New Platforms for Topological Superconductivity with Magnetic Atoms

Nodal-line semimetals are characterized by one-dimensional nodal rings in the bulk protected by symmetry. Projection of these nodal rings onto the surface of a three-dimensional topological semimetal leads to a new class of topological surface states known as drumhead surface states. Materials hosting these exotic features are expected to exhibit several quantum phenomena along with unusual transport characteristics and hence are promising candidates for device application and quantum information. We investigate the possibility of realizing drumhead surface states in a layered three-dimensional hexagonal lattice with the nodal-lines topologically protected. We examine these surface states in centrosymmetric systems with four-fold degeneracy, as well as, in the two-fold degenerate noncentrosymmetric systems with strong Rashba and Dresselhaus antisymmetric spin-orbit coupling. We find that in noncentrosymmetric systems, these nodal rings and their corresponding drumhead surface states are fully spin-polarized due to the spin-orbit coupling. These nontrivial topological surface states can be realized by the means of quasiparticle interference pattern, which we report for Weyl and Dirac nodal line semimetals considering the forthcoming experimental implications.

Motivated by their potential use as topological qubits, Majorana bound states (MBS) [1,2,3] have attracted an utmost interest. Theoretical calculations predict their occurrence in the combination of magnetic quasi-one-dimensional nanowire systems onto s-wave superconductors [2,3]. Here, we measure the spatial and electronic characteristics of topological, superconducting chains of iron atoms on Pb(110) to investigate the wave function and the localization length as fingerprint for MBSs [4]. After first observations by scanning tunneling microscopy (STM) [5,6], we demonstrate by combining STM and atomic force microscopy (AFM) at low temperature (< 5 K) that the Fe chains are mono-atomic, structured in a linear fashion, and exhibit zero-bias conductance peaks at their ends [7]. This can be interpreted as signature for a Majorana bound state [8]. Spatially resolved conductance maps of the atomic chains reveal that the MBSs are well localized at the chain ends (length below 25 nm). From these observations, we strongly support the idea of using MBSs in Fe chains on superconducting Pb as qubits for quantum computing devices. [1] Majorana, E.; Nuovo Cimento 14, 171 (1937) [2] Kitaev, A. Y.; Phys.-Usp. 44, 131 (2001) [3] Alicea, J.; Rep. Prog. Phys. 75, 07501 (2012) [4] Klinovaja J. & Loss, D.; Phys. Rev. B 86, 085408 (2012) [5] Nadj-Perge, S. et al.; Science 346, 602 (2014) [6] Ruby, M. et al.; Phys. Rev. Lett. 115, 197-204 (2015) [7] Pawlak, R. et al.; npj Quantum Information, 16035 (2016) [8] Mourik, V. et al.; Science 336, 1003 (2012)

Magnetic impurities on a superconductor act as scattering centers for the Cooper pairs of the latter and thereby induce a bound state, called Yu-Shiba-Rusinov state, in their vicinity. The spatial extent of these Yu-Shiba-Rusinov states is of several nanometers which allows for hybridization between them. Here, we observe the formation of a Kagome lattice after deposition of iron(III)-porphine-chloride molecules on Pb(111) and subsequent annealing. The lattice is composed of triangular units made of three iron-porphine molecules and one Cl adatom. By investigating small structures made of a few of these triangular units we can prove that the Yu-Shiba-Rusinov states induced by the molecules hybridize with one another within these structures. We are thus able to produce a two-dimensional coupled network by molecular self-assembly.

Magnetic skyrmions are nanometer-scale topological defects that are now easily created and manipulated experimentally [1]. Due to topological protection and easy manipulation they are expected to be at the core of next-generation devices for information storage and processing.     Recent theoretical work [2,3,4] has demonstrated that in a system where a skyrmion is coupled to an s-wave superconductor, a Majorana bound state can be realised at the core of the defect provided that both the interaction strength and the radial winding number of the skyrmion are large enough. In contrast, we study the flat band of bound states which appear localized at the outer edge of the skyrmion. We propose an interpretation of the flat band in terms of a chiral Majorana band. Using numerical calculations, we study the robustness of the band while we investigate analytically effective models that explain the protection of this band. Next, we discuss the possibility of bringing together two skyrmions to fuse or braid the Majorana fermions on the edge. Finally, we discuss the link between the skyrmion texture and defects in spin-orbit coupling. [1] N. Nagaosa and Y. Tokura, Nature Nanotechnology 8, 899-911 (2013). [2] S. Nakosai, Y. Tanak and N. Nagaosa, Phys. Rev. B. 88, 180503 (2013). [3] W. Chen and A. P. Schnyder, Phys. Rev. B 92, 214502 (2015). [4] G. Yang, P. Stano, J. Klinovaja and D. Loss, Phys. Rev. B 93, 224505 (2016).

In strongly correlated electronic systems, the electronic distribution favors local variations, often forming patterns that break the rotational symmetry of the crystal. Nematicity has special relevance in systems where competition of different phases plays an important role, and has been observed in the most prominent families of high temperature superconductors, i.e. cuprates and iron-pnictides. The Li doped NaFeAs samples allows a new approach to explore nematicity in Iron-pnictides, given the strong differences of the phase diagram in the parents compounds, NaFeAs and LiFeAs. In our work, we measure 3%, 4% and 5% Li doped NaFeAs samples, where long range order is suppressed, with a low temperature scanning tunneling microscope. We visualize the real-space distribution of the nematic fluctuations, which are found to ubiquitously exist, from the superconducting state to the normal state at higher temperatures.

We consider massless Dirac fermions in a graphene monolayer in the ballistic limit, subject to both a perpendicular magnetic field B and a proximity-induced pairing gap $\Delta$. When the chemical potential is at the Dirac point, our exact solution of the Bogoliubov-de Gennes equation yields $\Delta$-independent relativistic Landau levels. Since eigenstates depend on $\Delta$, many observables nevertheless are sensitive to pairing, e.g., the local density of states or the edge state spectrum. By solving the problem with an additional in-plane electric field, we also discuss how snake states are influenced by a pairing gap.

A junction between two boundaries of a topological superconductor, mediated by localized edge modes of Majorana fermions, is investigated. The tunneling of fermions across the junction depends on the magnetic flux. It breaks the time-reversal symmetry at the boundary of the sample. The persistent current is determined by the emergence of Majorana edge modes. The structure of the edge modes depends on the magnitude of the tunneling amplitude across the junction. It is shown that there are two different regimes, which correspond to strong and weak tunneling of Majorana fermions, distinctive in the persistent current behavior. In a strong tunneling regime, the fermion parity of edge modes is not conserved and the persistent current is a $2\pi$-periodic function of the magnetic flux. When the tunneling is weak the chiral Majorana states, which are propagating along the edges have the same fermion parity. They form a $4\pi$-phase periodic persistent current along the boundaries. The regions in the space of parameters, which correspond to the emergence of $2\pi$- and of $4\pi$-harmonics, are numerically determined. The peculiarities in the persistent current behavior are studied.

The host superconductors on which Yu-Shiba-Rusinov (YSR) typically form may contain a large amount of non-magnetic impurities, which may lead to fluctuations in the properties of the YSR states. In this work, we investigate this problem using a semi-classical scattering approach and reduce the effects of disorder to two contributions: disorder-induced normal reflection and a random phase of the amplitude for Andreev reflection. We find that both of these are small even for moderate amounts of disorder. In the dirty limit in which the disorder-induced mean free path is smaller than the superconducting coherence length, the variance of the energy of the Yu-Shiba-Rusinov state remains small in the ratio of the Fermi wavelength and the mean free path. This effect is more pronounced in three dimensions, where only impurities within a few Fermi wavelengths of the magnetic scatterer contribute. In two dimensions the energy variance is larger by a logarithmic factor because impurities contribute up to a distance of the order of the superconducting coherence length.

We study the influence of the inverse proximity effect on the superconductivity nucleation in hybrid structures consisting of the semiconducting nanowires placed in contact with a thin superconducting film and discuss the resulting restrictions on the operation of Majorana-based devices. A strong paramagnetic effect for electrons entering the semiconductor together with spin-orbit coupling and van Hove singularities in the electronic density of states in the wire are responsible for the suppression of superconducting correlations in the low field domain and for the reentrant superconductivity at high magnetic fields in the topologically nontrivial regime. The growth of the critical temperature in the latter case continues up to the upper critical field destroying the pairing inside the superconducting film due to either orbital or paramagnetic mechanism. The suppression of the homogeneous superconducting state near the boundary between the topological and non-topological regimes provides the conditions favorable for the Fulde-Ferrel-Larkin-Ovchinnikov instability.

Transition metal dichalcogenide (TMD) $2H-NbSe_{2}$ has a layered hexagonal structure consisting in the alternation of sheets of Nb atoms sandwiched between two layers of selenium atoms. It is superconducting under the critical temperature $T_{c}~7K$ and respects inversion symmetry. Monolayer $NbSe_{2}$ is also superconducting (ref 1) but contrary to bulk $2H-NbSe_{2}$ it breaks in-plane inversion symmetry. The presence of large spin orbit interaction due to the 4d electrons of Nb atoms combined with this broken inversion symmetry gives rise to spin-momentum locking in the out of plane direction and unconventional Ising pairing. One of the consequences of this Ising pairing is very high in-plane upper critical fields which were measured recently (ref 2). In the parent compound superconducting misfit TMD $LaNb_{2}Se_{5}$, which consists in alternation of bilayers of $NbSe_{2}$ and monolayers of insulating {LaSe}, very high in plane upper critical fields were also observed (ref 3). Because of this, one can assume that similar physics is at play. $LaNb_{2}Se_{5}$ is a good candidate for unconventional superconductivity and some recent ultra-high vaccum STM/STS results of our team seem to show signatures of 3D topological edge states at the surface of a cleaved in situ crystal of $LaNb_{2}Se_{5}$. Many studies are yet to be performed to confirm our observations. 1. Frindt, R. F. Superconductivity in ultrathin $NbSe_{2}$ layers. Phys. Rev. Lett. 28, 299-301 (1972).

A magnetic impurity adsorbed on a superconducting substrate yields so-called Yu-Shiba-Rusinov (YSR) states. These are low energy bound states inside the superconducting energy gap locally induced by magnetic exchange scattering. 2H -NbSe2 belongs to the class of transition metal dichalcogenides and is a layered van der Waals material with strong 2D character. In this material, superconductivity coexists with a charge density wave (CDW) at low temperatures. Here, we investigate YSR states of single transition metal atoms adsorbed on the surface of 2H -NbSe2 using low temperature scanning tunneling microscopy and spectroscopy. We observe variations in d state resonances as well as in the YSR excitations. We can link these variations to the adsorption in two distinct atomic adsorption sites. Furthermore, the energy of the YSR states and their spatial extend appear to be influenced by the CDW.

D. Longo, H.Cruguel, S. Royer, C. Brun, F. Debontridder, P. David, N.Witkowski, T.Cren Institut des NanoSciences de Paris, Sorbonne Université & CNRS, Paris, France Recent investigation of ultrathin metal films with atomically well-defined thickness and high crystallinity has shown the existence of 2D superconductivity down to a single atomic layer as shown in Pb/Si(111) monolayers [1]. Pb/Si(111) has different surface structures depending on conditions of preparation such as the Pb coverage and the annealing temperature. Since this system consists of heavy atoms, there also exists a strong Rashba spin-orbit coupling modifying the electronic properties. Combining the Pb/Si(111) system with local magnetism makes it an ideal platform in which topological superconductivity signatures can be searched [2]. Indeed, the introduction of magnetic impurities can induce the appearance of localized bound states in the 2D superconducting gap, known as Shiba states. Under some specific conditions, the magnetic interaction between several Shiba states in an either one-dimensional or two-dimensional array of magnetic impurities on the surface of a superconductor may realize new topological phases [3]. Magnetic phthalocyanines (Pcs) are very promising metallo-organic molecules which can be used to introduce the above-mentioned local magnetism [4]. Moreover, they can form 2D self-assemblies on various metallic surfaces [5] and they can make it possible to tailor two-dimensional ordered magnetic structures, as we have seen by STM on different Pb/Si(111) surfaces. 1. T. Zhang et al. Nature Physics 6, 104-108 (2010) 2. C. Brun et al. Supercond. Sci. Technol. 30 013003 (2017) 3. J. Röntynen et al. PRL 114, 236803 (2015) 4. Franke et al. Science 332, 940-944 (2011) 5. Gargiani et al. Phys. Rev. B 87, 165407 (2013)

We have studied TI Sb2Te3 nanowires synthesized by chemical vapor deposition with different cross-sectional areas. Sb2Te3 is one of the TI materials with a bulk band gap of 0.28 eV and simple surface states consisting of a single Dirac cone in the band gap. The magnetoresistance measurement is cross examined with the nano-ARPES measurement, unambiguously pointing out the quantum transport to be dominated by massless Dirac fermions at the surface of the nanowires. When a magnetic field is applied along the nanowire axis, the wavefunction of the surface states around the cross-sectional area of the nanowire, acquires a phase-shift of 2 $\pi$ $\Phi$/$\Phi0$, where $\Phi0$ defines the magnetic flux quantum h/e. In addition, the spin-momentum locking causes carriers encircling the wire to acquire a  Berry phase, which opens up a gap in the lowest-energy 1D surface sub-band. Inserting a half flux quanta through the nanowire cancels the Berry phase and restore the gapless 1D mode. Therefore, the topological nature of the 1D surface sub-bands is revealed via the behavior of the AB oscillation as a function of $\Phi$/$\Phi0$. In addition, we have recently explored the interface of TI Sb2Te3 nanowire with ferromagnet, and observed for the first time a non-local spin valve (NLSV) signal in a TI channel. We have determined that the magnetization of the two Py layers switch at nearly identical fields, so the antiparallel alignment of the injector and detector magnetic moments is never achieved in these devices. Nevertheless, a strong non-local spin valve signal is observed. Furthermore, the symmetry of this signal is dramatically different from that of a NLSV with a channel that lacks spin-momentum locking (such as graphene). Indeed, our data reveal that two parallel states of the injector and detector magnetic moments give rise to different non-local voltage values, which is never observed in conventional NLSVs. This unusual symmetry is a clear signature of the spin-momentum locking in the Sb2Te3 nanowire surface state.

We provide a scheme to investigate and identify systems with flat bands, like the Kagome and Lieb lattices, and address the sensitivity of the flat bands to the hopping parameters. A unique feature of the energy spectrum of a flat-band system, such as the Kagome lattice, is the parabolic band touching of the flat band and a dispersing one at the Γ-point. When the lattice sites are populated by hard core spins, it is known that the ground-state energy is determined solely by the occupancy of the flat-band up to a filling fraction of ν=1/9. Beyond this point, the parabolic band at higher energies begins to populate, leading to the physics of Bose-Einstein condensation. Upon fermionizing such a system, we can treat the bosonic excitations as fermions interacting with a Chern-Simons(CS) gauge field. These fermions have an interesting property that they occupy the flat-band up to ν=1/3. We investigate the precise role played by CS-gauge field which connects the fermionic ν=1/3 state to the bosonic ν=1/9 state. In the process we also address the formation of `moats’ in the electronic structure of such systems that prevents the formation of a condensate (which leads to a spin-liquid behavior).

We investigate the magnetic field dependence of an ideal superconducting vortex lattice in the parity-mixed pair-density wave phase of multilayer superconductors within a circular cell Ginzburg-Landau approach. In multilayer systems, due to local inversion symmetry breaking, a Rashba spin-orbit coupling is induced at the outer layers. This combined with a perpendicular paramagnetic (Pauli) limiting magnetic field stabilizes a staggered layer dependent pair-density wave phase in the superconducting singlet channel. The high-field pair-density wave phase is separated from the low-field BCS phase by a first-order phase transition. The motivating guiding question in this paper is: what is the minimal necessary Maki value $\alpha_M$ for the appearance of the pair-density wave phase of a superconducting trilayer system? To address this problem we generalize the circular cell method for the regular flux-line lattice of a type-II superconductor to include paramagnetic depairing effects. Then, we apply the model to the trilayer system, where each of the layers are characterized by Ginzburg-Landau parameter $\kappa_0$, and a Maki value $\alpha_M$. We find that when the spin-orbit Rashba interaction compares to the superconducting condensation energy, the orbitally limited pair-density wave phase stabilizes for Maki values $\alpha_M> 10$.

Olof Peters [1], Nils Bogdanoff [1], Gaël Reecht [1], Clemens B. Winkelmann [2], and Katharina J. Franke [1] — [1] Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany — [2] Univ. Grenoble Alpes, Inistitut Neél, 25 Avenue des Martyrs, 38042 Grenoble, France Besides its applications in quantum mechanical circuits the Josephson effect is a powerful tool to study the superconducting ground state. The combination of a Josephson junction with the atomic-scale precision of a scanning tunnelling microscope (STM) would enable the measurement of the superconducting order parameter around single magnetic defects. We present here a current-driven Josephson junction in an STM at $T=1.3 K$. Both tip and sample consist of Pb. Extending the setup with coaxial cables suitable for microwaves makes it possible to couple radiation up to $f=26\, GHz$ into the junction where the tip acts as an antenna. The Josephson current responds to the incident microwave radiation by showing multiple steps, with the step position and width depending on the amplitude and the frequency of the microwave radiation. Simulations of the step spacing reveals, that the current is dominated by Cooper pair tunnelling. The power dependence of the steps suggests that the Cooper pair tunnelling is a photon-assisted incoherent process similar as seen by [1]. [1] A. Roychowdhury et al., Phys. Rev. Applied 4, 034011 (2015)

Zero-energy Majorana bound state can be induced at the edge of the low dimensional systems. From practical point of view, an intentional creation and manipulation of this state is very important. These efforts have been realized in e.g. the quantum dot hybrid nanowire system [1,2]. Numerical calculations show that not every state of the quantum dot is resonance level of the Majorana quasiparticles, as a consequence of spin polarization of this state [3]. Here, we show a possible realization of the Majorana bound state in the periodic system composed by the nanoring-qunatum dot. We present how changing the gate potential can create (Andreev or Majorana) bound state. Moreover, we discuss these properties also in the context of the band structure of investigated system. [1] M. T. Deng et al., Science 354, 1557 (2016). [2] A. Ptok, A. Kobiałka and T. Domański, Phys. Rev. B 96, 195430 (2017). [3] D. Sticlet, C. Bena and P. Simon, Phys. Rev. Lett. 108, 096802 (2012).

I propose a general scheme to probe the compatibility of arbitrary pairing states with a given normal state Hamiltonian by the introduction of a concept called “Superconducting Fitness”. This quantity gives a direct measure of the suppression of the superconducting critical temperature in the presence of key symmetry-breaking fields and can be used as a tool to identify nontrivial mechanisms to suppress superconductivity in complex multi-orbital systems. This concept can also be used in order to engineer normal state Hamiltonians to favour or suppress different order parameters. I discuss the application of this idea to Sr2Ru04, CePt3Si and KFe2As2

Magnetic impurities on superconductors induce an exchange scattering potential that locally perturbs the pairing of the superconductor’s electrons. This leads to the presence of Yu-Shiba-Rusinov (YSR) states within the gap of the superconductors whose energy depends on the coupling strength between the impurity and Cooper pairs. In particular, upon increase of this coupling strength, the ground state of the system undergoes a quantum phase transition from a free to a screened spin state. Here, we investigate YSR states induced by Fe-porphin molecules on Pb(111). Upon tip approach, we are able to continuously tune their energy across the Fermi energy and thus study, on the single impurity level, such a quantum phase transition.

Yu-Shiba-Rusinov (YSR) states have become very important in artificial designer structures, e.g. in atomic iron chains on a superconducting substrate that gives rise to Majorana modes. It has been demonstrated that the strength of the magnetic coupling J between the impurity spin and the underlying superconducting substrate can be tuned by introducing molecular spacers. In this case, the YSR states migrate close to the main superconducting coherence peaks and new symmetric features with respect to $E_F$ are appear outside the superconducting gap. In this work, we experimentally and theoretically investigate the spectral evolution arising from the interplay between the YSR states and spin inelastic tunnelling in different spin systems (M$-$ phthalocyanine molecule with M: Co,Mn,Cu,Fe) on NbSe$_2$ surface. By controlling the distance between the central ion and the NbSe$_2$ substrate through approaching the STM tip to the ion, we can tune the exchange coupling strength J over the spin inelastic tunnelling, and demonstrate the spectral crossover from the YSR states to the intrinsic quantum spin states. The results pave the way to a detailed understanding of the low-energy quantum states and quantum phase transition induced by molecular flexibility. Our results further demonstrate that the structural flexibility of the molecular biased system can help to realize novel functionalities in nanostructures.