Weakly-coupled quasi-1D helical modes in disordered 3D topological insulator quantum wires

Disorder remains a key limitation in the search for robust signatures of topological superconductivity in condensed matter. Whereas clean semiconducting quantum wires gave promising results discussed in terms of Majorana bound states, disorder makes the interpretation more complex. Quantum wires of 3D topological insulators offer a serious alternative due to their perfectly-transmitted mode. An important aspect to consider is the mixing of quasi-1D surface modes due to the strong degree of disorder typical for such materials. Here, we reveal that the energy broadening γ of such modes is much smaller than their energy spacing Δ, an unusual result for highly-disordered mesoscopic nanostructures. This is evidenced by non-universal conductance fluctuations in highly-doped and disordered Bi2Se3 and Bi2Te3 nanowires. Theory shows that such a unique behavior is specific to spin-helical Dirac fermions with strong quantum confinement, which retain ballistic properties over an unusually large energy scale due to their spin texture. Our result confirms their potential to investigate topological superconductivity without ambiguity despite strong disorder.


I. CONDUCTANCE FLUCTUATIONS AND AHARONOV-BOHM OSCILLATIONS
Different quantum corrections to the conductance of a 3D topological insulator nanostructure can be studied in a finite magnetic field, the direction of which determines the exact nature of the quantum interference probed. In the diffusive regime, such a difference is best captured by a semi-classical approach 1 , considering all possible closed-loop trajectories (see Fig. 1). If the magnetic flux is trapped within closed loops with a well defined area (such as the transverse cross section for topological surface states -see Fig. 1:bottom,left), the conductance can show periodic oscillations with the magnetic field and their period only depends on the geometry of the nanostructure. If, instead, there is a size distribution of closed loops (such as for all other cases considered in Fig. 1), the conductance can show reproducible aperiodic fluctuations with the magnetic field and their quasi-period will depend on the longest quantum coherent paths within the plane perpendicular to the applied field, which are related to either the phase-coherence length or the geometry of the conductor, or to both of them. This intuitive description remains qualitatively relevant for a ballistic conductor and the nature of quantum corrections to the conductance depends on the size statistics of classical closed-loop trajectories (and therefore on the geometry of the conductor), a property related to the level statistics in the energy spectrum of quantum states 2 .
Importantly, the relative contribution of bulk carriers and topological surface states to the amplitude of quantum conductance fluctuations strongly depends on both the nature of charge transport (ballistic/diffusive), related to the transport length l tr , and the dimensionality of coherent transport, related to the phase coherence length L ϕ . For a 3D topological insulator with a typically strong disorder (mean-free path of about 30 nm), charge transport is 3D for bulk states (l BS tr ≈ l e < {w, h}), whereas it can be 2D (L p l SS tr ) or 1D (L p l SS tr ) for topological surface states (with l SS tr l BS tr ). Due to a different phase coherence length, quantum coherent transport is 1D over a wide temperature range for surface states (L SS ϕ > {w, h}) whereas it is 3D for bulk states (L BS ϕ < {w, h}), but at very low temperatures where it becomes 1D when L BS ϕ > {w, h}. Since the phase coherence lengths depend on temperature and since the decoherence mechanism can be different for bulk and surface states with quantum confinement, their relative contribution to conductance fluctuations depends on both the length of the mesoscopic conductor and temperature.
As a quantitative example at T =200mK, the phase coherence lengths are L SS ϕ ≈ 1 µm and L BS ϕ ≈390nm. The latter value is inferred from measurements on a wide Bi 2 Se 3 nanoribbon with similar bulk-transport properties 3 , assuming that decoherence is limited by electron-electron interactions. Besides, it is in good agreement with the quantitative values found here for the finite contribution to δG rms due to bulk states, giving a shift of its average < δG rms > over B , which is best visible at low temperature and for a short wire. Since quantum coherent transport is 1D for both bulk and surface states, the self averaging of δG rms varies as (L ϕ /L) 3 2 , and we find a relative contribution of bulk carriers of about 16% for L 2 = 1 µm and 50% for L 1 = 400nm. Most importantly, this bulk contribution to conductance fluctuations is universal (diffusive transport). Therefore, its standard deviation does not depend on the magnetic flux, contrary to the contribution of topological surface states in the quasi-ballistic regime. The latter gives rise to non-universal conductance fluctuations with a flux-periodic modulation of δG rms . This modulation is found to be as large as δG rms if L ≈ l SS tr or reduced in the ratio l SS tr /L if L > l SS tr , in very good quantitative agreement with the results shown in Fig.2a) in the main text and l SS tr ≈ 300 nm. Below, we consider the different contributions to quantum interference in a 3D topological insulator nanowire, probed by sweeping the magnetic field applied either along or perpendicular to the nanostructure. Figure 1: Quantum interference in a 3D topological insulator nanostructure. Top, Bulk carriers only contribute to universal conductance fluctuations, for both cases of a longitudinal field (left) and a perpendicular field (right). In the case considered here, L BS ϕ < {w, h} so that the correlation field is given by B BS C = Φ0/(L BS ϕ ) 2 ; Bottom, Topological surface states only contribute to periodic quantum oscillations for the case of a longitudinal field (left), with a period ∆B = Φ0/(w × h) (Aharonov-Bohm) or ∆B = 1 2 Φ0/(w × h) (Altshuler-Aronov-Spivak), whereas they only contribute to non-universal conductance fluctuations when a perpendicular field is applied (right), with a correlation field given by

A. Influence of a longitudinal field
Applying a longitudinal field B , bulk states only induce universal conductance fluctuations, whereas topological surface states do not give any aperiodic conductance fluctuations, since there is no flux trapped by surface closed loops, and they only result in periodic Aharonov-Bohm oscillations of the conductance, even at rather high temperatures (long phase coherence length L SS ϕ , with respect to the perimeter L p ). As sketched in Fig. 1, the different quantum corrections to the conductance are: • Aperiodic universal conductance fluctuations from bulk states. Since in general L BS ϕ < L, their amplitude has a power-law dependence with L BS ϕ /L (see ref. 1 for details), and their correlation field is given by • Periodic Aharonov-Bohm (AB, h/e flux periodicity) or Altshuler-Aronov-Spivak (AAS, h/2e flux periodicity) oscillations from surface states, with a period ∆B = Φ 0 /S (or Φ 0 /2S) related to the transverse cross section S = w × h. Such oscillations exist both in the ballistic regime and in the diffusive regime, also for rather long mesoscopic conductors due to the weak scattering by disorder.
i) In the diffusive regime, their amplitude has an exponential dependence with L SS ϕ . For long mesoscopic conductors, with L L SS ϕ > L p , only AAS oscillations should survive whereas AB oscillations are damped by ensemble averaging over uncorrelated coherent segments. However, this reduction is relatively small due to the large enhancement of the transport length for topological surface states. ii) In the ballistic regime, the amplitude of both AB and AAS oscillations depends on the transmission of transverse modes and their energy spectrum, which is periodically modified by the flux, in addition to the phase coherence length. Importantly, quantum confinement preserves AB oscillations even for L SS ϕ L (see ref. 4 for details).
In both cases, for very long L SS ϕ , the rich content of harmonics in periodic quantum oscillations of the conductance, combined with some frequency shifts due to disorder (see section II A) and to the finite quantum width of surface states, makes the flux-periodic evolution of the conductance with B very different from a pure sine function, as exemplified in Fig. 2. Therefore, at very low temperatures, a detailed study of Aharonov-Bohm oscillations is best done by performing a fast-Fourier transform (FFT) analysis. . Contributions of four harmonics to Aharonov-Bohm oscillations altered by disorder (the n th order of each harmonic is indicated after the number sign " "). The AB frequency is shifted by +20% for the first harmonic and by −20% for the second harmonic, with respect to the fundamental harmonics n = 0, accounting for the typical width of AB peaks in the FFT spectrum observed experimentally. Each harmonic has a different zero-flux phase. As a result, the total contribution (black line) strongly deviates from a pure sine function, and the periodic nature of conductance oscillations is best revealed in the FFT spectrum of a magneto-conductance trace.

B. Influence of a perpendicular field
Applying a perpendicular magnetic field B ⊥ , both bulk states and topological surface states lead to conductance fluctuations but their nature, as well as their amplitude and correlation field is very different. This is due to a couple of microscopic parameters that are different, such as the enhanced transport length and phase coherence length for topological surface states with respect to bulk states, but also to the diffusive nature of massive quasi-particles whereas the transport of spin-helical Dirac surface modes is quasi-ballistic.
As sketched in Fig. 1, the different quantum corrections to the conductance are: • Aperiodic universal conductance fluctuations from bulk states. Since in general L BS ϕ < L, their amplitude has a power-law dependence with L BS ϕ /L, and their relative contribution to conductance fluctuations is reduced by the self average between uncorrelated coherent segments.
• Aperiodic non-universal conductance fluctuations from topological surface states. As discussed in this paper, their amplitude depends on both L SS ϕ /L and l tr /L, as well as on the energy spectrum and transmissions of quantized transverse modes. Their dominant contribution to conductance fluctuations is a direct consequence of the enhancement of L SS ϕ , due to both anisotropic scattering and quantum confinement.

A. Aharonov-Bohm oscillations in 3D topological insulator quantum wires
The periodicity of Aharonov-Bohm oscillations is not necessarily seen in G(B ) traces directly. As shown in Fig. 3, it depends on how the phase coherence length L SS ϕ compares to the perimeter L p . In the narrow Bi 2 Se 3 quantum wire considered in this study, L SS ϕ L p , so that the conductance is modified by the interference of coherent paths corresponding to multiple windings around the perimeter. Aharonov-Bohm oscillations thus have a rich pattern of harmonics (see ref. 5 for details), the relative amplitude of which depends on disorder. As discussed in section I, G(B ) traces strongly deviate from a pure sine function. Since disorder modifies the relative contribution of Aharonov-Bohm harmonics to the conductance, this effect can be tuned by applying a constant transverse magnetic field, as shown in Fig. 3, Left). In the wider Bi 2 Te 3 quantum wire considered in this study, L SS ϕ ≈ L p , so that only the fundamental h/e Aharonov-Bohm harmonic contributes to quantum interference, and periodic oscillations can be directly seen in the conductance. For such a wide nanostructure, the slowly-varying background is not negligible, but periodic quantum oscillations can be easily separated since the Aharonov-Bohm period is small. The influence of disorder on the fundamental harmonic is evidenced by the small phase shifts induced by a transverse magnetic field (see Fig. 3, Right and inset). All mesoscopic conductors studied have a length L that is comparable to or much longer than the transport mean free path l tr , so that the longitudinal motion of helical Dirac fermions is diffusive. As expected from theory for quantum coherent transport in a mesoscopic conductor, the amplitude of conductance fluctuations is reduced in long wires when L > L ϕ (T ), due to averaging between uncorrelated coherent segments. The amplitude of nonuniversal conductance fluctuations is also reduced when the wire length is increased, but this is already happening for L ϕ (T ) > L > l tr . As discussed in the main text for a wire length L 2 /l tr 3, and as also seen in Fig. 4 (L 1 /l tr 1) and Fig. 5 (L 3 /l tr 20), a remarkable property of the modulation found in the standard deviation δG rms (B ) is its relatively weak temperature dependence, which does not depend on the wire length. It is determined by the transverse quantization and the condition that the energy level broadening Γ is much smaller than the large transverse energy quantization ∆, even for L > l tr . A direct consequence is that the phase coherence length does not directly control the temperature dependence of the modulation of δG rms (see also III D). Besides, this modulation has no more temperature dependence below T * ≈1 K. Since the disorder broadening Γ of energy levels should be independent of the length of a conductor when L ≥ l tr , the same crossover occurs when thermal broadening becomes smaller than Γ, independent of the wire length, which corresponds to our observations, as discussed in the main text.
It is also important to remark that the relative change in δG rms is much larger than the conductance change due to the Aharonov-Bohm effect, as clearly seen in Fig. 6 for the wire length L 2 = 1 µm, with a relative change of about 10% and 1%, respectively. Furthemore, the amplitude of conductance fluctuations does not scale with the conductance, as expected in the metallic limit (number of modes N = 80 1). For instance, the modulation of δG rms is very large in the range [3T-4T] whereas the conductance does not change much. A thorough analysis is given in section II D, confirming that the amplitude of non-universal conductance fluctuations has no correlation with the conductance. This result finds a simple explanation in the fact that all opened channels contribute to the surface conductance whereas only a limited number of nearly-opened or nearly-closed channels close to E F contribute to conductance fluctuations.  Longitudinal-field dependence of δGrms, measured at different temperatures. The modulation of the variance is strongly damped with respect to shorter wires, but it is still visible and it shows a temperature dependence similar to other wires.
C. Non-universal conductance fluctuations in a wider Bi2Te3 quantum wire Similar results are obtained with Bi 2 Te 3 quantum wires. Due to the weaker quantum confinement, the amplitude of non-universal conductance fluctuations is smaller, but still measured with a good accuracy. As shown in the main manuscript for a length L 1 = 740 nm, the relative change in δG rms is about 12% whereas it is only 0.1% for the conductance (due to a higher number of opened conduction modes). Moreover, it can be directly seen that there is no correlation between the conductance and the flux-modulation of its variance. The same behavior was found for the longer nanowires, as shown in Fig. 7.

D. Absence of correlations between the conductance and conductance fluctuations
Similarly to conductance fluctuations, both the mean conductance G and the conductance at a specific B ⊥ (labeled G B⊥ below) depend on B and are modulated by the introduction of an Aharonov-Bohm flux. However, and contrary to the case of a ballistic conductor with a small number of conductance channels 7 , their is no proportionality between δG rms and G or G B⊥ , as expected for the large number of modes considered here. A simple way to show this is to plot both δG rms (B ) and G (B ) on a full scale. If δG rms would be proportional to G , the relative fluctuations of both quantities ∆(δG rms )/δG rms and ∆ G / G , with · · · being the average value over the B range measured, should be of the same order of magnitude when changing the flux. This is obviously not the case, as seen in Fig. 6 and in Fig. 8, Left), and all our results give ∆(δG rms )/δG rms ∆ G / G , a situation which is specific to weakly-coupled quantized modes.
More generally, we could not find any correlation between δG rms and G (or G B⊥ ), as shown in Fig. 8, Right). In this figure, the dotted line refers to the proportionality between δG rms and G (or G B⊥ ). Such a scaling can be ruled out, and the large amplitude of the flux-induced modulation of δG rms rather gives a broad vertical line.
The zoom-in inset shows the absence of simple correlations between the conductance and its standard deviation. We stress that the size of the "cloud" of reproducible data points is much larger than the error bars shown in the upper right of the inset. In the case of weakly-coupled spin-helical Dirac modes, conductance fluctuations are dominated only by a small number of opened channels and their amplitude directly depends on the flux dependence of their transmissions, with little correlations with all other propagating modes, whereas the conductance is determined by all opened channels. As a consequence, for a large chemical potential, the relative change in the conductance variance can be much larger than that of the conductance, and an increase of the conductance can nevertheless result in a decrease of its disorder-induced fluctuations.
We also report the same analysis for the results obtained with the Bi 2 Te 3 nanoribbon. As seen in Fig. 9, similar conclusions can be drawn, confirming the absence of correlations between the conductance and its variance for our 3D topological insulator quantum wires. As discussed in the main text, our combined experimental and theoretical study reveals that this is a specific property of spin-helical Dirac fermions in presence of quantum confinement, retaining ballistic transport properties despite strong disorder (L > l tr ) and a high metallicity (N = E F /∆ 1).

E. Quantitative estimations of the transport length
To calculate the transport length, different reasonable assumptions about the Fermi energy and the bulk contribution to the conductance of Bi 2 Se 3 nanowires have to be made. The values obtained below are based on previous studies we realized with similar nanostructures ( 3,8 ) and the value of l SS tr used in the main text corresponds to an upper bound. An accurate measurement is made difficult by the finite contact resistance, but realistic values are found in the 150nm-300nm range.
A first method is to infer the value of l tr from the Drude formula in the 2D limit (large number of transverse modes) G = e 2 /h × π × E/∆ × l tr /L, with L being the length between the contact. This gives a transport length of 185 nm for L 2 = 1µm and of 135 nm for L 1 = 400 nm. Yet, even if relatively small, the contact resistance can significantly influence the estimation of the transport length, due to the rather large conductance of the nanostructures. Taking a typical 150 Ω contact resistance into account, the values of l tr become 255 nm for L 2 = 1µm and 270 nm for L 1 = 400 nm. Also, for such a Fermi energy (E F ∼ 250 meV is typical for Bi 2 Se 3 nanostructures, see 8 ), the contribution of bulk carriers to the total conductance cannot be neglected. In the nanowire studied here, it amounts to about half of the total conductance 3 so that the value found above are overestimated. Based the ratio G SS /G bulk ≈ 1.2 measured in 3 , we find l tr = 140 nm for L 2 = 1µm and and l tr = 150 nm for L 1 = 400 nm.
Another method is based on trans-conductance measurements, as reported in 3 , and it gives values close to the low estimations made above. Altogether, an upper bound of 300 nm for l tr is very reasonable, and all mesoscopic conductors studied here satisfy the condition L > l tr . A similar analysis for Bi 2 Te 3 nanowires gives l tr < 450 nm.

A. Theoretical model
To theoretically model our experiments we adapt a continuous Dirac fermion description of the surface state 4 , and take the bulk to be an inert insulator. Although a finite coupling to residual bulk states can increase the scattering of surface states, it remains a small energy that does not modify the energy spectrum of 1D spin-helical surface modes, and therefore does not change the conclusions obtained from our calculations. Explicitly, the surface Hamiltonian reads where v is the Fermi velocity, σ = (σ x , σ y ) are Pauli matrices, and the applied magnetic field B = ∇ × A. We take r = (x, y) with x the direction along the length of the wire, and y the periodic transverse direction. The spin of the Dirac fermion is constrained to lie in the tangent plane to the surface, and therefore rotates by 2π going once around the circumference of the wire. This leads to a Berry's phase of π that is taken into account via the boundary condition with W the wire circumference. Disorder is introduced through the time reversal invariant scalar potential V with correlator whereby g is a dimensionless measure of the disorder strength and ξ gives the characteristic length scale of potential variations. The Hamiltonian (1), together with metallic lead boundary conditions, defines a scattering problem that is solved via a transfer matrix technique 9 , giving the conductance through the Landauer-Büttiker equation. µ is the chemical potential in the wire.

B. Transmissions of weakly-coupled quantized surfaces modes and Quantum interference
The energy dependence of the transmissions of quantized transverse modes is shown in Fig. 10a, for a flux Φ/Φ 0 = 1 2 . The perfectly-transmitted mode (m = 0) has a constant transmission equal to one. Despite disorder, the transmissions of higher-energy modes also tend to unity when their longitudinal kinetic energy exceeds their confinement energy, so that the conductance is determined by all opened conduction channels. Small fast oscillations are due to Fabry-Pérot interference between metallic contacts, typical of quasi-ballistic transport, and some resonances are observed due to disorder (see section III D). An example of quantum corrections to the conductance calculated for a fullycoherent nanowire is shown in Fig. 10b. For a constant transverse field, the magneto-conductance traces G(B ) correspond to Aharonov-Bohm oscillations, as shown in Fig. 10c for B ⊥ = 1 T, which result from multi-harmonic interferences for every opened conduction modes. Similar to experiments, their peak-to-peak amplitude is found close to the conductance quantum e 2 /h. The nature of magneto-conductance traces G(B ⊥ ) is however different. Contrary to Aharonov-Bohm oscillations, which result from all opened modes, the statistics of conductance fluctuations is determined only by a limited number of modes, close to E F , which are nearly opened or nearly closed. As discussed in the main text, this statistics is not universal and δG rms has a periodic evolution with the flux that is typical of weakly-coupled Dirac fermions in presence of quantum confinement. C. Transverse-field dependence of the quantized energy spectrum As previously described in ref. 10 , we calculated the energy spectrum of a topological insulator quantum wire with a rectangular cross section (height h = 20 nm, width w = 170 nm). Fig. 11 shows the dependence of the quantized energy spectrum with a transverse magnetic field B ⊥ . Since a transverse magnetic fields breaks the initial symmetry of the Dirac Hamiltonian, it favors the mixing of transverse modes. Close to the Dirac point, this mixing can be so strong that edge states rapidly develop when B ⊥ is increased 10 , as seen in Fig. 11a). For a fixed value of B ⊥ , the degree of mixing is reduced when the energy of the modes becomes larger than the Zeeman energy. Therefore, B ⊥ has little influence on the spectrum of high-energy modes and, in the field and energy range studied (see Fig. 11b), the slow increase in the transverse energy remains smaller than the level spacing ∆. Therefore, conductance fluctuations can be studied from G(B ⊥ ) traces, independently from the Aharonov-Bohm physics. Besides, the small transverse field-induced change of the conductance cannot explain the large modulation of the conductance variance reported.

D. Transmission of a transverse mode and disorder broadening
Based on our calculations, we show here that the disorder broadening can be deduced from the energy dependence of the transmission T (E) of a transverse mode. As an exemple, we plot below the results obtained for the transmission for the transverse mode m = 9 (Fig. 12). Close to the onset energy E m on of a surface mode, with E m on = (m + 1/2)∆ at B = 0 (that is, for Φ AB = 0), the transmission is very sensitive to the disorder configuration and it shows a couple of resonances, as shown by vertical arrows in Fig. 12. In this low-energy range, ∂T /∂E is large, and the modulation of the conductance variance is significant, whereas it decreases at higher energy, so that the nearly opened channel does not contribute to any modulation of the conductance variance anymore. Much smaller oscillations of T (E) seen at higher energies correspond to Fabry-Pérot resonances between metallic contacts.
Importantly, the resonances induced by disorder allow us to estimate the broadening Γ, as inferred from their width at low energy. Whereas a strongly disordered system has broad overlapping resonances, a rather clean system has sharp resonances (and indeed, this case of weak disorder is realized in a disordered 3D topolgical insulator due to anisotropic Figure 11: Influence of a transverse field on quantized modes in a quantum wire. a, Transverse field dependence of the energy spectrum of a nanoribbon with a height h = 20 nm and width w = 170 nm. b, Zoom in the high-energy range, around the value of the Fermi energy for our Bi2Se3 quantum wires. scattering). Still, at high-enough temperatures, the temperature broadening of the Fermi-Dirac distribution (∼ 4k B T ) further smooths the resonances if 4k B T Γ. The temperature T * is then defined by 4k B T * = Γ. From the numerical calculations, we obtain a ratio Γ/∆ ≈0.2, a value different but rather close to the experimental result Γ/∆ ≈0.06.
Besides, this clarifies the origin of non-universal conductance fluctuations, which result from the nonmonotonous energy dependence of ∂T /∂E and involve a couple of slightly-opened modes at E F . Therefore, conductance fluctuations are dominated by the highest energy modes only, and the modulation of the conductance variance is determined by transverse modes with a small kinetic energy (that is, with E F close to E m on ). Near E F = 250meV, this corresponds to three or four partially opened channels. Conductance fluctuations show a maximum when the AB flux is such that E F coincides with the onset of the highest energy mode (∂T /∂E is maximum) and they decrease when the AB flux pushes E on away from E F (below or above) so that ∂T /∂E becomes smaller.
Note that for high-energy transverse modes, a B ⊥ sweep is equivalent to an energy scan, since the perpendicular magnetic field induces an overall shift of about ∆/2 in the field range studied. As a result, the fine structure of T (E) for slightly opened modes is well-probed by a B ⊥ sweeps, and the non-universal nature of conductance fluctuations affects individual magneto-conductance traces, even for a fixed chemical potential.