Publications by Karsten Flensberg

  • 2020
    • Parity-to-charge conversion in Majorana qubit readout - Abstract
      • We study the time-dependent effect of Markovian readout processes on Majorana qubits whose parity degrees of freedom are converted into the charge of a tunnel-coupled quantum dot. By applying a recently established effective Lindbladian approximation [1-3], we obtain a completely positive and trace preserving Lindblad master equation for the combined dot-qubit dynamics, describing relaxation and decoherence processes beyond the rotating-wave approximation. This approach is applicable to a wide range of weakly coupled environments representing experimentally relevant readout devices. We study in detail the case of thermal decay in the presence of a generic Ohmic bosonic bath, in particular for potential fluctuations in an electromagnetic circuit. In addition, we consider the nonequilibrium measurement environment for a parity readout using a quantum point contact capacitively coupled to the dot charge.
    • Morten I. K. Munk, Jens Schulenborg, Reinhold Egger, Karsten Flensberg
      2004.02123v2 [pdf]

    • Flux-induced topological superconductivity in full-shell nanowires - Abstract
      • We present a novel route to realizing topological superconductivity using magnetic flux applied to a full superconducting shell surrounding a semiconducting nanowire core. In the destructive Little-Parks regime, reentrant regions of superconductivity are associated with integer number of phase windings in the shell. Tunneling into the core reveals a hard induced gap near zero applied flux, corresponding to zero phase winding, and a gapped region with a discrete zero-energy state around one applied flux quantum, {\Phi}_0 = h/2e, corresponding to 2{\pi} phase winding. Theoretical analysis indicates that in the presence of radial spin-orbit coupling in the semiconductor, the winding of the superconducting phase can induce a transition to a topological phase supporting Majorana zero modes. Realistic modeling shows a topological phase persisting over a wide range of parameters, and reproduces experimental tunneling conductance data. Further measurements of Coulomb blockade peak spacing around one flux quantum in full-shell nanowire islands shows exponentially decreasing deviation from 1e periodicity with device length, consistent with Majorana modes at the ends of the nanowire.
    • S. Vaitiekėnas, G. W. Winkler, B. van Heck, T. Karzig, M. -T. Deng, K. Flensberg, L. I. Glazman, C. Nayak, P. Krogstrup, R. M. Lutchyn, C. M. Marcus
      Journal reference: Science 367, eaav3392 (2020) [ 2003.13177v1 ]
      DOI: 10.1126/science.aav3392

    • Long distance coherence of Majorana wires - Abstract
      • Theoretically, a pair of Majorana bound states in a topological superconductor forms a single fermionic level even at large separations, implying that the parity information is stored nonlocally. The nonlocality leads to a long-distance coherence for electrons tunneling through a Coulomb blockaded Majorana wire [Fu, Phys. Rev. Lett. 104, 056402 (2010)], an effect that can be observed, e.g., in an interferometer. Here, we examine theoretically the coherent electron transfer, taking into account that tunneling implies the long-distance transfer of charge, which is carried by one-dimensional plasmons. We show that the charge dynamics does not affect the coherence of the electron tunneling process in a topological superconductor consisting of a semiconductor wire proximitized by a single bulk superconductor. The coherence may be strongly suppressed, however, if the topological superconductivity derives from a semiconductor wire proximitized by a granular superconductor.
    • Zheng Shi, Piet W. Brouwer, Karsten Flensberg, Leonid I. Glazman, Felix von Oppen
      2002.01539v1 [pdf]

    • 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) [ 1910.04106v3 ]
      DOI: 10.1103/PhysRevB.101.014512

    • Weak Measurement Protocols for Majorana Bound State Identification - Abstract
      • We propose a continuous weak measurement protocol testing the nonlocality of Majorana bound states through current shot noise correlations. The experimental setup contains a topological superconductor island with three normal-conducting leads weakly coupled to different Majorana states. Putting one lead at finite voltage and measuring the shot noise correlations between the other two (grounded) leads, devices with true Majorana states are distinguished from those without by strong current correlations. The presence of true Majoranas manifests itself in unusually high noise levels or the near absence of noise, depending on the chosen device configuration. Monitoring the noise statistics amounts to a weak continuous measurement of the Majorana qubit and yields information similar to that of a full braiding protocol, but at much lower experimental effort. Our theory can be adapted to different platforms and should allow for clear identification of Majorana states.
    • Jan Manousakis, Carolin Wille, Alexander Altland, Reinhold Egger, Karsten Flensberg, Fabian Hassler
      Journal reference: Phys. Rev. Lett. 124, 096801 (2020) [ 1910.10591v2 ]
      DOI: 10.1103/PhysRevLett.124.096801

  • 2019
    • Current-Induced Gap Opening in Interacting Topological Insulator Surfaces - Abstract
      • Two-dimensional topological insulators (TIs) host gapless helical edge states that are predicted to support a quantized two-terminal conductance. Quantization is protected by time-reversal symmetry, which forbids elastic backscattering. Paradoxically, the current-carrying state itself breaks the time-reversal symmetry that protects it. Here we show that the combination of electron-electron interactions and momentum-dependent spin polarization in helical edge states gives rise to feedback through which an applied current opens a gap in the edge state dispersion, thereby breaking the protection against elastic backscattering. Current-induced gap opening is manifested via a nonlinear contribution to the system's $I-V$ characteristic, which persists down to zero temperature. We discuss prospects for realizations in recently discovered large bulk band gap TIs, and an analogous current-induced gap opening mechanism for the surface states of three-dimensional TIs.
    • Ajit C. Balram, Karsten Flensberg, Jens Paaske, Mark S. Rudner
      Journal reference: Phys. Rev. Lett. 123, 246803 (2019) [ 1901.08067v2 ]
      DOI: 10.1103/PhysRevLett.123.246803

    • 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) [ 1910.08420v2 ]
      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) [ 1905.05505v1 ]
      DOI: 10.1103/PhysRevLett.124.036802

    • Nonlocal Conductance Spectroscopy of Andreev Bound States: Symmetry Relations and BCS Charges - Abstract
      • Two-terminal conductance spectroscopy of superconducting devices is a common tool for probing Andreev and Majorana bound states. Here, we study theoretically a three-terminal setup, with two normal leads coupled to a grounded superconducting terminal. Using a single-electron scattering matrix, we derive the subgap conductance matrix for the normal leads and discuss its symmetries. In particular, we show that the local and the nonlocal elements of the conductance matrix have pairwise identical antisymmetric components. Moreover, we find that the nonlocal elements are directly related to the local BCS charges of the bound states close to the normal probes and we show how the BCS charge of overlapping Majorana bound states can be extracted from experiments.
    • Jeroen Danon, Anna Birk Hellenes, Esben Bork Hansen, Lucas Casparis, Andrew P. Higginbotham, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 124, 036801 (2020) [ 1905.05438v1 ]
      DOI: 10.1103/PhysRevLett.124.036801

    • Fidelity and visibility loss in Majorana qubits by entanglement with environmental modes - Abstract
      • We study the dynamics and readout of topological qubits encoded by zero-energy Majorana bound states in a topological superconductor. We take into account bosonic modes due to the electromagnetic environment which couple the Majorana manifold to above-gap continuum quasi-particles. This coupling causes the degenerate ground state of the topological superconductor to be dressed in a polaron-like manner by quasi-particle states and bosons, and the system to become gapless. Topological protection and hence full coherence is only maintained if the qubit is operated and read out within the low-energy spectrum of the dressed states. We discuss reduction of fidelity and/or visibility if this condition is violated by a quantum-dot readout that couples to the bare (undressed) Majorana modes. For a projective measurement of the bare Majorana basis, we formulate a Bloch-Redfield approach that is valid for weak Majorana-environment coupling and takes into account constraints imposed by fermion-number-parity conservation. Within the Markovian approximation, our results essentially confirm earlier theories of finite-temperature decoherence based on Fermi's golden rule. However, the full non-Markovian dynamics reveals, in addition, the fidelity reduction by a projective measurement. Using a spinless nanowire model with $p$-wave pairing, we provide quantitative results characterizing these effects.
    • Morten I. K. Munk, Reinhold Egger, Karsten Flensberg
      Journal reference: Phys. Rev. B 99, 155419 (2019) [ 1812.00254v3 ]
      DOI: 10.1103/PhysRevB.99.155419

  • 2018
    • Coulomb-interaction-induced Majorana edge modes in nanowires - Abstract
      • We show that Majorana edge modes appear in a strongly correlated phase of semiconducting nanowires with discrete rotational symmetry in the cross section. These modes exist in the absence of spin-orbit coupling, magnetic fields and superconductivity. They appear purely due to the combination of the three-dimensional Coulomb interaction and orbital physics, which generates a fermionic condensate exhibiting a topological ground state degeneracy in a sector of the spectrum which is gapped to continuum modes. The gap can be comparable in magnitude to the topological superconducting gap in other solid-state candidate systems for Majorana edge modes, and may similarly be probed via tunnel spectroscopy.
    • Tommy Li, Michele Burrello, Karsten Flensberg
      Journal reference: Phys. Rev. B 100, 045305 (2019) [ 1809.09564v1 ]
      DOI: 10.1103/PhysRevB.100.045305

    • Flux-induced topological superconductivity in full-shell nanowires - Abstract
      • We consider a new model system supporting Majorana zero modes based on semiconductor nanowires with a full superconducting shell. We demonstrate that, in the presence of spin-orbit coupling in the semiconductor induced by a radial electric field, the winding of the superconducting order parameter leads to a topological phase supporting Majorana zero modes. The topological phase persists over a large range of chemical potentials and can be induced by a predictable and weak magnetic field piercing the cylinder. The system can be readily realized in semiconductor nanowires covered by a full superconducting shell, opening a pathway for realizing topological quantum computing proposals.
    • Roman M. Lutchyn, Georg W. Winkler, Bernard van Heck, Torsten Karzig, Karsten Flensberg, Leonid I. Glazman, Chetan Nayak
      Journal reference: Science 367, eaav3392 (2020) [ 1809.05512v1 ]
      DOI: 10.1126/science.aav3392

    • Coulomb Blockade of a Nearly Open Majorana Island - Abstract
      • We consider the ground-state energy and the spectrum of the low-energy excitations of a Majorana island formed of topological superconductors connected by a single-mode junction of arbitrary transmission. Coulomb blockade results in $e$-periodic modulation of the energies with the gate-induced charge. We find the amplitude of modulation as a function of reflection coefficient ${\cal R}$. The amplitude scales as $\sqrt{\cal R}$ in the limit ${\cal R}\to 0$. At larger ${\cal R}$, the dependence of the amplitude on the Josephson and charging energies is similar to that of a conventional-superconductor Cooper-pair box. The crossover value of ${\cal R}$ is small and depends on the ratio of the charging energy to superconducting gap.
    • Dmitry I. Pikulin, Karsten Flensberg, Leonid I. Glazman, Manuel Houzet, Roman M. Lutchyn
      Journal reference: Phys. Rev. Lett. 122, 016801 (2019) [ 1809.01660v1 ]
      DOI: 10.1103/PhysRevLett.122.016801

    • Hybridization at Superconductor-Semiconductor Interfaces - Abstract
      • Hybrid superconductor-semiconductor devices are currently one of the most promising platforms for realizing Majorana zero modes. Their topological properties are controlled by the band alignment of the two materials, as well as the electrostatic environment, which are currently not well understood. Here, we pursue to fill in this gap and address the role of band bending and superconductor-semiconductor hybridization in such devices by analyzing a gated single Al-InAs interface using a self-consistent Schrodinger-Poisson approach. Our numerical analysis shows that the band bending leads to an interface quantum well, which localizes the charge in the system near the superconductor-semiconductor interface. We investigate the hybrid band structure and analyze its response to varying the gate voltage and thickness of the Al layer. This is done by studying the hybridization degrees of the individual subbands, which determine the induced pairing and effective $g$-factors. The numerical results are backed by approximate analytical expressions which further clarify key aspects of the band structure. We find that one can obtain states with strong superconductor-semiconductor hybridization at the Fermi energy, but this requires a fine balance of parameters, with the most important constraint being on the width of the Al layer. In fact, in the regime of interest, we find an almost periodic dependence of the hybridization degree on the Al width, with a period roughly equal to the thickness of an Al monolayer. This implies that disorder and shape irregularities, present in realistic devices, may play an important role for averaging out this sensitivity and, thus, may be necessary for stabilizing the topological phase.
    • August E. G. Mikkelsen, Panagiotis Kotetes, Peter Krogstrup, Karsten Flensberg
      Journal reference: Phys. Rev. X 8, 031040 (2018) [ 1801.03439v3 ]
      DOI: 10.1103/PhysRevX.8.031040

    • Four-Majorana qubit with charge readout: Dynamics and decoherence - Abstract
      • We present a theoretical analysis of a Majorana-based qubit consisting of two topological superconducting islands connected via a Josephson junction. The qubit is operated by electrostatic gates which control the coupling of two of the four Majorana zero modes. At the end of the operation, readout is performed in the charge basis. Even though the operations are not topologically protected, the proposed experiment can potentially shed light on the coherence of the parity degree of freedom in Majorana devices and serve as a first step towards topological Majorana qubits. We discuss in detail the charge-stability diagram and its use for characterizing the parameters of the devices, including the overlap of the Majorana edge states. We describe the multi-level spectral properties of the system and present a detailed study of its controlled coherent oscillations, as well as decoherence resulting from coupling to a non-Markovian environment. In particular, we study a gate-controlled protocol where conversion between Coulomb-blockade and transmon regimes generates coherent oscillations of the qubit state due to the overlap of Majorana modes. We show that, in addition to fluctuations of the Majorana coupling, considerable measurement errors may be accumulated during the conversion intervals when electrostatic fluctuations in the superconducting islands are present. These results are also relevant for several proposed implementations of topological qubits which rely on readout based on charge detection.
    • Tommy Li, William A. Coish, Michael Hell, Karsten Flensberg, Martin Leijnse
      Journal reference: Phys. Rev. B 98, 205403 (2018) [ 1807.05839v1 ]
      DOI: 10.1103/PhysRevB.98.205403

    • 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) [ 1804.09676v1 ]
      DOI: 10.1103/PhysRevLett.121.256803

    • Distinguishing Majorana bound states from localized Andreev bound states by interferometry - Abstract
      • Experimental evidence for Majorana bound states (MBSs) is so far mainly based on the robustness of a zero-bias conductance peak. However, similar features can also arise due to Andreev bound states (ABSs) localized at the end of an island. We show that these two scenarios can be distinguished by an interferometry experiment based on embedding a Coulomb-blockaded island into an Aharonov-Bohm ring. For two ABSs, when the ground state is nearly degenerate, cotunneling can change the state of the island and interference is suppressed. By contrast, for two MBSs the ground state is nondegenerate and cotunneling has to preserve the island state, which leads to $h / e$-periodic conductance oscillations with magnetic flux. Such interference setups can be realized with semiconducting nanowires or two-dimensional electron gases with proximity-induced superconductivity and may also be a useful spectroscopic tool for parity-flip mechanisms.
    • Michael Hell, Karsten Flensberg, Martin Leijnse
      Journal reference: Phys. Rev. B 97, 161401 (2018) [ 1710.05294v2 ]
      DOI: 10.1103/PhysRevB.97.161401

    • Probing electron-hole components of subgap states in Coulomb blockaded Majorana islands - Abstract
      • Recent tunneling spectroscopy experiments in semiconducting nanowires with proximity-induced superconductivity have reported robust zero-bias conductance peaks. Such a feature can be compatible with the existence of topological Majorana bound states (MBSs) and with a trivial Andreev bound state (ABS) near zero energy. Here, we argue that additional information, that can distinguish between the two cases, can be extracted from Coulomb-blockade experiments of Majorana islands. The key is the ratio of peak heights of consecutive conductance peaks give information about the electron and hole components of the lowest-energy subgap state. In the MBS case, this ratio goes to one half for long wires, while for short wires with finite MBS overlap it oscillates a function of Zeeman energy with the same period as the MBS energy splitting. We explain how the additional information might help to distinguish a trivial ABS at zero energy from a true MBS and show case examples.
    • Esben Bork Hansen, Jeroen Danon, Karsten Flensberg
      Journal reference: Phys. Rev. B 97, 041411 (2018) [ 1712.07972v2 ]
      DOI: 10.1103/PhysRevB.97.041411

  • 2017
    • Scaling of Majorana Zero-Bias Conductance Peaks - Abstract
      • We report an experimental study of the scaling of zero-bias conductance peaks compatible with Majorana zero modes as a function of magnetic field, tunnel coupling, and temperature in one-dimensional structures fabricated from an epitaxial semiconductor-superconductor heterostructure. Results are consistent with theory, including a peak conductance that is proportional to tunnel coupling, saturates at $2e^2/h$, decreases as expected with field-dependent gap, and collapses onto a simple scaling function in the dimensionless ratio of temperature and tunnel coupling.
    • Fabrizio Nichele, Asbjorn C. C. Drachmann, Alexander M. Whiticar, Eoin C. T. O'Farrell, Henri J. Suominen, Antonio Fornieri, Tian Wang, Geoffrey C. Gardner, Candice Thomas, Anthony T. Hatke, Peter Krogstrup, Michael J. Manfra, Karsten Flensberg, Charles M. Marcus
      Journal reference: Phys. Rev. Lett. 119, 136803 (2017) [ 1706.07033v2 ]
      DOI: 10.1103/PhysRevLett.119.136803

    • Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes - Abstract
      • We present designs for scalable quantum computers composed of qubits encoded in aggregates of four or more Majorana zero modes, realized at the ends of topological superconducting wire segments that are assembled into superconducting islands with significant charging energy. Quantum information can be manipulated according to a measurement-only protocol, which is facilitated by tunable couplings between Majorana zero modes and nearby semiconductor quantum dots. Our proposed architecture designs have the following principal virtues: (1) the magnetic field can be aligned in the direction of all of the topological superconducting wires since they are all parallel; (2) topological $T$-junctions are not used, obviating possible difficulties in their fabrication and utilization; (3) quasiparticle poisoning is abated by the charging energy; (4) Clifford operations are executed by a relatively standard measurement: detection of corrections to quantum dot energy, charge, or differential capacitance induced by quantum fluctuations; (5) it is compatible with strategies for producing good approximate magic states.
    • Torsten Karzig, Christina Knapp, Roman M. Lutchyn, Parsa Bonderson, Matthew B. Hastings, Chetan Nayak, Jason Alicea, Karsten Flensberg, Stephan Plugge, Yuval Oreg, Charles M. Marcus, Michael H. Freedman
      Journal reference: Phys. Rev. B 95, 235305 (2017) [ 1610.05289v4 ]
      DOI: 10.1103/PhysRevB.95.235305

    • Conductance spectroscopy on Majorana wires and the inverse proximity effect - Abstract
      • Recent experimental searches for signatures of Majorana-like excitations in proximitized semiconducting nanowires involve conductance spectroscopy, where the evidence sought after is a robust zero-bias peak (in longer wires) and its characteristic field-dependent splitting (in shorter wires). Although experimental results partially confirm the theoretical predictions, commonly observed discrepancies still include (i) a zero-bias peak that is significantly lower than the predicted value of $2e^2/h$ and (ii) the absence of the expected "Majorana oscillations" of the lowest-energy modes at higher magnetic fields. Here, we investigate how the inevitable presence of a normal drain lead connected to the hybrid wire can affect the conductance spectrum of the hybrid wire. We present numerical results using a one-band model for the proximitized nanowire, where the superconductor is considered to be in the diffusive regime, described by semi-classical Green functions. We show how the presence of the normal drain could (at least partially) account for the observed discrepancies, and we complement this with analytic results providing more insights in the underlying physics.
    • Jeroen Danon, Esben Bork Hansen, Karsten Flensberg
      Journal reference: Phys. Rev. B 96, 125420 (2017) [ 1706.03587v1 ]
      DOI: 10.1103/PhysRevB.96.125420

    • Coupling and braiding Majorana bound states in networks defined in two-dimensional electron gases with proximity-induced superconductivity - Abstract
      • Two-dimensional electron gases with strong spin-orbit coupling covered by a superconducting layer offer a flexible and potentially scalable platform for Majorana networks. We predict Majorana bound states (MBSs) to appear for experimentally achievable parameters and realistic gate potentials in two designs: either underneath a narrow stripe of a superconducting layer (S-stripes) or where a narrow stripe has been removed from a uniform layer (N-stripes). The coupling of the MBSs can be tuned for both types in a wide range (<1 nev to>10 $\mu$eV) using gates placed adjacent to the stripes. For both types, we numerically compute the local density of states for two parallel Majorana-stripe ends as well as Majorana trijunctions formed in a tuning-fork geometry. The MBS coupling between parallel Majorana stripes can be suppressed below 1 neV for potential barriers in the meV range for separations of about 200 nm. We further show that the MBS couplings in a trijunction can be gate-controlled in a range similar to the intra-stripe coupling while maintaining a sizable gap to the excited states (tens of $\mu$eV). Altogether, this suggests that braiding can carried out on a time scale of 10-100 ns.
    • Michael Hell, Karsten Flensberg, Martin Leijnse
      Journal reference: Phys. Rev. B 96, 035444 (2017) [ 1704.06427v1 ]
      DOI: 10.1103/PhysRevB.96.035444

    • 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) [ 1703.05643v1 ]
      DOI: 10.1103/PhysRevB.97.060508

    • 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) [ 1612.07989v2 ]
      DOI: 10.1126/science.aaf3961

  • 2016
    • Majorana box qubits - Abstract
      • Quantum information protected by the topology of the storage medium is expected to exhibit long coherence times. Another feature are topologically protected gates generated through braiding of Majorana bound states. However, braiding requires structures with branched topological segments which have inherent difficulties in the semiconductor-superconductor heterostructures now believed to host Majorana bound states. In this paper, we construct quantum bits taking advantage of the topological protection and non-local properties of Majorana bound states in a network of parallel wires, but without relying on braiding for quantum gates. The elementary unit is made from three topological wires, two wires coupled by a trivial superconductor and the third acting as an interference arm. Coulomb blockade of the combined wires spawns a fractionalized spin, non-locally addressable by quantum dots used for single-qubit readout, initialization, and manipulation. We describe how the same tools allow for measurement-based implementation of the Clifford gates, in total making the architecture universal. Proof-of-principle demonstration of topologically protected qubits using existing techniques is therefore within reach.
    • Stephan Plugge, Asbjørn Rasmussen, Reinhold Egger, Karsten Flensberg
      Journal reference: New J. Phys 19, 012001 (2017) [ 1609.01697v2 ]
      DOI: 10.1088/1367-2630/aa54e1

    • 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 {\mu}s) and sets a bound for a weakly coupled island (> 10 {\mu}s). 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) [ 1612.05748v1 ]
      DOI: 10.1103/PhysRevLett.118.137701

    • Two-Dimensional Platform for Networks of Majorana Bound States - Abstract
      • We model theoretically a two-dimensional electron gas (2DEG) covered by a superconductor and demonstrate that topological superconducting channels are formed when stripes of the superconducting layer are removed. As a consequence, Majorana bound states (MBS) are created at the ends of the stripes. We calculate the topological invariant and energy gap of a single stripe, using realistic values for an InAs 2DEG proximitized by an epitaxial Al layer. We show that the topological gap is enhanced when the structure is made asymmetric. This can be achieved by either imposing a phase difference (by driving a supercurrent or using a magnetic-flux loop) over the strip or by replacing one superconductor by a metallic gate. Both strategies also enable control over the MBS splitting, thereby facilitating braiding and readout schemes based on controlled fusion of MBS. Finally, we outline how a network of Majorana stripes can be designed.
    • Michael Hell, Martin Leijnse, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 118, 107701 (2017) [ 1608.08769v2 ]
      DOI: 10.1103/PhysRevLett.118.107701

    • 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) [ 1603.01852v2 ]
      DOI: 10.1038/ncomms12841

    • 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) [ 1611.00190v1 ]
      DOI: 10.1103/PhysRevB.95.035307

    • 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) [ 1607.08190v2 ]
      DOI: 10.1103/PhysRevB.94.144509

    • No-go theorem for a time-reversal invariant topological phase in noninteracting systems coupled to conventional superconductors - Abstract
      • We prove that a system of non-interacting electrons proximity coupled to a conventional s-wave superconductor cannot realize a time reversal invariant topological phase. This is done by showing that for such a system, in either one or two dimensions, the topological invariant of the corresponding symmetry class (DIII) is always trivial. Our results suggest that the pursuit of Majorana bound states in time-reversal invariant systems should be aimed at interacting systems or at proximity to unconventional superconductors.
    • Arbel Haim, Erez Berg, Karsten Flensberg, Yuval Oreg
      Journal reference: Phys. Rev. B 94, 161110 (2016) [ 1605.07179v2 ]
      DOI: 10.1103/PhysRevB.94.161110

    • Interaction-driven topological superconductivity in one dimension - Abstract
      • We study one-dimensional topological superconductivity in the presence of time-reversal symmetry. This phase is characterized by having a bulk gap, while supporting a Kramers' pair of zero-energy Majorana bound states at each of its ends. We present a general simple model which is driven into this topological phase in the presence of repulsive electron-electron interactions. We further propose two experimental setups and show that they realize this model at low energies. The first setup is a narrow two-dimensional topological insulator partially covered by a conventional s-wave superconductor, and the second is a semiconductor wire in proximity to an s-wave superconductor. These systems can therefore be used to realize and probe the time-reversal invariant topological superconducting phase. The effect of interactions is studied using both a mean-field approach and a renormalization group analysis.
    • Arbel Haim, Konrad Wölms, Erez Berg, Yuval Oreg, Karsten Flensberg
      Journal reference: Phys. Rev. B 94, 115124 (2016) [ 1605.09385v2 ]
      DOI: 10.1103/PhysRevB.94.115124

    • Signatures of Majorana Kramers pairs in superconductor-Luttinger liquid and superconductor-quantum dot-normal lead junctions - Abstract
      • Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of $4e^2/h$ when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions in the Luttinger liquid. We first calculate the phase diagram for the Luttinger liquid-Majorana Kramers pair junction and show that its low-energy properties are determined by Andreev reflection scattering processes in the spin-triplet channel, i.e. the corresponding Andreev boundary conditions are similar to that in a spin-triplet superconductor - normal lead junction. We also study here a quantum dot coupled to a normal lead and a Majorana Kramers pair and investigate the effect of local repulsive interactions leading to an interplay between Kondo and Majorana correlations. Using a combination of renormalization group analysis and slave-boson mean-field theory, we show that the system flows to a new fixed point which is controlled by the Majorana interaction rather than the Kondo coupling. This Majorana fixed point is characterized by correlations between the localized spin and the fermion parity of each spin sector of the topological superconductor. We investigate the stability of the Majorana phase with respect to Gaussian fluctuations.
    • Younghyun Kim, Dong E. Liu, Erikas Gaidamauskas, Jens Paaske, Karsten Flensberg, Roman M. Lutchyn
      Journal reference: Phys. Rev. B 94, 075439 (2016) [ 1605.02073v2 ]
      DOI: 10.1103/PhysRevB.94.075439

    • 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) [ 1606.08408v1 ]
      DOI: 10.1103/PhysRevB.94.174514

    • Quantum charge fluctuations of a proximitized nanowire - Abstract
      • Motivated by recent experiment, we consider charging of a nanowire which is proximitized by a superconductor and connected to a normal-state lead by a single-channel junction. The charge $Q$ of the nanowire is controlled by gate voltage $e{\cal N}_g/C$. A finite conductance of the contact allows for quantum charge fluctuations, making the function $Q(\mathcal{N}_g)$ continuous. It depends on the relation between the superconducting gap $\Delta$ and the effective charging energy $E^*_C$. The latter is determined by the junction conductance, in addition to the geometrical capacitance of the proximitized nanowire. We investigate $Q(\mathcal{N}_g)$ at zero magnetic field $B$, and at fields exceeding the critical value $B_c$ corresponding to the topological phase transition. Unlike the case of $\Delta = 0$, the function $Q(\mathcal{N}_g)$ is analytic even in the limit of negligible level spacing in the nanowire. At $B=0$ and $\Delta>E^*_C$, the maxima of $dQ/d\mathcal{N}_g$ are smeared by $2e$-fluctuations described by a single-channel "charge Kondo" physics, while the $B=0$, $\Delta<e^*_c$ case="case" is="is" described="described" by="by" a="a" crossover="crossover" between="between" the="the" kondo="Kondo" and="and" mixed-valence="mixed-valence" regimes="regimes" of="of" the="the" anderson="Anderson" impurity="impurity" model.="model." in="In" the="the" topological="topological" phase,="phase," $q(\mathcal{n}_g)$="$Q(\mathcal{N}_g)$" is="is" analytic="analytic" function="function" of="of" the="the" gate="gate" voltage="voltage" with="with" $e$-periodic="$e$-periodic" steps.="steps." in="In" the="the" weak="weak" tunneling="tunneling" limit,="limit," $dq/d\mathcal{n}_g$="$dQ/d\mathcal{N}_g$" has="has" peaks="peaks" corresponding="corresponding" to="to" breit-wigner="Breit-Wigner" resonances,="resonances," whereas="whereas" in="In" the="the" strong="strong" tunneling="tunneling" limit="limit" (i.e.,="(i.e.," small="small" reflection="reflection" amplitude="amplitude" $r$="$r$" )=")" these="these" resonances="resonances" are="are" broadened,="broadened," and="and" $dq/d\mathcal{n}_g-e="$dQ/d\mathcal{N}_g-e" \propto="\propto" r\cos(2\pi="r\cos(2\pi" \mathcal{n}_g)$.
    • Roman M. Lutchyn, Karsten Flensberg, Leonid I. Glazman
      Journal reference: Phys. Rev. B 94, 125407 (2016) [ 1606.06756v1 ]
      DOI: 10.1103/PhysRevB.94.125407

    • Time scales for Majorana manipulation using Coulomb blockade in gate-controlled superconducting nanowires - Abstract
      • We numerically compute the low-energy spectrum of a gate-controlled superconducting topological nanowire segmented into two islands, each Josephson-coupled to a bulk superconductor. This device may host two pairs of Majorana bound states and could provide a platform for testing Majorana fusion rules. We analyze the crossover between (i) a charge-dominated regime utilizable for initialization and readout of Majorana bound states, (ii) a single-island regime for dominating inter-island Majorana coupling, (iii) a Josephson-plasmon regime for large coupling to the bulk superconductors, and (iv) a regime of four Majorana bound states allowing for topologically protected Majorana manipulations. From the energy spectrum, we derive conservative estimates for the time scales of a fusion-rule testing protocol proposed recently [arXiv:1511.05153]. We also analyze the steps needed for basic Majorana braiding operations in branched nanowire structures.
    • Michael Hell, Jeroen Danon, Karsten Flensberg, Martin Leijnse
      Journal reference: Phys. Rev. B 94, 035424 (2016) [ 1601.07369v2 ]
      DOI: 10.1103/PhysRevB.94.035424

    • 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

    • Yu-Shiba-Rusinov states in phase-biased superconductor–quantum dot–superconductor junctions - Abstract
      • We study the effects of a phase difference on Yu-Shiba-Rusinov (YSR) states in a spinful Coulomb-blockaded quantum dot contacted by a superconducting loop. In the limit where charging energy is larger than the superconducting gap, we determine the subgap excitation spectrum, the corresponding supercurrent, and the differential conductance as measured by a normal-metal tunnel probe. In absence of a phase difference only one linear combination of the superconductor lead electrons couples to the spin, which gives a single YSR state. With finite phase difference, however, it is effectively a two-channel scattering problem and therefore an additional state emerges from the gap edge. The energy of the phase-dependent YSR states depend on the gate voltage and one state can cross zero energy twice inside the valley with odd occupancy. These crossings are shifted by the phase difference towards the charge degeneracy points, corresponding to larger exchange couplings. Moreover, the zero-energy crossings give rise to resonant peaks in the differential conductance with magnitude $4e^2/h$. Finally, we demonstrate that the quantum fluctuations of the dot spin do not alter qualitatively any of the results.
    • Gediminas Kiršanskas, Moshe Goldstein, Karsten Flensberg, Leonid I. Glazman, Jens Paaske
      Journal reference: Phys. Rev. B 92, 235422 (2015) [pdf]
      DOI: 10.1103/PhysRevB.92.235422

    • Milestones Toward Majorana-Based Quantum Computing - Abstract
      • We introduce a scheme for preparation, manipulation, and readout of Majorana zero modes in semiconducting wires with mesoscopic superconducting islands. Our approach synthesizes recent advances in materials growth with tools commonly used in quantum-dot experiments, including gate-control of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zero-mode detection and quantum computing that includes (1) detection of fusion rules for non-Abelian anyons using either proximal charge sensors or pumped current; (2) validation of a prototype topological qubit; and (3) demonstration of non-Abelian statistics by braiding in a branched geometry. The first two milestones require only a single wire with two islands, and additionally enable sensitive measurements of the system's excitation gap, quasiparticle poisoning rates, residual Majorana zero-mode splittings, and topological-qubit coherence times. These pre-braiding experiments can be adapted to other manipulation and readout schemes as well.
    • David Aasen, Michael Hell, Ryan V. Mishmash, Andrew Higginbotham, Jeroen Danon, Martin Leijnse, Thomas S. Jespersen, Joshua A. Folk, Charles M. Marcus, Karsten Flensberg, Jason Alicea
      Journal reference: Phys. Rev. X 6, 031016 (2016) [ 1511.05153v2 ]
      DOI: 10.1103/PhysRevX.6.031016

    • Phase-tunable Majorana bound states in a topological N-SNS junction - Abstract
      • We theoretically study the differential conductance of a one-dimensional normal-superconductor-normal-superconductor (N-SNS) junction with a phase bias applied between the two superconductors. We consider specifically a junction formed by a spin-orbit coupled semiconducting nanowire with regions of the nanowire having superconducting pairing induced by a bulk $s$-wave superconductor. When the nanowire is tuned into a topologically non-trivial phase by a Zeeman field, it hosts zero-energy Majorana modes at its ends as well as at the interface between the two superconductors. The phase-dependent splitting of the Majorana modes gives rise to features in the differential conductance that offer a clear distinction between the topologically trivial and non-trivial phases. We calculate the transport properties of the junction numerically and also present a simple analytical model that captures the main properties of the predicted tunneling spectroscopy.
    • Esben Bork Hansen, Jeroen Danon, Karsten Flensberg
      Journal reference: Phys. Rev. B 93, 094501 (2016) [pdf]
      DOI: 10.1103/PhysRevB.93.094501

    • Effects of spin-orbit coupling and spatial symmetries on the Josephson current in SNS junctions - Abstract
      • We present an analysis of the symmetries of the interference pattern of critical currents through a two-dimensional superconductor-semiconductor-superconductor junction, taking into account Rashba and Dresselhaus spin-orbit interaction, an arbitrarily oriented magnetic field, disorder, and structural asymmetries. We relate the symmetries of the pattern to the absence or presence of symmetries in the Hamiltonian, which provides a qualitative connection between easily measurable quantities and the spin-orbit coupling and other symmetries of the junction. We support our analysis with numerical calculations of the Josephson current based on a perturbative expansion up to eighth order in tunnel coupling between the normal region and the superconductors.
    • Asbjørn Rasmussen, Jeroen Danon, Henri Suominen, Fabrizio Nichele, Morten Kjaergaard, Karsten Flensberg
      Journal reference: Phys. Rev. B 93, 155406 (2016) [ 1510.05251v2 ]
      DOI: 10.1103/PhysRevB.93.155406

  • 2015
    • Environmental Coulomb blockade of topological superconductor-normal metal junctions - Abstract
      • We study charge transport of a topological superconductor connected to different electromagnetic environments using a low-energy description where only the Majorana bound states in the superconductor are included. Extending earlier findings who found a crossover between perfect Andreev reflection with conductance $2e^2/h$ to a regime with blocked transport when the resistance of the environment is larger than $2e^2/h$, we consider Majorana bound states coupled to metallic dots. in particular, we study two topological superconducting leads connected by a metallic quantum dot in both the weak tunneling and strong tunneling regimes. For weak tunneling, we project onto the most relevant charge states. For strong tunneling, we start from the Andreev fixed point and integrate out charge fluctuations which gives an effective low-energy model for the non-perturbative gate-voltage modulated cotunneling current. In both regimes and in contrast to cotunneling with normal leads, the conductance is temperature independent because of the resonant Andreev reflections, which are included to all orders.
    • Konrad Wölms, Karsten Flensberg
      Journal reference: Phys. Rev. B 92, 165428 (2015) [pdf]
      DOI: 10.1103/PhysRevB.92.165428

    • Nonlocal damping of helimagnets in one-dimensional interacting electron systems - Abstract
      • We investigate the magnetization relaxation of a one-dimensional helimagnetic system coupled to interacting itinerant electrons. The relaxation is assumed to result from the emission of plasmons, the elementary excitations of the one-dimensional interacting electron system, caused by slow changes of the magnetization profile. This dissipation mechanism leads to a highly nonlocal form of magnetization damping that is strongly dependent on the electron-electron interaction. Forward scattering processes lead to a spatially constant damping kernel, while backscattering processes produce a spatially oscillating contribution. Due to the nonlocal damping, the thermal fluctuations become spatially correlated over the entire system. We estimate the characteristic magnetization relaxation times for magnetic quantum wires and nuclear helimagnets.
    • Kjetil M. D. Hals, Karsten Flensberg, Mark S. Rudner
      Journal reference: Phys. Rev. B 92, 094403 (2015) [pdf]
      DOI: 10.1103/PhysRevB.92.094403

    • Local Adiabatic Mixing of Kramers Pairs of Majorana Bound States - Abstract
      • We consider Kramers pairs of Majorana bound states under adiabatic time evolution. This is important for the prospects of using such bound states as parity qubits. We show that local adiabatic perturbations can cause a rotation in the space spanned by the Kramers pair. Hence the quantum information is unprotected against local perturbations, in contrast to the case of single localized Majorana bound states in systems with broken time reversal symmetry. We give an analytical and a numerical example for such a rotation, and specify sufficient conditions under which a rotation is avoided. We give a general scheme for determining when these conditions are satisfied, and exemplify it with a general model of a quasi 1D time reversal symmetric topological superconductor.
    • Konrad Wölms, Ady Stern, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 113, 246401 (2014) [pdf]
      DOI: 10.1103/PhysRevLett.113.246401

    • Braiding properties of Majorana Kramers pairs - Abstract
      • We consider the braiding of Kramers pairs of Majorana bound states. We derive the most general transformation on the many-body ground state that is applied as the result of such a braiding process. The result is derived in the context of a simple toy model, but we will show that it has the most general form that is compatible with local and global conservation of electron parity. In accordance with earlier work the resulting transformation turns out to be path dependent, which shows that Kramers pairs of Majorana bound states cannot be used for topological quantum computation. We also discuss under which conditions the result is path independent and corresponds to two independent exchanges of pairs of Majorana bound states.
    • Konrad Wölms, Ady Stern, Karsten Flensberg
      Journal reference: Phys. Rev. B 93, 045417 (2016) [pdf]
      DOI: 10.1103/PhysRevB.93.045417

    • 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

    • Interaction effects on proximity-induced superconductivity in semiconducting nanowires - Abstract
      • We investigate the effect of electron-electron interactions on proximity-induced $s$-wave superconductivity in one-dimensional nanowires. We treat the interactions on a self-consistent mean-field level, and find an analytic expression for the effective pairing potential in the presence of interactions, valid for a weakly tunnel coupled wire. We show that for a set of two nanowires placed in parallel on a superconducting substrate, the interaction-induced reduction of the pairing energy could result in the effective interwire pairing potential exceeding the intrawire potential, which is one of the requirements for creating a time-reversal symmetric topological superconducting state in such a two-wire system.
    • Jeroen Danon, Karsten Flensberg
      Journal reference: Phys. Rev. B 91, 165425 (2015) [pdf]
      DOI: 10.1103/PhysRevB.91.165425

    • Spin-Lattice Order in One-Dimensional Conductors: Beyond the RKKY Effect - Abstract
      • We investigate magnetic order in a lattice of classical spins coupled to an isotropic gas of one-dimensional (1d) conduction electrons via local exchange interactions. The frequently discussed Ruderman-Kittel-Kasuya-Yosida (RKKY) effective exchange model for this system predicts that spiral order is always preferred. Here we consider the problem nonperturbatively, and find that such order vanishes above a critical value of the exchange coupling that depends strongly on the lattice spacing. The critical coupling tends to zero as the lattice spacing becomes commensurate with the Fermi wave vector, signalling the breakdown of the perturbative RKKY picture, and spiral order, even at weak coupling. We provide the exact phase diagram for arbitrary exchange coupling and lattice spacing, and discuss its stability. Our results shed new light on the problem of utilizing a spiral spin-lattice state to drive a one-dimensional superconductor into a topological phase.
    • Michael Schecter, Mark S. Rudner, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 114, 247205 (2015) [pdf]
      DOI: 10.1103/PhysRevLett.114.247205

    • 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

  • 2014
    • Tunnel spectroscopy of Majorana bound states in topological superconductor/quantum dot Josephson junctions - Abstract
      • We theoretically investigate electronic transport through a junction where a quantum dot (QD) is tunnel coupled on both sides to semiconductor nanowires with strong spin-orbit interaction and proximity-induced superconductivity. The results are presented as stability diagrams, i.e., the differential conductance as a function of the bias voltage applied across the junction and the gate voltage used to control the electrostatic potential on the QD. A small applied magnetic field splits and modifies the resonances due to the Zeeman splitting of the QD level. Above a critical field strength, Majorana bound states (MBS) appear at the interfaces between the two superconducting nanowires and the QD, resulting in a qualitative change of the entire stability diagram, suggesting this setup as a promising platform to identify MBS. Our calculations are based on a nonequilibrium Green's function description and is exact when Coulomb interactions on the QD can be neglected. In addition, we develop a simple pictorial view of the involved transport processes, which is equivalent to a description in terms of multiple Andreev reflections, but provides an alternative way to understand the role of the QD level in enhancing transport for certain gate and bias voltages. We believe that this description will be useful in future studies of interacting QDs coupled to superconducting leads (with or without MBS), where it can be used to develop a perturbation expansion in the tunnel coupling.
    • Guang-Yao Huang, Martin Leijnse, Karsten Flensberg, Hongqi Xu
      Journal reference: Phys. Rev. B 90, 214507 (2014) [pdf]
      DOI: 10.1103/PhysRevB.90.214507

    • Designing $π$-stacked molecular structures to control heat transport through molecular junctions - Abstract
      • We propose and analyze a new way of using $\pi$ stacking to design molecular junctions that either enhance or suppress a phononic heat current, but at the same time remain conductors for an electric current. Such functionality is highly desirable in thermoelectric energy converters, as well as in other electronic components where heat dissipation should be minimized or maximized. We suggest a molecular design consisting of two masses coupled to each other with one mass coupled to each lead. By having a small coupling (spring constant) between the masses, it is possible to either reduce, or perhaps more surprisingly enhance the phonon conductance. We investigate a simple model system to identify optimal parameter regimes and then use first principle calculations to extract model parameters for a number of specific molecular realizations, confirming that our proposal can indeed be realized using standard molecular building blocks.
    • Gediminas Kiršanskas, Qian Li, Karsten Flensberg, Gemma C. Solomon, Martin Leijnse
      Journal reference: Appl. Phys. Lett. 105, 233102 (2014) [pdf]
      DOI: 10.1063/1.4903340

    • Majorana fermions in superconducting nanowires without spin-orbit coupling - Abstract
      • We show that confined Majorana fermions can exist in nanowires with proximity induced s-wave superconducting pairing if the direction of an external magnetic field rotates along the wire. The system is equivalent to nanowires with Rashba-type spin-orbit coupling, with strength proportional to the derivative of the field angle. For realistic parameters, we demonstrate that a set of permanent magnets can bring a nearby nanowire into the topologically non-trivial phase with localized Majorana modes at its ends. Without the requirement of spin-orbit coupling this opens up for a new route for demonstration and design of Majorana fermion systems.
    • Morten Kjaergaard, Konrad Wölms, Karsten Flensberg
      Journal reference: Phys. Rev. B 85, 020503(R) (2012) [pdf]
      DOI: 10.1103/PhysRevB.85.020503

    • Majorana Bound States in Two-Channel Time-Reversal-Symmetric Nanowire Systems - Abstract
      • We consider time-reversal-symmetric two-channel semiconducting quantum wires proximity coupled to an s-wave superconductor. We analyze the requirements for a nontrivial topological phase and find that necessary conditions are 1) the determinant of the pairing matrix in channel space must be negative, 2) inversion symmetry must be broken, and 3) the two channels must have different spin-orbit couplings. The first condition can be implemented in semiconducting nanowire systems where interactions suppress intra-channel pairing, while the inversion symmetry can be broken by tuning the chemical potentials of the channels. For the case of collinear spin-orbit directions, we find a general expression for the topological invariant by block diagonalization into two blocks with chiral symmetry only. By projection to the low-energy sector, we solve for the zero modes explicitly and study the details of the gap closing, which in the general case happens at finite momenta.
    • Erikas Gaidamauskas, Jens Paaske, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 122, 126402 (2014) [pdf]
      DOI: 10.1103/PhysRevLett.112.126402

  • 2013
    • Coupling Spin Qubits via Superconductors - Abstract
      • We show how superconductors can be used to couple, initialize, and read out spatially separated spin qubits. When two single-electron quantum dots are tunnel coupled to the same superconductor, the singlet component of the two-electron state partially leaks into the superconductor via crossed Andreev reflection. This induces a gate-controlled singlet-triplet splitting which, with an appropriate superconductor geometry, remains large for dot separations within the superconducting coherence length. Furthermore, we show that when two double-dot singlet-triplet qubits are tunnel coupled to a superconductor with finite charging energy, crossed Andreev reflection enables a strong two-qubit coupling over distances much larger than the coherence length.
    • Martin Leijnse, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 111, 060501 (2013) [pdf]
      DOI: 10.1103/PhysRevLett.111.060501

  • 2012
    • Introduction to topological superconductivity and Majorana fermions - Abstract
      • This short review article provides a pedagogical introduction to the rapidly growing research field of Majorana fermions in topological superconductors. We first discuss in some details the simplest "toy model" in which Majoranas appear, namely a one-dimensional tight-binding representation of a p-wave superconductor, introduced more than ten years ago by Kitaev. We then give a general introduction to the remarkable properties of Majorana fermions in condensed matter systems, such as their intrinsically non-local nature and exotic exchange statistics, and explain why these quasiparticles are suspected to be especially well suited for low-decoherence quantum information processing. We also discuss the experimentally promising (and perhaps already successfully realized) possibility of creating topological superconductors using semiconductors with strong spin-orbit coupling, proximity-coupled to standard s-wave superconductors and exposed to a magnetic field. The goal is to provide an introduction to the subject for experimentalists or theorists who are new to the field, focusing on the aspects which are most important for understanding the basic physics. The text should be accessible for readers with a basic understanding of quantum mechanics and second quantization, and does not require knowledge of quantum field theory or topological states of matter.
    • Martin Leijnse, Karsten Flensberg
      Journal reference: Semicond. Sci. Technol. 27, 124003 (2012) [pdf]
      DOI: 10.1088/0268-1242/27/12/124003

    • Parity qubits and poor man's Majorana bound states in double quantum dots - Abstract
      • We study a double quantum dot connected via a common superconducting lead and show that this system can be tuned to host one Majorana bound state (MBS) on each dot. We call them "poor man's Majorana bound states" since they are not topologically protected, but otherwise share the properties of MBS formed in topological superconductors. We describe the conditions for the existence of the two spatially separated MBS, which include breaking of spin degeneracy in the two dots, with the spins polarized in different directions. Therefore, we propose to use a magnetic field configuration where the field directions on the two dot form an angle. By control of this angle the cross Andreev reflection and the tunnel amplitudes can be tuned to be approximately equal, which is a requirement for the formation of the MBS. We show that the fermionic state encoded in the two Majoranas constitutes a parity qubit, which is non-local and can only be measured by probing both dots simultaneously. Using a many-particle basis for the MBS, we discuss the role of interactions and show that inter-dot interactions always lift the degeneracy. We also show how the MBS can be probed by transport measurements and discuss how the combination of several such double dot systems allows for entanglement of parity qubits and measurement of their dephasing times.
    • Martin Leijnse, Karsten Flensberg
      Journal reference: Phys. Rev. B. 86, 134528 (2012) [pdf]
      DOI: 10.1103/PhysRevB.86.134528

    • Finite-bias conductance anomalies at a singlet-triplet crossing - Abstract
      • Quantum dots and single-molecule transistors may exhibit level crossings induced by tuning external parameters such as magnetic field or gate voltage. For Coulomb blockaded devices, this shows up as an inelastic cotunneling threshold in the differential conductance, which can be tuned to zero at the crossing. Here we show that, in addition, level crossings can give rise to a nearly vertical step-edge, ridge or even a Fano-like ridge-valley feature in the differential conductance inside the relevant Coulomb diamond. We study a gate-tunable quasidegeneracy between singlet and triplet ground states, and demonstrate how these different shapes may result from a competition between nonequilibrium occupations and weak (spin-orbit) mixing of the states. Our results are shown to be in qualitative agreement with recent transport measurements on a Mn complex [E. A. Osorio, et al., Nano Lett. 10, 105 (2010)]. The effect remains entirely general and should be observable in a wide range of Coulomb blockaded devices.
    • Chiara Stevanato, Martin Leijnse, Karsten Flensberg, Jens Paaske
      Journal reference: Phys. Rev. B 86, 165427 (2012) [pdf]
      DOI: 10.1103/PhysRevB.86.165427

    • Hybrid topological-spin qubit systems for two-qubit-spin gates - Abstract
      • We investigate a hybrid quantum system involving spin qubits, based on the spins of electrons confined in quantum dots, and topological qubits, based on Majorana fermions. In such a system, gated control of the charge on the quantum dots allows transfer of quantum information between the spin and topological qubits, and the topological system can be used to facilitate transfer of spin qubits between spatially separated quantum dots and to initialize entangled spin-qubit pairs. Here, we show that the coupling to the topological system also makes it possible to perform entangling two-qubit gates on spatially separated spin qubits. The two-qubit gates are based on a combination of topologically protected braiding operations, gate-controlled charge transfer between the dots and edge Majorana modes, and measurements of the state of the topological qubits.
    • Martin Leijnse, Karsten Flensberg
      Journal reference: Phys. Rev. B 86, 104511 (2012) [pdf]
      DOI: 10.1103/PhysRevB.86.104511

    • Cotunneling renormalization in carbon nanotube quantum dots - Abstract
      • We determine the level-shifts induced by cotunneling in a Coulomb blockaded carbon nanotube quantum dot using leading order quasi-degenerate perturbation theory within a single nanotube quartet. It is demonstrated that otherwise degenerate and equally tunnel-coupled $K$ and $K'$ states are mixed by cotunneling and therefore split up in energy except at the particle/hole-symmetric midpoints of the Coulomb diamonds. In the presence of an external magnetic field, we show that cotunneling induces a gate-dependent $g$-factor renormalization, and we outline different scenarios which might be observed experimentally, depending on the values of both intrinsic $KK'$ splitting and spin-orbit coupling.
    • Gediminas Kiršanskas, Jens Paaske, Karsten Flensberg
      Journal reference: Phys. Rev. B 86, 075452 (2012) [pdf]
      DOI: 10.1103/PhysRevB.86.075452

    • Emerging Dirac and Majorana fermions for carbon nanotubes with proximity-induced pairing and spiral magnetic field - Abstract
      • We study the low-energy bandstructure of armchair and small-bandgap semiconducting carbon nanotubes with proximity-induced superconducting pairing when a spiral magnetic field creates strong effective spin-orbit interactions from the Zeeman term and a periodic potential from the orbital part. We find that gapless Dirac fermions can be generated by variation of a single parameter. For a semiconducting tube with the field in the same plane, a non-degenerate zero mode at momentum k=0 can be induced, allowing for the generation of topologically protected Majorana fermion end states.
    • Reinhold Egger, Karsten Flensberg
      Journal reference: Physical Review B 85, 235462 (2012) [pdf]
      DOI: 10.1103/PhysRevB.85.235462

    • Spin-Orbit-Induced Strong Coupling of a Single Spin to a Nanomechanical Resonator - Abstract
      • We theoretically investigate the deflection-induced coupling of an electron spin to vibrational motion due to spin-orbit coupling in suspended carbon nanotube quantum dots. Our estimates indicate that, with current capabilities, a quantum dot with an odd number of electrons can serve as a realization of the Jaynes-Cummings model of quantum electrodynamics in the strong-coupling regime. A quantized flexural mode of the suspended tube plays the role of the optical mode and we identify two distinct two-level subspaces, at small and large magnetic field, which can be used as qubits in this setup. The strong intrinsic spin-mechanical coupling allows for detection, as well as manipulation of the spin qubit, and may yield enhanced performance of nanotubes in sensing applications.
    • András Pályi, P. R. Struck, Mark Rudner, Karsten Flensberg, Guido Burkard
      Journal reference: Phys. Rev. Lett. 108, 206811 (2012) [pdf]
      DOI: 10.1103/PhysRevLett.108.206811

    • Dephasing and hyperfine interaction in carbon nanotubes double quantum dots: Disordered case - Abstract
      • We study theoretically the \emph{return probability experiment}, used to measure the dephasing time $T_2^*$, in a double quantum dot (DQD) in semiconducting carbon nanotubes (CNTs) with spin-orbit coupling and disorder induced valley mixing. Dephasing is due to hyperfine interaction with the spins of the ${}^{13}$C nuclei. Due to the valley and spin degrees of freedom four bounded states exist for any given longitudinal mode in the quantum dot. At zero magnetic field the spin-orbit coupling and the valley mixing split those four states into two Kramers doublets. The valley mixing term for a given dot is determined by the intra-dot disorder and therefore the states in the Kramers doublets belonging to different dots are different. We show how nonzero single-particle interdot tunneling amplitudes between states belonging to different doublets give rise to new avoided crossings, as a function of detuning, in the relevant two particle spectrum, crossing over from the two electrons in one dot states configuration, $(0,2)$, to the one electron in each dot configuration, $(1,1)$. In contrast to the clean system, multiple Landau-Zener processes affect the separation and the joining stages of each single-shot measurement and they affect the outcome of the measurement in a way that strongly depends on the initial state. We find that a well-defined return probability experiment is realized when, at each single-shot cycle, the (0,2) ground state is prepared. In this case, valley mixing increases the saturation value of the measured return probability, whereas the probability to return to the (0,2) ground state remains unchanged. Finally, we study the effect of the valley mixing in the high magnetic field limit; for a parallel magnetic field the predictions coincide with a clean nanotube, while the disorder effect is always relevant with a magnetic field perpendicular to the nanotube axis.
    • Andres A. Reynoso, Karsten Flensberg
      Journal reference: Phys. Rev. B 85, 195441 (2012) [pdf]
      DOI: 10.1103/PhysRevB.85.195441

    • Magnetic-Field Dependence of Tunnel Couplings in Carbon Nanotube Quantum Dots - Abstract
      • By means of sequential and cotunneling spectroscopy, we study the tunnel couplings between metallic leads and individual levels in a carbon nanotube quantum dot. The levels are ordered in shells consisting of two doublets with strong- and weak-tunnel couplings, leading to gate-dependent level renormalization. By comparison to a one- and two-shell model, this is shown to be a consequence of disorder-induced valley mixing in the nanotube. Moreover, a parallel magnetic field is shown to reduce this mixing and thus suppress the effects of tunnel renormalization.
    • K. Grove-Rasmussen, S. Grap, J. Paaske, K. Flensberg, S. Andergassen, V. Meden, H. I. Jørgensen, K. Muraki, T. Fujisawa
      Journal reference: Phys. Rev. Lett. 108, 176802 (2012) [pdf]
      DOI: 10.1103/PhysRevLett.108.176802

  • 2011
    • Quantum Information Transfer between Topological and Spin Qubit Systems - Abstract
      • We propose a method to coherently transfer quantum information, and to create entanglement, between topological qubits and conventional spin qubits. Our suggestion uses gated control to transfer an electron (spin qubit) between a quantum dot and edge Majorana modes in adjacent topological superconductors. Because of the spin polarization of the Majorana modes, the electron transfer translates spin superposition states into superposition states of the Majorana system, and vice versa. Furthermore, we show how a topological superconductor can be used to facilitate long-distance quantum information transfer and entanglement between spatially separated spin qubits.
    • Martin Leijnse, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 107, 210502 (2011) [pdf]
      DOI: 10.1103/PhysRevLett.107.210502

    • Number conserving theory for topologically protected degeneracy in one-dimensional fermions - Abstract
      • Semiconducting nanowires in proximity to superconductors are among promising candidates to search for Majorana fermions and topologically protected degeneracies which may ultimately be used as building blocks for topological quantum computers. The prediction of neutral Majorana fermions in the proximity-induced superconducting systems ignores number-conservation and thus leaves open the conceptual question of how a topological degeneracy that is robust to all local perturbations arises in a number-conserving system. In this work, we study how local attractive interactions generate a topological ground-state near-degeneracy in a quasi one-dimensional superfluid using bosonization of the fermions. The local attractive interactions opens a topological quasiparticle gap in the odd channel wires (with more than one channel) with end Majorana modes associated with a topological near-degeneracy. We explicitly study the robustness of the topological degeneracy to local perturbations and find that such local perturbations result in quantum phase slips which split of the topological degeneracy by an amount that does not decrease exponentially with the length of the wire, but still decreases rapidly if the number of channels is large. Therefore a bulk superconductor with a large number of channels is crucial for true topological degeneracy.
    • Jay D. Sau, B. I. Halperin, K. Flensberg, S. Das Sarma
      Journal reference: Phys. Rev. B 84, 144509 (2011) [pdf]
      DOI: 10.1103/PhysRevB.84.144509

    • Scheme to measure Majorana fermion lifetimes using a quantum dot - Abstract
      • We propose a setup to measure the lifetime of the parity of a pair of Majorana bound states. The proposed experiment has one edge Majorana state tunnel coupled to a quantum dot, which in turn is coupled to a metallic electrode. When the Majorana Fermions overlap, even a small relaxation rate qualitatively changes the non-linear transport spectrum, and for strong overlap the lifetime can be read off directly from the height of a current peak. This is important for the usage of Majorana Fermions as a platform for topological quantum computing, where the parity relaxation is a limiting factor.
    • Martin Leijnse, Karsten Flensberg
      Journal reference: Phys. Rev. B 84, 140501(R) (2011) [pdf]
      DOI: 10.1103/PhysRevB.84.140501

    • Nonequilibrium Transport through a Spinful Quantum Dot with Superconducting Leads - Abstract
      • We study the nonlinear cotunneling current through a spinful quantum dot contacted by two superconducting leads. Applying a general nonequilibrium Green function formalism to an effective Kondo model, we study the rich variation in the IV-characteristics with varying asymmetry in the tunnel coupling to source and drain electrodes. The current is found to be carried respectively by multiple Andreev reflections in the symmetric limit, and by spin-induced Yu-Shiba-Russinov bound states in the strongly asymmetric limit. The interplay between these two mechanisms leads to qualitatively different IV-characteristics in the cross-over regime of intermediate symmetry, consistent with recent experimental observations of negative differential conductance and re-positioned conductance peaks in sub-gap cotunneling spectroscopy.
    • B. M. Andersen, K. Flensberg, V. Koerting, J. Paaske
      Journal reference: Phys. Rev. Lett. 107, 256802 (2011) [pdf]
      DOI: 10.1103/PhysRevLett.107.256802

    • Image charge effects in single-molecule junctions: Breaking of symmetries and negative-differential resistance in a benzene single-electron transistor - Abstract
      • Both experiments and theoretical studies have demonstrated that the interaction between the current carrying electrons and the induced polarization charge in single-molecule junctions leads to a strong renormalization of molecular charging energies. However, the effect on electronic excitations and molecular symmetries remain unclear. Using a theoretical framework developed for semiconductor nanostructure based single-electron transistors (SETs), we demonstrate that the image charge interaction breaks the molecular symmetries in a benzene based single-molecule transistor operating in the Coulomb blockade regime. This results in the appearance of a so-called blocking state, which gives rise to negative differential resistance (NDR). We show that the appearance of NDR and its magnitude in the symmetry-broken benzene SET depends in a complicated way on the interplay between the many-body matrix elements, the lead tunnel coupling asymmetry, and the bias polarity. In particular, the current reducing property of the blocking state causing the NDR, is shown to vanish under strongly asymmetric tunnel couplings, when the molecule is coupled stronger to the drain electrode. The calculated IV characteristic may serve as an indicator for image charge broken molecular symmetries in experimental situations.
    • K. Kaasbjerg, K. Flensberg
      Journal reference: Phys. Rev. B 84, 115457 (2011) [pdf]
      DOI: 10.1103/PhysRevB.84.115457

    • Dephasing and hyperfine interaction in carbon nanotube double quantum dots: The clean limit - Abstract
      • We consider theoretically ${}^{13}$C-hyperfine interaction induced dephasing in carbon nanotubes double quantum dots with curvature induced spin-orbit coupling. For two electrons initially occupying a single dot, we calculate the average return probability after separation into the two dots, which have random nuclear-spin configurations. We focus on the long time saturation value of the return probability, $P_\infty$. Because of the valley degree of freedom, the analysis is more complex than in, for example, GaAs quantum dots, which have two distinct $P_\infty$ values depending on the magnetic field. Here the prepared state and the measured state is non-unique because two electrons in the same dot are allowed in six different states. Moreover, for one electron in each dot sixteen states exist and therefore are available for being mixed by the hyperfine field. The return probability experiment is found to be strongly dependent on the prepared state, on the external magnetic field---both Zeeman and orbital effects - and on the spin-orbit splitting. The lowest saturation value, being $P_\infty$=1/3, occurs at zero magnetic field for nanotubes with spin-orbit coupling and the initial state being the groundstate, this situation is equivalent to double dots without the valley degree of freedom. In total, we report nine dynamically different situations that give $P_\infty$=1/3, 3/8, 2/5, 1/2 and for valley anti-symmetric prepared states in an axial magnetic field, $P_\infty$=1. When the groundstate is prepared the ratio between the spin-orbit splitting and the Zeeman energy due to a perpendicular magnetic field can tune the effective hyperfine field continuously from being three dimensional to two dimensional giving saturation values from $P_\infty$=1/3 to 3/8.
    • Andres A. Reynoso, Karsten Flensberg
      Journal reference: Phys. Rev. B 84, 205449 (2011) [pdf]
      DOI: 10.1103/PhysRevB.84.205449

  • 2010
    • Interaction-induced negative differential resistance in asymmetric molecular junctions - Abstract
      • Combining insights from quantum chemistry calculations with master equations, we discuss a mechanism for negative differential resistance (NDR) in molecular junctions, operated in the regime of weak tunnel coupling. The NDR originates from an interplay of orbital spatial asymmetry and strong electron-electron interaction, which causes the molecule to become trapped in a non-conducting state above a voltage threshold. We show how the desired asymmetry can be selectively introduced in individual orbitals in e.g., OPE-type molecules by functionalization with a suitable side group, which is in linear conjugation to one end of the molecule and cross-conjugated to the other end.
    • Martin Leijnse, Wei Sun, Mogens Brøndsted Nielsen, Per Hedegård, Karsten Flensberg
      Journal reference: J. Chem. Phys. 134, 104107 (2011) [pdf]
      DOI: 10.1063/1.3560474

    • Non-Abelian Operations on Majorana Fermions via Single-Charge Control - Abstract
      • We demonstrate that non-Abelian rotations within the degenerate groundstate manifold of a set of Majorana fermions can be realized by the addition or removal of single electrons, and propose an implementation using Coulomb blockaded quantum dots. The exchange of electrons generates rotations similar to braiding, though not in real space. Unlike braiding operations, rotations by a continuum of angles are possible, while still being partially robust against perturbations. The quantum dots can also be used for readout of the state of the Majorana system via a charge measurement.
    • Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 106, 090503 (2011) [pdf]
      DOI: 10.1103/PhysRevLett.106.090503

    • Nonequilibrium transport via spin-induced subgap states in superconductor/quantum dot/normal metal cotunnel junctions - Abstract
      • We study low-temperature transport through a Coulomb blockaded quantum dot (QD) contacted by a normal (N), and a superconducting (S) electrode. Within an effective cotunneling model the conduction electron self energy is calculated to leading order in the cotunneling amplitudes and subsequently resummed to obtain the nonequilibrium T-matrix, from which we obtain the nonlinear cotunneling conductance. For even occupied dots the system can be conceived as an effective S/N-cotunnel junction with subgap transport mediated by Andreev reflections. The net spin of an odd occupied dot, however, leads to the formation of sub-gap resonances inside the superconducting gap which gives rise to a characteristic peak-dip structure in the differential conductance, as observed in recent experiments.
    • V. Koerting, B. M. Andersen, K. Flensberg, J. Paaske
      Journal reference: Phys. Rev. B 82, 245108 (2010) [pdf]
      DOI: 10.1103/PhysRevB.82.245108

    • Spin-orbit effects in carbon-nanotube double quantum dots - Abstract
      • We study the energy spectrum of symmetric double quantum dots in narrow-gap carbon nanotubes with one and two electrostatically confined electrons in the presence of spin-orbit and Coulomb interactions. Compared to GaAs quantum dots, the spectrum exhibits a much richer structure because of the spin-orbit interaction that couples the electron's isospin to its real spin through two independent coupling constants. In a single dot, both constants combine to split the spectrum into two Kramers doublets, while the antisymmetric constant solely controls the difference in the tunneling rates of the Kramers doublets between the dots. For the two-electron regime, the detailed structure of the spin-orbit split energy spectrum is investigated as a function of detuning between the quantum dots in a 22-dimensional Hilbert space within the framework of a single-longitudinal-mode model. We find a competing effect of the tunneling and Coulomb interaction. The former favors a left-right symmetric two-particle ground state, while in the regime where the Coulomb interaction dominates over tunneling, a left-right antisymmetric ground state is found. As a result, ground states on both sides of the $(11)$-$(02)$ degeneracy point may possess opposite left-right symmetry, and the electron dynamics when tuning the system from one side of the $(11)$-$(02)$ degeneracy point to the other is controlled by three selection rules (in spin, isospin, and left-right symmetry). We discuss implications for the spin-dephasing and Pauli blockade experiments.
    • S. Weiss, E. I. Rashba, F. Kuemmeth, H. O. H Churchill, K. Flensberg
      Journal reference: Physical Review B 82, 165427 (2010) [pdf]
      DOI: 10.1103/PhysRevB.82.165427

    • Tunneling characteristic of a chain of Majorana bound states - Abstract
      • We consider theoretically tunneling characteristic of a junction between a normal metal and a chain of coupled Majorana bound states generated at crossings between topological and non-topological superconducting sections, as a result of, for example, disorder in nanowires. While an isolated Majorana state supports a resonant Andreev process, yielding a zero bias differential conductance peak of height 2e^2/h, the situation with more coupled Majorana states is distinctively different with both zeros and 2e^2/h peaks in the differential conductance. We derive a general expression for the current between a normal metal and a network of coupled Majorana bound states and describe the differential conductance spectra for a generic set of situations, including regular, disordered, and infinite chains of bound states.
    • Karsten Flensberg
      DOI: 10.1103/PhysRevB.82.180516
      1009.3533v1 [pdf]

    • Gate-dependent spin-orbit coupling in multi-electron carbon nanotubes - Abstract
      • Understanding how the orbital motion of electrons is coupled to the spin degree of freedom in nanoscale systems is central for applications in spin-based electronics and quantum computation. We demonstrate this coupling of spin and orbit in a carbon nanotube quantum dot in the general multi-electron regime in presence of finite disorder. Further, we find a strong systematic dependence of the spin-orbit coupling on the electron occupation of the quantum dot. This dependence, which even includes a sign change is not demonstrated in any other system and follows from the curvature-induced spin-orbit split Dirac-spectrum of the underlying graphene lattice. Our findings unambiguously show that the spin-orbit coupling is a general property of nanotube quantum dots which provide a unique platform for the study of spin-orbit effects and their applications.
    • Thomas Sand Jespersen, Kasper Grove-Rasmussen, Jens Paaske, Koji Muraki, Toshimasa Fujisawa, Jesper Nygård, Karsten Flensberg
      1008.1600v2 [pdf]

    • Exchange cotunneling through quantum dots with spin-orbit coupling - Abstract
      • We investigate the effects of spin-orbit interaction (SOI) on the exchange cotunneling through a spinful Coulomb blockaded quantum dot. In the case of zero magnetic field, Kondo effect is shown to take place via a Kramers doublet and the SOI will merely affect the Kondo temperature. In contrast, we find that the breaking of time-reversal symmetry in a finite field has a marked influence on the effective Anderson, and Kondo models for a single level. The nonlinear conductance can now be asymmetric in bias voltage and may depend strongly on direction of the magnetic field. A measurement of the angle dependence of finite-field cotunneling spectroscopy thus provides valuable information about orbital, and spin degrees of freedom and their mutual coupling.
    • J. Paaske, A. Andersen, K. Flensberg
      DOI: 10.1103/PhysRevB.82.081309
      1006.2371v1 [pdf]

    • Bends in nanotubes allow electric spin control and coupling - Abstract
      • We investigate combined effects of spin-orbit coupling and magnetic field in carbon nanotubes containing one or more bends along their length. We show how bends can be used to provide electrical control of confined spins, while spins confined in straight segments remain insensitive to electric fields. Device geometries that allow general rotation of single spins are presented and analyzed. In addition, capacitive coupling along bends provides coherent spin-spin interaction, including between otherwise disconnected nanotubes, completing a universal set of one- and two-qubit gates.
    • K. Flensberg, C. M. Marcus
      Journal reference: Phys. Rev. B 81, 195418 (2010) [pdf]
      DOI: 10.1103/PhysRevB.81.195418

    • Nonlinear thermoelectric properties of molecular junctions with vibrational coupling - Abstract
      • We present a detailed study of the non-linear thermoelectric properties of a molecular junction, represented by a dissipative Anderson-Holstein model. A single orbital level with strong Coulomb interaction is coupled to a localized vibrational mode and we account for both electron and phonon exchange with both electrodes, investigating how these contribute to the heat and charge transport. We calculate the efficiency and power output of the device operated as a heat to electric power converter and identify the optimal operating conditions, which are found to be qualitatively changed by the presence of the vibrational mode. Based on this study of a generic model system, we discuss the desirable properties of molecular junctions for thermoelectric applications.
    • M. Leijnse, M. R. Wegewijs, K. Flensberg
      Journal reference: Phys. Rev. B 82, 045412 (2010) [pdf]
      DOI: 10.1103/PhysRevB.82.045412

    • Transport via coupled states in a C60 peapod quantum dot - Abstract
      • We have measured systematic repetitions of avoided crossings in low temperature three-terminal transport through a carbon nanotube with encapsulated C60 molecules. We show that this is a general effect of the hybridization of a host quantum dot with an impurity. The well-defined nanotube allows identification of the properties of the impurity, which we suggest to be a chain of C60 molecules inside the nanotube. This electronic coupling between the two subsystems opens the interesting and potentially useful possibility of contacting the encapsulated molecules via the tube.
    • Anders Eliasen, Jens Paaske, Karsten Flensberg, Sebastian Smerat, Martin Leijnse, Maarten R. Wegewijs, Henrik I. Jørgensen, Marc Monthioux, Jesper Nygård
      DOI: 10.1103/PhysRevB.81.155431
      1002.0477v1 [pdf]

  • 2009
    • Electrical manipulation of spin states in a single electrostatically gated transition-metal complex - Abstract
      • We demonstrate an electrically controlled high-spin (S=5/2) to low-spin (S=1/2) transition in a three-terminal device incorporating a single Mn2+ ion coordinated by two terpyridine ligands. By adjusting the gate-voltage we reduce the terpyridine moiety and thereby strengthen the ligand-field on the Mn-atom. Adding a single electron thus stabilizes the low-spin configuration and the corresponding sequential tunnelling current is suppressed by spin-blockade. From low-temperature inelastic cotunneling spectroscopy, we infer the magnetic excitation spectrum of the molecule and uncover also a strongly gate-dependent singlet-triplet splitting on the low-spin side. The measured bias-spectroscopy is shown to be consistent with an exact diagonalization of the Mn-complex, and an interpretation of the data is given in terms of a simplified effective model.
    • Edgar A. Osorio, Kasper Moth-Poulsen, Herre S. J. van der Zant, Jens Paaske, Per Hedegard, Karsten Flensberg, Jesper Bendix, Thomas Bjornholm
      DOI: 10.1021/nl9029785
      0912.2640v1 [pdf]

    • Mesoscopic conductance fluctuations in InAs nanowire-based SNS junctions - Abstract
      • We report a systematic experimental study of mesoscopic conductance fluctuations in superconductor/normal/superconductor (SNS) devices Nb/InAs-nanowire/Nb. These fluctuations far exceed their value in the normal state and strongly depend on temperature even in the low-temperature regime. This dependence is attributed to high sensitivity of perfectly conducting channels to dephasing and the SNS fluctuations thus provide a sensitive probe of dephasing in a regime where normal transport fails to detect it. Further, the conductance fluctuations are strongly non-linear in bias voltage and reveal sub-gap structure. The experimental findings are qualitatively explained in terms of multiple Andreev reflections in chaotic quantum dots with imperfect contacts.
    • T. S. Jespersen, M. L. Polianski, C. B. Soerensen, K. Flensberg, J. Nygaard
      Journal reference: New J. Phys. 11, 113025 (2009) [pdf]
      DOI: 10.1088/1367-2630/11/11/113025

    • Superconductivity-enhanced bias spectroscopy in carbon nanotube quantum dots - Abstract
      • We study low-temperature transport through carbon nanotube quantum dots in the Coulomb blockade regime coupled to niobium-based superconducting leads. We observe pronounced conductance peaks at finite source-drain bias, which we ascribe to elastic and inelastic cotunneling processes enhanced by the coherence peaks in the density of states of the superconducting leads. The inelastic cotunneling lines display a marked dependence on the applied gate voltage which we relate to different tunneling-renormalizations of the two subbands in the nanotube. Finally, we discuss the origin of an especially pronounced sub-gap structure observed in every fourth Coulomb diamond.
    • K. Grove-Rasmussen, H. I. Jørgensen, B. M. Andersen, J. Paaske, T. S. Jespersen, J. Nygård, K. Flensberg, P. E. Lindelof
      Journal reference: Phys. Rev. B 79, 134518 (2009) [pdf]
      DOI: 10.1103/PhysRevB.79.134518

    • Electron–electron interaction effects in quantum point contacts - Abstract
      • We consider electron-electron interaction effects in quantum point contacts on the first quantization plateau, taking into account all scattering processes. We compute the low-temperature linear and nonlinear conductance, shot noise, and thermopower, by perturbation theory and a self-consistent nonperturbative method. On the conductance plateau, the low-temperature corrections are solely due to momentum-nonconserving processes that change the relative number of left- and right-moving electrons. This leads to a suppression of the conductance for increasing temperature or voltage. The size of the suppression is estimated for a realistic saddle-point potential, and is largest in the beginning of the conductance plateau. For large magnetic field, interaction effects are strongly suppressed by the Pauli principle, and hence the first spin-split conductance plateau has a much weaker interaction correction. For the nonperturbative calculations, we use a self-consistent nonequilibrium Green's function approach, which suggests that the conductance saturates at elevated temperatures. These results are consistent with many experimental observations related to the so-called 0.7 anomaly.
    • A. M. Lunde, A. De Martino, A. Schulz, R. Egger, K. Flensberg
      Journal reference: New J. Phys. 11, 023031 (2009) [pdf]
      DOI: 10.1088/1367-2630/11/2/023031

  • 2008
    • Critical and excess current through an open quantum dot: Temperature and magnetic-field dependence - Abstract
      • We present measurements of temperature and magnetic field dependence of the critical current and excess current in a carbon nanotube Josephson quantum dot junction. The junction is fabricated in a controlled environment which allows for extraction of the full critical current. The measurements are performed in the open quantum dot regime, and fitted to theory with good qualitative agreement. We also show how to extract level spacing, level broadening, and charging energy of an open quantum dot from a bias spectroscopy plot.
    • H. Ingerslev Jørgensen, K. Grove-Rasmussen, K. Flensberg, P. E. Lindelof
      Journal reference: Phys. Rev. B 79, 155441 (2009) [pdf]
      DOI: 10.1103/PhysRevB.79.155441

    • Relaxation and Dephasing in a Two-electron 13C Nanotube Double Quantum Dot - Abstract
      • We use charge sensing of Pauli blockade (including spin and isospin) in a two-electron 13C nanotube double quantum dot to measure relaxation and dephasing times. The relaxation time, T1, first decreases with parallel magnetic field then goes through a minimum in a field of 1.4 T. We attribute both results to the spin-orbit-modified electronic spectrum of carbon nanotubes, which suppresses hyperfine mediated relaxation and enhances relaxation due to soft phonons. The inhomogeneous dephasing time, T2*, is consistent with previous data on hyperfine coupling strength in 13C nanotubes.
    • H. O. H. Churchill, F. Kuemmeth, J. W. Harlow, A. J. Bestwick, E. I. Rashba, K. Flensberg, C. H. Stwertka, T. Taychatanapat, S. K. Watson, C. M. Marcus
      Journal reference: Phys. Rev. Lett. 102, 166802 (2009). [pdf]
      DOI: 10.1103/PhysRevLett.102.166802

    • Interplay between interference and Coulomb interaction in the ferromagnetic Anderson model with applied magnetic field - Abstract
      • We study the competition between interference due to multiple single-particle paths and Coulomb interaction in a simple model of an Anderson-like impurity with local-magnetic-field-induced level splitting coupled to ferromagnetic leads. The model along with its potential experimental relevance in the field of spintronics serves as a nontrivial benchmark system where various quantum transport approaches can be tested and compared. We present results for the linear conductance obtained by a spin-dependent implementation of the density matrix renormalization group scheme which are compared with a mean-field solution as well as a seemingly more advanced Hubbard-I approximation. We explain why mean-field yields nearly perfect results, while the more sophisticated Hubbard-I approach fails, even at a purely conceptual level since it breaks hermiticity of the related density matrix. Furthermore, we study finite bias transport through the impurity by the mean-field approach and recently developed higher-order density matrix equations. We find that the mean-field solution fails to describe the plausible results of the higher-order density matrix approach both quantitatively and qualitatively as it does not capture some essential features of the current-voltage characteristics such as negative differential conductance.
    • Jonas Nyvold Pedersen, Dan Bohr, Andreas Wacker, Tomas Novotny, Peter Schmitteckert, Karsten Flensberg
      Journal reference: Phys. Rev. B 79, 125403 (2009) [pdf]
      DOI: 10.1103/PhysRevB.79.125403

    • Strong Polarization-Induced Reduction of Addition Energies in Single-Molecule Nanojunctions - Abstract
      • We address polarization-induced renormalization of molecular levels in solid-state based single-molecule transistors and focus on an organic conjugate molecule where a surprisingly large reduction of the addition energy has been observed. We have developed a scheme that combines a self-consistent solution of a quantum chemical calculation with a realistic description of the screening environment. Our results indeed show a large reduction, and we explain this to be a consequence of both (a) a reduction of the electrostatic molecular charging energy and (b) polarization induced level shifts of the HOMO and LUMO levels. Finally, we calculate the charge stability diagram and explain at a qualitative level general features observed experimentally.
    • Kristen Kaasbjerg, Karsten Flensberg
      Journal reference: Nano Letters 8, 3809 (2008) [pdf]
      DOI: 10.1021/nl8021708

    • Gate-dependent tunneling-induced level shifts observed in carbon nanotube quantum dots - Abstract
      • We have studied electron transport in clean single-walled carbon nanotube quantum dots. Because of the large number of Coulomb blockade diamonds simultaneously showing both shell structure and Kondo effect, we are able to perform a detailed analysis of tunneling renormalization effects. Thus determining the environment induced level shifts of this artificial atom. In shells where only one of the two orbitals is coupled strongly, we observe a marked asymmetric gate-dependence of the inelastic cotunneling lines together with a systematic gate dependence of the size (and shape) of the Coulomb diamonds. These effects are all given a simple explanation in terms of second-order perturbation theory in the tunnel coupling.
    • J. V. Holm, H. I. Jørgensen, K. Grove-Rasmussen, J. Paaske, K. Flensberg, P. E. Lindelof
      Journal reference: Phys. Rev. B 77, 161406(R) (2008) [pdf]
      DOI: 10.1103/PhysRevB.77.161406

  • 2007
    • Singlet–triplet physics and shell filling in carbon nanotube double quantum dots - Abstract
      • An artifcial two-atomic molecule, also called a double quantum dot (DQD), is an ideal system for exploring few electron physics. Spin-entanglement between just two electrons can be explored in such systems where singlet and triplet states are accessible. These two spin-states can be regarded as the two states in a quantum two-state system, a so-called singlet-triplet qubit. A very attractive material for realizing spin based qubits is the carbon nanotube (CNT), because it is expected to have a very long spin coherence time. Here we show the existence of a gate-tunable singlet-triplet qubit in a CNT DQD. We show that the CNT DQD has clear shell structures of both four and eight electrons, with the singlet-triplet qubit present in the four-electron shells. We furthermore observe inelastic cotunneling via the singlet and triplet states, which we use to probe the splitting between singlet and triplet, in good agreement with theory.
    • H. Ingerslev Jørgensen, K. Grove-Rasmussen, K. -Y. Wang, A. M. Blackburn, K. Flensberg, P. E. Lindelof, D. A. Williams
      Journal reference: Nature Physics 4, 536 - 539 (2008) [pdf]
      DOI: 10.1038/nphys987

    • Coulomb Blockade of a Three-terminal Quantum Dot - Abstract
      • We study an interacting single-level quantum dot weakly coupled to three electrodes. When two electrodes are biased by voltages with opposite polarities, while keeping the third lead (the stem) grounded, the current through the stem is a measure of electron-hole asymmetry of the dot. In this setup we calculate the stem current for both metallic and ferromagnetic (collinearly polarized) leads and discuss how the three-terminal device gives additional information compared to the usual two-terminal setup. We calculate both the sequential and cotunneling contribution for the currents. For the latter part we include a regularization procedure for the cotunneling current, which enables us to also describe the behavior at the charge degeneracy points.
    • Robert Andrzej Zak, Karsten Flensberg
      DOI: 10.1103/PhysRevB.77.045329
      0711.1787v1 [pdf]

    • Critical Current 0−π Transition in Designed Josephson Quantum Dot JunctionsCritical Current 0-p Transition in Designed Josephson Quantum Dot Junctions - Abstract
      • We report on quantum dot based Josephson junctions designed specifically for measuring the supercurrent. From high-accuracy fitting of the current-voltage characteristics we determine the full magnitude of the supercurrent (critical current). Strong gate modulation of the critical current is observed through several consecutive Coulomb blockade oscillations. The critical current crosses zero close to, but not at, resonance due to the so-called 0-$\pi$ transition in agreement with a simple theoretical model.
    • H. Ingerslev Jørgensen, T. Novotný, K. Grove-Rasmussen, K. Flensberg, P. E. Lindelof
      Journal reference: NANO LETTERS 7 (8): 2441-2445 AUG 2007 [pdf]
      DOI: 10.1021/nl071152w

    • Electron-electron interaction effects in quantum point contacts - Abstract
      • We consider interaction effects in quantum point contacts on the first quantization plateau, taking into account all non momentum-conserving processes. We compute low-temperature linear and non-linear conductance, shot noise, and thermopower by perturbation theory, and show that they are consistent with experimental observations on the so-called "0.7 anomaly". The full temperature-dependent conductance is obtained from self-consistent second-order perturbation theory and approaches ~ e^2/h at higher temperatures, but still smaller than the Fermi temperature.
    • Anders Mathias Lunde, Alessandro De Martino, Reinhold Egger, Karsten Flensberg
      0707.1989v1 [pdf]

    • Kondo-Enhanced Andreev Tunneling in InAs Nanowire Quantum Dots - Abstract
      • We report measurements of the nonlinear conductance of InAs nanowire quantum dots coupled to superconducting leads. We observe a clear alternation between odd and even occupation of the dot, with sub-gap-peaks at $|V_{sd}|=\Delta/e$ markedly stronger(weaker) than the quasiparticle tunneling peaks at $|V_{sd}|=2\Delta/e$ for odd(even) occupation. We attribute the enhanced $\Delta$-peak to an interplay between Kondo-correlations and Andreev tunneling in dots with an odd number of spins, and substantiate this interpretation by a poor man's scaling analysis.
    • T. Sand-Jespersen, J. Paaske, B. M. Andersen, K. Grove-Rasmussen, H. I. Jørgensen, M. Aagesen, C. Sørensen, P. E. Lindelof, K. Flensberg, J. Nygård
      Journal reference: Phys. Rev. Lett. 99, 126603 (2007) [ cond-mat/0703264v1 ]
      DOI: 10.1103/PhysRevLett.99.126603

    • Three-particle collisions in quantum wires: Corrections to thermopower and conductance - Abstract
      • We consider the effect of electron-electron interaction on the electron transport through a finite length single-mode quantum wire with reflectionless contacts. The two-particle scattering events cannot alter the electric current and therefore we study the effect of three-particle collisions. Within the Boltzmann equation framework, we calculate corrections to the thermopower and conductance to the leading order in the interaction and in the length of wire $L$. We check explicitly that the three-particle collision rate is identically zero in the case of several integrable interaction potentials. In the general (non-integrable) case, we find a positive contribution to the thermopower to leading order in $L$. The processes giving rise to the correction involve electron states deep in the Fermi sea. Therefore the correction follows an activation law with the characteristic energy of the order of the Fermi energy for the electrons in the wire.
    • Anders Mathias Lunde, Karsten Flensberg, Leonid I. Glazman
      Journal reference: Phys. Rev. B 75, 245418 (2007) [ cond-mat/0703042v1 ]
      DOI: 10.1103/PhysRevB.75.245418

  • 2006
    • Spin-orbit induced spin-qubit control in nanowires - Abstract
      • We elaborate on a number of issues concerning our recent proposal for spin-qubit manipulation in nanowires using the spin-orbit coupling. We discuss the experimental status and describe in further detail the scheme for single-qubit rotations. We present a derivation of the effective two-qubit coupling which can be extended to higher orders in the Coulomb interaction. The analytic expression for the coupling strength is shown to agree with numerics.
    • Christian Flindt, Anders S. Sorensen, Karsten Flensberg
      Journal reference: J. Phys.: Conf. Ser. 61, 302 (2007) [ cond-mat/0612293v1 ]
      DOI: 10.1088/1742-6596/61/1/061

    • Spin-Orbit Mediated Control of Spin Qubits - Abstract
      • We propose to use the spin-orbit interaction as a means to control electron spins in quantum dots, enabling both single qubit and two qubit operations. Very fast single qubit operations may be achieved by temporarily displacing the electrons. For two qubit operations the coupling mechanism is based on a combination of the spin-orbit coupling and the mutual long-ranged Coulomb interaction. Compared to existing schemes using the exchange coupling, the spin-orbit induced coupling is less sensitive to random electrical fluctuations in the electrodes defining the quantum dots.
    • Christian Flindt, Anders S. Sorensen, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 97, 240501 (2006) [ cond-mat/0603559v2 ]
      DOI: 10.1103/PhysRevLett.97.240501

    • Interaction-Induced Resonance in Conductance and Thermopower of Quantum Wires - Abstract
      • We study the effect of electron-electron interaction on the transport properties of short clean quantum wires adiabatically connected to reservoirs. Interactions lead to resonances in a multi-channel wire at particular values of the Fermi energy. We investigate in detail the resonance in a two-channel wire. The (negative) conductance correction peaks at the resonance, and decays exponentially as the Fermi energy is tuned away; the resonance width being given by the temperature. Likewise, the thermopower shows a characteristic structure, which is surprisingly well approximated by the so-called Mott formula. Finally, four-fold splitting of the resonance in a magnetic field provides a unique signature of the effect.
    • Anders Mathias Lunde, Karsten Flensberg, Leonid I. Glazman
      Journal reference: Phys. Rev. Lett. 97, 256802 (2006) [ cond-mat/0609228v1 ]
      DOI: 10.1103/PhysRevLett.97.256802

    • Electron Transport in Single-Wall Carbon Nanotube Weak Links in the Fabry-Perot Regime - Abstract
      • We fabricated reproducible high transparency superconducting contacts consisting of superconducting Ti/Al/Ti trilayers to gated single-walled carbon nanotubes (SWCNTs). The reported semiconducting SWCNT have normal state differential conductance up to $3e^2/h$ and exhibit clear Fabry-Perot interference patterns in the bias spectroscopy plot. We observed subharmonic gap structure in the differential conductance and a distinct peak in the conductance at zero bias which is interpreted as a manifestation of a supercurrent. The gate dependence of this supercurrent as well as the excess current are examined and compared to a coherent theory of superconducting point contacts with good agreement.
    • H. I. Jørgensen, K. Grove-Rasmussen, T. Novotný, K. Flensberg, P. E. Lindelof
      Journal reference: Phys. Rev. Lett. 96, 207003 (2006) [ cond-mat/0510200v4 ]
      DOI: 10.1103/PhysRevLett.96.207003

  • 2005
    • Josephson current through a molecular transistor in a dissipative environment - Abstract
      • We study the Josephson coupling between two superconductors through a single correlated molecular level, including Coulomb interaction on the level and coupling to a bosonic environment. All calculations are done to the lowest, i.e., the fourth, order in the tunneling coupling and we find a suppression of the supercurrent due to the combined effect of the Coulomb interaction and the coupling to environmental degrees of freedom. Both analytic and numerical results are presented.
    • Tomas Novotny, Alessandra Rossini, Karsten Flensberg
      Journal reference: Phys. Rev. B 72, 224502 (2005) [ cond-mat/0508071v3 ]
      DOI: 10.1103/PhysRevB.72.224502

    • Nonequilibrium theory of Coulomb blockade in open quantum dots - Abstract
      • We develop a non-equilibrium theory to describe weak Coulomb blockade effects in open quantum dots. Working within the bosonized description of electrons in the point contacts, we expose deficiencies in earlier applications of this method, and address them using a 1/N expansion in the inverse number of channels. At leading order this yields the self-consistent potential for the charging interaction. Coulomb blockade effects arise as quantum corrections to transport at the next order. Our approach unifies the phase functional and bosonization approaches to the problem, as well as providing a simple picture for the conductance corrections in terms of renormalization of the dot's elastic scattering matrix, which is obtained also by elementary perturbation theory. For the case of ideal contacts, a symmetry argument immediately allows us to conclude that interactions give no signature in the averaged conductance. Non-equilibrium applications to the pumped current in a quantum pump are worked out in detail.
    • Piet W. Brouwer, Austen Lamacraft, Karsten Flensberg
      Journal reference: Phys. Rev. B 72, 075316 (2005) [ cond-mat/0502518v2 ]
      DOI: 10.1103/PhysRevB.72.075316

    • Weak Coulomb Blockade Effect in Quantum Dots - Abstract
      • We develop the general non-equilibrium theory of transport through a quantum dot, including Coulomb Blockade effects via a 1/N expansion, where N is the number of scattering channels. At lowest order we recover the Landauer formula for the current plus a self-consistent equation for the dot potential. We obtain the leading corrections and compare with earlier approaches. Finally, we show that to leading and next leading order in 1/N there is no interaction correction to the weak localization, in contrast to previous theories, but consistent with experiments by Huibers et al. [Phys. Rev. Lett. 81, 1917 (1998)], where N=4.
    • Piet W. Brouwer, Austen Lamacraft, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 94, 136801 (2005) [ cond-mat/0409244v2 ]
      DOI: 10.1103/PhysRevLett.94.136801

    • Electron–vibron coupling in suspended nanotubes - Abstract
      • We consider the electron-vibron coupling in suspended nanotube quantum dots. Modelling the tube as an elastic medium, we study the possible coupling mechanism for exciting the stretching mode in a single-electron-transistor setup. Both the forces due to the longitudinal and the transverse fields are included. The effect of the longitudinal field is found to be too small to be seen in experiment. In contrast, the transverse field which couple to the stretching mode via the bending of the tube can in some cases give sizeable Franck-Condon factors. However, the length dependence is not compatible with recent experiments [Sapmaz et al. cond-mat/0508270].
    • Karsten Flensberg
      Journal reference: New Journal of Physics 8, 5 (2006) [ cond-mat/0510044v1 ]
      DOI: 10.1088/1367-2630/8/1/005

    • On the Mott formula for the thermopower of non-interacting electrons in quantum point contacts - Abstract
      • We calculate the linear response thermopower S of a quantum point contact using the Landauer formula and therefore assume non-interacting electrons. The purpose of the paper, is to compare analytically and numerically the linear thermopower S of non-interacting electrons to the low temperature approximation, S^1=(pi^2/3e)k^2 T d(ln G(mu,T=0))/dmu, and the so-called Mott expression, S^M=(pi^2/3e)k^2 T d(ln G(mu,T))/dmu, where G(mu,T) is the (temperature dependent) conductance. This comparison is important, since the Mott formula is often used to detect deviations from single-particle behavior in the thermopower of a point contact.
    • Anders Mathias Lunde, Karsten Flensberg
      Journal reference: J. Phys.: Condens. Matter 17 (2005) 3879-3884. [ cond-mat/0506050v1 ]
      DOI: 10.1088/0953-8984/17/25/014

    • Noncollinear magnetoconductance of a quantum dot - Abstract
      • We study theoretically the linear conductance of a quantum dot connected to ferromagnetic leads. The dot level is split due to a non-collinear magnetic field or intrinsic magnetization. The system is studied in the non-interacting approximation, where an exact solution is given, and, furthermore, with Coulomb correlations in the weak tunneling limit. For the non-interacting case, we find an anti-resonance for a particular direction of the applied field, non-collinear to the parallel magnetization directions of the leads. The anti-resonance is destroyed by the correlations, giving rise to an interaction induced enhancement of the conductance. The angular dependence of the conductance is thus distinctly different for the interacting and non-interacting cases when the magnetizations of the leads are parallel. However, for anti-parallel lead magnetizations the interactions do not alter the angle dependence significantly.
    • Jonas N. Pedersen, Jesper Q. Thomassen, Karsten Flensberg
      Journal reference: Phys. Rev. B 72, 045341 (2005) [ cond-mat/0412145v2 ]
      DOI: 10.1103/PhysRevB.72.045341

    • Rectification in single molecular dimers with strong polaron effect - Abstract
      • We study theoretically the transport properties of a molecular two level system with large electron-vibron coupling in the Coulomb blockade regime. We show that when the electron-vibron coupling induces polaron states, the current-voltage characteristic becomes strongly asymmetric because, in one current direction, one of the polaron state blocks the current through the other. This situation occurs when the coupling between the polaron states is smaller than the coupling to the leads. We discuss the relevance of our calculation for experiments on C_140 molecules.
    • Gregers A. Kaat, Karsten Flensberg
      Journal reference: Phys. Rev. B 71, 155408 (2005) [ cond-mat/0411173v2 ]
      DOI: 10.1103/PhysRevB.71.155408

  • 2004
    • Vibrational Sidebands and the Kondo Effect in Molecular Transistors - Abstract
      • Electron transport through molecular quantum dots coupled to a single vibrational mode is studied in the Kondo regime. We apply a generalized Schrieffer-Wolff transformation to determine the effective low-energy spin-spin-vibron-interaction. From this model we calculate the nonlinear conductance and find Kondo sidebands located at bias-voltages equal to multiples of the vibron frequency. Due to selection rules, the side-peaks are found to have strong gate-voltage dependences, which can be tested experimentally. In the limit of weak electron-vibron coupling, we employ a perturbative renormalization group scheme to calculate analytically the nonlinear conductance.
    • Jens Paaske, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 94, 176801 (2005) [ cond-mat/0409158v1 ]
      DOI: 10.1103/PhysRevLett.94.176801

    • Intershell resistance in multiwall carbon nanotubes: A Coulomb drag study - Abstract
      • We calculate the intershell resistance R_{21} in a multiwall carbon nanotube as a function of temperature T and Fermi level (e.g. a gate voltage), varying the chirality of the inner and outer tubes. This is done in a so-called Coulomb drag setup, where a current I_1 in one shell induces a voltage drop V_2 in another shell by the screened Coulomb interaction between the shells neglecting the intershell tunnelling. We provide benchmark results for R_{21}=V_2/I_1 within the Fermi liquid theory using Boltzmann equations. The band structure gives rise to strongly chirality dependent suppression effects for the Coulomb drag between different tubes due to selection rules combined with mismatching of wave vector and crystal angular momentum conservation near the Fermi level. This gives rise to orders of magnitude changes in R_{21} and even the sign of R_{21} can change depending on the chirality of the inner and outer tube and misalignment of inner and outer tube Fermi levels. However for any tube combination, we predict a dip (or peak) in R_{21} as a function of gate voltage, since R_{21} vanishes at the electron-hole symmetry point. As a byproduct, we classified all metallic tubes into either zigzag-like or armchair-like, which have two different non-zero crystal angular momenta m_a, m_b and only zero angular momentum, respectively.
    • A. M. Lunde, K. Flensberg, A. P. Jauho
      Journal reference: Phys. Rev. B 71, 125408 (2005). [ cond-mat/0408112v1 ]
      DOI: 10.1103/PhysRevB.71.125408

    • Dissipative tunneling and orthogonality catastrophe in molecular transistors - Abstract
      • Transport through molecular devices with weak tunnel coupling to the leads but with strong coupling to a single vibrational mode is considered in the case where the vibration is damped by coupling to the environment. In particular, we investigate what influence the electrostatic coupling of the charge on the molecule to the vibrational modes of the environment has on the $I$-$V$ characteristics. We find that, for comparable characteristic length scales of the van-der-Waals and electrostatic interaction of the molecule with the environmental vibrational modes, the $I$-$V$ characteristics are qualitatively changed from what one would expect from orthogonality catastrophe and develop a steplike discontinuity at the onset of conduction. For the case of very different length scales, we recover dissipation consistent with modeling the electrostatic forces as an external influence on the system comprised of molecule and substrate, which implies the appearance of orthogonality catastrophe, in accord with previous results.
    • Stephan Braig, Karsten Flensberg
      Journal reference: Phys. Rev. B 70, 085317 (2004) [ cond-mat/0401347v2 ]
      DOI: 10.1103/PhysRevB.70.085317

  • 2003
    • Tunneling broadening of vibrational sidebands in molecular transistors - Abstract
      • Transport through molecular quantum dots coupled to a single vibration mode is studied in the case with strong coupling to the leads. We use an expansion in the correlation between electrons on the molecule and electrons in the leads and show that the tunneling broadening is strongly suppressed by the combination of the Pauli principle and the quantization of the oscillator. As a consequence the first Frank-Condon step is sharper than the higher order ones, and its width, when compared to the bare tunneling strength, is reduced by the overlap between the groundstates of the displaced and the non-displaced oscillator.
    • Karsten Flensberg
      Journal reference: Phys. Rev. B 68, 205323 (2003) [ cond-mat/0302193v3 ]
      DOI: 10.1103/PhysRevB.68.205323

    • Vibrational sidebands and dissipative tunneling in molecular transistors - Abstract
      • Transport through molecular devices with strong coupling to a single vibrational mode is considered in the case where the vibration is damped by coupling to the environment. We focus on the weak tunneling limit, for which a rate equation approach is valid. The role of the environment can be characterized by a frictional damping term $\mysig(\omega)$ and corresponding frequency shift. We consider a molecule that is attached to a substrate, leading to frequency-dependent frictional damping of the single oscillator mode of the molecule, and compare it to a reference model with frequency-independent damping featuring a constant quality factor $Q$. For large values of $Q$, the transport is governed by tunneling between displaced oscillator states giving rise to the well-known series of the Frank-Condon steps, while at small $Q$, there is a crossover to the classical regime with an energy gap given by the classical displacement energy. Using realistic values for the elastic properties of the substrate and the size of the molecule, we calculate $I$-$V$ curves and find qualitative agreement between our theory and recent experiments on $C_{60}$ single-molecule devices.
    • Stephan Braig, Karsten Flensberg
      Journal reference: Phys. Rev. B 68, 205324 (2003) [ cond-mat/0303236v3 ]
      DOI: 10.1103/PhysRevB.68.205324

  • 2002
    • Sign reversal of drag in bilayer systems with in-plane periodic potential modulation - Abstract
      • We develop a theory for describing frictional drag in bilayer systems with in-plane periodic potential modulations, and use it to investigate the drag between bilayer systems in which one of the layers is modulated in one direction. At low temperatures, as the density of carriers in the modulated layer is changed, we show that the transresistivity component in the direction of modulation can change its sign. We also give a physical explanation for this behavior.
    • Audrius Alkauskas, Karsten Flensberg, Ben Yu-Kuang Hu, Antti-Pekka Jauho
      Journal reference: Phys. Rev. B 66, 201304 (2002) [ cond-mat/0206340v1 ]
      DOI: 10.1103/PhysRevB.66.201304

  • 2001
    • Conductance of Rashba spin-split systems with ferromagnetic contacts - Abstract
      • We study theoretically the conductance of heterostructures with ferromagnetic conductors (F) and a two dimensional electron gas with Rashba spin-orbit interaction (R) using the Landauer-B\"{u}ttiker formalism. Assuming a one-dimensional model, we first find the $S$-matrix for the FR interface. This result is then applied to different devices such as a FRF structure, first suggested by Datta and Das[Appl. Phys. Lett. 56, 665 (1990)]. We find analytic results for the conductance for the case of collinear magnetization.
    • Morten Hogsbro Larsen, A. Mathias Lunde, Karsten Flensberg
      Journal reference: Phys. Rev. B 66, 033304 (2002) [ cond-mat/0112175v1 ]
      DOI: 10.1103/PhysRevB.66.033304

    • Mesoscopic fluctuations of Coulomb drag between quasiballistic one-dimensional wires - Abstract
      • Quasiballistic 1D quantum wires are known to have a conductance of the order of 2e^2/h, with small sample-to-sample fluctuations. We present a study of the transconductance G_12 of two Coulomb-coupled quasiballistic wires, i.e., we consider the Coulomb drag geometry. We show that the fluctuations in G_12 differ dramatically from those of the diagonal conductance G_ii: the fluctuations are large, and can even exceed the mean value, thus implying a possible reversal of the induced drag current. We report extensive numerical simulations elucidating the fluctuations, both for correlated and uncorrelated disorder. We also present analytic arguments, which fully account for the trends observed numerically.
    • Niels Asger Mortensen, Karsten Flensberg, Antti-Pekka Jauho
      Journal reference: Phys. Rev. B 65, 085317 (2002) . [ cond-mat/0108263v2 ]
      DOI: 10.1103/PhysRevB.65.085317

    • Coulomb drag in phase-coherent mesoscopic structures - Numerical study of disordered 1D-wires - Abstract
      • We study Coulomb drag between two parallel disordered mesoscopic 1D-wires. By numerical ensemble averaging we calculate the statistical properties of the transconductance G_21 including its distribution. For wires with mutually uncorrelated disorder potentials we find that the mean value is finite, but with comparable fluctuations so that sign-reversal is possible. For identical disorder potentials the mean value and the fluctuations nare enhanced compared to the case of uncorrelated disorder.
    • Niels Asger Mortensen, Karsten Flensberg, Antti-Pekka Jauho
      Journal reference: Springer Proceedings in Physics 87, 1347 (2001) [ cond-mat/0108204v1 ]

    • Coulomb Drag in the Mesoscopic Regime - Abstract
      • We present a theory for Coulomb drag between two mesoscopic systems which expresses the drag in terms of scattering matrices and wave functions. The formalism can be applied to both ballistic and disordered systems and the consequences can be studied either by numerical simulations or analytic means such as perturbation theory or random matrix theory. The physics of Coulomb drag in the mesoscopic regime is very different from Coulomb drag between extended electron systems. In the mesoscopic regime we in general find fluctuations of the drag comparable to the mean value. Examples are the vanishing average drag for chaotic 2D-systems and the dominating fluctuations of drag between quasi-ballistic wires with almost ideal transmission.
    • Niels Asger Mortensen, Karsten Flensberg, Antti-Pekka Jauho
      Journal reference: Physica Scripta T101, 177 (2002). [ cond-mat/0108203v1 ]
      DOI: 10.1238/Physica.Topical.101a00177

    • Diffusion equation and spin drag in spin-polarized transport - Abstract
      • We study the role of electron-electron interactions for spin polarized transport using the Boltzmann equation and derive a set of coupled transport equations. For spin polarized transport the electron-electron interactions are important, because they tend to equilibrate the momentum of the two spin species. This ``spin drag'' effect enhances the resistivity of the system. The enhancement is stronger the lower the dimension and should be measurable in for example a two dimensional electron gas with ferromagnetic contacts. We also include spin flip scattering which has two effects: it equilibrates the spin density imbalance and, provided it has a non s-wave component, also the current imbalance.
    • Karsten Flensberg, Thomas Stibius Jensen, Niels Asger Mortensen
      Journal reference: Phys. Rev. B 64, 245308 (2001). [ cond-mat/0107149v1 ]
      DOI: 10.1103/PhysRevB.64.245308

    • The anomalous 0.5 and 0.7 conductance plateaus in quantum point contacts - Abstract
      • The anomalous 0.5 and 0.7 conductance plateaus in quantum point contacts in zero magnetic field are analyzed within a phenomenological model. The model utilizes the Landauer-Buttiker formalism and involves enhanced spin correlations and thermal depopulation of spin subbands. In particular we can account for the plateau values 0.5 and 0.7, as well as the unusual temperature and magnetic field dependences of the 0.7 plateau. Finally, the model predicts the possibility of coexisting 0.5 and 0.7 plateaus.
    • Henrik Bruus, Vadim V. Cheianov, Karsten Flensberg
      Journal reference: Physica E 10, 97 (2001) [ cond-mat/0106504v1 ]
      DOI: 10.1016/S1386-9477(01)00061-3

  • 2000
    • Coulomb Drag in Coherent Mesoscopic Systems - Abstract
      • We present a theory for Coulomb drag between two mesoscopic systems. Our formalism expresses the drag in terms of scattering matrices and wave functions, and its range of validity covers both ballistic and disordered systems. The consequences can be worked out either by analytic means, such as the random matrix theory, or by numerical simulations. We show that Coulomb drag is sensitive to localized states, which usual transport measurements do not probe. For chaotic 2D-systems we find a vanishing average drag, with a nonzero variance. Disordered 1D-wires show a finite drag, with a large variance, giving rise to a possible sign change of the induced current.
    • Niels Asger Mortensen, Karsten Flensberg, Antti-Pekka Jauho
      Journal reference: Phys. Rev. Lett. 86, 1841 (2001). [ cond-mat/0007046v2 ]
      DOI: 10.1103/PhysRevLett.86.1841

    • Dephasing in semiconductor–superconductor structures by coupling to a voltage probe - Abstract
      • We study dephasing in semiconductor-superconductor structures caused by coupling to a voltage probe. We consider structures where the semiconductor consists of two scattering regions between which partial dephasing is possible. As a particular example we consider a situation with a double-barrier junction in the normal region. For a single-mode system we study the conductance both as a function of the position of the Fermi level and as a function of the barrier transparency. At resonance, where the double-barrier is fully transparent, we study the suppression of the ideal factor-of-two enhancement of the conductance when a finite coupling to the voltage probe is taken into account.
    • Niels Asger Mortensen, Antti-Pekka Jauho, Karsten Flensberg
      Journal reference: Superlattice Microstr. 28, 67 (2000). [ cond-mat/9909029v2 ]
      DOI: 10.1006/spmi.2000.0890

    • From mesoscopic magnetism to the anomalous 0.7 conductance plateau - Abstract
      • We present a simple phenomenological model which offers a unifying interpretation of the experimental observations on the 0.7 conductance anomaly of quantum point contacts. The model utilizes the Landauer-Buttiker formalism and involves enhanced spin correlations and thermal depopulation of spin subbands. In particular our model can account for the plateau value 0.7 and the unusual temperature and magnetic field dependence. Furthermore it predicts an anomalous suppression of shot noise at the 0.7 plateau.
    • Henrik Bruus, Vadim V. Cheianov, Karsten Flensberg
      cond-mat/0002338v1 [pdf]

  • 1999
    • Andreev scattering and conductance enhancement in mesoscopic semiconductor-superconductor junctions - Abstract
      • An inherent difficulty in studying mesoscopic effects in semiconductor--superconductor hybrid structures is the large Schottky barrier which often forms at the interface. A large technological effort has been invested in in improving the contact between the superconductor and the two-dimensional electron gas (2DEG) of a semiconductor heterostructure, and in recent years this has become possible for e.g. GaAs-Al, GaAs-In, and InAs-Nb junctions. This development motivates quantitative theoretical modeling of sample-specific transport properties. The aim of our work is to model the conducting properties of a ballistic 2DEG-S interfaces with a QPC in the normal region and also to take into account scattering due to a weak Schottky barrier and/or non-matching Fermi properties of the semiconductor and superconductor.
    • Niels Asger Mortensen, Antti-Pekka Jauho, Karsten Flensberg
      Journal reference: Extended abstracts of Electron Transport in Mesoscopic Systems, pp. 120-121 (1999) [ cond-mat/9911372v1 ]

    • Contact resistance of quantum tubes - Abstract
      • We consider the conductance of a quantum tube connected to a metallic contact. The number of angular momentum states that the tube can support depends on the strength of the radial confinement. We calculate the transmission coefficients which yield the conductance via the Landauer formula. We relate our results to armchair carbon nanotubes embedded in a metal. For Al and Au contacts and tubes with a realistic radial confinement we find that the transmission can be close to unity corresponding to a contact resistance close to h/2e^2 per band at the Fermi level in the carbon nanotube.
    • Niels Asger Mortensen, Kristinn Johnsen, Antti-Pekka Jauho, Karsten Flensberg
      Journal reference: Superlattices and Microstructures 26, 351 (1999). [ cond-mat/9907026v2 ]
      DOI: 10.1006/spmi.1999.0812

    • Screening, nonadiabaticity, and quantized acoustoelectric current - Abstract
      • Quantized single-electron transport driven by surface acoustic waves (SAW) through a pinched-off narrow constriction is studied theoretically. Long-range Coulomb interaction causes the tunneling coupling between the two-dimensional electron gas (2DEG) and the moving minimum of the SAW-induced potential to decay rapidly with time. The energy scale, associated with the characteristic time of this decay, controls both the width of the transition regions between the plateaus and the slope of the plateaus. This sets a limit for the accuracy of the quantization of acoustoelectric current at low temperature.
    • M. Pustilnik, K. Flensberg, Q. Niu
      Journal reference: J. Low Temp. Phys. 118, 571 (2000) [ cond-mat/9909187v2 ]

    • Nonadiabaticity and single-electron transport driven by surface acoustic waves - Abstract
      • Single-electron transport driven by surface acoustic waves (SAW) through a narrow constriction, formed in two-dimensional electron gas, is studied theoretically. Due to long-range Coulomb interaction, the tunneling coupling between the electron gas and the moving minimum of the SAW-induced potential rapidly decays with time. As a result, nonadiabaticiy sets a limit for the accuracy of the quantization of acoustoelectric current.
    • Karsten Flensberg, Qian Niu, Michael Pustilnik
      Journal reference: Phys. Rev. B 60, R16291 (1999) [ cond-mat/9908096v2 ]
      DOI: 10.1103/PhysRevB.60.R16291

      • We review the status of the understanding of single-electron transport (SET) devices with respect to their applicability in metrology. Their envisioned role as the basis of a high-precision electrical standard is outlined and is discussed in the context of other standards. The operation principles of single electron transistors, turnstiles and pumps are explained and the fundamental limits of these devices are discussed in detail. We describe the various physical mechanisms that influence the device uncertainty and review the analytical and numerical methods needed to calculate the intrinsic uncertainty and to optimise the fabrication and operation parameters. Recent experimental results are evaluated and compared with theoretical predictions. Although there are discrepancies between theory and experiments, the intrinsic uncertainty is already small enough to start preparing for the first SET-based metrological applications.
    • Karsten Flensberg, Arkadi A. Odintsov, Feike Liefrink, Paul Teunissen
      Journal reference: Int. J. Mod. Phys. B 13, 2651-2687 (1999) [ cond-mat/9908219v1 ]
      DOI: 10.1142/S0217979299002587

    • Conductance enhancement in quantum-point-contact semiconductor-superconductor devices - Abstract
      • We present numerical calculations of the conductance of an interface between a phase-coherent two-dimensional electron gas and a superconductor with a quantum point contact in the normal region. Using a scattering matrix approach we reconsider the geometry of De Raedt, Michielsen, and Klapwijk [Phys. Rev. B, 50, 631 (1994)] which was studied within the time-dependent Bogoliubov-de Gennes formalism. We find that the factor-of-two enhancement of the conductance G_NS compared to the normal state conductance G_N for ideal interfaces may be suppressed for interfaces with a quantum point contact with only a few propagating modes. The suppression is found to depend strongly on the position of the Fermi level. We also study the suppression due to a barrier at the interface and find an anomalous behavior caused by quasiparticle interference. Finally, we consider the limit of sequential tunneling and find a suppression of the factor-of-two enhancement which may explain the absence of conductance enhancement in experiments on metal-superconductor structures.
    • Niels Asger Mortensen, Antti-Pekka Jauho, Karsten Flensberg, Henning Schomerus
      Journal reference: Phys. Rev. B 60, 13762 (1999). [ cond-mat/9903205v2 ]
      DOI: 10.1103/PhysRevB.60.13762

  • 1998
    • Angle dependence of Andreev scattering at semiconductor–superconductor interfaces - Abstract
      • We study the angle dependence of the Andreev scattering at a semiconductor-superconductor interface, generalizing the one-dimensional theory of Blonder, Tinkham and Klapwijk. An increase of the momentum parallel to the interface leads to suppression of the probability of Andreev reflection and increase of the probability of normal reflection. We show that in the presence of a Fermi velocity mismatch between the semiconductor and the superconductor the angles of incidence and transmission are related according to the well-known Snell's law in optics. As a consequence there is a critical angle of incidence above which only normal reflection exists. For two and three-dimensional interfaces a lower excess current compared to ballistic transport with perpendicular incidence is found. Thus, the one-dimensional BTK model overestimates the barrier strength for two and three-dimensional interfaces.
    • Niels Asger Mortensen, Karsten Flensberg, Antti-Pekka Jauho
      Journal reference: Phys. Rev. B 59, 10176 (1999). [ cond-mat/9807049v3 ]
      DOI: 10.1103/PhysRevB.59.10176

    • Localized plasmons in point contacts - Abstract
      • Using a hydrodynamic model of the electron fluid in a point contact geometry we show that localized plasmons are likely to exist near the constriction. We attempt to relate these plasmons with the recent experimental observation of deviations of the quantum point contact conductance from ideal integer quantization. As a function of temperature this deviation exhibits an activated behavior, exp(-T_a/T), with a density dependent activation temperature T_a of the order of 2 K. We suggest that T_a can be identified with the energy needed to excite localized plasmons, and we discuss the conductance deviations in terms of a simple theoretical model involving quasiparticle lifetime broadening due to coupling to the localized plasmons.
    • Henrik Bruus, Karsten Flensberg
      Journal reference: Semicond. Sci. Technol. vol 13, A30-A32 (1998) [ cond-mat/9807342v1 ]

    • Coulomb Drag of Luttinger Liquids and Quantum Hall Edges - Abstract
      • We study the transconductance for two coupled one-dimensional wires or edge states described by Luttinger liquid models. The wires are assumed to interact over a finite segment. We find for the interaction parameter $g=1/2$ that the drag rate is finite at zero temperature, which cannot occur in a Fermi-liquid system. The zero temperature drag is, however, cut off at low temperature due to the finite length of the wires. We also consider edge states in the fractional quantum Hall regime, and we suggest that the low temperature enhancement of the drag effect might be seen in the fractional quantum Hall regime.
    • Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 81, 184 (1998) [ cond-mat/9802220v2 ]
      DOI: 10.1103/PhysRevLett.81.184

  • 1997
    • Frictional drag between quantum wells mediated by phonon exchange - Abstract
      • We use the Kubo formalism to evaluate the contribution of acoustic phonon exchange to the frictional drag between nearby two-dimensional electron systems. In the case of free phonons, we find a divergent drag rate ($\tau_{D}^{-1}$). However, $\tau_{D}^{-1}$ becomes finite when phonon scattering from either lattice imperfections or electronic excitations is accounted for. In the case of GaAs quantum wells, we find that for a phonon mean free path $\ell_{ph}$ smaller than a critical value, imperfection scattering dominates and the drag rate varies as $ln (\ell_{ph}/d)$ over many orders of magnitude of the layer separation $d$. When $\ell_{ph}$ exceeds the critical value, the drag rate is dominated by coupling through an electron-phonon collective mode localized in the vicinity of the electron layers. We argue that the coupled electron-phonon mode may be observable for realistic parameters. Our theory is in good agreement with experimental results for the temperature, density, and $d$-dependence of the drag rate.
    • Martin Bonsager, Karsten Flensberg, Ben Yu-Kuang Hu, Allan H. MacDonald
      Journal reference: Phys. Rev. B 57, 7085 (1998). [ cond-mat/9707111v1 ]
      DOI: 10.1103/PhysRevB.57.7085

    • Frictional Coulomb drag in strong magnetic fields - Abstract
      • A treatment of frictional Coulomb drag between two 2-dimensional electron layers in a strong perpendicular magnetic field, within the independent electron picture, is presented. Assuming fully resolved Landau levels, the linear response theory expression for the transresistivity $\rho_{21}$ is evaluated using diagrammatic techniques. The transresistivity is given by an integral over energy and momentum transfer weighted by the product of the screened interlayer interaction and the phase-space for scattering events. We demonstrate, by a numerical analysis of the transresistivity, that for well-resolved Landau levels the interplay between these two factors leads to characteristic features in both the magnetic field- and the temperature dependence of $\rho_{21}$. Numerical results are compared with recent experiments.
    • M. C. Bonsager, K. Flensberg, B. Y-K. Hu, A. -P. Jauho
      Journal reference: Phys. Rev. B 56, 10314 (1997). [ cond-mat/9704232v1 ]
      DOI: 10.1103/PhysRevB.56.10314

    • Correlation Effects on the Coupled Plasmon Modes of a Double Quantum Well - Abstract
      • At temperatures comparable to the Fermi temperature, we have measured a plasmon enhanced Coulomb drag in a GaAs/AlGaAs double quantum well electron system. This measurement provides a probe of the many-body corrections to the coupled plasmon modes, and we present a detailed comparison between experiment and theory testing the validity of local field theories. Using a perpendicular magnetic field to raise the magnetoplasmon energy we can induce a crossover to single-particle Coulomb scattering.
    • N P R Hill, J. T. Nicholls, E. H. Linfield, M. Pepper, D. A. Ritchie, G. A. C. Jones, Ben Yu-Kuang Hu, Karsten Flensberg
      Journal reference: Phys. Rev. Lett. 78, 2204 (1997). [ cond-mat/9703066v1 ]
      DOI: 10.1103/PhysRevLett.78.2204

    • Coherent-photon-assisted cotunneling in a Coulomb blockade device - Abstract
      • We study cotunneling in a double junction Coulomb blockade device under the influence of time dependent potentials. It is shown that the ac-bias leads to photon assisted cotunneling which in some cases may dominate the transport. We derive a general non-perturbative expression for the tunneling current in the presence of oscillating potentials and give a perturbative expression for the photon assisted cotunneling current.
    • Karsten Flensberg
      Journal reference: Phys. Rev. B 55, 13118 (1997). [ cond-mat/9607139v3 ]
      DOI: 10.1103/PhysRevB.55.13118

  • 1996
    • Observation of scaling behavior in a Coulomb blockade system - Abstract
      • We describe two experiments to study the influence of fluctuations in the electron charge on the transport properties of a quantum dot. First, we scan a device from single- to double quantum-dot behavior by varying the conductance of a connecting point contact. Second, we measure the dependence of the charging energy on the conductance of the barriers. The experiments are compared with traces obtained from a theory based on a Luttinger liquid type description. This theory predicts a scaling behaviour of the charging energy, in good agreement with our experiments.
    • K. Flensberg, L. W. Molenkamp
      cond-mat/9607138v1 [pdf]

    • Magneto-Coulomb Drag: Interplay of Electron-Electron Interactions and Landau Quantization - Abstract
      • We use the Kubo formalism to calculate the transresistivity $\rho_{21}$ for carriers in coupled quantum wells in a large perpendicular magnetic field $B$. We find that $\rho_{21}$ is enhanced by approximately 50--100 times over that of the B=0 case in the interplateau regions of the integer quantum Hall effect. The presence of both electron--electron interactions and Landau quantization results in (i) a twin-peaked structure of $\rho_{21}(B)$ in the inter-plateau regions at low temperatures, and, (ii) for the chemical potential at the center of a Landau level band, a peaked temperature dependence of $\rho_{21}(T)/T^2$.
    • M. C. Bonsager, K. Flensberg, B. Y. -K. Hu, A. -P. Jauho
      Journal reference: Phys. Rev. Lett. 77, 1366 (1996) [ cond-mat/9603110v1 ]
      DOI: 10.1103/PhysRevLett.77.1366

  • 1995
    • Electron-electron scattering in linear transport in two-dimensional systems - Abstract
      • We describe a method for numerically incorporating electron--electron scattering in quantum wells for small deviations of the distribution function from equilibrium, within the framework of the Boltzmann equation. For a given temperature $T$ and density $n$, a symmetric matrix needs to be evaluated only once, and henceforth it can be used to describe electron--electron scattering in any Boltzmann equation linear-response calculation for that particular $T$ and $n$. Using this method, we calculate the distribution function and mobility for electrons in a quantum-well, including full finite-temperature dynamic screening effects. We find that at some parameters which we investigated, electron--electron scattering reduces mobility by approximately 40\%.
    • Ben Yu-Kuang Hu, Karsten Flensberg
      Journal reference: Phys. Rev. B 53 (1996) 10072 [ cond-mat/9510044v1 ]
      DOI: 10.1103/PhysRevB.53.10072

    • Plasmon enhancement of Coulomb drag in double-quantum-well systems - Abstract
      • We derive an expression for the drag rate (i.e., interlayer momentum transfer rate) for carriers in two coupled two-dimensional gases to lowest nonvanishing order in the screened interlayer electron--electron interaction, valid for {\sl arbitrary} intralayer scattering mechanisms, using the Boltzmann transport equation. We calculate the drag rate for experimentally relevant parameters, and show that for moderately high temperatures ($T\gtrsim 0.2 T_F$, where $T_F$ is the Fermi temperature) the dynamical screening of the interlayer results in a large enhancement of the drag rate due to the presence of coupled plasmon modes. This plasmon enhancement causes the scaled drag rate to have a peak (i) as a function of temperature at $T \approx 0.5 T_F$, and (ii) as a function of the ratio of densities of the carriers in the two layers when their Fermi velocities are equal. We also show that the drag rate can be significantly affected by the {\sl intralayer} scattering mechanisms; in particular, the drag rate changes approximately by a factor of 2 when the dopant layer modulation doped structures are moved in from 400~\AA to 100~\AA.
    • Karsten Flensberg, Ben Yu-Kuang Hu
      Journal reference: Phys. Rev. B 52 (1995) 14796 [ cond-mat/9507059v1 ]
      DOI: 10.1103/PhysRevB.52.14796

    • Linear-response theory of Coulomb drag in coupled electron systems - Abstract
      • We report a fully microscopic theory for transconductivity, or, equivalently, momentum transfer rate, of Coulomb coupled electron systems. We use the Kubo linear response formalism, and our main formal result expresses the transconductivity in terms of two fluctuation diagrams, which are topologically related, but not equivalent to, the Aslamazov-Larkin diagrams known for superconductivity. Previously reported results are shown to be special cases of our general expression; specifically, for constant impurity scattering rates, we recover the Boltzmann equation results in the semiclassical clean limit, and the memory function results in dirty systems. Furthermore, we show that for energy dependent relaxation times, the final result is not expressible in terms of standard density-response functions. Other new results include: (i) at T=0, the frequency dependence of the transfer rate is found to be proportional to $\Omega$ and $\Omega^2$ for frequencies below and above the impurity scattering rate, respectively and (ii) the weak localization correction to the transconductivity is given by $\delta\sigma^{WL}_{21} \propto \delta\sigma^{WL}_{11} +\delta\sigma^{WL}_{22}$.
    • Karsten Flensberg, Ben Yu-Kuang Hu, Antti-Pekka Jauho, Jari M. Kinaret
      Journal reference: Phys. Rev. B 52, 14761 (1995) [ cond-mat/9504092v1 ]
      DOI: 10.1103/PhysRevB.52.14761

    • Scaling of the Coulomb Energy Due to Quantum Fluctuations in the Charge on a Quantum Dot - Abstract
      • The charging energy of a quantum dot is measured through the effect of its potential on the conductance of a second dot. This technique allows a measurement of the scaling of the dot's charging energy with the conductance of the tunnel barriers leading to the dot. We find that the charging energy scales quadratically with the reflection probability of the barriers. In a second experiment we study the transition from a single to a double-dot which exhibits a scaling behavior linear in the reflection probability. The observed power-laws agree with a recent theory.
    • L. W. Molenkamp, K. Flensberg, M. Kemerink
      Journal reference: Phys. Rev. Lett. 75, 4282 (1995) [ cond-mat/9503135v1 ]
      DOI: 10.1103/PhysRevLett.75.4282

  • 1994
    • Coulomb Drag as a Probe of Coupled Plasmon Modes in Parallel Quantum Wells - Abstract
      • We show theoretically that the Coulomb drag rate between two parallel quasi-two-dimensional electron gases is substantially enhanced by the coupled acoustic and optic plasmon modes of the system at temperatures $T \gtrsim 0.2T_F$ (where $T_F$ is the Fermi temperature) for experimentally relevant parameters. The acoustic mode causes a sharp upturn in the scaled drag rate as a function of temperature at $T \approx 0.2 T_F$. Other experimental signatures of plasmon-dominated drag are a $d^{-3}$ dependence on the well separation $d$, and a peak in the drag rate as a function of relative carrier densities at matched Fermi velocities.
    • Karsten Flensberg, Ben Yu-Kuang Hu
      Journal reference: Phys. Rev. Lett. 73, 3572 (1994). [ cond-mat/9406052v1 ]
      DOI: 10.1103/PhysRevLett.73.3572

  • 1993
    • Magnetoconductivity of quantum wires with elastic and inelastic scattering - Abstract
      • We use a Boltzmann equation to determine the magnetoconductivity of quantum wires. The presence of a confining potential in addition to the magnetic field removes the degeneracy of the Landau levels and allows one to associate a group velocity with each single-particle state. The distribution function describing the occupation of these single-particle states satisfies a Boltzmann equation, which may be solved exactly in the case of impurity scattering. In the case where the electrons scatter against both phonons and impurities we solve numerically - and in certain limits analytically - the integral equation for the distribution function, and determine the conductivity as a function of temperature and magnetic field. The magnetoconductivity exhibits a maximum at a temperature, which depends on the relative strength of the impurity and electron-phonon scattering, and shows oscillations when the Fermi energy or the magnetic field is varied.
    • Henrik Bruus, Karsten Flensberg, Henrik Smith
      DOI: 10.1103/PhysRevB.48.11144
      cond-mat/9303028v1 [pdf]