Highly skewed current-phase relation in superconductor-topological insulator-superconductor Josephson junctions

Three-dimensional topological insulators (TI's) in proximity with superconductors are expected to exhibit exotic phenomena such as topological superconductivity (TSC) and Majorana bound states (MBS), which may have applications in topological quantum computation. In superconductor-TI-superconductor Josephson junctions, the supercurrent versus the phase difference between the superconductors, referred to as the current-phase relation (CPR), reveals important information including the nature of the superconducting transport. Here, we study the induced superconductivity in gate-tunable Josephson junctions (JJs) made from topological insulator BiSbTeSe2 with superconducting Nb electrodes. We observe highly skewed (non-sinusoidal) CPR in these junctions. The critical current, or the magnitude of the CPR, increases with decreasing temperature down to the lowest accessible temperature (T ~ 20 mK), revealing the existence of low-energy modes in our junctions. The gate dependence shows that close to the Dirac point the CPR becomes less skewed, indicating the transport is more diffusive, most likely due to the presence of electron/hole puddles and charge inhomogeneity. Our experiments provide strong evidence that superconductivity is induced in the highly ballistic topological surface states (TSS) in our gate-tunable TI- based JJs. Furthermore, the measured CPR is in good agreement with the prediction of a model which calculates the phase dependent eigenstate energies in our system, considering the finite width of the electrodes as well as the TSS wave functions extending over the entire circumference of the TI.


Main text Introduction
Three-dimensional (3D) topological insulators are a new class of quantum matters and are characterized by an insulating bulk and conducting topological surface states (TSS). These TSS are spin-helical states with linear Dirac fermion-like energy-momentum dispersion [1,2]. The TSS of 3D topological insulators (TI's) in proximity to s-wave superconductors are among the top candidates proposed to realize topological superconductors [3], capable to support Majorana bound states and promising for future applications in topological quantum computing [4,5].
A Josephson junction (JJ) made of a TI with two superconducting contacts is one of the most common platforms to study the nature of the induced superconductivity in TIs and possible topological superconductivity. One of the fundamental properties of a JJ is its supercurrent ( ) as a function of the phase ( ) difference between two superconductors, referred to as the currentphase relation (CPR), where the maximum of ( ) is the critical current ( & ) of the JJ. Given the topological protection or the prohibited backscattering from non-magnetic impurities in the TSS of 3D TIs [1,2], superconductor-TI-superconductor (S-TI-S) junctions are expected to demonstrate novel features in their CPR. While for conventional junctions the CPR is 2p-periodic, for TI-based JJs the CPR is predicted to have an additional 4p-periodic component [3,6]. This 4p-periodicity originates from the zero-energy crossing (at = p) of the Andreev bound states (ABS) and is protected by the fermion parity conservation. However, if the temporal variation of is slower than the quasiparticle poisoning time, the 2p-periodicity of the CPR is restored, which masks the unique topological nature of the Josephson junctions [6][7][8]. Nonetheless, in this case the topologically protected modes can give rise to highly non-sinusoidal 2p-periodic CPR similar to a perfectly ballistic (scattering free) JJ [7,9].
Remarkably, the measured CPR in our S-TI-S junctions are highly skewed, revealing that the superconducting transport is carried by the ballistic TSS in our TI JJs. Furthermore, we observe that the skewness depends on the back-gate voltage ( ( ) and is the smallest close to the charge neutrality point (CNP). We present a theoretical model based on the induced superconductivity in the ballistic TSS of the TI. This model takes into account the finite-size (of both Nb and TI) and proximity effects and relates the induced supercurrent to the TSS that extend over the entire circumference of the TI. The calculated energy spectrum (energy vs. phase ) of the junction reveals the existence of extremely low-energy modes that exist over the entire range of phases, i.e. 0 ≤ < 2 . The computed CPR from the theory is in excellent agreement with the experimental results.

Materials and devices
High quality single crystals of BiSbTeSe2 were grown using the Bridgman technique as described elsewhere [24]. Such BiSbTeSe2 are among the most bulk-insulating 3D TIs, where the Fermi energy lies within the bulk bandgap and inside the topological surface states (TSS), as verified by the angle resolved photoemission spectroscopy (ARPES) and transport measurements [24].
Exfoliated thin films of this material exhibit ambipolar field effect as well as several signatures of topological transport through the spin-helical Dirac fermion TSS including half-integer quantum Hall effect and p Berry phase [24,25]. Furthermore, we have recently observed an anomalous enhancement of the critical current in BiSbTeSe2 nanoribbons-based Josephson junctions, demonstrating the induced superconductivity in the TSS [23]. We obtain BiSbTeSe2 flakes using the standard scotch-tape exfoliation technique and transfer them onto a 300-nm-thick SiO2/500µm-thick highly doped Si substrates, which are used as back gates. We then locate the BiSbTeSe2 flakes with different width ( ) and thickness ( ) using an optical microscope. Subsequently, an electron beam lithography is performed to define a SQUID consisting of a TI-based JJ and a reference (REF) junction. The electrode separation, , in the TI-based JJ is ~ 100 nm. Finally, a thin layer ( ~ 40 nm) of superconducting Nb is deposited in a DC sputtering system. Prior to the Nb deposition, a brief (~ 3 seconds) in situ Ar ion milling is used to clean the interface between Nb and the TI flake. Fig. 1a shows a scanning electron microscope (SEM) image of an asymmetric SQUID with a BiSbTeSe2 flake (sample A). The data presented here comes from two devices, sample A ( ~ 2 µm and ~ 40 nm) and sample B ( ~ 4 µm and ~ 13 nm). All our measurements are performed in a dilution refrigerator with a base temperature ( ) of ~ 20 mK.

CPR measurement and discussion
We adapt an asymmetric SQUID technique [26] to measure the CPR in our TI (BiSbTeSe2) based JJ. The asymmetric SQUID consists of two Josephson junctions in parallel as shown by the SEM image in Fig. 1a. The first JJ is the S-TI-S junction with an unknown CPR, ( ), and is highlighted    Fig. 2b. We observe that the CPR remains highly non-sinusoidal up to ~ 400 mK, but becomes nearly sinusoidal at higher = 1.3 K. Furthermore, & exhibits a strong dependence and increases as we decrease the temperature down to the lowest accessible = 20 mK. Such a temperature dependence is in contrast to that of conventional junctions, where & is expected to saturate at low temperatures [28]. The blue and black curves are predictions of a general model for ballistic junction and our model for TI junction, respectively, and will be discussed later. In order to describe the shape of the CPR in our samples, we define the total harmonic distortion ( ) as where j is the amplitude of the pK harmonic. Fig. 2d depicts , L / m , and q / m vs. in sample A at ( = 0 V. We observe that , L / m , and q / m are nearly temperature independent up to ~ 400 mK. Moreover, at = 1.3 K, q / m ~ 0 and ~ L / m , indicating that at this temperature, only the first and second harmonics are present in the CPR. Thus, the CPR of the TI junction is less skewed compared to that at the base temperature. and an ambipolar field-effect in its normal-state resistance. We also observe that in sample B the skewness changes as a function of ( . Fig. 3b plots the vs. ( for both sample A (red) and sample B (blue). We note that the CPR is most skewed in sample B at ( = 30 V, where the chemical potential is inside the bulk bandgap yet away from the CNP (see the inset of Fig. 3b).
The reduced skewness at ( ~ 0 V may be a result of the charge inhomogeneity and electron/hole puddles near the CNP.

Theoretical Model
In , and zero otherwise. Here and are the separation and width of the contacts, respectively. The wavefunction is subject to antiperiodic boundary conditions in [29]. In this simple model, we assume that the system is translationally invariant in the direction, so the wavefunction depends on as exp †ˆ‰ for some ˆ. This renders the problem effectively one-dimensional.
Our explanation of the CPR and temperature dependence of the critical current is based on an interplay between the finite-size and proximity effects. To compute the CPR, we first rewrite, following Ref. [3], into a matrix problem, which we diagonalize numerically for various values of the phase difference . {ℋ} has a particle-hole symmetry, which stems from using four fermionic components in place of two: at each , the energy levels come in ± pairs. In terms of the nonnegative levels • ≥ 0 (one from each pair), the total free energy at finite temperature is: and the current is obtained as ( ) = [ LM ℏ \ 3 d . As we increase the number of x (or ) participating in the expansion, ( ) suffers from an ultraviolet divergence, but the current does not. To calculate finite temperature properties, we replace Δ J above with the T-dependent superconducting gap Δ( ) modeled using the BCS self-consistent equation [30].
The energy spectrum (± • vs. ) for sample A for the modes within the gap, | • | ≤ J is shown in Fig. 4b. Interestingly, we observe modes with energies much smaller than J that extend over the entire range of , see red curves in Fig. 4b. These low-energy states lead to the non-saturation of the junction's critical current down to our lowest accessible temperature ( ~ 20 mK) as seen in Fig. 2b in the theoretical (blue) curve, consistent with the experimental data (symbols).
Because the wavefunction extends over the entire circumference x , while the Nb contacts occupy only a small part of it, the energy scale of the low-energy modes is only a fraction of the full J .
Our results can be understood qualitatively using the perturbation theory. For J = 0, energy = ±£ℏ 3 ¤ x L +L ± £ [3], so there is a strictly zero energy state whenever ′, defined above, equals one of the quantized free-fermion momenta where ≥ 0 is an integer. When J > 0, these states are gapped roughly by 2 J / x . For sample A with x ~ 6 µm and the contact width ~ 300 nm, this is about 0.1 J . Crucially, these lowenergy states exist for the entire range of phases, 0 ≤ < 2 , in contrast for instance to the case of a conventional ballistic junction (the Kulik-Omelyanchuk theory [31]), where the minimal excitation energy remains on the order J except for a narrow vicinity of = .
For a given , the condition (3) will be satisfied better for some ˆ than for others. In practice, the translational invariance in the direction is not precise, so ˆ is not an exact quantum number.
Nevertheless, because of the large size of the TI flake in the transverse ( ) direction, the quantization interval for ˆ is small, so unless is exceptionally close to zero, we expect there will be a significant number of modes for which (3) is satisfied to a good accuracy. We therefore adopt the simple model in which we calculate the supercurrent for ˆ = 0 only and multiply the result by an effective number of transverse channels rK to account for the contribution of all the modes. We determine rK by matching the overall magnitudes of the experimental and computed critical currents. We find rK ~ 19 and rK ~ 46 for sample A at ( = 0 V and sample B at ( = 30 V, respectively. We plot the computed CPR for sample A as solid curves in Fig. 2a, where an excellent agreement with the measured data is observed. The blue curve in Fig. 2c is the FFT calculated for the theoretical CPR (in the range −5 < /2 < 5 and at = 20 mK) of a perfectly ballistic short junction [28,32]: predicted for the fully ballistic junction (blue curve) and the = 0.46 extracted from our measured CPR at = 20 mK is within 20% of the theoretical ballistic limit ( = 0.55), indicating superconducting transport is nearly ballistic in sample A. The black curve in Fig. 2c plots the FFT of the CPR calculated using our theoretical model (Fig. 4). We observe that the FFT of the CPR, calculated using our model, is in reasonable agreement with the FFT of the measured CPR. In contrast, the perfectly ballistic model (blue curve) notably overestimates the higher harmonics ( q and above). The computed CPR for sample B is plotted with dashed curves in We note that in our previous experiments on S-TINR-S JJs [23], even though we have also observed evidence that the superconductivity is induced in the TSS, we only observe a sinusoidal CPR. A possible reason for this is a much smaller transverse size (ˆ) of the TINR compared to the flakes used in the current work. As a consequence, ˆ is quantized in larger units (i.e. 2 /ˆ), and the condition (3) is less readily satisfied. Effectively, the small transverse size generates a gap in the TSS spectrum, preventing occurrence of low-energy states and rendering the CPR more sinusoidal at our experimental temperatures [23]. A similar explanation may be relevant also for sample B of the present paper near the charge neutrality point.

Conclusions
We have measured the CPR, one of the fundamental properties of a Josephson junction, in a BiSbTeSe2-based JJ using an asymmetric SQUID technique. We observed highly forward-skewed CPR, indicating that the superconducting transport through the TSS of the TI junction was close to ballistic. Temperature and gate dependence of the CPR were also studied, where we observed that CPR became more sinusoidal at high temperatures ( ~ 1.3 K) and close to the CNP. The reduced skewness near CNP was an indication of diffusive transport and was associated with the existence of electron-hole puddles and charge inhomogeneity in the very thin TI. Moreover, we developed a theoretical model that considered induced superconductivity in the spin-helical TSS of TIs. Our model assumed that the surface states can extend over the entire circumference of the TI. The predicted skewness of the CPR and the dependence on the temperature were consistent with our experimental observations. Overall, the experiment and the theory pointed toward robust features that made our TI system an excellent candidate to observe topological superconductivity and Majorana bound states. SC0008630.