Topological insulators for thermoelectrics: A perspective from beneath the surface

Abstract

Thermoelectric properties of topological insulators have traditionally been examined in the context of their metallic surface states. However, recent studies have begun to unveil intriguing thermoelectric effects emerging from the bulk electronic states of topological insulators, which have largely been overlooked in the past. Charge transport phenomena through the bulk are especially important under typical operating conditions of thermoelectric devices, necessitating a comprehensive review of both surface and bulk transport in topological insulators. Here, we review thermoelectric properties that are uniquely observed in topological insulators, placing special emphasis on unconventional phenomena emerging from bulk states. We demonstrate that unusual thermoelectric effects arising from bulk states, such as band inversion-driven warping, can be discerned in experiments through a simple analysis of the weighted mobility. We believe that there is still plenty to uncover within the bulk of topological insulators, yet our current understanding can already inspire new strategies for designing and discovering new materials for next-generation thermoelectrics.

Keywords: band inversion; band warping; thermoelectrics; topology.

Graphical abstract

Figure 1

The “knowledge iceberg” of topological insulators in the context of thermoelectric properties and design While surface states and emergent surface properties of topological insulators are well known, there is much to uncover beneath the surface, i.e., unique thermoelectric effects arising from bulk electronic states.

Figure 2

Topological surface states and thermoelectric properties (A) Electronic structure of ${\text{Bi}}_2{\text{Te}}_3$ from angle-resolved photoemission spectroscopy (ARPES). The bulk valence bands and bulk conduction bands are separated by a band gap, yet the gapless surface bands cross the bulk band gap. Image adapted from Chen et al. with permission. Copyright 2009, American Association for the Advancement of Science. (B) Schematic of the electronic structure of a topological insulator. Surface carriers in the bulk band gap, which are protected against backscattering by time-reversal symmetry, have a longer mean free path than surface carriers in the bulk bands, which are subjected to bulk-surface interactions. (C) Seebeck coefficient and electrical conductivity (inset) of ${\text{Bi}}_2{\text{Se}}_3$ nanowires with varying surface-to-volume ratios. Error bars are indicated for the Seebeck coefficient. Within approximately the same Fermi level range (shaded region), the conductivity may increase, but the Seebeck coefficient drops as the surface states, with opposite signs of thermopower, contribute more. Image adapted from Shin et al. with permission. Copyright 2016, Royal Society of Chemistry.

Figure 3

Bulk band structures of topological insulators (A) Illustration of the bulk band structure and corresponding isoenergy Fermi surface near the band edge, and how they evolve with band inversion strength in a topological insulator. With stronger band inversion, the bands become warped, and the Fermi surface becomes multi-valleyed. (B–E) The band structure and Fermi surface calculated using ab initio methods are shown for (B) ${\text{Bi}}_2{\text{Se}}_3$, (C) ${\text{Bi}}_2{\text{Te}}_3$, (D) SnTe, and (E) rock-salt SnSe.

Figure 4

Thermoelectric properties of topological insulators (A) Maximum attainable $zT$ (optimized with respect to doping level) at room temperature, obtained using first-principles Boltzmann transport calculations. The maximum $zT$ is plotted against the $M_0$ parameter (defined in Equation 1) which, for topological insulators (TIs), represents the band inversion strength. (B) Weighted mobility $\left({\mu }_w\right)$ and lattice thermal conductivity $\left({\kappa }_{\mathrm{L}}\right)$, where TIs are represented by the red coloring (negative $M_0$) and normal insulators (NIs) are represented by the blue coloring (positive $M_0$). (C) Maximum attainable $zT$ from charge transport modeling using $k·p$ perturbation theory. The background coloring indicates the band structure shape (warped or single valleyed) and topology (inverted or noninverted). (D) Seebeck effective mass $\left(m_{\mathrm{S}}^{\ast }\right)$ and conductivity effective mass $\left(m_{\mathrm{C}}^{\ast }\right)$, where the red and blue coloring represent TIs and NIs, respectively. Images adapted from Toriyama and Snyder with permission. Copyright 2024, Royal Society of Chemistry.

Figure 5

Effects of strain and pressure on thermoelectric properties of topological insulators (A) Molecular orbital diagram for the L-point in IV–VI rock-salt phases, illustrating the inversion of the $L_6^{-}$ and $L_6^{+}$ states. Image adapted from Toriyama et al. with permission. Copyright 2022, Royal Society of Chemistry. (B) Calculated $n$-type power factor of undeformed and compressed SnTe. Data adapted from Dai et al. (C) Calculated maximum attainable $zT$ of ${\text{Bi}}_2{\text{Te}}_3$ at different strains, corresponding to varying degrees of band inversion. Image adapted from Toriyama and Snyder with permission. Copyright 2024, Royal Society of Chemistry. (D) Measured weighted mobility of ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ at different pressures. Data adapted from Bai et al.

Figure 6

Alloying topological insulators for thermoelectrics (A and B) Weighted mobility (A) and lattice constant (B) of SnSe alloyed with ${\text{AgSbSe}}_2$. The weighted mobility is calculated from the reported conductivity and Seebeck coefficient. Data adapted from Wang et al. and Luo et al. (C) Schematic of how alloying can affect the band inversion strength. Alloying-induced compression (i.e., chemical pressure) leads to stronger band inversion in the parent topological insulator phase.

Figure 7

Topological phase transition and thermoelectric properties (A) Schematic of a topological transition in an alloy between a normal insulator (NI) and a topological insulator (TI). In alloys of IV–VI compounds, the transition occurs between the $L_6^{+}$ and $L_6^{-}$ bands, where the band gap closes at composition $x_c$. (B) Band gap of ${\text{Pb}}_{1-x}{\text{Sn}}_x$Te and ${\text{Pb}}_{1-x}{\text{Sn}}_x$Se measured using optical techniques at different temperatures. By increasing the temperature $\left(T\right)$, $x_c$ also increases. Data adapted from Dimmock et al., Strauss, and Wu et al. (C) Schematic of how external factors, such as temperature, can affect $x_c$ by perturbing the $L_6^{+}$ and $L_6^{-}$ bands. When the energy separation between the $L_6^{+}$ and $L_6^{-}$ bands in the NI is large, $x_c$ is closer to the TI composition. (D) Room temperature weighted mobility of ${\text{Pb}}_{1-x}{\text{Sn}}_x$Te at different compositions, calculated using the reported conductivity and Seebeck coefficient. The blue shading is to guide the eye. Data adapted from Witting et al., Orihashi et al., Ortiz et al., Kim et al., Pang et al., and Freer et al.