Session W65: Topological Superconductivity I

From Majorana to parafermion corner states in second-order topological superconductors

Recently, the generalization of conventional topological insulators (TIs) and topological superconductors (TSCs) to so-called higher-order TIs and TSCs has raised significant interest. While conventional d-dimensional TIs and TSCs host gapless edge states at their (d-1)-dimensional boundaries, nth-order d-dimensional TIs and TSCs exhibit gapless edge states at their (d-n)-dimensional boundaries. In particular, a two-dimensional second-order TSC (SOTSC) hosts Majorana bound states at the corners of a rectangular sample. Here, we demonstrate how the presence of strong electron-electron interactions can ‘fractionalize’ such Majorana corner states [1,2] and promote them to even more exotic parafermion corner states. [3,4] For this, we construct an SOTSC from an array of weakly coupled one-dimensional wires, which allows us to treat the strong interactions analytically using bosonization techniques. As a possible platform to realize such a system, we discuss coupled helical channels in gated bilayer graphene.

 

[1] Y. Volpez, D. Loss, and J. Klinovaja, Phys. Rev. Lett. 122, 126402 (2019).

[2] K. Laubscher and J. Klinovaja, Journal of Applied Physics 130, 081101 (2021).

[3] K. Laubscher, D. Loss, and J. Klinovaja, Phys. Rev. Research 1, 032017(R) (2019).

[4] K. Laubscher, D. Loss, and J. Klinovaja, Phys. Rev. Research 2, 013330 (2020).   

Crossed Andreev reflection in the fractional quantum Hall state of graphene

We construct high-quality graphene-based van der Waals devices with narrow superconducting niobium nitride (NbN) electrodes, in which superconductivity and robust fqH can coexist. We find crossed Andreev reflection (CAR) across the superconductor separating two qH edges. In integer fillings, we find a filling-independent CAR probability robust to changes in magnetic field and temperature, which we attribute to spin-orbit coupling in NbN allowing for Andreev reflection between spin-polarized edges. Interestingly, while CAR in the hole-conjugate fractions has similar behavior, CAR probabilities in the particle-like fractional fillings are markedly higher and depend strongly on temperature and magnetic field. We investigate this behavior by tuning the doping of the NbN interface while keeping bulk filling constant. These results provide a route to realize novel topological superconducting phases in fqH–superconductor hybrid devices based on graphene and NbN.   

Skyrmion control of Majorana states in planar Josephson junctions

Planar Josephson junctions provide a versatile platform, alternative to the nanowire-based geometry, for the generation of the Majorana bound states, due to the additional phase tunability of the topological superconductivity. The proximity induction of chiral magnetism and superconductivity in a two-dimensional electron gas showed remarkable promises to manipulate topological superconductivity. In this talk, we shall present our proposal for a new geometry, involving a Josephson junction and a skyrmion crystal, that hosts the Majorana bound states. The chiral magnetism of the skyrmions can create and control the Majorana bound states without the requirement of an intrinsic Rashba spin-orbit coupling. Interestingly, the Majorana bound states in our geometry are realized robustly at zero phase difference at the junction. The skyrmion radius, being externally tunable by a magnetic field or a magnetic anisotropy, brings a unique control feature for the Majorana bound states that can be useful for topological quantum computing.   

Edelstein effect and induced topological phase transition in superconductor hybrids

we study topological phase transitions induced by the Edelstein effect in a one-dimensional spin-orbit-coupled wire in proximity to a noncentrosymmetric metallic superconductor. We showed that the Cooper pair flow in the substrate metallic superconductor, depending on the crystal symmetry of the metal, can be utilized to manipulate the Majorana fermions localized at the boundary of the topological phase in the wire. In a self-consistent calculation, we extended our results to finite temperatures and showed that our simple hybrid structure may be relevant to the coherent braiding of Majorana bound states and to further developing the fabrication of quantum gates in a web of semi-conducting nanowires.   

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We will present our recent results showing that finite current in superconductors can induce topological phase transitions, as a result of the deformation of the quasiparticle spectrum by a finite center-of-mass (COM) momentum of the Cooper pairs [1]. To show the wide applicability of this mechanism, we examine the topological properties of three prototypical systems, the Kitaev chain, s-wave superconductors, and d-wave superconductors. We introduce a finite COM momentum as an external field corresponding to supercurrent and show that all the models exhibit current-induced topological phase transitions. In this talk, we will also address the possibility of observing the phase transitions in experiments and the relation to the other finite COM momentum pairing states.   

Signatures of enhanced spin-triplet superconductivity induced by spin-orbit coupling

Through structural inversion asymmetry in superconducting junctions, the resulting interfacial spin-orbit coupling provides a versatile platform to realize proximity-induced spin-triplet superconductivity sought in superconductor spintronics and topological quantum computing. However, the signatures of such spin-triplet conductivity and the conditions for its enhancement remain debated. By calculating superconducting correlations in ferromagnet/s-wave superconductor junctions and zero-bias conductance contributions from equal-spin Andreev reflection, we reveal how very different signatures give a consistent picture for an enhanced spin-triplet superconductivity. Unlike the common expectation that strong spin-orbit coupling promotes the spin-triplet pairing or a small interfacial barrier is necessary for robust proximity effects, we reveal a more complex picture. An enhanced spin-triplet superconductivity, realized for intermediate values of spin-orbit coupling and interfacial barrier strength, is consistent with the experimentally observed huge increase in the conductance magnetoanisotropy.   

Effects of quantum dot-continuum coupling in topological superconductor-quantum dot-topological superconductor system: implications for quantum dot-based measurements of Majorana qubits and the 0-π transition

Quantum dot coupled to Majorana Zero Modes (MZMs) in one-dimensional topological superconductor-quantum dot-topological superconductor junctions can potentially serve two purposes for the needs of topological quantum computing. First, it is believed that absence of the 0-π transition in the junction can indicate presence of the MZMs and the topological phase. Second, quantum dot can be used as a tool to measure the state encoded by the MZMs. Considering proximitized Rashba nanowire with Zeeman energy as a model for a topological superconductor, here we investigate effect of continuum nanowire modes coupled to a quantum dot in topological superconductor-quantum dot-topological superconductor junctions. Using perturbation theory in the strength of the coupling between the quantum dot and the continuum modes, we find that for realistic parameters coupling to continuum modes does not considerably alter visibility of the quantum dot-based measurements of Majorana qubits. However, we also find that presence of this coupling can lead to a π-junction behavior even in the topological phase which would limit capabilities of the 0-π transition as a diagnostic tool for the presence/absence of MZMs.   

Scanning Microscopy Probes of Majorana States

Majorana bound states are difficult to distinguish from trivial near-zero energy states. Single-electron transistors (SETs) have been proposed as on-chip probes to solve this problem. Combining these devices with scanning probe methods allows us to extend the reach of these probes to test many of the systems that theoretically support Majorana bound states (vortices in a 2D topologogical superconductor, for example.) In this talk, we present progress on (i) scanning tunneling microscopy (STM) probes of Majorana states in Josephson junctions and (ii) development and testing of the Scanning Majorana Microscope (SMM).  We present a platform that implements a capacitance-based device constructed from a high-electron-mobility transistor that can measure the counting statistics of electrons tunneling into a quantum dot. Theoretically, the cumulants of the counting statistics change as the quantum dot hybridizes with a Majorana bound state. This device is integrated into a probe tip which fabricated by thermally evaporating two Al leads (one tunneling lead and one capacitance lead) onto a sharply-pointed optical fiber. Using a focused ion beam, we create a flat end of the pointed optical fiber where we deposit our quantum dot. For this talk, we will focus on key milestones that have been accomplished. The first, is a demonstration of the probes counting statistics in vacuum. Second, is confirmation that the probe can adequately perform the basic functions of an STM tip: approaching safely, and measuring the topography of the sample's surface. Additionally, we present progress on developing a device featuring planar Josephson junctions on the surface of a topological insulator. This device is fabricated with conventional electron-beam lithography and reactive ion etching. Such a device should theoretically support Majorana bound states in the presence of an external magnetic field. Our near-term goal is to probe this device with STM and the SMM.   

Transport of Majorana Zero Modes in 1D Topological Superconductors

We consider the transport of Majorana zero modes across a 1D topological superconductor by applying local gate voltages across sections of the superconductor. This “piano key” method allows for sections of the superconductor to switch between the trivial and topological phases, thereby facilitating the motion of a Majorana zero mode. As a single section, or piano key, undergoes a phase transition, it is possible for the ground state to experience excitations, especially near criticality. The excitation probability has been studied for a large piano key in Ref. [1] which casts the problem in terms of a simple Landau-Zener transition. In our work, we consider the excitation probability when a Majorana zero mode is transported using a series of smaller piano keys. We calculate the excitation probability numerically by simulating a sequence of piano keys. Furthermore, we demonstrate an analytical calculation of the excitation probability and compare this to the numerical results.

[1] B. Bauer et al., SciPost Phys. 5, 004 (2018)   

Direct Measurement of the Andreev Bound State Spin and Charge During the Singlet-Doublet Transition

The past decade has seen many proposals to measure Majorana quasi-particles definitively. One such measurement involves probing the bulk spin polarization of a system undergoing a topological phase transition. An Andreev bound state (ABS) is expected to similarly show both charge and spin reversal at the singlet-doublet quantum phase transition. 

In this talk, we demonstrate a direct measurement of the spin and charge of an ABS. We measure the spin of the ABS by coupling it to a spin-polarized quantum dot and track its evolution as a function of magnetic field and chemical potential. We also measure the charge of the ABS by using non-local spectroscopy. Considering the ABS as a model for a topological system, we observe the expected signal at the quantum phase transition. We propose to extend these techniques as complementary measures for the occurrence of a topological phase transition.     

Tuning singlet vs triplet Cooper-pair splitting in InSb nanowires

Cooper pairs are a natural source of entangled electrons provided by superconductors. Semiconducting-superconducting hybrid systems allow the splitting of Cooper pairs into separate single electrons and their subsequent manipulation. Using two few-electron InSb nanowire quantum dots separated by a narrow strip of proximitized superconducting Al, we achieve a high degree of control in Cooper-pair splitting via selecting for the single electron's desired charge as well as spin. The resulting Cooper-pair splitting signal is free of contributions from the competing process, elastic co-tunneling. Under an applied magnetic field parallel to the Rashba spin-orbit field of InSb, we observe complete blockade of the equal-spin splitting process due to spin-singlet superconducting pairing. Rotating the applied field to be perpendicular to the Rashba field introduces a triplet component in the effective superconducting pairing between electrons and lifts this blockade. The spin blockade results open new pathways to the precise manipulation and entanglement testing of the split electrons.   

Transport signatures of Majorana zero modes in magnetic insulator nanowire hybrid systems

Semiconducting nanowires with Rashba spin-orbit coupling and proximity-induced superconductivity are the forerunners as platforms to host Majorana Zero Modes (MZMs). Recent evidence of a topological transition in hybrid Rashba nanowires epitaxially coated by superconductor-magnetic insulator bilayers points to the formation of MZMs at significantly lower external magnetic fields than in conventional systems, making it experimentally viable for scaling networks of nanowires hosting MZMs. 

We employ the Keldysh non-equilibrium Green’s function (NEGF) formalism to model transport in such hybrid wires attached to metallic contacts. We find the zero bias conductance peaks (ZBCP) in the differential conductance in the topological regime consistent with the emergence of MZMs. However, this is insufficient to prove the existence of MZMs as similar signatures may be obtained from non-topological sources. We, therefore, probe the non-local conductance, which provides robust signatures of the topological transition and can probe bulk properties of the nanowire, as compared to local conductance measurements which often just probe the states adjacent to the lead. We thus demonstrate that one may find signatures of topological MZMs at smaller magnetic fields in magnetic insulator nanowire hybrid systems.