New Platforms for Topological Superconductivity with Magnetic Atoms

We present a theory of the spin-polarized STM experiments of the Fe magnetic chains on top of a Pb Superconductor. We show that there are some universal characteristics that distinguish the Majorana spin from that of a isolated Shiba state accidentally tuned at zero energy.

TBA

The goal of our work is to develop a theoretical description of the essentially nonequilibrium dynamics of Majorana states. We start from the time-dependent Bogolubov-de Gennes equations and show that the quasiparticle wavefunction evolving in time naturally consists of the contributions corresponding to both positive and negative energies of the stationary BdG Hamiltonian. As a result, the dynamics of two weakly overlapping Majorana states involves the hybridized wavefunctions corresponding to both positive and negative energy levels. This dynamics is substantially determined by the quantum mechanical beating phenomenon similar to the one well-known from the standard Shroedinger quantum mechanics. We suggest a microscopic model describing the nonlocal ac response of a pair of Majorana states in fermionic superfluids beyond the tunneling approximation. The time-dependent variation of quasiparticle transport parameters is shown to excite finite period beating of the wavefunction between the distant Majorana states with the period governed by the energy splitting of two Majorana states. We propose an experimental test to measure the characteristic time scales of quasiparticle transport through the pair of Majorana states defining, thus, quantitative characteristics of nonlocality known to be a generic feature of Majorana particles. We consider both the limits of weak and strong Coulomb effects and find that the beating phenomenon survives and is robust to the strong Coulomb blockade influence. The beating phenomenon is shown to impose important restrictions on the operation frequencies of various Majorana-based devices.

Recent experiments revealed that zero-energy bound state coalesces from the Andreev bound states. Such quasiparticle states can be controlled by the magnetic and electrostatic means as shifting of states can create the possibility of its existence and then allow Majorana bound states to leak to the other, non-topological parts of the system. In our investigation we use microscopic model of the nanowire structure applying the Bogoliubov–de Gennes technique. This is done by studying the gate voltage and magnetic dependence of density of states and band structure at zero energy. We show a way of understanding of Majorana quasiparticles emergence in said systems as a result of topological phase transitions

I will discuss the phase diagram of thin-film magnetically-doped topological insulators that are placed on a superconducting substrate and put under the influence of a gate-controlled displacement field. When viewed in the two-dimensional space of displacement field and carrier density control parameters and brought close to the parameter regime where the quantum anomalous Hall effect is observed, this system has broad regions of both trivial and topological superconducting phases. It follows that quasi-one-dimensional (1D) quantum wires can be written onto the surface of these magnetic topological insulator (MTI) thin films by gate arrays, without requiring exceptionally fine control, and used to control Majorana end state arrays.

Magnetic impurities in superconducting hosts induce bound states called Yu-Shiba-Rusinov (YSR) states. It has been recently revealed that the reduced dimensionality of the superconducting substrate can influence the spatial extent of such in-gap states. The spin-orbit coupling can additionally modify the density of states around an impurity, and even further increase the range of bound states. We will explore the behavior of YSR states induced by single magnetic impurities and dimers embedded in superconducting hosts with different lattices.

Gerbold C. Ménard1, Sébastien Guissart2, Christophe Brun1, Mircea Trif2, François Debontridder1, Raphaël T. Leriche1, Dominique Demaille1, Dimitri1,3, Pascal Simon2 and Tristan Cren1 1Institut des Nanosciences de Paris, Université Pierre et Marie Curie (UPMC),CNRS-UMR 7588, 4 place Jussieu, 75252 Paris, France 2Affiliation Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France 3Laboratoire de physique et d’étude des matériaux, LPEM-UMR8213/CNRS-ESPCI ParisTech-UPMC, 10 rue Vauquelin, 75005 Paris, France Email: Christophe.Brun@upmc.fr Key words: 2D superconductivity; atomic monolayers; topological superconductivity. Majorana states are predicted to appear as edge states of topological superconductors, in a similar way as Dirac surface states appears at the edge of topological insulators. Spectroscopic signatures of Majorana bound states were claimed to be observed in one-dimensional (1D) InAs nanowires proximity-coupled to a bulk superconductor [1]. Then Nadj-Perge et al. [2] have realized a chain of Fe adatoms on a Pb(110) crystal that is supposed to induce locally a 1D topological p-wave superconductivity. Zero-energy bound states were observed at the extremity of some the Fe chain and interpreted as Majorana excitations [2]. Nevertheless this interpretation is challenged by close to zero-energy Shiba states [3]. We have recently decided to follow a different strategy using a two-dimensional superconducting system consisting in a monolayer of Pb atoms grown on Si(111) [4]. We have shown that the strong spin-orbit coupling of Rashba type present in the superconductivity of the Pb/Si(111) monolayer can be revealed through the filling of in-gap quasiparticle states by scattering from non-magnetic disorder at 300 mK [5]. Following Rashba and Gor’kov this shows that a mixed singlet-triplet superconductivity exists in our Pb monolayer [6]. Nano-magnetic disks made of CoSi alloy grown below the Pb/Si(111) monolayer were used to induce locally 2D topological superconductivity by combining a strong local Zeeman field with a mixed singlet-triplet 2D superconductor. We have observed that dispersive edge states appear in the superconducting gap at the boundary of the magnetic domains [7]. We have interpreted these spectroscopic features as signatures of a locally induced 2D topological superconductivity. Indeed, we expect to get propagative dispersive Majorana edge states around 2D topological domains since the edges have a 1D character. References 1. V. Mourik et al., “Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices” Science 346, 602 (2012). 2. S. Nadj-Perge et al., “Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor” Science 336, 1003 (2014). 3. M. Ruby et al., “End States and Subgap Structure in Proximity-Coupled Chains of Magnetic Adatoms” PRL 115, 197204 (2015). 4. T. Zhang et al. “Superconductivity in one-atomic-layer metal films grown on Si(111) ” Nature Phys. 444, 10 (2014). 5. C. Brun et al., “Remarkable effects of disorder on superconductivity of single atomic layers of lead on silicon” Nature Phys. 444, 10 (2014). 6. E.I. Rashba and L.V. Gor’kov, ”Superconducting 2D System with Lifted Spin Degeneracy: Mixed Singlet-Triplet State” PRL 444, 10 (2010). 7. G. C. Ménard et al., “Two-dimensional topological superconductivity in Pb/Co/Si(111)”, Nature commun. 8, 2040 (2017) doi:10.1038/s41467-017-02192-x

Recent experiments have reported convincing evidence of 1d topological superconductivity in systems comprising of linear arrangements of magnetic atoms. In my talk I will explain why these developments could only be a starting point of engineering a myriad of topological states in 2d systems. The more exotic examples of these include topological superconductors with high Chern numbers and amorphous topological states in random systems. I also discuss prospects of fabricating Chern insulators with similar approach.

Recent microscopy experiments on superconducting monolayer of lead(Pb) grown over clusters of cobalt atoms have raised urgent questions about robust in-gap electronic states in presence of strong spin-orbit coupling and magnetism. Using analytics and exact diagonalization we find that a vortex defect in Rashba coupling fully explains puzzling features of this experiment: 1) Vortex on the magnetic island localizes a state at scale far smaller than superconducting coherence length; 2) The state is protected by large energy gap; 3) Island edge states are however localized on the coherence scale. The theory predicts that the defect state is a single Majorana state, well-localized, and remarkably well-isolated in energy. In contrast, a typical superconducting vortex would exhibit a tower of in-gap excited states. We also study the Majorana states in superconducting vortices in mixed singlet-triplet superconductors. We find that multiple Majorana states can be topologically protected in a single vortex, and explain how the Zeeman field can stabilize them. The multiple Majorana states still give non-trivial braiding properties to a vortex. Finally, we discuss the experimental feasibility of these Majorana states in a superconductor under magnetic field.

A variety of one-dimensional electronic systems can be engineered to host topological superconductivity and Majorana zero modes. In this talk, I’ll describe experiments on two different one-dimensional platforms. One is based on chains of magnetic atoms the work on which grew out of early efforts on the study of localized Bogoliubov quasi-particles near individual magnetic atoms on a superconductors. We have now performed a series of experiments to establish the presence of Majorana’s in this system, including a recent study of their spin polarization. A second system is based on introducing superconductivity on the hinge states of a high order topological insulator. Using combination of magnetism and superconductivity, we are exploring how Majorana zero mode can emerge in this system.

We demonstrate the fully-controlled bottom-up fabrication of artificial 1D atomic chains from individual magnetic Fe adatoms on high spin-orbit coupled superconducting Re(0001) substrates by STM-based atom-manipulation techniques at T=350 mK. Spin-polarized STM measurements indicate the presence of non-collinear spin textures, i.e. spin spiral ground states, stabilized by interfacial Dzyaloshinskii-Moriya interactions. Tunneling spectra measured spatially resolved on the Fe-atom chain on Re(0001) reveal the evolution of the local density of states inside the superconducting gap as well as the development of zero-energy bound states at the ends of the chain, which are distinguishable from trivial end states by systematically increasing the number of atoms within the Fe-atom chain. The experimental results will be compared with model-type calculations supporting the interpretation of the spectroscopic signatures at the ends of the chains as Majorana bound states. Such Majorana states can encode topological qubits and ultimately provide a new direction in topological quantum computation.

Traditionally, we regard the superconducting state as spatially homogeneous. Superconductors are not really prone to show domains or phase separation. An well-known exception occurs when applying a magnetic field to type I superconductors. There, a phase separated intermediate state builds up with normal and superconducting domains. However, this is just a geometric effect due to classical electrodynamics combined with the full expulsion of the Meissner state. Often, perturbations of clean superconducting samples occur at very small length scales. For instance, introducing an isolated magnetic impurity causes order parameter suppression at atomic scales, or a magnetic field produces vanishing order parameter just at the center of vortex cores. Such disturbances of the order parameter lead to coherent electron-hole excitations. Magnetic impurities show Yu-Shiba-Rusinov states (or shortly Shiba states) and vortex cores Caroli de-Gennes and Matricon states. I will first discuss experiments showing Shiba states in Fe impurities in the superconductor 2H-NbSe_1.8S_0.2 and their relation to Caroli de-Gennes Matricon states in vortex cores. Furthermore, I will show recent Atomic Force, Magnetic Force and Scanning Tunneling Microscopy experiments that have allowed us to identify a new intrinsic phase separated state, with normal and superconducting regions of nanometric sizes, in Co substituted CaFe$_2$As$_2$. Each phase is equivalent to what is found at either side of the first order phase transition between antiferromagnetic and superconducting phases, suggesting that first order quantum phase transitions might provide a new route to create nanometric size phase separation in superconductors.

In magnetic impurities with strong easy axis anisotropy, the two fully spin-polarized states are mixed by the in-plane anisotropy terms. This lifting of double degeneracy is also known as the quantum spin tunneling splitting. If the impurity is coupled to an environment, e.g. via Kondo exchange interaction, the splitting is renormalized downwards. A more complex situation arises when the fermionic environment is gapped so that sub-gap states are generated by partial or full Kondo screening in one or several orbital channels. The resulting spin multiplets inherit magnetic-anisotropy splitting for the parent (bare, unscreened) multiplet, but the renormalization is highly non-trivial. I will present new results obtained with an improved highly-efficient numerical renormalization group code which preserves fermionic-parity quantum numbers in separate screening channels, so that multi-channel calculations become feasible even in the absence of any spin symmetry. The main result is that (partially) Kondo-screened multiplets show opposite behavior as unscreened multiplets, i.e., enhancement of the splitting with increasing Kondo coupling.

Nematic fluctuations, a phenomenon that breaks the original rotational symmetry of an electronic system, can have surprising impact on the electronic properties of some of the iron-based superconductors. However, still little is known about its origin, and the role it plays in the superconductivity in those materials. Measurements of the nematic susceptibility has in recent years been developed to a new powerful tool which provides new insights into the role of symmetry breaking electronic states for the properties of iron-based superconductors. Here, I explore atomic scale imaging to search for symmetry breaking electronic states in the iron-based superconductor LiFeAs. In this talk, I will present a scanning tunneling microscopy and spectroscopy study of the superconductivity and electronic structure of LiFeAs, a material that has a tetragonal crystal structure, with a particular view to identify symmetry breaking electronic states.

Phase-sensitive measurements of the superconducting gap in Fe-based superconductors have proven more difficult than originally anticipated. While quasiparticle interference (QPI) measurements based on scanning tunneling spectroscopy are often proposed as definitive tests of gap structure, the analysis typically relies on details of the model employed. Here we point out that the temperature dependence of momentum-integrated QPI data can be used to identify gap sign changes in a qualitative way, and present an illustration for s± and s++ states in a system with typical Fe-pnictide Fermi surface.

For rationalizing the Cooper pairing mechanism in a superconductor, it is central to identify bosons interacting with the conduction electrons. Here we introduce a novel approach to determine the energy and momentum characteristics of such bosons which relies on the detection of a boson-assisted resonant amplification of Friedel oscillations by Fourier transform scanning tunneling spectroscopy (FT-STS). Upon focusing on the unconventional superconductor LiFeAs, our FT-STS data reveal bosonic states at small momentum and specific energy. We show that the FT-STS signatures of the bosonic states survive in the normal state, and, moreover, that they are in perfect agreement with well-known strong above-gap anomalies in independently measured tunneling spectra at the very same energy. Thus, the small-momentum bosonic modes are promising candidates for providing the pairing interaction in LiFeAs.

We study the symmetry of the potential superconducting order parameter in 5d Mott insulators with an eye toward Sr2IrO4. We investigate the potential existence of a superconducting phase in hole-doped Sr2IrO4, as an example of the new class of Mott insulators. Currently, remarkable attentions have been attracted to exotic physics driven by the interplay of the spin-orbit coupling and electronic correlations. Particularly, in 5d transition metals, neither the spin-orbit coupling nor the Coulomb interaction can merely lead to the insulating behavior. A very interesting question is whether these materials with very similar properties to the cuprates can clarify the essential microscopic physics of high Tc superconductivity? Using a mean-field method, a mixed singlet-triplet superconductivity, d+p, is observed due to the antisymmetric exchange originating from a quasi-spin-orbit-coupling. Our calculation on ribbon geometry shows possible existence of the topologically protected edge states, because of nodal structure of the superconducting gap. These edge modes are spin polarized and emerge as zero-energy at bands, supporting symmetry protected Majorana state, verified by evaluation of winding number and Z2 topological invariant [1]. [1] M.H. Zare, M. Biderang, A. Akbari, Phys. Rev. B 96, 205156 (2017).

Topological Josephson junctions designed on the surface of a 3D-topological insulator (TI) harbor a Majorana bound state (MBS) among a continuum of conventional Andreev bound states. The distinct feature of this MBS lies in the 4π-periodicity of its energy-phase relation that yields a fractional a.c. Josephson effect and a suppression of odd Shapiro steps under rf irradiation. In this talk we present measurements of Josephson junctions tailored on the large bandgap 3D TI Bi2Se3, which exhibit the fractional a.c. Josephson effect acting on the first Shapiro step only, accompanied by a residual supercurrent at the nodes between Shapiro lobes. We demonstrate with a modified RSJ model that the resilience of higher order odd Shapiro steps and the residual supercurrent at the nodes between Shapiro lobes can be accounted for by thermal poisoning of the MBS driven by Joule overheating. We will furthermore show that the residual supercurrent at the nodes between Shapiro lobes provides a novel signature of the current carried by the MBS.

I will present recent results on single and double nanowires with proximity gap and Rashba spin-orbit interaction hosting Majorana fermions but also parafermions arising from strong electron-electron interactions [1]. Typically, the topological phases are engineered by tuning the magnetic field to the topological threshold value of typically a few Teslas. However, the magnetic field has a detrimental effect on the host superconductor and so it is interesting to search for ways to achieve the topological phase without or with smaller B-fields. A particular way to achieve this goal is to exploit crossed Andreev pairing combined with strong correlations [2] in a double nanowire setup [1-3] which destructively interferes with the direct pairing, and thereby lowers the threshold for the B-field substantially [4]. In re-examining the proximity effect in such finite-size geometries we discovered that there is a strong shift of the chemical potential of the wires in the strong tunnel coupling regime (such as in Al-shell nanowires) which can be detrimental for topological phases [5]. [1] J. Klinovaja and D. Loss, PRL 112, 246403 (2014); PRB 90, 045118 (2014). [2] M. Thakurathi, P. Simon, I. Mandal, J. Klinovaja, and D. Loss, arXiv:1711.04682. [3] C. Reeg, J. Klinovaja, and D. Loss, PRB 96, 081301(R) (2017). [4] C. Schrade, M. Thakurathi, C. Reeg, S. Hoffman, J. Klinovaja, and D. Loss, PRB 96, 035306 (2017). [5] C. Reeg, D. Loss, and J. Klinovaja, PRB 96, 125426 (2017).

We consider densely packed Shiba chains with ferromagnetic or spiral alignment. on two-dimensional superconductors. In such a situation we show that all wavelengths must be taken into account in the treatment of the sub-gap properties along the chain. We show in particular that gap closings can occur at high momenta due to a mix of Shiba type and magnetic scattering states. Furthermore the sub-gap bands always connect to the continuum rather abruptly above the Fermi momentum such that it is impossible to single out a fully one-dimensional subband as required for distinct topological properties. To obtain the latter a finite spacing between the impurities, larger than the Fermi wavelength, needs to be reintroduced such that by zone folding the bands can disconnect from the continuum.

Topological superconductivity in low-dimensional systems is one of the amazing quantum states that can be accompanied by Majorana quasiparticles. Atomic chains of magnetic impurities (Shiba chains) on top of a superconducting substrate is one of the promising experimental set-ups that shows topological superconductivity in one dimension. Here, we address the problem of topological phase transition and emergence of Majorana bound states (MBS) at the wire end points when both conventional $s$-wave and unconventional $d$-wave superconductivity co-exist in the substrate, as there is evidence for such coexistence in high $T_c$ superconductors. Considering first a single magnetic impurity on a substrate with both $s$-wave and $d$-wave superconductivity, we observe that the $d$-wave order parameter plays a crucial role in the quantum phase transition. Next, we discuss how the topological phase transition in a Shiba chain is affected by $d$-wave order. In general, we assume Rashba spin-orbit coupling in the substrate and a gap function of the form $\Delta(k) = \Delta_d (\cos(k_x)-\cos(k_y)) + e^{i\phi} \Delta_s$ where $\phi$ can be either $\phi = n\pi$ or $\phi = (2n+1) \pi/2$ depending on whether time-reversal symmetry is broken or not in the superconducting substrate. Beside the emergence and robustness of MBS, another important aspect of such a system is the localization of the MBS. Our results show that a dense Shiba chain in the presence of both the $d$-wave and $s$-wave states leads to quite robust and more localized MBS for $d+is$-wave state, independent on the relative chain orientation on the substrate. However, the MBS in the case of $d+s$-wave state are fragile due to a small mini-gap. We also note that in this case, the MBS are sensitive to the the orientation of the chain with respect to the substrate. Furthermore, we identify an appropriate topological quantity to capture the topological phase transition for a one-dimensional chain embedded in a two-dimensional substrate. Here we necessarily have to keep the substrate to be two-dimensional, because of the $d$-wave order in the substrate. In particular, we find that the Berry phase ($\gamma_B$), based on a Wilson loop method for the occupied states,changes abruptly from $\gamma_B=0$ to $\gamma_B=\pi$ at the topological phase transition.

We show that bringing into proximity two topologically trivial systems can give rise to a topological phase. More specifically, we study a 1D metallic nanowire proximitized by a 2D superconducting substrate with a mixed s-wave and p-wave pairing, and we demonstrate both analytically and numerically that the phase diagram of such a setup can be richer than reported before. Thus, apart from the two 'expected' well-known phases (i.e. where the substrate and the wire are both simultaneously trivial or topological), we show that there exist two peculiar phases in which the nanowire can be in a topological regime while the substrate is trivial, and vice versa.

Majorana fermions (MF), predicted in 1937 by E. Majorana [1], are fermionic particles which are their own anti-particles. In analogy with electrons, MFs are half-spin integer particles and satisfy the Dirac equation. However, the MF wave-function is purely real in contrast to the electron (complex with positron as anti-particle) which has fundamental consequences on the intrinsic properties of the particle. In particular, its emergence in well designed condensed-matter systems opens new perspectives for quantum computation [2]. After more than 80 years search, this elusive particle has been recently observed by Scanning Tunneling Microscopy as zero-energy state at the end of Fe chains deposited on superconducting lead [3-4]. In this contribution, we will show our recent experiments combining scanning tunneling microscopy (STM) and atomic force microscopy (AFM) at low temperature. By revealing the atomic structure and electronic properties of such self-assembled chains on Pb(110), we confirmed the MF presence at the wire ends and extracted the decay length of its wave-function [5]. Our results, in good agreement with previous theory [6], show a short decay length (<25 nm) which supports its future use as qbits. Interestingly, AFM imaging also reveals an additional force signature at the MF location. [1] E. Majorana, Nuovo Cimento 14, 171 (1937). [2] A. Kitaev, Ann. Phys. 303, 2 (2003). [3] V. Mourik et al., Science 336, 1003 (2012). [4] S. Nadj-Perge et al. Science 346, 602 (2014). [5] R. Pawlak et al., npj Quantum Info, 2, 16035 (2016). [6] J. Klinovaja, D. Loss, Phys. Rev. B 86, 085408 (2012).

We analyze the formation of Majorana zero-modes at the edge of a two-dimensional topological superconductor. In particular, we study a time-reversal-invariant triplet phase that is likely to exist in doped Bi2Se3 . Upon the introduction of an in-plane magnetic field to the superconductor, a gap is opened in the surface modes, which induces localized Majorana modes. The position of these modes can be simply manipulated by changing the orientation of the applied field, yielding novel methods for braiding these states with possible applications to topological quantum computation. We shall discuss the approach to aqdiabacity and braiding of such systems.