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

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

Dirac point lies deep into the valence band, whereas the Fermi level is pinned near or above

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 large-

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

Se3

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

and Bi2

Te3

). Their spin texture also contribute

Se3

, but only high-energy surface quasi-particles can be studied because the

, the Dirac point lies within the bulk band gap, but the

reduced in Bi2

angle scattering).

highly-degenerate semiconductors, such as Bi2

obtained from magneto-transport measurements below 15 T [18].

Te3

30 Heterojunctions and Nanostructures

the top of the valence band. In Bi2

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

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

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, such as Bi2 Se3 , with a bulk carrier density as high as 5·1019 cm−3 (which roughly corresponds to about 1% of Se vacancies, acting as double donors), the cross over thickness tc for which the surface conductance becomes comparable to the bulk conductance is roughly given by tc = 2(ns μs )/(nb μb ). For a strong disorder typical for Bi2 Se3 , common to both surface and bulk states, the mobility of 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 for tc (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 nanostructures stable in air can be randomly grown on or transferred onto SiOx /Si substrates, appropriate to create an electric field at the interface by applying a backgate voltage.

In our work, we mostly investigated the charge transport properties of *single-crystalline atomicflat Bi<sup>2</sup> Se3 and Bi<sup>2</sup> Te3 nanostructures* grown by catalyst-free vapor transport (**Figure 2a**), with *faceted* 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.

different Fermi energies), and the bulk carrier density is large enough to control charge transfer at interfaces and to induce upward band bending. Due to efficient Coulomb screening, an electrostatic gate is solely influencing the population of a single topological interface nearby, so that an independent tuning of both topological states is only achieved in dual-gate devices [38]. In a backgate geometry, the applied voltage is modifying the band bending at the bottom interface only, which results in the tuning of the electro-chemical potential of the interface topological states but not of the surface topological states since, in most cases, the electrical field is totally screened by bulk carriers nearby the bottom interface [17]. In all cases, the backgate-voltage dependence of the conductance nevertheless remains an efficient way to probe the interface topological states and study their properties by transport measurements. A striking example is

a low-enough bulk-carrier density to favor downward band bending at interfaces [36]. For a large interface carrier density, massless Dirac fermions coexist with a Rashba-type massive 2D electron gas. In this case, the back-gate voltage is changing the carrier concentration of different electronic states located at the bottom interface (shift of some peaks in the Fourier transform of Shubnikov de Haas oscillations, **Figure 3b**). The related band profile shown in **Figure 3c** is in

Further important information on the scattering of spin-helical Dirac fermions by a non-magnetic disorder can be obtained from transconductance G(V\_G) measurements [17]. It is indeed important to distinguish between two different scattering times and, accordingly, between

**1.** The quantum lifetime of quasi-particles can be inferred from Shubnikov-de Hass measure-

**2.** The transport time of carriers corresponds to the timescale for momentum backscattering, over a length called the transport length Ltr, related to the transport mobility, which determines the classical conductance. Whereas direct backscattering is forbidden by spinmomentum locking (giving dissipationless states in a 2D TI), it is allowed by multiple scat-

band bending, induced by a rather large bulk carrier density, at the bottom interface ("lower surface state", LSS). All microscopic transport parameters could be inferred from quantum magnetotransport (**Figure 4c**) and trans-conductance (**Figure 4d**) measurements [17]. Contrary to bulk

short-range disorder potential), the transport length of both upper and lower topological surface states was found to be much larger than the mean-free path, despite a limitation at about 200 nm probably due to the finite coupling with bulk carriers [larger values are expected for decoupled

topological states and/or in materials with a larger disorder correlation length, such as Bi2

tering processes in a disordered 3D TI, resulting in a finite transport length.

(thus to the microscopic disorder correlation length) and this momentum scattering time

nanoribbon (**Figure 4b**), the backgate voltage allowed us to modify the upward

~ Ltr is given by the disorder correlation length ξ (isotropic scattering due to a

nanoribbon patterned in a Hall-bar geometry, with

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

http://dx.doi.org/10.5772/intechopen.76152

33

between two successive scattering centers

Te3 ].

Se3

very good quantitative agreement with a triangular potential at this interface.

**2.3. Anisotropic scattering and charge transport length (=spin diffusion length)**

shown in **Figure 3**, for a rather thick Bi2

two different length scales (**Figure 4a**):

relates to the quantum mobility.

Se3

Using a thin Bi2

carriers, for which le

ments. It is associated with the mean-free path le

**Figure 2.** (a) Growth of Bi2 Se3 nanostructures on p++-Si/SiO2 substrates by vapor transport in a closed quartz ampoule. The sublimation of Bi2 Se3 crystals generates a flow of molecular species (BiSe and Se<sup>2</sup> ) toward the lower temperature area, where the recombine to form single crystalline nanostructures *in the plane* of the substrate. (b) Nanostructures with different aspect ratios (thin platelets, wide nanoribbons, and narrow nanowires) are distributed randomly onto amorphous SiO2 . After [37].

## **2.2. Band bending and interface charge transfers**

The electrical properties of 2D topological surface states can be investigated by quantum magnetotransport G(B) studies and transconductance G(Vg) measurements at low temperatures, which give access to all microscopic parameters (carrier density, mobility, and effective mass) for all carriers (topological interface states, topological surface states, and bulk states). A careful analysis of Shubnikov-de Haas oscillations due to the energy quantization of Landau levels in high magnetic fields gave detailed insights into the electronic band profiles in the thickness of wide Bi2 Se3 nanostructures (nanoplatelets and nanoribbons) [36]. Important results are summarized as follows:


For usual as-grown Bi2 Se3 nanostructures exposed to air, a typical band profile is defined from the contribution of three electronic populations (bulk carriers and two topological states with different Fermi energies), and the bulk carrier density is large enough to control charge transfer at interfaces and to induce upward band bending. Due to efficient Coulomb screening, an electrostatic gate is solely influencing the population of a single topological interface nearby, so that an independent tuning of both topological states is only achieved in dual-gate devices [38]. In a backgate geometry, the applied voltage is modifying the band bending at the bottom interface only, which results in the tuning of the electro-chemical potential of the interface topological states but not of the surface topological states since, in most cases, the electrical field is totally screened by bulk carriers nearby the bottom interface [17]. In all cases, the backgate-voltage dependence of the conductance nevertheless remains an efficient way to probe the interface topological states and study their properties by transport measurements. A striking example is shown in **Figure 3**, for a rather thick Bi2 Se3 nanoribbon patterned in a Hall-bar geometry, with a low-enough bulk-carrier density to favor downward band bending at interfaces [36]. For a large interface carrier density, massless Dirac fermions coexist with a Rashba-type massive 2D electron gas. In this case, the back-gate voltage is changing the carrier concentration of different electronic states located at the bottom interface (shift of some peaks in the Fourier transform of Shubnikov de Haas oscillations, **Figure 3b**). The related band profile shown in **Figure 3c** is in very good quantitative agreement with a triangular potential at this interface.

### **2.3. Anisotropic scattering and charge transport length (=spin diffusion length)**

**2.2. Band bending and interface charge transfers**

Se3

Se3

. After [37].

nanostructures on p++-Si/SiO2

**Figure 2.** (a) Growth of Bi2

32 Heterojunctions and Nanostructures

The sublimation of Bi2

amorphous SiO2

the conductance is never negligible.

Se3

For usual as-grown Bi2

The electrical properties of 2D topological surface states can be investigated by quantum magnetotransport G(B) studies and transconductance G(Vg) measurements at low temperatures, which give access to all microscopic parameters (carrier density, mobility, and effective mass) for all carriers (topological interface states, topological surface states, and bulk states). A careful analysis of Shubnikov-de Haas oscillations due to the energy quantization of Landau levels in high magnetic

crystals generates a flow of molecular species (BiSe and Se<sup>2</sup>

area, where the recombine to form single crystalline nanostructures *in the plane* of the substrate. (b) Nanostructures with different aspect ratios (thin platelets, wide nanoribbons, and narrow nanowires) are distributed randomly onto

> Se3 nano-

substrates by vapor transport in a closed quartz ampoule.

) toward the lower temperature

fields gave detailed insights into the electronic band profiles in the thickness of wide Bi2

structures (nanoplatelets and nanoribbons) [36]. Important results are summarized as follows:

**1.** Due to the large residual bulk density and the pinning of the Fermi energy in the conduction or valence band of materials with a large dielectric constant, the bulk contribution to

**2.** Besides, it usually controls the upward band bending at interfaces of the topological insulator (charge transfer of bulk carriers to empty gapless topological surface states).

**3.** Downward band bending can, however, exist if a massive charge transfer from another origin is also present (surface/interface disorder, surface adsorbents, and electrostatic gate). Such a situation is more likely to happen for materials with a small bulk carrier density.

**4.** If the surface/interface density is very large (typically for ns > 1013 cm−2), a Rashba chargeaccumulation 2DEG of massive quasi-particles coexist with Dirac topological states.

the contribution of three electronic populations (bulk carriers and two topological states with

nanostructures exposed to air, a typical band profile is defined from

Further important information on the scattering of spin-helical Dirac fermions by a non-magnetic disorder can be obtained from transconductance G(V\_G) measurements [17]. It is indeed important to distinguish between two different scattering times and, accordingly, between two different length scales (**Figure 4a**):


Using a thin Bi2 Se3 nanoribbon (**Figure 4b**), the backgate voltage allowed us to modify the upward band bending, induced by a rather large bulk carrier density, at the bottom interface ("lower surface state", LSS). All microscopic transport parameters could be inferred from quantum magnetotransport (**Figure 4c**) and trans-conductance (**Figure 4d**) measurements [17]. Contrary to bulk carriers, for which le ~ Ltr is given by the disorder correlation length ξ (isotropic scattering due to a short-range disorder potential), the transport length of both upper and lower topological surface states was found to be much larger than the mean-free path, despite a limitation at about 200 nm probably due to the finite coupling with bulk carriers [larger values are expected for decoupled topological states and/or in materials with a larger disorder correlation length, such as Bi2 Te3 ].

spin-momentum locking, which results in strongly anisotropic scattering, as observed by local probe microscopy. This means that forward scattering is favored for such quasi-particles

confirmed by trans-conductance measurements [17]. Such a situation never happens in other materials within which charge carriers directly coexist with disorder (including Dirac fermions in graphene), where the scattering of quasi-particles by any kind of disorder is isotropic

scattering. In this case, however, this is due to the spatial separation of free carriers and localized ionized donors, and it is not possible to vary the degree of disorder over a wide range, so that there is little room to investigate quasi-ballistic transport (in other words, the transition from ballistic to diffusive transport is rather abrupt and happens already when a small amount of impurities are introduced in the system). This is not the case for spin-helical Dirac fermions, and this property is at the origin of their unique transport properties, particularly in nanostructures, with an extended range of parameters to study quasi-ballistic transport, and

This enhanced transport length for topological surface states is also important for spin transport studies, as it gives a lower limit for the size of functional spintronic devices making use of the spin-momentum locking property. Indeed, due to the strong spin-orbit coupling, there is a direct correspondence between the momentum-backscattering transport length and the

[17] so that, for instance, lateral spin valves could only be realized in the short-junction limit. This also shows that the true potential for the spin-to-charge conversion in highly-disordered

The enhancement of the transport length for topological surface states also has two fundamental consequences for the quantum transport properties of 3D TI nanostructures and the

**1.** The phase coherence length of TSS is also enhanced in the same ratio for diffusive 2D surface states in nanoplatelets or wide nanoribbons, so that mesoscopic transport can be studied in rather wide and long conductors (well beyond the micron size), despite relatively strong disorder. Since Lϕ >> Ltr (Lϕ being determined by inelastic scattering), quantum corrections to the conductance due to diffusive phase-coherent transport can thus be revealed

**2.** The condition for ballistic transport in the transverse motion of surface carriers along the perimeter is more restrictive (Lp < 2 Ltr), but it can be fulfilled for rather long (L<sup>p</sup> ~ 500 nm), and

carbon nanotubes, magnetic flux-dependent periodic phenomena in these *quantum wires* (such as the Aharonov-Bohm interference) can therefore be studied in rather small fields, below 1 T.

by magneto-transport measurements with magnetic fields as small as 100 mT.

therefore with nanostructures having a large cross section (S ~ 0.2 μm2

 could still give an improvement in the conversion efficiency by two orders of magnitude with respect to state-of-the-art records, with an inverse Edelstein length lIEE determined by the intrinsic transport length of the 3D topological insulator, whereas it presently remains limited by the spin/momentum relaxation below metallic ohmic contacts (with lIEE ~ 2 nm).

e

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

nanostructures, this scale is rather short (~200 nm),

). Contrary to the case of

). Only high-mobility AlGaAs 2DEGs can realize such a situation of anisotropic

), as predicted by theory [8] and

http://dx.doi.org/10.5772/intechopen.76152

35

and that the transport length is strongly enhanced (Ltr >> l

therefore the ballistic-to-diffusive crossover in mesoscopic conductors.

Se3

spin relaxation length. With wide Bi2

**2.4. Dimensionalities of transport**

dimensionality of surface charge transport:

(that is, Ltr ~ le

Bi2 Se3

**Figure 3.** (a) Scanning-electron microscope image of a Hall-bar patterned Bi<sup>2</sup> Se3 thick nanoribbons, (b) fast-Fourier transform of the longitudinal magneto-resistance for two different back-gate voltages, and (c) evolution of the electronic band profile from the top surface to the bottom interface, typical for a low bulk-carrier density (downward band bending) and a large interface carrier density (coexistence of topological states with a Rashba 2DEG). After [36].

**Figure 4.** (a) Anisotropic scattering by disorder. The backscattering transport length Ltr can be much longer than the mean-free path le ; (b) scanning electron microscope image of a thin Bi<sup>2</sup> Se3 nanoribbon with traversing ohmic contacts; (c) longitudinal magneto-conductance for two different backgate voltages. Inset: Shubnikov-de Haas quantum oscillations; (d) backgate voltage dependence of the conductance and linear fit from which the transport length of the lower surface states (bottom interface) is inferred. After [17].

The average number of scattering centers involved in a backscattering process is directly related to the ratio Ltr/le , and it can give some important information on the nature of both the scattering potential and the quasi-particles. Importantly, we revealed the long-range nature of disorder for spin-helical surface Dirac fermions, due to efficient electrostatic screening and spin-momentum locking, which results in strongly anisotropic scattering, as observed by local probe microscopy. This means that forward scattering is favored for such quasi-particles and that the transport length is strongly enhanced (Ltr >> l e ), as predicted by theory [8] and confirmed by trans-conductance measurements [17]. Such a situation never happens in other materials within which charge carriers directly coexist with disorder (including Dirac fermions in graphene), where the scattering of quasi-particles by any kind of disorder is isotropic (that is, Ltr ~ le ). Only high-mobility AlGaAs 2DEGs can realize such a situation of anisotropic scattering. In this case, however, this is due to the spatial separation of free carriers and localized ionized donors, and it is not possible to vary the degree of disorder over a wide range, so that there is little room to investigate quasi-ballistic transport (in other words, the transition from ballistic to diffusive transport is rather abrupt and happens already when a small amount of impurities are introduced in the system). This is not the case for spin-helical Dirac fermions, and this property is at the origin of their unique transport properties, particularly in nanostructures, with an extended range of parameters to study quasi-ballistic transport, and therefore the ballistic-to-diffusive crossover in mesoscopic conductors.

This enhanced transport length for topological surface states is also important for spin transport studies, as it gives a lower limit for the size of functional spintronic devices making use of the spin-momentum locking property. Indeed, due to the strong spin-orbit coupling, there is a direct correspondence between the momentum-backscattering transport length and the spin relaxation length. With wide Bi2 Se3 nanostructures, this scale is rather short (~200 nm), [17] so that, for instance, lateral spin valves could only be realized in the short-junction limit. This also shows that the true potential for the spin-to-charge conversion in highly-disordered Bi2 Se3 could still give an improvement in the conversion efficiency by two orders of magnitude with respect to state-of-the-art records, with an inverse Edelstein length lIEE determined by the intrinsic transport length of the 3D topological insulator, whereas it presently remains limited by the spin/momentum relaxation below metallic ohmic contacts (with lIEE ~ 2 nm).

#### **2.4. Dimensionalities of transport**

The average number of scattering centers involved in a backscattering process is directly

**Figure 4.** (a) Anisotropic scattering by disorder. The backscattering transport length Ltr can be much longer than the

(c) longitudinal magneto-conductance for two different backgate voltages. Inset: Shubnikov-de Haas quantum oscillations; (d) backgate voltage dependence of the conductance and linear fit from which the transport length of the

; (b) scanning electron microscope image of a thin Bi<sup>2</sup>

**Figure 3.** (a) Scanning-electron microscope image of a Hall-bar patterned Bi<sup>2</sup>

34 Heterojunctions and Nanostructures

transform of the longitudinal magneto-resistance for two different back-gate voltages, and (c) evolution of the electronic band profile from the top surface to the bottom interface, typical for a low bulk-carrier density (downward band bending)

and a large interface carrier density (coexistence of topological states with a Rashba 2DEG). After [36].

lower surface states (bottom interface) is inferred. After [17].

scattering potential and the quasi-particles. Importantly, we revealed the long-range nature of disorder for spin-helical surface Dirac fermions, due to efficient electrostatic screening and

, and it can give some important information on the nature of both the

Se3

Se3

thick nanoribbons, (b) fast-Fourier

nanoribbon with traversing ohmic contacts;

related to the ratio Ltr/le

mean-free path le

The enhancement of the transport length for topological surface states also has two fundamental consequences for the quantum transport properties of 3D TI nanostructures and the dimensionality of surface charge transport:


In the case of highly-disordered Bi2 Se3 nanostructures (le ~ 30 nm), large values of Ltr (>200 nm) [17] and of L<sup>φ</sup> (>2 μm) [39] were found. This implies that the dimensionality of surface-state transport is reduced in narrow nanostructures (mostly nanowires) and that their band structure is modified due to quantum confinement (which even further reduces the scattering by disorder). It thus becomes important to distinguish between three different situations for the dimensionality of charge transport:

the weak coupling between quantum states, a necessary condition for their manipulation by radio-frequency fields, as well as for the study of specific properties related to a single topologically-protected low-energy mode, such as 1D chiral edge states or Majorana bound states [42].

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

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37

In quantum wires (Lp < 2 Ltr), the surface-state band structure is modified by periodic boundary conditions imposed in the transverse direction of the nanostructure, leading to the quantization of the transverse momentum k⊥ (**Figure 5b**). This situation is equivalent to the energy quantization of quasi-particles confined into a quantum well with infinitely-high potential barriers, and for Dirac fermions, the transverse energy becomes quantized with a constant energy-level spac-

along the perimeter (curvature) and to the spin-momentum locking of helical Dirac fermions, an additional Π Berry phase suppresses the topological protection of all energy modes in zero magnetic field (pairs of gapped states). However, if a magnetic field is applied along the nanowire axis, the Aharonov-Bohm flux modifies the periodic boundary condition (which gives an overall shift of transverse quantization planes, thus tuning the energy spectrum). Importantly, this flux dependence can restore the topological protection periodically when the Aharonov-

1/Lp between successive transverse modes. Due to the winding of the wave function

Se3

lines separate different facets. Inset: schematics of the cross section and coherent winding of topological surface states; (b) transverse-impulse quantization planes intersecting the spin-helical Dirac cone. By applying a magnetic flux, all planes are continuously shifted in the k⊥ direction; (c) resulting band structures for two different values of the magnetic flux ϕ.

with linear dispersion appears (quantization plane intersecting the Dirac point); (d) energy broadening Γ of the quantized transverse energy due to disorder [for 3DTI quantum wires, Γ is smaller than Δ, even for relatively strong disorder].

For ϕ = 0, quantum confinement gives pairs of modes with a finite energy gap Δ. For ϕ = 1/2ϕ<sup>0</sup>

nanowire, with a perimeter Lp = 300 nm. Dashed

, a single topological mode

**3.1. Quantum confinement: 1D Dirac spectrum**

**Figure 5.** (a) Scanning electron microscope image of a narrow Bi2

ing Δ = hv<sup>F</sup>


We remark that the dimensionality of quantum coherent transport is another quantity determined by comparing the dimensions of a mesoscopic conductor to the phase coherent length L<sup>φ</sup> . Since L<sup>φ</sup> is longer than Ltr, nanoribbons with a width W, such as Ltr < W < L<sup>φ</sup> , have 2D spin-helical surface Dirac fermions but quantum coherent transport is 1D (which modifies the self-averaging of quantum interference in long conductors, for which the length L is longer than L<sup>φ</sup> ).
