Publications by Karsten Flensberg

  • 2022
    • Evidence for spin-polarized bound states in semiconductor–superconductor–ferromagnetic-insulator islands - Abstract
      • We report Coulomb blockade transport studies of semiconducting InAs nanowires grown with epitaxial superconducting Al and ferromagnetic insulator EuS on overlapping facets. Comparing experiment to a theoretical model, we associate cotunneling features in even-odd bias spectra with spin-polarized Andreev levels. Results are consistent with zero-field spin splitting exceeding the induced superconducting gap. Energies of subgap states are tunable on either side of zero via electrostatic gates.
    • S. Vaitiekėnas, R. Seoane Souto, Y. Liu, P. Krogstrup, K. Flensberg, M. Leijnse, C. M. Marcus
      Journal reference: Phys. Rev. B 105, L041304 (2022) [pdf]
      DOI: 10.1103/PhysRevB.105.L041304
  • 2021
    • Topological superconductivity in semiconductor–superconductor–magnetic-insulator heterostructures - Abstract
      • Hybrid superconductor-semiconductor heterostructures are promising platforms for realizing topological superconductors and exploring Majorana bound states physics. Motivated by recent experimental progress, we theoretically study how magnetic insulators offer an alternative to the use of external magnetic fields for reaching the topological regime. We consider different setups, where: (1) the magnetic insulator induces an exchange field in the superconductor, which leads to a splitting in the semiconductor by proximity effect, and (2) the magnetic insulator acts as a spin-filter tunnel barrier between the superconductor and the semiconductor. We show that the spin splitting in the superconductor alone cannot induce a topological transition in the semiconductor. To overcome this limitation, we propose to use a spin-filter barrier that enhances the magnetic exchange and provides a mechanism for a topological phase transition. Moreover, the spin-dependent tunneling introduces a strong dependence on the band alignment, which can be crucial in quantum-confined systems. This mechanism opens up a route towards networks of topological wires with fewer constraints on device geometry compared to previous devices that require external magnetic fields.
    • A. Maiani, R. Seoane Souto, M. Leijnse, K. Flensberg
      Journal reference: Phys. Rev. B 103, 104508 (2021) [pdf]
      DOI: 10.1103/PhysRevB.103.104508
  • 2020
    • Absence of supercurrent sign reversal in a topological junction with a quantum dot - Abstract
      • Experimental techniques to verify Majoranas are of current interest. A prominent test is the effect of Majoranas on the Josephson current between two wires linked via a normal junction. Here, we study the case of a quantum dot connecting the two superconductors and the sign of the supercurrent in the trivial and topological regimes under grand-canonical equilibrium conditions, explicitly allowing for parity changes due to, e.g., quasi-particle poisoning. We find that the well-known supercurrent reversal for odd occupancy of the quantum dot (pi-junction) in the trivial case does not occur in the presence of Majoranas in the wires. However, we also find this to be a mere consequence of Majoranas being zero energy states. Therefore, the lack of supercurrent sign reversal can also be caused by trivial bound states, and is thus not a discriminating signature of Majoranas.
    • J. Schulenborg, K. Flensberg
      Journal reference: Phys. Rev. B 101, 014512 (2020) [pdf]
      DOI: 10.1103/PhysRevB.101.014512
    • Timescales for charge transfer based operations on Majorana systems - Abstract
      • In this article we analyze the efficiency of operations based on transferring charge from a quantum dot (QD) to two coupled topological superconductors, which can be used for performing nonabelian operations on Majorana bound states (MBSs). We develop a method which allows us to describe the full time-evolution of the system as the QD energy is manipulated. Using a full counting statistics analysis, we set bounds to the operation time scales. The lower bound depends on the superconducting phase difference due to a partial decoupling of the different MBSs parity sectors, while the upper bound is set by the tunneling of quasiparticles to the MBSs. Using realistic parameters, we find the existence of a regime where the operation can be carried out with a fidelity close to unity. Finally, we propose the use of a two operations protocol to quantify the effect of the dephasing and accumulated dynamical phases, demonstrating their absence for certain superconducting phase differences.
    • R. Seoane Souto, K. Flensberg, M. Leijnse
      Journal reference: Phys. Rev. B 101, 081407 (2020) [pdf]
      DOI: 10.1103/PhysRevB.101.081407
    • Conductance-Matrix Symmetries of a Three-Terminal Hybrid Device - Abstract
      • We present conductance-matrix measurements of a three-terminal superconductor-semiconductor hybrid device consisting of two normal leads and one superconducting lead. Using a symmetry decomposition of the conductance, we find that the antisymmetric components of pairs of local and nonlocal conductances match at energies below the superconducting gap, consistent with expectations based on a non-interacting scattering matrix approach. Further, the local charge character of Andreev bound states is extracted from the symmetry-decomposed conductance data and is found to be similar at both ends of the device and tunable with gate voltage. Finally, we measure the conductance matrix as a function of magnetic field and identify correlated splittings in low-energy features, demonstrating how conductance-matrix measurements can complement traditional tunneling-probe measurements in the search for Majorana zero modes.
    • G. C. Ménard, G. L. R. Anselmetti, E. A. Martinez, D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm, K. Flensberg, C. M. Marcus, L. Casparis, A. P. Higginbotham
      Journal reference: Phys. Rev. Lett. 124, 036802 (2020) [pdf]
      DOI: 10.1103/PhysRevLett.124.036802
  • 2018
    • Hybridization of Subgap States in One-Dimensional Superconductor-Semiconductor Coulomb Islands - Abstract
      • We present measurements of one-dimensional superconductor-semiconductor Coulomb islands, fabricated by gate confinement of a two-dimensional InAs heterostructure with an epitaxial Al layer. When tuned via electrostatic side gates to regimes without sub-gap states, Coulomb blockade reveals Cooper-pair mediated transport. When sub-gap states are present, Coulomb peak positions and heights oscillate in a correlated way with magnetic field and gate voltage, as predicted theoretically, with (anti) crossings in (parallel) transverse magnetic field indicating Rashba-type spin-orbit coupling. Overall results are consistent with a picture of overlapping Majorana zero modes in finite wires.
    • E. C. T. O'Farrell, A. C. C. Drachmann, M. Hell, A. Fornieri, A. M. Whiticar, E. B. Hansen, S. Gronin, G. C. Gardener, C. Thomas, M. J. Manfra, K. Flensberg, C. M. Marcus, F. Nichele
      Journal reference: Phys. Rev. Lett. 121, 256803 (2018) [pdf]
      DOI: 10.1103/PhysRevLett.121.256803
    • Anharmonicity of a superconducting qubit with a few-mode Josephson junction - Abstract
      • Coherent operation of gate-voltage-controlled hybrid transmon qubits (gatemons) based on semiconductor nanowires was recently demonstrated. Here we experimentally investigate the anharmonicity in epitaxial InAs-Al Josephson junctions, a key parameter for their use as a qubit. Anharmonicity is found to be reduced by roughly a factor of two compared to conventional metallic junctions, and dependent on gate voltage. Experimental results are consistent with a theoretical model, indicating that Josephson coupling is mediated by a small number of highly transmitting modes in the semiconductor junction.
    • A. Kringhøj, L. Casparis, M. Hell, T. W. Larsen, F. Kuemmeth, M. Leijnse, K. Flensberg, P. Krogstrup, J. Nygård, K. D. Petersson, C. M. Marcus
      Journal reference: Phys. Rev. B 97, 060508 (2018) [pdf]
      DOI: 10.1103/PhysRevB.97.060508
  • 2017
    • Transport Signatures of Quasiparticle Poisoning in a Majorana Island - Abstract
      • We investigate effects of quasiparticle poisoning in a Majorana island with strong tunnel coupling to normal-metal leads. In addition to the main Coulomb blockade diamonds, "shadow" diamonds appear, shifted by 1e in gate voltage, consistent with transport through an excited (poisoned) state of the island. Comparison to a simple model yields an estimate of parity lifetime for the strongly coupled island (~ 1 μs) and sets a bound for a weakly coupled island (> 10 μs). Fluctuations in the gate-voltage spacing of Coulomb peaks at high field, reflecting Majorana hybridization, are enhanced by the reduced lever arm at strong coupling. In energy units, fluctuations are consistent with previous measurements.
    • S. M. Albrecht, E. B. Hansen, A. P. Higginbotham, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, J. Danon, K. Flensberg, C. M. Marcus
      Journal reference: Phys. Rev. Lett. 118, 137701 (2017) [pdf]
      DOI: 10.1103/PhysRevLett.118.137701
    • Anomalous Fraunhofer interference in epitaxial superconductor-semiconductor Josephson junctions - Abstract
      • We investigate patterns of critical current as a function of perpendicular and in-plane magnetic fields in superconductor-semiconductor-superconductor (SNS) junctions based on InAs/InGaAs heterostructures with an epitaxial Al layer. This material system is of interest due to its exceptionally good superconductor-semiconductor coupling, as well as large spin-orbit interaction and g-factor in the semiconductor. Thin epitaxial Al allows the application of large in-plane field without destroying superconductivity. For fields perpendicular to the junction, flux focusing results in aperiodic node spacings in the pattern of critical currents known as Fraunhofer patterns by analogy to the related interference effect in optics. Adding an in-plane field yields two further anomalies in the pattern. First, higher order nodes are systematically strengthened, indicating current flow along the edges of the device, as a result of confinement of Andreev states driven by an induced flux dipole; second, asymmetries in the interference appear that depend on the field direction and magnitude. A model is presented, showing good agreement with experiment, elucidating the roles of flux focusing, Zeeman and spin-orbit coupling, and disorder in producing these effects.
    • H. J. Suominen, J. Danon, M. Kjaergaard, K. Flensberg, J. Shabani, C. J. Palmstrøm, F. Nichele, C. M. Marcus
      Journal reference: Phys. Rev. B 95, 035307 (2017) [pdf]
      DOI: 10.1103/PhysRevB.95.035307
  • 2016
    • Majorana bound state in a coupled quantum-dot hybrid-nanowire system - Abstract
      • Hybrid nanowires combining semiconductor and superconductor materials appear well suited for the creation, detection, and control of Majorana bound states (MBSs). We demonstrate the emergence of MBSs from coalescing Andreev bound states (ABSs) in a hybrid InAs nanowire with epitaxial Al, using a quantum dot at the end of the nanowire as a spectrometer. Electrostatic gating tuned the nanowire density to a regime of one or a few ABSs. In an applied axial magnetic field, a topological phase emerges in which ABSs move to zero energy and remain there, forming MBSs. We observed hybridization of the MBS with the end-dot bound state, which is in agreement with a numerical model. The ABS/MBS spectra provide parameters that are useful for understanding topological superconductivity in this system.
    • M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, C. M. Marcus
      Journal reference: Science 354, 1557-1562 (2016) [pdf]
      DOI: 10.1126/science.aaf3961
    • Quantized conductance doubling and hard gap in a two-dimensional semiconductor–superconductor heterostructure - Abstract
      • The prospect of coupling a two-dimensional (2D) semiconductor heterostructure to a superconductor opens new research and technology opportunities, including fundamental problems in mesoscopic superconductivity, scalable superconducting electronics, and new topological states of matter. For instance, one route toward realizing topological matter is by coupling a 2D electron gas (2DEG) with strong spin-orbit interaction to an s-wave superconductor. Previous efforts along these lines have been hindered by interface disorder and unstable gating. Here, we report measurements on a gateable InGaAs/InAs 2DEG with patterned epitaxial Al, yielding multilayer devices with atomically pristine interfaces between semiconductor and superconductor. Using surface gates to form a quantum point contact (QPC), we find a hard superconducting gap in the tunneling regime, overcoming the soft-gap problem in 2D superconductor-semiconductor hybrid systems. With the QPC in the open regime, we observe a first conductance plateau at 4e^2/h, as expected theoretically for a normal-QPC-superconductor structure. The realization of a hard-gap semiconductor-superconductor system that is amenable to top-down processing provides a means of fabricating scalable multicomponent hybrid systems for applications in low-dissipation electronics and topological quantum information.
    • M. Kjaergaard, F. Nichele, H. J. Suominen, M. P. Nowak, M. Wimmer, A. R. Akhmerov, J. A. Folk, K. Flensberg, J. Shabani, C. J. Palmstrom, C. M. Marcus
      Journal reference: Nat. Commun. 7, 12841 (2016) [pdf]
      DOI: 10.1038/ncomms12841
    • Spiral magnetic order and topological superconductivity in a chain of magnetic adatoms on a two-dimensional superconductor - Abstract
      • We study the magnetic and electronic phases of a 1D magnetic adatom chain on a 2D superconductor. In particular, we confirm the existence of a `self-organized' 1D topologically non-trivial superconducting phase within the set of subgap Yu-Shiba-Rusinov (YSR) states formed along the magnetic chain. This phase is stabilized by incommensurate spiral correlations within the magnetic chain that arise from the competition between short-range ferromagnetic and long-range antiferromagnetic electron-induced exchange interactions, similar to a recent study for a 3D superconductor [M. Schecter et al. Phys. Rev. B 93, 140503(R) 2016]. The exchange interaction along diagonal directions are also considered and found to display behavior similar to a 1D substrate when close to half filling. We show that the topological phase diagram is robust against local superconducting order parameter suppression and weak substrate spin-orbit coupling. Lastly, we study the effect of a direct ferromagnetic exchange coupling between the adatoms, and find the region of spiral order in the phase diagram to be significantly enlarged in a wide range of the direct exchange coupling.
    • M. H. Christensen, M. Schecter, K. Flensberg, B. M. Andersen, J. Paaske
      Journal reference: Phys. Rev. B 94, 144509 (2016) [pdf]
      DOI: 10.1103/PhysRevB.94.144509
    • Roadmap to Majorana surface codes - Abstract
      • Surface codes offer a very promising avenue towards fault-tolerant quantum computation. We argue that two-dimensional interacting networks of Majorana bound states in topological superconductor/semiconductor heterostructures hold several distinct advantages in that direction, both concerning the hardware realization and the actual operation of the code. We here discuss how topologically protected logical qubits in this Majorana surface code architecture can be defined, initialized, manipulated, and read out. All physical ingredients needed to implement these operations are routinely used in topologically trivial quantum devices. In particular, we show that by means of quantum interference terms in linear conductance measurements, composite single-electron pumping protocols, and gate-tunable tunnel barriers, the full set of quantum gates required for universal quantum computation can be implemented.
    • S. Plugge, L. A. Landau, E. Sela, A. Altland, K. Flensberg, R. Egger
      Journal reference: Phys. Rev. B 94, 174514 (2016) [pdf]
      DOI: 10.1103/PhysRevB.94.174514
    • Self-organized topological superconductivity in a Yu-Shiba-Rusinov chain - Abstract
      • We study a chain of magnetic moments exchange coupled to a conventional three dimensional superconductor. In the normal state the chain orders into a collinear configuration, while in the superconducting phase we find that ferromagnetism is unstable to the formation of a magnetic spiral state. Beyond weak exchange coupling the spiral wavevector greatly exceeds the inverse superconducting coherence length as a result of the strong spin-spin interaction mediated through the subgap band of Yu-Shiba-Rusinov states. Moreover, the simple spin-spin exchange description breaks down as the subgap band crosses the Fermi energy, wherein the spiral phase becomes stabilized by the spontaneous opening of a $p-$wave superconducting gap within the band. This leads to the possibility of electron-driven topological superconductivity with Majorana boundary modes using magnetic atoms on superconducting surfaces.
    • M. Schecter, K. Flensberg, M. H. Christensen, B. M. Andersen, J. Paaske
      Journal reference: Phys. Rev. B 93, 140503 (2016) [pdf]
      DOI: 10.1103/PhysRevB.93.140503
  • 2015
    • Quantum transport in carbon nanotubes - Abstract
      • Carbon nanotubes are a versatile material in which many aspects of condensed matter physics come together. Recent discoveries, enabled by sophisticated fabrication, have uncovered new phenomena that completely change our understanding of transport in these devices, especially the role of the spin and valley degrees of freedom. This review describes the modern understanding of transport through nanotube devices. Unlike conventional semiconductors, electrons in nanotubes have two angular momentum quantum numbers, arising from spin and from valley freedom. We focus on the interplay between the two. In single quantum dots defined in short lengths of nanotube, the energy levels associated with each degree of freedom, and the spin-orbit coupling between them, are revealed by Coulomb blockade spectroscopy. In double quantum dots, the combination of quantum numbers modifies the selection rules of Pauli blockade. This can be exploited to read out spin and valley qubits, and to measure the decay of these states through coupling to nuclear spins and phonons. A second unique property of carbon nanotubes is that the combination of valley freedom and electron-electron interactions in one dimension strongly modifies their transport behaviour. Interaction between electrons inside and outside a quantum dot is manifested in SU(4) Kondo behavior and level renormalization. Interaction within a dot leads to Wigner molecules and more complex correlated states. This review takes an experimental perspective informed by recent advances in theory. As well as the well-understood overall picture, we also state clearly open questions for the field. These advances position nanotubes as a leading system for the study of spin and valley physics in one dimension where electronic disorder and hyperfine interaction can both be reduced to a very low level.
    • E. A. Laird, F. Kuemmeth, G. Steele, K. Grove-Rasmussen, J. Nygård, K. Flensberg, L. P. Kouwenhoven
      Journal reference: Rev. Mod. Phys. 87, 703 (2015) [pdf]
      DOI: 10.1103/RevModPhys.87.703
    • Parity lifetime of bound states in a proximitized semiconductor nanowire - Abstract
      • Quasiparticle excitations can compromise the performance of superconducting devices, causing high frequency dissipation, decoherence in Josephson qubits, and braiding errors in proposed Majorana-based topological quantum computers. Quasiparticle dynamics have been studied in detail in metallic superconductors but remain relatively unexplored in semiconductor-superconductor structures, which are now being intensely pursued in the context of topological superconductivity. To this end, we introduce a new physical system comprised of a gate-confined semiconductor nanowire with an epitaxially grown superconductor layer, yielding an isolated, proximitized nanowire segment. We identify Andreev-like bound states in the semiconductor via bias spectroscopy, determine the characteristic temperatures and magnetic fields for quasiparticle excitations, and extract a parity lifetime (poisoning time) of the bound state in the semiconductor exceeding 10 ms.
    • A. P. Higginbotham, S. M. Albrecht, G. Kirsanskas, W. Chang, F. Kuemmeth, P. Krogstrup, T. S. Jespersen, J. Nygard, K. Flensberg, C. M. Marcus
      Journal reference: Nature Physics 11, 1017 (2015) [pdf]
      DOI: 10.1038/nphys3461