Ab initio thermoelectric calculations of ring-shaped bands in two-dimensional ${\mathrm{Bi}}_2{\mathrm{Te}}_3,{\mathrm{Bi}}_2{\mathrm{Se}}_3$, and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$: Comparison of scattering approximations

Materials with ring-shaped electronic bands are promising thermoelectric (TE) candidates since their unusual dispersion shape is predicted to give large power factors. In this paper, we use density functional theory to investigate single and double quintuple-layer ${\mathrm{Bi}}_2{\mathrm{Te}}_3,{\mathrm{Bi}}_2{\mathrm{Se}}_3$, and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ and to compare the TE properties using three scattering approximations: constant relaxation time, constant mean free path, and scattering rates proportional to the density of states (the so-called DOS model). Focus is placed on elucidating how these particular dispersion shapes influence the TE characteristics and on understanding how each scattering model impacts TE transport. The single quintuple-layer materials possess two ring-shaped valence-band maxima that provide an abrupt increase in conducting channels, which benefits the power factor. Below the band edge a ring-shaped minimum is found to further enhance TE performance within the DOS model due to a sharp drop in the DOS and, thus, scattering. An analytic “octic” dispersion model, designed to capture the observed characteristics of the band structure, is introduced and shown to qualitatively reproduce the first-principles results. The double quintuple-layer materials display notably worse TE properties since their dispersions are significantly modified compared to the single quintuple-layer case and lose much of the ring-shaped character. Our analysis shows that the benefits of ring-shaped bands are sensitive to the alignment of the two ring maxima and to the degree of ring anisotropy. Moreover, the predicted TE performance can vary significantly depending on the choice of scattering approximation, which could benefit from further study to assess the accuracy of these simple scattering models.

Figure 1

Single quintuple-layer ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Electron dispersion along high-symmetry points. (b) Energy contour plot of the valence band. (c) Distribution of modes $M\left(E\right)$, (d) average velocity $V_{\lambda }\left(E\right)$, and (e) density of states $D\left(E\right)$ versus energy for the valence states. Shaded regions indicate the energies of the rings and the moat feature.

Figure 2

Analytic octic band model. (a) Electron dispersion $\epsilon \left(k\right)$ versus $k$. (b) Distribution of modes $M\left(E\right)$, (c) average velocity, $V_{\lambda }\left(E\right)$, and (d) density of states $D\left(E\right)$ versus energy relative to the valence-band edge. To illustrate the shape of the distributions, we used ${\epsilon }_0,=0.15\mathrm{eV},k_a=2$, and $k_b=1$, which gives a moat energy of $-48\mathrm{meV}$.

Figure 3

Single quintuple-layer ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Transport distribution $\mathrm{\Sigma }\left(E\right)$ of the valence states versus energy. (b) PF and (c) figure-of-merit $ZT$ versus the Fermi level relative to the valence-band edge. Comparison of cmfp-, crt-, and DOS-scattering approximations. $T=300\mathrm{K}$.

Figure 4

Single quintuple-layer ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Electron dispersion along high-symmetry points. (b) Energy contour plot of the valence band. (c) Distribution of modes $M\left(E\right)$, (d) average velocity $V_{\lambda }\left(E\right)$, and (e) density of states $D\left(E\right)$ versus energy for the valence states. Shaded regions indicate the energies of the rings and the moat feature.

Figure 5

Single quintuple-layer ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Transport distribution $\mathrm{\Sigma }\left(E\right)$ of the valence states versus energy. (b) PF and (c) figure-of-merit $ZT$ versus Fermi level relative to the valence-band edge. Comparison of cmfp-, crt-, and DOS-scattering approximations. $T=300\mathrm{K}$.

Figure 6

Single quintuple-layer ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. (a) Electron dispersion along high-symmetry points. (b) Energy contour plot of the valence band. (c) Distribution of modes $M\left(E\right)$, (d) average velocity $V_{\lambda }\left(E\right)$, and (e) density of states $D\left(E\right)$ versus energy for the valence states. Shaded regions indicate the energies of the rings and the moat feature.

Figure 7

Single quintuple-layer ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. (a) Transport distribution $\mathrm{\Sigma }\left(E\right)$ of the valence states versus energy. (b) PF and (c) figure-of-merit $ZT$ versus Fermi level relative to the valence-band edge. Comparison of cmfp-, crt-, and DOS-scattering approximations. $T=300\mathrm{K}$.

Figure 8

Double quintuple-layer ${\mathrm{Bi}}_2{\mathrm{Te}}_3,{\mathrm{Bi}}_2{\mathrm{Se}}_3$, and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. (a)–(c) Electron dispersion along high-symmetry points. (d)–(f) Energy contour plot of the valence band.

Figure 9

Double quintuple-layer ${\mathrm{Bi}}_2{\mathrm{Te}}_3,{\mathrm{Bi}}_2{\mathrm{Se}}_3$, and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. (a) Distribution of modes $M\left(E\right)$, (b) average velocity $V_{\lambda }\left(E\right)$, and (c) density of states $D\left(E\right)$ versus energy for the valence states. (d) Transport distribution $\mathrm{\Sigma }\left(E\right)$ versus energy. (e) PF and (c) figure-of-merit $ZT$ versus Fermi level. $\mathrm{\Sigma }\left(E\right)$, PF, and $ZT$ are calculated using the DOS-scattering model. $T=300\mathrm{K}$.