Isoelectronic tuning of heavy fermion systems: Proposal to synthesize ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pd}}_3$

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 straightforward, 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 ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$, we consider, within band theory, isoelectronic substitutions of the pnictogen ($\mathrm{Bi}\to \mathrm{Sb}$) and/or the precious metal ($\mathrm{Pt}\to \mathrm{Pd}$). We show for the isovolume series ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\left({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x\right)}_3$ 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 ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$. Our analysis motivates studying the so-far unknown Kondo insulator ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pd}}_3$, for which we predict $m^{*}/m_{\mathrm{band}}=O\left(10\right)$.

Figure 1

Trends in the charge gap. Top: experimental gaps of our focus compounds (see also Fig. 2); the value for ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pd}}_3$ is unknown. Bottom: theoretical gaps within band theory. Arrows indicate trends for pnictogen ($\mathrm{Bi}\leftrightarrow \mathrm{Sb}$) and precious-metal ($\mathrm{Pt}\leftrightarrow \mathrm{Pd}$) substitution. We seek to identify the microscopic control parameters that these substitutions engage. Replacing the precious metal was shown [7, 8, 9] (is predicted, see Sec. 4a) to be isovolume for the Bi (Sb) compounds. Note that the gap ${\mathrm{\Delta }}_N$ is measured from the Fermi level to the $N$ point, for reasons explained in the text.

Figure 2

Survey of experimental data. Shown are the resistivities from Refs. [21, 22] (top panel) and magnetic susceptibilities from Refs. [7, 20] (bottom panel) of ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$ and ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pt}}_3$, respectively. The susceptibility of ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$ is from Ref. [12]. Using phenomenological fits as indicated (in gray, see also Ref. [4]), the charge and spin gaps are extracted. For $\chi$, the fitting ignores low-temperature Curie-Weiss contributions likely to derive from impurities. The gap in ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$ (no data shown) has been estimated as $\sim 2\mathrm{meV}$ [17].

Figure 3

Experimental and theoretical lattice constants. Shown is the relative energy difference $\mathrm{\Delta }E$ for varying lattice constant $a$, using the PBE (symbols and solid lines) and the LDA (dashed lines) potential, where lines are parabolic fits. Arrows indicate experimental lattice constants (see text). The colors are chosen as follows: ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$ (red), ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$ (blue), ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pt}}_3$ (yellow), ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pd}}_3$ (green). In that order, the computed equilibrium lattice constants are 10.12, 10.10, 9.90, and 9.88 Å with PBE and 9.89, 9.85, 9.70, and 9.66 Å using LDA.

Figure 4

Band-theory results. Band structures of (a) ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$ and ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$, (b) ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pt}}_3$ and ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pd}}_3$. The gray arrows in (a) indicate the fundamental indirect gap $\mathrm{\Delta }$ and the gap ${\mathrm{\Delta }}_N$ used as a proxy for the apparent gap in the density of states (DOS) shown, for all materials, in (c). (d) Charge gaps: fundamental indirect gap $\mathrm{\Delta }$ and ${\mathrm{\Delta }}_N$ as defined in (a). Selectively turning off the spin-orbit coupling (SOC) for the pnictogen and precious-metal atoms demonstrates that within DFT the SOC does not control the overall trends within this family of compounds. Additionally shown are gap values of fictitious ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$ that use the lattice constant/volume of ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pt}}_3$.

Figure 5

Hybridization nature of the charge gap. Shown is the relative magnitude of the gap $\mathrm{\Delta }$ when scaling the hybridization between the Ce-$4f$ and the $M-d$ ($M=$ Pt, Pd) or $A-p$ ($A=$ Bi,Sb) orbitals in a maximally localized Wannier basis with a factor $\alpha \le 1$ for (a) ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{M}_3$ and (b) ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{M}_3$.

Figure 6

The radial extent of orbitals. (a), (b) Visualize (to scale) the average radial extent (square root of the Wannier spread) of the Ce-$f$, M-$d$ ($M=$ Pt,Pd), $A-p$ ($A=$ Bi,Sb) orbitals in relation to interatomic nearest-neighbor distances. Also shown in (a) is the van der Waals radius $R_{\text{vdW}}$ of Pt (1.75 Å) and Pd (1.63 Å) [51] in solid (dashed) gray, as well as the metallic radii $R_M$ of Pt (1.39 Å), Pd (1.37 Å), and Ni (1.24 Å) [52]. (c) Displays the average radial extent ${\overline{R}}_L$ relative to the respective lattice constant $a$. (d) Shows the summed volume $V_{\text{Wannier}}$ of all Wannier orbital spheres of radius $R_L$ in the unit cell. (e) Shows the same as (c) but divided by the unit-cell volume $V_0$. (f) Displays the average radial extent ${\overline{R}}_f$ of the Ce-$4f$ orbitals; the relative extent ${\overline{R}}_f/a$ is shown in Fig. 7.

Figure 7

Control parameters. The figure shows a (linearly interpolated) intensity map of the charge gap ${\mathrm{\Delta }}_N$ of ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$ (red), ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$ (blue), ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pt}}_3$ (yellow), and ${\mathrm{Ce}}_3{\mathrm{Sb}}_4{\mathrm{Pd}}_3$ (green) as a function of the lattice constant $a$ and the relative extent ${\overline{R}}_f/a$ of the Ce-$4f$ orbitals. The substitution of the precious-metal $\mathrm{Pt}\leftrightarrow \mathrm{Pd}$ (horizontal arrow) only changes the size of the orbitals, while the pnictogen replacement $\mathrm{Bi}\leftrightarrow \mathrm{Sb}$ (vertical arrow) dominantly engages the unit-cell volume.

Figure 8

Hybridization function $\mathrm{\Delta }\left(\omega \right)$. (a) Shows the imaginary part of the DFT hybridization function $\mathrm{\Delta }\left(\omega \right)$, resolved into its $m_J$ components. (b) Indicates the trends in the maximal amplitude of the $m_J=\frac{1}{2}$ component of $\mathrm{\Delta }\left(\omega \right)$ and the frequency ${\omega }_{\max }$ where the latter occurs. The origin of energy corresponds to the valence-band maximum.