**1. Introduction**

The topology of electronic band structures in crystals can have profound consequences at their interfaces with materials having a trivial topology, resulting in novel electronic states in reduced dimension [1, 2]. A striking example is the realization of gapless metallic states with a Dirac band

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

structure at the interface between two insulators, if one is a topological insulator (TI). For a 2D crystal, these are dissipationless 1D spin-helical edge states, an electronic phase known as the quantum spin Hall state [3–5]. For a 3D crystal, the metallic phase develops as 2D surface/interface states [6, 7] with a spin-helical band structure that limits their scattering by non-magnetic disorder [8]. This novel type of insulators was discovered in materials having a strong spin-orbit coupling (broken rotational symmetry) and a Hamiltonian invariant by time-reversal symmetry (TRS), belonging to the so-called ℤ2 topological class [7, 9]. The spin-orbit interaction can lead to a band inversion that gives rise to novel quasi-particles having a Dirac-like band structure with spin-momentum locking, at the interface with a trivial insulator (i.e., with non-inverted bands) where the energy gap must close (see **Figure 1a**). A remarkable property is that such metallic states always exist (no gap), even in the presence of perturbations, as long as the main symmetries are preserved. Thus, regardless the strength of a non-magnetic disorder, the condition for strong (Anderson) localization is never fulfilled. Still, an electronic gap (an important property for the operation of classical electronic devices) can be created, for instance by using a perturbation breaking TRS or by coupling topological states at opposite interfaces. The non-trivial nature of such gapped topological states further gives rise to novel exotic phases with a lower dimension (Majorana bound states [10] and quantum anomalous Hall state [11]) of particular interest for the quantum manipulation of coherent states weakly coupled to their environment.

orbital effect). Another important discovery was that of 3D topological insulators [14–16], which realize a 2D electron gas of massless Dirac fermions (different from those in graphene due to their helical spin texture) at the surfaces/interfaces of a *single* crystalline film (**Figure 1b**), whereas the creation of a charge-accumulation 2D gas of massive quasi-particles in conventional semiconductors often requires a more complex stacking of epitaxial layers of materials with different energy gaps and electronic-band offsets. In practice, the control of surface electronic states in a 3D topological insulator also requires that of band bending at interfaces (due to charge transfer from the environment) and a significant limitation to investigate topological surface-state transport came from that most materials are not true bulk insulators. Still, an important difference with massive quasi-particles in semiconductors, or Dirac fermions in graphene, is the reduced scattering rate by disorder due to the spin-helical texture of surface Dirac fermions [8, 17]. This favors the quasi-ballistic transport of such quasi-particles, even if they directly coexist with a large density of scattering centers, and it gives a new possibility to study such a transport regime [18] that is otherwise only accessible for massive quasi-particles in rather high-mobility AlGaAs 2DEGs (for which the 2DEG is spatially separated from Coulomb scatterers). Besides, the spin-momentum locking property of the Dirac cone favors the efficient spin-charge conversion in spintronic devices [19–25], still with some potential for improvement due to the enhanced transport length [17], if interface scattering could be further reduced. Contrary to the case of 2D TIs, with only rare examples found to date, a number of 3D TI materials were discovered, both by theory and experiments, often using ab initio calculations or electron spectroscopy techniques to directly unveil the Dirac band structure of topological surface states (TSS). This offered an important playground to investigate [26, 27], but it is also a source of confusion due to the diversity of these materials and the complexity of some of them. In particular, their study by charge transport measurements proved to be difficult because many materials identified as topological insulators are also semiconductors with a rather small gap, often with a large finite bulk conductivity due to disorder (electrical doping) and a strong electronic polarizability [28]. The search for large-gap 3D topological insulators remains challenging. Indeed, beyond the need for specific symmetries of the crystal structure, it is fundamental to search for materials with a band inversion. With conventional semiconductors, this inversion is usually induced by a strong intrinsic spin-orbit coupling. This is typical for materials with large-Z elements, which also have a rather small bulk energy gap induced by Coulomb interactions. This combination is therefore favorable to realize a band inversion, and since it results from opposite contributions, most Z2 topological insulators tend to have a rather small gap that barely exceeds 300 meV in most cases. Natural point defects, such as vacancies (which are thermodynamically stable and therefore always exist in crystals), can be massively present (small activation energy) and they often act as donors or acceptors, thus easily leading to metallic-like bulk properties typical of degenerate semiconductors (sometimes approaching the dirty-metal limit). Among these semiconductors, Bi-based chalcogenides play an important role for that the continuous tuning of their chemical composition, between different stoichiometric compounds, can change a trivial insulator (no band inversion) into a topological insulator (band inversion), with a varying bulk gap and a relative change of the topological Dirac degeneracy point with respect to the bulk band structure

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29

[29]. Binary 3D topological insulators, such as Bi2

Se3

and Bi2

Te3

, usually have a large residual

The importance of topological insulators also comes from that they represent a promising alternative to conventional semi-conducting hetero-structures with, in principle, a reduced complexity on the materials' side and an increased stability of their electronic phases (over a large range of physical parameters: temperature, magnetic field, chemical potential, and disorder/inhomogeneities). The most striking example came with the discovery of the quantum spin Hall edge states in a 2D topological insulator, first evidenced with HgCdTe quantum wells [12]. This novel electronic phase shares some similarities with the magnetic-field induced integer quantum Hall state in high-mobility low-density 2D electron gases (2DEGs) of massive quasi-particles but was predicted in zero field [3, 4, 13] (no magnetic-field induced

**Figure 1.** (a) Evolution of the band gap between a trivial insulator and a 3D topological insulator (with an inverted bulk band structure). At the interface, the gap closes and metallic states with a linear dispersion form. (b) These interface/ surface states are 2D spin-helical Dirac fermions, with opposite spin helicity for opposite surfaces (the simplest case of a single Dirac cone in the center of the Brillouin zone is sketched).

orbital effect). Another important discovery was that of 3D topological insulators [14–16], which realize a 2D electron gas of massless Dirac fermions (different from those in graphene due to their helical spin texture) at the surfaces/interfaces of a *single* crystalline film (**Figure 1b**), whereas the creation of a charge-accumulation 2D gas of massive quasi-particles in conventional semiconductors often requires a more complex stacking of epitaxial layers of materials with different energy gaps and electronic-band offsets. In practice, the control of surface electronic states in a 3D topological insulator also requires that of band bending at interfaces (due to charge transfer from the environment) and a significant limitation to investigate topological surface-state transport came from that most materials are not true bulk insulators. Still, an important difference with massive quasi-particles in semiconductors, or Dirac fermions in graphene, is the reduced scattering rate by disorder due to the spin-helical texture of surface Dirac fermions [8, 17]. This favors the quasi-ballistic transport of such quasi-particles, even if they directly coexist with a large density of scattering centers, and it gives a new possibility to study such a transport regime [18] that is otherwise only accessible for massive quasi-particles in rather high-mobility AlGaAs 2DEGs (for which the 2DEG is spatially separated from Coulomb scatterers). Besides, the spin-momentum locking property of the Dirac cone favors the efficient spin-charge conversion in spintronic devices [19–25], still with some potential for improvement due to the enhanced transport length [17], if interface scattering could be further reduced.

structure at the interface between two insulators, if one is a topological insulator (TI). For a 2D crystal, these are dissipationless 1D spin-helical edge states, an electronic phase known as the quantum spin Hall state [3–5]. For a 3D crystal, the metallic phase develops as 2D surface/interface states [6, 7] with a spin-helical band structure that limits their scattering by non-magnetic disorder [8]. This novel type of insulators was discovered in materials having a strong spin-orbit coupling (broken rotational symmetry) and a Hamiltonian invariant by time-reversal symmetry (TRS), belonging to the so-called ℤ2 topological class [7, 9]. The spin-orbit interaction can lead to a band inversion that gives rise to novel quasi-particles having a Dirac-like band structure with spin-momentum locking, at the interface with a trivial insulator (i.e., with non-inverted bands) where the energy gap must close (see **Figure 1a**). A remarkable property is that such metallic states always exist (no gap), even in the presence of perturbations, as long as the main symmetries are preserved. Thus, regardless the strength of a non-magnetic disorder, the condition for strong (Anderson) localization is never fulfilled. Still, an electronic gap (an important property for the operation of classical electronic devices) can be created, for instance by using a perturbation breaking TRS or by coupling topological states at opposite interfaces. The non-trivial nature of such gapped topological states further gives rise to novel exotic phases with a lower dimension (Majorana bound states [10] and quantum anomalous Hall state [11]) of particular interest

28 Heterojunctions and Nanostructures

for the quantum manipulation of coherent states weakly coupled to their environment.

The importance of topological insulators also comes from that they represent a promising alternative to conventional semi-conducting hetero-structures with, in principle, a reduced complexity on the materials' side and an increased stability of their electronic phases (over a large range of physical parameters: temperature, magnetic field, chemical potential, and disorder/inhomogeneities). The most striking example came with the discovery of the quantum spin Hall edge states in a 2D topological insulator, first evidenced with HgCdTe quantum wells [12]. This novel electronic phase shares some similarities with the magnetic-field induced integer quantum Hall state in high-mobility low-density 2D electron gases (2DEGs) of massive quasi-particles but was predicted in zero field [3, 4, 13] (no magnetic-field induced

**Figure 1.** (a) Evolution of the band gap between a trivial insulator and a 3D topological insulator (with an inverted bulk band structure). At the interface, the gap closes and metallic states with a linear dispersion form. (b) These interface/ surface states are 2D spin-helical Dirac fermions, with opposite spin helicity for opposite surfaces (the simplest case of a

single Dirac cone in the center of the Brillouin zone is sketched).

Contrary to the case of 2D TIs, with only rare examples found to date, a number of 3D TI materials were discovered, both by theory and experiments, often using ab initio calculations or electron spectroscopy techniques to directly unveil the Dirac band structure of topological surface states (TSS). This offered an important playground to investigate [26, 27], but it is also a source of confusion due to the diversity of these materials and the complexity of some of them. In particular, their study by charge transport measurements proved to be difficult because many materials identified as topological insulators are also semiconductors with a rather small gap, often with a large finite bulk conductivity due to disorder (electrical doping) and a strong electronic polarizability [28]. The search for large-gap 3D topological insulators remains challenging. Indeed, beyond the need for specific symmetries of the crystal structure, it is fundamental to search for materials with a band inversion. With conventional semiconductors, this inversion is usually induced by a strong intrinsic spin-orbit coupling. This is typical for materials with large-Z elements, which also have a rather small bulk energy gap induced by Coulomb interactions. This combination is therefore favorable to realize a band inversion, and since it results from opposite contributions, most Z2 topological insulators tend to have a rather small gap that barely exceeds 300 meV in most cases. Natural point defects, such as vacancies (which are thermodynamically stable and therefore always exist in crystals), can be massively present (small activation energy) and they often act as donors or acceptors, thus easily leading to metallic-like bulk properties typical of degenerate semiconductors (sometimes approaching the dirty-metal limit). Among these semiconductors, Bi-based chalcogenides play an important role for that the continuous tuning of their chemical composition, between different stoichiometric compounds, can change a trivial insulator (no band inversion) into a topological insulator (band inversion), with a varying bulk gap and a relative change of the topological Dirac degeneracy point with respect to the bulk band structure [29]. Binary 3D topological insulators, such as Bi2 Se3 and Bi2 Te3 , usually have a large residual bulk conductivity (despite a very small bulk-carrier mobility), so that the total conductance of wide and/or thick nanostructures is dominated by bulk transport. This contribution is slightly reduced in Bi2 Te3 , but only high-energy surface quasi-particles can be studied because the Dirac point lies deep into the valence band, whereas the Fermi level is pinned near or above the top of the valence band. In Bi2 Se3 , the Dirac point lies within the bulk band gap, but the Fermi level is high above the bottom of the conduction band (due to too many Se vacancies), so that it can only be modified over a small energy range by an electrical gate (efficient electrostatic screening), unless the nanostructure is ultra thin. Ternary compounds gave the most advanced results to optimize the surface-to-bulk ratio to the conductance and this approach was successfully used to investigate surface-state transport in the quantum Hall regime at low magnetic fields (close to the Dirac point) [30]. It remains, however, difficult to prepare nanostructures (thin films, nanoribbons, and nanowires) of ternary or even quaternary materials with optimized compositions [31]. Novel heterostructures, such as core-shell nanowires, could thus be important to develop functional devices based on 3D topological insulators, not only to achieve dominant surface transport and use the spin-momentum locking property of 2D topological surface states, but also to realize novel low-dimensional quantum devices at the sub-micron scale despite disorder, due to anisotropic scattering, which are different from "conventional" mesoscopic conductors with either massive quasi-particles in semiconductors or Dirac fermions in graphene (both with quantum transport properties controlled by largeangle scattering).

**2. Transport properties of 2D topological surface states**

To understand the physics of 3DTI quantum wires, it is first necessary to take a closer look at the scattering of 2D spin-helical Dirac fermions by a non-magnetic disorder. In this case, backscattering is not suppressed since it remains possible by successive small-angle scattering events, over a scale given by the transport length, ltr. However, the spin texture favors forward scattering, so that the transport length of topological surface state is largely enhanced with respect to the disorder correlation length. As a consequence, the condition for ballistic transport can already be realized in nanostructures with dimensions smaller than a micron, and

Due to their residual bulk doping, the study of surface-state transport in disordered 3D topological insulators is not straightforward. Considering the case of a highly-degenerate semiconductor,

topological surface states is about one order of magnitude larger than that of bulk states (enhancement due to the anisotropic scattering of spin-helical Dirac fermions) [17]. Taking band bending into account [36], typical values for the surface and bulk carrier densities are ns = 5·1012 cm−2 and nb = 5·1019 cm−3. This gives a value tc = 20 nm. Based on a realistic approximation (ignoring details of band bending due to a very short Thomas-Fermi screening length λTF ≲ 3 nm), this lower bound

 (which is likely be larger for smaller bulk carrier densities) clearly shows that the control of topological surface-state transport in disordered 3D TIs requires the use of thin nanostructures. These were successfully grown by different bottom-up methods, each technique having both advantages and disadvantages. Ultra-thin layers of high quality can be prepared by molecular beam epitaxy and are well adapted to tune the interface band structure by electrical fields, but films thicker than 20 nm tend to grow 3D and have more defects. High-quality single-crystalline nanostructures with large aspect ratios can be grown by catalyst-assisted molecular-beam epitaxy or chemical vapor deposition, as well as by catalyst-free vapor transport, but ultra-thin structures are seldom. Very narrow quantum wires can also be prepared by electrodeposition in nanomembranes, presently with some intrinsic limitations related to small diameter fluctuations in ultimate nanomembranes that give quantum confinement inhomogeneities. In all cases, individual nano-

In our work, we mostly investigated the charge transport properties of *single-crystalline atomic-*

*eted* shapes of different aspect ratios (nanoplatelets, nanoribbons, nanowires; **Figure 2b**) [37]. As explained below, this allowed us to study the physics of spin-helical Dirac fermions in 2D or in 1D, that is, without or with quantum confinement, respectively. Despite strong disorder, it was shown that momentum scattering is reduced in all cases, due to the spin texture (2D surface states) and quantum confinement (1D surface modes). This has important consequences for both spin transport and quantum coherent transport, as discussed in detail below.

 *nanostructures* grown by catalyst-free vapor transport (**Figure 2a**), with *fac-*

, with a bulk carrier density as high as 5·1019 cm−3 (which roughly corresponds to

, common to both surface and bulk states, the mobility of

Spin-Helical Dirac Fermions in 3D Topological Insulator Quantum Wires

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31

for which the surface

/Si substrates, appropri-

μs )/(nb μb ).

quantum devices with a simple geometry can be built from individual nanowires.

conductance becomes comparable to the bulk conductance is roughly given by tc = 2(ns

**2.1. Nanostructures of 3D topological insulators for surface-transport studies**

about 1% of Se vacancies, acting as double donors), the cross over thickness tc

Se3

structures stable in air can be randomly grown on or transferred onto SiOx

ate to create an electric field at the interface by applying a backgate voltage.

such as Bi2

for tc

*flat Bi<sup>2</sup> Se3*

 *and Bi<sup>2</sup>*

*Te3*

Se3

For a strong disorder typical for Bi2

Unique properties of spin-helical Dirac fermions in disordered 3D topological insulators also arise from their strong quantum confinement in narrow nanowires (quantum wires). The Dirac nature results in a large energy quantization, as compared to the confinement energy of massive quasi-particles, which further reduces their scattering by disorder and favors the quasiballistic transport of quasi-1D surface modes [18]. Thus, their quantized band structure gives some specific signatures of topological surface-state transport, unveiled by quantum corrections to the conductance [18, 32, 33], which can be easily distinguished from bulk transport (even for highly-degenerate semiconductors, such as Bi2 Se3 and Bi2 Te3 ). Their spin texture also contribute to reduce their coupling to the environment, so that decoherence due to electronic interactions is also further reduced. All in all, 3D topological insulator quantum wires offer new possibilities to investigate mesoscopic transport in the quasi-ballistic regime over a large range of parameters (dimensionality, aspect ratio, and disorder strength), whereas it can hardly be investigated otherwise. Rare studies based on high-mobility AlGaAs 2DEGs were limited to the near-clean limit [34, 35], with little possibilities to modify some important physical parameters, such as disorder or the kinetic energy, over a broad range. Disordered semiconducting or metallic nanowires are always diffusive conductors. Disordered semiconducting or metallic nanowires are always diffusive conductors. Most similar systems are actually carbon nanotubes, which are clean systems with ballistic-transport properties and very large confinement energies. Disordered 3D TI quantum wires represent an intermediate situation that corresponds to quasi-ballistic transport (due to anisotropic scattering rather than to a weak disorder) and their quantized surface states can be manipulated with rather small magnetic fields, due to larger diameters than for carbon nanotubes. As a consequence, the specific properties of quantized surface Dirac modes can be revealed by the study of different quantum corrections to the conductance (Aharonov-Bohm oscillations and non-universal conductance fluctuations), with good statistical information obtained from magneto-transport measurements below 15 T [18].
