Electronic structure of ${\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$

${\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ belongs to a class of complex chalcogenides that shows potential for superior thermoelectric performance. This compound forms in two distinct phases, $\alpha$ and $\beta$. The $\beta$ phase, which has several sites with mixed $\mathrm{K}∕\mathrm{Bi}$ occupancy, is a better thermoelectric. To understand the origin of this difference between the two phases we have carried out electronic structure calculations within ab initio density functional theory using the full potential linearized augmented plane wave (FLAPW) method. Both the local spin density approximation (LSDA) and the generalized gradient approximation (GGA) were used to treat the exchange and correlation potential. The spin-orbit interaction (SOI) was incorporated using a second variational procedure. The $\alpha$ phase is found to be a semiconductor with an indirect LSDA/GGA band gap of $0.38\mathrm{eV}∕0.46\mathrm{eV}$ compared to $0.76\mathrm{eV}$ for the observed direct optical gap. For the $\beta$ phase we have chosen two different ordered structures with extreme occupancies of K and Bi atoms at the “mixed sites.” The system is found to be a semimetal for both the ordered structures. To incorporate the effect of mixed occupancy we have chosen a $1\times 1\times 2$ supercell with an alternative $\mathrm{K}∕\mathrm{Bi}$ occupancy at the mixed sites. The system is a semiconductor with an indirect LSDA/GGA band gap of $0.32\mathrm{eV}∕0.41\mathrm{eV}$. We find that the mixed occupancy is crucial for the system to be a semiconductor because the Bi atoms at the mixed sites stabilize the $p$ orbitals of the nearest-neighbor Se atoms by lowering their energy. We also find a strong anisotropy in the effective mass near the conduction band minimum, with the smallest effective mass along the mixed $\mathrm{K}∕\mathrm{Bi}$ chains (parallel to the $c$ axis). This large anisotropy suggests that $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ has a great potential for an $n$-type thermoelectric.

Figure 1

Projection of the crystal structure of $\alpha \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ viewed down the $c$ axis ($z$ axis). ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, $\mathrm{Cd}{\mathrm{I}}_2$ and ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ building blocks in the structure are highlighted by the shaded areas.

Figure 2

Projection of the crystal structure of $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ viewed down the $c$ axis ($z$ axis). ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, $\mathrm{Cd}{\mathrm{I}}_2$ and NaCl building blocks in the structure are highlighted by the shaded areas.

Figure 3

Brillouin zone of $\alpha \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$.

Figure 4

Band structure of $\alpha \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ at $V_{\mathit{opt}}$, (a) LSDA without SOI, (b) LSDA with SOI, (c) GGA without SOI, and (d) GGA with SOI.

Figure 5

Brillouin zone of $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$.

Figure 6

Band structure of $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ configuration I with SOI. Orbital character of (a) Se9 $p$, (b) Se10 $p$, (c) Bi8 $p$, (d) Se4 $p$. Band structure of $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ configuration I with SOI for partially optimized calculations. Orbital character of (e) Se9 $p$, (f) Se4 $p$. The size of the circles is directly proportional to the strength of the orbital character.

Figure 7

Band structure of $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ configuration II with SOI. Orbital character of (a) Se4 $p$, (b) Bi9 $p$.

Figure 8

Band structure of the $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ supercell at $V_{\mathit{opt}}$ (a) LSDA without SOI, (b) LSDA with SOI, (c) GGA without SOI, and (d) GGA with SOI.

Figure 9

Charge density of $\beta \text{−}{\mathrm{K}}_2{\mathrm{Bi}}_8{\mathrm{Se}}_{13}$ supercell in planes parallel to the $c$ axis corresponding to (a),(b) CBM states and (c),(d) VBM states. The charge density is represented by closed lines from low charge density regions to high charge density regions in steps of $5\times {10}^{-4}\text{electrons}∕{\mathrm{\AA }}^3$.