Universal conductance fluctuations in a Bi1.5Sb0.5Te1.8Se1.2 topological insulator nano-scaled Hall bar structure

We present low-temperature magnetotransport measurements characterizing the promising quaternary Bi1.5Sb0.5Te1.8Se1.2 topological insulator material. The measurements performed on a nano-Hall bar grown by selective-area molecular beam epitaxy revealed pronounced universal conductance fluctuations. It is shown that these fluctuations originate from phase-coherent loops within the topologically protected surface states. Furthermore, the decay of the fluctuation amplitude with increasing temperatures suggests a quasi one-dimensional transport regime.


Introduction
Three-dimensional topological insulators (TI) such as Bi 2 Te 3 or Bi 2 Se 3 are a new material class that is attracting increasing interest as they are characterized by a bulk electronic gap and a metallic surface behaviour that is caused by a nontrivial topology of the electronic band structure [1][2][3]. Due to the nonzero Berry phase of the spin texture, TI nanoribbons in proximity to an s-wave superconductor are supposed to hold Majorana zero modes. As a matter of fact, these hybrid systems are considered to be excellent candidates for topological quantum * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. computing [4][5][6]. Although current research towards this goal is promising, one finds that material issues are still a major challenge.
Particularly in binary three-dimensional TIs the contributions from the bulk are dominant in transport, i.e. Bi 2 Se 3 is found to be n-type doped owing to Se vacancies while in Sb 2 Te 3 antisite defects result in a p-type behaviour. Consequently, a lot of effort is put into reducing or even eliminating the bulk contribution so that the transport properties are governed by the topological surface channel. An obvious route is to combine two binary materials with opposite type of conductance to form a ternary compound to reduce the bulk carrier concentration by compensation [7][8][9][10] or by means of a pn junction in an epitaxial heterostructure [11]. A further improvement and tuning of the material properties can be achieved by moving over to quaternary compounds. For a large proximity-induced topological superconducting energy gap that may hold Majorana zero modes, a TI with a free-standing surface-state Dirac cone and a Fermi level centered at the Dirac point would be ideal [12]. In fact, in BiSbTeSe 2 a topological surface state quantum Hall effect was observed indicating the intrinsic conduction properties of the material system [13][14][15]. Further amendment of the suppression of bulk conductance was achieved by optimizing the composition of the quaternary alloy Bi 2−x Sb x Te 3−y Se y , e.g. Bi 1.5 Sb 0.5 Te 1.7 Se 1.3 , proved to have a low bulk carrier density [16][17][18].
Searching for an optimization of the ratio between transport contributions of the surface states and the bulk, different attempts of using quaternary TI systems like Bi 2−x Sb x Te 3−y Se y show promising results [16,18,[32][33][34]. Nevertheless, for most stoichiometries there is only little information about the magnetotransport properties. In this context the main goal of the present paper is further elucidation of quantum transport in a quaternary TI.
In order to assess the suitability of quaternary TI nanostructures for topological quantum circuits we focus on phasecoherent transport in a nano-scaled Bi 1.5 Sb 0.5 Te 1.8 Se 1.2 Hall bar structure. The choice of composition was motivated by previous work of Ren et al [16] where this composition of the bulk BiSbTeSe crystal, grown by the Bridgman technique proved to have intrinsically low conduction contributions from the bulk. By performing accurate analysis of conductance fluctuations detailed information on the phasecoherence is gained. In this work, we employ molecular beam epitaxy (MBE) for the sample fabrication, as it offers several advantages with respect to the Bridgman growth technique. In MBE the fabrication of atomically thin films with a welldefined composition, as well as selective-area growth on prestructured substrates is possible.

Experimental
Using an MBE setup, TI nano-Hall bars are fabricated by selective-area growth [35,36]: First, 5 nm of SiO 2 and 20 nm of Si 3 N 4 are deposited via low pressure chemical vapor deposition on a Si(111) substrate. Subsequently, trenches in the Si 3 N 4 layer are patterned using electron-beam lithography and etched by reactive ion etching. After a cleaning procedure with Piranha solution and 1% hydrofluoric acid, which removes the SiO 2 and passivates the Si surface, the substrate is introduced into the MBE chamber and heated to 700 • C to desorb the hydrogen passivation. The Bi 1.5 Sb 0.5 Te 1.8 Se 1.2 layer is grown selectively into the trenches at 355 • C substrate temperature with a flux ratio of 3:1:90:7. Due to the amorphous structure, the TI material diffuses on the Si 3 N 4 surface and only crystallizes in form of quintuple layers at the periodic Si(111) substrate surface in the trenches. The composition was confirmed by Rutherford-backscattering measurements (see supplementary material). The thickness of the TI layer is 20 nm. The sample is capped in-situ below 0 • C with 3 nm of Al, which oxidizes at ambient conditions and forms a capping layer [37]. Finally, the structure was contacted with 50/100 nm Ti/Au contacts via ex-situ electron beam evaporation.
The magnetoresistance measurements are conducted on a selectively-grown nano-Hall bar structure comprising a length of 400 nm between the outer Hall contacts (3 and 4 in figure 1(a)) and a width of 80 nm. The cryogenic magnetoconductance measurements are performed in a variable temperature insert cryostat that includes a superconducting 13 T magnet. In order to change the tilt angle of the magnetic field from parallel along the current path (θ = 0 • ) to perpendicular (θ = 90 • ), the sample is mounted into a high-precision rotational sample holder. The Hall bar is measured between a temperature of 1.5 K and 30 K. For the electrical measurements, using standard lock-in technique an AC current of 10 nA is applied. All measurements are performed in a four-terminal setup, where the contacts for the measurements lay on different sides of the Hall bar due to some broken contacts. The longitudinal and Hall signal are gained by symmetrizing and anti-symmetrizing the data:

Results and discussion
From Hall measurements at a magnetic field B between −13 T and 13 T and a temperature T of 1.5 K an overall charge carrier concentration of 2.26 × 10 13 cm −2 is derived. The transport is identified to be n-type. This is in agreement with the angle-resolved photo-emission spectroscopy (ARPES) measurements shown in figure 1(b) (see also supplementary material), i.e. the Dirac point is located below the Fermi level. From the energy difference between Fermi level and Dirac point an electron concentration of 6.9 × 10 12 cm −2 is roughly estimated for each surface channel. Thus, although not directly visible in the ARPES data, a small contribution of bulk carriers is still present in our sample owing to a non-complete compensation. Probably, there is some band bending present in the Bi 1.5 Sb 0.5 Te 1.8 Se 1.2 layer, so that at the surface the Fermi level lies in the bulk band gap as seen in the ARPES measurements, while deeper in the layer the Fermi level crosses the bottom part of the conduction band. Nevertheless, compared to binary TIs prepared by MBE the total carrier concentration found here is significantly reduced [38,39]. In fact, our value is comparable to compensated (Bi 1−x Sb x ) 2 Te 3 layers [10]. The relatively low total mobility of µ = 155 cm 2 V −1 s −1 derived from the transport data indicates that a bulk channel contributes to the transport. A low mobility is expected because of the statistically distributed atom positions in a quaternary composition as compared to the regular lattice structure with translational symmetry of a binary material [40]. Below 7 K an insulating behaviour is observed, i.e. an increase of resistance with decreasing temperature, as shown in the supplementary material. This behaviour is in agreement with previous studies on quaternary TIs [16,18,33,34,41].
The longitudinal magnetoresistance measurements for θ between 0 • and 90 • in steps of 2 • are shown in figure 2(a). The conductance fluctuations dominate the signal, so that they overlay other interference features such as WAL. The angular behaviour of a representative maximum is highlighted exemplarily. The shift in magnetic field of the maximum under rotation for θ > 20 • is fitted in figure 2(b) and it shows a 1/ sin(θ) dependency. Very similar results are achieved with other extrema. This justifies the assumption that the observed modulations can be attributed to UCFs that are basically only sensitive to the vertical component of the magnetic field. Furthermore, the data is decomposed into the respective components by interpolating and plotting it on a 2D grid in figure 3. Some vertical lines are visible that have the same color and thus a similar resistance. These features show the reproducibility of the UCFs with respect to vertical component B ⊥ of the magnetic field. The high amplitude of the UCFs allows to analyse the Fourier transformation of the signal in more detail [30]. A schematic drawing of phase-coherent loops on the surface of a TI that are associated with the UCF signal is depicted in figure 4(c). The UCFs observed in the magnetoresistance measurements are only sensitive to the angular dependent projection of the loop area. In figure 4(a) the Fourier transformed signal from figure 2(a) is shown. The curves for different orientations of the magnetic field are presented and the amplitude is logarithmized and color coded. In general, the UCF amplitude scales inversely with the corresponding oscillation frequency and thus with the loop size, so that smaller loops result in a higher amplitude. For θ > 20 • distinguishable paths show up that have a sinusoidal angular dependency. Those paths are traced by hand in figure 4(b) and fitted by assum- [30]. Here, f B is the frequency of the magnetic field resulting from the Fourier transformation of the data, f 0 is the maximal oscillation frequency for each loop and θ 0 is the angular offset with respect to the 90 • -orientation. The corresponding results are shown in figure 4(d), where the loop area S 0 is calculated from the fit parameter f 0 by multiplying it with the magnetic flux quantum Φ 0 = h/e. Here, h is the Planck constant and e is the elementary charge.
A first estimate of the phase-coherence can be made by assuming a rectangular shape of the closed loop for the highest maximum shown in the figure at f 0 = 2.92 T −1 . One finds that l ϕ is at least l ϕ = 230 nm assuming a width of 80 nm due to the geometry of the sample. A more precise value of the phase-coherence length is obtained from the correlation field B c of the fluctuations in the magnetoconductance G(B). The correlation field is derived from the autocorrelation function [42]: The correlation field B c corresponds to half of the maximum of F(∆B). For a temperature of T = 1.5 K, a correlation field of B c = 0.20 T is calculated. Assuming a diffusive quasi onedimensional transport regime, the phase-coherence length is estimated from [43]: where w is the width of the sample. The result for the phasecoherence length is l ϕ = 243 nm. This value is in the same order as l ϕ estimated before from the largest phase-coherence loop and it is in the range reported for binary and ternary threedimensional TIs [30,44,45]. When having a closer look at the angular offset parameter θ 0 of the loops in the insert of figure 4(d), one finds that the displacement with respect to the perpendicular orientation of the magnetic field and the Hall bar is only a few degrees for all fits. This indicates that the loops are predominantly located in the horizontal plane, i.e. in the topologically protected surface states. Indeed, for the highest frequency feature at f 0 = 2.92 T −1 at θ = 90 • , corresponding to a lateral extension of the loop of 155 nm, one estimates a maximum possible tilt angle θ of the loop of about 7 • when assuming that the loop extends from the top to the bottom surface. For smaller frequencies f 0 the maximum tilt angle would be even larger, e.g. as large as 79 • for the lowest frequency of f 0 = 0.077 T −1 . Obviously, the values of θ 0 shown in figure 4(d) (inset) are considerably smaller than the values estimated here supporting our statement above that the loops are located in the topologically protected surface states. Rosenbach et al [20] unambiguously showed a non-zero Berry phase, so that the quantum transport phenomena in binary TIs must be attributed to transport in the topologically protected surface states. We transfer this information to quaternary TIs, since the experimental transport data is very similar. Moreover, no pronounced contributions of AB type oscillations arising from electrons travelling on closed loops around the nanowire can be seen in the magnetotransport signal as it was seen in binary or ternary materials [22][23][24][25][26][27][28]. One possible reason could be the amount of defects that may cause local disorder or even charge puddles [10,40]. In addition to that, for the presence of AB oscillations the surface states on the whole circumference need to contribute and form a closed loop. We conclude that possibly at least one of the four surfaces is not hosting suitable surface states e.g. due to strain that could origin from the lattice mismatch between the TI and the Si(111) substrate or the SiO 2 /Si 3 N 4 sidewalls [46]. In order to gain information about phase breaking mechanisms, temperature-dependent measurements were performed. With increasing temperature the phase-coherence length is affected by electron-phonon and electron-electron interaction [47]. The impact is a decrease of the UCF amplitude and the phase-coherence length, where l ϕ is proportional to T −2/3 for a three-dimensional system, to T −1/2 for a twodimensional system and it is proportional to T −1/3 for onedimensional systems [22]. Since the phase-coherence length derived above is larger than the width of the sample, a quasi one-dimensional behaviour and thus a proportionality of T −1/3 is expected. In figure 5(a) the temperature dependent UCF data is plotted, whereas in (b) the corresponding temperature dependent phase-coherence length calculated from the correlation field (see equation (2)) of the UCFs is shown. The curves for T ⩽ 8.5 K have a fine stepping of 0.5 K and from 10 K to 30 K a stepping of 5 K is selected. In figure 5(b) only temperatures up to 8.5 K are considered as the UCFs vanish for measurements at higher temperatures and thus the information about the phase-coherence length. The curve is fitted and it shows a T −0.34 behaviour, which is in good agreement with the expected quasi one-dimensional transport.

Conclusion
In this article it was shown that it is possible to grow high-quality quaternary TIs by MBE By performing lowtemperature magnetotransport measurements on a nano-Hall bar with a Bi 1.5 Sb 0.5 Te 1.8 Se 1.2 stoichiometry pronounced UCFs were observed. By observing the UCF behaviour under various tilt angles it was shown that the origin lays in quasi one-dimensional in-plane loops that prove the phase-coherent electrical transport in this material. The charge carrier concentration is comparable to that of ternary TIs and might be overestimated due to the diagonal arrangement of the contacts, so that an alternative to e.g. a global gate-tuning of the Fermi energy is approached.
From the decay behavior at higher temperatures, a twodimensional transport behavior of the properties was deduced. The presented measurements are another step on the long road to topological quantum computing. We recommend this type of TI material for further investigation, as the results discussed above are not only comparable to those observed for common TI compositions, but are even an improvement in some respects. In particular, as a next step towards a topological qubit, the interaction between this material and an s-wave superconductor needs to be investigated.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https://doi.org/10.26165/ JUELICH-DATA/XOCEBN.