Topological insulators

Recently, research interest has shifted from quantum size effects in conventional/classical semiconductors to new types of quantum materials (e.g., topological insulators [TIs], Weyl and Dirac semimetals) characterized by their nontrivial electronic topological order. Topological order is a classification of matter by the internal structure of its quantum ground state. The class of TI materials hereby represents the most relevant quantum materials for thermoelectricity. Conventional semiconductors are characterized by a bulk band structure that is constituted by a valence band and a conduction band with parabolic bands and an energy bandgap E _gap < 3 eV. TE materials for near-room-temperature applications have this type of band structure, typically with E _gap around 0.1 eV to 0.25 eV for high performance in the low- to mid-temperature regime. TIs also exhibit this conventional bulk band structure of a semiconductor, but feature additional topological surface states with a linear dispersion relation. As in graphene, this results in relativistic carrier transport with extremely high carrier mobility, µ, and a small effective carrier mass, m _eff, which, in the ideal case, equals zero. Hence, in a primitive picture, TIs are characterized by graphene-like states on the surface of a conventional semiconductor. TIs therefore provide two different transport channels, one of graphene-like surfaces and the other of the (conventional) semiconducting bulk inner part.

Ordinary crystals are three-dimensional (3D) objects. If the crystal exhibits topological surface states completely surrounding the outer surface in all facet directions, this is called a 3D TI, also classified as a strong TI. If only certain or high-symmetry facets exhibit TI properties, the material is called a weak TI or topological crystalline insulator (TCI).

Topological surface states are protected from backscattering between equal and opposite momentum by time-reversal symmetry.^{Reference Seo, Roushan, Beidenkopf, Hor, Cava and Yazdani6} Therefore, the TI properties are robust, and they not only are of theoretical interest, but also contribute to the materials’ performance optimization. If, for instance, a surface of a 3D TI is oxidized, the TI states move deeper into the crystal, rather than being annihilated. Breaking of the TI state can be triggered by interaction with a magnetic dipole.^{Reference Buchenau, Sergelius, Wiegand, Bäßler, Zierold, Shin, Rübhausen, Gooth and Nielsch7} To probe for TI transport properties, experiments at high magnetic fields and low temperatures are performed to separate the contribution from the semiconducting bulk inner part of the material from the oscillatory behavior of the quantum transport caused by the mobile TI carriers of the surface states. By analyzing several quantum transport phenomena, such as weak antilocalization, the Aharonov–Bohm effect, Schubnikov–de Haas oscillations (SdH),^{Reference Gooth, Hamdou, Dorn, Zierold and Nielsch8} in addition to Fermi-level tuning, information about m _eff and µ can be obtained and the linear dispersion relation proven.

The link between TI and thermoelectricity is that good TE materials are usually narrow bandgap semiconductors (E _gap < 0.3 eV) and are constituted by heavy elements that suppress phonon propagation. The heavy elements also result in strong spin–orbit coupling. This is a prerequisite for the formation of topological surface states since it adds an amount of energy to the energy levels. If this amount is higher than the distance between the levels that define the bandgap, the band structure becomes inverted, and the topologically insulating state occurs.

Some of the best TE materials are also strong 3D TIs, for example, Bi₂Te₃, Bi₂Se₃, Sb₂Te₃,^{Reference Zhang, Liu, Qi, Dai, Fang and Zhang9–Reference Xu, Xu and Zhu11} and their solid solutions. Their macroscopic TE-transport properties exhibit signatures of the superposition of the surface and the bulk-transport channels. This is most clearly seen by studying transport properties in one-dimensional and two-dimensional model systems such as nanowires and thin films of Bi₂Te₃, Bi₂Se₃, and Sb₂Te₃.

In the last several years, substantial efforts have been invested in the development of TE Bi₂Te₃ nanowires and thin films to study the influence of a reduced dimension on the TE performance. The full ZT characterization of Bi₂Te₃ nanowires revealed values of ZT in the range of 0.02^{Reference Zhou, Jin, Seol, Li and Shi12} up to 0.45,^{Reference Chen, Chen, Lin, Ho, Lee, Chen and Harutyunyan13} considerably lower than the corresponding bulk ZT values (typically ZT > 1)^{Reference Snyder and Toberer14} instead of higher as was predicted due to the beneficial effect of the reduced dimension.^{Reference Hicks and Dresselhaus5} This is not a general characteristic of nanowires. For instance, silicon nanowires exhibit ZT values between 0.6 and 1 at room temperature,^{Reference Hochbaum, Chen, Delgado, Liang, Garnett, Najarian, Majumdar and Yang15,Reference Boukai, Bunimovich, Tahir-Kheli, Yu, Goddard and Heath16} and perform therewith approximately one order of magnitude better than their bulk counterparts,^{Reference Bux, Blair, Gogna, Lee, Chen, Dresselhaus, Kaner and Fleurial17,Reference Schierning18} mostly due to increased scattering at the rough nanowire boundaries.^{Reference Liu and Chen19} In InAs nanowires, a strong decrease in κ originates from increased phonon scattering at stacking faults between wurtzite and zincblende segments, accompanied by a reduction of σ attributed to a reduction of µ as a consequence of surface charges.^{Reference Karg, Troncale, Drechsler, Mensch, Das Kanungo, Schmid, Schmidt, Gignac, Riel and Gotsmann20,Reference Zhou, Moore, Bolinsson, Persson, Fröberg, Pettes, Kong, Rabenberg, Caroff, Stewart, Mingo, Dick, Samuelson, Linke and Shi21}

In contrast, Bi₂Te₃, Bi₂Se₃, and Sb₂Te₃ exhibit a more metallic character with increasing surface-to-volume (s/V) ratio, as shown in Figure 1.^{Reference Gooth, Gluschke, Zierold, Leijnse, Linke and Nielsch22–Reference Hinsche, Zastrow, Gooth, Pudewill, Zierold, Rittweger, Rauch, Henk, Nielsch and Mertig24} With higher s/V, a decrease in S, an increase in σ (not shown in Figure 1), and hence a decrease in ZT was evidenced. This can be interpreted to be a consequence of the surface states contributing charge carriers that exhibit superior µ carried under the regime of the linear dispersion relation. By analyzing Shubnikov-de Haas oscillations in Bi₂Te₃ nanowires,^{Reference Hamdou, Gooth, Dorn, Pippel and Nielsch23} it was found that the surface states have m _c = 0.04 m ₀ (the free electron mass), and µ = 21,000 cm² V^–1s^–1. Shubnikov-de Haas oscillations are oscillations in the electrical resistivity of a material that occur at low temperatures in the presence of intense magnetic fields. The Shubnikov–de Haas effect is a macroscopic manifestation of the inherent quantum mechanical nature of matter. The transport properties can be described by a phenomenological model that uses two in-parallel connected transport channels for surface and bulk and also by ab initio calculations.^{Reference Hinsche, Zastrow, Gooth, Pudewill, Zierold, Rittweger, Rauch, Henk, Nielsch and Mertig24}

Figure 1.

Compilation of thermoelectric (TE) transport data of the 3D topological insulator (TI) TE materials Bi₂Te₃, Sb₂Te₃, and Bi₂Se₃ obtained using 1D and 2D model systems (nanowires and thin films, respectively) with varying surface-to-volume (s/V) ratios. (a) ZT data of Bi₂Te₃ nanowires and thin films show a clear trend: The smaller s/V, the higher is ZT. Inset: Sketch of the band structure of a TI. It is constituted by a bulk valence (green) and conduction (blue) band that exhibits a parabolic dispersion, and features additional topological surface states that exhibit a linear dispersion. (b) Absolute Seebeck coefficients (S) of Bi₂Se₃ nanowires and normalized Seebeck coefficients of Sb₂Te₃ thin films show more metallic behavior with higher s/V.^{Reference Gooth, Gluschke, Zierold, Leijnse, Linke and Nielsch22–Reference Hinsche, Zastrow, Gooth, Pudewill, Zierold, Rittweger, Rauch, Henk, Nielsch and Mertig24} Note: S _tot, measured total Seebeck coefficient with contributions from the bulk and the surface states; S _{160 nm}, Seebeck coefficient measured using a thin film with 160-nm film thickness.

To date, the impact of topological surface states on nanograined bulk materials has not been fully understood. But some trends can be extracted. A breakthrough in TE materials development was the use of the ball-milling and hot-pressing method to achieve nanograined bulk with significantly improved ZT, compared to microstructured bulk, demonstrated for Bi_2–xSb_xTe₃-alloys.^{Reference Poudel, Hao, Ma, Lan, Minnich, Yu, Yan, Wang, Muto, Vashaee, Chen, Liu, Dresselhaus, Chen and Ren25} Today, this approach has been successfully applied to all known TE materials, systematically resulting in improved ZT values. It is remarkable that in Bi_2–xSb_xTe₃ nanograined bulk, σ is higher than that for the microstructured bulk counterpart (see Figure 2a), although often only the lower κ is discussed.

Figure 2.

Signatures of topological insulator properties in nanograined bulk thermoelectric BiSbTe alloy and Bi₂Se₃. (a) Nanograined bulk of ball-milled and hot-pressed Bi_2–xSb_xTe₃-alloy ingot resulted in an improved ZT value due to the often discussed reduction of the lattice thermal conductivity (not shown here), but also an enhancement in the electrical conductivity σ, accompanied by a reduction of the Seebeck coefficient S, despite the small grain sizes.^{Reference Poudel, Hao, Ma, Lan, Minnich, Yu, Yan, Wang, Muto, Vashaee, Chen, Liu, Dresselhaus, Chen and Ren25} (b) Grain-size dependence of the electrical conductivity σ and the Seebeck coefficient S of Bi₂Se₃ nanograined bulk also demonstrates more metallic behavior with smaller grain sizes. The analysis reveals exceptionally high values of the charge-carrier mobility μ for average grain sizes below 80 nm.^{Reference Sun, Li, Qin, Li, Zou, Xin, Ren, Zhang, Li and Li26}

Sun and coworkers^{Reference Sun, Li, Qin, Li, Zou, Xin, Ren, Zhang, Li and Li26} found an indication of the validity of interpreting this behavior as due to topological surface states. They analyzed the grain-size dependence of TE transport data in nanograined Bi₂Se₃. Due to its relatively high bandgap of 0.3 eV with an almost ideal linear electronic dispersion, called a Dirac cone, which results in exceptionally robust TI properties well above room temperature, Bi₂Se₃ is a perfect model TI.^{Reference Zhang, Liu, Qi, Dai, Fang and Zhang9,Reference Xia, Qian, Hsieh, Wray, Pal, Lin, Bansil, Grauer, Hor, Cava and Hasan27} As shown in Figure 2b, below an average grain size of 80 nm, S is lower and σ higher than in samples exhibiting an average grain size above this threshold. Most remarkable is the grain size-dependence of µ, with values of 400–600 cm² V^–1 s^–1 evidenced in nanograined Bi₂Se₃ for grain sizes below 80 nm, compared to around 12–15 cm² V^–1 s^–1 for samples with larger grain sizes. A higher σ of grain boundaries than that of the grain interiors is sometimes connected to a higher concentration of charge carriers. A higher µ of grain-boundary charge carriers is absolutely unique and can only be explained by the exceptional robustness of the surface states of strong 3D TIs that even survive the strongly distorted grain-boundary environment of a nanograined bulk.

TCIs are also found in the Pb_1–xSn_xTe²⁸ (Figure 3) and Pb_1–xSn_xSe²⁹ material families known for their massive Dirac states in three dimensions and exceptional TE properties^{Reference Heremans, Jovovic, Toberer, Saramat, Kurosaki, Charoenphakdee, Yamanaka and Snyder30,Reference Pei, Gibbs, Gloskovskii, Balke, Zeier and Snyder31} with record ZT values of 2.6, as in SnSe.^{Reference Zhao, Lo, Zhang, Sun, Tan, Uher, Wolverton, Dravid and Kanatzidis32} These materials can exhibit metallic surface states on high-symmetry crystal surfaces such as [111], [110], and [100] with an even number of Dirac cones.

Figure 3.

Nontrivial electronic topology in the Pb_1–xSn_xTe materials family. For x = 0.4, Pb_0.6Sn_0.4Te is a Dirac semimetal. At the composition x _c, a trivial-to-nontrivial topological quantum phase transition between the topological crystalline insulator (TCI) phase and trivial phases occurs. This is also consistent with the band inversion taking place at x _c, visualized by the coloring of the valence band (VB, green in the trivial phases, yellow in the TCI phase) and conduction band (CB, yellow in the trivial phases, green in the TCI phase). Reprinted with permission from Reference Tanaka, Sato, Nakayama, Souma, Takahashi, Ren, Novak, Segawa and AndoReference 28. © 2013 Elsevier. Note: SS, surface state.

Dirac and Weyl semimetals

Dirac^{Reference Li, He, Lu, Zhang, Liu, Ma, Fan, Shen and Wang33–Reference Xiong, Kushwaha, Liang, Krizan, Hirschberger, Wang, Cava and Ong35} and Weyl^{Reference Shekhar, Nayak, Sun, Schmidt, Nicklas, Leermakers, Zeitler, Skourski, Wosnitza, Liu, Chen, Schnelle, Borrmann, Grin, Felser and Yan36–Reference Niemann, Gooth, Wu, Bäßler, Sergelius, Hühne, Rellinghaus, Shekhar, Süß, Schmidt, Felser, Yan and Nielsch39} semimetals, for instance, Bi_1–xSb_x,^{Reference Shin, Lee, Sasaki, Jeong, Weickert, Betts, Kim, Kim and Kim40,Reference Kim, Kim, Wang, Sasaki, Satoh, Ohnishi, Kitaura, Yang and Li41} TaAs,^{Reference Peng, Zhang, Shao, Lu, Zhang and Zhu42} NbP,^{Reference Gooth, Niemann, Meng, Grushin, Landsteiner, Gotsmann, Menges, Schmidt, Shekhar, Süß, Hühne, Rellinghaus, Felser, Yan and Nielsch43,Reference Stockert, Dos Reis, Ajeesh, Watzman, Schmidt, Shekhar, Heremans, Felser, Baenitz and Nicklas44} or Cd₃As₂,^{Reference Hosseini, Yavarishad, Alward, Kouklin and Gajdardziska-Josifovska45–Reference Zhou, Zhang, Zhang, Xiu and Yang48} are two recently discovered classes of 3D topological materials in which conduction and valence bands touch linearly close to the Fermi energy. In Dirac semimetals, both time-reversal and inversion symmetries are preserved, and the Dirac cone is degenerated (Figure 4a). Weyl states emerge when one of these symmetries is broken. The discovery of Weyl semimetals has revitalized interest in Weyl fermions that were first discovered in high-energy physics experiments. The investigation of the TE transport properties of Weyl semimetals opens the door for an advanced understanding of these exotic states of matter^{Reference Jia, Li, Li, Shi, Liao, Yu and Wu46} in which even Ohm’s Law is fundamentally violated.^{Reference Shin, Lee, Sasaki, Jeong, Weickert, Betts, Kim, Kim and Kim40}

Figure 4.

Dirac and Weyl semimetals. (a) Sketch of the band structure of a Dirac and (b) of a Weyl semimetal. The cones in a Dirac semimetal are degenerated in chirality χ (direction of the charge carrier’s spin relative to its momentum). The Weyl semimetal hosts separated pairs of opposite chirality χ = –1 and χ = +1. E is the energy, and k _x, k _y are the momentum in the x and y directions, respectively. (c–e) show the real space scheme of the charge-carrier flow (left panels) and the associated band structure occupation (right panels). (c) Usually, the chirality is conserved. Consequently, net current flow is suppressed. However, quantum mechanical fluctuations can break the conservation law of chirality under (d) parallel magnetic B and electric fields E_el (chiral anomaly) and (e) parallel magnetic field B and temperature gradient ΔT (axial-gravitational anomaly). Under these circumstances, electrical and thermal current flow is allowed and can be tuned by the applied magnetic field.

As sketched in Figure 4b, the band touching point splits in momentum space into a pair of separated nodes (Weyl nodes) with opposite chirality, which refers to the handedness of the spin of electrons with respect to linear momentum. The conservation law of chirality dictates that the number of left-handed and right-handed electrons must always be symmetric, in other words, more electrons of one chirality than of the other cannot exist (Figure 4c). This symmetry has important consequences for the TE performance of Dirac and Weyl semimetals, because current flow is essentially the difference between the amount of left and right moving electrons of opposite chirality. Consequently, the net electrical current flow can be strongly suppressed due to the chiral conservation law. The thermopower, arising from the balance between the thermal diffusion current from hot to cold and its counter current from the accumulated charges at the cold side, is also strongly suppressed. Moreover, in both Dirac and Weyl semimetals, the energy spectrum is gapless, which results in ambipolar transport coefficients^{Reference Lundgren, Laurell and Fiete49} such as a small S and enhanced κ_el. Therefore, Dirac and Weyl semimetals are naturally poor TE materials.

However, when a magnetic field is applied in parallel to an electric field, the chiral symmetry can be violated at the quantum level.^{Reference Xiong, Kushwaha, Liang, Krizan, Hirschberger, Wang, Cava and Ong35,Reference Lundgren, Laurell and Fiete49–Reference Hirschberger, Kushwaha, Wang, Gibson, Liang, Belvin, Bernevig, Cava and Ong51} Applying electric and magnetic fields simultaneously to a Weyl semimetal leads to chiral charge pumping between the two branches of chiral energy levels (Landau levels), with a flow rate that is proportional to the product of the parallel magnetic and electric field components (Figure 4d). Likewise, a temperature gradient parallel to the magnetic field leads to an additional contribution that breaks the chiral symmetry (Figure 4e), originating from the so-called axial-gravitational anomaly.^{Reference Gooth, Niemann, Meng, Grushin, Landsteiner, Gotsmann, Menges, Schmidt, Shekhar, Süß, Hühne, Rellinghaus, Felser, Yan and Nielsch43,Reference Landsteiner, Megías and Pena-Benitez52,Reference Lucas, Davison and Sachdev53} The term “gravitational” refers to the fact that this quantum anomaly was originally thought to only occur in strong gravitational fields.^{Reference Witten54} For all electronic TE transport coefficients, these quantum anomalies lead to an additional topological current in parallel to the bias aligned magnetic fields,^{Reference Lundgren, Laurell and Fiete49,Reference Lucas, Davison and Sachdev53} increasing S, σ, and κ_el of a Weyl semimetal with the square of the increasing magnetic field. Consequently, the TE performance of Dirac and Weyl semimetals may be strongly enhanced when magnetic fields are present. While the existence of the chiral^{Reference Xiong, Kushwaha, Liang, Krizan, Hirschberger, Wang, Cava and Ong35,Reference Lundgren, Laurell and Fiete49–Reference Hirschberger, Kushwaha, Wang, Gibson, Liang, Belvin, Bernevig, Cava and Ong51} and the axial-gravitational anomaly^{Reference Gooth, Niemann, Meng, Grushin, Landsteiner, Gotsmann, Menges, Schmidt, Shekhar, Süß, Hühne, Rellinghaus, Felser, Yan and Nielsch43} have recently been experimentally confirmed, a full ZT characterization of a Weyl semimetal in a magnetic field is still missing to date.

In addition to the anomalous longitudinal transport contributions, it has been found that the linear dispersion relations in Dirac and Weyl semimetals gives rise to a large perpendicular TE voltage, the Nernst signal,^{Reference Liang, Gibson, Xiong, Hirschberger, Koduvayur, Cava and Ong29,Reference Sharma, Goswami and Tewari55–Reference Ferreiros, Zyuzin and Bardarson57} which can exceed their thermopower by two orders of magnitude and is significantly larger than the thermopower of traditional TE materials. Such multiterminal TE devices are of special interest, because the heat reservoirs become spatially separated from the electrical circuitry, while at the same time performing at high efficiencies. In Weyl semimetals with tilted cones, so-called Weyl type II semimetals,^{Reference Ferreiros, Zyuzin and Bardarson57} the Berry curvature, a local gauge field associated with the Berry phase of the topological band structure, is predicted to cause an anomalous Nernst effect even in the absence of a magnetic field, providing a platform for highly efficient TE energy converters. The Berry phase occurs in a quantum mechanical system during slow (adiabatic) traversing of a closed path in momentum space. The system does not return to its initial state, but its wave function acquires a phase factor, namely the Berry phase.