**4. Conclusion and perspectives**

high-energy modes, so that this is not a good strategy for highly-doped quantum wires (as

**2.** In the small-N limit (less than about 4–5 modes populated), the relative influence of the topological mode on the quantum magneto-conductance will be larger, also in short wires. This condition is rather restrictive and requires either to bring the electro-chemical poten-

With the goal to investigate the physics of 3D TI quantum wires close to the Dirac point, best results could be obtained in long and ultra-narrow nanostructures [46, 47], since lowenergy modes other than the topological mode have a reduced transmission due to disorder

transport of surface modes is quasi-ballistic, it will become important to optimize/control the coupling between metallic contacts and the transverse wave function of a given mode. In particular, the amplitude of probability can have an azimuthal angle dependence, which varies from one mode to another, so that quantum transport properties will ultimately depend on the exact geometry of the mesoscopic conductor. Also, the low-energy spectrum can be modified by a large transverse magnetic field. For a rectangular cross section (**Figure 13**), a striking property is related to the evolution of the topological mode from a helical state to a chiral edge state, when a moderate transverse magnetic field is applied [42]. The specific orbital response of such 3DTI quantum wires correponds to an intermediate situation between the quantum

The control of low-energy quantum states in 3DTI nanostructures would offer novel opportunities for their quantum manipulation as well as for spin filtering, tuning the quantum states with an electric or a magnetic field. When coupled to metallic electrodes with gapped excitations, the topological mode generates novel quantum states with an intrinsic topological protection, such as Majorana bound states or spin-polarized edge states in the quantum

**Figure 13.** Energy spectrum of a 3DTI quantum wire with a rectangular cross section (h = 40 nm; w = 160 nm) for η = ϕ/ϕ<sup>0</sup> = 0 (left) and η = ϕ/ϕ<sup>0</sup> = ½ (center; right), low-energy band structure in the presence of a large transverse magnetic induction B⊥ = 2 T, showing the emergence of chiral edge states without dispersion over a wide range of impulse,

). Furthermore, since the

this is the case for Bi2

46 Heterojunctions and Nanostructures

independent of η. After [42].

Se3

nanostructures).

tial close to the Dirac point or to achieve very large values of Δ.

(minimum of the backscattering length, so that Ltr << L and G<<G0

spin Hall in a 2D TI and the Quantum Hall effect in 2DEGs.

The weak coupling of surface states in 3D topological insulator quantum wires, due to both their spin texture and the quantum confinement of Dirac fermions, gives unique opportunities to control novel quantum states in mesoscopic conductors, despite non-magnetic disorder. Yet, it remains difficult to control a small number of transverse quantized states close to the Dirac degeneracy point, mostly due to intrinsic limitations in conventional 3DTIs materials. Whereas the Bi2 Se3 family offers many advantages (tunable band structure in solid solutions of ternary compounds and high-quality single-crystalline nanostructures), it remains difficult to achieve surface transport only, and, most important, to control low-energy surface quasi-particles (large residual bulk doping or interface charge transfer, due to disorder).

Therefore, the next generation of electronic devices based on 3D topological insulators will necessarily be developed from advanced functional nanostructures and heterostructures. One of the most important challenge will be the full control of interface band bending, with a high-enough interface quality so as to optimize the coupling between metallic contacts and spin-helical surface Dirac fermions. For instance, this is particularly true for spin transport experiments, which require to minimize the momentum/spin relaxation below the contacts in order to make use of the intrinsic potential of electronic states with spin-momentum locking.

Toward this goal, new growth and nanofabrication methods need to be envisioned, in combination with those already used to prepare high-quality single-crystalline nanostructures (vapor transport, vapor-liquid-solid epitaxy, and chemical-vapor deposition). Novel techniques, such as the in-situ stencil lithography of metallic contacts combined the growth of ultra-thin films by molecular beam epitaxy, already gave some promising results, for instance to realize highlytransparent superconducting contacts and investigate topological superconductivity [48]. Also, atomic layer epitaxy holds promises to realize core-shell lateral nanostructures adapted to the control of the electro-chemical potential at the interface with a topological insulator [49–51].
