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  • 1.
    Awoga, Oladunjoye A.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Bouhon, Adrien
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Domain walls in a chiral d-wave superconductor on the honeycomb lattice2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 1, article id 014521Article in journal (Refereed)
    Abstract [en]

    We perform a fully self-consistent study of domain walls between different chiral domains in chiral d(x2-y2) +/- id(xy) -wave superconductors with an underlying honeycomb lattice structure. We investigate domain walls along all possible armchair and zigzag directions and with a finite global phase shift across the domain wall, in addition to the change of chirality. For armchair domain walls we find the lowest domain wall energy at zero global phase shift, while the most favorable zigzag domain wall has a finite global phase shift dependent on the doping level. Belowthe van Hove singularity the armchair domain wall is most favorable, while at even higher doping the zigzag domain wall has the lowest energy. The domain wall causes a local suppression of the superconducting order parameter, with the superconducting recovery length following a universal curve for all domain walls. Moreover, we always find four subgap states crossing zero energy and well localized to the domain wall. However, the details of their energy spectrum vary notably, especially with the global phase shift across the domain wall.

  • 2.
    Awoga, Oladunjoye
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Björnson, Kristofer
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Disorder robustness and protection of Majorana bound states in ferromagnetic chains on conventional superconductors2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 95, no 18, article id 184511Article in journal (Refereed)
    Abstract [en]

    Majorana bound states (MBSs) are well established in the clean limit in chains of ferromagnetically aligned impurities deposited on conventional superconductors with finite spin-orbit coupling. Here we show that these MBSs are very robust against disorder. By performing self-consistent calculations we find that the MBSs are protected as long as the surrounding superconductor show no large signs of inhomogeneity. We also find that longer chains offer more stability against disorder for the MBSs, albeit the minigap decreases, as do increasing strengths of spin-orbit coupling and superconductivity.

  • 3.
    Awoga, Oladunjoye
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Probing unconventional superconductivity in proximitized graphene by impurity scattering2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 21, article id 214515Article in journal (Refereed)
    Abstract [en]

    We demonstrate how potential impurities are a very powerful tool for determining the pairing symmetry in graphene, proximity coupled to a spin-singlet superconductor. All d-wave states are characterized by subgap resonances, with spatial patterns clearly distinguishing between nodal and chiral d-wave symmetry, while s-wave states have no subgap resonances. We also find strong supergap impurity resonances associated with the normal state Dirac point. Subgap and supergap resonances only interact at very low doping levels, then causing suppression of the supergap resonances.

  • 4.
    Awoga, Oladunjoye
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Cayao, Jorge
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Supercurrent Detection of Topologically Trivial Zero-Energy States in Nanowire Junctions2019In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 123, no 11, article id 117001Article in journal (Refereed)
    Abstract [en]

    We report the emergence of zero-energy states in the trivial phase of a short nanowire junction with a strong spin-orbit coupling and magnetic field, formed by strong coupling between the nanowire and two superconductors. The zero-energy states appear in the junction when the superconductors induce a large energy shift in the nanowire, such that the junction naturally forms a quantum dot, a process that is highly tunable by the superconductor width. Most importantly, we demonstrate that the zero-energy states produce a re shift in the phase-biased supercurrent, which can be used as a simple tool for their unambiguous detection, ruling out any Majorana-like interpretation.

  • 5.
    Banerjee, S.
    et al.
    Los Alamos Natl Lab, Inst Mat Sci, Los Alamos, NM 87545 USA.;KTH Royal Inst Technol, Ctr Quantum Mat, Nordita, Roslagstullsbacken 23, S-10691 Stockholm, Sweden.;Stockholm Univ, Roslagstullsbacken 23, S-10691 Stockholm, Sweden.;Royal Inst Technol, Div Theoret Chem & Biol, SE-10691 Stockholm, Sweden..
    Fransson, Jonas
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Ågren, H.
    Royal Inst Technol, Div Theoret Chem & Biol, SE-10691 Stockholm, Sweden..
    Balatsky, A. V.
    Los Alamos Natl Lab, Inst Mat Sci, Los Alamos, NM 87545 USA.;KTH Royal Inst Technol, Ctr Quantum Mat, Nordita, Roslagstullsbacken 23, S-10691 Stockholm, Sweden.;Stockholm Univ, Roslagstullsbacken 23, S-10691 Stockholm, Sweden..
    Granular superconductor in a honeycomb lattice as a realization of bosonic Dirac material2016In: PHYSICAL REVIEW B, ISSN 2469-9950, Vol. 93, no 13, article id 134502Article in journal (Refereed)
    Abstract [en]

    We examine the low-energy effective theory of phase oscillations in a two-dimensional granular superconducting sheet where the grains are arranged in a honeycomb lattice structure. Using the example of graphene, we present evidence for the engineered Dirac nodes in the bosonic excitations: the spectra of the collective bosonic modes cross at the K and K' points in the Brillouin zone and form Dirac nodes. We show how two different types of collective phase oscillations are obtained and that they are analogous to the Leggett and the Bogoliubov-Anderson-Gorkov modes in a two-band superconductor. We show that the Dirac node is preserved in the presence of an intergrain interaction, despite induced changes of the qualitative features of the two collective modes. Finally, breaking the sublattice symmetry by choosing different on-site potentials for the two sublattices leads to a gap opening near the Dirac node, in analogy with fermionic Dirac materials. The Dirac node dispersion of bosonic excitations is thus expanding the discussion of the conventional Dirac cone excitations to the case of bosons. We call this case as a representative of bosonic Dirac materials (BDM), similar to the case of Fermionic Dirac materials extensively discussed in the literature.

  • 6.
    Björnson, Kristofer
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, Alexander V.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Superconducting order parameter π-phase shift in magnetic impurity wires2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 95, no 10, article id 104521Article in journal (Refereed)
    Abstract [en]

    It has previously been found that a magnetic impurity in a conventional s-wave superconductor can give rise to a local π-phase shift of the superconducting order parameter. By studying a finite wire of ferromagnetic impurities, we are able to trace the origin of the π-phase shift to a resonance condition for the Bogoliubov-de Gennes quasiparticle states. When non-resonating states localized at the impurity sites are pulled into the condensate for increasing magnetic strength, the superconducting order parameter is reduced in discrete steps, eventually resulting in a π-phase shift. We also show that for a finite spin-orbit coupling, the π-phase shift is preserved and occurs in a large portion of the topologically non-trivial phase.

  • 7.
    Björnson, Kristofer
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory. Univ Copenhagen, Niels Bohr Inst, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark..
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Probing chiral edge states in topological superconductors through spin-polarized local density of state measurements2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 14, article id 140504Article in journal (Refereed)
    Abstract [en]

    We show that spin-polarized local density of states (LDOS) measurements can uniquely determine the chiral nature of topologically protected edge states surrounding a ferromagnetic island embedded in a conventional superconductor with spin-orbit coupling. The spin-polarized LDOS show a strong spin polarization directly tied to the normal direction of the edge with opposite polarizations on opposite sides of the island and with a distinct oscillatory pattern in energy.

  • 8.
    Björnson, Kristofer
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Probing vortex Majorana fermions and topology in semiconductor/superconductor heterostructures2015In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 91, no 21, article id 214514Article in journal (Refereed)
    Abstract [en]

    We investigate the local density of states, spectral function, and superconducting pair amplitudes for signatures of Majorana fermions in vortex cores in ferromagnetic and spin-orbit coupled semiconductor/superconductor heterostructures. We show that the Majorana fermion quasiparticle momentum distribution is always symmetrically distributed at a finite radius around a high-symmetry point, thereby providing a necessary condition for a low-energy state to be a Majorana fermion. In real-space profiles of the local density of states through the vortex core the Majorana fermion, together with other finite-energy vortex states, forms a characteristic X-shape structure only present at nontrivial topology. Moreover, we find that the Mexican hat band structure property of the topologically nontrivial phase translates into multiple high-intensity band edges and also vortex core states located above the superconducting gap in the local density of states. Finally, we find no strong correlation between odd-frequency pairing and the appearance of Majorana fermions, but odd-frequency pairing exists as soon as ferromagnetism is present. In fact, we find that the only vortex superconducting pair amplitude directly related to any phase transition is the appearance of certain spin-triplet p-wave pairing components in the vortex core at a pre-topological vortex core widening transition.

  • 9.
    Björnson, Kristofer
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Skyrmion spin texture in ferromagnetic semiconductor-superconductor heterostructures2014In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 89, no 13, p. 134518-Article in journal (Refereed)
    Abstract [en]

    We provide a derivation of a spin Skyrmion number classification for two-dimensional topological superconductors constructed from ferromagnetic and Rashba spin-orbit coupled semiconductor-superconductor heterostructures. We show that in the nontrivial topological phase, characterized by a nonzero Chern number, there is always a topological spin texture in the occupied bands represented by a Skyrmion number. The Skyrmion number has the advantage of being both physically intuitive and directly measurable using spin-sensitive band structure imaging techniques. In addition, we show that the Skyrmion classification can be extended to the equivalent one-dimensional topological superconductors.

  • 10.
    Björnson, Kristofer
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory. Univ Copenhagen, Niels Bohr Inst, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Solid-state Stern-Gerlach spin splitter for magnetic field sensing, spintronics, and quantum computing2018In: Beilstein Journal of Nanotechnology, ISSN 2190-4286, Vol. 9, p. 1558-1563Article in journal (Refereed)
    Abstract [en]

    We show conceptually that the edge of a two-dimensional topological insulator can be used to construct a solid-state Stern-Gerlach spin splitter. By threading such a Stern-Gerlach apparatus with a magnetic flux, Aharanov-Bohm-like interference effects are introduced. Using ferromagnetic leads, the setup can be used to both measure magnetic flux and as a spintronics switch. With normal metallic leads a switchable spintronics NOT-gate can be implemented. Furthermore, we show that a sequence of such devices can be used to construct a single-qubit SU(2)-gate, one of the two gates required for a universal quantum computer. The field sensitivity, or switching field, b, is related to the characteristic size of the device, r, through b = h/(2 pi qr(2)), with q being the unit of electric charge.

  • 11.
    Björnson, Kristofer
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Vortex states and Majorana fermions in spin-orbit coupled semiconductor-superconductor hybrid structures2013In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 88, no 2, p. 024501-Article in journal (Refereed)
    Abstract [en]

    We study the energy spectrum of a vortex core in a two-dimensional semiconductor with Rashba spin-orbit interaction and proximity coupled to a conventional superconductor and a ferromagnetic insulator. We perform self-consistent calculations using the microscopic tight-binding Bogoliubov-de Gennes method on a lattice and confirm the existence of Majorana fermions in the nontrivial topological phase. We also find two different topologically trivial bulk superconducting phases, only differing in the type of vortex core structure they support and separated by a zero-energy excitation. Furthermore, we find an asymmetry in the energy spectrum with respect to both Zeeman splitting and vortex rotation direction and explain its physical origin.

  • 12.
    Björnson, Kristofer
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Pershoguba, Sergey S.
    KTH Royal Inst Technol, Ctr Quantum Mat, Nordita, S-10691 Stockholm, Sweden.;Stockholm Univ, S-10691 Stockholm, Sweden..
    Balatsky, Alexander V.
    KTH Royal Inst Technol, Ctr Quantum Mat, Nordita, S-10691 Stockholm, Sweden.;Stockholm Univ, S-10691 Stockholm, Sweden.;Los Alamos Natl Lab, Inst Mat Sci, Los Alamos, NM 87545 USA..
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Spin-polarized edge currents and Majorana fermions in one- and two-dimensional topological superconductors2015In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 92, no 21, article id 214501Article in journal (Refereed)
    Abstract [en]

    We investigate the persistent currents, spin-polarized local density of states, and spectral functions of topological superconductors constructed by placing ferromagnetic impurities on top of an s-wave superconductor with Rashba spin-orbit interaction. We solve self-consistently for the superconducting order parameter and investigate both two-dimensional blocks and one-dimensional wires of ferromagnetic impurities, with the magnetic moments pointing both perpendicular and parallel to the surface. We find that the topologically protected edge states of ferromagnetic blocks give rise to spin-polarized edge currents, but that the total persistent current flows in opposite direction to what is expected from the dispersion relation of the edge states. We also show that the Majorana fermions at the end points of one-dimensional wires are spin polarized, which can be directly related to the spin polarization of the edge currents of two-dimensional blocks. Connections are also made to the physics of the Yu-Shiba-Rusinov states for zero-dimensional impurities.

  • 13.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Chiral d-wave superconducting state in the core of a doubly quantized s-wave vortex in graphene2013In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 88, no 10, p. 104506-Article in journal (Refereed)
    Abstract [en]

    We show that the intrinsic chiral (d(x2-y2) +/- id(xy))-wave superconducting pairing in doped graphene is significantly strengthened in the core region of a doubly quantized s-wave superconducting vortex produced in a graphene-superconductor hybrid structure. The chiral d-wave state is induced by the proximity effect, which transfers the center-of-mass angular momentum of the s-wave vortex to the orbital angular momentum of the chiral d-wave Cooper pairs. The proximity effect is enhanced by the circular geometry of the vortex and we find a [1 + (T -T-c,T-J)(2)](-1) temperature dependence for the chiral d-wave core amplitude, where T-c,T-J is its intrinsic bulk transition temperature. We further propose to detect the chiral d-wave state by studying the temperature dependence of the low-energy local density of states in the vortex core, which displays a sudden radial change as a function of the strength of the d-wave core state.

  • 14.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Edge Properties and Majorana Fermions in the Proposed Chiral d-Wave Superconducting State of Doped Graphene2012In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 109, no 19, p. 197001-Article in journal (Refereed)
    Abstract [en]

    We investigate the effect of edges on the intrinsic d-wave superconducting state in graphene doped close to the van Hove singularity. While the bulk is in a chiral d(x2-y2) + id(xy) state, the order parameter at any edge is enhanced and has d(x2-y2)-symmetry, with a decay length strongly increasing with weakening superconductivity. No graphene edge is pair breaking for the d(x2-y2) state, and there are no localized zero-energy edge states. We find two chiral edge modes which carry a spontaneous, but not quantized, quasiparticle current related to the zero-energy momentum. Moreover, for realistic values of the Rashba spin-orbit coupling, a Majorana fermion appears at the edge when tuning a Zeeman field.

  • 15.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, A. V.
    Fransson, Jonas
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Filling of magnetic-impurity-induced gap in topological insulators by potential scattering2015In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 91, no 20, article id 201411Article in journal (Refereed)
    Abstract [en]

    We show that the energy gap induced by ferromagnetically aligned magnetic impurities on the surface of a topological insulator can be filled, due to scattering off the nonmagnetic potential of the impurities. In both a continuum surface model and a three-dimensional tight-binding lattice model, we find that the energy gap disappears already at weak potential scattering as impurity resonances add spectral weight at the Dirac point. This can help explain seemingly contradictory experimental results as to the existence of a gap.

  • 16.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, Alexander V.
    Odd-frequency superconducting pairing in multiband superconductors2013In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 88, no 10, p. 104514-Article in journal (Refereed)
    Abstract [en]

    We point out that essentially all multiband superconductors have an odd-frequency pairing component, as follows from a general symmetry analysis of even- and odd-frequency pairing states. We show that odd-frequency superconducting pairing requires only a finite band hybridization, or scattering, and nonidentical intraband order parameters, of which only one band needs to be superconducting. Under these conditions odd-frequency odd-interband pairing is always present. From a symmetry analysis we establish a complete reciprocity between parity in band index and frequency.

  • 17.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, Alexander V.
    Odd-frequency superconducting pairing in topological insulators2012In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 86, no 14, p. 144506-Article in journal (Refereed)
    Abstract [en]

    We discuss the appearance of odd-frequency spin-triplet s-wave superconductivity, first proposed by Berezinskii [J. Exp. Theor. Phys. 20, 287 (1974)], on the surface of a topological insulator proximity coupled to a conventional spin- singlet s-wave superconductor. Using both analytical and numerical methods, we show that this disorder robust odd-frequency state is present whenever there is an in-surface gradient in the proximity induced gap, including superconductor-normal state junctions. The time-independent order parameter for the odd-frequency superconductor is proportional to the in-surface gap gradient. The induced odd-frequency component does not produce any low-energy states.

  • 18.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, Alexander V.
    Proximity-induced unconventional superconductivity in topological insulators2013In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 87, no 22, p. 220506-Article in journal (Refereed)
    Abstract [en]

    We study and classify the proximity-induced superconducting pairing in a topological insulator (TI)-superconductor (SC) hybrid structure for SCs with different symmetries. The Dirac surface state gives a coupling between spin-singlet and spin-triplet pairing amplitudes as well as pairing that is odd in frequency for p-wave SCs. We also find that all SCs induce pairing that is odd in both frequency and orbital (band) index, with oddness in frequency and orbital index being completely interchangeable. The different induced pairing amplitudes significantly modify the density of states in the TI surface layer.

  • 19.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, Alexander V.
    Strong potential impurities on the surface of a topological insulator2012In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 85, no 12, p. 121103-Article in journal (Refereed)
    Abstract [en]

    Topological insulators (TIs) are said to be stable against nonmagnetic impurity scattering due to suppressed backscattering in the Dirac surface states. We solve a lattice model of a three-dimensional TI in the presence of strong potential impurities and find that both the Dirac point and low-energy states are significantly modified: Low-energy impurity resonances are formed that produce a peak in the density of states near the Dirac point, which is destroyed and split into two nodes that move off center. The impurity-induced states penetrate up to ten layers into the bulk of the TI. These findings demonstrate the importance of bulk states for the stability of TIs and how they can destroy the topological protection of the surface.

  • 20.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, Alexander V.
    Subsurface impurities and vacancies in a three-dimensional topological insulator2012In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 86, no 11, p. 115433-Article in journal (Refereed)
    Abstract [en]

    Using a three-dimensional microscopic lattice model of a strong topological insulator (TI) we study potential impurities and vacancies in surface, subsurface, and bulk positions. For all impurity locations we find impurity-induced resonance states with energy proportional to the inverse of the impurity strength, although the impurity strength needed for a low-energy resonance state increases with the depth of the impurity. For strong impurities and vacancies as deep as 15 layers into the material, resonance peaks will appear at and around the Dirac point in the surface energy spectrum, splitting the original Dirac point into two nodes located off-center. Furthermore, we study vacancy clusters buried deep inside the bulk and find zero-energy resonance states for both single and multiple-site vacancies. Only fully symmetric multiple-site vacancy clusters show resonance states expelled from the bulk

  • 21.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Golubev, Dmitri S.
    Bauch, Thilo
    Lombardi, Floriana
    Fogelstrom, Mikael
    Model Evidence of a Superconducting State with a Full Energy Gap in Small Cuprate Islands2013In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 110, no 19, p. 197001-Article in journal (Refereed)
    Abstract [en]

    We investigate subdominant order parameters stabilizing at low temperatures in nanoscale high-T-c cuprate islands, motivated by the recent observation of a fully gapped state in nanosized YBa2Cu3O7-delta [D. Gustafsson et al., Nature Nanotech. 8, 25 (2013)]. Using complementary quasiclassical and tight-binding Bogoliubov-de Gennes methods, we show on distinctly different properties dependent on the symmetry being d(x2-y2) + is or d(x2-y2) + id(xy). We find that a surface-induced d(x2-y2) + is phase creates a global spectroscopic gap which increases with an applied magnetic field, consistent with experimental observation.

  • 22.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Honerkamp, Carsten
    Chiral d-wave superconductivity in doped graphene2014In: Journal of Physics: Condensed Matter, ISSN 0953-8984, E-ISSN 1361-648X, Vol. 26, no 42, p. 423201-Article, review/survey (Refereed)
    Abstract [en]

    A highly unconventional superconducting state with a spin-singlet d(x2-y2) +/- id(xy)-wave, or chiral d-wave symmetry has recently been suggested to emerge from electron-electron interactions in doped graphene. It has been argued that graphene doped to the van Hove singularity at 1/4 doping, where the density of states diverge, is particularly likely to be a chiral d-wave superconductor. In this review we summarize the currently mounting theoretical evidence for the existence of a chiral d-wave superconducting state in graphene, obtained with methods ranging from mean-field studies of effective Hamiltonians to angle-resolved renormalization group calculations. We further discuss the multiple distinctive properties of the chiral d-wave superconducting state in graphene, as well as its stability in the presence of disorder. We also review the means of enhancing the chiral d-wave state using proximity-induced superconductivity. The appearance of chiral d-wave superconductivity is intimately linked to the hexagonal crystal lattice and we also offer a brief overview of other materials which have also been proposed to be chiral d-wave superconductors.

  • 23.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Le Hur, K.
    Ecole Polytech, CNRS, Ctr Phys Theor, F-91128 Palaiseau, France..
    Topological superconductivity in two dimensions with mixed chirality2015In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 92, no 14, article id 140503Article in journal (Refereed)
    Abstract [en]

    We find a mixed chirality d-wave superconducting state in the coexistence region between antiferromagnetism and interaction-driven superconductivity in lightly doped honeycomb materials. This state has a topological chiral d + id-wave symmetry in one Dirac valley but d - id-wave symmetry in the other valley and hosts two counterpropagating edge states, always protected in the absence of intervalley scattering. A first-order topological phase transition, with no bulk gap closing, separates the chiral d-wave state at small magnetic moments from the mixed chirality d-wave phase.

  • 24.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Linder, Jacob
    Majorana fermions in spin-orbit-coupled ferromagnetic Josephson junctions2011In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 84, no 18, p. 180509(R)-Article in journal (Refereed)
    Abstract [en]

    We study all possible Majorana modes in two-dimensional spin-orbit-coupled ferromagnetic superconductor-normal state-superconductor (SNS) Josephson junctions and propose experiments to detect them. With the S region in a nontrivial topological phase and a superconducting phase difference phi = pi across the junction, two delocalized Majorana fermions with no excitation gap appear in the N region. In addition, if S and N belong to different topological phases and have well separated the Fermi surfaces, localized Majorana fermions with a finite excitation gap also emerge at both SN interfaces for all phi.

  • 25.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Wu, Wei
    Le Hur, Karyn
    Chiral d-wave superconductivity on the honeycomb lattice close to the Mott state2014In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 90, no 5, p. 054521-Article in journal (Refereed)
    Abstract [en]

    We study superconductivity on the honeycomb lattice close to the Mott state at half filling. Due to the sixfold lattice symmetry and disjoint Fermi surfaces at opposite momenta, we show that several different fully gapped superconducting states naturally exist on the honeycomb lattice, of which the chiral d + id'-wave state has previously been shown to appear when superconductivity appears close to the Mott state. Using renormalized mean-field theory to study the t-J model and quantum Monte Carlo calculations of the Hubbard-U model we show that the d + id'-wave state is the favored superconducting state for a wide range of on-site repulsion U, from the intermediate to the strong coupling regime. We also investigate the possibility of a mixed chirality d-wave state, where the overall chirality cancels. We find that a state with d + id'-wave symmetry in one valley but d - id'-wave symmetry in the other valley is not possible in the t-J model without reducing the translational symmetry, due to the zero-momentum and spin-singlet nature of the superconducting order parameter. Moreover, any extended unit cells result either in disjoint Dirac points, which cannot harbor this mixed chirality state, or the two valleys are degenerate at the zone center, where valley hybridization prevents different superconducting condensates. We also investigate extended unit cells where the overall chirality cancels in real space. For supercells containing up to eight sites, including the Kekule distortion, we find no energetically favorable d-wave solution with an overall zero chirality within the restriction of the t-J model.

  • 26.
    Black-Schaffer, Annica M.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Yudin, Dmitry
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Spontaneous gap generation on the surface of weakly interacting topological insulators using nonmagnetic impurities2014In: Physical Review B Condensed Matter, ISSN 0163-1829, E-ISSN 1095-3795, Vol. 90, no 16, p. 161413-Article in journal (Refereed)
    Abstract [en]

    Strong nonmagnetic impurities on the surface of three-dimensional topological insulators (TIs) generate localized resonance peaks close to the Dirac point. We show that this results in a strongly reduced critical Coulomb interaction strength to reach a magnetic surface state, following a Stoner-like criterion. Thus even weakly interacting TIs host a finite (local) magnetization around strong nonmagnetic impurities. The local magnetization gives rise to a global energy gap, linearly dependent on the maximum value of the magnetization but decreasing with reduced impurity concentration.

  • 27.
    Borla, Umberto
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy. Tech Univ Munich, Dept Phys, D-85748 Garching, Germany..
    Kuzmanovski, Dushko
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Tuning Majorana zero modes with temperature in pi-phase Josephson junctions2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 1, article id 014507Article in journal (Refereed)
    Abstract [en]

    We study a superconductor-normal-state-superconductor Josephson junction along the edge of a quantum spin Hall insulator with a superconducting pi phase across the junction. We solve self-consistently for the superconducting order parameter and find both real junctions, where the order parameter is fully real throughout the system, and junctions where the order parameter has a complex phase winding. Real junctions host two Majorana zero modes (MZMs), while phase-winding junctions have no subgap states close to zero energy. At zero temperature we find that the phase-winding solution always has the lowest free energy, which we establish as being due to a strong proximity effect into the N region. With increasing temperature this proximity effect is dramatically decreased and we find a phase transition into a real junction with two MZMs.

  • 28.
    Bouhon, Adrien
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Global band topology of simple and double Dirac-point semimetals2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 95, no 24, article id 241101Article in journal (Refereed)
    Abstract [en]

    We combine space group representation theory together with the scanning of closed subdomains of the Brillouin zone with Wilson loops to algebraically determine the global band-structure topology. Considering space group No. 19 as a case study, we show that the energy ordering of the irreducible representations at the high-symmetry points {Gamma, S, T, U} fully determines the global band topology, with all topological classes characterized through their simple and double Dirac points.

  • 29.
    Bouhon, Adrien
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Slager, Robert-Jan
    Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany;Harvard Univ, Dept Phys, Cambridge, MA 02138 USA.
    Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 100, no 19, article id 195135Article in journal (Refereed)
    Abstract [en]

    We present a general methodology toward the systematic characterization of crystalline topological insulating phases with time-reversal symmetry. In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle, we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries. Our approach finds a prominent use in elucidating and quantifying the recently proposed "topological quantum chemistry" concept. Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase. For this we first show that in addition to the Fu-Kane-Mele Z(2) classification, there is C2T-symmetry-protected Z classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e., forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs that are independent of the parametrization of the flow of Wilson loops. Then, by systematically embedding all combinatorial four-band phases into six-band phases, we find a refined topological feature of split EBRs. Namely, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial," i.e., necessarily nontrivial in the nonstable (few-band) limit but possibly trivial in the stable (many-band) limit. This clarifies the nature of fragile topology that has appeared very recently. We then argue that in the many-band limit, the stable Wilson loop winding is only determined by the Fu-Kane-Mele Z(2) invariant implying that further stable topological phases must belong to the class of higher-order topological insulators.

  • 30.
    Bouhon, Adrien
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Schmidt, Johann
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Topological nodal superconducting phases and topological phase transition in the hyperhoneycomb lattice2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 10, article id 104508Article in journal (Refereed)
    Abstract [en]

    We establish the topology of the spin-singlet superconducting states in the bare hyperhoneycomb lattice, and we derive analytically the full phase diagram using only symmetry and topology in combination with simple energy arguments. The phase diagram is dominated by two states preserving time-reversal symmetry. We find a line-nodal state dominating at low doping levels that is topologically nontrivial and exhibits surface Majorana flatbands, which we show perfectly match the bulk-boundary correspondence using the Berry phase approach. At higher doping levels, we find a fully gapped state with trivial topology. By analytically calculating the topological invariant of the nodal lines, we derive the critical point between the line-nodal and fully gapped states as a function of both pairing parameters and doping. We find that the line-nodal state is favored not only at lower doping levels but also if symmetry-allowed deformations of the lattice are present. Adding simple energy arguments, we establish that a fully gapped state with broken time-reversal symmetry likely appears covering the actual phase transition. We find this fully gapped state to be topologically trivial, while we find an additional point-nodal state at very low doing levels that also break time-reversal symmetry and has nontrivial topology with associated Fermi surface arcs. We eventually address the robustness of the phase diagram to generalized models also including adiabatic spin-orbit coupling, and we show how all but the point-nodal state are reasonably stable.

  • 31.
    Cayao, Jorge
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Finite length effect on supercurrents between trivial and topological superconductors2018In: The European Physical Journal Special Topics, ISSN 1951-6355, E-ISSN 1951-6401, Vol. 227, no 12, p. 1387-1396Article in journal (Refereed)
    Abstract [en]

    We numerically analyze the effect of finite length of the superconducting regions on the low-energy spectrum, current-phase curves, and critical currents in junctions between trivial and topological superconductors. Such junctions are assumed to arise in nanowires with strong spin-orbit coupling under external magnetic fields and proximity-induced superconductivity. We show that all these quantities exhibit a strong dependence on the length of the topological sector in the topological phase and serve as indicators of the topological phase and thus the emergence of Majorana bound states at the end of the topological superconductor.

  • 32.
    Cayao, Jorge
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Odd-frequency superconducting pairing and subgap density of states at the edge of a two-dimensional topological insulator without magnetism2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 15, article id 155426Article in journal (Refereed)
    Abstract [en]

    We investigate the emergence and consequences of odd-frequency spin-triplet s-wave pairing in superconducting hybrid junctions at the edge of a two-dimensional topological insulator without any magnetism. More specifically, we consider several different normal-superconductor hybrid systems at the topological insulator edge, where spin-singlet s-wave superconducting pairing is proximity induced from an external conventional superconductor. We perform fully analytical calculations and show that odd-frequency mixed spin-triplet s-wave pairing arises due to the unique spin-momentum locking in the topological insulator edge state and the naturally nonconstant pairing potential profile in hybrid systems. Importantly, we establish a one-to-one correspondence between the local density of states (LDOS) at low energies and the odd-frequency spin-triplet pairing in NS, NSN, and SNS junctions along the topological insulator edge; at interfaces the enhancement in the LDOS can directly be attributed to the contribution of odd-frequency pairing. Furthermore, in SNS junctions we show that the emergence of the zero-energy LDOS peak at the superconducting phase phi = pi is associated purely with odd-frequency pairing in the middle of the junction.

  • 33.
    Cayao, Jorge
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Odd-frequency superconducting pairing in junctions with Rashba spin-orbit coupling2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 98, no 7, article id 075425Article in journal (Refereed)
    Abstract [en]

    We consider normal-superconductor (NS) and superconductor-normal-superconductor (SNS) junctions based on one-dimensional nanowires with Rashba spin-orbit coupling and proximity-induced s-wave spin-singlet superconductivity and analytically demonstrate how both even- and odd-frequency and spin-singlet and -triplet superconducting pair correlations are always present. In particular, by using a fully quantum mechanical scattering approach, we show that Andreev reflection induces mixing of spatial parities at interfaces, thus being the unique process which generates odd-frequency pairing; on the other hand, both Andreev and normal reflections contribute to even-frequency pairing. We further find that locally neither odd-frequency nor spin-triplet correlations are induced, but only even-frequency spin-singlet pairing. In the superconducting regions of NS junctions, the interface-induced amplitudes decay into the bulk, with the odd-frequency components being generally much larger than the even-frequency components at low frequencies. The odd-frequency pairing also develops short- and long-period oscillations due to the chemical potential and spin-orbit coupling, respectively, leading to a visible beating feature in their magnitudes. Moreover, we find that in short SNS junctions at pi-phase difference and strong spin-orbit coupling, the odd-frequency spin-singlet and -triplet correlations strongly dominate with an alternating spatial pattern for a large range of subgap frequencies.

  • 34.
    Cayao, Jorge
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Prada, Elsa
    Univ Autonoma Madrid, Dept Fis Mat Condensada, Condensed Matter Phys Ctr IFIMAC, E-28049 Madrid, Spain;Univ Autonoma Madrid, Inst Nicolas Cabrera, E-28049 Madrid, Spain.
    Aguado, Ramon
    CSIC, ICMM, Madrid 28049, Spain.
    Andreev spectrum and supercurrents in nanowire-based SNS junctions containing Majorana bound states2018In: Beilstein Journal of Nanotechnology, ISSN 2190-4286, Vol. 9, p. 1339-1357Article in journal (Refereed)
    Abstract [en]

    Hybrid superconductor-semiconductor nanowires with Rashba spin-orbit coupling are arguably becoming the leading platform for the search of Majorana bound states (MBSs) in engineered topological superconductors. We perform a systematic numerical study of the low-energy Andreev spectrum and supercurrents in short and long superconductor-normal-superconductor junctions made of nanowires with strong Rashba spin-orbit coupling, where an external Zeeman field is applied perpendicular to the spin-orbit axis. In particular, we investigate the detailed evolution of the Andreev bound states from the trivial into the topological phase and their relation with the emergence of MBSs. Due to the finite length, the system hosts four MBSs, two at the inner part of the junction and two at the outer one. They hybridize and give rise to a finite energy splitting at a superconducting phase difference of p, a wellvisible effect that can be traced back to the evolution of the energy spectrum with the Zeeman field: from the trivial phase with Andreev bound states into the topological phase with MBSs. Similarly, we carry out a detailed study of supercurrents for short and long junctions from the trivial to the topological phases. The supercurrent, calculated from the Andreev spectrum, is 2 pi-periodic in the trivial and topological phases. In the latter it exhibits a clear sawtooth profile at a phase difference of pi when the energy splitting is negligible, signalling a strong dependence of current-phase curves on the length of the superconducting regions. Effects of temperature, scalar disorder and reduction of normal transmission on supercurrents are also discussed. Further, we identify the individual contribution of MBSs. In short junctions the MBSs determine the current-phase curves, while in long junctions the spectrum above the gap (quasi-continuum) introduces an important contribution.

  • 35.
    Cayao, Jorge
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    San-Jose, Pablo
    CSIC, ICMM, Cantoblanco, Madrid 28049, Spain..
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Aguado, Ramon
    CSIC, ICMM, Cantoblanco, Madrid 28049, Spain..
    Prada, Elsa
    Univ Autonoma Madrid, Dept Fis Mat Condensada, Condensed Matter Phys Ctr IFIMAC, E-28049 Madrid, Spain.;Univ Autonoma Madrid, Inst Nicolas Cabrera, E-28049 Madrid, Spain..
    Majorana splitting from critical currents in Josephson junctions2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 20, article id 205425Article in journal (Refereed)
    Abstract [en]

    A semiconducting nanowire with strong Rashba spin-orbit coupling and coupled to a superconductor can be tuned by an external Zeeman field into a topological phase with Majorana zero modes. Here we theoretically investigate how this exotic topological superconductor phase manifests in Josephson junctions based on such proximitized nanowires. In particular, we focus on critical currents in the short junction limit (L-N << xi, where L-N is the junction length and xi is the superconducting coherence length) and show that they contain important information about nontrivial topology and Majoranas. This includes signatures of the gap inversion at the topological transition and a unique oscillatory pattern that originates from Majorana interference. Interestingly, this pattern can be modified by tuning the transmission across the junction, thus providing complementary evidence of Majoranas and their energy splittings beyond standard tunnel spectroscopy experiments, while offering further tunability by virtue of the Josephson effect.

  • 36.
    Dutta, Paramita
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Signature of odd-frequency equal-spin triplet pairing in the Josephson current on the surface of Weyl nodal loop semimetals2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 100, no 10, article id 104511Article in journal (Refereed)
    Abstract [en]

    We theoretically predict proximity-induced odd-frequency (odd-omega) pairing on the surface of a Weyl nodal loop semimetal, characterized by a nodal loop Fermi surface and drumheadlike surface states (DSSs), attached to conventional spin-singlet s-wave superconducting leads. Due to the complete spin polarization of the DSS, only odd-omega equal-spin triplet pairing is present, and we show that it gives rise to a finite Josephson current. Placing an additional ferromagnet in the junction can also generate odd-omega) mixed-spin triplet pairing, but the pairing and current are not affected if the magnetization is orthogonal to the DSS spin polarization, which further confirms the equal-spin structure of the pairing.

  • 37.
    Fransson, Jonas
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, A. V.
    Engineered near-perfect backscattering on the surface of a topological insulator with nonmagnetic impurities2014In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 90, no 24, p. 241409-Article in journal (Refereed)
    Abstract [en]

    We show how to engineer enhanced skew scattering and gaplike openings in the surface state of three-dimensional topological insulators using only nonmagnetic impurities. Enhanced skewscattering off nonmagnetic impurities is obtained as a finite size effect of the scattering potential. Intimately related to the generated skew scattering is the emergence of a gaplike density of electron states locally around the impurities and surrounded by sharp resonances, with an extended energy gap appearing in engineered impurity structures.

  • 38.
    Komendova, Lucia
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, A. V.
    KTH Royal Inst Technol, Ctr Quantum Mat, NORDITA, SE-10691 Stockholm, Sweden.;Stockholm Univ, SE-10691 Stockholm, Sweden.;Los Alamos Natl Lab, Inst Mat Sci, Los Alamos, NM 87545 USA..
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Experimentally observable signatures of odd-frequency pairing in multiband superconductors2015In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 92, no 9, article id 094517Article in journal (Refereed)
    Abstract [en]

    We investigate how hybridization (single-quasiparticle scattering) between two superconducting bands induces odd-frequency superconductivity in a multiband superconductor. An explicit derivation of the odd-frequency pairing correlation and its full frequency dependence is given. We also find that the density of states is modified, at higher energies, from the sum of the two BCS spectra to also include additional hybridization gaps with strong coherence peaks when odd-frequency pairing is present. These gaps constitute clear experimentally measurable signatures of odd-frequency pairing in multiband superconductors.

  • 39.
    Komendova, Lucia
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Odd-Frequency Superconductivity in Sr2RuO4 Measured by Kerr Rotation2017In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 119, no 8, article id 087001Article in journal (Refereed)
    Abstract [en]

    We establish the existence of bulk odd-frequency superconductivity in Sr2RuO4 and show that an intrinsic Kerr effect is direct evidence of this state. We use both general two-and three-orbital models, as well as a realistic tight-binding description of Sr2RuO4 to demonstrate that odd-frequency pairing arises due to finite hybridization between different orbitals in the normal state, and is further enhanced by finite interorbital pairing.

  • 40.
    Kuzmanovski, Dushko
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Multiple odd-frequency superconducting states in buckled quantum spin Hall insulators with time-reversal symmetry2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 17, article id 174509Article in journal (Refereed)
    Abstract [en]

    We consider a buckled quantum spin Hall insulator (QSHI), such as silicene, proximity coupled to a conventional spin-singlet s-wave superconductor. Even limiting the discussion to the disorder-robust s-wave pairing symmetry, we find both odd-frequency (omega) spin-singlet and spin-triplet pair amplitudes, both of which preserve time-reversal symmetry. Our results show that there are two unrelated mechanisms generating these different odd-omega pair amplitudes. The spin-singlet state is due to the strong interorbital processes present in the QSHI. It exists generically at the edges of the QSHI, but also in the bulk in the heavily doped regime if an electric field is applied. The spin-triplet state requires a finite gradient in the proximity-induced superconducting order along the edge, which we find is automatically generated at the atomic scale for armchair edges but not at zigzag edges. In combination these results make superconducting QSHIs a very exciting venue for investigating not only the existence of odd-omega superconductivity but also the interplay between different odd-omega states.

  • 41.
    Löthman, Tomas
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Defects in the d+id-wave superconducting state in heavily doped graphene2014In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 90, no 22, article id 224504Article in journal (Refereed)
    Abstract [en]

    A chiral time-reversal symmetry breaking d+id-wave superconducting state is likely to emerge in graphene doped close to the Van Hove singularity. As heavy doping procedures are expected to introduce defects, we investigate here the effects of microscopic defects on the d+id-wave superconducting state at the Van Hove singularity. We find that, while the superconducting order is reduced near a defect, the d+id-wave state remains intact and recovers in an exponential manner away from the defect. The recovery length is found to be on the order of one lattice constant for weak couplings, and, as we show, this is comparable to the recovery length of a conventional s-wave state on the graphene honeycomb lattice, thereby demonstrating that the unconventional d+id-wave state is quite resilient to defects. Moreover, we find no significant changes between a single site defect and more extended defects, such as a bivacancy. While the d+id-wave state is fully gapped, we also show that defects introduce localized midgap states with nonzero energies, which should be accessible via scanning probe experiments.

  • 42.
    Löthman, Tomas
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Universal phase diagrams with superconducting domes for electronic flat bands2017In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 96, no 6, article id 064505Article in journal (Refereed)
    Abstract [en]

    Condensed matter systems with flat bands close to the Fermi level generally exhibit, due to their very large density of states, extraordinarily high critical ordering temperatures of symmetry-breaking orders, such as superconductivity and magnetism. Here we show that the critical temperatures follow one of two universal curves with doping away from a flat band depending on the ordering channel, which completely dictates both the general order competition and the phase diagram. Notably, we find that orders in the particle-particle channel (superconducting orders) survive decisively farther than orders in the particle-hole channel (magnetic or charge orders) because the channels have fundamentally different polarizabilities. Thus, even if a magnetic or charge order initially dominates, superconducting domes are still likely to exist on the flanks of flat bands. We apply these general results to both the topological surface flat bands of rhombohedral ABC-stacked graphite and to the Van Hove singularity of graphene.

  • 43.
    Mashkoori, Mahdi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Björnson, Kristofer
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Impurity bound states in fully gapped d-wave superconductors with subdominant order parameters2017In: Scientific Reports, ISSN 2045-2322, E-ISSN 2045-2322, Vol. 7, article id 4107Article in journal (Refereed)
    Abstract [en]

    Impurities in superconductors and their induced bound states are important both for engineering novel states such as Majorana zero-energy modes and for probing bulk properties of the superconducting state. The high-temperature cuprates offer a clear advantage in a much larger superconducting order parameter, but the nodal energy spectrum of a pure d-wave superconductor only allows virtual bound states. Fully gapped d-wave superconducting states have, however, been proposed in several cuprate systems thanks to subdominant order parameters producing d + is- or d + id'-wave superconducting states. Here we study both magnetic and potential impurities in these fully gapped d-wave superconductors. Using analytical T-matrix and complementary numerical tight-binding lattice calculations, we show that magnetic and potential impurities behave fundamentally different in d + is- and d + id'-wave superconductors. In a d + is-wave superconductor, there are no bound states for potential impurities, while a magnetic impurity produces one pair of bound states, with a zero-energy level crossing at a finite scattering strength. On the other hand, a d + id'-wave symmetry always gives rise to two pairs of bound states and only produce a reachable zero-energy level crossing if the normal state has a strong particle-hole asymmetry.

  • 44.
    Mashkoori, Mahdi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Majorana bound states in magnetic impurity chains: Effects of d-wave pairing2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 2, article id 024505Article in journal (Refereed)
    Abstract [en]

    We consider an atomic chain of magnetic impurities on the surface of a spin-orbit coupled superconductor with a dominating d-wave and subdominating s-wave order parameters. In particular, we investigate the properties of the Majorana bound states (MBSs) emerging at the chain end points in the topological phase and how MBSs are affected by the d-wave order parameter. We provide a comprehensive picture by both studying time-reversal invariant and breaking superconducting substrates as well as chains oriented in different directions relative to the d-wave rotation. We show that increasing the d-wave order parameter significantly enhances the localization of MBSs and their protective minigap, as long as the direction along which the impurity chain is oriented does not cross any nodal lines of the gap function. Moreover, we find an extra gap-closing for a specific condensate and chain orientation within the topological phase, which we are able to attribute to simple geometrical effects in the corresponding two-dimensional limit. These results show how high-temperature d-wave superconductors can be used to significantly enhance the properties and stability of MBSs.

  • 45.
    Mashkoori, Mahdi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Moghaddam, A. G.
    IASBS, Dept Phys, Zanjan 4513766731, Iran;IASBS, Res Ctr Basic Sci & Modem Technol RBST, Zanjan 4513766731, Iran;Inst Res Fundamental Sci IPM, Sch Phys, Tehran 193955531, Iran.
    Hajibabaee, M. H.
    IASBS, Dept Phys, Zanjan 4513766731, Iran.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Parhizgar, Fariborz
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory. Inst Res Fundamental Sci IPM, Sch Phys, Tehran 193955531, Iran.
    Impact of topology on the impurity effects in extended s-wave superconductors with spin-orbit coupling2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 1, article id 014508Article in journal (Refereed)
    Abstract [en]

    We investigate the impact of topology on the existence of impurity subgap states in a time-reversal-invariant superconductor with an extended s-wave pairing and strong spin-orbit coupling. By simply tuning the chemical potential, we access three distinct phases: topologically trivial s-wave, topologically nontrivial s(+/-)-wave, and nodal superconducting phase. For a single potential impurity, we find subgap impurity bound states in the topological phase, but notably no subgap states in the trivial phase. This is in sharp contrast with the expectation that there would be no subgap state in the presence of potential impurities in s-wave superconductors. These subgap impurity states have always finite energies for any strength of the potential scattering and, subsequently, the superconducting gap in the topological s(+/-)-wave phase survives but is attenuated in the presence of finite disorder. By creating islands of potential impurities, we smoothly connect the single impurity results to topological edge states of impurity islands. On the other hand, magnetic impurities lead to the formation of Yu-Shiba-Rusinov states in both the trivial and topological phases, which even reach zero energy at certain scattering strengths. We thus propose that potential impurities can be a very valuable tool to detect time-reversal-invariant topological superconductivity.

  • 46.
    Parhizgar, Fariborz
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory. IPM, Inst Res Fundamental Sci, Sch Phys, Tehran 193955531, Iran..
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Highly tunable time-reversal-invariant topological superconductivity in topological insulator thin films2017In: Scientific Reports, ISSN 2045-2322, E-ISSN 2045-2322, Vol. 7, article id 9817Article in journal (Refereed)
    Abstract [en]

    We study time-reversal-invariant topological superconductivity in topological insulator (TI) thin films including both intra-and inter-surface pairing. We find a nontrivial topology for multiple different configurations. For intra-surface pairing a p-phase difference between the intra-surface pairing states is required. We show that in this case the resulting topological phase is highly tunable by both an applied electric field and varied chemical potential. For spin-singlet inter-surface pairing, a sign-changing tunnel coupling present in many TI thin films is needed, and again, the topology can be tuned by electric field or doping. Notably, we find that the required inter-surface pairing strength for achieving nontrivial topology can still be subdominant compared to the intra-surface pairing. Finally, for spin-triplet intersurface pairing we prove that the superconducting state is always topological nontrivial. We show that thin films of Cu-doped Bi2Se3 will likely host such spin-triplet inter-surface pairing. Taken together, these results show that time-reversal-invariant topological superconductivity is common in superconducting TI thin films and that the topological phase and its Kramers pair of Majorana edge modes is highly tunable with an applied electric field and varied chemical potential.

  • 47.
    Parhizgar, Fariborz
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Unconventional proximity-induced superconductivity in bilayer systems2014In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 90, no 18, p. 184517-Article in journal (Refereed)
    Abstract [en]

    We study the proximity-induced superconducting state in a general bilayer-conventional s-wave superconductor hybrid structure. For the bilayer we include a general parabolic dispersion, Rashba spin-orbit coupling, and finite-layer tunneling as well as the possibility to apply a bias potential and a magnetic Zeeman field, in order to address experimentally relevant bilayer systems, ranging from topological insulator thin films to generic double quantum well systems. By extracting the proximity-induced anomalous Green's function in the bilayer, we show a very rich structure for the superconducting pairing, including different spin states and odd-frequency pairing. Equal-spin spin-triplet (p(x) +/- ip(y))-wave pairing is induced in both layers in the presence of a finite spin-orbit coupling and opposite-spin spin-triplet s-wave pairing with odd-frequency dependence appears for an applied magnetic Zeeman field. Finite interlayer pairing is also generally present in the bilayer. The interlayer pairing can be either even or odd in the layer index, with a complete reciprocity between parity in frequency and in layer index. We also find that a bilayer offers the possibility of sign reversal of the superconducting order parameters, both between the two layers and between multiple Fermi surfaces.

  • 48.
    Pershoguba, Sergey S.
    et al.
    KTH Royal Inst Technol, Ctr Quantum Mat, Nordita, S-10691 Stockholm, Sweden.;Stockholm Univ, S-10691 Stockholm, Sweden..
    Björnson, Kristofer
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Balatsky, Alexander V.
    KTH Royal Inst Technol, Ctr Quantum Mat, Nordita, S-10691 Stockholm, Sweden.;Stockholm Univ, S-10691 Stockholm, Sweden.;Los Alamos Natl Lab, Inst Mat Sci, Los Alamos, NM 87545 USA..
    Currents Induced by Magnetic Impurities in Superconductors with Spin-Orbit Coupling2015In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 115, no 11, article id 116602Article in journal (Refereed)
    Abstract [en]

    We show that superconducting currents are generated around magnetic impurities and ferromagnetic islands proximity coupled to superconductors with finite spin-orbit coupling. Using the Ginzburg-Landau theory, T-matrix calculation, as well as self-consistent numerical simulation on a lattice, we find a strong dependence of the current on the direction and magnitude of the magnetic moment. We establish that in the case of point magnetic impurities, the current is carried by the induced Yu-Shiba-Rusinov (YSR) subgap states. In the vicinity of the phase transition, where the YSR states cross at zero energy, the current increases dramatically. Furthermore, we show that the currents are orthogonal to the local spin polarization and, thus, can be probed by measuring the spin-polarized local density of states.

  • 49.
    Schmidt, Johann
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Scherer, Daniel D.
    Univ Copenhagen, Niels Bohr Inst, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark..
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Topological superconductivity in the extended Kitaev-Heisenberg model2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 1, article id 014504Article in journal (Refereed)
    Abstract [en]

    We study superconducting pairing in the doped Kitaev-Heisenberg model by taking into account the recently proposed symmetric off-diagonal exchange Γ. By performing a mean-field analysis, we classify all possible superconducting phases in terms of symmetry, explicitly taking into account effects of spin-orbit coupling. Solving the resulting gap equations self-consistently, we map out a phase diagram that involves several topologically nontrivial states. For Γ<0, we find a competition between a time-reversal symmetry-breaking chiral phase with Chern number ±1 and a time-reversal symmetric nematic phase that breaks the rotational symmetry of the lattice. On the other hand, for Γ≥0 we find a time-reversal symmetric phase that preserves all the lattice symmetries, thus yielding clearly distinguishable experimental signatures for all superconducting phases. Both of the time-reversal symmetric phases display a transition to a Z2 nontrivial phase at high doping levels. Finally, we also include a symmetry-allowed spin-orbit coupling kinetic energy and show that it destroys a tentative symmetry-protected topological order at lower doping levels. However, it can be used to tune the time-reversal symmetric phases into a Z2 nontrivial phase even at lower doping.

  • 50.
    Teixeira, Raphael L. R. C.
    et al.
    Univ Sao Paulo, Inst Fis, Caixa Postale 66318, BR-05315970 Sao Paulo, SP, Brazil.
    Kuzmanovski, Dushko
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Black-Schaffer, Annica M.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    da Silva, Luis G. G. V. Dias
    Univ Sao Paulo, Inst Fis, Caixa Postale 66318, BR-05315970 Sao Paulo, SP, Brazil.
    Gap oscillations and Majorana bound states in magnetic chains on superconducting honeycomb lattices2019In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 99, no 3, article id 035127Article in journal (Refereed)
    Abstract [en]

    Magnetic chains on superconducting systems have emerged as a platform for realization of Majorana bound states (MBSs) in condensed-matter systems with possible applications to topological quantum computation. In this work, we study the MBSs formed in magnetic chains on two-dimensional honeycomb materials with induced superconductivity. We establish chemical potential vs Zeeman splitting phase diagrams showing that the topological regions (where MBSs appear) are strongly dependent on the spiral angle along the magnetic chain. In some of those regions, the topological phase is robust even for large values of the local Zeeman field, thus producing topological regions which are, in a sense, "unbounded" in the large-field limit. Moreover, we show that the energy oscillations with magnetic field strength due to MBS splitting can show very different behaviors depending on the parameters. In some regimes, we find oscillations with increasing amplitudes and decreasing periods, while in the other regimes the complete opposite behavior is found. We also find that the topological phase can become dependent on the chain length, particularly in topological regions with a very high or no upper bound. In these systems, we see a very smooth evolution from MBSs localized at chain end points to in-gap Andreev bound states spread over the full chain.

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