Cooper-pair splitting in two parallel InAs nanowires

Topologically protected electronic states in nanostructures have recently attracted wide attention, as they may provide fundamental building blocks for quantum computation [1, 2]. Recent advances in material science and device fabrication resulted in considerable progress towards the generation and detection of topologically protected bound states in topologically non-trivial semiconducting nanowires (NWs), so called Majorana fermions (MF) [3–7]. Two MFs can combine together into a regular fermion that is why MFs are also known as Z₂ fermions. MFs are predicted to have non-Abelian braid statistics and may provide a platform for topological quantum computation [8, 9]. To implement MF based qubits, one strategy is to use double NWs in the box-qubit architecture [10]. However, one cannot implement all required operations for universal quantum computation in quantum bits (qubits) based on MFs by using topologically protected braiding. In this respect, Z₄ fermions, also known as Parafermions (PFs), are better as they allow for a larger set of operations [11]. Recently, it has theoretically been predicted that PFs can be generated in a system based on two NWs with different spin orbit interaction coupled to a common superconducting electrode [12–14]. The superconductor (SC) induces both a pairing interaction within each NW and between the two NWs due to Cooper-pair splitting (CPS). In order to realize PFs, the interwire coupling must dominate [13]. It has been shown that this is possible in systems with strong electron–electron interaction, such as nanoscaled semiconducting wires [15–17]. Another advantage of the parallel two-wire approach, even if PFs are not formed, is the fact, that for large interwire pairing an external magnetic field is not required or only a small field is enough to reach the topological phase [18]. A large magnetic field is a limiting factor, because of the critical magnetic field of the SC. It is therefore crucial both for MF based charge qubits and for the realization of PFs to demonstrate an appreciable magnitude of interwire pairing interaction mediated by a SC. This coupling is also known as crossed-Andreev reflection or CPS [15, 19, 20]. In the last few years, several CPS experiments have been performed using different platforms, mainly single NWs, [21–25] carbon nanotubes [26, 27] and graphene [28, 29]. Splitting efficiencies close to 100% have been reported, [27] demonstrating that intrawire pairing can exceed local Cooper pair tunneling. Up to now, all NW based CPS devices consisted of a single NW contacted by two normal metal electrodes and one superconducting contact in between. In order to assess the interwire pairing in double NW structures, it is essential to investigate CPS in such a system. In this work we demonstrate CPS in a parallel double NW device, which is an important first step towards topological quantum computation with PFs.

We investigate a device shown schematically in figure 1(a). Two InAs semiconducting NWs with large spin orbit interaction are placed in parallel (NW₁ green, NW₂ red) and electrically coupled by a common superconducting electrode S (blue). Both NWs are contacted by individual normal metal leads N_1/2 (yellow). Sidegates SG_1/2 (yellow) are located on each side of the NWs, in order to separately tune the chemical potentials of the quantum dots (QDs), which form between N_1/2 and S. We note, that the exact location of the QDs is not known, since we do not use additional barrier gates to terminate the QDs [30]. However, it is clear that both N_1/2 and S induce a potential step from which (partial) electron reflection is possible and QD bound states can form. We also point out already here that the electronic boundary conditions on the S side may change if S is in the normal or superconducting state, due to the proximity effect. Besides local Cooper pair tunneling from S to N_1/2 [31], Cooper pairs (white circle with red/black dot) can be split, resulting in a non-local current consisting of entangled single electrons. This process is expected to be large if both QDs, QD_1/2, are in resonance and the electrons can sequentially tunnel from the SC to the two normal metal leads.

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2.1. Fabrication and characterization

In figure 2(a) and (b) the simultaneously measured differential conductance G₁ through QD₁ and G₂ through QD₂ are shown, both as a function of the side gate voltages V_SG1 and V_SG2. These measurement are done at zero bias and without an external magnetic field. Varying the sidegate voltage V_SG1 tunes QD₁ through several Coulomb blockade resonances, resulting in conductance peaks in G₁. Similarly V_SG2 tunes the resonances of QD₂ in NW₂. Each resonance signifies a change in charge state of the respective QD. The charge on one QD can be sensed by the other QD, due to the capacitive coupling between QD₁ and QD₂. In our experiment, QD₂ acts as a good sensor for the charge on QD₁, as the Coulomb blockade resonance lines of QD₁ shift whenever the charge on QD₂ changes by one electron, see figure 2(b). Due to capacitive crosstalk from V_SG1 on QD₂ (V_SG2 on QD₁ respectively) the resonance positions are slightly tilted in both graphs. At certain gate voltages, when both QDs are in resonance, an increase of conductance can be observed on both sides. This can be seen more clearly in the cross sections (see figure 2(c) and (d)) indicated by black and white dashed lines in figure 2(a), (b). We observe an enhancement of G₁ along the black dashed line in figure 2(a) at the peak positions of G₂ for the two resonances at V_SG2 ≈ 1.87 V and V_SG2 ≈ 1.91 V (see arrows), while other possible correlations are less clear and disappear in the background noise. A similar characteristics can be found in cross sections of G₂ along the white dashed line in figure 2(b) at peak positions of G₁. Here, the positive correlation is very clear at V_SG1 ≈ 0.32 V (arrow), while there is only a weak quite broadened correlation visible for the other two resonances at V_SG1 ≈ 0.4 V and V_SG1 ≈ 0.49 V. Hence, we observe clear positive correlation between G₁ and G₂ on three resonances, which we assign to CPS from S into QD₁ and QD₂. We repeat the same measurement in the absence of superconductivity by applying an out of plane magnetic field of 250 mT, which is larger than the critical field of the Al contact. In this case the previously positive correlations between G₁ and G₂ disappear fully, proofing that the positive correlation in conductance originates from CPS (see figure A2 in appendix). We define the CPS efficiency as 2G_CPS/G_total resulting in a maximum efficiency of ≈20%, similar to the largest reported values on single NW devices.

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In the following we investigate the same type of measurement as discussed in figure 2, for a different gate voltage region. The data is shown in figure 3. Here, we observe two sets of resonances for QD₁. Besides the QD levels, we already discussed in figure 2 (indicated with I), we detect a second set of resonances (referred to II). For QD₂ we observe only one set of QD resonances. The resonances denoted with II are significantly different from the one denoted with I. First, the amplitude of type II resonances is only half of the value of type I. Furthermore, type II have a different slope than type I, indicating different capacitive coupling ratio between SG₁ and SG₂. The broadening differs roughly by a factor of two: Γ_I = 0.5 meV whereas Γ_II = 1.2 meV. Most strikingly, the resonances of type II vanish completely when an external magnetic field is applied, i.e. these resonances are only present in the superconducting state (not shown). In addition, QD₂ only reacts on a change in charge state of QD₁ for resonances of type I (e.g. green arrow in 3(b)). For the resonances of type II, there seems to be no change in the charge state, as QD₂ does not sense these resonances (e.g. yellow arrows in 3(b)). We therefore conclude that the resonance lines of type II are not Coulomb blockade resonances. They must have their origin within the superconducting phase, most likely near S. Though these features are not Coulomb blockade resonances, they can still be used to test CPS by conductance correlations. We observe a clear positive correlation between the conductances G₁ and G₂ in figure 3(d) as well as in 3(c) for both types of resonances. We estimate a splitting efficiency of about ≈13% for type II, which is of similar magnitude as the one obtained for resonances of type II in the gate region of figure 2.

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The measurements of type II resonances suggest the existence of subgap states which are not located in QD₁, but rather in the lead connecting to S. Since these states are gate-tunable, they are not fully screened by S. We therefore propose that a proximitized region is formed in NW₁ that extends to some distance out from S. QD₁ is coupled to this proximitized lead. Within the lead, bound states can form due to potential fluctuations and residual disorder in at least one NW. There are two kinds of bound states, Andreev bound states [34–36] or Yu-Shiba Rusinov (YSR) [37] states. These states do not usually occur at zero energy, but can be tuned electrically to zero energy, signaling a ground state transition between the proximitized lead region and the bulk of the SC. This gives in effect rise to a density-of-state peak in the gap of the SC, enhancing the subgap conductance which we measure. In this case the two electrons that are launched by CPS are transmitted in a different way to the respective drain electrodes. The electron that takes the path through QD₂ is transferred by the usual (resonant) sequential tunneling, while the one that takes the path through QD₁ is transferred by co-tunneling. One might expect that this suppresses CPS as the latter process corresponds to a low probability for out-tunneling into the drain contact. However, due to the subgap resonance in the proximitized lead, this process is enhanced and one can therefore reach almost similar CPS efficiencies. In spite of disorder in at least one NW we demonstrate a significant CPS efficiency, the first fundamental effect necessary for the formation of NW based PFs. Future devices with less disorder might even exhibit larger CPS efficiencies, which is relevant for the non-local superconducting correlation.

In summary, we demonstrate the fabrication of an electronic device, consisting of two closely placed parallel InAs NWs, contacted by a common superconducting lead and individual normal metal leads. By addressing individual sidegates we can separately tune the QDs formed in each NW. When both QDs are in resonance, we observe CPS with efficiencies up to 20%. For certain gate voltages, we detect a second set of QD resonances in one of the NWs, which only appears in the superconducting state. The second set of resonances do not correspond to Coulomb blockade resonances, hence, are not related to a change in the charge state of the QD. Since they only appear in the superconducting state, we tentatively assign the second set to subgap states in the lead that connects to the SC, which are most likely caused by superconducting bound states (Andreev and or YSR states). For the first time we provide a platform, suitable to implement the next milestone in topological quantum computation, namely PFs. In the present device CPS is probably enforced by the Coulomb charging energy of the QDs formed in the NW segments not covered by the SC. It will be interesting to investigate whether these interactions, or alternatively intrawire electron–electron interactions are sufficient to also generate non-local superconducting correlations between the segments below the SC in order to generate PFs in such device architectures.

This work was partially supported by a Grant-in-Aid for Young Scientific Research (A) (Grant No. JP15H05407), Grant-in-Aid for Scientific Research (B) (No. JP18H01813), Grant-in-Aid for Scientific Research (A) (Grant No. JP16H02204), Grant-in-Aid for Scientific Research (S) (Grant No. JP26220710), JSPS Research Fellowship for Young Scientists (Grant No. JP14J10600 and Grant No. JP18J14172), and the JSPS Program for Leading Graduate Schools (MERIT) from JSPS, Grants-in-Aid for Scientific Research on Innovative Area 'Nano Spin Conversion Science' (Grant No. JP17H05177) and a Grant-in-Aid for Scientific Research on Innovative Area 'Topological Materials Science' (Grant No. JP16H00984) from MEXT, JST CREST (Grant No. JPMJCR15N2 and JPMJCR1675), and the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). HK acknowledges support from RIKEN Incentive Research Projects, and JSPS Early-Career Scientists (Grant No. JP18K13486). CJ acknowledges support from JSPS Postdoctoral Fellowship for Overseas Researchers. Part of this research was performed within the Nanometer Structure Consortium/NanoLund-environment, using the facilities of Lund Nano Lab, with support from the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF) and from Knut and Alice Wallenberg Foundation (KAW). It was also supported by the Swiss National Science Foundation, the Swiss Nanoscience Institute (SNI), the NCCR on Quantum Science and Technology and the H2020 project QuantERA. The authors would like to thank D Chevallier for fruitful discussions.