**3. Transport properties of quasi-1D topological surface states**

Signatures of the quasi-ballistic transport of topological surface states in 3D TI quantum wires can be revealed by the study of quantum corrections to the conductance. In a bulky nanostructure, spin-helical Dirac fermions propagate on the surface in a hollow-type electrical geometry, in analogy to carbon nanotubes (ballistic transport) or to the Sharvin-Sharvin metallic tubes (diffusive transport). Phase-coherent transport in the transverse direction (along the perimeter of the nanowire) thus gives rise to periodic Aharonov-Bohm oscillations in the longitudinal magnetoconductance, determined by a well-defined cross section S = LP 2 /(4\ϕ), and their observation in wide Bi2 Te3 nanoribbons gave the first robust evidence of surface states by transport measurements [40]. Their topological nature was then confirmed by a study of decoherence at very low temperatures in narrow (quantum) nanowires [39], which revealed the unusual weak coupling to the environment and the ballistic motion in the transverse direction. Later, phase-coherent transport in the longitudinal direction was investigated in a study of conductance fluctuations [18], which revealed the subtle influence of disorder in the quasi-ballistic regime, leading to a non-universal behavior of quantum interference. A detailed understanding of the propagation of spin-helical 1D surface modes showed that both the spin texture of Dirac fermions and their quantum confinement are responsible for the weak scattering by disorder [41], thus leading to 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].

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

In the case of highly-disordered Bi2

36 Heterojunctions and Nanostructures

dimensionality of charge transport:

helical Dirac-cone band structure.

[17] and of L<sup>φ</sup>

Since L<sup>φ</sup>

wide Bi2

Te3

Se3

nanostructures (le

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

**1.** No quantum confinement [Ltr is shorter than every dimensions]. Surface-state transport is diffusive and quasi-particles are 2D spin-helical Dirac fermions with a continuous spin-

**2.** Transverse quantum confinement [the perimeter Lp becomes shorter than 2 Ltr]. Surfacetransport is quasi-ballistic if the distance between contacts is longer than Ltr and ballistic otherwise. The length L remains much larger than Ltr, so that surface modes are quasi-1D

**3.** Full quantum confinement [all dimensions are shorter than Ltr]. Spin-helical Dirac fermions are then fully localized in a *short* nanowire, which becomes a 0D quantum dot.

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>

surface Dirac fermions but quantum coherent transport is 1D (which modifies the self-averaging

Signatures of the quasi-ballistic transport of topological surface states in 3D TI quantum wires can be revealed by the study of quantum corrections to the conductance. In a bulky nanostructure, spin-helical Dirac fermions propagate on the surface in a hollow-type electrical geometry, in analogy to carbon nanotubes (ballistic transport) or to the Sharvin-Sharvin metallic tubes (diffusive transport). Phase-coherent transport in the transverse direction (along the perimeter of the nanowire) thus gives rise to periodic Aharonov-Bohm oscillations in the longitudinal magneto-

ments [40]. Their topological nature was then confirmed by a study of decoherence at very low temperatures in narrow (quantum) nanowires [39], which revealed the unusual weak coupling to the environment and the ballistic motion in the transverse direction. Later, phase-coherent transport in the longitudinal direction was investigated in a study of conductance fluctuations [18], which revealed the subtle influence of disorder in the quasi-ballistic regime, leading to a non-universal behavior of quantum interference. A detailed understanding of the propagation of spin-helical 1D surface modes showed that both the spin texture of Dirac fermions and their quantum confinement are responsible for the weak scattering by disorder [41], thus leading to

nanoribbons gave the first robust evidence of surface states by transport measure-

2

channels in such *long* nanowires (becoming truly 1D only when they close).

is longer than Ltr, nanoribbons with a width W, such as Ltr < W < L<sup>φ</sup>

of quantum interference in long conductors, for which the length L is longer than L<sup>φ</sup>

**3. Transport properties of quasi-1D topological surface states**

conductance, determined by a well-defined cross section S = LP

(>2 μm) [39] were found. This implies that the dimensionality of surface-state

~ 30 nm), large values of Ltr (>200 nm)

.

, have 2D spin-helical

).

/(4\ϕ), and their observation in

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 spacing Δ = hv<sup>F</sup> 1/Lp between successive transverse modes. Due to the winding of the wave function 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-

**Figure 5.** (a) Scanning electron microscope image of a narrow Bi2 Se3 nanowire, with a perimeter Lp = 300 nm. Dashed 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. For ϕ = 0, quantum confinement gives pairs of modes with a finite energy gap Δ. For ϕ = 1/2ϕ<sup>0</sup> , a single topological mode 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].

Bohm phase compensates the curvature-induced Berry phase, giving rise to a single gapless and linear mode with perfect transmission (**Figure 5c**), independent of disorder [43].

*3.2.1. Case of wide nanoribbons (long-perimeter limit, with Lϕ ~ Lp/2)*

is the number of populated transverse modes, with N = EF

E\_F). As seen in **Figure 6** for a wide Bi2

confinement (see Section 3.4).

**Figure 6.** (a) Scanning electron microscope image of a Bi2

For wide nanoribbons, periodic Aharonov-Bohm oscillations are directly visible in G(B//) traces, with a rather small amplitude typical for the large-N limit in a mesoscopic conductor, where N

the Aharonov-Bohm period, δBAB = 150 mT, directly relates to the electrical cross section of the nanostructure, with a value being slightly smaller than that given by its physical dimensions (the topological surface states being "burried" below a thin native oxide layer, typically 5 nm thick). The fast-Fourier transform of the G(B//) trace thus gives a single peak at the AB frequency. According to theory, the overall phase shift of this sine evolution due to the AB quantum interference depends on both the degree of disorder and the energy of Dirac quasi-particles [43]. In most cases, the Fermi energy is very large and AB oscillations are phase locked with a conductance maximum in zero magnetic field, as found in many experiments and confirmed by theory. Yet, theory predicts the opposite situation (conductance minimum for a zero magnetic flux) when the chemical potential is near the Dirac point. The overall energy dependence of this phase shift can be quantitatively obtained from models taking explicitly disorder into account, and it allowed us to reveal an oscillatory behavior that is directly related to quantum

For lower temperatures (longer Lφ) or for narrower nanoribbons, roughly when Lφ ~ Lp, additional Altshuler-Aronov-Spivak (AAS) oscillations develop in addition. These correspond to quantum interference related to the complete winding of coherent paths along the perimeter, with time-reversed coherent loops so that this contribution is never damped by disorder,

For narrow nanostructures, the conductance modulation due to both AB and AAS interferences results from a complex mixing of high-order harmonics (multiple windings of coherent loops),

Te3

harmonics), with a period δB\_AB that directly relates to the nanowire's cross section. After [18].

*w* = 400 nm, height *h* = 70 nm) and ohmic CrAu contacts; (b) Aharonov-Bohm periodic oscillations (fundamental

which is the usual situation found in (diffusive) mesoscopic conductors.

*3.2.2. Case of narrow (quantum) nanowires (short-perimeter limit, Lp < 2 Ltr << L<sup>φ</sup>*

nanoribbon (EF

Te3

/2Δ, since Δ = hv<sup>F</sup>

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

~ 120 meV; Lp = 940 nm; Δ = 2 meV),

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

*)*

nanowire with a rather large perimeter Lp = 940 nm (width

/Lp is smaller than

39

In recent years, a couple of interesting studies suggested the influence of such a topological mode on quantum transport properties, particularly the Aharonov-Bohm (AB) oscillations [32, 33, 44]. These results raised some important questions since it was not possible to give a quantitative interpretation of the physical phenomena observed (Aharonov-Bohm oscillations and non-universal conductance fluctuations) solely based on the contribution of this perfectly transmitted mode to the conductance. In particular, the amplitude of these quantum corrections to the conductance was always found much smaller than the conductance quantum G0 . It thus remained unclear whether these properties were a signature of a topological transition or were rather induced by all spin-textured modes, including dominant contributions from high-energy quasi-1D modes. Actually, the quantum magneto-conductance is mostly due to a limited number of modes, those partially-opened modes with a quantized transverse energy close to the Fermi energy. Since most studies were conducted in the large-N limit (EF >> Δ), the relative contribution of the topological surface mode is therefore rather small. A full quantitative understanding required to describe the energy dependence of the transmissions for all surface modes, considering both disorder and interfaces with metallic contacts (see Section 3.4 for details). In particular, the scattering of surface modes by disorder results in the energy broadening Γ of quantized modes (**Figure 5d**). We evidenced that the quasi-ballistic regime is closely related to the condition Γ << Δ, which is satisfied over an unusual broad parameters range (disorder strength, energy) in 3D topological insulator nanostructures [41].

#### **3.2. Quantum coherence I: Aharonov-Bohm oscillations**

The quantum coherent transport of topological surface states in the transverse direction of a 3D TI nanostructure results in conductance oscillations when a *longitudinal magnetic induction* B// is applied (hence a magnetic flux ϕ = B//\*Sel, where Sel is the effective electrical cross section of metallic surface states). This is due to the flux-periodic evolution of the Aharonov-Bohm quantum interference giving successive conductance maxima (constructive interference) and minima (destructive interference). Because the phase coherence length can be as large as a couple of micrometers (at very low temperatures), two different situations must be considered for coherent transport in the *transverse motion* of surface states:

