table 1. Table 1: Collection of experimental T χ m ​ a ​ x subscript superscript 𝑇 𝑚 𝑎 𝑥 𝜒 T^{max}_{\chi} (see Fig.  2 ), estimated Kondo temperatures T K = 4 ​ T χ m ​ a ​ x / ( 2 ​ J + 1 ) subscript 𝑇 𝐾 4 subscript superscript 𝑇 𝑚 𝑎 𝑥 𝜒 2 𝐽 1 T_{K}=4T^{max}_{\chi}/(2J+1) with J = 5 / 2 𝐽 5 2 J=5/2 , m J subscript 𝑚 𝐽 m_{J} -averaged peak magnitude of the hybridization function − Im ​ Δ ​ ( ω m ​ a ​ x ) Im Δ subscript 𝜔 𝑚 𝑎 𝑥 -\hbox{Im}\Delta(\omega_{max}) , and anticipated mass renormalization 1 / Z 1 𝑍 1/Z estimated via k B ​ T K = − ( π / 4 ) ​ Im ​ Δ ​ ( ω m ​ a ​ x ) × Z / ( 2 ​ J + 1 ) subscript 𝑘 𝐵 subscript 𝑇 𝐾 𝜋 4 Im Δ subscript 𝜔 𝑚 𝑎 𝑥 𝑍 2 𝐽 1 k_{B}T_{K}=-(\pi/4)\,\hbox{Im}\Delta(\omega_{max})\times Z/(2J+1) ; the gap Δ Δ \Delta is predicted via Z × Δ D ​ F ​ T 𝑍 subscript Δ 𝐷 𝐹 𝑇 Z\times\Delta_{DFT} .

	exp		theory	prediction
material	$T_{\chi }^{max}$	$T_K$	${⟨-\text{Im}\Delta \left({\omega }_{max}\right)⟩}_{m_J}$	$m^{\ast }/m_{band}$	$\Delta$
	[K]	[K]	[meV]	$=1/Z$	[meV]
Ce3Bi4Pt3	80	53	400	11	17
Ce3Bi4Pd3	10	7	290	66	2.6
Ce3Sb4Pt3	400	265	480	2.7	90
Ce3Sb4Pd3	?	?	300	?	?

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I Introduction.

The physics of materials with $f$-electrons is determined by a delicate interplay of energy scales. The competition between the latter can be tuned by various means—including external fields, pressure, carrier doping—resulting in enormously rich phase diagramsGegenwart et al. (2008); Si and Paschen (2013); Wirth and Steglich (2016); Tomczak (2018). Of particular interest are stimuli that control the (Kondo) coupling strength while leaving (nominal) valence states inert. In the case of heavy-fermion metals this allows, for instance, to directly explore Doniach’s phase diagramDoniach (1977). Isoelectronic tuning can be achieved, e.g., through external or chemical pressure. In the latter case, elemental constituents are (at least partially) replaced with atoms having the same valence configuration but a different principle quantum number $n$.

In binary compounds, such isoelectronic substitutions directly control the unit-cell volume, as $n$ typically correlates with the element’s bonding radius. Furthermore, calculations of the hybridization of the $f$-states with their surrounding display—in binary compounds—an empirical one-to-one correspondence to the experimental Kondo temperatureHerper et al. (2017). Hence the notion that, both, external and chemical pressure manipulate the Kondo coupling $J_{K}$ primarily through the change in lattice constants.

For more complex Kondo systems—ternary compounds and above—we here demonstrate that the unit-cell volume (perhaps unsurprisingly) is not a good proxy for the size of the Kondo coupling. Consider that in isoelectronic substitutions both the shape and the radial extent of the wavefunction is different as the principle quantum number changes. The will affect the hybridization and bonding not just with the atoms hosting the $4f$-orbitals but also between all other atoms. Therefore it is not a priori clear whether isovalent substitutions modify (i) cohesive properties (largely dominated by hybridizations not involving the $f$-orbitals), (ii) the Kondo coupling $J_{K}$ (controlled by the hybridization of the $f$-electrons with their surrounding), or (iii) both of them.

Here, we explore how isoelectronic tuning affects the itineracy of $f$-electrons in the family of Kondo insulatorsFisk et al. (1995); Riseborough (2000); Tomczak (2018) Ce3A4M3 (A=Bi,Sb, M=Pt,Pd), which is of high current interestDzsaber et al. (2017, 2018); Tomczak (2019); Dzsaber et al. (2019); Cao et al. (2019); Kushwaha et al. (2019); Campbell et al. (2019). We consider both possible isovalent substitutions: that of the pnictogen and that of the precious metal. We find that the difference between the Bi- and the Sb-derived materials are mainly driven by the unit-cell volume—a conventional case of isoelectronic tuning. Interchanging the precious metal element in the Bi-compounds, on the other hand, does not change the unit-cell volumeHundley et al. (1990); Kwei et al. (1992); Hermes et al. (2008); Dzsaber et al. (2017). Nonetheless, we recently establishedTomczak (2018) that the Kondo couplings of Ce3Bi4Pt3 and Ce3Bi4Pd3 differ significantly and surmised that the main driver is the different radial extent of the 5d (Pt) and 4d (Pd) orbitals. In the current work, we confirm this proposal and investigate the detailed interplay of volume, orbital extent, hybridization, and charge gaps within band-theory. In other words, we elucidate the microscopic parameters that tune Ce3A4M3 (A=Bi,Sb, M=Pt,Pd) along the arrows in Fig. 1(bottom). Finally, by combining insights from our band-structure calculations with experimental magnetic susceptibilities, we predict the magnitude of mass renormalizations (Tab. 1): We find (i) excellent agreement with recent many-body simulations for Ce3Bi4Pt3Tomczak (2019); (ii) the Pd-based system have significantly larger renormalizations than the Pt-ones; (iii) the Sb-members have larger Kondo scales than their Bi-counterparts. The latter highly motivates synthesizing and characterizing the so far unknown compound Ce3Sb4Pd3 as well as the series Ce3Sb4(Pt1-xPdx)3, in which the isovalent substitution can be studied at more accessible temperatures than for the Bi-series.

II Survey of experimental results and previous interpretations.

We summarize pertinent experimental findings and their interpretation for the materials under consideration (for a broader perspective see Refs. Riseborough, 2000; Tomczak, 2018): As seen in the collection of resistivities (top) and magnetic susceptibilities (bottom) in Fig. 2, the charge and spin gap, $\Delta$ and $\Delta_{s}$, and the crossover temperature, $T^{\hbox{\tiny max}}_{\chi}$, indicative of the onset of Kondo screening, vary significantly in these materials: Interchanging of pnictogen and/or precious metal elements markedly tunes the energy scales. The tendencies of the charge gap $\Delta$—on which we will focus our attention—are condensed in Fig. 1(top): The Pt-based compounds are prime examples of Kondo insulatorsHundley et al. (1990); Fisk et al. (1995); Kasaya et al. (1991a); Riseborough (2000), with the Sb-member having a slightly larger charge gap than its Bi-counterpart. Exchanging Pt with Pd in Ce3Bi4Pt3 diminishes the charge gapHundley et al. (1990); Katoh and Takabatake (1998), while the volume remains constantHundley et al. (1990); Kwei et al. (1992); Hermes et al. (2008). The isovolume nature of the precious-metal substitution was recently confirmed by studying the continuous series Ce3Bi4(Pt1-xPdx)3Dzsaber et al. (2017). The end member Ce3Bi4Pd3Hermes et al. (2008) has been characterized as a semi-metalDzsaber et al. (2017), while more recent experimentsKushwaha et al. (2019) extract a gap $\Delta\sim 2$meV from the Hall resistivity and the critical magnetic field. Under the assumption of a direct correspondence between unit-cell volume and the Kondo coupling $J_{K}$, it was initially assertedDzsaber et al. (2017) that $J_{K}$ is independent of $x$ in isovolume Ce3Bi4(Pt1-xPdx)3. It was then concludedDzsaber et al. (2017) that the decrease of energy scales with increasing $x$ is driven by the change in the spin-orbit coupling, owing to the mass differences of Pt and Pd—an exciting perspective in view of the recent proposals of topological Kondo insulatorsDzero et al. (2016); Chang et al. (2017). Here, instead, we argue in favour of a substantial evolution of the Kondo coupling in the series Ce3Bi4(Pt1-xPdx)3. The compound Ce3Sb4Pd3 has—to the best of our knowledge—not yet been synthesized.

III Crystal-structure and Methods.

The compounds Ce3A4M3 (A=Bi,Sb, M=Pt,Pd) are a family of cubic intermetallics that crystallize in the body-centred Y3Sb4Au3 structureDwight (1977) which has the non-symmorphic space-group I-43d, two formula units per unit-cell and a lattice constant $a=10.051$-$10.058$ÅHundley et al. (1990); Kwei et al. (1992); Hermes et al. (2008); Dzsaber et al. (2017) ($a=9.814$ÅJones et al. (1999)) for Ce3Bi4Pt3 and Ce3Bi4Pd3 (Ce3Sb4Pt3). In our band-structure calculations we used $a=10.051$Å ($a=9.814$Å) for Ce3Bi4M3 (Ce3Sb4M3), as motivated below in Section IV.1. In all four compounds, the Ce atoms occupy in the Wyckoff position 12a (3/8, 0, 1/4), Pt/Pd sits at site 12b (7/8, 0, 1/4), and Bi/Sb is found at (u,u,u). Here, we have followed Ref. Takegahara et al., 1993 and used the ideal position $u=1/12$, which is very close to $u=0.0857$ found for Ce3Bi4Pt3Kwei et al. (1992) and $u=0.084$ of Ce3Bi4Pd3Hermes et al. (2008).

In this setting, we performed electronic structure calculations using the wien2k packageBlaha et al. (2001). We employed the PBE functional, and included spin-orbit coupling (SOC) in the scalar relativistic approximation for all atoms. Volume optimizations were performed with $12^{3}$ reducible k-points, self-consistency for experimental lattice constants was reached with a k-mesh of $20^{3}$ points. We use the auxiliary Kohn-Sham spectrum as proxy for the electronic structure of the studied materials. Statements about inter-atomic hybridizations are based on: (a) maximally localized Wannier functionsMarzari et al. (2012) constructed in the total angular momentum $J$-basis for the subspace of Ce-$4f$, A-$p$ and M-$d$ orbitals with Wannier90Mostofi et al. (2008) via wien2wannierKunes et al. (2010) and using a tool of J. Fernandez AfonsoFernandez Afonso (2019) to rotate the local spin coordinate system; (b) a local projection formalism due to Haule et al.Haule et al. (2010). The investigation of the electronic structure of the Pt-members has been pioneered by Takegahara et al. Takegahara et al. (1993). The band-structures and density of states of the two Bi-compounds have been previously shown in the review article Ref. Tomczak, 2018.

III.1 Comment: Band-theory for Kondo insulators

Given that effective one-particle theories fail in most respects for systems with $f$-electrons, a few comments are in order. Indeed, the presence of quasi-localized states makes the use of genuine many-body methods a prerequisite for an adequate description, see, e.g., Refs. Held et al., 2001; Amadon et al., 2006; Haule et al., 2010; Tomczak et al., 2013; Pourovskii et al., 2014; Shick et al., 2015; Wissgott and Held, 2016; Plekhanov et al., 2018; Tomczak, 2019; Cao et al., 2019; Seong et al., 2019. Nonetheless, conventional electronic structure theory can give meaningful insights into at least the trends of properties in families of $f$-electron compoundsZwicknagl (2016); Herper et al. (2017); Hafiz et al. (2018).

This is particularly true for (Ce-based) Kondo insulators: In the language of the Kondo lattice model (KLM), the charge gap in Kondo insulators is interaction-driven through the Kondo coupling $J_{K}$. However, this spin-fermion interaction is not a fundamental force, but a quantity that emerges in the renormalization group-like flow to a low-energy description. Indeed, it is well known that the KLM can be obtained as a limiting case of the periodic Anderson model (PAM), which contains the Hubbard $U$ interaction as a remnant of the fundamental Coulomb force: The strong $U$-coupling limit of the PAM yields the weak $J_{K}$-coupling regime of the KLM with $J\propto V^{2}/U$111Actually, there exists an exact mapping of the PAM to the KLM beyond the Kondo regimeSinjukow and Nolting (2002).. The above illustrates that (bare) hybridizations $V$ between conduction electrons and the $f$-level and the Hubbard interaction $U$, together, generate the Kondo coupling. Note that, while the KLM is metallic for vanishing interaction, $J=0$, in the absence of Hubbard $U$-interactions, the (symmetric, half-filled) PAM is a band-insulator. In this sense, the Kondo insulating gap in the PAM is not interaction-driven.

Similar to the non-interacting PAM, band-theory finds Ce3A4M3 to be covalent band-insulators. Band-theory of course grossly overestimates the hybridization $V$ of $f$-states with their surroundings. Yet, it is sensitive to changes in the Lanthanide environment (external pressure, isoelectronic substitutions) and can therefore qualitatively capture modifications in the essential inter-atomic hybridizations and the Kondo coupling. Hence, trends in the hybridization gap of Ce3A4M3 within DFT can be used as a gauge for the Kondo coupling.

Let us note that this situation is facilitated by (i) the nominal $4f^{1}$ configuration in Ce-based compoundsFuggle et al. (1983); Strigari et al. (2012, 2015), as multiplet effects are less crucial than in systems with more electrons in the $f$-shell222See, however, e.g., Refs. Temmerman et al., 2007; Pourovskii et al., 2014 for multiplet effects in Ce-compounds under external pressure.; (ii) the $f$-electrons in Ce-materials are more delocalized than, e.g., in the hole-analogues based on trivalent Yb; (iii) contrary to Mott insulators, the self-energy of Kondo insulators is benign near the Fermi levelTomczak (2018), yielding a renormalized band-structure below the Kondo temperature.

IV Results & Discussion.

IV.1 Volume optimization.

We assess the isovolume nature of Ce3Bi4(Pt1-xPdx)3 by optimizing the volume of the end members ($x=0,1$) within DFT for fixed fractional atomic positions. As can be seen in Fig. 3 the theoretical lattice constants of Ce3Bi4Pt3 and Ce3Bi4Pd3—identified as the minima in the energy curves—are indeed very similar. We find, using PBE and the Pt-compound as reference: $\Delta a/a=0.2\%$. Experiments find instead that Ce3Bi4Pd3 is minutely larger than Ce3Bi4Pt3: $\Delta a/a=-0.07\%$Dzsaber et al. (2017). We repeat the same calculations for the pair Ce3Sb4M3 (M=Pt,Pd). We find that exchanging Bi with Sb yields significantly smaller compounds. Crucially, the exchange Pt$\leftrightarrow$Pd leads again only to very small changes: $\Delta a/a=0.2\%$.333All relative changes are slightly larger when using LDA instead of PBE. We interpret the overall small changes in lattice constant, incurred upon precious-metal substitution, to warrant the label ’isovolume’. For Ce3Bi4M3 this supports previous experimental findingsHundley et al. (1990); Kwei et al. (1992); Hermes et al. (2008); Dzsaber et al. (2017), while the isovolume nature of Ce3Sb4(Pt1-xPdx)3 is a prediction.

Let us remark that replacing Pt with Pd is not affecting the lattice constant is not a generic behaviour. Indeed, the difference in lattice constant between CePtSnTakabatake et al. (1993) and CePdSnMalik et al. (1989), amounting to $\Delta a/a=0.8\%$, is experimentally an order of magnitude larger than in Ce3Bi4(Pt1-xPdx)3. We can speculate that in the more complex unit-cell of Ce3A4M3 (in which, e.g., every “M” has 4 nearest “A” neighbours) small atomic differences can be better compensated than in CeMB (where each “M” has only 3 neighbouring “B”).

We further note that—in line with common experience—PBE(LDA) over(under)estimates the lattice constant. In the following electronic structure calculations we have therefore settled to employing experimental values: we use the lattice constant of Ce3Bi4Pt3 for Ce3Bi4M3, and the one of Ce3Sb4Pt3 for Ce3Sb4M3, where M=Pt,Pd.

IV.2 Band-structures.

Fig. 4 displays the band-structures of all Ce3A4M3 compounds considered here, while Fig. 5(a) shows the corresponding density of states (DOS). For a given pnictogen “A”, the valence states are rather similar, irrespective of the precious metal “M”. Changing the pnictogen, results in perceptible changes of the valence states, in particular near the Brillouin zone centre. Overall, the valence states are—congruent with photoemission spectroscopyBreuer et al. (1998); Takeda et al. (1999)—comparatively dispersive as they are dominantly of $pd$-character. Indeed, as expected with only one nominal $4f$-electron, quasi-atomic narrow bands are prevalent only above the Fermi level, where also the DOS exhibits pronounced peaks.

In this work, we are mostly interested in the trends of the charge gap under isovalent substitutions. The gaps $\Delta$ extracted from the band-structures are reported in Fig. 5(b). All gaps are indirect and spread, for most compounds, from around the H-point to the zone centre $\Gamma$. An exception is Ce3Bi4Pt3, where the top of the valence bands emerges between the P and the $\Gamma$-point. Another speciality of Ce3Bi4Pt3 is the fact that the $\Gamma$-point conduction-band minimum is particularly pronounced. While the DOS—dominated by the $f$-spectral weight—suggests that the charge gap increases when substituting Pd for Pt, the fundamental gap of Ce3Bi4Pd3—controlled by the dispersive band at $\Gamma$—is actually larger than in Ce3Bi4Pt3 within DFT. Since we believe the physics of these compounds to be dominated by the $f$-electrons, we introduce—as a proxy for the trends in the $f$-band position—the quantity $\Delta_{N}$ which measures the gap between the Fermi level and onset of the conduction states at the $N$-point. To disentangle putative control parameters of the charge gap, we report in Fig. 5(b) also the values of $\Delta$ and $\Delta_{N}$ when turning off the SOC on selected atoms. Furthermore, so as to demonstrate the effect of the unit-cell volume onto the charge gap, we also show results for a fictitious variant of Ce3Bi4Pt3 that has the lattice constant of Ce3Sb4Pt3.

IV.3 Analysis of electronic structure

In this discussion section, we asses the influence of the spin-orbit coupling, the unit-cell volume, and the radial extent of orbitals onto the charge gap.

IV.3.1 Spin-orbit coupling

As seen in Fig. 5(b), the SOC is—within DFT—irrelevant for the qualitative trends of the charge gap in this family of compounds: Turning selectively off the SOC for Pt in Ce3Bi4Pt3—to simulate the change in atomic mass of the precious metal atoms—does not result in a band gap comparable to isovolume Ce3Bi4Pd3. In fact, this procedure yields a $\Delta$ and $\Delta_{N}$ that surpass the values obtained with SOC on all atoms. Hence, the difference in masses of Pt and Pd and the resulting change in the magnitude of relativistic effects is likely not the driver of the Kondo insulator to semi-metal transition in Ce3Bi4(Pt1-xPdx)3 of Ref. Dzsaber et al., 2017 (see also the discussion in Ref. Tomczak, 2018). Also, eliminating the SOC of Bi in Ce3Bi4Pt3—to simulate a smaller pnictogen mass when going to Ce3Sb4Pt3—cannot bridge the disparity of band-gaps in these two materials. The spin-orbit coupling propels the properties neither along the vertical (Bi$\leftrightarrow$Sb) nor the horizontal (Pt$\leftrightarrow$Pd) arrow in Fig. 1.

IV.3.2 Hybridization nature of the gap

A far more common tuning parameter in heavy fermion systems is the unit-cell volume. While the precious metal substitution series Ce3Bi4(Pt1-xPdx)3 is isovolumeHundley et al. (1990); Kwei et al. (1992); Hermes et al. (2008); Dzsaber et al. (2017), changing the pnictogen atom from Bi to Sb decreases the lattice constant by 2%Hundley et al. (1990); Kwei et al. (1992); Jones et al. (1999). To isolate the effect of the lattice, we report, in Fig. 5(b), the charge gap of fictitious Ce3Bi4Pt3 where the lattice constant has been artificially shrunken to the value realized in Ce3Sb4Pt3. With respect to true Ce3Bi4Pt3, we find a significantly increased gap. In fact, the change in the unit-cell volume accounts for almost 80% of the gap difference between Ce3Sb4Pt3 and Ce3Bi4Pt3. This finding identifies the unit-cell volume as the leading protagonist for the trends between the Bi and the Sb pairs of materials.

That the lattice has such a large effect onto the electronic structure finds its origin in the hybridization nature of the gap in these systems. Indeed, as has been demonstrated in Ref. Tomczak, 2018 it is the hybridization of the Ce-$4f$ orbitals to their surrounding that accounts for the gap opening: Treating the Ce-$f$ (J=5/2) as core-states results in metallic ground-states, in analogy to a periodic Anderson model with vanishing hybridization between localized $f$ and conduction states.

Here, we want to further distinguish the individual contributions of atoms/orbitals to the hybridization gap $\Delta$. To this end we constructed maximally localized Wannier functions (see method section) and expressed the projected DFT Hamiltonian in that basis. Fig. 6 shows what happens to the relative magnitude of the gap in (a) Ce3Bi4M3 and (b) Ce3Sb4M3 when the hybridization (off-diagonal elements in the Hamiltonian) between the Ce-$4f$ orbitals and the M$-d$ (M=Pt,Pd) or the Bi/Sb$-p$ orbitals is scaled with a factor $\alpha\leq 1$.444See also Ref. Tomczak, 2018 where such a construction helped distinguishing the gap nature in covalent FeSi and ionic LaCoO3. We find the gap to be most sensitive to changes in the hybridization between the $4f$-orbitals of Ce and the $p$-orbitals of Bi and Sb. For the Sb compounds, a smaller scaling factor $\alpha$ is needed to collapse the gap, congruent with Ce3Sb4M3 having larger hybridizations than their Bi-counterparts. The gap is much less sensitive to hybridizations between the Ce-$f$ and the precious metal M-$d$-orbitals. We note that for both A=Bi and A=Sb the gap is more quickly diminished with $\alpha$ if the precious metal is M=Pt. The relative suppression of $\Delta(\alpha)$ when scaling the precious metal-hybridization is similar irrespective of the type of the pnictogen “A”. This finding supports the assertion that the anticipated diminishing of the gap in Ce3Sb4(Pt1-xPdx)3 with $x$ is not volume driven—akin to what happens in Ce3Bi4(Pt1-xPdx)3.

IV.3.3 Radial extent of orbitals

Now we will disentangle changes in the inter-atomic hybridization mediated by (i) the unit-cell volume and (ii) the radial extent of pertinent orbitals. To this end we and analyse the orbitally ($l$-)resolved spread $\Omega_{l}=\langle l|R^{2}|l\rangle-\langle l|R|l\rangle^{2}$ of the Wannier functions, the sum of which is minimized in the maximally localization procedureMarzari et al. (2012). We define the mean extent of a type $L$ of orbitals as the average of the square-root of the $l$-resolved Wannier spread: $\overline{R}_{L}=1/N_{L}\sum_{l\in L}\sqrt{\Omega_{l}}$. For example, in case of the Ce-$4f$ orbitals, i.e. $L=4f$, $l$ runs over the $N_{L}=14$ spin-orbitals per Ce-atom. Fig. 7 visualizes the thus measured size of orbitals to scale with nearest-neighbour atomic distances for (a) Ce3Bi4M3, (b) Ce3Sb4M3 where M=Pt,Pd. Panel (c) displays the trends of $\overline{R}_{L}$ for precious metal and pnictogen substitutions relative to the materials’ lattice constant $a$.

First, we discuss the trends upon interchanging Pt $\leftrightarrow$ Pd. Congruent with the precious metals’ van der WaalsBondi (1964) and metallicGre (1997) radii indicated in Fig. 7(a), we find the Pd-$4d$ orbitals to be smaller than the Pt-$5d$ ones, irrespective of the pnictogen. The Bi and Sb-$p$ orbitals instead are slightly larger when the precious metal is palladium, which in part compensates for the chemical pressure of the smaller Pd. It is thus suggestive to ascribe this expansion of the pnictogen to play a role in the Ce3Bi4(Pt1-xPdx)3 series being isovolume. Nonetheless, computing the volume of all “Wannier spheres” (of radii $\overline{R}_{L}$) in the unit-cell, $V_{Wannier}=4/3\pi\times\left(6\times 14{\overline{R}_{f}}^{3}+4\times 6{\overline{R}_{p}}^{3}+3\times 10{\overline{R}_{p}}^{3}\right)$, we find for Ce3Bi4Pt3 ($V_{Wannier}=4340\AA^{3}$) a value still larger by 3% than for Ce3Bi4Pd3 ($V_{Wannier}=4203\AA^{3}$), see Fig. 7(d). The relative difference is similar for the Sb-based compounds Ce3Sb4Pt3 ($V_{Wannier}=3843\AA^{3}$) and Ce3Sb4Pd3 ($V_{Wannier}=3732\AA^{3}$). This suggests that the crystal-structure is held together dominantly by the (much larger) pnictogen $p$-orbitals, while the contribution to bonding from the precious metal $d$-orbitals is sub-leading. The small dependence of the hybridization gap on the Ce-$4f$ to Pt/Pd-$d$ hopping (see Fig. 6) supports this statement.

The above is further substantiated when looking at the exchange of the pnictogen: Bi $\leftrightarrow$ Sb. There, we find the smaller orbitals of the Sb-based compounds to be in line with the smaller unit-cell volume as compared to their Bi-counterparts (lattice constants: $10.051\AA$ for Ce3Bi4M3 and $9.814\AA$ for Ce3Sb4Pt3, see Section IV.1). Indeed, the Wannier volume (as defined above) decreases by more than 10% when replacing Bi with Sb, see Fig. 7(d). Note, that this trend with (pnictogen) chemical pressure is opposite to the generic behaviour of Wannier functions under external pressure, where the spread increases when the volume shrinksTomczak et al. (2009, 2010). When rescaling $V_{Wannier}$ with the unit-cell volume $V_{0}=a^{3}$, see Fig. 7(e), we find the relative differences between the pairs with fixed precious metal “M” to more than halve. Hence, for the Bi $\leftrightarrow$ Sb substitution, the intuitive correspondence between the size of ($p$-)orbitals that mediate bonding and the resulting lattice constant holds to a large extent.

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Finally, we discuss the extent of the Ce-$4f$ orbitals. The differences in $\overline{R}_{f}$ between the four compounds may seem minute on the scale of Fig. 7(a,b,c), yet their trends are revealing, see Fig. 7(f): Indeed, while for a given precious metal, a swapping of the pnictogen does not change the size of the $4f$-shell by much, replacing the precious metal Pt with Pd significantly decreases $\overline{R}_{f}$ despite leaving the unit-cell volume inert. This suggest that—for the chosen compounds—the impact of (i) the unit-cell volume and (ii) the relative extent of the $4f$-orbitals can be approximately disentangled, as discussed in the next section.

IV.4 Isoelectronic tuning: Summary & Perspective

Fig. 8 visualizes the central result of this paper: It characterizes the charge gap of our four compounds as a function of their lattice constant $a$ and the relative extent $\overline{R}_{f}/a$ of their $f$-orbitals: $\Delta_{N}=\Delta_{N}(a,\overline{R}_{f}/a)$. Quite intuitively, $\Delta_{N}$ increases with shrinking volume and larger relative orbital extent, as both lead to larger orbital overlaps/hybridizations. Most importantly, from this representation it is apparent, that the isoelectronic pnictogen replacement primarily acts on the lattice constant, while the relative extent of the $f$-shell is well-neigh untouched. The isoelectronic tuning mediated by an exchange of the precious metal, on the other hand, leaves the volume fixed, but expands or shrinks the $f$-orbitals’ reach relative to the surrounding atoms.

We have thus identified a viable choice of essentially independent variables that connect our four materials in parameter space: Fig. 8 mirrors Fig. 1 in a one-to-one fashion (if rotated by 40∘to the right).

Let us put these findings into perspective: It is a well-known fact that the Kondo coupling in $f$-electron materials can be dramatically affected by (i) pressure or by (ii) isoelectronic substitutions with atoms of varying sizeSi and Paschen (2013). Here, we demonstrated that these two routes can be linearly independent: Besides the effect of exercising chemical pressure, the change in the principle quantum number $n$ of substituted atoms—here Pt: $n=5$, Pd: $n=4$—may largely change the charge gap while having little or no effect on cohesive properties. The isoelectronic tuning scenario that we propose extents the empirical volume picture (valid in binary compounds) to include orbital effects at constant volume. In this regard, the Ce3A4M3 system is an ideal case, in that the changes in the lattice constant induced by different “A” atoms and the relative orbital extent of “M”-atoms form an essentially orthogonal coordinate system. In general, volume and $\overline{R}_{f}/a$ will provide linear independent yet skewed coordinates. Near orthogonality of the control parameters $\overline{R}_{f}/a$ and the volume are likely realized for replacements of atoms—like Pt$\leftrightarrow$Pd—that have comparable electronegativities and (empirical, van der Waals, or metallic) atomic radii (see below for a “non-orthogonal” example). It also stands to reason that an isovolume tuning of inter-atomic hybridizations, and hence the Kondo coupling, is more prevalent in complex materials—ternary compounds and above—and for substitutions of atoms that contribute only subleadingly to the charge gap/bonding (as is the case for the precious metal “M” in Ce3A4M3, cf. Fig. 6 and the discussion above).

The proposed orbital scenario will be relevant in other heavy fermion systems. A putative example is the Ce(Ni,Pd)Sn system: CePdSn is an antiferromagnetic Kondo metal with a Néel temperature of about 7KKasaya et al. (1988); Malik et al. (1989); Iga et al. (1993), while CeNiSn is a paramagnetic anisotropic Kondo insulatorTakabatake et al. (1990); Mason et al. (1992). In the standard Doniach phase diagramDoniach (1977), CePdSn falls into the regime dominated by the Ruderman–Kittel–Kasuya–Yosida interaction, whereas CeNiSn realizes a larger Kondo coupling $J_{K}$ that impedes long-range magnetic order. This interpretation is congruent with the “volume scenario”: CeNiSn has a unit-cell volume (263.6Å3) smaller by 6% than CePdSn (281.7Å3)Iga et al. (1993) arguably leading to larger hybridizations and, hence, bigger $J_{K}$. However, it has been highlightedSi and Paschen (2013) that there is a significant difference between the isoelectronic substitution series, Ce(Pd1-xNix)Sn, that connects the two compounds, and CePdSn under hydrostatic pressure. Indeed, while the Néel temperature is fully suppressed for $x\geq 0.75$Kasaya et al. (1991b), the pressure needed to achieve the same is at least 6GPaIga et al. (1993), which—using the bulk modulus 55.6GPa of CeNiSnIkushima et al. (1999)—corresponds to a volume reduction of more than 10%: an external compression of almost twice as much is required to equal the effect of chemical pressure. With the “orbital scenario” proposed above, this large quantitative discrepancy is readily explained: As suggested by the metallic radii $R_{M}$ of Pd and NiGre (1997) (both are indicated in Fig. 7(a)), the $3d$-orbitals of Ni will be smaller than the $4d$ of Pd. We therefore hypothesize that for growing $x$ in CeNixPd1-xSn the shrinking of the unit-cell volume and the decrease of the orbital extent conspire to jointly reduce the Kondo coupling—driving the system faster towards smaller $J_{K}$-coupling in the Doniach phase diagram. The scenario might also be of relevance in the comparison of CeRu4Sn6Das and Sampathkumaran (1992) under externalSengupta et al. (2012); Zhang et al. (2018) and chemical (Sn$\rightarrow$Ge)Zhang et al. (2018) pressure. In fact, it will be interesting to gain systematic insights into larger groups of materials to assess the described physics. Indeed, we believe that $\overline{R}_{f}/a$ could be a useful quantity in databases for heavy-fermion material propertiesHerper et al. (2017); Hafiz et al. (2018). Its approximation $R_{M}/a$, using tabulated metallic radii $R_{M}$, could even be used as descriptor in high-throughput surveys.555For non-cubic compounds, such as CeNiSn and CeRu4Sn6, the lattice parameter $a$ might have to be replaced with relevant nearest neighbour distances.

IV.5 Anticipated many-body renormalizations

Finally, we transition to a more many-body oriented language. In the spirit of the Anderson impurity model, the embedding of the correlated Ce-$4f$ orbitals into the solid is often described by a so-called hybridization function $\Delta(\omega)=\omega+\mu-H^{loc}_{ff}-(G_{loc}^{-1})_{ff}$. Here, $\mu$ is the chemical potential, and $H^{loc}$ and $G_{loc}$ are the local projectionsHaule et al. (2010) of the Hamiltonian and the Greensfunction, respectively, while the subscripts $f$ denotes the evaluation within the $f$-orbital subspace. Indeed, $\Delta(\omega)$ is the back-bone of dynamical mean-field theory (DMFT)Georges et al. (1996), where it is promoted to a self-consistent Weiss field. However, already on the level of DFT, the hybridization function can divulge important physical insight, in particular regarding trends among material familiesHerper et al. (2017); Hafiz et al. (2018). Moreover, $\Delta(\omega)$ can be used to classify insulatorsTomczak (2019), e.g., distinguishing between Mott and Kondo insulators.

For our four materials, Fig. 9(a) displays the imaginary parts of the DFT hybridization function, while panel (b) shows the corresponding positions $\omega_{max}$ and peak amplitudes $-\hbox{Im}\Delta(\omega_{max})$. In both cases, the origin of energy corresponds to the top of the valence bands, and we have limited the presentation to the $m_{J}=1/2$ and $m_{J}=5/2$ components of the $J=5/2$ multiplet (the $m_{J}=3/2$ components are the smallest).

Overall, the Bi-compounds exhibit peaks that are sharper and closer to the Fermi level, indicating that these materials are closer to the flat-band limit. In the latter, an $f$-level hybridizes with a dispersionless conduction band, yielding $-\hbox{Im}\Delta(\omega)=\pi V^{2}\delta(\omega+\mu-\omega_{max})$, where $V_{\mathbf{k}}=V$ is the amplitude of the hybridization between the conduction band and the Ce-$4f$ orbitals and $\epsilon_{\mathbf{k}}=\omega_{max}$ the position of the conduction band. For conduction-states $\epsilon_{\mathbf{k}}$ and couplings $V_{\mathbf{k}}$ with finite momentum-dependence, the hybridization strength is spread over larger energies. The latter is the case for the Sb-members of the Ce3A4M3 family. For a given pnictogen “A”, the hybridization function is much more pronounced for Pt than for Pd, as could be anticipated from the size of orbitals ($5d$ vs. $4d$) as discussed in the previous section.

Using the DFT hybridization function in conjunction with the experimental magnetic susceptibility, we can estimate the magnitude of many-body renormalizations, see Tab. 1. We first assess the Kondo temperature from the spin degrees of freedom using the Kondo lattice expression, $T_{K}=4T^{max}_{\chi}/(2J+1)$Coleman (1983) with $J=5/2$ and $T^{max}_{\chi}$ from Fig. 2. Note that the estimate for $T_{K}$ of Ce3Bi4Pd3 is of the same order of magnitude of what has been deduced from the experimental specific heatDzsaber et al. (2017). The Kondo temperature can also be determined from the charge degrees of freedom: $k_{B}T_{K}=-(\pi/4)\,\hbox{Im}\Delta(\omega_{max})\times Z/(2J+1)$ Hewson (1993); Coleman (1983). With the $T_{K}$ from above, we then extract the effective mass enhancement $1/Z$. For Ce3Bi4Pt3, we find $Z\approx 1/10$, in excellent agreement with recent realistic DMFT calculationsTomczak (2019). While for the other compounds, a quantitative theoretical assessment has to await the application of many-body electronic structure methodsGeorges et al. (1996); Tomczak et al. (2017),666 For the temperature dependence of $\Delta(\omega)$ in Ce3Bi4Pt3, see Ref. Tomczak, 2019; for spectral properties of Ce3Bi4Pt3 and Ce3Bi4Pd3, see Refs. Tomczak, 2019; Cao et al., 2019. , we believe the following simple estimates to reflect at least the hierarchy of effective masses in the Ce3A4M3 family: The largely reduced Kondo temperature when substituting Pd for Pt accounts for Ce3Bi4Pd3 having the largest effective mass ($1/Z=66$), as well as for Ce3Sb4Pt3 being less correlated than Ce3Bi4Pt3. Naively applying the mass renormalization of Ce3Bi4Pd3 to its DFT band-gap yields 2.6meV, in good agreement with the experimental estimate of $1.8\pm 0.5$meVKushwaha et al. (2019). Doing the same for Ce3Sb4Pt3, however, significantly overestimates the gap. We can surmise that the Sb-compounds are further from a description in terms of fully localized $f$-states, which renders the Kondo lattice formula used in the determination of $T_{K}$ less reliable.

Given the trends in the band gaps, hybridization functions, and the Kondo temperature (that itself depends on the two former), we nonetheless gauge the Ce3Sb4Pd3 compound—that we propose experimentalists to synthesize—to exhibit mass enhancements of the order of 10.

V Conclusions.

We have shown the suppression of coherence and energy scales in isoelectronic and isovolume Ce3Bi4(Pt1-xPdx)3—previously attributed to changes of the spin-orbit couplingDzsaber et al. (2017)—to derive from a reduced Kondo coupling. The latter is tunable—even at constant volume—through the principle quantum number of orbitals that host conduction electrons. We have identified the relative extent, $\overline{R}_{f}/a$, of the lanthanide’s $f$-orbitals as (indirect) control parameter of the Kondo coupling and demonstrated it to be linearly independent from changes mediated by the unit-cell volume $V_{0}$: In the Ce3A4M3 system, the precious metal Pt$\leftrightarrow$Pd (pnictide Bi$\leftrightarrow$Sb) substitution only changes the $f$-orbitals’ hybridizations through $\overline{R}_{f}/a$ ($V_{0}$). Our “orbital scenario” generalizes the empirical “volume scenario” (applicable to binary compounds) to isoelectronic tuning in complex heavy-fermion materials. In particular, the two parameter dependency $J_{K}=J_{K}(a,\overline{R}_{f}/a)$ explains why there can be notable differences between external and chemical pressure, cf. the Ce(Ni,Pd)Sn system.

Since Ce3Bi4Pd3 has a very small coherence scale, its ground-state remains controversialDzsaber et al. (2017); Kushwaha et al. (2019). We therefore propose to synthesise the new material Ce3Sb4Pd3. We predict it to have a unit-cell volume very close to that of Ce3Sb4Pt3, and a mass enhancement of the order of 10. Anticipating larger gaps in Ce3Sb4(Pt1-xPdx)3, the series provides an attractive testing ground for the proposed “orbital scenario” of isoelectronic (and isovolume) tuning of energy and coherence scales.

VI Acknowledgements

The author is indebted to Juan Fernandez Afonso. Exchanges with S. Dzsaber, G. Eguchi, K. Held, J. Kuneš, S. Paschen, and Q. Si are acknowledged. This work has been supported by the Austrian Science Fund (FWF) through project “LinReTraCe” P 30213-N36. Some calculations were performed on the Vienna Scientific Cluster (VSC).