Kondo Insulator to Semimetal Transformation Tuned by Spin-Orbit Coupling

Recent theoretical studies of topologically nontrivial electronic states in Kondo insulators have pointed to the importance of spin-orbit coupling (SOC) for stabilizing these states. However, systematic experimental studies that tune the SOC parameter ${\lambda }_{\mathrm{SOC}}$ in Kondo insulators remain elusive. The main reason is that variations of (chemical) pressure or doping strongly influence the Kondo coupling $J_K$ and the chemical potential $\mu$—both essential parameters determining the ground state of the material—and thus possible ${\lambda }_{\mathrm{SOC}}$ tuning effects have remained unnoticed. Here, we present the successful growth of the substitution series ${\mathrm{Ce}}_3{\mathrm{Bi}}_4({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x{)}_3$ ($0\le x\le 1$) of the archetypal (noncentrosymmetric) Kondo insulator ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$. The Pt-Pd substitution is isostructural, isoelectronic, and isosize, and it therefore is likely to leave $J_K$ and $\mu$ essentially unchanged. By contrast, the large mass difference between the $5d$ element Pt and the $4d$ element Pd leads to a large difference in ${\lambda }_{\mathrm{SOC}}$, which thus is the dominating tuning parameter in the series. Surprisingly, with increasing $x$ (decreasing ${\lambda }_{\mathrm{SOC}}$), we observe a Kondo insulator to semimetal transition, demonstrating an unprecedented drastic influence of the SOC. The fully substituted end compound ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$ shows thermodynamic signatures of a recently predicted Weyl-Kondo semimetal.

Figure 1

Relative change of lattice parameter $a$ as function of the Pd content $x$ in ${\mathrm{Ce}}_3{\mathrm{Bi}}_4({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x{)}_3$ (red, stars), on a scale relevant to other archetypal Kondo systems [31, 32, 33, 34] (the open symbols). The lines are guides for the eye.

Figure 2

(a) Temperature-dependent electrical resistance $R(T)$, normalized to its room temperature value $R(300\mathrm{K}$), on a log-log scale for all investigated ${\mathrm{Ce}}_3{\mathrm{Bi}}_4({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x{)}_3$ samples. Labels reflect order of the curves at low temperatures. An exponentially activated resistivity, fitted to the high-temperature regime of the $x=0$ sample, is shown as a blue line. (b),(c) Arrhenius plots for data above 50 K. The solid lines are the fits (see the text). The data for the $x=1$ sample were taken at 2 T to suppress a spurious superconducting resistance drop at 3 K that has no correspondence in magnetization and specific heat measurements. This magnetic field has no effect on the resistance data above 3 K.

Figure 3

Characteristic temperature scales for ${\mathrm{Ce}}_3{\mathrm{Bi}}_4({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x{)}_3$ as a function of $x$. $T_{\max }^{\chi }$ and ${\mathrm{\Theta }}_C$ are taken from Fig. 4 and the Curie-Weiss fits in Fig. 4, respectively. $\mathrm{\Delta }/k_B$ is taken from the Arrhenius fits in Figs. 2 and 2. The Kondo temperature $T_K$ is taken from the entropy analysis in Fig. 5. The full lines are guides for the eye. The shaded pink (light gray) area represents the fact that, for $x=1$, the Arrhenius fit loses significance and $R(T)$ becomes compatible with a gapless state (see the text). The dashed green line indicates that the maximum in $\chi (T)$ is suppressed to below 2 K for $x=0.37$. The arrow on the top axis indicates that an increase of the conduction-electron ${\lambda }_{\mathrm{SOC}}$ (corresponding to a decrease of $x$) drives the system from a semimetallic to an insulating state. The structure sketches illustrate the unit cell (bottom), the environment of Ce with the four nearest $\mathrm{Pt}/\mathrm{Pd}$ (3.01 Å) and eight next-nearest Bi (3.41 Å) neighbors (top), and the lack of a center of inversion for (selected) Ce and Pt atoms (left).

Figure 4

(a) Temperature-dependent magnetic susceptibility $\chi (T)$ of all investigated ${\mathrm{Ce}}_3{\mathrm{Bi}}_4({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x{)}_3$ samples, together with the Curie-Weiss fit for the $x=1$ data [the blue line, from (b)]. Labels reflect inverse order of the data sets at low temperatures. (b) ${\chi }^{-1}$ vs $T$ along with the Curie-Weiss fits that determine the paramagnetic Weiss temperature ${\mathrm{\Theta }}_C$. The data for $x>0$ are successively shifted by $255\mathrm{mol}\mathrm{Ce}\mathrm{Oe}/\mathrm{emu}$ for clarity. Unless specified, the field for the data in (a) and (b) was 100 mT. (c) Selected magnetization vs field $M(B)$ isotherms. The slight nonlinearities at low fields indicate that the low-temperature upturn of $\chi (T)$, seen most clearly for the $x=1$ sample in (a), is readily saturable by small fields.

Figure 5

(a) Temperature-dependent specific heat $C(T)$ of ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pd}}_3$ below 100 K. (b) $C/T$ vs $T^2$ below 14 K, in zero field and 7 T. The slope $\beta$ of the black straight line corresponds to the phonon contribution $C_{\mathrm{ph}}$; the red and blue (steepest) lines with steeper slope ${\beta }^{\prime }$ are interpreted as Weyl contributions (see the text). (c) Entropy of electronic specific heat $C_{\mathrm{el}}=C-C_{\mathrm{ph}}$ vs temperature. (d) With the lowest-$T$ CDW and NFL contributions (see Fig. S3 in the Supplemental Material [20]) subtracted, linear-in-$T^2$ behavior (the black line) of $C_{\mathrm{el}}/T$ prevails from 0.4 to 5.5 K.