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

Bohm phase compensates the curvature-induced Berry phase, giving rise to a single gapless and

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

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

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 consid-

**1.** When Lϕ ~ Lp/2, clear periodic oscillations of the conductance are directly visible in the longitudinal magneto-conductance G(B//). In this case (wide nanoribbons), only the fundamental-harmonic Aharonov-Bohm interference modifies the conductance, a behavior which already reveals that the phase averaging due to disorder is not efficient, despite a

**2.** When Lϕ >> Lp, that is, either at very low temperatures or for short-perimeter nanowires (quantum wires), the periodic AB behavior is usually hidden in complex G(B//) traces. This is due to the multiple-harmonic contributions to the transverse quantum interference (related to the multiple winding of coherent trajectories along the perimeter), and to the influence of disorder (phase shifts). The periodic behavior can, however, be revealed by a

close to the Fermi energy. Since most studies were conducted in the large-N limit (EF

range (disorder strength, energy) in 3D topological insulator nanostructures [41].

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

ered for coherent transport in the *transverse motion* of surface states:

high point-defect density in most 3DTI materials.

Fourier transform analysis.

.

>> Δ), the

linear mode with perfect transmission (**Figure 5c**), independent of disorder [43].

38 Heterojunctions and Nanostructures

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 is the number of populated transverse modes, with N = EF /2Δ, since Δ = hv<sup>F</sup> /Lp is smaller than E\_F). As seen in **Figure 6** for a wide Bi2 Te3 nanoribbon (EF ~ 120 meV; Lp = 940 nm; Δ = 2 meV), 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 confinement (see Section 3.4).

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, 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> )*

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),

**Figure 6.** (a) Scanning electron microscope image of a Bi2 Te3 nanowire with a rather large perimeter Lp = 940 nm (width *w* = 400 nm, height *h* = 70 nm) and ohmic CrAu contacts; (b) Aharonov-Bohm periodic oscillations (fundamental harmonics), with a period δB\_AB that directly relates to the nanowire's cross section. After [18].

with harmonic-dependent phase shifts induced by disorder and varying relative amplitudes due to quasi-ballistic transport. The periodic-flux dependence of the longitudinal magneto-conductance is therefore hardly visible in most G(B//) traces, though it can still be when low-order harmonics remain dominant (as shown in **Figure 7b**). Since this periodic behavior is specific to topological surface states (with a flux-periodic energy spectrum), it can always be unveiled by a careful fast-Fourier transform analysis, provided that enough oscillations are measured (that is, when the field range largely exceeds the fundamental AB period). For a micron-long Bi<sup>2</sup> Se3 quantum wire with perimeter Lp = 280 nm, up to six harmonics were clearly resolved at very low temperature, as seen in **Figure 7c**) [39].

**3.3. Quantum coherence II: non-universal conductance fluctuations**

absolute amplitude of the conductance variance varG = δGrms

**Figure 8.** (a) Quantum corrections to the conductance of a Bi2

all transverse modes. After [18].

More information about the weak scattering of quantized surface modes by a non-magnetic disorder can be obtained by studying the (static) conductance fluctuations due to the *longitudinal motion* of surface carriers in a highly-disordered 3DTI quantum wire. Contrary to the case of a diffusive mesoscopic conductor, their statistical properties such as the conductance variance are not universal and they can vary when the quantized Dirac band structure is modified by an Aharonov-Bohm flux [18]. Using a 3D vector magnet, we could vary independently the longitudinal field (tuning of the energy spectrum; transverse motion) and the transverse magnetic field (probing the aperiodic conductance fluctuations due to disorder; longitudinal motion). This provides the complete mapping of quantum interference, as seen in **Figure 8a**). It was shown that the

lation with the magnetic flux (**Figure 8b**), a property specific to surface transport (well-defined cross section). This behavior is well captured by numerical calculations (**Figure 8c** and **d**), which also reproduce the correct amplitude of this modulation mod(varG). We evidenced that nonuniversal conductance fluctuations are the signature of the weak coupling between transverse quantized modes induced by disorder, and we inferred the amplitude of the disorder broadening Γ from the temperature dependence of the modulation mod(VarG), in rather good agreement

Se3

magnetic fields (magnetic flux) and transverse magnetic fields, at very low temperature; (b) flux-dependence of the conductance variance revealing a non-universal behavior and a periodic evolution, a signature of the quasi-ballistic transport of Dirac surface modes; (c and d) numerical calculations of the energy (c) and flux (d) dependences of the conductance variance in a disordered 3DTI quantum wire, showing periodic evolutions due to quantum confinement of

quantum wire mapped over a large range of longitudinal

<sup>2</sup> = <G-<G>>2

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

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41

has a periodic modu-

For short wires (L ~ L<sup>φ</sup> BS), we also remark that a complication comes further from that aperiodic conductance fluctuations due to bulk carriers coexist with surface periodic AB oscillations [although L<sup>φ</sup> BS < L<sup>φ</sup> SS, the self-averaging of coherent bulk transport is reduced at very low temperatures due to their charge transport dimensionality d = 3 and to longer L<sup>φ</sup> BS values]. Besides, because G(B//) curves are measured over a finite field range, the FFT of bulk aperiodic conductance fluctuations often results in a non-monotonous background, possibly giving "peaks" but with no relation to a periodic behavior, contrary to that of G(B//) changes due to the AB interference.

The ballistic nature of the transverse motion in such quantum wires results in an unusual temperature dependence of the phase coherence length L<sup>φ</sup> SS, with a 1/T behavior observed for all harmonics. This is the signature of both ballistic transport (Lφ = vF τφ) and a decoherence time τφ ~ 1/T limited by a weak coupling to fluctuations of the environment [39]. All other scenarios based on decoherence limited by either the Nyquist noise or the thermal noise give a very different power-law dependence.

An extra signature of the quasi-ballistic regime is also found when considering the relative amplitude of AB harmonics. Contrary to the case of a diffusive mesoscopic conductors, their amplitudes are not increasingly small for higher orders n and they cannot be described by an exponential damping behavior related to the ratio Lφ/Ln, where Ln = n\*Lp [39]. This is due to disorder and to both geometric and contact effects, which all influence details of the quantum interference for different quantum coherent paths, in the quasi-ballistic regime [41]. In general, it thus remains difficult to investigate details of the AB oscillations in this regime, due to the complex mixing of all harmonics in the presence of disorder, which varies for different configurations of the microscopic disorder (as obtained by thermal cycling at room temperature of a given mesoscopic conductor).

**Figure 7.** (a) Scanning electron microscope image of a narrow Bi2 Se3 nanowire with a rather short perimeter Lp = 280 nm and ohmic Al contacts; (b) Aharonov-Bohm periodic oscillations, with the first two harmonics directly visible in the G(B//) trace; (c) fast-Fourier transform revealing higher-order harmonics, up to n = 6 at very low temperature. After [39].

#### **3.3. Quantum coherence II: non-universal conductance fluctuations**

with harmonic-dependent phase shifts induced by disorder and varying relative amplitudes due to quasi-ballistic transport. The periodic-flux dependence of the longitudinal magneto-conductance is therefore hardly visible in most G(B//) traces, though it can still be when low-order harmonics remain dominant (as shown in **Figure 7b**). Since this periodic behavior is specific to topological surface states (with a flux-periodic energy spectrum), it can always be unveiled by a careful fast-Fourier transform analysis, provided that enough oscillations are measured (that is, when the field range largely exceeds the fundamental AB period). For a micron-long Bi<sup>2</sup>

quantum wire with perimeter Lp = 280 nm, up to six harmonics were clearly resolved at very low

odic conductance fluctuations due to bulk carriers coexist with surface periodic AB oscilla-

Besides, because G(B//) curves are measured over a finite field range, the FFT of bulk aperiodic conductance fluctuations often results in a non-monotonous background, possibly giving "peaks" but with no relation to a periodic behavior, contrary to that of G(B//) changes due

The ballistic nature of the transverse motion in such quantum wires results in an unusual

time τφ ~ 1/T limited by a weak coupling to fluctuations of the environment [39]. All other scenarios based on decoherence limited by either the Nyquist noise or the thermal noise give

An extra signature of the quasi-ballistic regime is also found when considering the relative amplitude of AB harmonics. Contrary to the case of a diffusive mesoscopic conductors, their amplitudes are not increasingly small for higher orders n and they cannot be described by an exponential damping behavior related to the ratio Lφ/Ln, where Ln = n\*Lp [39]. This is due to disorder and to both geometric and contact effects, which all influence details of the quantum interference for different quantum coherent paths, in the quasi-ballistic regime [41]. In general, it thus remains difficult to investigate details of the AB oscillations in this regime, due to the complex mixing of all harmonics in the presence of disorder, which varies for different configurations of the microscopic disorder (as obtained by thermal cycling at room tempera-

Se3

and ohmic Al contacts; (b) Aharonov-Bohm periodic oscillations, with the first two harmonics directly visible in the G(B//) trace; (c) fast-Fourier transform revealing higher-order harmonics, up to n = 6 at very low temperature. After [39].

temperatures due to their charge transport dimensionality d = 3 and to longer L<sup>φ</sup>

BS), we also remark that a complication comes further from that aperi-

SS, the self-averaging of coherent bulk transport is reduced at very low

temperature, as seen in **Figure 7c**) [39].

BS < L<sup>φ</sup>

a very different power-law dependence.

ture of a given mesoscopic conductor).

**Figure 7.** (a) Scanning electron microscope image of a narrow Bi2

temperature dependence of the phase coherence length L<sup>φ</sup>

all harmonics. This is the signature of both ballistic transport (Lφ = vF

For short wires (L ~ L<sup>φ</sup>

40 Heterojunctions and Nanostructures

to the AB interference.

tions [although L<sup>φ</sup>

Se3

BS values].

SS, with a 1/T behavior observed for

nanowire with a rather short perimeter Lp = 280 nm

τφ) and a decoherence

More information about the weak scattering of quantized surface modes by a non-magnetic disorder can be obtained by studying the (static) conductance fluctuations due to the *longitudinal motion* of surface carriers in a highly-disordered 3DTI quantum wire. Contrary to the case of a diffusive mesoscopic conductor, their statistical properties such as the conductance variance are not universal and they can vary when the quantized Dirac band structure is modified by an Aharonov-Bohm flux [18]. Using a 3D vector magnet, we could vary independently the longitudinal field (tuning of the energy spectrum; transverse motion) and the transverse magnetic field (probing the aperiodic conductance fluctuations due to disorder; longitudinal motion). This provides the complete mapping of quantum interference, as seen in **Figure 8a**). It was shown that the absolute amplitude of the conductance variance varG = δGrms <sup>2</sup> = <G-<G>>2 has a periodic modulation with the magnetic flux (**Figure 8b**), a property specific to surface transport (well-defined cross section). This behavior is well captured by numerical calculations (**Figure 8c** and **d**), which also reproduce the correct amplitude of this modulation mod(varG). We evidenced that nonuniversal conductance fluctuations are the signature of the weak coupling between transverse quantized modes induced by disorder, and we inferred the amplitude of the disorder broadening Γ from the temperature dependence of the modulation mod(VarG), in rather good agreement

**Figure 8.** (a) Quantum corrections to the conductance of a Bi2 Se3 quantum wire mapped over a large range of longitudinal magnetic fields (magnetic flux) and transverse magnetic fields, at very low temperature; (b) flux-dependence of the conductance variance revealing a non-universal behavior and a periodic evolution, a signature of the quasi-ballistic transport of Dirac surface modes; (c and d) numerical calculations of the energy (c) and flux (d) dependences of the conductance variance in a disordered 3DTI quantum wire, showing periodic evolutions due to quantum confinement of all transverse modes. After [18].

with numerical calculations [18]. Because Γ << Δ, quasi-ballistic transport is a common property to all populated surface modes, each of them giving a significant contribution to the conductance (close to G0 ) as compared to that of the perfect transmission case (G0 ). Besides, the sharp evolution of varG with the energy of successive transverse modes (**Figure 8c**) suggested that only a limited number of partially-opened modes (nearby EF ) contribute to the conductance fluctuations, due to a rapid energy dependence of the transmission for all channels.

nearby mode and because disorder broadening remains smaller than the energy level spacing. This can be directly seen in the energy dependence of the transmission of a high-energy mode

**Figure 9.** Energy dependence of the surface-mode transmissions in disordered 3D topological insulator quantum wires,

induced by a random disorder potential δV; (b) energy dependence of the transmission of the m = 9 quantized mode for ϕ = 0, showing resonances due to disorder-induced inter-mode mixing, with an energy broadening Γ. After [41].

and inter-mode scattering

for three values of the disorder strength g=0.02 (a), g=0.2 (b), and g=1 (c). After [41].

**Figure 10.** (a) Quantized band structure of surface modes for a magnetic flux ϕ = 1/2ϕ<sup>0</sup>

values of g, Γ exceeds Δ and charge transport becomes diffusive.

quantum wire (using a realistic value of ξ), as shown in **Figure 10b**). For very large

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43

in a Bi2

Se3

#### **3.4. Quasi-ballistic transport: disorder and transmissions**

To evidence that this interpretation is actually very general to all quantum corrections to the conductance in 3DTI quantum wires, it is important to calculate the energy dependence of the transmissions for all modes, taking disorder into account but also interfaces with metallic contacts. This was done in a comparative study of numerical calculations with an analytical model that captures the main property common to all quasi-1D spin-helical surface modes, that is, the suppression of scattering due to quantum confinement [41]. The set of transmissions Ti represents the mesoscopic code of a coherent conductor, from which all important quantities can be calculated [45], the simplest one being the total conductance G = ∑TiG<sup>0</sup> . Importantly, the transmissions were found to nearly reach unity for all modes when their longitudinal kinetic energy exceeds Δ (**Figure 9**). The same (rapid) evolution was found even for high-energy modes, though over a slightly broader energy window, thus explaining why quasi-ballistic transport properties exist for many modes over a broad energy range. This also shows that diffusive longitudinal transport is realized only for conductor lengths that largely exceed the transport length. Contrary to the case of 2D quasi-particles with isotropic scattering for which the transition from the ballistic to the diffusive regime is rather abrupt (le < L < Ltr, with Ltr ≤ 2 l<sup>e</sup> ), the quasi-ballistic regime in 3DTI quantum wires exists over a wider parameter range (Ltr/2 < L < α Ltr, with Ltr >> l e and α is related to the aspect ratio L/Lp). This unusual behavior, related to the enhanced transport length, is due to both the spin texture of Dirac modes (anisotropic scattering) and to their large confinement energy in quantum wires, both favoring the weak scattering of quantized modes by disorder.

#### *3.4.1. Scattering by disorder and contacts*

Considering the scattering by disorder as due to a random potential of energy barriers (Gaussian disorder, with a correlation length ξ, see **Figure 10a**), it is possible to give an analytical description of the transmissions of high-energy modes propagating between two transparent ohmic contacts, for different degrees of disorder from the clean limit (ballistic, Fabry-Pérot) to the dirty limit (diffusive) [41]. In the quasi-ballistic regime, we found that the conductance is determined by the interfaces with metallic contacts (similarly to a clean conductor) and not by details of the microscopic disorder in the quantum wire. Due to the quantum confinement of Dirac fermions with evenly-spaced energy levels, the energy dependence of the conductance can oscillate at low energies (whereas its has a linear dependence at high energy, as for the 2D limit) and the average transmission per mode only depends on the nature of the contacts. For an intermediate disorder strength g, Fabry-Pérot interferences are suppressed by efficient phase averaging, and inter-mode scattering results in an oscillatory energy dependence of the transmission, due to the increased density of states at the onset of a nearby mode and because disorder broadening remains smaller than the energy level spacing. This can be directly seen in the energy dependence of the transmission of a high-energy mode in a Bi2 Se3 quantum wire (using a realistic value of ξ), as shown in **Figure 10b**). For very large values of g, Γ exceeds Δ and charge transport becomes diffusive.

with numerical calculations [18]. Because Γ << Δ, quasi-ballistic transport is a common property to all populated surface modes, each of them giving a significant contribution to the conductance

of varG with the energy of successive transverse modes (**Figure 8c**) suggested that only a limited

To evidence that this interpretation is actually very general to all quantum corrections to the conductance in 3DTI quantum wires, it is important to calculate the energy dependence of the transmissions for all modes, taking disorder into account but also interfaces with metallic contacts. This was done in a comparative study of numerical calculations with an analytical model that captures the main property common to all quasi-1D spin-helical surface modes, that is, the suppression of scattering due to quantum confinement [41]. The set of transmis-

quantities can be calculated [45], the simplest one being the total conductance G = ∑TiG<sup>0</sup>

Importantly, the transmissions were found to nearly reach unity for all modes when their longitudinal kinetic energy exceeds Δ (**Figure 9**). The same (rapid) evolution was found even for high-energy modes, though over a slightly broader energy window, thus explaining why quasi-ballistic transport properties exist for many modes over a broad energy range. This also shows that diffusive longitudinal transport is realized only for conductor lengths that largely exceed the transport length. Contrary to the case of 2D quasi-particles with isotropic scattering for which the transition from the ballistic to the diffusive regime is rather abrupt

e

unusual behavior, related to the enhanced transport length, is due to both the spin texture of Dirac modes (anisotropic scattering) and to their large confinement energy in quantum wires,

Considering the scattering by disorder as due to a random potential of energy barriers (Gaussian disorder, with a correlation length ξ, see **Figure 10a**), it is possible to give an analytical description of the transmissions of high-energy modes propagating between two transparent ohmic contacts, for different degrees of disorder from the clean limit (ballistic, Fabry-Pérot) to the dirty limit (diffusive) [41]. In the quasi-ballistic regime, we found that the conductance is determined by the interfaces with metallic contacts (similarly to a clean conductor) and not by details of the microscopic disorder in the quantum wire. Due to the quantum confinement of Dirac fermions with evenly-spaced energy levels, the energy dependence of the conductance can oscillate at low energies (whereas its has a linear dependence at high energy, as for the 2D limit) and the average transmission per mode only depends on the nature of the contacts. For an intermediate disorder strength g, Fabry-Pérot interferences are suppressed by efficient phase averaging, and inter-mode scattering results in an oscillatory energy dependence of the transmission, due to the increased density of states at the onset of a

represents the mesoscopic code of a coherent conductor, from which all important

), the quasi-ballistic regime in 3DTI quantum wires exists over a wider

and α is related to the aspect ratio L/Lp). This

). Besides, the sharp evolution

.

) contribute to the conductance fluctuations, due to

) as compared to that of the perfect transmission case (G0

a rapid energy dependence of the transmission for all channels.

**3.4. Quasi-ballistic transport: disorder and transmissions**

number of partially-opened modes (nearby EF

(close to G0

42 Heterojunctions and Nanostructures

sions Ti

(le < L < Ltr, with Ltr ≤ 2 l<sup>e</sup>

parameter range (Ltr/2 < L < α Ltr, with Ltr >> l

*3.4.1. Scattering by disorder and contacts*

both favoring the weak scattering of quantized modes by disorder.

**Figure 9.** Energy dependence of the surface-mode transmissions in disordered 3D topological insulator quantum wires, for three values of the disorder strength g=0.02 (a), g=0.2 (b), and g=1 (c). After [41].

**Figure 10.** (a) Quantized band structure of surface modes for a magnetic flux ϕ = 1/2ϕ<sup>0</sup> and inter-mode scattering induced by a random disorder potential δV; (b) energy dependence of the transmission of the m = 9 quantized mode for ϕ = 0, showing resonances due to disorder-induced inter-mode mixing, with an energy broadening Γ. After [41].

To understand why the quasi-ballistic regime exists over a broad range of parameters, it is important to consider the energy dependence of the transport length [see [41] for details], as shown in **Figure 11** for different g values. Contrary to the case of massive quasi-particles, Ltr does not vanish at low energy for Dirac fermions in 1D. Instead, it diverges and a similar behavior occurs at high energy, due to the anisotropy of scattering. As a consequence, the transport length has a minimum value that depends on the strength of disorder. For a given disorder correlation length ξ, this minimum value is obtained for kξ ≈ 1 and the values of Ltr min can be much larger than the transverse dimensions of the nanostructure for all energies, for a broad range of g values, so that the condition for quasi-ballistic transport is always fulfilled for such highly-disordered 3D topological insulator quantum wires. Good agreement was found between this simplified analytical model and numerical calculations, for that details of the microscopic disorder do not affect the conductance in this regime, which is mostly determined by metallic contacts.

### *3.4.2. Quantitative derivation of the Aharonov-Bohm amplitude*

Based on the transmissions calculated for different values of the magnetic flux (corresponding to different quantized energy spectra), it is possible to calculate the energy dependence of Aharonov-Bohm oscillations. As seen in **Figure 11**, their amplitude decreases with the energy of surface modes and a good quantitative agreement with experiments are found at high energies. The oscillatory behavior reported in experiments is also well reproduced, as well as energy-periodic phase shifts [33] which are actually due to the quantized band structure (**Figure 12**) [41].

**Figure 11.** Energy dependence of the transport length l in a quantum wire with a perimeter W = Lp and a transverse energy quantization Δ is calculated for three different strengths of the disorder potential. Red dotted lines show asymptotic behaviors related to the divergence of l = Ltr at low or high energies, due to the density of states or to the anisotropy of scattering, respectively. In all cases, the transport of all surface modes is ballistic or quasi-ballistic. After [41].

*3.4.3. Perfectly-transmitted topological mode*

(contribution of the topological mode only) to a fraction of G0

G0

sary to set the mesoscopic conductor in specific conditions:

The influence of the topological mode is seen only for very low energies. After [41].

To evidence the influence of the topological mode on the conductance, it is therefore neces-

**Figure 12.** (a) Energy dependence of the Aharonov-Bohm amplitude, rapidly decreasing from the conductance quantum

dependence of the conductance for different energies, from 0 to 4.5 Δ (successive thin lines correspond to an energy change 0.05 Δ and thick lines to multiple values of 1/2 Δ). Phase shifts are due to the quantized energy spectrum of surface modes.

(contribution of higher-energy transverse modes); (b) flux

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**1.** In long wires, the transmission of all modes but the topological one should be reduced. However, the spin texture of surface Dirac states prevents the strong localization of

**Figure 12.** (a) Energy dependence of the Aharonov-Bohm amplitude, rapidly decreasing from the conductance quantum G0 (contribution of the topological mode only) to a fraction of G0 (contribution of higher-energy transverse modes); (b) flux dependence of the conductance for different energies, from 0 to 4.5 Δ (successive thin lines correspond to an energy change 0.05 Δ and thick lines to multiple values of 1/2 Δ). Phase shifts are due to the quantized energy spectrum of surface modes. The influence of the topological mode is seen only for very low energies. After [41].

#### *3.4.3. Perfectly-transmitted topological mode*

To understand why the quasi-ballistic regime exists over a broad range of parameters, it is important to consider the energy dependence of the transport length [see [41] for details], as shown in **Figure 11** for different g values. Contrary to the case of massive quasi-particles, Ltr does not vanish at low energy for Dirac fermions in 1D. Instead, it diverges and a similar behavior occurs at high energy, due to the anisotropy of scattering. As a consequence, the transport length has a minimum value that depends on the strength of disorder. For a given disorder cor-

larger than the transverse dimensions of the nanostructure for all energies, for a broad range of g values, so that the condition for quasi-ballistic transport is always fulfilled for such highly-disordered 3D topological insulator quantum wires. Good agreement was found between this simplified analytical model and numerical calculations, for that details of the microscopic disorder do not affect the conductance in this regime, which is mostly determined by metallic contacts.

Based on the transmissions calculated for different values of the magnetic flux (corresponding to different quantized energy spectra), it is possible to calculate the energy dependence of Aharonov-Bohm oscillations. As seen in **Figure 11**, their amplitude decreases with the energy of surface modes and a good quantitative agreement with experiments are found at high energies. The oscillatory behavior reported in experiments is also well reproduced, as well as energy-periodic phase shifts [33] which are actually due to the quantized band structure

**Figure 11.** Energy dependence of the transport length l in a quantum wire with a perimeter W = Lp and a transverse energy quantization Δ is calculated for three different strengths of the disorder potential. Red dotted lines show asymptotic behaviors related to the divergence of l = Ltr at low or high energies, due to the density of states or to the anisotropy of

scattering, respectively. In all cases, the transport of all surface modes is ballistic or quasi-ballistic. After [41].

min can be much

relation length ξ, this minimum value is obtained for kξ ≈ 1 and the values of Ltr

*3.4.2. Quantitative derivation of the Aharonov-Bohm amplitude*

(**Figure 12**) [41].

44 Heterojunctions and Nanostructures

To evidence the influence of the topological mode on the conductance, it is therefore necessary to set the mesoscopic conductor in specific conditions:

**1.** In long wires, the transmission of all modes but the topological one should be reduced. However, the spin texture of surface Dirac states prevents the strong localization of high-energy modes, so that this is not a good strategy for highly-doped quantum wires (as this is the case for Bi2 Se3 nanostructures).

anomalous Hall regime. These could be important for quantum dynamics studies, with limited decoherence. Non-topological low-energy modes are also interesting for their energy tuning, by a gate voltage or a magnetic flux, which is associated with a continuous change of their spin state between nearly-orthogonal states. Besides, these can be either 1D extended states (long quantum wires) or 0D localized states (short quantum wires, that is, for L < Ltr).

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 materi-

tions 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].

This work was supported by the German Research Foundation DFG through the SPP 1666 Topological Insulators program. This book chapter reviews our previous work on 3D topological insulator quantum wires, for which we acknowledge the contributions of our collaborators: J. H. Bardarson, B. Büchner, J. Cayssol, B. Dassonneville, B. Eichler, W. Escoffier, H. Funke, S. Hampel, C. Nowka, O.G. Schmidt, J. Schumann, L. Veyrat, K. Wruck, and E. Xypakis.

family offers many advantages (tunable band structure in solid solu-

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

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**4. Conclusion and perspectives**

Se3

als. Whereas the Bi2

**Acknowledgements**

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

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 (minimum of the backscattering length, so that Ltr << L and G<<G0 ). Furthermore, since the 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 spin Hall in a 2D TI and the Quantum Hall effect in 2DEGs.

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, independent of η. After [42].

anomalous Hall regime. These could be important for quantum dynamics studies, with limited decoherence. Non-topological low-energy modes are also interesting for their energy tuning, by a gate voltage or a magnetic flux, which is associated with a continuous change of their spin state between nearly-orthogonal states. Besides, these can be either 1D extended states (long quantum wires) or 0D localized states (short quantum wires, that is, for L < Ltr).
