Superconductivity in a chiral nanotube

Chirality of materials are known to affect optical, magnetic and electric properties, causing a variety of nontrivial phenomena such as circular dichiroism for chiral molecules, magnetic Skyrmions in chiral magnets and nonreciprocal carrier transport in chiral conductors. On the other hand, effect of chirality on superconducting transport has not been known. Here we report the nonreciprocity of superconductivity—unambiguous evidence of superconductivity reflecting chiral structure in which the forward and backward supercurrent flows are not equivalent because of inversion symmetry breaking. Such superconductivity is realized via ionic gating in individual chiral nanotubes of tungsten disulfide. The nonreciprocal signal is significantly enhanced in the superconducting state, being associated with unprecedented quantum Little-Parks oscillations originating from the interference of supercurrent along the circumference of the nanotube. The present results indicate that the nonreciprocity is a viable approach toward the superconductors with chiral or noncentrosymmetric structures.

C hirality of crystal or magnetic structures in solids was recently recognized as a powerful source of unique optical and electronic properties and novel functionalities. For instance, it is well known that polarization of light or spin of electron are sensitive to the chirality of lattice or magnetic structure [1][2][3][4] and electric transport reflecting the chiral structure are also reported [5][6][7] . Among them, effects of chiral structures on superconductivity have not been investigated so far because of the lack of suitable materials. One of the interesting superconducting materials with chirality is carbon nanotube (NT) 8,9 . Chiral structures and their relations to the electronic properties in carbon NTs have been well studied by Raman scattering 10,11 , scanning tunnel microscope 12 and even magneto-chiral transport 6 . However, since superconductivity in carbon NTs [13][14][15][16] has been investigated only in the assembled form of single, double or multi-walled NTs, relations between superconducting transport and the chirality in individual tubes have remained elusive.
Tungsten disulfide (WS 2 ) is a member of transition metal dichalcogenides (TMDs), which are now attracting significant attention as two dimensional (2D) materials beyond graphene with the potential application for electronics, photonics, spintronics, mechanics, as well as valleytronics 17,18 . Recent systematic studies clarified many TMDs including WS 2 , which are semiconductors without carrier doping, exhibit superconductivity under the ionic gating 19,20 . Importantly, TMD can form tubular structures with noncentrosymmetric chiral structures [21][22][23][24][25][26] . The semiconducting property of the WS 2 NT was indeed demonstrated using the field effect transistor devices 24 . Superconductivity in such a noncentrosymmetric chiral cylinder, once it is realized, is a potential candidate for searching the exotic quantum phenomena and nontrivial Cooper pairing 27,28 . One of the manifestations of the chiral structure in the electronic transport is the unidirectional resistance. As shown in Fig. 1a,b, the two directions of current injection are not identical due to the chiral nature of the conducting substance when the magnetic field is applied parallel to the tube. Such nonreciprocity is highly anticipated to yield nontrivial quantum transport particularly in the superconducting states.
In this study, the transport properties of individual WS 2 NT have been investigated by using the ionic liquid gating technique and resistance measurement on both first and second harmonic signals in alternative current (AC) mode. We have observed ambipolar transfer curve in electrostatic doping region and the emergence of superconductivity by electrochemical doping. The superconducting properties of individual WS 2 NT have been further investigated, in which the observed anisotropy of the superconductivity and Little-Parks (LP) oscillation 29 are consistent with tubular structure of WS 2 NT. More importantly, we have experimentally discovered nonreciprocal superconducting transport via the second harmonic signal, being suggestive of chirality effect on superconductivity. Such nonreciprocal signal is largely enhanced in the superconducting state and affected by the magnetic flux quantum, showing periodic oscillations. The present study paves a route for studying the interplay between superconductivity and chirality or noncentrosymmetry.

Results
Sample characterization. WS 2 NTs were synthesized following the literature [21][22][23] . Figure 1c shows a transmission electron microscope (TEM) image of a single WS 2 NT (see Supplementary  Fig. 1). The tube has a multi-walled structure, with the outer/ inner diameters estimated as 132/107 nm, respectively, indicating that the layer number is B20. According to the literature 21-23 , the tube part has a 2H-polymorph-layered structure of WS 2 , where each tungsten atom is surrounded by six sulfur atoms in a trigonal biprism coordination (space group P63/mmc). The outer diameter distribution of tubes in the batch used for this measurement takes a broad maximum around 100 nm (See Supplementary Fig. 4). An electron diffraction pattern of a single WS 2 NT is displayed in Fig. 1d. The red arrow and yellow hexagon represent the direction of tube axis and diffraction pattern from the zigzag type NT, respectively. The different walls of the tube can have different chirality. In addition to the contribution of the zigzag type NT, we can see the pair of tilted hexagonal pattern which confirms the co-existence of chiral structures in this NT. The TEM analysis indicates that the tubes used for the transport measurement are multi-walled WS 2 NTs with chirality, having the outer diameter of nearly 100 nm (See Supplementary Fig. 4).
Ionic liquid gating on WS 2 NT and superconductivity. We fabricated an individual tube device as shown in Fig. 1e,f, and measured gate responses of the transport characteristics. Based on our previous research on the systematic study of superconductivity in TMDs 19 , we used KClO 4 /polyethylene glycol electrolyte as the gate medium to facilitate electrochemical intercalation of potassium ions into the layered structure of WS 2 . Figure 2a displays the source-drain current (I DS ) of the individual WS 2 NT device against the gate voltage (V G ) between À 2 and 3 V. The device nicely operates in an ambipolar mode, in a similar manner to the 2D devices, showing marked contrast with the unipolar response of WS 2 NTs in the solid gated field effect transistor 24 . This indicates the strong gate coupling of the presently used ionic medium. The transistor operation is most likely in the electrostatic mode in this regime, considering the ambipolar behaviour is reversible and repeatable. When V G was increased to 8 V at a constant rate of 50 mV s À 1 , we found a saturation of I DS , similarly to the case of 2D WS 2 (ref. 19). When V G was kept at 8 V for a couple of minutes, we encountered another dramatic increase of I DS by more than two orders of magnitude as shown in Fig. 2b. This I DS increase is presumably attributed to intercalation of K þ ions into WS 2 NTs.
When we cooled down the device to 2 K keeping V G at 8 V, superconductivity appeared at T c ¼ 5.8 K, defined as the temperature corresponding to the half of normal state resistance (Fig. 2c). In contrast to the K-intercalated 2D WS 2 multilayer with T c of 8.6 K (ref. 19), the superconducting transition here is shifted to lower temperature and considerably broadened, potentially due to the reduced dimensions or lack of commensurability between the different walls.  We then investigated the anisotropy of the observed superconductivity for sample 3 (V G ¼ 6 V). Figure 2d,e displays the temperature variation of the resistance under magnetic field H for H>z and H||z, respectively. Here z represents the tube axis direction. In the case of the H||z, the superconductivity is robust against the magnetic field and remains undefeated even under m 0 H ¼ 9 T at T ¼ 2 K, while the superconducting phase rapidly disappears for the H>z configuration with increase of magnetic field. The anisotropic superconductivity was also confirmed by the angular dependence (Fig. 2f) and temperature dependence (Fig. 2g) of the upper critical magnetic field. In Fig. 2f, the estimated upper critical field at T ¼ 3.5 K are well fitted by the anisotropic Ginzburg-Landau model m 0 H C2 ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a cos y ð Þ 2 þ b sin y ð Þ 2 p with fitting parameters a ¼ 0.13 T À 1 and b ¼ 0.75 T À 1 . The temperature dependence of the critical magnetic field (Fig. 2g) cannot be explained either by the simple 2D or 3D models, implying that the system is of intermediate dimension.
Here we should note that the upper critical field at T ¼ 0 K for the H||z configuration seemingly exceeds the Pauli paramagnetic limit m 0 H BCS P ¼ ffiffi Little-Parks oscillations. Figure 3a shows the AC magnetoresistance of sample 4 (V G ¼ 12 V) at various temperatures around T c in H||z configuration. In addition to the robustness of the superconductivity discussed above, the magnetoresistance observed via the first harmonic signals in AC resistance (R o ) shows periodically oscillating behaviour in the low-magnetic field region. These oscillations during the superconducting transition known as LP effect 29  ¼ h 2e (Fig. 3c). This is consistent with the diameter distribution histogram of the same batch shown in Supplementary Fig. 4. These results provide firm evidence that superconductivity occurs in the tubular region in the present WS 2 sample.

Nonreciprocal superconducting transport in chiral WS 2 NT.
To clarify the characteristic properties due to the chiral structure, we have measured the second harmonic signals in the AC resistance (R 2o ). In noncentrosymmetric systems, the cross term of magnetic field H and electric current I in the resistance is allowed on the basis of the symmetry argument, which indicates the difference between the forward and backward transports under magnetic field [5][6][7] . This term generates the nonlinear voltage response, which can be measured as the second harmonic components in the AC resistance (See Supplementary Note 3).
Especially in chiral systems, this nonreciprocal electric transport called magneto-chiral anisotropy has been reported in several materials with different chiral degrees of freedom [5][6][7] . Phenomenologically, magneto-chiral anisotropy can be expressed as where both the magnetic field H and electric current I are parallel to the chiral axis and g is the ratio of R 2o to the normal resistance R 0 . So far, there has been no report of such a phenomenon in superconducting phase and it is of great interest whether excitation or quasiparticle in superconducting state without inversion symmetry generates such an unidirectional magnetoresistance. Figure 4a shows the magnetic field dependence of second harmonic components R 2o in the AC resistance measured in sample 4. Finite antisymmetric R 2o signals were observed in the superconducting region, which unambiguously indicate the unidirectional electrical transport due to the chiral symmetry (Fig. 1a,b). In sharp contrast, R 2o signals are negligibly small in the normal state (see Fig. 4c), indicating that magneto-chiral anisotropy signals are significantly enhanced in the superconducting phase due to the coherent nature of superconductivity.
The observed R 2o signal has the two characteristic structures: that is, the broader antisymmetric components and the periodically oscillating terms in the low-magnetic field region. In the broader antisymmetric components, there are characteristic minima indicated by small triangles in Fig. 4a, which are dependent on temperature and enhanced at low temperature. The temperature dependence of the characteristic minima show similar behaviour as the critical magnetic field (Fig. 4b), strongly suggesting that the observed signals are related to the superconducting transition. (We show the details of temperature dependence of the antisymmetrized signals in the Supplementary  Fig. 12 and discuss the possible co-existence of different chiral types in WS 2 NTs in the Supplementary Note 3) The oscillating terms, on the other hand, show the periodicity of f 0 (Fig. 4c-e) and enhancement around T c (Fig. 4f) similar to the LP oscillations (Fig. 3b), indicating that both have the same origin. Interestingly, this term shows the stepwise behaviour at low temperature (Fig. 4e, from m 0 H ¼ À 1 T to 1 T) distinct from the conventional linear relation of the external magnetic field according to equation (1). This analysis indicates that the nonreciprocal supercurrent also has the interference nature and is affected by the flux quanta passing through the NT.

Discussion
The observed second harmonic signals in AC resistance, together with LP oscillations in the first harmonic signals, are the direct manifestations of superconductivity in chiral NTs, however, the detailed mechanisms of the asymmetric electric transport and pairing symmetry (parity mixing) in the superconducting state needs to be further pursued. In light of this work, we expect various superconducting materials with broken inversion symmetry offer a similar transport, which provides a powerful   (c-e) Comparative plots of R o and R 2o in low-magnetic field region at T ¼ 6, 4 and 2 K. Periodic oscillations were observed for both signals in the superconducting state (2 and 4 K), which disappears rapidly in the normal state (6 K). R 2o shows the stepwise behaviour at low temperature (2 K). (f) Temperature variation of the resistance and magnitude of the quantum oscillations in R o and R 2o . For R 2o oscillations, we estimate the magnitude of the jumps which are observed in low-field region. Both oscillating signals are enhanced around T c . approach for probing the exotic superconducting state in a variety of noncentrosymmetric systems.

Methods
Sample preparation. The WS 2 NTs were synthesized following the literature 22,23 . The starting materials for the WS 2 NT synthesis route were spherical tungsten oxide nanoparticles, which were sulfurized by solid-gas reaction with hydrogen and hydrogen sulfide at elevated temperatures (4800°C). During this one-pot reaction, tungsten suboxide whiskers grow and are subsequently sulfurized into WS 2 NTs of B100 nm in diameter and up to 20 micron in length. These two main steps of the reaction-oxide whiskers growth and sulfurization-occur under the same H 2 S/H 2 gas flow regime and are not separated in space following each other in a self-controlled mechanism.
Device fabrication. WS 2 NTs were dispersed in isopropyl alcohol solvent by ultrasonication for 20 min. A droplet of the suspension was spin-coated on a Si/SiO 2 (3,000 Å) substrate, and immediately covered by polymethyl methacrylate. The isolated WS 2 NTs were subsequently chosen by an optical microscope. The device pattern was designed via standard electron beam lithography techniques and developed by mixed solution of methyl isobutyl ketone and isopropyl alcohol with the ratio of methyl isobutyl ketone:isopropyl alcohol ¼ 1:3. After the deposition of Cr/Au (5 nm/90 nm), pads and gate electrode were covered by photoresist and Cr/SiO 2 (5 nm/20 nm) was subsequently deposited to protect electrodes from the chemical reaction by ionic gating. Finally, we removed the redundant polymethyl methacrylate and gold by dipping the substrate into acetone for more than 1 h. KClO 4 /polyethylene glycol was selected as a gate medium 19 .
Transport measurements. All the transport properties have been measured in a Quantum Design Physical Property Measurement System with a horizontal rotator probe under He-purged and high-vacuum environments. High-vacuum mode was used when the temperature was higher than 200 K, while the system was kept under He-purged condition below 200 K. Gate voltage was applied by a Keithley 2400 sourcemeter at 300 K with a sweeping rate of 50 mV s À 1 under high-vacuum condition. Both the first and the second harmonic signals of the AC resistance have been measured by a lock-in amplifier (Stanford Research Systems Model SR830 DSP) with a frequency of 13 Hz. During the AC resistance measurements, the phase of the first (or second) harmonic signal was kept around 0 or p 2 À Á , which is consistent with the theoretical expectation. All of the R o (or R 2o ) signals discussed in the main text were obtained as x (or y)-component of the lock-in measurement.
Data availability. All of the experimental data supporting this study are available from the corresponding author.