Two distinct topological phases in the mixed-valence compound ${\text{YbB}}_6$ and its differences from ${\text{SmB}}_6$

We discuss the evolution of topological states and their orbital textures in the mixed-valence compounds ${\text{SmB}}_6$ and ${\text{YbB}}_6$ within the framework of the generalized gradient approximation plus onsite Coulomb interaction $(\mathrm{GGA}\hspace{0.17em}+\hspace{0.17em}U)$ scheme for a wide range of values of $U$. In ${\text{SmB}}_6$, the topological Kondo insulator (TKI) gap is found to be insensitive to the value of $U$, but in sharp contrast, Kondo physics in isostructural ${\text{YbB}}_6$ displays a surprising sensitivity to $U$. In particular, as $U$ is increased in ${\text{YbB}}_6$, the correlated TKI state in the weak-coupling regime transforms into a $d-p$-type topological insulator phase with a band inversion between $\text{Yb-5}d$ and $\text{B-2}p$ orbitals in the intermediate coupling range, without closing the insulating energy gap throughout this process. Our theoretical predictions related to the TKI and non-TKI phases in ${\text{SmB}}_6$ and ${\text{YbB}}_6$ are in substantial accord with recent angle-resolved photoemission spectroscopy experiments.

Figure 1

Crystal structure and the first BZ of ${\text{Sm(Yb)B}}_6$.

Figure 2

(a) Band structure of ${\text{SmB}}_6$ from GGA calculations. Sizes of yellow, blue, and green dots denote weights of Sm-$4f$, Sm-$5d$, and B-$2p$ orbitals in various bands. (b) Zoomed in view of (a) near $E_F$. Red dots denote metallic Sm-$5d$ orbitals without hybridization with Sm-$4f$ orbitals. (c) Low-energy band structure around $E_F$ based on GGA and GGA + $U$ ($U=8\mathrm{eV})$ computations.

Figure 3

Band structure of ${\text{YbB}}_6$ for (a) $U=0$, (b) $U=5\mathrm{eV}$, and (c) $U=8\mathrm{eV}$. Black solid lines and light orange circles in (a) give GGA + SOC and GGA + SOC + GW results, respectively (d)–(f) Zoom-in views of band structures around the $E_F$ corresponding to (a)–(c). Sizes of yellow, blue, and green dots give weights of $\text{Yb-4}f, \text{Yb-5}d$, and $\text{B-2}p$ orbitals in the band structures. (g)–(i) Schematic band structures corresponding to the results of (a)–(c).

Figure 4

(a) Weights of Yb-$4f$ (red dots) and B-$2p$ (blue dots) orbitals at the bottom of the conduction band at the $X$ point as a function of $U$ in ${\text{YbB}}_6$. Lines are drawn through the computed points to guide the eye. (b) Corresponding direct band gaps at the $X$ point.